There is only one mathematical object
A challenge to Platonism, which is IMO one of the more serious ones, is that mathematical objects lack clear identity conditions. What if there's only one object, Math, and the existence of individual objects is determined by whether they are validly derived from Math?
The logical problem here is determining what is a valid derivation. Naively, I say that a valid derivation is one that is logically coherent. So any logically consistent set of axioms from which one derives a mathematics is a valid derivation from Math, which means that those objects it posits are "real" in the sense that they are validly derived from math. However, they do not have independent existence, i.e. they are not separate entities in any sense but the conceptual. However, Math is mind-independent and our familiar mathematical objects are derivations therefrom.
If anybody in phil of math has already posited something like this, lemme know.
The logical problem here is determining what is a valid derivation. Naively, I say that a valid derivation is one that is logically coherent. So any logically consistent set of axioms from which one derives a mathematics is a valid derivation from Math, which means that those objects it posits are "real" in the sense that they are validly derived from math. However, they do not have independent existence, i.e. they are not separate entities in any sense but the conceptual. However, Math is mind-independent and our familiar mathematical objects are derivations therefrom.
If anybody in phil of math has already posited something like this, lemme know.
Comments (228)
But maybe those maths already exist in the possible space of all Math defined by inherent logic. We could say all possible games of chess or life worlds similarly exist. That requires a realist commitment to possible worlds or at least possibility space. But what happens when one modifies the logic? Does that cause a new possibility space to come into existence? You can always create derivatives of chess, or your own cellular automata.
Or the case of math, add new concepts like infinity or imaginary numbers with their own derivation of the rules.
All you establish with Quoting Pneumenon
is that it is possible for this object to be real. It exists in the sense that you can imagine it as a unity (a single object).
I'll set aside the obvious objection that each of these derived-from-math objects would imply there's more than one object, since you seem to think they are really just aspects of the single object, Math, as it appears in different ways.
By the way, some countries in the world don't say "Math" they say "Maths".
If it helps: I'm primarily interested in answering the identity-condition objection to Platonism. The dialectic goes kinda like this, where P is a Platonist and AP is an anti-Platonist:
P: The triangle is a mathematical object that exists.
AP: All of the triangles, or just one?
P: All of them.
AP: Okay. So how do we tell the difference between two triangles? Do all acute triangles answer to the form of the acute triangle, or are there multiple acute triangles?
And so on. The two big objections to Platonism that arise from conversations like this are that Platonic objects lack clear identity conditions and that the ontology is profligate, a crowded slum, what Quine called Plato's Beard. Reducing every object to Math should answer both objections.
Quoting Marchesk
This is a good question. I would say that modifications to the logic introduce new subdomains. Provided that translation functions can be constructed between those domains, there shouldn't be any problem with all of this being and expression of Math.
EDIT: or perhaps, instead of translation functions, we can say that any logical space that gives us valid derivations from Math is constructed via identity statements.
I think the point is that there are numerous different types of triangles. And if you want to argue that there is only one type, "the triangle", then why isn't the triangle just one type of polygon? And the polygon is a type of geometrical figure, and so on.
I would have thought that the identity conditions of integers was abundantly obvious. I mean, any integer is distinct from all other integers - how does that not constitute an 'identity condition'?
As for the triangle, it's 'a flat plane bounded by three intersecting straight lines'. That applies to any triangle. The 'form' is not the shape.
We mathematicians pull threads from the rich fabric, imagining ourselves creating mathematics, when in fact we only uncover wisps of a majestic and largely unknowable tapestry - a single entity beyond our wildest speculations.
:100:
The concrete world is an abstract object: it's just the one that we're a part of.
Because you can make abstract objects from collections of other abstract objects, then yes OP, you can say that there is just one abstract object of which all other objects are parts, and that one all-encompassing abstract object is the entirety of existence in the broadest possible sense.
How can we say that 2 represents a unique and particular object? That is the identity condition. To have an identity is to be identifiable as a unique and particular individual. But every time that there are two objects, the number 2 is represented, so 2 represents something universal, rather than something unique and particular. Therefore it appears like the number signified by the numeral 2 cannot fulfill identity conditions.
Quoting Wayfarer
We can attempt to provide identity through the means of a definition, but the definition always allows that more than one thing can be identified as fulfilling the identity conditions of the definition. So a definition cannot serve to give us adequate identity conditions because it allows that more than one thing might have the same identity.
:100:
Sub specie aeternitatis: natura naturans. (Spinoza)
Right. So 'identity condition' pertains to individual identity, something unique and particular. What is the source or definition of 'identity condition'?
Quoting Metaphysician Undercover
Be that as it may, a triangle will never have other than three sides.
BTW there is a current article in Smithsonian Magazine about the ongoing appeal of mathematical platonism. https://www.smithsonianmag.com/science-nature/what-math-180975882/ . It says while there is some support for mathematical platonism, that:
which basically says it all.
I don't know what that means. Your example was triangles. Triangles are identified up to similarity by their angles; and up to congruence by the lengths of their sides; and identified uniquely by their congruence class and position and orientation in space. It's perfectly simple to define a similarity class of triangles or a congruence class of triangles or a particular triangle. I don't follow your examples.
A classic Platonic object in math is the unit circle. It's a circle of radius one centered at the origin. Now you're right, coordinate systems are arbitrary so in fact we could locate the unit circle anywhere on the plane. In that sense "the" unit circle is arbitrary. But once you fix a rectangular coordinate system, you have a unique unit circle and you can then define the trigonometric functions, Fourier series, topological groups, and all the rest of the interesting mathematical concepts that generalize or abstract from the unit circle.
Math isn't concerned at all with particular objects; only with abstract forms. It doesn't matter what numbers "are," only how they behave and how they relate to other numbers. This is mathematical structuralism. As David Hilbert said, "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs."
That quote answers your concern that we can't identify particular things. We don't care about identifying particular things. Rather, we only care about structural and logical relationships among particular things; and those relations are independent of the things themselves.
To those throwing rocks at mathematical Platonism (as I do when I'm taking the other side of this debate), was 3 prime before there were intelligent life forms in the universe? If that's too easy (I don't think it is), were there infinitely many primes? Or at least no largest prime?
I don't know where Pneumenon gets this specific terminology, but I know the identity condition as the "law of identity", which states that a thing is the same as itself. Leibniz' posited the "identity of indiscernibles" which stipulates that each individual thing is unique.
Quoting Wayfarer
That's true, but "triangle" does not suffice as a thing's identity because many things are said to be triangles. What I believe to be Pneumenon's point, is that whatever "triangle" refers to, it cannot be a thing, because all things have a distinct and unique identity according to the law of identity, or what Pneumenon calls "identity condition". Pneumenon uses this argument against any Platonists who believe that ideas exists as objects, arguing that ideas such as 'triangle" do not fulfil the "identity conditions to be rightly called "objects".
This is an ontological issue, and I don't think it's meant as a simple attack against dualism. When using this argument we do not mean to imply that immaterial things do not qualify as "objects", therefore we ought to reject the existence of the immaterial. What I believe, is that the argument is meant to elucidate the fundamental, and radical difference between the immaterial and the material, demonstrating that it is a basic misunderstanding to portray the immaterial as objects.
If we accept this perspective, that the immaterial ought not be represented as consisting of objects, there is far reaching consequences. Many mathematical axioms such as those of set theory rely on the assumption of mathematical objects. Further, we can see the trend in physics, to translate the immaterial wave fields to objects (particles), though things like virtual particles appear as immaterial. I see this reduction of the immaterial to objects, as a real problem. It basically says that the immaterial has to be like the material, existing as objects. But when we see that the immaterial does not fulfill the "identity conditions" which is required for existence as objects, we need to move on and recognize that the immaterial is radically different from the assumption of immaterial objects.
But the most succinct formulation of 'the law of identity' is 'a=a'. So are you saying that 'a' doesn't have an identity?
Quoting Metaphysician Undercover
Many things can be triangles, but that is only insofar as those things assume that form. The form itself is not a thing. Only three-sided flat planes bounded by lines constitute a triangle but that covers an endless variety of things. That's the 'thing' about universals. ;-)
--
Quoting fishfry
That's a very interesting question. I generally defend mathematical Platonism, on the grounds that numbers are real but immaterial, in that they can only be grasped by a rational mind, but they're the same for anyone who can count.
But if this is true, then it falsifies the idea that only material things are real; hence the passage I quoted above from the Smithsonian article. Platonism is generally resisted on exactly those kinds of grounds. Hence fictionalism etc.
There was an essay by Paul Benacereff, Mathematical Truth, which has apparently been quite influential, pointing out that mathematical intuition is not empirically verifiable, and so philosophically at odds with empiricism. In an analysis from an article in IEP we read:
I find it amusing that modern philoosphy has to come up with arguments (or tie itself in knots) to find an alternative to rational insight, especially as so many popular intellectuals blather endlessly about the 'rationalism of science'.
In any case, getting back to your question: my solution to it is that in humans, the mind has evolved to the point where it can grasp the nature of prime numbers. But that means neither that prime numbers are 'out there somewhere' (which is naturalism, again) or 'in the mind' (which is subjectivism and relativism). They can only be grasped by a mind, but they're the same for all minds. Which is pretty much a definition of 'objective idealism'. We discover the deep laws and regularities which 'govern' how phenomena behave, but science itself can't explain those laws - as you said elsewhere, science 'describes but doesn't explain' on that level. If that was understood it would solve many a philosophical conundrum.
That is why I think mathematical Platonism still has a pretty strong hand.
You know that this is just a symbolic representation of the law. So the symbols need to be interpreted. What 'a=a' represents is that for any object, represented as 'a', that object is the same as itself.
Therefore it's not saying that 'a' doesn't have an identity, it's saying that the identity of the object represented by 'a', is the object represented by 'a'. In other words, an object is its own identity. Aristotle found it necessary to formulate the law of identity in this way, to recognize the difference between the identity we assign to an object, and the object's true identity. Sophistry had demonstrated that the identity which we give an object is sometimes incorrect. So we need a way to allow that the human assigned identity is incorrect, yet the object still has a true identity which the human beings have not determined. Therefore Aristotle posited that the identity of any object is within itself.
Quoting Wayfarer
Yes, that's a feature of universals. the word "triangle" covers an endless variety of things. But this same feature indicates to us, that a universal is not itself an individual thing. A universal is not a thing because it does not conform to the law of identity. It cannot be identified as a thing, because it has no identity as a thing. But, a thing necessarily has an identity, itself. Therefore a universal is not a thing.
This does not imply that there is no immaterial existence, it just means that in our understanding of immaterial existence we have to get beyond the idea that immaterial existence is in the form of things. That is just a sort of mistake which has developed from human beings relating what they know of the material world, to the immaterial, in an attempt to understand the immaterial. Because the material world consists of things, we want to apply the same principles to the immaterial, and portray the immaterial world as consisting of things. But the law of identity is interjected to demonstrate to us, the fundamental difference between material and immaterial, and that this would be a misunderstanding.
Quoting fishfry
The word "prime" was created by human beings, and has a meaning according to what human beings think. It does not make any sense at all to ask about the meaning of the word "prime", or any word for that matter, at a time before the word existed. (What did the word "prime" mean before it existed?) I'm sure you can understand that. We can however use the word to refer to something that we believe existed before the word. Like "earth" for example is supposed to have existed before the word. Your question therefore asks, whether there was something which we refer to with "3", and something which we refer to with "prime", which existed prior to the existence of these words, and that is a difficult metaphysical question without a straight forward answer.
So, you're a relativist after all?
Can you name one such? Structuralism is in these days. It doesn't matter if you call sets "beer mugs" as Hilbert pointed out. It's the properties and relations that matter, not the nature of individual things.
Quoting Metaphysician Undercover
Hey, something on which we agree!
No, not necessarily, I just recognize that it is impossible for a word to have a meaning before that word exists.
Quoting fishfry
I think we've been through this before. You insisted on an unreasonable separation between "objects" and "mathematical objects", such that mathematical objects are not a type of object.
We could start with the axiom of extensionality. Any axiom which treats numbers as elements of a set, treats the numbers as objects.
Quoting fishfry
The issue though, is that set theory treats them as "individual things", therefore Platonism is implied. Set theory relies on Platonism because it cannot proceed unless what 2, 3, 4, refer to are objects, which can be members of a set.
Good point, I knew better than to start. Don't know what I'm thinking. This can't end well.
Quoting Metaphysician Undercover
Well, mathematical objects are abstract objects. But I agree that numbers aren't like rocks. That doesn't mean that numbers don't exist. It only means that numbers are abstract. And, per structuralism and Benacerraf's famous essay, What Numbers Cannot Be, numbers are not any particular thing. They're not actually sets, even though they are typically represented as sets. Numbers are the abstract things represented by sets. I suspect you and I might be in violent agreement on this point, but I'm not sure.
Quoting Metaphysician Undercover
Only sets can be elements of sets in pure set theory. So we can represent numbers by sets and say that the [math]\in[/math] relation holds between a pair of sets; but you are reading more into that than is intended. In high school set theory we would say that the students are members of the school, and we'd call the school a set and the students elements of that set. And I doubt that you'd disagree. But i formal set theory we are not reifying, I think that's the word, the things that are elements of sets. We're just saying that the membership relation holds between the abstract things represented by the symbols. You can, if you like, view the entire enterprise as an exercise in formal symbol manipulation that could be carried out by computer, entirely devoid of meaning. It would not make any difference to the math.
Quoting Metaphysician Undercover
Maybe you'd like category theory better. In categorical set theory there are no elements at all, only relationships among sets. But in category theory they call things "objects" and that might make you unhappy.
We could go down a rabbit hole here but just tell me this. Do you believe that if E is the set of even positive integers, then E = {2, 3, 4, 6, ...}. Do you agree with that statement? Or do you deny the entire enterprise? I'm trying to put a metric on your mathematical nihilism. Is it naughty to put numbers into sets? Why? Some numbers are prime, some are even, some are solutions to various equations. Why can't we collect them into sets? Either conceptually or, if you don't like that, formally?
Quoting Metaphysician Undercover
The Platonist explanation is that these 'things' - they're not actually things, which is part of the point - are discerned by the rational intellect, nous. They transcend individual minds, but they're constituents of rational thought because thought must conform to them in order to proceed truly. In which case they precede their discovery by the mind, not in a temporal sense, but in the sense that they must already be the case in order for thought to be rational in the first place.
Why this is so murky, and so controversial, is because it all harks back to the disputes between realists and nominalists in the late medieval period. Nominalism was the chief precursor to today’s empiricism, and it has so permeated the public discourse that we don’t know how to think any other way. History is written by the victors, and they were indubitably victorious in this matter. That’s why discussion of universals generally only draws blank stares in modern culture. And also because it thinks ‘mind’ is the product of undirected matter and that number is ‘a product’ of this undirected process.
Well, I wouldn't go so far as to call it a violent agreement. The point of the op I believe, is that it's incorrect to call mathematical objects "objects" at all, because they do not fulfill the requirement of identity. And so, if we start talking about them as if they are objects, and believe that they have identities as objects, and treat them that way, when they do not, there is bound to be problems which arise.
Where we agree is that they are "abstract", but the problem is in where we go from here.
Quoting fishfry
Here is where the difference between us appears to arise. You are saying that there are "abstract things represented by the symbols". That's Platonism plain and simple, the "abstract things" are nothing other than Platonic Ideas, or Forms. See, you even allow that there are relations between these things. But from my perspective, a symbol has meaning, and meaning is itself a relation between a mind and the symbol. So I see that you've jumped to the conclusion that this relation between a symbol and a mind, is itself a thing, and you then proceed to talk about relations between these supposed things which are really just relations, and not things at all, in the first place.
Quoting fishfry
If you can follow what I said above, then I'll explain why there's a real problem here. The relation between a symbol and a mind, which is how I characterized the abstract above, as meaning, is context dependent. When you characterize this relation, the abstract, as a thing, you characterize it as static, unchangeable. This is what allows you to say that it is the same as manipulating symbols devoid of meaning, the symbol must always represent the exact same thing. But here is where this thing represented, the abstract, fails the law of identity, the meaning, which is the relation between the mind and the symbol, is context dependent and does not always remain the exact same.
Quoting fishfry
I think we've discussed this enough already, for you to know that I denounce all set theory as ontologically unsound, fundamentally. It doesn't mean a whole lot though, only that I think it's bad, like if I saw a bunch of greedy people behaving in a way I thought was morally bad, I might try to convince them that what they were doing is bad. However, if it served them well, and made their lives easy, I'd have a hard time convincing them.
Quoting Wayfarer
I think that modern Platonism treats the abstract as things. This is what allows them to define static relations between these things, as fishfry describes in set theory. But if the abstract is not really things, then this static nature is unsupported, and so are the static relations unsupported.
Quoting Wayfarer
It appears, that what you are saying here is that there are some sort of ways of thinking, which thought must conform to in order to be rational. I would agree, but the specifics of the correct way is something to be determined. So I will ask you a question to see if you might have an answer. Wouldn't the way of thinking, which would be judged as the rational way, be itself dependent on and therefore determined by the particular situation, or context? So for example, there is some sort of way of thinking which is the correct way for a particular situation, and this transcends all individual minds, making it the correct way for any mind, in that situation. But that correctness, and the way of thinking itself which is correct, is determined by and specific to the particular situation itself. Would you agree with this? And what about the inverse? Is it possible that there is one correct way of thinking which transcends all situations as the correct way of thinking no matter what the situation?
But, the point about universals is that they're universally applicable, isn't it? They're applicable in any context. Think about scientific laws, which I think must in some sense be descended from such ideas. Water doesn't sometimes flow uphill, for instance. Think also about Kant's deontological ethics, which individuals are obliged to conform to if their actions are to be ethically sound.
Furthermore, I can't arbitarily designate the rules of math or the laws of logic, I have to conform to them, as much as I'm able (which in my case, is not very much). I can adapt them to my situation, I can use them to advantage, but I can't change them. (Again, the clearest exposition of these ideas are in the Cambridge Companion to Augustine, on the passage on Intelligible Objects.)
Quoting Metaphysician Undercover
Modern thought treats everything as a thing. (Who's paper is it, 'What is a thing'? Heidegger, I think.) Anyway, the point is, the modern mentality is so immersed in the sensory domain, that it can only reckon in terms of 'things'. Things are 'what exists' - which is what throws us off about mathematical concepts, they're not things, but they seem real, so 'what kind' of reality do they have? In our world, real things can only be 'out there', the only alternative being 'in the mind'. But in reality, 'out there' and 'the mind' are not ultimately separable - hence, as I say, the logic of objective idealism. But it takes a shift in perspective to see it.
I confess I didn't understand the OP at all. They seemed to be saying that there is only one mathematical object, all of math. I couldn't parse that. I didn't join the thread till you were talking about something I could at least understand.
But now you say, "it's incorrect to call mathematical objects "objects" at all, because they do not fulfill the requirement of identity." When a while back you disagreed that 2 + 2 and 4 represent the same mathematical object (regarding which you are totally wrong but nevermind), that was one thing. But now you seem to be saying that 4 = 4 is not valid to you because mathematical objects don't fulfill the law of identity. Am I understanding you correctly? Do you agree that 4 = 4 and that both sides represent the same mathematical object? Or are you saying that since there aren't any mathematical objects, 4 = 4 does not represent anything at all?
Quoting Metaphysician Undercover
4 = 4 is true by the law of identity, yes or no? I can't believe I'm even having this conversation. You've never convinced me that your mathematical nihilism isn't an elaborate troll.
Quoting Metaphysician Undercover
Probably nowhere, as I knew this would. I imagine you knew it too.
Quoting Metaphysician Undercover
On my Platonist days I say that. But if you object, I am perfectly willing to do exactly the same math, but regarding it as a purely formal game of symbol manipulation, no different in principle than chess. Then you have no philosophical objections, any more than you would to formal games like chess or Go or Parcheesi. 4 + 4 and 2 + 2 = 4 are legal moves in my game. It doesn't matter to me. Do you at least accept that math can be regarded as a formal game without regard to meaning? It's actually often helpful to think of it that way even if you secretly believe otherwise. One can be pragmatic regarding one's philosophy.
Quoting Metaphysician Undercover
Well then you are the Platonist. What is the meaning of the way the knight moves in chess? Clearly there is no real world referent. Nor is there any real world referent for many of the constructs of higher math. I'm willing to stipulate, for purposes of this discussion, that there are no real world referents for any of the constructs of math. So what? As long as the rules are consistent and the game is fun, we can all play.
Quoting Metaphysician Undercover
No, you are the one saying that. I'm saying that if you don't believe 4 represents an abstract mathematical object, then it's perfectly ok to regard it as a meaningless symbol subject to the laws of arithmetic, which can be mindlessly encoded in a computer program like your calculator. When you punch '4' into your pocket calculator, the circuitry doesn't know what 4 means, but it perfectly manipulates 4 according to the rules with which it's programmed.
Quoting Metaphysician Undercover
Ok fine, it's all a meaningless formal game. It makes no difference to me. But if you don't think 4 is a thing, you are most definitely a mathematical nihilist. When you go to the store and buy a dozen eggs, do you make these same points at the checkout stand?
Quoting Metaphysician Undercover
Truly I stopped following you back when you claimed that 2 + 2 and 4 don't represent the same mathematical object, when in fact they do. And when I gave you a purely formal syntactic proof that they represent the same thing, and you refused to even engage with my argument. You didn't say, "I reject the Peano axioms," or "I have it on good authority that Giuseppe Peano cheated at cribbage and is therefore not to be trusted," or "You made a mistake on line 3," or anything like that. You simply ignored the argument entirely, despite my asking you several times to respond. You have yet to demonstrate that you're having a serious conversation with me.
Quoting Metaphysician Undercover
I have no trouble with time-dependent assignment of meaning. But, are you claiming that 4 means one thing to you today and other thing tomorrow? What ever are you talking about?
Quoting Metaphysician Undercover
Well yes, that's your nihilism speaking again. If I say 4 = 4 and you assert that the symbol '4' may have different meanings on each side of the equation, you are the crazy one. I don't mean that in pejorative sense, it's an accurate description.
Of course a symbol like 'x' may mean one thing in one context and a different thing in another, but I truly hope you are not thinking that this is a very deep or significant point. Within a given context, a symbol only has one particular meaning; otherwise we can't do math, we can't do science, we can't even get on the bus. "Oh, today the #4 bus goes to Liverpool. You must want yesterday's #4 bus that went to Bristol." Come on, how can you expect me to take you seriously when you assert such nonsense?
Quoting Metaphysician Undercover
In the string 4 = 4, does the symbol '4' refer to the same thing on each side of the equation? Is this or is this not an instance of the law of identity?
Quoting Metaphysician Undercover
Well ok you're a mathematical nihilist and you don't deny it. But I doubt you actually live like that. You couldn't pay your bills or read the paper, if the meaning of the symbols keeps changing for you.
Quoting Metaphysician Undercover
Set theorists are morally bad people? Who need to be shown the error of their ways? Wow you are far gone my friend. Are you a type theorist? A category theorist? Or do you feel that those people are morally bad too? Not just wrong, but morally bad. I'm genuinely curious about this. Is it just set theory? Or do you feel this way about all historical attempts at mathematical systemization and formalization, from Euclid on down to the present?
Quoting Metaphysician Undercover
You haven't convinced me that you're serious about anything you write. At least when you converse with me.
Ok to make this short, can you please just respond to these two questions:
* Is 4 = 4 an instance of the law of identity; or does the symbol '4' have a different meaning on each side?
* Is Euclid morally bad by virtue of attempting mathematical synthesis?
I agree, but the universality of universals is exactly what makes them incompatible with identity which is what particulars have. This produces a hole, or gap in human understanding, because the material reality, to which we apply these universals in our attempts to understand, consists of unique particulars. This means that there is always a deficiency in human understanding. The point in enforcing a "law of identity", is to recognize and adhere to this understanding, that this gap exists, so that we do not push Platonism to its extremes, claiming that the physical universe is composed of mathematical objects. This is impossible, because mathematical objects are universals, but the universe is composed of unique particulars. The gap of incompatibility between these two demonstrates that such extreme Platonism, better known as Pythagorean Idealism, cannot be true.
Quoting Wayfarer
This goes both ways. There are people who literally make up, or create axioms of mathematics, it's what fishfry calls pure mathematics. We must ensure that the mathematical axioms which we employ conform to reality or else they will lead us astray. Therefore it is actually necessary that we do change mathematical axioms as we try and test them. And if you look at the history of them you will see that they evolve, just like knowledge evolves, and living beings evolve. This implies that we must accept such things as evolving properties of living beings, rather than eternal immutable objects.
That the rules of math, and laws of logic appear to you as something which you cannot change, is the result of many years of usage by many different people. They are tried and tested so they are what we use. If, in your occupation there are rules which must be applied in order for you to fulfill your job, then you cannot change those rules or else your job would not get done. However, the world is full of innovative and creative people who could come up with new rules, which fulfill an end different from what you are doing, but is judged to be better than yours, and renders yours obsolete. Therefore this idea of "I can't change them" is just an illusion. Yes, if you want to keep doing what you are doing, you cannot change them, but if you quit what you are doing, and adopt other rules which are conducive to something else instead, which renders what you were doing before as obsolete, you really do change them.
Quoting Wayfarer
This I completely agree with, but again we have to be aware of where things go the other way. Philosophers looking toward the reality of Ideas describe them in terms of objects to facilitate understanding. However, we can see that thinking is an activity and it doesn't really exist as objects. On the other side of the coin, we see physicists who look at objects and use mathematical ideas to describe them in terms of activity. So we can see that the world of physical objects gets reduced to the activity of energy, because this is compatible with the realm of thought, mathematics. Now we have no adequate principles to separate objects from activities, and it's an ontological mess.
Quoting fishfry
I guess you don't remember the key points (from my perspective) of or previous discussions. What I objected to was calling things like what is represented by 4, as "objects". I made this objection based on the law of identity, similar to the op here. You insisted it's not an "object" in that sense of the word, it's a "mathematical object". And I insisted that it ought not be called an object of any sort. So you proceeded with an unacceptable interpretation of the law of identity in an attempt to validate your claim. What I believe, is that "mathematical object" is an incoherent concept.
Quoting fishfry
This depends on what = represents. Does it represent "is the same as", or does it represent "is equal to"? From our last discussion, you did not seem to respect a difference between the meaning of these two phrases. If you're still of the same mind, then there is no point in proceeding until we work out this little problem. This is why I say context of the symbol is important. When the law of identity is represented as a=a, = symbolizes "is the same as". But when we write 2+2=4, = symbolizes "is equal to". If we assume that a symbol always represents the very same thing in every instance of usage, we are sure to equivocate. Clearly, "is the same as" does not mean the same thing as "is equal to".
Quoting fishfry
No, of course not, that's clearly a false representation of what math is. That would be like saying that 2+2=4 could be considered to be valid regardless of what the symbols mean. That's nonsense, it's what the symbols mean which gives validity to math.
Quoting fishfry
Yes, that is exactly the case it can even mean something different in the same sentence. When some says "I reserved a table for 4 at 4", each instance of 4 means something different to me. And, as I explained to you already, when someone says 2+2=4, each instance of 2 must refer to something different or else there would not be four, only two distinct instances of the very same two, and this would not make four.
This is the point of hylomorphic dualism, is it not?
Thomistic Psychology: A Philosophic Analysis of the Nature of Man, Brennan.
Quoting Metaphysician Undercover
I say it does. I think you're splitting hairs for the sake of argument.
Good one! :rofl:
This might be the key to their salvation: { } = {N} :cool:
My mind has blocked them out as traumatic experiences.
Quoting Metaphysician Undercover
What's incoherent is you objecting to 4 = 4 as an instance of the law of identity; and claiming that 2 + 2 and 4 refer to different mathematical objects, when in fact I showed you a formal proof that they represent exactly the same mathematical object. A proof that you refuse to this day to even acknowledge, let alone refute or discuss. That's to your shame. You pretend to be an honest conversant but you are not.
Quoting Metaphysician Undercover
Bill Clinton on his slimiest day never reached such heights of bullshit. Pardon my French.
Quoting Metaphysician Undercover
There's no difference. And you have failed to articulate even a plausible argument for a difference.
Quoting Metaphysician Undercover
You got that right.
Quoting Metaphysician Undercover
A point on which we agree. There's no point in proceeding. You're trolling me as you have been for two years now.
Quoting Metaphysician Undercover
Bullpucky to da max.
Quoting Metaphysician Undercover
So you're right and Hilbert and Euclid are wrong. Ok. Forgive me if I choose to disagree.
Quoting Metaphysician Undercover
LOL. That's a good example and I take your point. Natural language is often ambiguous. In a given mathematical context, a given symbol holds the exact same meaning throughout. Dinner isn't a mathematical context. But yeah that's a good example of the problem with ambiguity of natural language. And thanks for the chuckle.
Quoting Metaphysician Undercover
That's utter nonsense. Interestingly though, I came across this identical line of reasoning this morning. I don't know what the source or full context of this page is, but you either have a kindred spirit, or you wrote this. Author is objecting to the equation 2 = 1 + 1. He says:
Hard to believe there are two people who assert this nonsense, not just you alone. Unless as I say you are the author. But I concede that IF you are not the author, then you are not unique in your confusion regarding mathematical objects.
http://www.henryflynt.org/meta_tech/that1=2.html
By the way there is a standard formalism for obtaining multiple copies of the same object, you just Cartesian-product them with a distinct integer. So if you need two copies of the real line [math]\mathbb R[/math], you just take them as [math]\mathbb R \times \{1\}[/math] and [math]\mathbb R \times \{2\}[/math]. It's not that mathematicians haven't thought about this problem. It's that they have, and they have easily handled it. As usual you confuse mathematical ignorance with philosophical insight.
In any event, you avoided (as you always do when presented with a point you can't defend) my question. If set theorists are not only wrong but morally bad, is Euclid equally so? You stand by your claim that set theorists are morally bad? Those are your words. Defend or retract please.
Quoting Wayfarer
Thanks man, sometimes @Meta makes me question my own sanity.
In Plato's dialogues, whenever metaphysical ideas are encountered, they are nearly always referred to as 'likely stories'and treated with a degree of diffidence. Plato was never close to a naive or even scientific realist, many of the ideas in Platonic dialogues are obviously mythological, symbolic, or allegorical. Accordingly, I don't claim to know whether the Platonic forms are real. But I will say, that this 'traditionalist' account, preserved mainly in various forms of Christian Platonism, is a plausible and coherent philosophical view which makes much more sense than modern irrationalism. (Of course, from the modern p-o-v, this is simply clinging to a nostalgic and bygone philosophy, but I'm ok with that.)
Are you serious? As human beings, you and I are equal, based in a principle of equality. Clearly we are not the same. A judgement of equality is based in a principle of measurement, volume, weight, temperature, species, whatever. It allows that two distinct things are equal, by the precepts of the principle. They are the same volume, or the same weight, the same temperature, or the same species. Notice how in the concept of "equal", "the same" is qualified so that it is what is attributed to the thing, volume, weight, etc., which is said to be the same, not the thing itself. Under the law of identity, a thing is the same as itself. So it is impossible that two distinct things are the same thing, as we say that two distinct things are equal. By that law, we can only use "the same" to refer to one and the same thing, the very same thing. This is not a matter of splitting hairs, there's a fundamental difference between two distinct things which are the same in some way (equal by that principle), and one thing, of which no other thing can be said to be the same thing as.
Quoting fishfry
On the basis of that statement I am concluding that proceeding with any discussion with you on this matter is pointless because your mind is liable to block out anything I write.
Quoting fishfry
You obviously do not know the law of identity. I had to spell it out for you already. You objected, and offered some axiom of equality which is obviously not the law of identity. This statement above, indicates that you clearly did not take the time to learn it yet. From what I learned last time, until we agree as to what the law of identity stipulates, further discussion on this issue is pointless.
Quoting fishfry
So the issue is, in the context of mathematics, does = mean equal to, or does it mean the same as? I'm sure you can grasp the fact that you and I are equal, as human beings, but we are not the same as each other. Therefore, I'm sure you can accept that equal to has a different meaning from the same as. Which does = symbolize in the context of mathematics?
Quoting fishfry
Reason is contagious, it tends to catch on. Notice that the op agrees with me as well. And, I think jgill agreed with me on this point in that other thread as well. I don't understand why the obvious appears as nonsense to you. It's very clear, that if 1 always referred to the same object we could not make 2 out of two instances of 1, we'd always have just one object symbolized.
Quoting fishfry
It doesn't matter how you formalize it, the point is that it violates the law of identity. "Multiple copies of the same object" is exactly what is outlawed. You can rationalize your violation of any law however you like, but it doesn't change the fact that you violate the law. You can show me a thousand objects, and insist that according to your axioms they are all one and the same object. So what? All this indicates is that your axioms are faulty.
Quoting fishfry
Did I say set theorists are morally bad? No, it was an analogy. The analogy was that if I saw set theorists doing something I thought was wrong (bad), I might be inclined toward explaining to them how I thought what they are doing is wrong, just like if I saw someone behaving in a greedy way which I thought was morally wrong, I might be inclined to explain to them why I thought what they were doing is morally wrong. The point being that it really doesn't make very much difference to me, in my life personally, if these people, either the set theorists, or the greedy immoral people, continue along their misguided pathways. Nevertheless, I might take it upon myself to make an attempt to point out to them how their pathways are misguided.
Quoting Wayfarer
This is where Aristotle parts from Plato. In Plato's Timaeus particulars are supposed to be in some way derived from universal forms. But Plato is incapable of describing the mechanism by which a universal form could create the existence of a particular individual. He found the need to posit "matter" as the recipient of the form, in order to account for the particularities of the individual. The peculiarities of the individual are due to the matter. But when Aristotle developed this idea he discovered that matter itself cannot account for any of the properties of an object, and so each individual thing must have a unique form proper to itself. That's his hylomorphism
This was the fundamental question of his metaphysics, why is a thing the unique thing which it is, rather than something else. He said the commonly asked question of why there is something rather than nothing cannot be answered, and is therefore a fruitless question. So he suggested the proper question to ask of being qua being, is why is there what there is instead of something else. Why is each thing the unique and particular thing that it is, instead of something else. This led him to the conclusion that there is a unique and particular form which is responsible for each thing being the particular thing which it is. Hence the law of identity as formulated, each thing has a unique identity, it is the same as itself, and nothing else. For Aristotle, this is the reality of the particular.
Now who’s equivocating? Social equality means ‘treating everyone the same’. In that sense it means treating them as ‘equal’ but that is a specific use in a specific context. When we’re discussing the ‘=‘ sign we are by definition discussing a symbol which denotes strict identity. So in the expressions a=a and 2+2 = 4, it is meaningless to say that each occurrence of the symbol ‘a’ or ‘2’ refers to a specific object. It refers to a symbol, which is useful precisely because it can be applied universally.
Quoting Metaphysician Undercover
I’m aware that Aristotle rejected the Platonic doctrine of forms, but he still maintained a role for universals. ‘In Aristotle's view, universals are incorporeal and universal, but only exist only where they are instantiated; they exist only in things.’ So they don’t exist in another domain or realm, which is what Platonism appears to propose. I don’t think there’s any ‘mechanism’ by which ‘forms’ are instantiated as individuals. Perhaps one way of conceiving it would be that for a creature to fly, then wings have to assume a certain form; you can’t fly using weight-bearing limbs. So wings evolve towards a certain form, because of the function that they need to realise, not because there’s an ‘ideal wing factory’ situated off in la-la land. The form is what something has to take in order to exist. (Actually there’s a really good Kelly Ross essay on this, Meaning and the Problem of Universals.)
Quoting Metaphysician Undercover
I’d agree with that, although I’d be surprised if it was really an FAQ in his day.
That's not really true. Wikipedia says the sign is "used to indicate equality in some well-defined sense... In an equation, the equal sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value." Clearly the '=' sign is not used in mathematics to denote strict identity. It might be defined in some axiom of set theory, as denoting strict identity, but that definition would not reflect how it is used, therefore that definition ought to be rejected.
Familiarize yourself with the law of identity. It states that "same" identifies one thing and only one thing. There cannot be two things which are the same. Then, take a look at how the = sign is employed in an equation. Clearly the right and left side of an equation cannot both represent the exact same thing, or else the equation would be completely useless. We'd have to ask, if there's something represented on the right side, and the exact same thing is represented on the left side, and we already know that we are just representing the exact same thing in two different ways, because use of '=' indicates that we know that the two are the exact same thing, then what are we doing with the equation? The equation would be doing absolutely nothing for us. But the fact is, that we represent something different on each side, we say that the two are equal, not that the exact same thing is represented twice. So '=' does not indicate strict identity. Therefore if someone proposes to you that "=" denotes strict identity, as a mathematical proposition, an axiom to be used as a premise, you ought to reject that premise as false because it will lead to unsound conclusions.
Quoting Wayfarer
Do you recognize that in Aristotelian physics, each individual material object has a particular form which is unique to itself, and this is expressed in the law of identity? If so, then the point of the op is that universal forms do not have such a particular form, this would be incoherent. Therefore the law of identity is not applicable to universal forms, nor can we say that universal forms are particular objects.
That is exactly what I said.
Quoting Metaphysician Undercover
It's not a matter of 'recognising it', this is something that I have only ever read in your posts. If you provide a reference I'd be obliged.
There is something really absurd here. So, you're saying, that in the expression A=A, that this expression only refers to particular instances of 'A'? That in order for 'A' to be 'A' then we have to refer to a particular instance of 'A'? That when we say, 2 + 2 = 4, that you're saying 'hang on! Which individual instances of '2' are you referring to?'
The point about 'the law of identity' is that both sides of the equation absolutely represent the exact same thing, namely, 'A'. That doesn't make the equation 'useless', it is why it is meaningful.
I'm not going to pursue this, as I think life's too short to argue about the meaning of the very first element of logic.
From the perspective of appearances of symbols you have a point. Clearly, 2+2=3+1 displays symbols on either side that are not the same as symbols on the other side. So the two sides are not "the same" in this sense. But this is a triviality among mathematicians - and the general public - who associate with each side a mathematical entity, the number 4. Likewise, Four=4 shows different symbols representing the same mathematical item. However, I believe your position exceeds these parameters and is somehow more "fundamental".
This seems like a silly game of distinction without a difference that could only appeal to intellectual descendants of medieval scholasticism. But I could be wrong.
I will say that logic, like mathematics, like Shannon information, is not about meaning - meaningfulness is assumed upon use. It’s about the relation between signs (not things) within a specific value system. The equation is ‘possibly meaningful’ only within that system, in which both sides represent the exact same value, regardless of any particular instance, and regardless of its possible meaning. So long as you assume a perfect alignment in instances of value structure and possible meaning, then both sides of the equation 4=4 are ‘the same’. In reality, it’s more like a six-dimensional ratio (0, 0, 0, 0, 4x, 0) = (0, 0, 0, 0, 4x, 0), with only some of the redundancy removed - this equation 4=4 is entirely redundant in logic, mathematics and Shannon information theory. It has meaning only when the sides are NOT identical.
That is exactly the meaning of ‘abstraction’.
Well, it is an answer, but why is it the answer? Why one object and not two or 42 or all of them? Why do you elect to be a lumper and not a splitter?
I think you need to back up a bit and tell us why the question matters. What difference would an answer make?
Obviously not. You said:
Quoting Wayfarer
"Strict identity" is what is defined by the law of identity. The "=" in mathematics signifies that two distinct things have the same value. It does not signify that what is on the right is the same thing as what is on the left, as "strict identity" indicates.
Quoting Wayfarer
I think you ought to take some time to study the law of identity. Check Wikipedia, Stanford, and Internet Encyclopedia of Philosophy to get a good consensus. Remember, "A=A" is just a formal representation, and the true representation of the law is stated as a proposition, one of the three fundamental laws of logic, including also noncontradiction, and excluded middle. What the law of identity says is that a thing is the same as itself. What this means, is that a thing is unique to itself, and is not identical to anything else. Check the Leibniz interpretation. He says that if we try to assert that two distinct things have the exact same properties, they are in fact one and the same thing.
So, what I am saying is that when '=' is used to symbolize the law of identity, it has a completely different meaning from when it is used in '2+2=4'. The representation, 'A=A' is just a convenience. The 'A' symbolizes an object, any object. The '=' symbolizes 'is the same as'. So the representation means that an object is the same as itself. I argue that it is a misinterpretation of what '=' symbolizes in mathematics, to assert that it means 'is the same as', as it does in this representation of the law of identity. Therefore we ought to respect the fact that what '=' symbolizes in the expression 'A=A', which is meant to represent the law of identity, is not the same as what '=' symbolizes in its mathematical context.
Anyway, let's leave this issue for now, and I'll answer your other question which is much more interesting to me.
Quoting Wayfarer
OK, but this is a complicated issue, Aristotle is somewhat ambiguous, and there are numerous interpretations, so it will take some work on your part. I'll take the time to take you through a number of references which support my interpretation, I can only hope that you'll take the time to try and understand.
First, we look at Physics Bk.2 Ch.3, where "form" is defined in relation to the four causes. The form of a thing is said to be the thing's essence, or definition. At this point you need to adhere to Aristotle's description of "essence" and not be swayed by later interpretations which attempt to decisively remove accidentals from a thing's essence, giving us the term "essential".
Now let's proceed to Aristotle's extensive description of the particular individual, in Metaphysics, Bk.7. I suggest you read the section of Ch.4-11 numerous times, because it's not easy reading. However, it's very important, and the ambiguity will make you lean one way at one time, and another way at another time, possibly allowing previous biases to sway your overall interpretation. The goal I think ought to be to understand what is written, interpret it in a way which makes sense to you. There is really a need to refer to other writings, like Categories, and On the Soul, to fully understand his use of terms, but I'll try to guide you on these other references.
Starting at Ch.4. "The essence of each thing is what it is said to be 'propter se"'. The footnote to my translation (W.D. Ross) states that it is convenient to translate 'propter se' as "in virtue of itself". If we proceed, we find in Ch.5 how "essence" is related to "substance". The closing sentence of the chapter reads "Clearly, then, definition is the formula of the essence, and essence belongs to substances either alone or chiefly and primarily and in the unqualified sense." Referring to "Categories" Ch.5, Aristotle distinguishes primary and secondary substance. In the truest and primary sense substance is the individual. In the secondary sense it is the species within which the primary substances are included.
Proceeding to Ch.6 of Bk.7 Metaphysics, he question whether a thing and its essence are the same "for each thing is thought to be not different from its substance, and the essence is said to be the substance of each thing". So the problem of accidentals is now brought up, and it appears like a thing cannot be the same as its essence. But in the case of supposed self-subsistent Ideas, Forms, it is shown to be impossible, as incoherent, that a Form's essence could be different from the Form itself. "Each thing itself, then, and its essence are one and the same in no merely accidental way, as is evident both from the preceding arguments and because to know each thing, at least, is just to know its essence, so that even by the exhibition of instances it becomes clear that both must be one." 1031b,18. "Clearly, then, each primary and self-subsistent thing is one and the same as its essence. The sophistical objections to this position, and the question whether Socrates and to be Socrates are the same thing, are obviously answered by the same solution; for there is no difference either in the standpoint from which the question would be asked, or in that from which one could answer it successfully." 1032a,5.
However, Aristotle leaves the door open to ambiguity here, by allowing that in an accidental way, a thing is not the same as its essence. So we need to proceed further, and understand the nature of accidentals, the existence of which appears to drive a wedge between a thing and its essence. So Ch.7 proceeds to question the nature of "comings to be" with a comparison made between natural things and artificial things. To fully apprehend Aristotle's position here it is necessary to understand how he defines "soul" in "On the Soul". The question here is the relationship between a thing's matter and its form. I had an extensive discussion with dfpolis a year or two ago, on this chapter. It is described by Aristotle, that in artificial things, the form of the thing which will come to be as a material thing, exists in the soul of the artist, and is then put into the matter. Aristotle compares this to natural things, and concludes that the process must be similar. The form of the individual thing must be prior to its material existence, and some how put into the matter. Df argued against this point, insisting that Aristotle's position is that the particular form which the individual thing will have, is already intrinsic to the matter. However, careful interpretation will reveal that this is rejected because of infinite regress.
In any case, the line we need to follow, is the idea that the "formula" precedes the existence of the material thing. Now the question is asked, to what extent is the matter a part of the formula (1032b-1033a). When this occurs, the suffix "en" is used to determine the matter, "brazen", "wooden", etc.. "The brazen circle, then, has its matter within its formula" 1033a,4. At this point, we can see how the accidentals of the individual, which are commonly attributed to the matter, may be transferred to the form, when the matter becomes part of the form. Notice however, that this is how artificial production is represented, and it is necessary that the form be supported by the soul of the artist. Without this soul, we lose the ability to separate the form completely from the matter, the separation which allows that the matter itself is part of the formula, and we are left with df's argument that the uniqueness supplied by the accidentals inheres within the matter.
So, I referred df to Ch.8, which ties together natural things, and artificial things, as being the same type of process. At 1033b, we can see that if the accidents of the thing which will come to be are accounted for as being within the matter, we have an infinite regress. Therefore we must assume a separation between the form and the matter, in both natural and artificial things, such that the form comes from someplace else, the soul in the case of artificial things. And, I'll interject here to remind you (as relevant to the subject we are discussing), that the matter may become part of the formula, so that the accidentals which are commonly attributed to the matter, are in reality, part of the form.
"It is obvious, then, from what has been said, that that which is spoken of as form or substance is not produced, but the concrete thing which gets its name from this is produced, and that in everything which is generated matter is present, and one part of the thing is matter and the other form."1033b 17.
The remainder of this section which I recommended, up to and including Ch.11, deals with the difficulties, which are abundant, involved in trying to understand this relation between form and matter. Read the following section carefully, and recall the distinction made in "Categories" between "primary substance" referring directly to the individual (Callias for example), and secondary substance, the species (man). Notice how he says that there is no formula which includes the matter because the matter is indefinite, but with reference to primary substance itself, (which is the individual itself, therefore the application of the law of identity), there is a formula which includes the matter.
"What the essence is and in what sense it is independent has been stated universally in a way which is true of every case, and also why the formula of some things contains the parts of the thing defined, while that of others does not. And we have stated that in the formula of the substance the material parts will not be present (for they are not even parts of the substance in that sense, but of the concrete substance; but of this, there is in a sense a formula, and in a sense there is not; for there is no formula of it with its matter, for this is indefinite, but there is a formula of it with reference to its primary substance---e.g. in the case of man the formula of the soul---for the substance is the indwelling form, from which and the matter the so-called concrete substance is derived; e.g. concavity is a form of this sort, for from this and the nose arise 'snub nose' and 'snubness'); but in the concrete substance, the matter will also be present, e.g. a snub nose or Callias, the matter will also be present." 1037a 21-32.
What seems to be neglected in mathematical Platonism, is that '2+2' signifies an operation, and '3+1' signifies an operation. The two operations are clearly not the same, though they are in some sense equal. The more complicated the equation is, the more complicated are the operations which are signified. The difference between equal operations can be quite significant. To reduce these complex mathematical operations to simple mathematical objects, and assert that substantially different operations are actually the same mathematical object is a dreadful misrepresentation of what mathematics really is.
Quoting jgill
I think it's a matter of metaphysics, ontology. But what is really at stake here is the meaning behind mathematical symbols, and therefore an understanding of what mathematicians are actually doing. I think that fishfry for example, demonstrates a very naive understanding of what mathematicians are actually doing by insisting that mathematical operations could be carried out in the same way which they are, even if the symbols signified nothing. This may be true of some formal logic, but in mathematics, the possible operations are determined by the meaning of the symbols. So it is impossible to separate the operations from what the symbols signify, as is done in formal logic. Therefore the attempt to represent mathematics as a type of formal logic which provides a separation between the operations which are performed with the symbols, as distinct from what is represented by the symbols, is a common misunderstanding of the nature of what is actually symbolized by the symbols in mathematics. In reality, what is signified by the symbols is operations, not objects. Even the most simple mathematical symbols like 4, can be understood as denoting an operation of grouping four individuals, and the symbol 1 denotes an operation of individualization. So we cannot get beyond the fact that operations are intrinsic within, and essential to, the mathematical symbols. Therefore the attempt to separate symbols from operations would leave us no access to any operations, and no mathematics.
Quoting Possibility
What's your opinion on my reply to jgill above, Possibility? Do you agree that what the mathematical symbols represent are operations? So when we have an equation, we say that the operation on the right side has the same value as the operation on the left side. And when we say that 4=4, the symbol 4 refers to a grouping of individuals, and we say that one grouping of four has the same mathematical value as another grouping of four. Therefore relative to mathematical value, a grouping of four is "the same" as any other grouping of four, but relative to identity, the two groups are clearly not the same. Notice how I refer to the "grouping" of four, because this is an activity, an operation, carried out by the sentient being which apprehends the four individuals as a group of four. Likewise, to apprehend one thing as an object, an individual unity, is an operation (individuation) carried out by the sentient being which perceives it that way. This fundamental act of individuation is the basic premise for mathematics. Therefore the axioms of mathematics need to be well grounded in the law of identity which stipulates the criteria for being an individual.
So, again, you're saying that every occurence of 'A' is unique? I still think you're confusing the law of identity, with the meaning of individual identity, which are different subjects even if related.
I did take the time to read your argument on essence, substance and so on. As you note it is replete with difficulties, ambiguities and aporia. This is a deep problem with Aristotelian metaphysics, generally - the difficulty of arriving at any ultimate definition of the fundamental terminology, I think due to the inherent limitations in reason itself. But, it's still worth studying and I appreciate the time you've taken to spell it out. It's one of the subjects I'm trying to find time to understand better.
The reason I mentioned the (non)distinction between sameness and identity is because it is pertinent to the question of mathematical objects. If we begin by thinking of identity as
Quoting Metaphysician Undercover
We can only investigate identity by looking at the qualities, properties, parts, etc. of an object in order to identify it. The essence of the object, or its form, is never what differentiates it for this purpose. Instead, it is what is accidental to it that allows it to be identified.
Plato, I think, takes identity "all the way" and so sees this process of identification as moving these accidents (which allowed identification) into the essence of the categorically more specific object that is identified. For instance, I see a person, then by perceiving certain accidents of that person (beard, tall, male, etc.) I realize the person is my father. What was accidental to the person (and to fathers in general) is actually essential to the individual that is my father. But if we admit an essence that is my father, he loses his individuality since some other person with the same (identical?) properties would also be my father.
Certain proofs in mathematics hinge on the dissolution of separate identities. For instance, the proofs on this page about lines tangent to a circle presuppose the existence of points with certain accidents. It is through this method that the contradiction necessary for the proof is shown. This reflexively shows that the points themselves cannot have the accidents which were assigned to them and thus the essence of the points of tangency is grasped. The proof equivalently amounts to showing that these points are the same.
Plato's mistake, it seems, is not noticing that identity only arises insofar as objects are not the same. It is an instrument of abstraction or speculation. Its persistence indicates an indefinite understanding. This implies it is never really present in complete understanding, actuality, truth, etc. Perhaps he was disturbed by the thought that his own philosophy suggested that we do not really have individuality or self-ness. It may have also threatened some of his assumptions about Ethics.
Kant also comments on this in Critique of Pure reason, Transcendental Doctrine of Method, Chapter I, Section I:
(my emphasis)
Kantian intuition therefore must involve this process of construction and dissolution of identity, not as sameness but as arbitrary differences which ultimately prove insubstantial for the concept.
Later, in section 4, he writes
Kant seems to use Identity to mean sameness, or more specifically that to deduce two things as the same is to show that they share the same identity. This is further supported by Division I, Endnote 1. So even Kant doesn't really distinguish sameness from identity.
I have more to say but I've run out of brain juice...
Plato was concerned with the identity of the transcendent soul, the identity of Forms in relation to particulars, and the identity of abstract parts with the whole. Accidents, essence, object are not in Plato.
You appear to suggest that mathematical axioms are similar to theory in physics. String theory, however, seems un-testable at present. Does it then lead us astray? If you were to say it does, how could you possibly know? How might you test the Axiom of Choice?
It's interesting to read perspectives of mathematics that I suppose could be called pre-foundational to distinguish them from formal foundation theory that fishfry is good at explaining. These are notions I never entertained while active as a mathematician. Of course, I didn't spend time looking into formal foundations either.
fishfry refers to math as a game, and it certainly is that. But a practicing mathematician may lose that perspective and math may assume a kind of non-physical solidity and seem "real", even when it's not obvious that it may be related to physical phenomena. Similarly chess probably seems "real" to serious devotees. Incidentally, MU, "pure mathematics" simply means not immediately applicable to the physical world. I've dabbled in this sort of math for decades.
It seems like you're not familiar with the law of identity.
Here's Wikipedia: "In logic, the law of identity states that each thing is identical with itself. It is the first of the three laws of thought, along with the law of noncontradiction, and the law of excluded middle."
Now here's Wikipedia on Leibniz' identity of indiscernibles: " The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa; to suppose two things indiscernible is to suppose the same thing under two names."
The law of identity refers to all things, and I'm not sure what you mean by "individual identity". The law of identity is a fundamental ontological principle which represents the uniqueness of a thing. So, if 'A=A' is meant to represent the law of identity, then A represents an object, and both instance of A represent the exact same object, and '=' signifies "is the same as". A thing is the same as itself.
Quoting Wayfarer
I strongly recommend reading Aristotle's "On the Soul". At first glance, it appears outdated, but it is very well written, not difficult, but quite informative, giving good examples of Aristotle's usage of fundamental ontological terms like matter, form, actual, and potential. Then read his "Metaphysics", because much in his metaphysics will seem incomprehensible without the background provided for by "On the Soul".
Quoting Garth
The point though, which makes identity a real and true principle, is that there will not be some other person with the same, identical properties. This is what Leibniz' principle says, such is impossible. And, for simplicity sake, we can see that if spatial-temporal location are considered as properties of a thing, it is truly impossible that two distinct things have the exact same properties.
Quoting Garth
Clearly, each point on the circumference of a circle is distinct, not the same, having a different identity from every other point. If this were not the case, then there would be no definable angles between distinct radii.
Quoting Garth
There are two ways of looking at this. You suggest that identity is not ever an aspect of "complete understanding, actuality, truth, etc.". You might believe that human beings possess "complete understanding...", and therefore identity is not anything real. I believe that identity is real, and human beings cannot ever possess "complete understanding...".
Quoting Garth
This is right, the accidental differences are insubstantial for the concept, but for Kant, we cannot ever know the thing in itself. So Kant is consistent with me, identity is real, but human knowledge will always be incomplete, because those particular aspects of the thing in itself cannot enter into the concept.
Quoting Garth
By the law of identity, identity is sameness, but two distinct things cannot share the same identity because it is incoherent to say that two distinct things are the same thing.
Quoting jgill
I think that some theories and axioms take a long time to test. The problem is that our testing capacity is very limited compared to the wide range of possible situations for application. So we test theories within a very limited range, and when they work within that range, we proceed to apply them to a much wider range where we do not have the capacity to adequately test the results. So for example, we take theories like Einstein's relativity, and we tested them around earth in a very limited range of spatial temporal relations, which we might call the midrange. Then we apply them to the furthest spatial distances in the universe, and the tiniest temporal durations in quantum physics, where we haven't tested them nor can we test them. We have no reason to believe that the theories are giving us accurate results in these conditions because we are operating on the assumption that what is true at the midrange is also true at the extremes. This is explained well in physicist Lee Smolin's book, "Time Reborn", in the chapter "Doing Physics in a Box".
So, in my analogy which suggests mathematical axioms are similar to theories in physics, we could consider the same principle. The axioms might prove themselves very well in all sorts of common applications, but when we get to the extremes, like when infinities enter the equations, they might really be failing us. Boundary conditions for example are very curious things. We could stipulate them arbitrarily, and when we apply them they confirm themselves, through the act of application, as long as we see no reason to doubt them. So it appears like there is a real boundary anywhere that the boundary conditions are imposed, because the mathematics is designed to treat the observations that way through the application of the boundary conditions. .
Quoting jgill
I don't consider any such human activity as a game. Games are played for entertainment, and in general, the goal is to win. You must use a different definition of "game". I'd say you're playing a different game from me, and that would serve to demonstrate that we are not playing a game, because how could we be engaged in the same game, yet playing different games?
Anyway, fishfry goes beyond your definition of "pure mathematics" to claim that "You can, if you like, view the entire enterprise as an exercise in formal symbol manipulation that could be carried out by computer, entirely devoid of meaning. It would not make any difference to the math." This is what I disputed above, because the possible manipulations are determined by the meaning of the symbols. The most fundamental being the meaning of the unit, 1. Furthermore, the meaning of the symbols has been developed over years of application. So there is really no such thing as "pure mathematics" by your definition, unless one is starting with all new symbols never before used, because the symbols employed have already derived there meaning through application.
I’ve found that the term ‘object’ - denoting a consolidated focus of thought or feeling - is often freely applied to physical objects, events or concepts. I find this ambiguity leads to much confusion, and I’ve had numerous discussions with other contributors to this forum regarding the dimensional distinctions between the relation of self-consciousness to, say, an actual object, an operation/event (eg. grouping), a symbol for the concept that represents the value/significance of an event, and meaning prescribed to that symbol.
But mathematics and logic, like computer information systems, are often treated as closed conceptual systems, with any qualitative relations (necessary for the system to be understood) assumed and consolidated: ignored, isolated and excluded. So a ‘mathematical object’ refers to the ‘individual’ symbol for a concept that represents consolidated value/significance of an event - any instance of which is a subjective, temporally-located relation between an observer/measuring device and qualitative relational structures of measurement/observation. But within the isolated conceptual system of mathematics (which effectively assumes and then ignores an alignment of underlying relational structure by abstraction), a ‘mathematical object’ would abide by the law of identity. This from the Wikipedia entry on Law of Identity, referring to violation:
“we cannot use the same term in the same discourse while having it signify different senses or meanings without introducing ambiguity into the discourse – even though the different meanings are conventionally prescribed to that term.”
The Law of Identity applies only in a logical, abstract (closed) system of thought or language. Any ‘mathematical object’ is interpretable in reality only by a self-conscious observer in a qualitative potential relation to both the symbol (to prescribe qualities of meaning) and the event (to attribute qualities of sense or affect). The moment you relate the Law of Identity to anything outside of logic - ie. once you cannot assume an alignment of sense or meaning in discussion - you risk violation.
Sometimes games are played for money or prestige. The professional mathematician finds his activities entertaining, frequently fascinating, and he definitely likes to arrive at a result before others. He likes to win.
Quoting Metaphysician Undercover
Nope
Quoting Metaphysician Undercover
That's one way of looking at it. It's not the way I perceive the discipline.
I wish you would stop saying that. I think it's your interpretation that is idiosyncratic.
Quoting Metaphysician Undercover
Precisely. 'A' represents an object. So, it's an abstraction. As are all symbols and numbers.
Where this all started was your nonsensical claim that:
Quoting Metaphysician Undercover
When it obviously does, despite your obfuscation.
Quoting Metaphysician Undercover
What I mean by ‘individual identity’ is ‘the identity of individual particulars’. It’s exactly what you mean by it. So take two individuals, Peter and Paul. They’re equal in one sense, i.e. equal before the law, but that doesn’t mean they’re the same. You can’t say that Peter and Paul 'are equal', but you can say 'they’re equals'. (Unless they’re being graded for a sport, and Paul is graded higher than Peter, in which case Paul is ‘greater than’ Peter in respect of that ranking i.e. a better tennis player.)
But that has no bearing on the symbolic representation 'A=A' because in that case, we're not referring to particular beings, but to symbols. Same with mathematics. Symbols are abstractions, but due to our rational ability, they have bearing on the world.
The law of identity is a statement concerning the identity of particulars. Here's a quote from Stanford: "Numerical identity is our topic. As noted, it is at the centre of several philosophical debates, but to many seems in itself wholly unproblematic, for it is just that relation everything has to itself and nothing else – and what could be less problematic than that?"
This is what we are discussing, the identity of individual particulars. The point was that an abstraction, a Platonic Form, does not fulfill the conditions of the law of identity ("the identity of individual particulars), therefore it is not an individual particular, and ought not be called an "object". A Platonic Form does not have an identity as an individual particular. An object has an identity as an individual particular. Therefore a Platonic Form is not an object.
Quoting Wayfarer
We are referring to particular objects, you agreed above, 'A' represents an object. An object is an individual, a particular. The phrase 'A=A' is commonly used to represent the law of identity ("the identity of particulars"), which states that a thing is the same as itself. This is what Stanford refers to as "numerical identity", it means 'the very same', 'absolute identity', the "relation everything has to itself and nothing else".
You'll see that Stanford also refers to "qualitative identity", which means that things share properties. The law of identity is not concerned with qualitative identity. When we say two things are "equal" we are using qualitative identity.
An abstraction is a concept, an idea, which is within a mind, as a product of a mind, unless we provide for transcendent existence such as Forms which allows the abstraction to exist outside the mind. So symbols are not themselves abstractions, they are physical things which are meant to signify something. Symbols cannot provide for that transcendent existence of the Forms unless we show that the meaning, the abstraction, somehow inheres within the symbol itself.
But a symbol might signify an object, like a name or a proper noun does, or a symbol might represent an abstraction, as '2' does. Now, the case of 'A=A' is tricky. 'A' represents an abstraction meaning an object, any object. It does not directly represent a particular object. So in that sense, it represents an abstraction. However, that abstraction is as a universal law, a general statement such as an inductive conclusion. The statement is a proposition about particular individuals. Now, 'A=A' is a symbolic representation of a proposition concerning particular individuals.
Therefore, strictly speaking 'A' represents a specific part of an abstraction, it represents all objects. And, we are employing qualitative identity to make a statement about what all objects have in common. However, that statement concerns what distinguishes all objects as different from each other. Therefore the quality which all objects share (they are the same in this respect) is that they are different from each other. This is why I described in the other thread, that identity is a special type of equality. It is an equality which a thing has with itself, and all others are excluded from. It may take on the appearance of contradiction to some (Hegel), but what it really does is separate "same" as categorically different from, rather than opposed to "different". In a sense then it appears as an irrational equality, by that appearance of contradiction, but this is precisely why the accidental properties of objects are unintelligible to us. And it isn't really contradictory, as I said.
Now, the problem arises if we try to exclude those accidental properties as differences which don't make a difference, or something like that. When we say that '2', as a symbol, represents a "number", and that number is itself an object, we have assumed that the abstraction is an intelligible object. But the accidentals which distinguish one instance of '2' from another have been excluded because we assume that each instance of '2' refers to the very same object (as per identity principle). Therefore we do not have a true representation of an object here, because the difference between one instance and another has been excluded for the purpose of claiming "the same object". So, because the fact, or inductive truth about objects, which is pointed to by the law of identity is circumvented, to claim that '2' represents an object, we avoid a true representation of what '2' means. And in reality, "the number" does not fulfill the identity conditions of the law of identity, which are required of every object, as individual particulars.
This allows for the possibility of equivocation. Anyone who insists that "a number" is an object would likely proceed to equivocation. Since '2' does not refer to an object with a particular identity, it will have a distinct meaning depending on the context of usage. But if someone insists that it refers to an object, then it is asserted that it necessarily has the same identity in each instance of usage, as referring to the same object. To insist that the symbol '2' refers to the same object in each application, when it really has a distinct meaning, is to equivocate.
As I described at the end of the last post, misuse of "object" allows freedom for equivocation.
Quoting Possibility
It appears like modern information theory and systems theory have converted our concepts of "information", such that the word now refers to the symbols directly, rather than what the symbols mean. This was discussed in the other thread on information.
Quoting Possibility
I think that the law of identity is actually an attempt to produce a closed system of thought. It is a prescriptive rule as to how we ought to use terms. Of course, as soon as a rule is imposed, there will be violations, that's the point of producing the rule, to distinguish violation from non-violation, and attempt to clear things up. But without the law of identity being enforced, there is freedom of ambiguity, and equivocation, as you describe.
Quoting jgill
What would you say is the "object" of that game, the goal? To get a "result" does not qualify as the object of the game, because anything could be construed as a result. If people are playing the same game, then they hold the same goal as the object of that game. If all mathematicians do not have the same goal, then they are not playing the same game, and we cannot describe mathematics as "a game"
Pure mathematics is more like an art. And art cannot be described as a game, because it breaks the rules which attempt to constrain one's goals. It's actually quite contrary to game play.
Quoting jgill
Sure looks like it to me. I think that a game has a clearly defined goal, and play without a clearly defined goal is not properly called a game. Mathematics does not have a clearly defined goal and is therefore not a game. I guess we'll have to just disagree.
Is there a distinct Platonic form of the isosceles triangle, or just of triangles in general? Take the exact shape that my shoe has at some instant (since the particles in it move around). Is there a Platonic form of that shape?
I understand what you mean by mathematical structuralism. But relations between objects are only identifiable if you have identity conditions for the objects between which the relations obtain.
How pleasantly wrong you are, MU. There are cliques within the broad structure of math in which participants work towards common goals. I was in such a clique.
Quoting Metaphysician Undercover
Since leaving my clique years ago, this is how I perceive math. I was never a good game player since I enjoyed going off in imaginative directions and doing my own thing.
The quote you provided does not support the point you made.
Quoting Metaphysician Undercover
We've been discussing the nature of symbolic expressions, such as a=a, with some tangential discussion of the platonic forms.
However, I do agree that numbers and the like are not actually 'objects', but that the use of the term 'object' is metaphorical in this context.
Quoting Pneumenon
Platonic forms are not shapes per se. Triangles and circles are used as examples because they're simple.
Feser, Some Brief Arguments for Dualism
Those in such a clique might be said to be playing a game. Therefore the game is limited to that clique. Bit that's insufficient for the the claim that mathematics in general is a game.
Quoting jgill
So you recognize that mathematics in general cannot be said to be a game then? Maybe we do have a similar definition of "game".
Quoting Wayfarer
We have clearly been on different pages here. The discussion as far as I am concerned, has been the law of identity, and how it relates to so-called "mathematical objects", it has not been "the nature of symbolic expressions". The law of identity is not concerned with symbolic expressions, because it stipulates that the identity of a thing is within the thing itself. Symbolic expression is excluded from identity, as not an aspect of identity. Hence the quote from Stanford: "that relation everything has to itself and nothing else". The relation between object and symbol has been exclude from identity by the law of identity, and the symbolic expression rendered irrelevant to identity. This is how Aristotle dealt with the false identity asserted by the sophists, by removing the assumption that the identity of an object is something we create with a symbol.
Yes, but that is a much deeper problem, in some ways. You're talking about ontology, the nature of being. But the debate started with the argument over whether, in the expression a=a, that the 'a' on both sides of the '=' is the same. I'm saying, of course it is, and that the identity of 'a' is fully explained by its definition. I'm not talking about the being or essential nature of a, because 'a' is a symbol.
I've gone back to the previous page. You said:
Quoting Metaphysician Undercover
The question I was asking at the time was whether numbers (etc) meet 'identity conditions'. And actually your answer was 'yes, but this is not relevant to the 'law of identity'. And that's because you're treating the 'law of identity' as an ontological issue concerning the 'essential nature of beings.' The point remains, however, that in the domain of symbolic logic, maths, and everyday speech, the identify of the symbols used - letters, numbers and so on - is fixed in relation to a domain of discourse. Therefore, letters, numbers, and so on, have an identity, which is fixed by their meaning. That was the only point at issue in my view.
Quoting Metaphysician Undercover
Well, glad that is cleared up. :-)
I'm going to start at the heart of our misunderstanding:
Quoting Wayfarer
The law of identity is an ontological issue concerning the nature of all things. Did you not read the Wikipedia, or Stanford quote I provided? Here's Wikipedia:
"In logic, the law of identity states that each thing is identical with itself."
See, the law of identity makes a statement about the nature of things.
Quoting Wayfarer
I don't think anyone was ever talking about the status of the symbol, 'A', I sure wasn't. You must have misunderstood. Sorry if I didn't express myself well.
However this is what is inconsistent with the law of identity: "the identity of 'a' is fully explained by its definition". The whole point of the law of identity is to affirm that the identity of a thing is within the thing itself, not some description or definition which someone gives. That's what gave Aristotle argumentative power over the sophists. If the identity of a thing is guaranteed by the definition employed, then we have no defense against unsound logic carried out on faulty definitions. Therefore we need a law of identity which takes identity away from the definition, to protect us against, and expose the inevitable false identities which will arise if the identity of a thing is whatever someone claims that it is with a definition.
Quoting Wayfarer
No, numbers do not meet identity conditions, that's the whole point. Identity conditions are stipulated by the law of identity. Since whatever it is which is referred to by the numeral 2 varies from one application to another, as 2 is meant to have universal application, then whatever it is which this symbol signifies, does not fulfill the identity conditions of the law of identity. That's the point, numbers do not fulfill identity conditions. Objects fulfill identity conditions, numbers do not. The conclusion which we need to make is that we have to look toward some other principles to understand what a number is. This is the highest division in Plato's divisions of knowledge, addressing directly, and attempting to understand the nature of, the so-called 'intelligible objects', philosophy. The second category is using these 'intelligible objects', such as mathematics.
Quoting Wayfarer
This is very obviously not true, and I've argued it extensively elsewhere, enough to know that if a person is not inclined to see the reality of this, they are not likely to change. However, I'll provide a brief explanation. I gave an example of "I reserved a table for 4 at 4", already in this thread, but here's something a little more technical for you. Consider that there are natural numbers, rational numbers, real numbers, and we could even throw in imaginary numbers. In each different case, what the numeral means is slightly different because the operations which can be carried out are different. If mathematics is "a domain", then clearly what the symbols mean is not fixed within the domain. You might start breaking down mathematics into multiple domains, and say that the meaning is fixed within a specific domain, but this is not true, because people cross the boundaries, and this is why there is such a thing as equivocation.
The real existence of equivocation ought to be enough evidence for you to see that what the symbol signifies is in no way fixed by the domain of discourse. And if one is under the illusion that it is fixed, and even accepts the premise that there are fixed objects of meaning like Platonic Forms, that person will no doubt be deceived by the equivocation when it occurs. The first step in the defence against the malicious form of equivocation is to understand how it is enabled. This allows one to be wary of the conditions.
I was, although the original question was about numbers, not letters.
All three laws of logic aim to produce a closed system of thought - that’s what logic is. Quantum physics demonstrates the process of accurately aligning the significance of physical event structures within the same logical system, and the qualitative uncertainty that necessarily exists at this level.
If we go back to your simple example of “I reserved a table for 4 at 4”, the symbol 4 is the same, but the physical event structures they represent as a value are not. For this to be a logical statement, the symbols need to be expanded out to include a qualitative relation to their represented physical event structures: “I reserved a table for 4 people at 4pm AEST.” What results is akin to a wavefunction: describing the significance of a four-dimensional relational structure between the significance of two measurement events within the same logical system. The more effort and attention required to potentially align the senses and meanings of sender and receiver, the more accurately the significance of the relational structure must be described in the information to reduce uncertainty (eg. What date? What restaurant? What town?). Because the receiver of the message needs the most accurate information to align the potential of their own physical event structure to that of the sender, in order to produce a genuinely closed system of thought.
I think what Wayfarer keeps trying to point out is what I’ve highlighted in bold: the law of identity makes a statement about the nature of things within a closed system of thought. I don’t agree that the law of identity is meant to be ontological.
I think that many of the problems of interpretation of quantum mechanics are the results of the culture of non-conformity to the law of identity within the mathematical community, which is highly evident in this forum. If some energy is assigned a quantitative value, and the same quantity of energy is allowed to be interpreted as "the same object", regardless of the form in which it exists, then there are no features to distinguish it from any other energy of the same value. It is impossible to maintain the identity of any particular quantity of energy through a temporal extension, if one quantity of energy which has the same value as another quantity of energy, can be asserted to be "the same" energy. A photon is an object defined as a particular quantity of energy. If any energy of equal quantity can be said to be "the same" photon, because the law of identity is violated in the way that it is in mathematical axioms, then it's very obvious that temporal continuity of a photon, as an object cannot be maintained.
Quoting Possibility
This is a mistake, and to make this assumption is a problem. Logical statements exist independently, and are valid independently, of the physical structure which they are applied to. That is why we have a distinction between being valid and being sound. The judgement as to the truth or falsity of the premises, which are the grounds by which the logic is actually related to physical structures, is a completely different type of judgement from the judgement as to whether the statement is "logical". That judgement of truth or falsity, is outside the so-called "closed system of thought" (logical system). Nevertheless, it is a crucial part of soundness, though not a part of logical validity.
So in relation to what we're discussing here, we can take the natural numbers as a "closed logical system", which provides the rules for counting objects. However, the system does not give rules for what constitutes "an object". Therefore, strictly speaking, the fact that the count is valid, cannot guarantee that the count is sound, or correct. The person counting might have had to make some judgements along the way, and there might have been some ambiguity within the criteria of what constitutes a countable object. Therefore the so-called "closed system" is not actually completely closed because ambiguity cannot be excluded from the defining principles.
Quoting Possibility
So, when we're dealing with numbers, the fundamental "meaning" which must be aligned between sender and receiver, is the meaning of "1", a unit, or object which counts as a unit. Numbers inherently deal with individual units. We could say that they were designed that way, how they got designed is another question we can put aside, and just respect the fact that numbers deal with units. Because of this, we need a very clear, and rigorous definition of what constitutes "a unit", which is understood all around, and adhered to, or else work done with numbers becomes unsound due to ambiguity. Hence we have "the law of identity".
Quoting Possibility
To understand, and judge this statement we need to understand what comprises a "closed system of thought". The problem here is that no system of thought is truly closed, as demonstrated above. A system of thought is a feature of a living system, and living systems are fundamentally open, as evidenced by evolution. This is why Platonism (eternal unchanging, closed, rules) is often contrasted with, as being inconsistent with, evolution (changing rules). A closed system cannot evolve.
So we might understand a system of thought as consisting of different levels of rules, none of the rules, neither those at the bottom, the top, or middle, ought to attempt to close the system, as this would be unnaturally stifling to the evolutionary process. If we go to the rules at the base of epistemology, upon which logic and mathematics are constructed, we find the three fundamental rules of logic. The soundness, or veracity of these rules must be judged in relation to something outside the epistemological system which they support. These are the premises of the system, which need to be judged for truth or falsity to make sure that the system is sound. So the judgement of these rules which form the foundation of epistemological principles, must be an ontological judgement. Ontology supports epistemology. That's why I represent them as ontological. A premise is always in some sense a conclusion, being a judgement. So the three laws of logic are epistemological premises, but they are ontological conclusions.
[quote=Feser] to grasp something with the intellect is not the same as to form a mental image of it. For any mental image of a triangle is necessarily going to be of an isosceles triangle specifically, or of a scalene one, or an equilateral one; but the concept of triangularity that your intellect grasps applies to all triangles alike. Any mental image of a triangle is going to have certain features, such as a particular color, that are no part of the concept of triangularity in general. A mental image is something private and subjective, while the concept of triangularity is objective and grasped by many minds at once. [/quote]
[quote=Bertrand Russell, Problems of Philosophy, The World of Universals]It is largely the very peculiar kind of being that belongs to universals which has led many people to suppose that they are really mental. We can think of a universal, and our thinking then exists in a perfectly ordinary sense, like any other mental act. Suppose, for example, that we are thinking of whiteness. Then in one sense it may be said that whiteness is 'in our mind'.... In the strict sense, it is not 'whiteness' that is in our mind, but 'the act of thinking of whiteness'. The connected ambiguity in the word 'idea ...also causes confusion here. In one sense of this word, namely the sense in which it denotes the object of an act of thought, whiteness is an 'idea'. Hence, if the ambiguity is not guarded against, we may come to think that whiteness is an 'idea' in the other sense, i.e. an act of thought; and thus we come to think that whiteness is mental. But in so thinking, we rob it of its essential quality of universality. One man's act of thought is necessarily a different thing from another man's; one man's act of thought at one time is necessarily a different thing from the same man's act of thought at another time. Hence, if whiteness were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts.[/quote]
Hope that helps. Not wanting to come across as pedantic, I'm in the middle of trying to understand all this myself.
In my book, they don’t exist independently. It’s a claim to independence - a closed system of thought is never really closed, but exists in potential relation to sentient beings, who are ignoring the potentiality of its relation to physical structures in order to examine only its structure of logical possibility. Nevertheless, my use of the term ‘logical statement’ was only to highlight the ambiguity of the original statement, which might make sense in conversation, but would not be useful in logic - it was not a comment on the validity or soundness of the statement itself. Sorry for the confusion - I can quickly get out of my depth in discussions on logic, but I think we are on the same page here.
Quoting Metaphysician Undercover
I agree with this - my point was that those who consider themselves ‘logical beings’ do not typically ground their system in a larger ontological structure. This is a problem I encounter often. They don’t recognise or acknowledge a necessary relation to the broader structure of reality in which logic, for instance, does not reign supreme. So it seems that what you’re referring to is not so much logic’s Principle of Identity, but Leibniz’s Principle of the Identity of Indiscernibles, as a principle of analytic ontology?
But what if somebody else comes up to me and says, "I, too, have contemplated mathematical objects, and I apprehend that there is no form of specific polygons, only a form of The Polygon, and all shapes, like squares and triangles, participate in it."
Then another person says, "I have also contemplated mathematical objects, and the opposite is true: there are separate forms for scalene and right triangles, and I have perceived them both."
Without positing some kind of identity conditions for abstracta, how do I even begin arguing with those two?
Hmmm. :chin: Physicist around?
The point I would argue here, is that I would agree with Russell, that no two men think the exact same whiteness, exactly as described. And, many people think of whiteness at many different times. These I accept as true premises. Along with the law of identity as another premise, the proper conclusion, is that whiteness is necessarily not an object.
That which the many different thoughts of whiteness have in common, is similarity in the conditions of use. Russell's conclusion is wrong, he has been influenced by the Platonic realism inherent in the mathematics he has studied. There is nothing to indicate the existence of such an object. As Wittgenstein argues, it would have to be some sort of paradigm and none exist. However, evidence shows similar usage. Therefore we can conclude that what the thoughts of whiteness have in common is a similar application. This similarity is simplified in the notion of conventions.
Quoting Possibility
Logic's principle of identity is the one put forward by Aristotle as the law of identity, commonly expressed as "a thing is the same as itself". This is consistent with the Leibniz principle which says that if two named things have the exact same properties, they are in fact one and the same thing. If you study them both, you'll see that one is a sort of inversion of the other. Aristotle says that the only thing which is the same as a thing is the thing itself. Therefore the thing itself is the thing's own identity. Leibniz says that if you claim to have two things which are the very same in terms of properties, they are really one thing.
What I think is the important aspect of the law of identity, is that it places the true identity of a thing within itself, as the Stanford article I quoted says, identity is a relation which a thing has with itself, and nothing else. However, in our modern way of talking about identity, we think of identity as something we assign to the thing, we say that a person's identity, for example, is the name that we give them. As I explained earlier in the thread, Aristotle formalized the law of identity to get us away from this notion of identity, because it was being abused in sophistry. Here's the quote I produced from Aristotle's Metaphysics Bk.7:
Quoting Metaphysician Undercover
In many modern schools of logic, the law of identity is simply expressed as A=A. Since it is often not explained exactly what the law of identity really is, it is sometimes simply assumed, that the meaning here is that the symbol A must always symbolize the same thing. But that is not an accurate representation of the law of identity. The law of identity stipulates that symbols cannot give the true identity of an object. The true identity is within the thing itself.
I would think that = is appropriate for equivalence in physics for symbols or quantities with mixed implicit or explicit units attached, as in E=mc^2. In some computer languages = might stand for arbitrary assignment of value to a variable, like x=3. Clearly, neither is an identity in either mathematical or philosophical meaning. ? might be too strict for philosophy?
:up:
You’re missing the point of being able to abstract. Abstraction is at the basis of language, and you’re not getting it. Logic and language relies on representation, representing some [x] in symbolic form. You’re mistaking logic for soteriology
Logic and language relies not just on representation, but on a potential relation to the possible existence of some [x] as it is. Otherwise what IS the point of being able to abstract?
The problem that I’m having with Metaphysician Undiscovered’s posts in this thread, is that he’s referring to ‘identity conditions’ in terms of ‘what really make some particular what it is’. He’s talking about the metaphysics of identity. Whereas I and others are saying that ‘a = a’ purely on the basis of abstraction, or in terms of the meaning of symbols. I’m leaving aside the metaphysical question of ‘what makes [some particular] what it really is.’ The question I asked was, doesn’t ‘the number seven’ have an identity? Which was a rhetorical question, in that I take the meaning of ‘7’ to be precisely ‘ the number that is not equal to everything that is not 7’, or, ‘7 = 7’. But somehow, this has given rise to pages and pages of metaphysical speculation.
Photons and other sub-atomic units of matter~energy are obviously ‘indiscernible’, in that they have no individual identity. All those with the same attributes - spin, polarity, etc - are indistinguishable from one another. They belong to the domain of the unmanifest, the unrealised. That is why ‘the observer’ plays a role - when you ‘see’ one, then it becomes particularised; hence the ‘observer problem’. ‘It from bit’ - Wheeler.
This relates to the point that he’s making, though: ‘the number seven’ is not identical to its value, so 7=7 risks equivocation. It reminds me of the children’s trick: ‘one plus one equals window’. It’s all very well to insist on a closed system of thought in which abstraction is all that matters, but it isn’t, and equivocating symbols with their value/potential leads to inaccuracy in terms of the meaning of symbols, and all sorts of interpretation issues when applying logic to both physics and philosophy. We need to be more conscious of methodologies employed in abstraction and interpretation that carelessly assume a closed system of thought.
Maybe your question is not well formed? To Plato, there ought to be only three forms of number, namely none, one, many.
7 is not a platonic form capable of formal identity but is derived from iterated copies of the One. An issue is that if 7 then why not 77 or 777 which lead to an explosion of copies of the One. But still, there is only one Form for One.
Quoting Possibility
Just to clarify the "potential relation to the possible existence of some [x] as it is", what is abstract and what exists in the following identities ?
A=A :: Cloud=Cloud :: Knowledge=Knowledge :: 9bananas=2apples :: Virtue=Wisdom
We're talking fundamental laws of logic. This is not soteriology. How is that even relevant?
Despite the fact that the first law of logic is expressed in language, and is an abstraction, stating a general rule, a universal, it clearly makes a statement about particular things. Do you apprehend a difference between a universal rule, and a representation? Physics for example, is full of universal rules. Being universals, they are rules for the application of logical processes, just like mathematical axioms. Strictly speaking, they are not representations, they are rules of procedure. In the case of the law of identity, it is not the case that there is some [x] (thing) represented in symbolic form. If I stated it that way earlier, this was a mistake of sloppiness on my part. What there is, is a universal statement, a law, which makes a statement about any, and every [x] (thing) which might be represented in symbolic form.
When we move to the second law, there is another statement, another universal law, concerning what we can say about that [x] (thing) which is represented in symbolic form. This law is a statement concerning how we represent that thing, or object. We are forbidden from representing the object as both having and not having the same property.
The fundamental laws of logic are meant to ground logic in fundamental realities of the world, truth about substance, in Aristotle's terms. This is why they are ontological. The judgement of truth or falsity of the laws themselves is an ontological judgement.
One might argue, the ontological position that a universal is itself a type of object. From this perspective the law of identity is violated because we assume an object (the universal) which has no particular identity. This is the route which Peirce takes, and he proceeds to argue how these universals, as things, require exceptions to the laws of logic, resulting in his philosophy of vagueness. The boundaries which we assume to define an object, as an object, are extremely unclear when a universal is looked at as an object, (i.e. when a type is an object) and so there are various reasons to violate the second and third laws of logic. I see this as the approach of process philosophy in general, which is heavily influenced by Mathematical Platonism. An "object" is what mathematical axioms say an object is, and there is an incompatibility between this and the physical world of "becoming" such that the boundaries are necessarily vague, and there is no such thing as an object in the physical world. This renders the law of identity as completely ineffectual.
Hegel also argued that the law of identity ought to be rejected, as somewhat incoherent, and you can see my argument with Jersey Flight on this subject in the debates column of this forum. He approaches the law of identity from a slightly different ontology, which he called dialectics, arguing that the law is fundamentally incoherent. He also proposes that the distinct logical separation between being and not being are subsumed within "becoming", I think the term is "sublate". This renders the separation between opposing properties as a temporal separation. But there's a trend in the modern scientific community to look at time as an illusion, so Hegel's rejection of the law of identity leads to dialectical materialism, and dialetheism which openly propose violation of the law of non-contradiction.
Quoting Wayfarer
What I argue is that you misinterpret the law of identity, which does not say anything about the meaning of symbols. It says something about particular things. Look at the Stanford quote again:
[quote=SEP]Numerical identity is our topic. As noted, it is at the centre of several philosophical debates, but to many seems in itself wholly unproblematic, for it is just that relation everything has to itself and nothing else – and what could be less problematic than that? [/quote]
You are making "identity" into something other than it is, as stated by the law of identity. In common parlance there might be such a thing as identity "in terms of the meaning of symbols", but this is not what "identity" refers to in logic, or philosophy in general.
Quoting Wayfarer
The number seven does not have an identity, if we adhere to the law of identity. Only particulars have an identity and 7 refers to a universal. That is the point. You are appealing to something other than the law of identity, some colloquialism of "identity", to justify your claim that it does have an identity. And when I point out that you misunderstand the law of identity you get flustered, as if it would be a significant embarrassment, if true. It's not an embarrassment, because the vast majority of human beings, including high level mathematicians, and most physicists, do not understand it at all, having no respect for it. They do not understand it because it is a high level principle of ontology, or metaphysics, which requires that discipline to apprehend, and this is a very specialized field which is not taught in most university courses.
Quoting Wayfarer
This is the point I made earlier. Failure to adhere to the law of identity in the mathematical, and scientific communities is what has resulted in the interpretation problem of quantum mechanics. It's a real problem, because without something real, a grounding in substance, the designation of "a unit", entity, or in this case "a quantum", as a photon, is based on a judgement of value rather than on a principle of identity. If equal value means the same entity, this is a failure in the rigours of logic. This is what I wrote:
Quoting Metaphysician Undercover
Quoting Possibility
The issue I see is that the law of identity has been openly challenged in modern metaphysics, starting with Hegel. Kant exposed the separation between phenomena, as what's in our minds, and the thing itself. When he designated the thing itself as absolutely inaccessible and unknowable (contrary to Plato), he rendered the law of identity as irrelevant, outside the domain of knowledge, as a statement about the thing itself. This allowed Hegel to unabashedly abuse and violate that law, because it appears like the law really doesn't make any difference. This is the modern attitude toward metaphysics and ontology in general, it really doesn't make any difference. But what has happened is that a huge gap has opened up between reality and what is represented in models, because the true nature of reality is seen as inconsequential, due to the Kantian belief that we have no access to it anyway (model-dependent realism for example). The Aristotelian concept of "substance", as that which substantiates logic, has been rejected due to this ontology which stipulates that logic cannot be substantiated.
Point taken, wrong choice of words on my part. I meant ‘metaphysic’.
Quoting Metaphysician Undercover
I see the difference, but I also believe that representation would not be possible without abstraction, and abstraction in turn relies on generalisations that are grounded in universals. That is why I think nominalism is fallacious. Universals are basic to the mechanisms of meaning.
Quoting Metaphysician Undercover
Hence, the role of mathematics in quantification which is fundamental to scientific method. It singles out specifically those attributes of any object which are measurable and quantifiable as the ‘primary attributes’. That becomes the basis of physicalism.
Quoting Metaphysician Undercover
I had the idea Plato regards the sensory domain as inherently unknowable as lacking in real being, which only inheres in the formal domain.
Quoting Metaphysician Undercover
Not ‘flustered’ - at the beginning of this entire can of worms, the topic whether the two symbols in the expression a=a were the same and you saying ‘it depends on what they refer to’
Quoting Metaphysician Undercover
So, you’re saying that ‘identity’ is the same as ‘esse’?
Quoting magritte
I’m still reading on the Forms, it’s a very deep and difficult topic. But generally speaking I’m arguing for ‘small-p platonism’ which is not necessarily the philosophy of Plato, but realism with respect to certain classes of ideas, which originated with Plato but was refined and qualified by subsequent generations.
Platonism in the Philosophy of Mathematics, Stanford Encyclopedia of Philosophy.
The way I see it, rational judgements make constant reference to mathematical and logical concepts. Such judgements are ‘subjectivised’ in a lot of modern philosophy by being treated as being ‘in the mind’ or the products of human minds. But the key idea I take from Platonism is that there are ‘objects’ (using that term metaphorically) which can only be grasped by a mind, but are not the product of the individual mind. Whenever we say that something is ‘the same as’ something else, we’re abstracting the similar attributes of two or more different particulars and making a judgement. But that is internal to the nature of reason, so ‘transcendental’ in the Kantian sense.
I’m curious to know how you would address the following scenario via the law of identity:
--The concept of tree is the same as (is equal to; i.e., is identical to) the concept of tree … and is different from (is not equal to; i.e., is not identical to) the concept of rock.
Here, I’m addressing conceptual forms (which are naturally devoid of perceivable shapes: for, as a concept, i.e. as a generalized idea, it can take on multiple concrete, perceivable shapes …. None of which individually specifies what the concept, a generality, itself consists of in full). This, to me, is very much in tune to how a triangle, a geometric concept, can take on innumerable perceivable shapes without any such concrete shape being in and of itself the universal, abstract form of triangle per se. Likewise to how any number, itself an abstract concept, is identical to itself as number but not to any other number.
But my main interest here is in how you'd address the concept of tree as having, or as not having, an identity (albeit an inter-subjective one) as a concept - this as per the example mentioned. To be explicit, an identity via which it as concept can be identified.
(1) Ordinary mathematics, formally and informally, uses the law of identity. This is the use of first order logic with identity (sometimes called 'identity theory') that has the built-in semantics:
If 'T' and 'S' are terms then
T = S
is true if and only if 'T' and 'S' stand for the same object.
Said another way:
T = S
is true if and only if the denotation of 'T' is identical with the denotation of 'S'.
Said another way:
In any interpretation of a language, '=' maps to the identity relation on the domain.
Moreover, for proofs, identity theory is axiomatized by an axiom schema.
This is the very precise sense of the identity.
(2) Leibniz's law is taken as either the principle of the indiscernibility of indenticals (if x and y are identical then x and y share all properties) or as the conjunction of two principles - the principle of the identity of indiscernibles (if x and y share all properties, then x and y are identical) and the aforementioned principle of the indiscernibility of indenticals.
Identity theory, hence mathematics, adheres to both principles:
* If T = S, then T and S have all the same properties.
This is the indiscerbility of identicals and is expressible formally in a first order schema:
For any formula F,
x=y -> (Fx <-> Fy)
* If T and S have all the same properties, then T=S.
This is not expressible in a first order schema unless the number of non-logical constants in the language is finite. However, even if the number of logical constants is infinite, the principle is still upheld by the semantics of identity theory.
(3) Formulations of set theory may be based on first order logic with identity (so '=' is taken as primitive). And this holds even if you take out the axiom of extensionality. The axiom of extensionality is needed to prevent urelements, but it does not contradict identity theory, rather it is an addition to identity theory.
So there are three ways to handle identity and have extensionality in set theory:
1. '=' is primitive from identity theory, so we have:
theorem: x=y -> Az((zex <-> zey) & (xez <-> yez))
and we add the axiom of extensionality:
axiom: Az(zex <-> zey) -> x=y
2, dispense with identity theory and stipulate:
definition: x=y <-> (zex <-> zey)
axiom: x=y -> Az(zex -> zey)
3. dispense with identity theory and stipulate:
definition: x=y <-> Az(zex <-> zey)
axiom: Az(zex <-> zey) -> Az(xez -> yez)
In all three cases, we have the same set of theorems and the identity of indiscernibles and the indiscernibility of identicals. So we have Leibniz's principles to exact specification. Both syntactically and semantically.
(3) It was claimed that '=' has two different senses, for example:
2=2
vs,
2x=x+3
But those aren't different senses of '='. Rather they are examples of the difference between a formula with no free variables and a formula with at least one free variable. The first is true and the second is true or false depending on what value is assigned to the free variable 'x'. That doesn't entail that '=' has two different senses. It's a matter of understanding variables here, not any supposed difference (there is not one) in the meaning of '='.
These represent two types of equations in mathematics. But you are correct in a fundamental sense of the symbol "=". (just another small reason I've stayed away from phil of math - angels dancing you know where)
T=S
where 'T' and 'S' are terms.
The equations mentioned differ in that one has no free variables and the other has occurrences of free variables. One of them happens to be satisfied in all structures, while the other is satisfied in some structures and assignments for the variables but not in others.
It's as simple as that. There are no "angels on pins" involved.
Quoting GrandMinnow
Ah, but this is a philosophy forum. We like those kinds of problems. I read about the origin of that 'urban myth' about angels 'dancing on the head of a pin'. The original dispute was about whether two angelic (i.e. incorporeal) intelligences could occupy the same spatial location - which really is not such a daft thing to ponder, if you believe that there could be immaterial beings. (I began to wonder whether there was an analogy of sorts with the concept of 'super-position' which is the notion that a quantum entity can be in more than one location simultaneously - an inverse of the medieval's conundrum. One thing in two places, rather than two beings in one place. ;-)
I don't see how this relates to nominalism, but I don't agree that generalizations are grounded in universals. I think that they are both of the same category, essentially the same type of thing, and grounding requires reference to another category. So for instance, we can't ground the concept of red by reference to another colour. We might try to ground it by reference to colour, but understanding colour requires reference to something outside the concept of colour. This is the problem I had with the idea of a closed system of thought, mentioned above. A closed system would be ungrounded. By going outside we avoid the vicious circle, but then the possibility of an infinite regress appears. So Aristotle grounded his logic in substance.
Quoting Wayfarer
I don't think Plato regarded the sensory realm as completely unknowable. Recall the divided line in The Republic. The visible realm is one half, so it does have some epistemological status. If I remember correctly, the higher knowledge of the visible realm is belief, and the lower is opinion, or something like that. More importantly though, for Plato, the visible objects partake in the Forms. A beautiful thing has beauty through partaking in the Idea of beauty. So we can come to know the visible objects through the means of the Forms, because we know the Forms, and the objects partake in the Forms.
Quoting Wayfarer
I can't answer this because I'm not familiar with the word esse. I don't think it's English and it doesn't enter my translations. I am familiar with 'essence' and with 'essential' and they both have a range of usage. Even if you mean 'to be' by esse, it's not that straight forward. 'Being' sometimes is used as a verb, and sometimes as a noun.
With the law of identity, we are talking about a thing, not an activity, and that is what we assign the uniqueness of particularity to the thing. In naming it, the thing is represented as the grammatical subject. When we talk about activities, it's always types, universals, because activities are properties. An activity only becomes particular when we assign it to a specified thing, just like other properties. So we have to be careful when we use the word "being", to clarify whether we are talking about a thing, a being, or some activity which beings have in common.
Quoting javra
The argument I've made, is that a concept is not an object, therefore the law of identity does not apply. The concept of tree is not the same as the concept of tree, because there are accidental differences in each instance that it occurs, therefore it violates the law of identity and cannot be an object.
Quoting javra
Because the law of identity applies to objects only, and a concept is not an object, I don't think there is a valid way to say that a concept might be identified. Instead, we define concepts. If we proceed to state that a definition identifies the concept, then we are in violation of the law of identity. A definition exists as words, symbols, so now we'd be saying that the identity of the concept is in the words, but by the law, the identity must be in the thing itself. That's why a concept does not have an identity. However, if we assume an ideal, as the perfect, true definition of tree, an absolute which cannot change, then this ideal concept could exist as an object. Every time "tree" is used, it would be used in the exact same way, to refer to the very same conceptual object. But I don't think that this is realistic.
Quoting GrandMinnow
This is only true, if numbers are objects. And we've seen already in this thread that they do not qualify as objects because in mathematical usage the law of identity is violated. Since the law of identity is violated in mathematical usage of numbers, numbers cannot be objects. So your formula just begs the question. You assume that a number is an object, therefore '=' means identity. But of course, as I've already demonstrated, '=' is not actually used that way. So your question begging premise is actually false.
Quoting GrandMinnow
OK, show me how T and S necessarily refer to the exact same object, as required by the law of identity. Please don't beg the question by asserting that the '=' means that they refer to the exact same object, because we already know that this is not true in the common usage of '=' in equations.
If two distinct things occupied the exact same space at the exact same time, I think we'd have a true violation of the law of identity.
I didn't say that they necessarily refer to the same object. I said the formula is satisfied when they refer to the same object.
The fixed semantics for '=' is given in the method of structures for languages in mathematical logic. If you are not familiar with that method and subject matter, then you won't follow what I'm saying here.
Quoting Metaphysician Undercover
In common, pervasive usage in mathematics, as I mentioned, a formula
T = S
is true (or satisfied) if and only if 'T' and 'S' refer to the same object.
The reason you are not familiar with that fact is that you are not familiar with rigorous mathematics and especially as mathematics is treated in mathematical logic.
No, the laws of identity are not violated in mathematics, no matter what we take the ontological status of the referents to be. Or, please state precisely a mathematical text in which you find violation of the laws of identity. I mean a specific piece of mathematical writing; not just something that you imagine someone has meant somewhere or another.
Quoting Metaphysician Undercover
What specific formula are you referring to? A formula is just a formula and it's not an argument, while question begging refers to arguments, so a formula itself is not question begging. And the rest of what you wrote there is double-talk ignorant of the subject of actual mathematics. The remedy for you is to get a book on beginning symbolic logic then on mathematical logic.
Yes, it's always the case, that if the very same thing is referred to on the right and the left, use of the '=' is valid. That is because a thing cannot be unequal to itself. But since there are many instance when the right and the left refer to something different, we cannot conclude that '=' signifies identity.
Quoting GrandMinnow
This is a false statement. It is very evident from the common use of mathematics, and even your example of "free variables", that the right and left side usually do not signify the very same thing.
You're confused. If the left and the right refer to the same thing, then the formula is true (or satisfied). And when the left and the right refer to different things, then formula is false (or not satisfied. The fact that we can write a false identity formula doesn't vitiate that.
0 = 0 is a true equation.
0 = 1 is a false equation.
That doesn't contradict identity principles.
Quoting Metaphysician Undercover
You don't know anything about it. You've never read a single page in a textbook on the subject. What I wrote is correct. You may have your own philosophy about things, but when you make claims about what happens in mathematics, you are prone to be flat out wrong and posting disinformation.
The left and the right may refer to the same thing or to different things, but the equation is TRUE if and only if the left and right refer to the same thing; and the equation is FALSE if and only if the left and right refer to different things.
Here's a simple example GrandMinnow: '2+2=4'. On the left side there is a specific operation represented. On the right side there is no operation represented. Therefore it is very obvious that what is represent on the right is not the same thing as what is represented on the left, and '=' does not signify identity.
The value of the operation + applied to the operand pair <2 2> is 4. Thus the equation 2+2=4 is true.
That's the way it works in mathematics. Your philosophy about things does not refute mathematics. Meanwhile, if you wish to continue to ignore how mathematics actually works and instead insist on your philosophy, then you would do better to present a systematic development of the subject with your alternative premises, definitions, and notations listed, and not continue to post disinformation about mathematics you know nothing about.
Gosh, you are so confident!
Wikipedia on Equation: "There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variable. A conditional equation is only true for particular values of the variables."
This is how most practicing mathematicians (possibly not those engaged in foundations) see things. Show us your background in mathematics, please.
By this argument, no continuity of (the Aristotelian notion of) any substance can occur, for any physical object will have accidental differences between itself at any time t and t'. Yet (the Aristotelian notion of) substance - as I best understand it - is precisely that with is identical relative to itself over time; more precisely, that which survives accidental changes (implicitly, over time). In much the same way, the concept of tree remains identical relative to itself over time; i.e., it survives accidental changes, or differences, over time.
The issue becomes even more problematic when considering personal identity over time.
Quoting Metaphysician Undercover
When we say “tree” and a Spaniard says “arbol” are not the concepts denoted by each different term identical - this despite possible accidental differences in the two term’s connotations? As in: the concept of tree, T, is the same as the concept of arbol, A. Hence T = A.
Given that the definitions of each will utilize different words, the English definition of “tree” and the Spanish definition of “arbol” might very well not be identical; but both definitions will define an identical concept. Again, one that survives accidental changes, including those of possible differences in connotations.
(1) There is nothing incorrect in what you quoted.
(2) Since not just Wikipedia (which itself is not a reliable source on mathematics and certain other subjects) mentions a usage that distinguishes between an 'identity' and a 'conditional equation', fair enough, I should not have allowed an impression that I claim that such usage does not exist, and I was incorrect to dispute that some people use that basis of distinction. But from a brief perusal on the Internet, I see that that usage is found mainly in high school level algebra texts (and "college" level that is seen in the examples to be really review of high school level). It is often wise to be wary of high school level explanations and terminology that need to be made rigorous and even corrected by rigorous mathematical treatments (for a salient example, the definition of 'function'). Meanwhile, I have never seen that rubric mentioned by Wikipedia used in rigorous mathematics at upper division and early graduate level, including the basic ordinary subjects: mathematical logic, set theory, abstract algebra, analysis or topology. In such subjects, the notion of an equation is as I have mentioned it, and it is at least implicit in texts that include introductions recognizing the logical and set theoretical foundations. I can't claim to a certainty that the rubric is not found anywhere in serious mathematics, but I am skeptical that it is.
(3) The distinction that Wikipedia mentioned is different from the example you gave. You gave an example of a true statement with no variables versus an equation with variables that is not true on all values for the variables. The Wikipedia article refers to a distinction between formulas that are true for all values for the variables versus formulas that are true only for certain values for the variables.
The distinction you mentioned is correctly given a precise explanation in my previous posts. For the Wikipedia sense, we can somewhat expand to make explicit the distinction between:
Valid formulas. Formulas are valid if and only if they are true (or satisfied) in all structures/variable_assignments.
and
Formulas that are not validities. Those include those that are true in some structures/variable_assignments and those that are logically false as they are not true in any structure/variable_assignment.
Sorry, I was referring rather poetically to the Aristotelian 'essence'.
I think that you are equating, or conflating, ‘essence’ and ‘identity’.
OK, on the left side is an operation with a value, and on the right side is something which is not an operation, which is assumed to have the same value. Therefore it is very obvious that the right and left side represent different things which are assumed to have the same value through some principles, or mathematical axioms. Clearly, '=' does not represent identity, it represents equal value according to those principles, in a way very similar to the way that you and I are equal, as human beings, according to some principles of value, but we are clearly not the same..
Quoting GrandMinnow
You do not seem to know much about philosophy. I do not need to present a better system to expose problems in the existing system. Finding deficiencies, and resolving them, are two distinct activities. A single person might not be adept at both finding the privation and fulfilling the need. That's the way it works in philosophy we apprehend the value of the "division of labour".
Quoting javra
In Aristotelian physics temporal continuity is provided for by matter. Matter is what persists, unchanged as the form of a thing changes, and substance contains matter. Today, this is represented by conservation laws, energy and mass. Accidentals are formal, as part of a thing's essence. The problem with representing "the concept" in the same way, as having temporal continuity, is that it seems to be immaterial. So it seems like we need a principle other than the physical "matter" to account for any temporal continuity of a concept. We might try 'information' to account for the identity of a concept, but that doesn't remain constant over time, so identity of the concept would be completely different from identity of an object, if we were to develop such a principle.
.
Quoting javra
No, the concept denoted must be different, because the Spaniard and the Anglophone are two distinct people, with two distinct backgrounds, so the meaning will be different to each, just like the concept of 'tree' is different for you and me.. No two people would have the exact same idea of what "tree" is. We assume that there is such a thing as "the concept", for simplicity sake, because we do not understand the complexities of the mind. This allows us to carry on in our linguistic endeavours as if we know what we're talking about, when we use "concept", when we really don't. In philosophy we approach these issues with the intent of understanding, so we cannot just gloss over the complexities of these mental activities assuming that the mental activities exist as 'concepts'.
Quoting javra
That there is "an identical concept", is just an assumption made to facilitate communication. So it is justified only on a pragmatic basis. It allows us to group together a whole lot of distinct (mental) activities, without any understanding of them, and talk about them as "the concept". So this idea, that there is such a thing as "the concept" is supported only because it facilitates, in that respect. Relative to the goal of understanding the true nature of reality, it is a hinderance.
Quoting GrandMinnow
Actually, what we need to be wary about, is when we learn the fundamentals, the basics, within a field, in high school, and then we proceed to the higher levels in that field, and find that what is taught in the fundamentals is contradicted in the higher levels. This happens in physics for example, when we learn about wave motion, as activity within a medium. Then we get to the higher levels and they want you to believe that there's wave motion without a medium. Such discrepancies are good cause for healthy skepticism.
Quoting Wayfarer
If there is a conflation of 'essence' and 'identity', it is Aristotle who makes this conflation. And, since Aristotle is often consider the author of the law of identity, then the so-called conflation is what is intended by the law of identity. Therefore the mistake is on your part, in rejecting it.
Maybe you could reread that post I made concerning Aristotle's Metaphysics Bk.7. Or even better, read the primary source, perhaps a couple of times because it's quite difficult. Also, it might be necessary to read "On the Soul" to have adequate background information.
Quoting Metaphysician Undercover
And this is supported by reference to Plato's Socratic discussion of 'snub nose' and form of 'Snubness' at 1037a?
Unfortunately, Aristotle was a logician and not a foundational mathematician like Plato, and distinctions implicit in Plato's discussions directed at Pythagorean mathematicians were lost in the translation.
My take is that Aristotle's metaphysics requires flat single level 'nominalist' logic as further developed in the first half of the 20th century. More recently this has been implemented as relational database systems. Plato used two-level hierarchical logic where higher level forms inform many lower-level particulars. In the Dialogues, Plato attempts to define Forms by induction from bottom up, and also conducts pathetic witch hunts for sophists from top down in a hierarchical database schema.
I can try to formalize this, as what is A=A for an Aristotelian is just A for Plato's forms and A>{a1,a2,a3,...} for his particulars.
I'm going to push this issue a little.
Quoting https://en.wikipedia.org/wiki/Hylomorphism#Matter_and_form
From such quotes I interpret Aristotelian matter to be fairly synonymous with composition. A material cause is a compositional cause, for instance, one whose effects are bottom-up and concurrent with the composition as cause.
So Aristotelian matter need not be physical (as we moderns interpret it to be). For a somewhat easier example by comparison to a concept, a paradigm's Aristotelian matter is, or at least can be, the sum of ideas from which it is composed. This in the same way that a syllable's matter is the sum of letters from which it is composed.
I know this breaks with common and traditional interpretations, but how do you find that Aristotle himself would have disagreed with what I've just outlined in relation to matter?
Quoting Metaphysician Undercover
As someone who speaks two languages fluently, I wholeheartedly disagree with this. Yes, some concepts do not translate in a single word, if at all. But basic concepts (again, generalized ideas), such as that of "tree", are the same across multiple cultures regardless of the language via which they are addressed (given that the populace is exposed to concrete instantiations of trees in its environment).
Quoting Metaphysician Undercover
As to the concept of "tree" being different for you and me: these visual scribbles we term letters, syllables, and words are meaningless in the absence of the concepts they convey. The complexities of language aside, if no such scribble could convey the same (essential) concept between two different people, how would communication of anything be possible?
Of course. In my years as a prof the only times I recall discussing the equal symbol is in elementary set theory. If we had had an actual course in foundations or even set theory It's not likely I would have taught the course. I looked in an advanced text, Introduction to Topology and Modern Analysis by Simmons, which I consider an exemplary work of clarity, and he only gives the definition of "=" at the beginning for sets.
Quoting GrandMinnow
Thanks for the advice. A bit late, however. :cool:
What is included in the category of 'primary and self-subsistent things'? Do tools and artefacts belong in that category? Do they have 'an identity' according to this criteria?
Plato a mathematician? Come on, have you read Plato's dialogues? What are you going by, the doctrine of recollection? I guess we're all mathematicians then.
Quoting javra
I don't think that you can call the parts of a thing as the cause of its composition. Notice Aristotle's full description of material cause, "that out of which a thing comes to be, and which persists...e.g. the bronze of the statue". The cause of the bronze being composed as a statue would be something other than the bronze itself.
Quoting javra
I agree, that matter need not be physical, but what is at issue is temporal continuity. Notice that the same matter which a thing is composed of, exists prior to the thing coming to be, and after it has come to be. This is why matter is described as potential, it provides the potential for the object, and even after the object exists the matter has the potential to be something else. So the problem I see with positing ideas as 'the matter' of a concept, is that ideas come and go; if they are forgotten, or replaced by something better, they disappear forever. Furthermore, within the Aristotelian conceptual structure, ideas are formal, so if we could find some temporal continuity within the ideas which make up a concept, as its parts, there is still a big inconsistency here.
Therefore, I believe we need to go much deeper than conscious ideas, right through the emotions, into the subconscious level of human existence, to find the true 'content', the 'subject matter' of ideas and concepts, the underlying substance. Since we haven't been able to determine this element which could account for temporal continuity in concepts, thereby providing that a concept has identity, we haven't been able to demonstrate that a concept is an object. In the case of a material object, we say that the object continues to be the same object because it is made of the same matter, despite the fact that its form undergoes changes as time passes. However, we need to respect the fact, that the belief that there is an underlying matter is just an assumption. Aristotle assumed that there was matter so that he could say that an object has an identity, and to insist that it continues to be the same object despite changes to it. This was an argument against philosophers like Heraclitus who would say that all is flux, becoming, disputing the idea that there even is any real objects.
Quoting javra
The point is, that if I were to define "tree", and you were to define "tree", I'm quite sure that we would not define it in the exact same way. In fact, I'm quite sure that I would define it differently myself, depending on the circumstances. Remember, that to be the same object, by the law of identity requires that it is the very same. Now, we know that with temporal continuity, an object changes as time passes, and the law of non-contradiction states that it cannot have contradictory properties at the same time, but what about the difference between your concept of "tree", and mine, which exist at the same time? You might say that these are not contradictory, but surely the concept of "identity" which I have contradicts the concept of "identity" which some others in this thread have. And even in numbers we see contradictions between natural numbers, rational numbers, real numbers, imaginary numbers. Therefore even the concepts of numbers cannot be objects, because defining any particular number in a way which would encompass all usage of that number, would involve contradiction.
Quoting javra
I get this question asked of me over and over again on this forum, and the answer is very simple. Communication clearly does not require that different people convey the same concept. All sorts of animals communicate without the use of concepts. We develop concepts to facilitate a higher understanding, but these concepts are developed through the use of communication, not vise versa. I believe that this is an important part of the issue, communication came first, then concepts were developed. And this is why it is so hard to establish the temporal continuity of ideas and concepts, they are forms, and forms come into existence and go out of existence.
Quoting Wayfarer
I believe a self-subsistent thing is a thing which is primary, having nothing which accounts for its existence, as prior to it, like Forms. In Bk.7 Ch.6, it is said that a substance is the same as its essence, but he then proceeds to question "self-subsistent" Forms to see if this would be true for them. Things even Forms, if they are self-subsistent as some Platonists argued, would have to be the same as there essence. Otherwise the essence of good would be different than the good-itself, and the good-itself would not have the essence of good and without the essence of good, it could not be good. So the conclusion is that all things, whether a material thing as substance, or a self-subsistent Form, must be the same as their essence.
The question of the difference between the coming-to-be of artificial things, and the coming-to-be of natural things is addressed in in Ch.7-8. This is where it gets quite complicated. He is at this point carrying on with the problem of trying to account for the existence of accidentals, which is what that discussion of self-subsistent things was related to. So he now comes to matter, and says that in an artificial thing, the matter to be used might be stipulated as part of the formula, a bronze sphere for example. Therefore the form of the artificial thing comes from somewhere other than within the matter, and this is the soul of the artist. He then proceeds in Ch.8 to compare natural things to artificial things, and concludes that the process must be similar, the form which the material thing will have, must come from somewhere other than within the matter. But he claims there is no need for a self-subsistent Form, only that the thing which begets is of a similar type to the thing begotten.
Here, you’ve misconstrued what I was saying. I wasn’t saying that a given’s summation of parts *causes* the given’s composition/matter. What I was suggesting is that its summation of parts, or constituents, *is* its composition/matter. This such that “matter” and “composition” can be used interchangeably. Hence the reason why the bronze statue can dent—for one example—rather than shatter or burn, is its composition/matter of bronze (rather than the same statue-form being composed of stone (which can shatter) or wood (which can burn)). Reworded, the bronze statue is dent-able (rather than shatter-able or burnable) due to its composition as the cause of its dent-ability. Again, such that composition and matter are in the addressed Aristotelean context interchangeable.
Quoting Metaphysician Undercover
As regards continuity, forms, and matter, I think it’s a complex minefield. However, this is to me an interesting Aristotelian tidbit that might (?) clash with the gist of your affirmations regarding the impermanence of forms vs matter: the teleological unmoved mover is taken to be pure form sans any and all matter. And, as the unmoved mover, it neither comes into being nor goes from being, remaining as permanent as permanence can get, this while being deemed the mover of everything hylomorphic, with the latter taken to be in states of change, i.e. flux.
So, in my quirks of interpreting Aristotle, if we’re looking to affix identity strictly to that which is permanent, unchanging, then this cannot be matter but instead can only be form: specifically, that matter-less/composition-less form which specifies the identity of the unmoved mover as telos.
But again, to me this is a complex field to enquire into. And, for the record, no, I’m not denying that forms (other than that form which specifies Aristotle's teleological prime mover) change over time, including by appearing and disappearing. I simply don’t associate identity to that which is necessarily unchanging.
As regards concepts, it looks like we disagree on a lot of small points that, in short, add up to a large disagreement overall. I won’t nitpick, preferring the let this issue of concepts be for the time being.
Sure, all things have an identity, by the law of identity. The point of the law of identity is that the identity of a thing is within the thing itself, it is not the word "hammer", nor the humanly applied definition of the word, because the thing's true identity needs to include all the accidentals proper to each individual hammer. To us, the accidentals appear to inhere in the matter of the thing which is the principle that we use to account for a thing's indefiniteness.
I'm sorry I made a mess of my reply when you said that I equate essence and identity, with those quotes I produced. I believe there are two distinct senses of "form" in my interpretation of Aristotle. One refers to the universals, which are the ideas by which we understand things, and the other refers to the form of the individual. "Essence" and "form" are very similar in meaning, so "essence" is ambiguous being used commonly in both ways.
The reason why we need to assume a form or essence which is proper to the individual, comes out earlier in the "Metaphysics". It is explained that when a thing comes into being it must necessarily be the thing which it is, or else it would be something other than it is; which is impossible, that a thing is something other than it is. This is the basis for the law of identity. It is impossible that the thing is something else, something other than the thing that it actually is. And, the important point is that a thing is not random matter, it is matter which is structured, 'organized' in a particular way. This means that the form of the particular, individual thing, must be prior in time to the thing's material existence, to ensure that it is the thing that it is, and not something else.
We have a good example of this in the case of living beings. The soul of the individual must be temporally prior to one's material existence, to account for the organization of the material body. This is the sense in which the matter must be included in the formula. This is the passage: "... for there is no formula of it with its matter, for this is indefinite, but there is a formula of it with reference to its primary substance---e.g. in the case of man the formula of the soul---for the substance is the indwelling form, from which and the matter the so-called concrete substance is derived;..."
If we proceed further in his "Metaphysics", understanding his so-called cosmological argument is the real key to making sense of all this. The cosmological argument demonstrates how "form" through its defined nature as actual, must be prior in time to "matter" in its defined nature of potential, in an absolute sense. So, according to what is described previously in the book (above), this Form (I capitalize it to signify its independence from matter) is the form of the individual.
You'll see that this is consistent with the Neo-Platonists who assign individuality to the independent Forms, "One", "the soul". Further, in Aquinas there is described a complete separation between the forms as universals which are dependent on the human mind, and therefore not separate from matter, and the independent or separate immaterial Forms such as God and the angels. Understanding this separation is important to understanding Aquinas. When I first started reading Aquinas he was talking about how the forms, as universals, intelligible objects are not immaterial, being dependent on matter, and I could not understand what he was talking about. I had to go back and reread a lot of Aristotle, specifically Metaphysics Bk.9 which contains the cosmological argument. Then it all made sense.
Quoting javra
What I did, was argue against what you said here. We cannot equate "matter" (or parts), with "composition". This is because the particular arrangement of the parts is just as important to the composition as is the parts themselves. So I am disagreeing with your use of "composition". I think it is misleading, implying that we can remove the particular arrangement of the parts as inessential to the composition.
Quoting javra
"Matter" in the Aristotelian context, can always be broken down, given a formula. So if we want to compare wood, and bronze, we proceed to the form of "wood", and the form of "bronze". Then we see that each of these consists of parts, atoms which are arranged as molecules with a molecular structure which is the form of wood, or bronze. Therefore the "shatter-able", "burnable", or "dent-ability", of the substance is still accounted for, by giving what was supposed to be "the matter", a form. In modern physics, they bring the form to deeper and deeper levels, in an attempt to understand the lower levels.
What is inevitable with this process of reduction of the matter, is the appearance of infinite regress. This is because "matter" is inherently indefinite, by its definition, so we must determine its form to understand it. But the conceptual structure which we're locked into is that if there is a form, there must be an underlying matter. So whenever we determine a lower level form, it is necessary that there is matter underlying this form. But the matter is necessarily only intelligible by its form, so we need to determine another level of form, and this appears like it might go on ad infinitum.
This is why we need to turn things around, as Aristotle did with the cosmological argument, and recognize the necessity of assuming immaterial forms which are prior to material existence. This is a way to break the infinite regress. However, it requires a distinctly different definition of "form", hence a dualism of form. The bottom-up form, which is properly an immaterial form, as responsible for the cause of material objects, is the form of an individual, rather than a universal form.
Quoting javra
Yes, I believe that this is the right direction to take in understanding Aristotle. The teleological form, associated with intention and final cause is the bottom-up cause. We find this explained in "On the Soul". The living being, as an organized material body, must come into being as organized matter. So the matter of this body is organized to the lowest levels of its existence. This implies that even when this matter comes into existence as "matter", it must already be organized by a form which has prior existence, the soul.
What this principle does, is that it takes the indefiniteness away from matter. Matter is necessarily organized, there is no such thing as "prime matter" according to what the conceptual structure dictates. The cosmological argument demonstrates that the reality of prime matter is impossible. Now, the reason why "matter" is designated as indefinite is because that is the way that it appears to us, human beings who have deprived, or imperfect intellects. In our attempts to understand the parts of objects, there is always something at the bottom which appears unintelligible to us, as indefinite. Aristotle assigns "matter" to this. However, as "indefinite" is just the way that it appears to us, in reality there are immaterial Forms which underlie the matter, making it really organized and structured. This is what validates the law of identity. But with our deficient intellects we do not have the capacity to grasp these immaterial forms, and the true identity of material things.
No such implication was intended:
Quoting https://en.wiktionary.org/wiki/composition
One one hand, these definitions are in accord with Aristotle's definition of matter as ""that out of which" X is made". One the other hand, my current more formal definition of composition is "a given's synergy of parts"
My reason for using "composition" rather than "matter" as a determinant (as in: material cause) - this in what I'm currently working on - is that "matter" nowadays commonly denotes that which constitutes the physical, whereas composition does not. The latter being more in-tune with what Aristotle meant. So, the synergy of ideas which constitutes a paradigm (say evolutionism rather than creationism, or vice versa) is the paradigms composition (its matter in Aristotelian terms). This synergy of parts, in this case of ideas, then sets the limits or bounds of what form the paradigm can take and of what changes it may or may not undergo so as to remain the same paradigm. This synergy of parts is then the paradigm's compositional determinant (the paradigm's material cause). This even though, in today's terminology, neither the paradigm nor the ideas from which it is composed are material - rather, both, to most, are deemed immaterial.
Quoting Metaphysician Undercover
Here, and in related passages, we seem to agree in full.
Quoting Metaphysician Undercover
This part to me is a bit confusing. Are you saying that formal causation is a bottom-up causation? Or that a hylomorphic given's form is the result of material causation, with the latter being bottom-up? Or something other?
As a little bit of background: To me a form, as the term was traditionally intended, can be reexpressed as a whole, as a given's entirety (of being). So, in a maybe oversimplified manner, one can contrast a given whole with the same given whole's synergy of parts. (Yes, each individual part is its own whole ... but this leads into different avenues of investigation, ones you've already touched upon). The given's whole, or form, results in the given's formal cause upon its synergy of parts. Whereas the given's synergy of parts results in the given's material cause upon its form (upon that which makes the given a whole given).
At least when viewed this way, formal causation is to me always top-down, rather than bottom-up, even if it is deemed to be of primary importance relative to any identity: for it is the whole's determination of the synergy of parts from which it as a whole is constituted. (And I acknowledge this is a very complex subject to embark upon; but, notwithstanding, it still strikes me as a synchronic top-down determination). Whereas it is material causation - the synergy of parts' determination of what the whole is - that strikes me as bottom-up causation.
So, here, everything both material and immaterial (modern usage) is hylomorphic, save for the unmoved mover. But the latter is a telos which moves everything teleologically. So its being as form need not have a synergy of parts. Not being itself a hylomorphic being, it holds neither top-down (downward) nor bottom-up (upward) determinations, but instead is solely composed of teleological (what I currently term "pull-ward") determinations ... which affects everything (including all efficient causes) either directly or indirectly.
I likely expressed more than a mouthful. Of course, feel free to critique my interpretations, but I am curious why you express formal causes to be bottom-up rather than top-down IF this is indeed what you here intended.
Quoting javra
Minefield, indeed. Buddhism says that there is nothing that constitutes an 'ultimate identity' in this sense whatever. Perhaps we could venture that Kant's antinomies are a corrective to the tendency in Aristotelian metaphysics to try and absolutise 'essence' etc.
I also question the tendency to 'absolutize' the forms. I think they're real on a specific level, viz, that of the 'formal realm' which 'underlies' the phenomenal realm but they can't be pinned down or ultimately defined.
Quoting Metaphysician Undercover
I query this analysis. That would make Aquinas a conceptualist - 'Conceptualism is a doctrine in philosophy intermediate between nominalism and realism that says universals exist only within the mind and have no external or substantial reality.' Aquinas was not a conceptualist, but a scholastic realist, whom by definition accepts the reality of forms.
From Thomistic Psychology: A Philosophical Analysis of the Nature of Man, by Robert E. Brennan, O.P.; Macmillan Co., 1941.
I don't think so. #1 refers to an efficient cause. #2 refers to the complete substance, matter and form. #3 is a form, or formula. And #4, referring to a "general" make up, must also be formal. I really don't see how "composition" can refer to matter. I can see "what a thing is composed of" as being consistent with "matter", but neither the noun or verb form of "composition" seems to be the same.
Quoting javra
I don't agree, I think "that which constitutes" is closer to what Aristotle meant than "composition".
Quoting javra
What I'm saying is that I believe that final causation, intention, will, is bottom-up. Formal cause, which we apprehend as acting top-down, is distinct from final cause. What I tried to explain in the last post, is that material cause appears unintelligible to us ultimately, as leading to infinite regress. We can assign intelligibility to it, claim that matter is necessarily intelligible, by positing an immaterial Form as the cause of matter. The example Aristotle gives is the soul, which is the first cause of existence of living matter. This type of Form can be apprehended as acting from within the matter, teleologically, as final cause, and is distinct from formal cause. It must be a bottom-up cause.
Quoting Wayfarer
I was of that mind as well, until I actually read Aquinas. In reality, he argues that the forms which we know, as universals, are the product of abstraction, whereby the intellect uses the body, through the means of sensation, to produce the "intelligible species". Notice the use of "species" here, as he is very thorough to follow Aristotelian terminology, instead of the Platonic "intelligible objects". Because the human intellect is dependent on the body for its knowledge, it cannot grasp separate, or immaterial forms.
If you go to the Summa Theologica, the section on "Man", you'll find a part on the Knowledge of Bodies, At Q.84. Art.4, you'll see "Whether the Intelligible Species are Derived by the Soul from Certain Separate Forms". Following "I answer that..." you'll see a significant writing about Platonic Forms, followed by a discussion of Aristotle, Avicenna, and the active intellect. At the end: "We must therefore conclude that the intelligible species, by which our soul understands, are not derived from separate forms". This conclusion is brought about by the fact that the intellect is united to the body as a power of the soul. He explains this further in the following articles of Q.84. In Q.85, we have "Of the Mode and Order of Understanding". Here he explains abstraction, referring to the role of sensation, "phantasms", and the reality that the soul is united to the body. In Q.85 Art. 1 he explains that Plato did not properly respect the consequences or conclusions derived from the intellect being united to the body. Further along, you'll find Q.88, "How the Human Soul knows What is Above Itself". Here he looks into the possibility that we could have knowledge of separate substances, immaterial forms. In Art. 1, you'll see a number of detailed arguments and the conclusion: "Hence in the present state of life we cannot understand separate immaterial substances in themselves, either by the passive or the active intellect."
This is all a product of the separation which Aristotle established between the forms which the human intellect understands, (abstractions, universals, concepts), and the separate, immaterial Forms, which are proper to the divine realm. For the very same reason that we cannot properly know God, we cannot know the other immaterial forms, the angelic forms.
Quoting Wayfarer
I quoted the two sentences from the passage above, which are misleading. The active intellect does not receive the form from the sensible object, reception is a passivity. The active intellect actually creates a representation of the sensible object. This is the importance of phantasms and imagination in the intellect: Q.84 Art.7."And, therefore, for the intellect to understand actually its proper object, it must of necessity turn to the phantasms in order to perceive the universal nature existing in the individual."
So it is incorrect to say that the intellect receives a form from the sensible object. The form of the sensible object is a particular, and is united to that object, just like the soul is united to the man. But the intellect produces a universal form, through the use of phantasms, by means of which it understands the individual. That's why there is a separation between these two types of "form".
Thanks. Briefly skimmed some of it for now. Will look further in it in a few days. Looks to be up my alley.
Quoting Wayfarer
I have a great deal of respect for Buddhism in many regards, this being one of them.
That said, the Buddhist notion of Nirvana, though different in many ways to that of Aristotle's unmoved mover, to me does share a number of similarities. The utterly, literally, selfless state of awareness (hence, a state of awareness devoid of all duality) which is Nirvana - more correctly in this context, "nirvana without residue" - seems to be interpretable, to me at least, as the ultimate identity of all sapient beings (and at least some schools of Buddhism seem to hold of all sentient beings; this being an inference gathered from those Buddhists that take an oath to enlighten all sentient beings). And - again imo - it is from this vantage of what our ultimate identity is that the no-self principle of Buddhism can be derived.
Don't know if its just me, but I take it that anything which can be identified holds an identity, a discernible form. Though we're accustomed to thinking of all identities as being finite forms that are constituted of parts, the state of being which is Nirvana certainly is utterly devoid of parts, is stated to be infinite in the sense of being devoid of limits or boundaries, and is identifiable. Hence, Nirvana is a discernible form of being. So while this will probably be a bit irksome, I can interpret Nirvana to of itself be the ultimate identity. This in parallel to what was previously discussed about the Aristotelian unmoved mover as the ultimate identity ... or of what can be said in relation to the Neo-platonic notion of "the One".
(The perennial philosophy parts of me like to believe that all three are different interpretations of the same metaphysical given.)
Curious to hear your thoughts on the just given musings.
Quoting Wayfarer
Right. I tend to agree in this for all forms save for "the One".
Quoting Metaphysician Undercover
In truth I can't find much of any meaningful different between a constituent and a component of a given, so I'm perfectly fine with rephrasing material causes as a "constitutional determination" rather than a "compositional determination". I was initially hesitant in so doing due to "constitution" being so readily interpretable in the senses of government and law. But I suppose the term's contextual use would suffice to clarify the intended meaning. Thanks for that.
Quoting Metaphysician Undercover
While I find reasons to disagree, thanks for the explanation.
To illustrate what I mean, the goal, or objective, one has in mind while engaged in making a decision will be the decision's final cause: it determines what will be chosen in so far as what will be chosen will be so chosen for the reason of best obtaining the objective. Bottom-up addresses synchronic occurrences, yet the goal pulls the momentary act of choice making toward a potential future that has yet to be objectified. So, to me, there is a type of temporality involved with a telos. This to me stands in contrast to both bottom-up and top-down determinants, both or which strictly occur in the specified moment of time without any temporal extensions into the future.
I don't think we can correctly say that anything occurs in a moment of time without any temporal extension. All occurrences require duration. Therefore I do not think we can exclude "bottom-up" and "top-down" from a temporal analysis.
The problem I see is in the way that we commonly represent space. In simple conception we see space as representable with a 3-d coordinate system. This makes the thing represented, (i.e. the reified space, and there necessarily is a reified space to allow for the reality of motion), the same everywhere. So all of space is fundamentally exactly the same everywhere, under this conception, no matter how big or how small, and it might be infinitely big or infinitely small. This is a problem because it provides us with no principles to distinguish the real spatial difference between inside and outside, which is implied by the concept of spatial expansion. And, such a principle is required if we want to validate the real existence of objects. An object is defined by its boundaries, and this allows us to distinguish properties of the object itself from what is external to the object, in order to maintain consistency with the three basic laws of logic.
Because we have no such principles, we cannot properly differentiate between a force which acts from the inside, and a force which acts from the outside of an object. So I see the top-down/bottom-up distinction, when it's applied to causation, as based in the global/local distinction which is applied in physics. The problem though is that there are no real principles to distinguish inside from outside, so these distinctions are somewhat arbitrary, and gauge theory for example is just a mess. It's as if the whole of gauge theory is an attempt to deal with anomalies brought about by failing spatial conceptions.
In one sense I agree, but in this sense all four of Aristotle's causes co-occur (an Aristotelian variant of codependent arising). Which is not the case when each cause-type is addressed individually.
To address this via example, if a wooden table’s burnability holds as its material cause the wood out of which the table is constituted, what duration occurs between a) the material cause of wood and b) the table’s intrinsic potential to burn?
So far, to me (a) and (b) seem to be necessarily simultaneous, with no duration in-between, while standing in a bottom-up relation.
Nice concepts on the issue of space, btw.
Matter being potential in Aristotle's conceptual structure, I think that (a) and (b) are just different ways of saying the same thing.
There is an issue with matter, which I sort of explained earlier. When we start saying anything about matter, we are always referring to a specific form of matter. That is because matter is defined as an indefinite aspect of things. We can try to get away from this by making the most general statements possible about matter, but since matter is indefinite such attempts would still end up being in a sense statements about forms. For example, "an object is composed of matter", and "matter has inertia", each in its own way states something formal, despite being attempt to say something strictly about matter. Even when I say "matter is the indefinite aspect of things", I say something formal, by referring to a privation of form.
To do a Galileo like thing: but still the table can burn due to being constituted of wood rather than marble.
You are of course correct in respect to the notion of primary matter. Yet in practice we, for example, have to build houses whose bricks are constituted of solid matter rather than, say, some sponge-like material. I was addressing this more practical view of a hylomorphic given's constituency when addressing bottom-up determinacy.
I'm with you on that. Not that they're all saying the same, or heading to the same destination, but that they're agreeing, and disagreeing, about the ultimate state.
Quoting javra
Obviously a major digression from the OP (for which I'm responsible) - but, yes, this is the East Asian teaching of the 'Buddha Nature', tathagathagarbha. A very beautiful philosophy.
Finally got around to reading Kelly Ross's manuscript which you linked to. I’m envious of the clarity and simplicity with which complex concepts are expressed. I don’t fully agree with some of the concluding inferences. But, for the sake of this thread, I’ll skip all of this. (And for what its worth, despite his many shortcomings as a philosopher (what philosophy can ever be “perfect”?), I continue to greatly admire Hume for many of his insights. :razz: But anyways …)
For anyone interested in furthering the issues of identity already discussed in this thread, taken from about a third of the way in in Ross's manuscript:
Quoting https://www.friesian.com/universl.htm
If my interpretation of it is valid, to me Leibniz's principle of "identity of indiscernibles" can equally apply to substance theory and to bundle theory - the latter standing in contrast to the former, with the former being typified by the first portion of the quoted passage.
If so, curious to hear what would be wrong with the following: an individual object's identity of itself consists of a gestalt form that results from the synergy between all relevant properties as parts. In this manner, hybridizing substance theory with bundle theory in relation to identity. (I'm toying around with this notion at present). Hence, there here would be no inherent, independent substance (primary or secondary): all substances being emergent byproducts of properties. On the other hand, the gestalt is that to which all its properties are predicates of.
An individual apple's identity would then be the gestalt that results from all of the individual apple's properties, including those of its spatiotemporal placement (which is a predicate of the apple).
An individual number's identity (say, the number 2) would then likewise be the gestalt that results from all of its properties: this gets far more tricky due to the degree of abstraction, but maybe including those of duality, its placement within the appropriate context of other numbers (e.g., greater than 1 but lesser then 3), and so forth.
Edit: I'm aware that the Wikipedia article on bundle theory makes a skimpy mention of "bundle theory of substance". More musings on this issue can be found here https://plato.stanford.edu/entries/substance/#BundTheoTheiProb. All the same, if anyone is interested in debating the notion of identity as a gestalt form emerging from a bundle of properties as parts, I'm curious to see in which ways this would be critiqued.
The problem here is that the number 2 is a property itself. We take a group of two and look at it as a single thing, and say that this thing has the property of consisting of two. There's no fundamental problem in saying that the group is not a true object, it's arbitrary, and arguing therefore that the only true object is the property which is assigned. However, if we have no way to distinguish a true object, then the number 1 is invalidated as a false property because it cannot be truthfully assigned, and so the falsity of 2, as a property follows, being dependent on the truth of 1.
A very similar problem will appear with bundle theory. If an object is a bundle of properties, then there is no real principle whereby we might judge if this property is part of a specified bundle, or another bundle. Then we could not claim any objects as real objects, except perhaps a property itself. But this will prove to be completely incoherent because contradictory properties, as objects themselves, will be all over the place, and if the contradictory properties are not properties of the same thing, then we can't reject them by way of the law of non-contradiction. So we'd have all sorts of contradictory properties with no way to reject contradictions.
Given what we've been through in terms of prime matter being pure potential and all givens being identified by their forms, why would the abstract form of "2" be deemed arbitrary rather than a "true (abstract) object"? Seems to me that basic numbers are not arbitrary, despite their very abstract nature; else, for example, 2 + 2 could equal 5 in certain cases.
Any concept which cannot be substantiated (grounded in substance) is an arbitrary concept. Unless we have a principle as to what constitutes a whole, an entity, or an object, all concepts with numbers would be arbitrary. If there is nothing to distinguish one object from two objects, then 2+2 might just as well equal 5 as 4, or any other random number, because it really doesn't make any difference, as number itself would be fundamentally meaningless.
You're using substance to denote something different than what I'm denoting by it: for you, it seems, substance is only that which is empirically cognized via the physiological senses. For me it is any whole that can be cognized - perceptually or otherwise, such as via the understanding - which is constituted of parts, any hylomorphic given. In this latter sense, then, every concept is itself a substance. This as per Aristotle's philosophy, wherein concepts are secondary substances. Even so:
Quoting Metaphysician Undercover
I don't yet understand why you presume that basic numbers are not substantiated via that which is empirically cognized? We perceive quantities. And we express these perceptions of quantity via numbers. Thereby making basic numbers (e.g., 2), as well as their basic relations (e.g., 2 + 2 = 4), non-arbitrary.
I haven't followed the posts in this thread for a while but just so you know, @Meta doesn't think 2 + 2 and 4 refer to the same mathematical object. You may be assuming too much if you think he agrees with the rest of the world about this.
Mathematics is not the entire reality but only an minor aspect of it. Trying to define the concrete reality as made up of numbers is as dumb as defining it as colors. Thanks Kant for that.
Any "mathematical model" of anything - and especially regarding human society - is just a metonym, not a substantive description. Those who do not realize this, even if they have a Nobel Prize, are illiterate.
Good luck.
I think "substance" has its meaning relative to logic, and it refers to whatever grounds any particular system of logic, as what underlies it to support it. So it is quite clear to me, that "any whole that can be cognized" is not an acceptable definition of substance, because it allows that fictitious objects may be substance, or have substantial existence. And this is clearly inconsistent with any logically rigorous definition of "substance", as that which provides truth to the logic.
In Aristotle's logic, secondary substance is what grounds the logic as the most specific within the system. So if we claim a concept, the genus, "animal", that genus might be grounded in the more specific, concepts "man", and "horse" for example. That is secondary substance. The primary substance is the individual, the particular horse or man, while the secondary substance is the type that the individual is. If, for example, there was proposed a species, like "man" and no particular example of that species could be found, the proposal would be unsubstantiated (in the sense of primary substance). Likewise, if someone proposed a genus, "animal", and no species could be shown to be a member of that genus, the genus would be unsubstantiated (in the sense of secondary substance).
Aristotle's conception of "secondary substance" does not allow that "every concept is itself a substance". Only a more specific concept, which grounds a more general concept is a secondary substance. But the secondary substance itself still needs to be grounded in particular individuals, which is primary substance.
Quoting javra
It is only when you disavow the object to which "2" is attributed as a property, that numbers are necessarily arbitrary. That's what you've been talking about isn't it, claiming that there is not need for the physical object which substantiates the number? Didn't you mention bundle theory? Physical groups of two things, is what substantiates the non-arbitrariness of 2, just like physical instances of animals substantiates the genus "animal". The physical group, which consists of two, is the physical object, the particular, the substance in this instance, and "2" is a property of that physical object.
If we remove that object, that physical group of two, as the primary substance, we might substantiate the concept "2", in secondary substance, with the concept of "1", and the concept of "+", or something like that, like we might substantiate the concept "animal" in the species of "horse" (secondary substance). However, we still need to substantiate "1" in primary substance, like we need to substantiate "horse" in individual horses, or else any designations of one, or unity, or whole, are arbitrary. And, since "2" as a concept has been grounded in the secondary substance of "1" as a concept, and "1" is arbitrary if it's not grounded in primary substance, then so is "2"
That's the problem with Platonic Realism, in general, which is more appropriately called Pythagorean Idealism. Unless we provide the required separation between the model, and that which is modeled, then we are misguided into the rather silly and naive notion that the universe is constituted of mathematical objects.
We cannot provide for the separation simply by referring to the symbols, as if the symbols themselves are the model. In reality, we have the two levels of separation. We have the reality, we have the mathematical model which represents reality, and we have the symbols which represent the model. So, we have two levels of interpretation to work through, 1) to understand the model based on an interpretation of the symbols, and 2) to understand how well the model represents reality. We cannot deny the importance of either one, nor can one be reduced to the other.
I see your point here. A unicorn as concept is not substantiated by primary substances (which I still maintain can only be empirically known). The claim that “unicorns are objectively real” thereby being unsubstantiated. Yet the claim that “the concept of unicorns occurs within western thought” would be substantiated. Granted. Yet for me the concept of unicorns is itself hylomorphic and as such has an identity as a whole given, as a form, which is itself composed of parts, i.e. has a constituency.
Quoting Metaphysician Undercover
In relation to both quotes:
For what it’s worth, I personally don’t take the laws of thought, the law of identity included, to be grounded in anything physical. I instead interpret these to be grounded in metaphysical aspects of reality that then, via awareness, govern how we interpret that which is physical. This in a Kantian-like manner. It’s a can of worms - the details of which I’d rather skip - but, for instance, the absolute unity which can be conveyed by the numeral “1” cannot be found in physical givens: for any one physical given is itself less than perfectly integral—being, instead, in constant flux, change, regarding its constituency, with smaller components always coming in and out, with these leading all the way down into zero point energy. So I take it that the integrity, wholeness, of physical givens is only relative to their context, rather than absolute, and that a perfect wholeness, or unit, is what we experientially project onto the world perceptually. In short, to me, the law of identity isn’t substantiated by physical reality; instead, it of itself governs, and in this sense substantiates, that which we deem to be integral wholes within physical reality.
I mention this because, at the end of the day, our different takes on the law of identity - and maybe on laws of thought in general - seems to play a crucial role in why we disagree about the nature of mathematical objects.
I think I get where you’re coming from, however. More or less, the position that all concepts need to be substantiated in empirically known to be real particulars in order for the concepts to be non-arbitrary, and thereby true to reality. If I’m indeed interpreting you correctly, when applied to most contexts, I would be in agreement with this. The only main, but subtle, disagreement would be that the empirical itself is, to me, governed by metaphysical properties (these including what is formalized as the law of identity, in addition to other Kantian categories such as those of space and causation): thereby making the empirically known reality of the physical itself, in one sense, substantiated by that which is purely metaphysical.
Noticed this yesterday. I think this is the book I have been looking for in regard to philosophy of arithmetic.
https://play.google.com/store/books/details/Edmund_Husserl_Philosophy_of_Arithmetic?id=lxftCAAAQBAJ
Quoting javra
That is basically modern realism. It calibrates all philosophy against the a priori presumption of the reality of the empirical domain; whereas what you're advocating is much nearer Platonic or scholastic realism or Kantian transcendental idealism.
OK, this is a good starting point. What has the capacity to govern how we interpret "that which is physical"? Suppose we could interpret the physical in absolutely any possible way. Then our interpretations would be arbitrary, or random. But we want to say that there are real constraints on the way that we may interpret the physical, so that our interpretations are truly consistent with the physical reality. So we proceed to make some metaphysical or ontological conclusions about physical reality. These are what we use to govern how we interpret the physical. Don't the three fundamental laws of logic qualify here, as fundamental conclusions concerning "that which is physical"?
Quoting javra
But "1" does not signify any absolute unity. It is divisible, and infinitely so, by the accounts of many. So how could it signify an absolute unity?
Quoting javra
I can't understand what you're trying to say here. If I said that the law of identity is substantiated by physical reality, I would mean that it is made true by the conditions present in the physical reality. So, if you say that the law of identity governs us as to what we can deem "a whole", aren't we really both saying a very similar thing, in slightly different ways? You are saying that the law of identity governs what we can say about physical reality, and I am saying that the reason why it governs what we can say about physical reality is that it already says something true about physical reality. The only difference is that you are not moving along to see the reason why the law of identity has the capacity to govern what we say about physical reality. It gains that capacity to govern, by saying something true about physical reality.
Quoting javra
I don't see how you can say this. The "empirical" is fundamentally sense experience. Therefore it is a very base level of knowledge. How could it be "governed by metaphysical properties" which is a principled, and therefore higher level of knowledge? The most basic must always govern the higher, as the most basic has a higher degree of certainty. The lower substantiates the higher, and the empirical is the lowest. So the metaphysical cannot substantiate the empirical, it must be vise versa.
Agreed.
Quoting Metaphysician Undercover
For me they apply to all forms, including fictional ones, and not only to that which is physical. That Harry Potter is not a unicorn is true - addresses a reality that stands in its own context of fictional concepts - this via the laws of thought, including the law of identity.
Quoting Metaphysician Undercover
"Oneness" can be readily defined as the state of being undivided, of being a whole. As to 1's infinite divisibility, remember that I take the concept of one to be a hylomorphic whole, a form endowed with constituents. But one constituent does not of itself equate to the given whole. A whole given is taken to be undivided as form, hence - for me at least - can be represented by the number 1. As one example, one horse can only be represented by the number "1", and not by any division. Yes, a horse can be divided into parts ad nauseam, all the way down into zero point energy. But its multiple parts are not the horse as a whole, which is in a state of being undivided. As a more abstract example, one grouping of two or more givens is, as a grouping, itself one whole. As one example, "animal" can be conceived of as a grouping of givens, yet the concept of "animal" is itself one whole - distinct, for instance, from the concept of "plant".
As to the adjective "absolute" we neither innately perceive nor contemplate "a horse", for example, to be a relative whole - a whole that is only so due to its relativity to some other given(s). We innately identify it as a complete, unmitigated, whole.
Quoting Metaphysician Undercover
Again, the law of identity pertains to all conceivable givens, and not just those of physical reality. One three-headed dragon - say one that a person saw in an REM dream - cannot at the same time and in the same respect be both green and not-green. This, to my mind, is so because it would then break with the law of identity. At any rate, a three-headed dragon holds an identity despite it not being a physical given.
Quoting Metaphysician Undercover
Two points:
The empirical is just one aspect of awareness, not the only, nor, imv, the most important. Take the sense of understanding. Without an understanding of that which is perceived via the physiological senses, that which is perceived would be meaningless. We also experientially know of things such as being ourselves happy or sad, and some such states of personal being of which we are aware are in no way obtained via the physiological senses. Hence, I maintain that awareness, and not that which is empirical, is fundamental to knowledge.
Secondly, knowledge of metaphysical realities has nothing to do with whether or not these metaphysical properties occur. Same as with physical reality. Take a preadolescent child or a lesser animal as example. Their awareness operates via the law of identity without them having any knowledge of the law of identity. Or else take adult humans prior to Aristotle's formulation of the principle. They too where governed by the law of identity thought they had no propositional knowledge of it.
That said, to me these metaphysical realities are intrinsic aspects of awareness - again, irrespective of whether the awareness addressed has propositional knowledge of them. We do not, and cannot, create them. We can only discover them. As such, we do not govern metaphysical realities, this just as we don't govern physical realities. We, as aware beings, are predetermined by the former. And, though in different ways, we are likewise determined - bounded/limited - by the latter.
Giving the game away, MU. No Platonist - or Aristotelian - worth his/her [s]salt[/s] ink would say such a thing.
It is the universal view of ancient philosophy that the 'empirical realm' which is taken by moderns as the sine qua non of the real, is in fact a treacherous illusion, which the hoi polloi do not see as their minds are contaminated by worldly passions, which blind them to the higher truths. The real can only be grasped by reason, and its truths are invariant and never subject to decay. Whereas everything in the sensory domain is subject to constant change and degradation through the ravages of time.
We're right back to the initial problem. If we apply the law of identity to all forms, we see that universal forms cannot have an identity. And by that law, "identity" is something that "things" have, so we can use the law to exclude universal forms from the category of "things". This is the intent of that law, to distinguish two distinct types of form, and this is why Aristotle is properly called dualist. There is a categorical separation between primary substance and secondary substance. In your example, the "reality" you refer to is that of secondary substance. Not being "a thing", secondary substance has no identity.
Quoting javra
The issue here, is whether the numeral 1 can represent a unified whole, which is not a physical object. In your example, I assume one horse is meant to be a physical object. As I explained when we refer to one horse, we are stating something about it, that it is one, so "one" is used as a property of that thing. Now you need to justify your assertion that the symbol "1" refers to a unified whole, independently of the object which is said to be one. My argument is that since there are numerous number systems, natural numbers, rational numbers, real numbers, imaginary numbers, and so forth, there are numerous different conceptions of "one", and no single mathematical system unifies these into one concept. Therefore one as a concept, is not a unified whole.
Quoting javra
You don't seem to be grasping the fact, that when we apply the law of identity to forms which are other than material particulars, we are forced by the precept of that law, to exclude certain forms from the category of having an identity. This is because we can use language to refer things without any identity. You don't seem to be grasping the intent of the law, which is to prevent the situation where we assume that just because we can talk about it, it is a thing with an identity. You completely misinterpret the law if you claim that a fictitious thing has an identity, because the law of identity puts the identity of a thing into the thing itself, rather than what we say about the thing. The fictitious thing has no existence independent from what we say about it, therefore it cannot have an identity.
Sure, you can say that "the law of identity pertains to all conceivable givens", but unless you abide by that law, and acknowledge that some conceivable givens do not have an identity, then you step outside that law and you enter into hypocrisy.
Quoting javra
This is not a breaking of the law of identity, it is an issue with the law of non-contradiction. And, if we do not limit what we can logically predicate of a subject, to what is actually possible in the physical world, there is no need for a law of non-contradiction at all. If relevance to the physical world is completely unnecessary, why not allow all sorts of contradictions?
Quoting javra
"Awareness" implies being aware of something. I believe that being aware of the external is prior to being aware of the internal, and being aware of the external requires some form of sense. Of course this is a difficult issue to discuss logically because the terms are naturally slanted in my direction. To "be aware of" implies being informed, and this seems to refer to information from the external. But if we allow that information may come to us from an internal source, then ultimately the empirical is not fundamental. This is why Aristotle defined "the soul", as the first principle of actuality, or first form, of the living body. But if the soul is the first principle of actuality, and it is also the inner most thing, as well as the thing which is "aware", then there is nothing more inner that it could be aware of, and primary awareness is necessarily of the external.
Quoting javra
I don't agree with this at all. I think it is incoherent, so perhaps I misunderstand. First, how could one have knowledge of something which is independent of whether that something occurs? If the something does not occur, yet someone is claimed to have knowledge of it, this is not knowledge at all. It is misunderstanding masquerading as understanding. Second, the law of identity is extremely difficult even for human beings to understand (as evidenced by this thread), it is set up as a defence against sophism. So I don't see how children or lesser animals could be applying the law of identity as a defence against sophism. I believe you continue to misrepresent "the law of identity".
Quoting javra
This is the pivotal point of how Aristotle applies the cosmological argument against Pythagorean Idealism (and some forms of Platonism). He analyzes what is involved in "discovering" such principles. Would you agree with Aristotle, that when the geometer produces geometrical constructs, and discovers geometrical principles, this is an act which is properly described as the mind actualizing the principles. The principles exist in potential, prior to being actualized by the mind.
Quoting Wayfarer
You've already demonstrated how you misinterpret Aristotle. This is quite understandable because he has a massive volume of material and some is quite difficult. I was in the same position until I read Aquinas, and found that his interpretation of Aristotle was inconsistent with mine. Then I had to go back and read much of Aristotle (Metaphysics, On the Soul) all over again, some of which I had already read two or three times, to see what I was missing.
Have you looked at any of that material which I referred you to in Aquinas, how the human intellect is united with the body, and how this union with the body effects the way that the intellect understands? These principles are taken directly for Aristotle's "On the Soul". The intellect is a power of the soul, just like the other powers of the soul such as self-subsistence, self-movement, and sensation. As such, it operates through the means of the bodily organs. Therefore there is necessarily a privation in the knowledge which the human intellect holds. We can say that the bodily organs taint out knowledge because the intellect is dependent on them for the acquisition of knowledge.
If we ignore, or circumvent this reality, we can assert that the human intellect has direct access to the independent, immaterial Forms. But then we have no principles by which we might demonstrate that any claimed a priori knowledge is actually, truly deficient. We can understand this in Kantian terms. The a priori intuitions of space and time are responsible for the tainting of our knowledge. These are manifestations of the physical constitution of the human body, which serve as the conditions for any ideas, concepts, or knowledge in general. Unless we acknowledge that what Kant calls a priori intuitions are properties of the human body, and are therefore fallible principles rather than eternal, immutable, truths, we have no approach toward arguing the deficiencies of them.
Plato himself demonstrated the deficiencies of the theory of participation, which provides the ontological support for the independent Ideas of Pythagorean Idealism. This is why he turned to "the good" to support the existence of Ideas. Notice that it is "the good" which supports intelligible objects for Plato, not "the idea of good", as often represented. It is sometimes argued that Plato himself was not a Platonist, and this is due to the common misrepresentation of Plato's philosophy in modern discussion which produces the common notion of "Platonism". It is very important to approach this ancient philosophy, as much as possible, without bias, if one is intending to develop a true understanding.
Quoting Wayfarer
In this passage you demonstrate significant ambiguity. The "empirical realm", if we use Kantian terminology, is phenomena. And this is how the mind apprehends the sensible objects. However, this refers to all ideas, and concepts which may be employed by reason toward apprehending truths. So all understanding is necessarily "a treacherous illusion" according to what you have stated. Therefore if a philosopher desires a true understanding, one must find a way out of this trap. The way out has been discovered by Plato, through the means of "the good", (implying final cause), and it has been developed by Aristotle, and carried forward by others like Augustine and Aquinas.
You ought to accept that "reason" refers to an activity, not a thing. It is carried out by the human intellect through the use of tools, ideas, and concepts. The human intellect is fundamentally deficient, necessarily so, because it is dependent on the human body. Therefore the ideas and concepts which the human intellect grasps are not the real, higher, invariant truths, which we might assume are out there somewhere. The things which the human intellect grasps receive their intelligibility relative to "the good", which is not necessarily the truth.
All the same I'll give a reply.
Quoting Metaphysician Undercover
The conceptual form of "griffin" is not the same as the conceptual form of "unicorn". I take it we agree in this. How could this be so if neither has an identity? (an issue further addressed below)
Quoting Metaphysician Undercover
You would need to establish how the concept of "one" holds a different meaning in each of these systems to make this affirmation. What I find is that - even though they use the foundational concept of "one" in different ways - the concept of "one" remains the same. It's a given whole, a concept requisite for any such system of mathematics to manifest.
Quoting Metaphysician Undercover
You are conflating identity with primary substances (with empirically known to be physically existent givens).
If I were to ask you for an example of a thing language can refer to that is devoid of any identity, you would likely identify givens that are not "empirically known to be physically existent" ... but you would be identifying them all the same, i.e. disclosing their identity. This in the same breath with which you'd affirm that they lack any identity.
If you believe you can sidestep this contradiction, please provide an example.
Quoting Metaphysician Undercover
It is commonly accepted that the law of noncontradiction is a derivative of the law of identity, and that the former is meaningless without the latter.
Quoting Metaphysician Undercover
We again disagree. A different issue, though. But by this I take it that to you the laws of thought can only be external to awareness. A view which stands in utter contradiction to my own.
Quoting Metaphysician Undercover
Yes, my statement was misunderstood: What is real is regardless of whether or not it is known. One does not need to know what is real in order for what is real to be. This applies to what is metaphysically real just as it applies to what is physically real.
Quoting Metaphysician Undercover
My dog can identify me (e.g., as not being another member of the household or some stranger). Nor does my dog behave as though me is not the same as me. My dog doesn't need to have a cognitive understanding of the law of identity in order to do so. He just does. This is what I meant by saying that the law of identity is intrinsic to awareness, i.e. that it governs all awareness - irrespective of whether there is propositional knowledge of it.
Quoting Metaphysician Undercover
For the example just provided, I would not agree - though there might be other examples for which I would agree. The geometric principles, say those pertaining to a triangle, exist as actuality prior to their discovery. Awareness of them is a potentiality that becomes actualized. But the geometric principles can only be so, in actuality, prior to their discovery.
MU, since we disagree on so many issues, I'm OK with leaving things as they are. Of course, feel free to critique my reply, but I might not reply in turn. Benefited from the discussion all the same. Thanks.
Quoting javra
There's a very straight forward answer to this. A conceptual form is not an "identity", by the law of identity. That's what I've been arguing is the intent of the law of identity, to elucidate this difference.
My belief is that the law of identity puts "same" into a different category from "different". Identity, which is indicated by "same", allows for difference, so that "same" in the context of this law, does not mean lack of difference. This is evident, because the same thing undergoes changes, becoming different from what it was, while still maintaining its identity as the same thing. This is the temporal extension of a thing, and the law of identity allows it to have the property X at one time, and not have the property X at another time, yet still be the same thing. So difference is allowed within the meaning of "same", not excluded by it.
Relating this to your example, what is dictated by the law of identity is what 'same" means, not what "different" means, and different is not opposed to "same" in this definition. So in your example, the fact that the two forms are different, doesn't say anything about identity. Difference is a concept which is related to something other than "same", when "same" is defined by the law of identity.
Quoting javra
That's very simple as well. For instance, in the natural numbers "one" is not divisible, in the real numbers it is. It's easy to say "one" represents a "whole", but the divisibility of that whole is part of the concept. And then there is the possibility of having a negative whole, which is part of the concept as well. It's very clear that the meaning of "one" is quite different in these different systems, and you do not accurately represent the concept of "one" when you say it represents a whole. Using "one" in different ways implies necessarily different meanings, because the meaning is dependent on the use.
Quoting javra
It appears like you are not quite grasping the law of identity clearly, and you are equivocating between two senses of identity, sometimes known as "numerical identity" and "qualitative identity" (check Stanford for an explanation). A type is identified by qualitative identity, but that is not what the law of identity refers to, which indicates one and the same thing. When language refers to 'a thing' which is devoid of identity (by the law of identity), it is referring to a type of thing, so it has qualitative identity, but this is distinct from "identity" as signified by the law of identity. So there is no contradiction, only a misrepresentation as to what "identity" means according to the law of identity, and possibly replacing this with "identity" in the sense of qualitative identity.
Quoting javra
Ok, maybe we have grounds for agreement here, and if we try we might be able to proceed. Do you believe, that just like "what is real is regardless of whether or not it is known", a real thing also has an identity regardless of whether or not the thing is known. This is what I am arguing that the law of identity states, the identity of the thing is within the thing itself, regardless of what is known about the thing. Can you agree with that? Further now, when we use "identity" in reference to what we know about a thing, this is "identity" in the other sense, qualitative identity. We describe the thing in terms of qualities, which are types.
This forms two distinct uses of identity which are consistent with the two distinct senses of "same". We might say that two different people have the same car, meaning the same make, model, year, colour, etc.. This is qualitative identity, reducible to type, really meaning the same type of car. On the other hand, we have to maintain that each car is distinct, because we know that the two cars are not the same car. This is numerical identity, what the law of identity refers to. If we are allowed to say, in strict logic, that two things are the same thing, because we have described them both in the same way, this creates many false conclusions.
Since you’ve pointed me to SEP, you’ll notice that the entry on identity is in no way unequivocal about what identity is. But taken from the introduction:
Quoting https://plato.stanford.edu/entries/identity/#1
Given this distinction - and the plasticity of the term "thing", which can reference a concept – how is your own concept of “griffin” not numerically identical?
(That it might change over time equally applies to any physical thing. Moreover, it can only change so much as concept while remaining the same concept of “griffin” – this, again, in parallel to the numerical identity of any physical object: e.g. the numerical identity of a flower between the time it is a bud, or earlier, and its full wilting, or later.)
Notice that qualitative identity requires the sameness of qualities that pertain to two or more things. By comparison, the concept of “griffin” is one thing - a given whole that as form is undivided - and not two or more. It is a hybridization of different animals – an eagle and a lion – true; but the hybridized given is nevertheless singular.
----------
As to the ontology of identity, my own views are fringe. In all fairness, I’d rather not get into them right now.
I cannot locate, or in any way, point to this supposed concept, within me, to demonstrate to myself, it's existence as a thing. I might conjure up an image, but that's not really a concept. The closest I can come to finding a concept, is definition. I might assume the definition is "a thing", but the definition is just a bunch of words, which in that form is a multitude of things, each needing to be interpreted as signifying a concept itself. So this is really going in the wrong direction. It's heading toward an infinite regress of words associated with an assumed "concept", but no actual concepts. To find "the concept", if it exists as a thing, I need to take a different approach.
I could look for the boundaries of that thing, and this is how we distinguish things with our senses, or I could look for a unity of meaning, signified by the definition, or just the unity of meaning signified by the word "griiffin". Each of these, boundaries and unity, can be an indication of "a thing", and ultimately the most conclusive evidence would be to find both. But based on my understanding of the difference between sensible objects and concepts, I think the latter would be the most productive course. This is regardless of the modern trend in philosophy to look at definitions as boundaries created by rules of language, which I think is a dead end route. It is a dead end for the reasons stated above, the rules need to be interpreted, which would require rules for interpretation, resulting in an infinite regress of words, without finding any real boundaries. So the proper course appears to be to seek the unity of meaning, if I want find the object which is "the concept".
This is where the real difficulty begins, requiring an understanding of the use of words to signify meaning. Notice that we have to differentiate between using words to name a thing, as proper nouns, and using words to signify meaning. So to make a long story short (I could give you a vast multitude of examples of words referring to properties, which indicate meaning rather than an object), it is my understanding that when we use words to indicate meaning, what is signified is a type, rather than a thing. We might signify a type of property, or a type of thing.
So, we have a unity principle indicated here, which designates a type, as a category, a classification principle, like a "set" in mathematical terms. To understand this unity, and determine whether it is "a thing", we need to grasp the meaning of the principle which is supposed to render the unity. I suggest that the intent of this unity principle is to group numerous things together, which have some property in common, to aid in understanding. We could say that we attempt to create "a thing" in this way, and the thing which would be created exists as the category itself, which allows us to group numerous other "things" within that category.
Notice that I have determined two very distinct usages of "thing" now. One refers to the category itself, the principle of classification, the other refers to the objects to be classified. In effect, I have developed a principle of classification to distinguish two very different types of "thing". Therefore I need principles which distinguish them as separate. The one category I assign "identity" to. The things which will be classed, and categorized have an identity. The other category, the principles by which things are categorized, or classified, I cannot assign "thing" to, nor can I assign "identity" to, because I've already used those terms in the other category, and this is a philosophical endeavour, and my goal is to maintain logical rigour. It is true that there are many conventions in common vernacular which would call these categories "things", and say that they have "identity", but the goal here (being philosophy) is to maintain the validity of logic, and therefore avoid the equivocation and category mistake, which would inevitably result from such a duality of meaning.
The conclusion now, is that those so-called 'things", which I've now excluded from the category of "things", those concepts, which are more properly referred to as principles by which a unity is to be created, are not themselves properly called unities. This is because each one fits into a higher unity, and is not itself a proper whole, receiving meaning from the higher whole. So Socrates refers to man, man refers to animal, refers to living being, etc.. Therefore no individual concept is a complete unity, it always refers to something outside as a source for meaning. It is a part which is not itself a whole, because it is wholly dependent on something external to it for its meaning. I believe this is the point of the op, the meaning of the axiom is always derived from something else, so the object is not complete as an object until we determine the whole.
This is why we need the highest unity, "the Ideal", what the Neo-Platonists called "the One", to allow that a concept is a thing. This highest unity, this Ideal, is supposed to be the complete conceptual unity, being deferred to no higher whole to complete its meaning and unification. It is by that assumption, "whole", and therefore a thing with identity. This Ideal, "the One", is assumed as a concept which is also a thing with identity.
Quoting javra
I don't think this is quite true though. You are simply assuming that the griffin concept within your mind is one thing. But if you analyze this concept within your mind, you'll see that "griffin" refers to a type of thing. And if you assert that this "type of thing" is a thing itself, that assertion needs to justified. That is what I tried to do above. It leads to an infinite regress of meaning, requiring the assumption of a fundamental unity such as "the One", which unites all conceptual structure as one united whole. This is why we ought not allow any contradictions within knowledge, no matter how far apart the fields of study are, it negates the possibility of a united whole,
A very informative post. Thanks for it. To let you know a little more of where I’m coming from:
There is the philosophical notion of holons: givens that are simultaneously both wholes and parts. Although my views are not identical to those addressed in the article, I do have great empathies toward the views therein expressed.
Any animal - as a whole token - is itself in part determined by its environment: from that of its ecological environment to that of the world’s natural laws as environmental givens. As one example, a mammal would not be in the absence of air it inhabits just as a fish would not be in the absence of water it inhabits; in both cases the occurrence of the former is *in part* determined by the occurrence of the latter. An individual animal can thereby be construed to be a part-holon of its environmental-holon.
It’s a complex ontological approach, but then an animal's parts, say its lungs, has an identity, just as the animal itself has an identity, just as the animal’s environment, say a particular forest, has an identity.
Using the notion of holons, then, to me each concept is itself a holon - constituted of parts that are themselves holons, and is itself a part of greater concepts that are themselves holons.
While this synopsis will not address all conceivable issues related to this approach, I get that we will likely disagree in our basic approaches. No harm in that though. Save for a few disagreements here and there. :grin:
I'll see if I can interpret this in a way which is consistent with what I said in the last post. Let's say that there are two aspects of the holon, one which makes it in some sense an individual, free and independent, and another aspect which makes it necessarily dependent on a larger whole, so that it is united to other concepts by this aspect. You can see that these two aspects are fundamental incompatible, the one making it a distinct individual with its own independent identity, the other wanting to negate this existence as a distinct individual, giving it an identity, as a part of a larger whole. So they must be distinct aspects of the concept. This would mean that the holon partakes in both the categories I described, as a material individual, and also as part of a type, receiving meaning from a higher order. This is comparable to Aristotle's matter and form.
If a concept is an object, as a holon, then it cannot be purely immaterial, it must also have a material aspect, which is responsible for it being, in some sense, an independent individual. If a concept is strictly formal, then its complete existence is dependent on the larger whole. If it has any sort of individual, independent existence, then it must have a material aspect to account for this separate identity.
What Aristotle does, is seek the source of this material aspect, as that which accounts for the existence of individuals. In many ways, the material aspect is indefinite and therefore unintelligible. So the concept, being essentially something intelligible, must also have an aspect of it which is unintelligible, just like any material thing. This is the deficiency which Aquinas spoke of. Due to the fact that the intellect is united with the material body, and is not properly a separate substance, the intelligible forms which it deals with are also deficient.. I don't mind looking at concepts in this way. It makes more sense than saying that a concept is a completely separate, and immaterial object.
Seems like we’re approaching a common ground in respect to the hylo-morphology of concepts. Cool.
BTW, to me there’s a parallel between Aristotle’s prime matter and today’s notion of zero-point energy. Both seeming to hold the properties of pure potentiality and unintelligibility while underlying all that is intelligible matter. As we were previously discussing, the intelligibility of actualized identity is always brought about by forms - including the forms of intelligible matter. And, in Aristotelian terms, the ultimate form is that of the teleological unmoved mover, which is singular as form in being devoid of constituents and, therefore, devoid of matter. Please remind me if there were any disagreements between us in the aforementioned.
What criticism would you give to the proposition that every intelligible form is, and can only be, cognized as a whole (for context, where every whole - save for the unmoved mover - is itself a hylomorphic holon). Thereby making the concept of a whole, i.e. of an entirety, and the concept of a form fully synonymous.
As background, I find this issue to be pertinent to the context of the Aristotelian category of formal causation. Which is distinct from, though entwined with, teleological causation (as might be evidenced in Aristotle’s coinage of entelechy as term for addressing actualized things).
I used to hold a very similar understanding, but I've progressed toward what I consider to be a deeper understanding. If we maintain that concepts have a material aspect, then we need to account for the manifestation of that material aspect. In Aristotle's hylomorphic structure, matter accounts for the temporal continuity of the object, its capacity to persist, and therefore its identity as a continuation of being the same object. This is described as mass or inertia. This is also the aspect of the object which is unintelligible to us, in physical objects we simply take it for granted.
The temporal continuity of a concept can be seen to be provided for by the physical existence of the symbols. This is how we communicate, allowing the same concept to exist for generation after generation, and amongst different individuals. Also, we can see a similar situation within the mind of an individual, where memory relies on symbols, or some other sort of representation such as an image. The representation is essential to the temporal continuity of memory. However, this implies that the symbol is an aspect of the concept itself, as its material element. It is an accidental, as we see that the symbol may be arbitrary, but still the medium employed (as the matter) has an effect on the essence of the concept through the capacity for temporal continuity. So for example, there is a big difference in the potential for temporal continuity, between written symbols (as we find in mathematics), and spoken words. The idea that mathematical symbols represent eternal objects is derived from this capacity for extended temporal continuity. But when we look at spoken words we find that the concepts represented by symbols are actually quite fluid. So the forms (concepts) are changing as time passes, and even the mathematical ones change as knowledge evolves, but through a slower process.
Regardless of those revelations, as philosophers we may have a commitment toward further analysis of "the concept". If a concept is composed of matter and form, then to fully understand the nature of a concept we need to produce a proper separation between the matter and the form, so as to distinguish the intelligible from the unintelligible Matter has been designated as fundamentally indefinite and unintelligible.. Allowing the unintelligible (matter) to enter into the intelligible in our apprehension of "the concept", assuming that it is an inherent part of the intelligible (as a concept is fundamentally intelligible), is to provide for misunderstanding. So we still need a technique to identify and expel misunderstanding. Therefore we still need to proceed toward separating the material aspect (the representation), out, and attempting to understand strictly the immaterial, intelligible aspect of the concept.
Quoting javra
The cosmological argument, and the teleological unmoved mover, are the means by which Aristotle brings matter itself into the realm of the intelligible. In Bk.7 Metaphysics, the part I referred to already, you can see how he begins this process by showing how the matter is a part of the formula in creative art and production. Ultimately, in the cosmological argument, Bk.9, it is shown that actuality must be temporally prior to potentiality, therefore form is prior to matter. This is the form of final causation which determines the matter in creation. Accordingly, there is no such thing as "prime matter", because matter is created as required by the purpose, so all matter is inherently formed by the purposeful act which creates it, therefore it is fundamentally intelligible. The fact that matter is by definition indefinite, and unintelligible, is an epistemological fact. The deficient human intellect does not have the capacity to understand temporal continuity, so this part of reality is designated as indefinite and unintelligible (matter). But this is the fault of the human intellect, temporal continuity, matter, is actually quite intelligible to a higher intellect, according to Aristotelian principles.
Quoting javra
This I think, is the problem evident in the hylomorphic approach to concepts. In the case of conception, such a whole is never quite complete, therefore an invalid "whole". This is the example I provided with the regress into unclarity: the concept of "Socrates" refers to "man", which refers to "mammal" which refers to "animal" which refers to "living being", and so on. The more specific is defined by the more general, and the more general becomes increasingly vague and ill-defined, such that we can never claim completion of a whole in conception. And, we see the same thing in describing the existence of physical objects. An object on the earth requires reference to the earth, and the earth requires reference to the sun, the milky way, etc.. We do not get the closure of a whole, even in our descriptions of physical objects.
So we might produce an Ideal, "the One", as a proposal of a valid whole. The One is what both closes the conceptual expansion into vagueness, and also validates a particular physical whole as everything, the universe. Aristotle proposed circular motion, and the divine thinking, which is thinking on thinking, as the closure of wholeness, but this proved to be insufficient. Aquinas stipulates "God", but it is asserted by him, that the human intellect, being dependent on the material body, cannot obtain a proper understanding of God.
So I would say that wholeness is what is required by the intelligible form in order to be completely and absolutely intelligible, but human conceptions lack this. This is quite evident in the most fundamental mathematical principles. The natural numbers are infinite. The spatial point is infinitely small. A line is infinitely long, etc. This is evidence that human conceptual forms, as intelligible objects, are fundamental lacking in wholeness. This is why I prefer not to call them "objects". However, as I said above, in our attempts to understand physical objects we are met with the same deficiency of wholeness. But this might just be due to the deficiency in our capacity to understand. We clearly sense boundaries of closure, we see objects as closed wholes. So it may be the case that we simply misunderstand what we see, and there may actually be some truth to the wholeness of an object which we perceive, but we just have not developed a proper understanding of it.
"Refers" is an inadequate term here. "Socrates" refers to Socrates, and not just any man. Likewise "animal" refers to animals, and not just any living being (plants, for example).
Quoting Metaphysician Undercover
I'm glad that this is evident. In short, when in search of absolutes - such as in a complete and absolute intelligibility, to paraphrase from this quote - absolute wholeness does not occur for givens, be they conceptual or physical. Nevertheless we cognize givens as bounded entireties. For example, a rock is cognized as a bounded entirety, as a whole given. Not as two or more givens; and not as an amorphous process. Even "a process" is cognized as a bounded entirety, and can thereby be discerned to be one of two or more processes.
Maybe you're looking for the absolute, fundamental nature of individual things that dwells behind our awareness of them, so to speak. Whereas I'm addressing the very nature of how we cognize givens: by cognizing each individual given to hold the attribute of oneness.
But I find that this following statement might be pivotal to our disagreements in large:
Quoting Metaphysician Undercover
What then do you make of formal causation?
I also note that while a flower is neither an unopened bud nor the stem off of which all petals have fallen, it yet remains the same (numerically identical) flower throughout the time period in-between, despite considerable changes in its matter over this span of time. Its identity nevertheless remains static in its form - again, despite the changes in its matter - such that form accounts for the temporal continuity of the object, and therefore its identity.
You misunderstand. The word refers to the object in some circumstances, but we're talking about the concept here, the concept of "Socrates". To say what Socrates is, the concept of Socrates, is to refer to "man". To say what a man is, is to refer to "animal". The issue was whether or not a concept is a whole. And since each concept is defined by something more general, which makes the definition get more and more vague, without closure, as we proceed in defining the terms, a concept cannot properly be called a whole.
Quoting javra
The problem with this is that we see a physical object as a bounded whole. The spatial boundaries are very evident to our vision, so we have some reason to believe that there is some reality to these boundaries. Therefore we assume that physical objects are wholes. In the case of concepts we cannot find those boundaries. One concept is defined in relation to another, which is defined in relation to another, and so on, and we cannot find any true boundaries. So we might insert arbitrary ones. This is the same with "a process". If there is a temporal duration, we cannot really discern when one process stops and the next starts, so we insert arbitrary boundaries. Temporal processes are like that. When we look for the beginning or ending of a particular, identifiable object, the precise moment that it starts to exist or ends existing is arbitrary.
So I think that our conceptions of boundaries are based in spatial concepts, being derived from our sensations of spatial boundaries, and we don't really have any real, applicable principles toward understanding temporal boundaries. We can readily understand that the present makes a boundary between the past and future, but we have very little if any understanding of what this means.
Quoting javra
I must admit that I am not familiar with your use of "givens", and I don't think I understand what you mean with it. Maybe you can explain.
Quoting javra
I would describe formal causation as the restriction imposed on the possibility of change, by the actual physical conditions present at the time. So at any given time, any situation is describable in formal terms. The describable physical conditions which are present act as a constraint on the possibility of future situations, therefore this present form, is in that sense, a cause of future situations.
Quoting javra
I think this is incorrect. The flower changes its form. The form is what is describable, and the changed form is describable. The matter only changes to the extent that such changes, material changes, are describable, but if they are describable, then they are formal changes. So by definition, the matter does not change. As Aristotle says in his Physics, it is what a thing comes from, and persists afterward. So by definition it is what does not change. Only the form changes. If there was a prime matter, it would be the fundamental elements or particles out of which all existing things are made. These fundamental particles would never themselves change, they would just keep existing in different configurations, and this would account for all possible change. However, this is the conception which Aristotle demonstrates as fundamentally incoherent in his Metaphysics. He shows how matter itself must come to be from some type of teleological form, therefore we need to seek the Divine Will, as the cause of matter and temporal continuity.
Quoting javra
The form of a thing does not remain static, it is always changing, and is by definition what is "actual". The matter, as "potential" is static, because despite changes the potential remains the same (conservation laws in modern physics). And by definition, the matter is what existed prior to a change, and persists after the change.
We are oceans apart. A culture's form (imperfectly) determines the nature of the individual, constituent, human psyches it, as a culture, is composed of - language and its semantics as one example. But nowhere does a culture have "describable physical conditions".
Quoting Metaphysician Undercover
This puts a big damper on things for me. I cannot logically appraise Aristotle's teleological unmoved mover to be "Divine Will" - in part because will itself is always teleological motivated by an outcome it seeks to accomplish, and it is thus always in motion. Maybe this is a/the primary source of our disagreements - with most other issues regarding identity being derivatives.
In any case, I'm respectfully bowing out of the conversation.
I thought we were sticking to "form" in the sense of Aristotle's hylomorphism.
Four years later, I had a whim to come back here. I just wanted to explain why this is wrong.
Take graph theory. I show you a graph with two vertices and no edges. By hypothesis, the two vertices are two different things. Those two vertices, however, are structurally indiscernible. Which makes them the same vertex, according to structuralism. Contradiction.
Therefore, either mathematical objects are not identified by their structure, or the stated graph can't be defined. The latter is false. So mathematical objects are not identified by their structure.
That's the fundamental problem with structuralism. You cannot escape the need for identity conditions by focusing solely on relations rather than particulars, because relations are particulars. "I don't have to count objects, because I go by kinds" – and if I ask how many kinds there are...?
Ah the good old daze ...
Quoting Pneumenon
This is the famous two spheres argument against the identity of indiscernibles.
[quote=Wiki]
Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it is sufficient that one provide a model in which there are two distinct (numerically nonidentical) things that have all the same properties. He claimed that in a symmetric universe wherein only two symmetrical spheres exist, the two spheres are two distinct objects even though they have all their properties in common.[/quote]
If this resonates with you, all to the good. I think this is the same argument you are making.
Quoting Pneumenon
I don't think anyone is saying there can't be two of something in math. There's only one set of real numbers, defined axiomatically as the unique (up to isomorphism) Dedekind-complete ordered field.
Yet we have no trouble taking two "copies" of the real numbers, placing them at right angles to each other, and calling it the Cartesian plane. Or n copies to make Cartesian n-space.
For that matter we say that a line is determined by two points. But one point is exactly like any other, except for their location, yet nobody thinks there's only one point. There are lots of points. And if we wanted to have this conversation, we could argue that there aren't any points at all, a point being a "zero-dimensional location in space," whatever that ultimately means.
Still, two points that aren't the same point determine a unique straight line, and nobody from Euclid to the present has every argued otherwise.
I don't know if this is a violation of structuralism. Structuralism is about deeper matters, not about whether there can be two of the same thing. And if they're the same, how can there be two of them? I feel @Metaphysician Undercover about to take me to task. I don't know the answer to any of this.
Quoting Pneumenon
I'm not the self-appointed defender of structuralism. As I say, structuralism is about deeper matters. There is only one set of real numbers, even though there are many ways to define the real numbers.
The set {0, 1, 2, 3} with addition mod 4, and the complex numbers {1, i, -1, -i} with complex number multiplication, are isomorphic as groups. We say they are two different instances of the "same" group even though they are manifestly not the same thing. There is only one cyclic group of order 4, even though we have here two vastly different instances of it. Like Superman and Clark Kent. The same, but very much not the same.
I note that you opened this thread with
Quoting Pneumenon
So are you arguing against Platonism? Or constructivism? They're different philosophies, right? And perhaps even my Hilbert beer mug quote was a little off the mark, since AFAIK he was explaining how axiomatics should work, and not specifically structuralism.
Or perhaps Hilbert was a pre-structuralist. Saunders MacLane, the originator of Category theory, attended Hilbert's lectures. I found a fascinating-looking article, "Saunders Mac Lane: From Principia Mathematica through Göttingen to the Working Theory of Structures" in an anthology called, "The Prehistory of Mathematical Structuralism." So perhaps I intuited more than I realized when I mentioned the beer mug remark. Hilbert was thinking about structures all along.
Perhaps you can glean some clues from this essay, which I haven't yet read but which mentions Carnap, Husserl, and other philosophical luminaries of that era. Evidently these ideas were in the air in the 1930s.
https://academic.oup.com/book/41041/chapter/349348878
(edit) -- I skimmed through the article. Not too many clues in there of an elementary nature directly on point to your concern. In any event, MacLane learned from Hilbert; and Hilbert was indeed a pre-structuralist. The beer mug remark goes deep.
I have a question, and sorry for my basic knowledge on this, but I want to learn about this very interesting topic.
My question: according to your points, are you stating that mathematical objects are dependent upon identity conditions rather than structure? If I remove those conditions, the mathematical object doesn't exist?
I'm curious about the fundamental core of each mathematical object's existence.
I want to start with this: I'm defending Platonism, bro. The objection based on identity conditions really bothers me, so I want to say find some way to get around it and still be a Platonist.
I think it's analogous to Quine's observation about modality:
I think this is a deep problem. Deep problems manifest in multiple places – that's how you know they're deep. The fact that something similar happens with modality as with math ought to tip us off that this is not a surface-level concern.
So how does it manifest for math? Like this:
Antiplatonist: "We cannot say that mathematical objects exist in a mind-independent sense. This is because there are no clear identity conditions for them."
Platonist: "Identity is structural. Two mathematical objects are numerically identical iff they are structurally identical."
Antiplatonist: "Here are two objects. They are numerically distinct and structurally identical. Therefore, numerical identity is not structural identity. Therefore, you have not answered my objection."
My solution to this dilemma, per the first post, is simple: no two mathematical objects are numerically distinct. There's only one, Math. Any two apparently distinct ones are just logically valid observations of the same object. A duck and a rabbit, if you like Wittgenstein. Attributes, if you like Spinoza. Emanations, if you like Plotinus.
(I like Spinoza. )
I agree, with a caveat.
Black's argument is this: "Two things can be indiscernible, in every single respect, and still numerically distinct". He's saying that two things can be indiscernible in every possible way and still be two things.
My argument is a lot more constrained. I'm saying, "Two things can be indiscernible, in terms of mathematical structure, and still be numerically distinct". And this is proved, IMO, by the graph example.
Quoting fishfry
I mean, you're not wrong. Everybody agrees that there can be two of something in math. As you note, points differ by their location. Location is structural. Therefore, points are structurally distinct. And I agree with you on that. I'm not saying, "Ha! Here are two different points. Suck it, structuralists!"
What I am saying is, "Here are two objects. They are numerically distinct, but structurally identical. Therefore, structural identity is not numerical identity".
If it is possible to individuate two objects without appeal to structure, then we can find two such objects that are structurally identical. Which sinks structuralism. So the burden of the structuralist is to do math in such a way that you must appeal to structure in order to individuate two objects.
You did this for points, I think. If points are individuated by location, and location is structural, then numerical identity for points is structural identity. It works.
But this doesn't apply to the graph-theoretic example. You can simply say, "There exists a graph with two vertices and no edges". If any graphs exist, this one does. And now we've individuated two vertices without appeal to structure. The structuralist must show that one vertex has a structural property that the other doesn't. By definition, this cannot be done.
You can't make structure itself a primitive notion, by the way. That defeats the point of structuralism. The whole point was to abstract away from particulars and deny intrinsic properties. If you make structure primitive, then you've basically made it an intrinsic property.
One last point: when I say, "Therefore, the two vertices are numerically distinct", I'm speaking ex hypothesi. In fact, I do not think the vertices are numerically distinct. I don't think any two mathematical objects are numerically distinct. There's only one. Any talk of distinction is just a way of talking about how that one object relates to itself.
I'm not sure how to cash that out, exactly. Perhaps I should say: math relates to itself in every logically valid way, for every coherent system of logic. Therefore, every coherent logical system expresses math.
Oh I see. In your OP you were stating an objection to it only to knock down the objection. That was not clear to me, and I did not read through this entire four year old thread to discern that. My most humble apologies. Bro'.
Quoting Pneumenon
You know, you might have missed the mark in tagging me for your resurrection of this thread. I made an offhand remark about Hilbert's famous beer mug quote, and I may have mentioned the turn towards structuralism of latter 20th century math. But I am not proclaiming myself the Lord High Defender of these ideas, nor am I particularly knowledgeable about these matters, nor do I even have any particularly strong opinions about them. You don't like structuralism, that's ok by me. You are a Platonist, so was Gödel, and so are most working mathematicians.
So I am poorly positioned to KNOW anything about these matters, nor particularly CARE about them. That is, I am both ignorant and apathetic about the subject at hand.
That said, I did see a bit of modern math a long time ago; and in particular, I was trained in the doctrine of isomorphism; that is, that two things that have the same structure, may as well be regarded as the same thing, in a given context.
So I believe I can perhaps explain some of these ideas, and put them into mathematical context.
But I can't offer any kind of spirited rebuttal to your Platonism, since "I don't know and I don't care."
With that said, I will plunge in and stumble on.
One more thing ...
Quoting Pneumenon
You've used that phrase a few times, and I should admit that I have no idea what it means.
What is an identity condition? Can you explain what you mean? Can you give me an example or two, say with regard to some familiar mathematical examples like the set of natural numbers, or the cyclic group of order 4, or the Riemann sphere, or any other example you care to name.
I just have no idea what you mean by an "identity condition," and how this idea stands in opposition to the idea of structuralism.
Do you mean perhaps that the two representations of the cyclic group of order 4 I gave earlier should be properly regarded as two separate things? But of course they are. They also happen to have the same group-theoretic structure, so they represent the same group. It's all about the context in which we use the word "same." Nobody is arguing with this. Clark Kent and Superman are the same and they are different. It all depends on the context.
Quoting Pneumenon
Now here you are barking up the wrong tree. I know little of Quine and even less about modal logic. I'm singularly unqualified to even think about what you wrote. I simply do not have any place in my mind to hang these concepts. I plead abject ignorance.
Quoting Pneumenon
I simply fail to see the relevance to any ideas I may have about mathematical structuralism, modern concepts of isomorphism, or anything else. This is exactly the kind of question that, if it were to appear in a thread on this forum, I'd simply ignore. I have nothing at all to say.
Quoting Pneumenon
I respect that. I just wonder why you are drawing my personal attention to it. I'm singularly unqualified to have an opinion. I have no knowledge, no interest, and I have no referents in my conceptual scaffolding. I have to just read, ignore, and move on. In some other life I was a big Quine fan and knew something of modal logic. In this life, no.
Quoting Pneumenon
Perhaps you can give me a mathematical example so I can know what you are talking about.
"something similar happens with modality as with math" -- I just have no idea.
Remember, all I did was toss out the famous beer mug quote. Perhaps you think too much of me.
Quoting Pneumenon
Ah! Ok. Thanks. I'll gratefully take the lifeline.
Quoting Pneumenon
I can't relate that to any aspect of math I've ever encountered. You threw me an anchor, not a lifeline.
Give me a mathematical example. Something involving the number 5, say,
Again, you are arguing a thesis -- clearly one you've thought deeply about -- that I just have no knowledge of and little interest in.
Why me? All I did was quote-check the beer mug. I might as well have worn Hilbert's famous hat.
Now if you had specific questions about how the concept of isomorphism is used in math, I could definitely be of help. But I did give you a concrete example, the two representations of the cyclic group of order 4, and you didn't engage.
Quoting Pneumenon
What does "numerically distinct" mean? I'm willing to agree that 5 and 6 are numerically distinct.
Past that, I have no idea what you mean. Explain please?
Quoting Pneumenon
Didn't he get excommunicated by the Jewish faith? That's all I know about the guy. I hope you see what a philosophical ignoramus you are talking to. I wish I could engage. And I wish to hell you would give me some concrete mathematical examples of what you are talking about.
You wrote this long post to the wrong person. I can't respond to any of it and I have no idea what you are referring to. But some mathematical examples would help. And when you say you're going to give a mathematical example, you don't. You just give some vague hypothetical dialog that has nothing to do with math as I understand it. I wish I could help. Maybe some of the other participants in this thread can chime in.
Quoting Pneumenon
Do you mean that the two representations of the cyclic group of order 4 are "numerically distinct" yet have the same group-theoretic structure? If that's what you mean, I heartily agree.
Quoting Pneumenon
That's high praise around this place :-)
Quoting Pneumenon
You are using "numerically distinct" in some kind of crazy way. 5 and 6 are numerically distinct. Past that I have no idea what you mean.
But if you are saying that two mathematical objects may have the same structure yet be presented vastly differently, well duh. That's a commonplace observation that nobody would disagree with.
I think you are misconstruing the hell out of mathematical structuralism; flailing at a straw man; and expecting me to engage or to be an enthusiastic advocate of something or other.
Quoting Pneumenon
I couldn't parse that. You are talking in such vague generalities. Give me a specific mathematical example.
But the article about MacLane that I recommended to you actually explained that. It said that we can't call two objects isomorphic until we say which category we're in. So the two representations of the cyclic group of order 4 are the same as groups, and different as mathematical objects.
What is the point of making such a trivial observation that nobody disagrees with?
Quoting Pneumenon
Ok. Fine. Whatever. Tell me why you think I am the right target for this conversation. I can't hold up my end. I'm defenseless.
Quoting Pneumenon
Have I ever done so? Are you arguing against Category theory? If structure is not a primitive notion, you must be against abstract algebra. I'm actually laughing as I write this. That is such a naive and inaccurate argument. I just don't know what you mean.
Quoting Pneumenon
Who made structure primitive? Are they in the room with us right now? You're playing word games. And I gather from your discourse that you don't have much idea what structuralism in math is really about.
Quoting Pneumenon
What the heck does numerically distinct mean? You mean that one is 5 and the other is 6? I don't know what you mean by that phrase.
For one thing, they're unequal as grade school numbers. For another, they're unequal as sets. And for a third reason, 6 is the Peano successor of 5, and the successor of a number is always distinct from the number.
So you're just mathematically wrong here.
Quoting Pneumenon
Only one number? What the heck are you talking about? You are making demonstrably false claims. Euclidean 4-space is mathematically distinct from the Lorentzian 4-space of general relativity. They're entirely distinct mathematical objects. They both consist of exactly the same underlying set of points; but one has the Euclidean metric and the other has the Lorentzian metric. That makes them utterly distinct; as distinct as Newton's and Einstein's conceptions of gravity.
You just made the claim that there are no distinct mathematical objects.
Do you stand by that? At least it's a claim you made that I am qualified to dispute.
Quoting Pneumenon
Which tells us what?
Anyway I find this conversation interesting, but I am not really understanding what you're getting at. I could really use some mathematical examples. Do you really claim that 5 and 6 are not "numerically distinct" mathematical objects?
In any event, the observation that two mathematical objects can be
1) Very different in form and nature; and yet
2) Have the same structure with respect to some abstract properties;
is a commonplace barely worthy of note.
Superman and Clark Kent. The same in one context, utterly different in another.
I just wanted to bother you 'cause I thought you'd be fun to talk to. I figured you'd just ignore me if you weren't into it. I didn't think you'd start screaming "OH GOD WHY DID YOU CHOOSE ME I WANT NO PART IN THIS MADNESS" like I jumped out of the bushes and throttled you or something.
It's okay. You can just ignore me if you want. But I think you'll have fun.
Okay, so some philosophers use the terms "qualitative identity" and "quantitative identity" (or numerical identity). Qualitative identity is where you can't tell them apart. Quantitative is where they're the exact same thing like Superman and Clark Kent. "Quantitative" is kinda misleading, there, so I'll drop these terms.
Instead, I'll use "indiscernible" and "identical". Two things are indiscernible if you cannot tell them apart. They're identical if they are one and the same thing. So two "identical" twins can be indiscernible. But they're not identical, strictly speaking, cause they're two different people. Do I still sound like raving lunatic?
An identity condition is a statement that only applies to one thing. The statement "a prime number between 2 and 5" is an identity condition, because it applies only to one mathematical object, which is 3.
That sounds very simple, right? But it gets complicated. See below.
(Apropos of nothing, have you ever read Augustin Rayo's The Construction of Logical Space? You might enjoy it.)
I think we agree that the set {0, 1, 2, 3} is not identical to {1, i, -1, -i}. And we agree that both of those are instances of the cyclic group with order 4, if they have the right operations defined on them.
Now, here are some questions:
1. Is "{0, 1, 2, 3} under addition mod 4" identical to the cyclic group with order 4?
2. Is "{0, 1, 2, 3} under addition mod 4" an instance of said group?
They're only "different" in a figurative sense. I think Clark Kent is identical to Superman at all times. They're both one dude. "Clark Kent" shares a referent with "Superman".
I don't think that identity is dependent on context like that. There's nothing you can do to make something stop being identical to itself.
Nah, man. It was just a tangential aside. It's okay. I'm not trying to throttle you. I'm just talking to you. I'm just writing this on my laptop. I'm just under your floorboards. Ominously cracking my long, gnarled fingers. And salivating.
Yes ok all good. Having fun. Late now and have some things to do tomorrow morning so I'll get to this in the afternoon.
Oh gosh I hope I haven't disappointed you ... now I feel pressure to be entertaining!
Quoting Pneumenon
Not at all. Just on the one hand, feeling totally inadequate to discuss the philosophical side of this ... Quine and Spinoza for example. And on the other hand, feeling that you are mischaracterizing mathematical structuralism and attacking a strawman.
But if it's good for you it's good for me.
Quoting Pneumenon
I'm having fun.
Quoting Pneumenon
I'm more confused than ever now regarding those terms. As modern math shows us, concepts like identical and same and "can't tell them apart" depend on the context. They are not absolute. I'll give examples in what follows.
Quoting Pneumenon
Big trouble, as the two sphere example shows.
Quoting Pneumenon
No, but the counterexamples come up fast. Especially the twins. Identical twins do not have the same fingerprints. Interestingly when I typed "Do identical twins ..." into my Chrome search bar, it autocompleted "have the same fingerprints," showing that tens or hundreds of thousands of people have asked the same question. So identical twins are not identical and we CAN tell them apart. Forget the twins.
But just take some mathematical examples. Is the natural number 3 identical to the real number 3? They are quite different as sets, but they are exactly the same via the usual embedding maps; which is to say that the reals contain a copy of the naturals that respects all the order and arithmetic information, so we regard the naturals as a subset of the reals, even though set-theoretically, this is false.
Consider the x-axis of the Cartesian plane in high school analytic geometry class (or Algebra II in the US). Consider the y-axis. Are they the same? Clearly not, they're different lines in the plane with only one point of intersection. Yet they are both "copies," whatever that means, of the real numbers. But in set theory there's only one copy of any given set. Where'd we get another copy? Are the x-axis and the y-axis the same?
I don't accept the distinction you're trying to make. Too many corner cases.
Quoting Pneumenon
You sure? Does it apply to the real number 3, which has an entirely different set-theoretic representation?
But ok. An identity condition is a statement that uniquely characterizes a mathematical object. Like "the unique Dedekind-complete linearly ordered set" uniquely characterizes the real numbers.
Ok I'll accept that definition. But note that it's a structural condition, not a Platonic one.
I better call this out to make sure you note it.
By your definition, an identity condition is a structural condition, not a Platonic one.
I can construct the real numbers many different ways, but they are all the same real numbers, because they all satisfy the identity condition of being a Dedekind-complete, linearly ordered set.
Am I understanding your meaning of identity condition?
And am I making a valid point that this is a structural condition, which seems to negate your point that identity conditions are the opposite of structuralism?
Quoting Pneumenon
No, not simple. Already complicated, and undermines your point that identity conditions are the opposite of structuralism. If I'm understanding you.
Quoting Pneumenon
No, but I'm a huge fan of professor Rayo. For a long time he put on a MOOC called Paradox and Infinity, which I took several times because I love the material so much. He went over omega paradoxes (what we'd call supertasks around here), ordinals, the nonmeasurable set, even gave an accessible proof of the Banach-Tarski theorem. This was in a course aimed at total mathematical novices. Fantastic course. He doesn't do it anymore. Last time the attendance was very low. I think everyone is into machine learning and AI these days, if you can judge by the MOOC offerings.
Quoting Pneumenon
But we can NOT make that agreement. That statement has no truth value till we declare what category we're in; or less technically, till we specify a context.
If we are group theorists, they ARE identical, because there is only one cyclic group of order 4. This point is even more strict if we are in the category of groups. Then there is literally only one such group, and there is no such thing as a representation or presentation or instance of it.
If we aren't in the category of groups, or if we are just beginning to learn group theory, these are two distinct groups that happen to be isomorphic.
The claim that "isomorphism is identity" is part of homotopy type theory. I'm probably lying through ignorance here but that's my sense of the matter.
So no, we do NOT agree that these two isomorphic groups are identical. In beginning abstract algebra class, they are NOT identical, even as groups. They are "merely" isomorphic.
It's only when we take the more abstract point of view of Category theory that we can't tell them apart.
So no. No agreement on this point until you tell me the context.
Quoting Pneumenon
In beginning abstract algebra class, yes. In the category of groups, no, because there is exactly one cyclic group of order 4, and I have no idea what it's "made of." I have no concept of its "elements" or its "operations." It has no internal structure at all. Objects in categories do not have any internal structure. All they have is maps to other objects. This is categorical structuralism.
So, no, we do not agree that those are "instances" of anything except in the most casual sense of perhaps beginning group theory. After all, they are the exact same structure.
Would you say that 4 and 2 + 2 are two "instances" of the same number? Good question.
In fact now that I think of it, I have no idea if I've got any of this right. In the category of groups are these two separate objects that have an isomorphism between them? Or is there only one object that has no instances and no internal structure? I'm not entirely sure.
I should mention in passing that earlier today I ran my eyeballs over the SEP article on mathematical structuralism. They made a clear distinction between the philosophical development of mathematical structuralism via Benacerraf and Putnam, on the one hand; and categorical structuralism via Mac Lane and Grothendieck. So categorical structuralism is a branch or aspect of mathematical structuralism, and shouldn't be identified with it as I've been doing.
Quoting Pneumenon
In the category of groups I don't think the question makes sense. Any more than if you pressed me on whether 4 and 2 + 2 are each instances of the same thing. Is instance even the right word? Maybe representation, or presentation.
Quoting Pneumenon
Easily falsified in-universe. Lois Lane has no idea. It's the glasses, apparently. A little willing suspension of disbelief on that point. How can she possibly not see that they're the same guy? But she can't tell.
Quoting Pneumenon
But of course a thing is identical to itself. That's the law of identity. But recognizing when two things are the same is one of the fundamental problems in mathematics!
Now I'm reminded of Barry Mazur's famous essay (pdf link)
When is one thing equal to some other thing, which delves into some of these categorical issues.
Quoting Pneumenon
This was in reference to your straw man argument that structuralists make structure a primitive, or some such. I don't think the question was on point but I could be wrong.
Quoting Pneumenon
All good, but I'm at the limit of my knowledge and not sure I'm even telling the truth about a few things.
ps - Poincaré said that "mathematics is the art of giving the same name to different things." Another early instance of pre-structuralism.
In fact see the entire context of Poincaré's quote here ... good reading.
https://ncatlab.org/nlab/show/isomorphism
He even extends the idea to physics.
"The physicists also do it just the same way. They invented the term ‘energy’, a word of very great fertility, because through the elimination of exceptions it established a law; because it gave the same name to things different in substance, but alike in form."
pps -- I'm wrong about there not being separate isomorphic instances of groups in the category of groups. That implies that I'm wrong about a few other things in this post. So in the end I'll agree that the two representations of the cyclic group of order 4 are indeed distinct objects, even in the category of groups.
https://math.stackexchange.com/questions/2041417/are-objects-in-the-category-grp-actually-groups-or-isomorphism-classes-of-groups
ppps -- I have resolved my confusion.
So there's a concept called the skeleton of a category, which contains exactly one copy of each isomorphism class of objects. So in the skeleton of the category of groups, there is only one cyclic group of order 4; and there aren't any variations on its representation.
So if we're in the category of groups, ({0, 1, 2, 3}, +) and ({1, i, -1, -i}, *), both exist as distinct, but isomorphic, objects.
But in the skeleton of the category of groups, they do not exist as separate objects; there's only one such group.
So once again, when asking when two mathematical objects are the "same," the answer is always that it depends on the category. And also -- the structuralist view -- we should not use the word same, but only isomorphic, because isomorphism is the only thing that matters.
Math itself is full of systems, like the system of the natural numbers, or the system of addition. So, in a sense, you might say it is a system itself, but a very complex system. @fishfry will give a much more sophisticated answer to your question.
I'll just give the laziest answer possible. I typed "is math a system" into Google. The Google AI responded:
[quote=GoogleAI]
Yes, mathematics can be considered a system, as it is a structured set of rules, axioms, and concepts that are interconnected and used to reason about and describe patterns and relationships, often through symbols and operations; essentially forming a logical framework for understanding abstract concepts
[/quote]
Quoting ssu
Google AI thinks so. Pretty much everything is a system, from indoor plumbing to the National Football League.
There's a discipline called general systems theory, which is ...
[quote=Wiki]
... the transdisciplinary[1] study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or artificial. Every system has causal boundaries, is influenced by its context, defined by its structure, function and role, and expressed through its relations with other systems. A system is "more than the sum of its parts" when it expresses synergy or emergent behavior.
[/quote]
The formal context I mention here is formal Z set theory and its Z based variants.
Quoting fishfry
Formally, no. As you mention, they are different kinds of sets but there is an embedding of the naturals in the reals.
Informally, yes.
Quoting fishfry
"whatever that means" indeed. I'd have to hear someone's definition of "copy".
Formally, the x-axis is {
p in R} and the y-axis is {<0 p> | p in R}. So the domain of the x-axis = R = the range of y-axis. And we can define less than relations: for all p and q in R ,
less than
Proof: Take any two ordinals with different cardinalities. The standard ordering on an ordinal is a strict linear ordering with the least upper bound property.
Even ridiculously trivial: The membership relation e_0 on 0 is a strict linear ordering with the least upper bound property. The membership relation e_1 on 1 is a strict linear ordering with the least upper bound property. But <0 e_0> and <1 e_1> are not isomorphic since there is no 1-1 function from 0 onto 1.
So, unless I've overlooked something, I think we need to mention that there are fields involved. (Or maybe it's tacitly understood that there are fields involved.)
This bears on another thread lately.
Two definitions of 'the continuum'"
the continuum = R
the continuum =
Definition of 'is a continuum':
A topological space S is a continuum if and only if S is connected, compact and Hausdorff
Let z be the standard ordinal ordering on 2^aleph_0.
z is a strict linear ordering on 2^aleph_0 and with the least upper bound property.
But it is not the case that
So, it seems to me that the claim that R is unique to within isomorphism, or that
And that serendipitously ties in with the topic of "systems", as a reminder that another sense of 'system' is that of a tuple with a carrier set and operations and relations on that carrier set.
(Another sense I'd mention is 'logistic system'.)
Quoting fishfry
Hence the question was stupid, as I assumed.
It being a logical system would be perhaps more fruitful, but the notion would still be in the set of self-evident "So what?" truths about mathematics. Just what kind of logical systems math has inside it would be the more interesting question. Now when mathematics has in it's system non-computable, non-provable but true parts (as it seems to have), this would be a question of current importance (comes mind the Math truths aren't orderly but chaotic -thread).
If so, perhaps the old idea of math being a tautology comes to mind: something being random and non-provable but true is... random and non-provable but true. Yet how do we then stop indoor plumbing and the National Football League being math? That the two aren't tautologies, even if indoor plumbing ought to be designed logically(?) Would it be so simple?
That seems right to me.
Lois Lane doesn't know that they are the same, but that doesn't entail that they're not the same. What are not the same is Lois Lane's notion of Kal-El as he is in the guise of a person named 'Clark Kent' and Lois Lane's notion of Kal-El as he is the super-person called 'Superman'.
This reminds me somewhat of:
the set of natural numbers = N = the ordinal w = the cardinal aleph_0.
N, w and aleph_0 are the same thing, even though we think of it differently depending on context.
We think of N with its property of being the carrier set for the system of natural numbers.
We think of w with its property as being the first infinite ordinal.
We think of aleph_0 with its property as being the first infinite cardinal.
But still they are the same thing.
Meanwhile, Lois Lane thinks of Clark Kent with his property of being a mild mannered newspaper reporter, and she thinks of Superman with his property of being a possessor of superpowers dressed in tights and a cape. She doesn't know that Kal-El = Clark Kent = Superman, but still it is the case that Kal-El = Clark Kent = Superman. She doesn't know that the mild mannered newspaper reporter is the same being as the possessor of superpowers dressed in tights and a cape, but still there is only one being involved. Suppose she learns that Clark Kent is Superman. So it had to have been true for her to learn it. It's not the case that the sameness of Clark Kent and Superman happened only upon Lois Lane learning it. Meanwhile, Kal-El very well knows that he is Clark Kent is Superman. What happens in the phone booth is a change of apparel not a change of being.
Statements devoid of content? (Frege)
I think not.
Look, I'm not educated. At all. I have neither a degree nor a high school diploma. I have what Americans call a "GED", and that's it. My job is to sit at home in my underwear and write code. If I tried to show up at an academic conference, they'd kick me out just based on smell. I can make computers go bleep bloop, though!
Although I am wearing this shirt right now:
If that counts for anything.
Quoting fishfry
Let me make sure I understand you. You're saying this:
"There's this thing called identity. Math tells us that identity is contextual. So we should also say that the identity of people is contextual, since math tells us about identity. For example, Clark Kent and Superman are contextual identities."
Is that right?
Quoting fishfry
I don't understand. "Lois is unaware of the fact that Clark Kent is Superman. Therefore, Clark Kent is not Superman". That's not what you're saying, is it? If I put on clown makeup and then kill someone, I'm still guilty of a crime, even after I wash off the makeup.
Reading over your other points, I have a hunch about your position. You have used the term "representation" a few times. And you seem to think that, if A and B both represent C, then A and B are identical insofar as they represent C. You seem to identify how something is represented with what it is.
Two things.
1. Do Clark Kent and Superman represent one thing? If so, what is that one thing?
2. Would horses have four legs if nobody counted them? If so, which four would it be? The natural number 4? The rational number 4? The real number 4?
ok ...
Quoting Pneumenon
You opened by saying you wanted to discuss mathematical structuralism; specifically, that you wanted to oppose it.
I'd be happy if we could focus on your ideas. You clarified what you meant by an identity condition, and I pointed out that it was structural, and seemed to undermine your own thesis. You didn't engage and didn't agree or disagree. I'd find it helpful if we could focus in.
Quoting Pneumenon
I made the point that Clark Kent and Superman are two representations, or guises if you will, of the same entity. Just as two isomorphic groups are really the same group. I don't think the Superman analogy bears too much weight. Better to come back to your ideas about mathematical structuralism; since I had in the past some training in the fundamentals of contemporary math, including set theory and a little category theory. Then again I have a pretty good background in Superman comics.
Quoting Pneumenon
They're not identical. They're isomorphic. Identity is a tricky thing in this context. Structuralism is all about isomorphism. "Sameness" and "identity" don't really come up.
If you would clarify your thesis -- perhaps recall why you initially started this thread and resurrected it -- we could be more focussed.
In modern math, isomorphism is important. Identity and sameness aren't as important; as as we're seeing, they're kind of slippery.
Quoting Pneumenon
I wish we could get back to your ideas about Platonism versus mathematical structuralism. The Superman analogy does not bear too much scrutiny in this context.
In any event, if it wasn't clear, in my previous long post I eventually talked myself out of my own point. The additive group of integers mod 4 and the multiplicative group of the integer powers of the complex number i are indeed distinct groups, and distinct objects in the category of groups; although they are isomorphic. I believe that actually supports a point you were making, so if you like we could go back to that before I confused myself on that point.
I didn't say that at all. Only that calling math a system is far too general. It doesn't tell us anything about math.
Quoting ssu
Yes ok.
Quoting ssu
I suppose you'd need a more specific definition if you wanted to uniquely characterize math. It's tricky, since the nature of math is historically contingent. In the old days they didn't even have negative numbers or fractions, or algebra; let alone the wild abstractions we have today.
I'm not sure how important it is to nail down a particular definition. Math is what mathematicians do.
Isn't this the definition you gave earlier?
Was this post for me?
If I mis-stated the definition, my bad.
My error if I gave the wrong definition. Which I always have to look up since I always forget it. Sometimes the Archimedean property is mentioned, other times not.
The OP is making some point about mathematical structuralism, which was the context for the discussion of instances. I wasn't entirely sure how your remarks related to that.
It's not a definition. What definiendum do you have in mind?
Quoting fishfry
It's for whomever wishes to read it.
Quoting fishfry
What definition? I didn't take issue with a definition.
I didn't intend my posts to comment on structuralism.
I'm gonna quit while I'm behind here.
Quoting TonesInDeepFreeze
My point. That was the subject of the conversation. So I wasn't sure how to respond to your remarks without going off on extraneous tangents.
People may view this in different ways. But, for me, as far as the bare bones context of extensional set theoretic mathematics: Two different, but isomorphic, groups are not the same object, not the same set. They may be regarded as interchangeable as far as their common aspects are concerned, but still they are different objects. Informally they may be regarded as the same, but formally they are different.
/
Clark Kent is Superman. Clark Kent is not a representation. Superman is not a representation. He is a being. But, of course, 'Clark Kent' and 'Superman' are different things. And Lois Lane has two different notions, one of a human named 'Clark Kent' and one of a superhuman named 'Superman', but Lois Lane's notions don't negate that Clark Kent is Superman.
The proverbial example: One might have a notion of the Morning star and a notion of the Evening star and believe that they are two different celestial objects. But believing that they are different doesn't make them different. There is only Venus, which is seen at different times of the day. It's not a different object when seen in the morning than it is when seen at night.
Kar-El is seen sometimes wearing a suit and glasses and with people calling him 'Clark Kent' and seen sometimes wearing tights and a cape with people calling him 'Superman'. He's not a different being when seen wearing a suit and glasses and being referred to as 'Clark Kent' than when he is seen wearing tights and a cape and being referred to as 'Superman'.
More like that the truths in mathematics are tautologies: a statement that is true by necessity or by virtue of its logical form. Wouldn't that description fit to mathematics?
Quoting fishfry
For example to the question in the OP it is something. Also do note that it does influence on how people see mathematics and how the field is understood and portrayed. Do people see it from the viewpoint of Platonism (numbers are real), logicism (it's logic) or from formalism (it's a game) or something else?
An apt description works wonders here.
Quoting fishfry
Oh, and this is a perfect example why it is important: with that you'll give here not only your little finger, but your left leg to the worst kind of post-modernists and sociologists that then can proclaim that math is only a societal phenomenon and a power play that a group of people (read men) do. That all this bull of math being something different, having it's own logic or being something special that tells something about reality is nonsense (or in their discourse, an act in that power play) when it's just what mathematicians declare doing. That's obviously not what you meant, but how can your statements be used is important.
These kinds of ideas, which I myself oppose, is like taking Thomas Kuhn totally out of context and using him (or the study of science as a human enterprise and interaction) as a way to claim that in this case mathematics is nothing special (perhaps from plumbing or playing a sport).
It's not clear to me whether you're suggesting that my remarks were not pertinent. But in case you are:
The conversation has had many subjects. You mentioned certain isomorphisms. I was interested in that. My remarks about that don't have to comment on structuralism.
You said you wanted to focus on my ideas. Okay, then. Let me explain my motivation for the thread. Please read the following carefully. If I lose you, say where and why. I promise, it's relevant.
So, I am really really interested in whether or not mathematical objects exist in a mind-independent way. Would there be numbers even if we weren't here? I want to say yes. I think that numbers would be here even if we weren't here. I think that brontosaurus had 4 legs long before we counted them. I can't fathom what it would mean to say that it didn't. So that makes me a Platonist, because I don't think that numbers depend on our minds. At least, not in that way.
But there's a problem with Platonism. If I say that something exists, I need to identify it. If I say, "the Blarb exists", then I need to say what the Blarb is. I need an identity condition that picks out the Blarb and only the Blarb.
This is a counter to Platonism, because it confounds the motivation. Would horses have 4 legs if nobody counted them? I, the Platonist, say yes. But if you ask the me for an identity condition, I'm in trouble. See, if say that the legs of a horse are the set {0, 1, 2, 3}, then I've said that the number of horse legs is the natural number four. But is that even true? What if the number of horse legs is the set of all rationals smaller than 4, i.e. the real number 4? How would we know which one it is?
So it comes out to this:
1. To say that a certain thing exists, you need an identity condition for it.
2. You can't always get those identity conditions for mathematical objects.
3. Therefore, we can't say that mathematical objects exist.
Uh-oh!
My hare-brained solution to that was "maybe there's only one mathematical entity". Math is just a single thing. Because then math can be Platonically real without needing identity conditions. You just say that different mathematical objects are what happens when you analyze that one single object in different ways.
I know that sounds weird. Maybe an analogy will help.
So, for example, you know the duckrabbit?
It looks like a duck if you look at it one way. It also looks like a rabbit, though not both at once. However, you can't just see it however you want; it's not a duckrabbitgorilla. It's not a duckrabbithouse. It's not a duckrabbithitler. So there are two valid ways of seeing it, but only one can be used at a time. And some ways of seeing it are invalid.
What if math is kind of like that? There's a single mathematical reality, but it looks different depending on how you analyze it. So if you bring a certain set-theoretic framework, math gives you real numbers, and if you bring a different one, math gives you rational numbers. But the rationals and reals aren't separate, self-identical objects. They're just representations, ways of representing one underlying reality.
Basically, you don't need identity conditions for a representation, because representations don't need to be self-identical in that sense. We can answer the question, "Did horses have 4 legs before anyone counted them?" with "yes" (which is what I wanted). That's because the underlying mathematical reality is Platonic and never changes. But then we have this question: "Was that the rational number 4, or the real number 4, or what?" That's the question that initially flummoxed us. But if there's only one mathematical object, that question is no longer sensible. Represent the number of its legs however you like. You can represent it as rational 4, or real 4, or natural 4. The underlying reality is the same.
I'll come back to that in a moment. Now for structuralism.
The SEP article on structuralism tells me that there's a methodological kind of structuralism, which is basically just a style of doing math. Then there's a metaphysical structuralism, which is an ontology. The former is a style of mathematical praxis and the latter is a philosophical position.
You seem to waver between methodological and metaphysical structuralism, and it confuses me. On the one hand, you take a stance like, "I'm no philosopher. I just find this to be an interesting way of doing math". On the other hand, you do seem interested in the philosophical implications of structuralism, e.g. when you said that modern mathematics tells us new things about the notion of identity. And you're on a philosophy forum discussing it, rather than a math forum.
So let's put the discussion on these questions:
1. Does methodological structuralism imply metaphysical structuralism? Or at least, enable it?
2. Is metaphysical structuralism compatible with Platonism? If so, is it compatible with my idea that there is, in a sense, only one mathematical object?
P.S. the SEP article has it,
We ask, "Is the number of horse legs the natural number 4 or the rational number 4?". Well, for Shapiro, the number of horse legs occupies a position. That same position is filled by {1, 2, 3} in the naturals and { x ? Q : x < 4 } in the reals. So the answer is, "The number of horse legs is both of those".
The identity condition, then, is "That unique office occupied by {1, 2 3} in the natural numbers". But there are plenty of other identity conditions that pick out the same object: "That unique office occupied by { x ? Q : x < 4 } in the reals" picks it out as well. At this point, the numbers themselves are identity conditions for offices!
Maybe I don't need to reduce math to one object after all...
I hope I didn't wander off too far. And I hope that this is, at least, interesting.
In America women make up 25-30% of PhD students. 15-20% of math faculties. Not entirely men.
Quoting ssu
To me this seems like a word game. Describing a theorem, if A then B, requires specific terms, going beyond its "logical form". The idea of categorizing math as a tautology contributes nothing to its practice.
Quoting ssu
Years ago the maid for a prominent mathematician was asked what her employer did. She replied, after some thought, "He scribbles on pieces of paper, grumbles, then wads the papers up and throws them in the trashcan."
I wouldn't say that.
(1) the set has only three members, not four, (2) the legs of the horse is not that set; rather the cardinality of the set of legs of the horse is that set.
So, I would say: The cardinality of the set of legs of the horse is 4 = {0 1 2 3}.
Quoting Pneumenon
You could say that there are different forms of "number of":
cardinality - every cardinality is an ordinal (a finite ordinal, i.e. a natural number if the set is finite).
cardinality_q - every finite cardinality is the embedded natural number in Q
cardinality-r - every finite cardinality is the embedded natural number in R
Then each appropriate statement, couched in each form, about the set of horse's legs is true,
Quoting Pneumenon
If we're talking about formal mathematics:
(1) To define a constant symbol, we need first an existence_&_uniqueness theorem.
That's syntactical.
There are only countably many constant symbols, so only countably many such definitions.
(2) To define an object b in a model M (hope I've not made a mistake):
we need a formula F, whose only free variable is x, such that M satisfies F iff x is assigned to b.
Then b = the p such that M satisfies F iff x is assigned to p.
/
Perhaps, this might work (I don't know, it only occurred to me just now):
Let the mysterysaurus be a dinosaur that we inferred existed but don't know the number of its legs. To say that the mysterysaurus had four legs might not require a mathematical definition of 'four'. Rather, you could be saying "If you were an observer when the mysterysaurus was extant, then when you looked at one leg after another, and said the words, 'one', 'two', 'three', 'four' - one of those for each leg - then you would stop at 'four'". Yes, observation is mentioned, but isn't the statement independent of my knowledge? Does that hold up?
Was it Einstein that said something to the affect that God gave us the natural numbers, all else of mathematics are mans' ? Or something like that. When I think of one of my theorems about the indifferent fixed point of an infinite composition of LFTs that converge to a parabolic LFT I never stop and wonder if all that preexists in a timeless realm beyond the thoughts on man. I do believe it exists in a potential way.
I'm always amused by this common philosophical example, since Venus isn't a star at all.
Wow. I'm kind of stunned. I have no idea what I said that triggered such a strong reaction.
Before dealing with that, let me expand on my remark.
Math is a historically contingent creative activity of humans. What was considered math by the Babylonians was different than what was considered math by the Greeks. When Cantor came up with his revolutionary theory of infinite sets and transfinite numbers, Kronecker famously said, "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there." Yet today, set theory is a basic part of the undergraduate math major curriculum.
So "math is what mathematicians do," is on the one hand glib, trivial, and superficial; and on the other hand, somewhat deep; since it encapsulates the idea that what is regarded as math changes from one generation to the next; and in the end, math is literally what mathematicians do.
The same goes for art. When abstract art began supplanting representational art, I'm sure critics howled. (I can't claim to know much about art history, but I'm guessing). The French impressionists changed people's ideas about what art is; as did the cubists. So in the end, art is what artists do.
Music is what musicians do. Same combination of triviality and depth. People think they know what music is, then Mozart or Chuck Berry come along to expand and alter people's idea about what is music.
So "mathematics is what mathematicians do" was intended by me to capture the historical truth that what one generation considers radical heresy, the next considers orthodoxy. The universe was Euclidean till it wasn't. Physics used to be about the nature of the universe; now it's "shut up and calculate." Physics is what physicists do.
My remark was anodyne, a truism at once trivial yet expressing the idea of historical progress within a creative discipline.
Ok. And now you react strongly. I wish you'd tell me what you mean.
"... you'll give here not only your little finger, but your left leg to the worst kind of post-modernists and sociologists that then can proclaim that math is only a societal phenomenon and a power play that a group of people (read men) do."
I intended to do nothing of the sort. Can you explain what you mean? Surely you know that what we considered math in 1900 was completely supplanted by 1950; and then again by 2000. "What math is" changes all the time. What's wrong with saying that?
"That all this bull of math being something different, having it's own logic or being something special that tells something about reality is nonsense (or in their discourse, an act in that power play) when it's just what mathematicians declare doing.'
I can read the words, but I honestly haven't got a parser for that language. Can you explain to me what you want me to take from this?
"That's obviously not what you meant,"
Well thanks for at least giving me credit for that.
" but how can your statements be used is important."
Who is doing these terrible things with an anodyne statement like, "Math is what mathematicians do?" And what are they doing?"
I'd think that's the most trivial of truths there is. "Why's that guy building a table out of wood?" "He's a carpenter." "Oh, that makes perfect sense then. He's a carpenter, so he does carpentry."
You take exception? I was literally stunned by your response to what I wrote.
What philosophical taboo have I crossed?
ps -- On the other hand, if you object to my implication that math is a social process, I'll be happy to defend that thesis. Perhaps you've heard of the Mochizuki affair. For the past twelve years the highest-end world class analytic number theorists have been arguing about the validity of a published proof of a famous problem. The current Western consensus is that he doesn't have a proof. In Japan, the consensus is that he does. Math is very much a social process.
Quoting ssu
Most people are just glad to get out of high school algebra alive. Anyone who cares enough to think about these philosophical issues will certainly not be scarred for life by my little remark.
Writing papers in social science. Though I've not seen the specific attitude for maths, I've seen the attitude recently for medicine. Medicine is what medical doctors do, thus making it a principally discursive phenomenon. About words rather than bodies.
If you go looking you can find papers on boolean logic being a colonialist abstraction. I just don't want to go searching for this brainrot again.
I was only explaining that I preferred not to engage.
Quoting TonesInDeepFreeze
Isomorphisms have everything to do with structuralism. An isomorphism says that two things are the same that are manifestly not the same. That's structuralism.
Quoting fishfry
Just to be clear, the example doesn't assert that Venus is a star.
In set theory, 'isomorphism' is not 'two things are the same that are not the same'. Rather, two things are similar; they have structures that are similar. But it's not the case that different objects that are not the same are the same.
Here's the most trivial example:
<{0} 0> and <{1} 0> are isomorphic but <{0} 0> not= <{1} 0> and {0} not= {1}.
I didn't say isomorphism isn't relevant to discussion about structuralism.
Is this not perfectly true? Applying leeches used to be medicine. Now it's not. Removing the perfectly healthy breasts of emotionally troubled 12 year old girls used to not be medicine. Now it is. (Please don't bother to tell me that doesn't happen, I have the facts and figures at hand. Just using an obvious contemporary example).
You know the story of Ignaz Semmelweis. Austrian obstetrician, told doctors to scrub and disinfect their hands before delivering babies to prevent fatal sepsis in mothers. They all laughed at him. "What is the scientific mechanism?" He got increasingly frustrated to the point that his family sent him to an asylum to relax. He got into a fight with the guards and was beaten to death the first week.. Then Pasteur came out with the germ theory of disease and everyone said, Oh yeah old Ignaz was right after all.
So goes scientific progress. A substantially social enterprise. It was Planck who noted that science progresses one funeral at a time. Meaning that the old guard dies off and the young scientists are more open to new ideas.
It's sort of true. The move also denies that eg 2+2=4 is true. It's just valid as a statement of mathematics. The medical equivalent I saw was that... I think it was heroin wasn't addictive, it was addictive in the context of current medical theory.
The annoying bit isn't recognising that stuff is discursively mediated, the annoying bit is saying that because it's discursively mediated it isn't true, or accurate, or real or whatever. The book about heroin I read made the point about medicine and ontology thusly: "there is no ontological distinction between discourse and reality" - IE, what we say about things and things.
Whereas there is such a distinction for maths objects. You can say that 2+2=5, but it isn't.
I once audited a class with an infuriating social anthropology lecturer who wrote 2+2<4 on the board. But hid he fact he was adding the left two numbers as noise sources in decibels and treating the right as a natural number. That was him, by his reckoning, demonstrating the above point. That 2+2 doesn't have to equal 4.
If you've not interacted with these people I envy you.
Science is sometimes but not always a social process.
Then again I could defend the more radical framing. Science is done by people. How could it not be a social process?
Medicine is massively a social process. How did corporations come to control our entire health care system? That's not natural and it's not particularly scientific. The covid response was substantially political, not scientific.
In Orwell's 1984, the fact that 2 + 2 = 4 was shown to be ultimately political. In the end, Winston Smith didn't just agree that 2 + 2 = 5 to stop the pain. He came to truly believe it. State coercion is effective that way.
I don't think you are making your point.
Quoting fdrake
On the contrary. I did interact with these people. They closed down the beach near me during covid. There is no healthier place to be during a respiratory disease epidemic than the beach. That's the day I knew they were insane, and that we were dealing with authoritarian politics and not science.
2 + 2 = 4 isn't always political. It's usually not. But Orwell taught us that it sometimes can be.
Och, I've made it. I imagine you've never had the pleasure of interacting with these people, so you're able to do the sensible thing and see "science is done by people, how could it not be a social process?" as completely separate from "2+2=4 isn't demonstrably true". Unfortunately it is often held that the fact that some practice is socially or discursively mediated undermines any truth claim in the practice. If that seems totally absurd, yes it is, but it is the attitude your comment resembles and @ssu reacted to.
I'm not imputing that set of beliefs to you. Just highlighting what that phrase could suggest if you read it from a certain angle. But I don't think that is an angle you wanted to suggest, or did suggest.
I resemble that remark?
My sense is that we are only arguing about a matter of degree. You don't entirely deny the social component of science. How could anyone? And I don't deny the objective component of science. iow could anyone?
I don't think the social relativist or postmodern or whatever social critics are getting their cues from me. And there's a lot to be said for their points. We saw rational scientific dissent get crushed by "the science," politics masquerading as science. That made an impression on me. It's more important than ever to distinguish between science and scientism; between rational skeptical inquiry, and authoritarian crushing of dissent. That's one of the major themes of the age in which we live, as in the current argument over free speech and "disinformation."
I'm in favor of rational inquiry and skepticism; and opposed to authoritarian crushing of dissent in the name of scientism and political power. You may have a different sensibility regarding these issues.
You cease believing in objective properties, that's one of the steps to the conclusion.
Quoting fishfry
You don't. If I read your remark out of context, and didn't know your post history, I could read it that way. But I know you didn't mean it like that.
I am wondering who these people are that you and @ssu think I'm giving comfort to.
Do you mean cultural relativists, postmodernists, etc.? People who think that objectivity and merit are tools of the cis white patriarchy?
If so, I oppose these people. But they're not waiting for the likes of me to give them encouragement.
I'm big on scientific objectivity. However I'm also aware of the social component of science. And I did recently witness massive authoritarian suppression of legitimate scientific skepticism and dissent in the NAME of science. The political reaction to covid was anything but scientific. Epidemiologists warned that lockdowns were contra-indicated for respiratory infections. And like I say, they closed the beach in my little coastal community. You remember when they arrested some guy paddle boarding by himself on the ocean. That wasn't science, but it was done in the name of science.
So when it comes to the forces of science versus anti-science, I'm firmly on the side of science, objectivity, reason, merit, data, and all that.
But I'm opposed to scientism, and the use of the NAME of science to enforce political, anti-scientific orthodoxy.
Hope that's clear.
But I did want to make sure I understand who you and ssu are referring to. The postmodernists and "merit is racist" types? Those folks, I oppose.
But being someone who often sees too many sides of an issue; I will agree with those postmodernists who say that scientific objectivity and "reason" have often been used by colonizers to oppress the colonized. It's a matter of historical record.
So I do have some intellectual sympathy for the postmodernists in that regard.
For me it's a particular set of cultural theory tropes. They're generally working in paradigms like "subtle realism", "new materialism" or the less nebulous actor network theory these days. I can't name any contemporary academic names, a couple of friends' colleagues in academia are full of that stuff, and a few old friends (grad students at the time) and their supervisors bought into that hook line and sinker.
Those are new to me. Evidently they all have Wiki pages.
I surely didn't intend to give aid and comfort to social theorists whose ideas I've never heard of.
That said, I'll stand behind "Math is what mathematicians do." It's not original with me, I read it somewhere. I never intended for it to be a point of conflict, I though it was harmless. Apparently nothing's harmless these days.
If these ideas are flavors of cultural relativism or postmodernism or whatever, I'll be happy to mildly oppose them. But I'm not dying on any of those hills.
I did read it carefully. It was interesting. I have some minor remarks but no great insights.
Quoting Pneumenon
Yes, interesting question. Was 5 prime before there were any intelligent agents in the universe? Hard to say. Isn't the question man-made? But weren't there 5 things? I myself go back and forth on this question. I could argue either way.
Quoting Pneumenon
Hmmm, identity conditions again. As you know, in math there are existence proofs that show a thing exists without being able to construct or specify it. Before there were people, the earth existed, but nobody had descriptions or words for it.
Quoting Pneumenon
Well if you're a structuralist the natural number 4 and rational number 4 are the same thing.
By the way, if there were numbers before there were people, were there sets? Topological spaces? Complex numbers, quaternions? All the high-powered gadgets of modern math?
Quoting Pneumenon
Well mathematical objects surely don't require "identity conditions," whatever they are. I know you explained them but I'm not sure I believe your definition. But in any event, a Vitali set is a standard example of a set that we can show exists, but we don't know which elements are in it, nor can we specify any particular Vitali set uniquely.
Quoting Pneumenon
Well math is one thing, although exactly what it is, is historically contingent. But it has subthings. Algebra and analysis. Real numbers and complex numbers. It's a system as @ssu noted, with many subsystems.
Quoting Pneumenon
This didn't do much for me. It's a simple optical illusion. Or are we back to Clark Kent and Superman again?
Quoting Pneumenon
Ok. Not following your point about math. Math is one thing, though it's hard to say what it is. I'd say math is what mathematicians do, but evidently that anodyne statement got a fair amount of pushback. I believe a basic knowledge of the history of math support the statement.
Quoting Pneumenon
Like an optical illusion?
Quoting Pneumenon
Ok, if this is meaningful for you. Not doing much for me. As Poincaré said, math is the art of giving the same name to different things. That's early pre-structuralism.
Quoting Pneumenon
Now you're a structuralist! The natural, rational, and real 4 are the exact same number; even though their set-theoretic representations are quite different. That's because there's a copy of the naturals in the rationals and a copy of the rationals in the reals. So we make those structural identifications, and then we can say that there's only one number 4.
Quoting Pneumenon
Ok I should go back to SEP. I've skimmed the article a couple of times but evidently didn't catch this distinction. I was more interested in the distinction of philosophical versus categorical structuralism.
Quoting Pneumenon
As I say, I failed to catch this particular distinction in the SEP article, but I'll go back and look.
Quoting Pneumenon
I think the structuralist view is that there are mappings, say from 1 to 2 and from 2 to 3, that let us capture the order relations. That might be a little different than Benecerraf's original concept of positions. Category theory is all about the mappings. But categorical structuralism and philosophical structuralism are not the same, and I'm way out of my depth at this point.
Quoting Pneumenon
Don't understand the referents. No idea what a realist version of structuralism compared to nominalism and constructivism.
Quoting Pneumenon
As I say, the mappings are more important than the slots. But this might represent different views on the matter.
Quoting Pneumenon
Ok. But that is the structuralist view in terms of mappings. The integers have a natural embedding in he reals that preserves all their algebraic and order properties, so we can "identify" them; which is to say, we can consider them the same.
Quoting Pneumenon
Ok. Nobody says identity conditions are unique. For example in math texts it's common to introduce some mathematical object by listing several different characterizations, showing they're all equivalent, and then giving a name to anything that satisfies any of those conditions.
Quoting Pneumenon
I think your one-object idea is murky to me. I don't follow it.
Quoting Pneumenon
Yes, interesting. Can't add much at my end.
Perhaps at the more prestigious schools, and maybe at less elite institutions as well. I think I checked on this for Harvard and found such a course, but the branch of the state university where I taught doesn't seem to offer such a course, although set theory is mentioned in a couple of conglomerate classes.
Quoting fishfry
Absolutely. Although many mathematicians work alone much of the time, with social contact allowing critiques by colleagues. It gets very social when one publishes a paper, with regard to referees.
Quoting fishfry
I might comment that whereas isomorphisms are very important in mathematics, not all practitioners are heavily involved with them. I have written many papers and notes without mentioning the word. However, the trend for the past how many years, maybe 70 or 100 or so, has been to rise above the nitty gritty of much of classical material and look for generalities that show how one subject in one area is "isomorphic" (use here in a more general sense) to another subject in another area. Or create generalities that when applied specifically to a lower level collection of results show them to be instances of one higher outcome.
I didn't go in this "modern" direction, and my late advisor would speculate that where he and I explored (ground troops, not aviators) would at some future time return to fashion. These days there is such a plethora of subject matter I'm not sure what is fashionable.
I didn't mean to react strongly. And do note that I said I opposed the kind of thinking. I tried to give the example that in many social sciences (and humanities), the people talking about mathematics (or sciences in general) use simply their own fields discourse and focus, yet then make conclusions of mathematics itself based on these findings. And that's what I referred to giving the left leg to them when saying "mathematics is simply what mathematicians do". That's true of course, math is made by mathematicians, but how that is understood can be quite different.
I should clarify what I mean by this.
The teaching of mathematics in the school system is an educational question, how well the educational system works, not actually about the subject itself, mathematics. Or if as @jgill commented, women make up 25-30% of PhD students in America and 15-20% of math faculties. Thus the make up isn't at all close to the natural 50/50 divide, hence mathematics is male dominated. Yet there being more male mathematicians than female isn't a question about math itself again. It might be a question how faculties work and what kind of groups there are and who has control of the faculty, but that isn't mathematics itself. It would be hilarious to argue the male and female mathematicians would make different kinds of mathematics. There obviously isn't "feminine" or "masculine" mathematics, just as there's no difference in the study of some specific field of mathematics done today in the US, Europe or Asia. "European math" and "Asian math" fit together quite well.
Yet exactly these kinds of differences, who is doing math and where is it done, interest the social sciences. Those questions can be indeed interesting, but they aren't about mathematics itself or the philosophy of mathematics. In areas like literature, art and many fields of human activity etc. there is obviously a difference in where and who does it to the end result. Yet if mathematics is extensively studied in some part of the World or another, I would argue that the pure mathematics would be similar. Yet I know that many would disagree with this, seeing mathematics totally similar to these other endeavors.
In fact @fdrake makes the point perhaps even better than me. Hard to talk about mathematical insights if the other person just focuses that your basically just referring to white European males (that are dead). But the fact is that many people in social sciences are simply so mesmerized by the findings of their own field, that they go too far viewing everything as a social construct, a tool of the society to control people and so on. If mathematics is a social construct, then it can change as the society changes. And here we get back to the discussion of this thread: Is math different? If mathematics is a logical system that studies statements (usually about numbers, geometry and so on), that are true by necessity or by virtue of their logical form, wouldn't this mean then that mathematics is different from being just a social construct of our time?
Hopefully this clarified my position.
But they do have an effect. Well, It's not like the Catholic Church going against Galileo Galilei and others (or what happened to scientific studies in Islamic societies, that had no renaissance), but distantly it resembles it.
Quoting fishfryQuoting fishfry
Good luck finding anyone here that doesn't share your views.
But some things are political, just like the response to the COVID pandemic. Lock downs would be the obvious political move: the government has to do something and show it's doing something, that it cares about citizens dying. Taking the stance that Sweden did would take a lot of courage, but there the chief scientific authority was against lock downs, so it was easy for the politicians to do so. How you respond to natural disasters or pandemics is a political decision.
One thing is to keep politics out of things like mathematics. Sounds totally obvious, but we live in strange times...
Here's a personal anecdote that may be telling: My PhD class had several women. One dropped out for health reasons, and another was the top student, by far. Shortly after graduation she married a forest ranger and became a housewife.
In the international research clique I joined there were several women, but more men than women. A fairly close colleague, a European woman who had left behind a role as housewife, became the holder of an endowed chair at a major Scandinavian university.
Quoting ssu
Mathematics can be thought of as a structure or system or whatever, but certainly it is not "just a social construct". Sociologists are a little irritating.
I said math is what mathematicians do. I stand by the remark. I reiterate my literal shock that this anodyne and obvious statement generated pushback from two people. FWIW Galileo got in trouble for insulting his former buddy the Pope. That incident was about politics too.
Quoting ssu
Maybe.
Quoting ssu
Correct. And calling skeptics "anti-science" is politics too. You're agreeing with me, not disagreeing.
Quoting ssu
Pure math, maybe, though the Mochizuki affair is of interest.
But "mathematics" can be interpreted as the use of data for political purposes. Mathematics is highly political. The NSA employs more number theorists than academia does.
Nothing involving humans is above politics. Professor John Kelley had to leave U.C. Berkeley when he refused to sign a loyalty oath. Fields medal winner Steve Smale was subpoenaed by the House Unamerican Activities committee over his anti-Vietnam war comments. Many other examples could be cited.
Quoting ssu
I appreciate the clarification. It doesn't change my opinion that math is what mathematicians do; nor my surprise at receiving pushback over the matter. Clearly I can't do anything about what various social theorist, postmodernists, et. al. might make of my remark.
Quoting ssu
I used to think so. I probably still do. Still ... Newton and Kant's absolute space and time are reflections of the European paradigm of society in their day. Some philosophers have so argued. I'm not prepared to go into that too deeply. But I don't see anything humans do as above politics, even math. Even if the radical social philosophers are wrong, they're not 100% wrong.
Of course I oppose this kind of thinking when it comes to the math is racist curriculum revisionism these days. Public school math curricula are a disaster. I would say that publicly I agree with the purity of math; but privately, I'm not willing to totally dismiss the critics.
More of an algebra thing, as is category theory.
Quoting jgill
Reminds me of Tim Gowers's distinction between problem solvers and theory builders.
https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf (pdf link)
Somewhat similar, but not quite the same thing. I've never particularly enjoyed solving problems, but rather exploring where certain specific ideas in classical analysis lead. The celestial aviators can cruise the heavens taking us ground troops on ethereal adventures.
Sounds interesting. Life in the complex plane. By the way have you seen much of the modern graphing software that's so good at representing complex functions and Riemann surfaces and the like? Don't you wish you'd had that back in the day? I wish they'd had LaTeX, I always had bad writing.
Have I got that right?
Also I realized they talked about the 2-node edgeless graph, which has two absolutely indistinguishable objects. They quoted a paper by a mathematical philosopher named Hannes Leitgeb in Germany, whose name I know because he ran a fabulous MOOC a few years ago on the uses of mathematics in philosophy. Not the philosophy of mathematics; but rather the applications of mathematics in philosophy. We even did the Monty Hall problem, which was when I finally understood it once and for all.
I read his paper, and he talked in depth about the 2-element connectionless graph, and he related it to the famous two-sphere problem. So I had the right instinct. In fact the paper relates Indiscernibles of identicals to mathematical structuralism.
So I just want to say that I have a somewhat better understanding of where you are coming from and what questions you are asking. I'm beginning to understand what this thread is about.
From now on you can assume that I have a more nuanced understanding of your questions, and more of a conceptual framework in which to process them.
I'd have to say that if I have to pick a side, I'm more for structuralism as a handy way of doing math. It's a tool. I don't have any strong ontological feelings. When we study the number 5 or the natural numbers in general, we are treating them as a conceptual primitive. Everyone knows what the natural numbers are.
But I don't care that they're encoded within set theory one way or another; or if they're conceptually put into "slots," and their position in the line of slots determines their nature ... that is, we know the number 5 is the number 5 because it sits in slot 5 in our "slots not set" model of the natural numbers.
I regard that only as a conceptual idea. A proof of concept that if we wanted to, we could encode the natural numbers in ZF, or in "slot theory," if they've worked ou the details. I don't think the number 5 is really a set. I the number 5 is some kind of deep archetype in the human brain ... it's out there somehow. It's not its formal model. It's the abstract thingie "pointed to" by the formal model.
I suppose this makes me a Platonist.
But I think I did make that point earlier. Most mathematicians are Platonists. They conceptualize the mathematical objects they work with as "real," as having an independent existence. A number theorist is interested that 5 is prime and has no interest in how the nature of 5 is modeled by set theorists or philosophers. Likewise the group theorists and complex analysts and everyone else. Most working mathematicians never have a single thought about any of this in their entire professional lives.
We can all believe this. And since people are mathematicians, they can understand the effect when the dean or the higher ups in the universtity or research establishment simply demand that "there should be more women". The obvious reason is simply viewed as toxic. That women have babies and do still become housewives. My wife wrote and finished her PhD when she was nurturing our first baby. When we had second child, she decided to stay home. My income made it possible.
Of course the present culture is in no way as hostile as was the Catholic Church before (to science in general) or the Islamic religion after the brief spell when the Muslim World upheld Western knowledge. Hence if someone cries after the "decolonization of Mathematics", it actually isn't a threat in any way to the study of mathematics.
Quoting fishfry
Confused really why you would be in "literal shock" and why talk of having pushback.
The statement "Math is what mathematicians do" can be interpreted totally differently by for example social sciences. Totally differently what you mean. I do understand your point, but what I'm trying to say here that all do not share your perspective and they will use a totally different discourse. The conclusion and the counterargument isn't that "If then all mathematicians sleep, is sleeping then mathematics?", no, it's not so easy. It's that if mathematics is just what mathematicians do, then we just can just focus on the mathematicians as group and in their social behavior and interactions and workings as a group. Because what mathematicians do is what is mathematics, we can take out any consideration of things like mathematics itself or the philosophy of math. What the schools of math disagree on isn't important. I'll repeat it: all you need is to look at mathematicians as a group of people and the behavior and interactions. And in the end some can then talk about "decolonization of mathematics", because the study will notice that it's all about "dead white European males". This is just the way some people think.
Hopefully you get my point.
Quoting fishfry
Good that you used the word "interpreted". It's crucial here.
Cryptography and secure communications are important, and it's quite math related. And Wall Street uses quants, quantitative analysts, who do also know their math. Would then mathematics be capitalist? Of course not. I myself disagree with these kinds of interpretations.
Quoting fishfry
You don't have to, it's all quite simple. Thomas Kuhn came up with the term "scientific paradigm" and note that Kuhn isn't any revolutionary and he doestn't at all question science itself. He's basically a historian of science. It's simply a well thought and researched book that states that basically everybody everybody is a child of their own time, even scientists too. And so is the scientific community, it has these overall beliefs until some important discoveries change the underlying views of the community. And that's basically it.
For the philosophy of mathematics or the to the question of just what math is, Kuhnian paradigms don't give any answer and actually aren't important. What is important is the questions in mathematics... that perhaps in the end can get a response like a Kuhnian paradigm shift. So hopefully you still think that way, not only probably.
No I haven't. And I have little interest in Riemann surfaces. I have used MathType for years with Microsoft Word for writing purposes. For imaging, I have found BASIC is excellent for what I want to do, and have written many math programs. The image of the Quantum Bug on my info page was done with a simple program. Higher, more sophisticated languages seem to be directed toward what is popular in math, and what I do is virtually unknown.
Oh. Interesting.
Retired for 24 years. Lots of things slip by. Hard enough to persist along the lines of mathematical thought I know about.
I wish I'd been able to visualize complex functions the way people can these days.
But you seem to be using visualization software in your images. They didn't have that stuff when I was in school.
It's just fairly simple BASIC programming that I enjoy creating. I tried Pascal, Fortran, Mathematica, C++, and one or two others, but by the mid 1990s I returned to BASIC. I use Liberty Basic now. Microsoft's Visual 6 was excellent, but one morning I turned on my computer and it was gone. Instead Microsoft tried to get me to use some new programming language you had to employ at their servers. I've never quite forgiven them. I've written 3D programs, but haven't been happy with them. I'm a 2D guy.
Question: what is the one object being represented?
I'm a little bit at a loss as to how to respond. I find myself defending a hill that I'm definitely not willing to die on. If it made a difference to anyone, I'd gladly deny, renounce, disavow, and forswear my earlier claim that "Math is what mathematicians do." It was a throwaway line, a triviality, a piece of fluff. I can see that someone could use it to argue for woke math or decolonized math or whatever. So I abjure my former heresy. if it will help.
But you put some thought into your response, so I'll do my best to type in some words. Just please be aware that my heart's not in it. Not a hill I want to die on, not even a hill I want to get a hangnail on.
Quoting ssu
Yes I know these people. How bad has it gotten when Scientific American, of all outlets, publishes Modern Mathematics Confronts Its White, Patriarchal Past.
So I hear you loud and clear on this issue.
I truly don't think any of these people are waiting for the likes of me to give them aid and comfort with an anodyne remark like "Math is what mathematicians do." But if you disagree, I'll withdraw the remark. It's not going to help. Scientific American didn't call to ask my opinion before they published that article.
Quoting ssu
Well, math is what mathematicians do when they're doing math. An even more mindless slogan, therefore even more easily used by the enemies of rationality and merit, I suppose.
Quoting ssu
I can't do much about those folks. And I can't censor my opinions (and mindless slogans) just in case one of them overhears me and takes comfort in my words.
It's kind of like in politics. Sometimes there's a candidate with flaws. Some of their supporters deny the flaws entirely. Others admit the flaws but affirm their support for the candidate anyway. I'd be in the latter category. I'll tell the truth even if it undermines my own case. Perhaps I've done that here. So be it.
Quoting ssu
Not entirely, that last paragraph went over my head a bit. I'll agree that I'd be hard pressed to give a definition of mathematics that transcended historical contingency.
Quoting ssu
You can't separate math from its uses. If I was making the point that math can be political, I'd agree with myself.
Quoting ssu
Ok. Math is a historically contingent human activity. How is that any better than "math is what mathematicians do?" Maybe anti-racist math is the nex big paradigm.
[url=https://www.nctm.org/uploadedFiles/Conferences_and_Professional_Development/Webinars_and_Webcasts/Webcasts/4-19-22_WebinarSlides.pdf]“Grades and Test Scores Do Not Define
Us as Math Learners”: Cultivating Transformative Spaces for Anti-Racist Math Education[/url] [pdf link]
It's always the math "educators" and not the mathematicians promoting this stuff. Of course they started with a "land acknowledgment." I've noticed that they never give their real estate back, though.
So ok, you say I'm giving aid and comfort to these people. But how else should I say that math is a historically contingent human activity? Kuhn's paradigm theory says the same thing. One day someone comes along and changes everyone's view of the subject.
Kuhn is subject to the exact same criticism you level at what I said.
Quoting ssu
I never gave any thought to "what math is." It just like what Justice Potter Stewart said about pornography. "I know it when I see it."
Well that's the best I can do today by way of response. But do tell me if you think Kuhn might be subject to your criticism, for noting that the nature of science changes radically from time to time.
Exactly what it was intended to be. How about my previous statement about a mathematician is one who scribbles on paper, curses, then wads the paper up and throws it into the trash. Nobody seemed offended by that. Folksy I guess.
Quoting fishfry
That is a pretty bad article. It paints a picture of an entire profession based on a few incidents.
Quoting fishfry
Well, that article basically states what this is about: attempt to get job positions. What better way is there to accuse a field of study, mathematics, to itself be "white and patriachial", or whatever. But it works. What can the head of a mathematics department say when accused that there are too few if any women or minorities represented in the staff? Stop hiring your male buddies and follow the implemented DEI rules!
With this short interlude to social discourse, I would like to go back to the actual topic of this thread.
One more slight digression from the original topic. I have been in this position. Rules of Affirmative Action applied and the dean asked for the top three candidates. There was a woman, but no minorities. The dean then placed a minority in with our recommendations. When the time came to decide to make an offer the dean picked the minority. It did not work out well in the long run.
Quoting ssu
May I suggest focusing on math objects having several representations (like my example four days ago) and speculating on what the object "really" is or looks like. Or where it lies in a metaphysical sense. To say math is one object is absurd IMO.
Yes, the similarities don't define the object, however. Is an "object" its' representation picked at random? Or, is there a more metaphysical meaning of the one object having representations? Is there an Object Theory? Just thinking of a way this thread might proceed.
Let's try an example to clarify this idea:
What would be the mathematical object behind/described by the "well ordering theorem", which can be described as every nonempty set of positive integers contains a smallest member?
How is this object comparable to the axiom of choice? They are basically equivalent to each other. Are they still two different mathematical objects? Do they differ and if they do, how?
Thoughts?
You're conflating non-equivalent theorems.
Theorem 1. The set of natural numbers is well ordered by the standard less than relation on the set of natural numbers.
That does not require the axiom of choice.
Theorem 2. Every set has a well ordering.
That is equivalent with the axiom of choice.
/
I don't know what is being asked by "what is the object behind/described by" a theorem.
Do you want to know what object a theorem is? A theorem of a theory is a sentence provable in the theory. A sentence per a language is a certain kind of sequence of symbols. A sequence is a certain kind of function. And symbols themselves may be taken to be certain mathematical objects.
Though, when I say 'object' I profess no particular metaphysical sense. I only mean 'object' in the ordinary sense of 'a thing' or 'something mentioned' or 'the referent of a pronoun such as 'it'', as I don't profess to be able to explicate that notion more than as a basic presupposition for talking about, well, things, as in "the number 1 is something different from the number 2".
Sure.
Quoting ssu
This brings up an interesting question: If two things are equivalent, A<->B, does that mean they represent the same math object? In the example I gave there are two ways of representing the object, Mobius transformation, analytical or matrix, without a non-trivial equivalence argument. The WOT and AofC require a logical argument to verify. Representation theory, in general, includes equivalences.
All of this gets technical and may not be suitable for TPF. Also, set theory quickly moves beyond my levels of proficiency. More appropriate for and
In set theory, equivalence does not imply equality. Here's the most trivial example:
{{0 1}} is a partition of {0 1}. And that partition induces the equivalence relation {<0 0> <1 1> <0 1> <1 0>}. And per that equivalence relation, 0 and 1 are equivalent. But 0 and 1 are not equal.
But, of course, one may posit a different mathematical approach by which certain equivalences imply equality. That's a matter of stipulation.
Though if a metaphysical or philosophical ruling on the question is sought, then that is yet another matter and would not be settled by mere mathematical formulations or stipulations.
Ok, I perhaps I could have better defined the axiom of choice, but in the latter you get the point.
Quoting jgill
That basically was my question. And I think comes to this thread's main question, because mathematics is quite connected.
Perhaps our questions define what we treat as objects. Could logical mathematics in it's entirety be an object compared to let's say something different like poetry.
Barry Mazur has a really neat paper on this question, and at least parts of it are quite accessible. He ends up advocating (maybe just "showing the benefits of" is a better term) of an approach grounded in category theory.
https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://people.math.osu.edu/cogdell.1/6112-Mazur-www.pdf&ved=2ahUKEwjZhPfV0qCJAxW9vokEHYuQCMwQFnoECB4QAQ&sqi=2&usg=AOvVaw0j1f7DfoQP7OKuvRZ37rIU
In a very loose sense, there is a neat parallel here to Roveli's Relational Quantum Mechanics or some forms of process metaphysics. I am less hot on those than I used to be, coming around to views that still include a role for the nature/essence of objects (e.g. Aristotle, or some interpretations of Hegel—things might be defined by their relations, but they are not [i]just[/I] collections of atomic relations, rather relations are defined by what a thing is as a whole), or Deely's Scholastic-informed semiotic view of things existing in a "web of relations," which still holds on to "realist" intuitions re essence—a "balancing act." (Well, that's all vague I know, but the paper IS interesting!)
And there is also a neat parallel to St. Maximus the Confessor's philosophy and the Patristic philosophers' conception of number, which I will perhaps return to elucidate if I have more time. But basic idea is that things are not intelligible in themselves (although they do have intelligible natures, logoi). For instance, the idea "tree" is only fully intelligible in terms of other ideas such as the sun and water that are necessary for the tree, the soil it grows in, etc. You cannot explain what it is in isolation. Numbers and figures (following the old division between magnitude[discrete] and multitude[continuous]) are included here, in that they only exist where instantiated, in minds or things, and are not wholly intelligible on their own.
This makes even number dynamic in an important sense. To be fully intelligible, things must exist in the absolute unity of the Logos (Christ as Divine Word, but due to divine simplicity we might say God as a whole as well—on this view the entire cosmos is incarnational).
Anyhow, this sort of relates back to the OP. The idea is that, yes, there is a sense in which everything must be one (i.e. unity in the Doctrine of Transcedentals), but there is obviously also differentiation and intelligibility in the many (the old problem of the "One and the Many").
Another interesting thing is how this relates to knowledge. In Metaphysics, IX 10, Aristotle distinguishes between two kinds of knowledge/truth:
-Asytheta: truth as the conformity of thought and speech to reality (whose opposite is falsity); and
-Adiareta, truth as the grasping of a whole, apprehension (whose opposite is simply ignorance).
Obviously, we follow relations discursively, through asytheta. But then it is by coming to grasp the whole (via adiareta), the principles by which these relations obtain, that we gain a more full understanding (St. Maximus gets at this in Ad Thalassium 60). Likewise, in the Arithmetic Diophantus*, although generally dealing with problems whose principles he cannot discover, makes the case that solutions are virtually present in the principles that will allow for solutions (the "many" contained in the "one," e.g. many shapes, with their own distinct quiddity/whatness, flowing from Euclid's postulates).
*Interesting bit of math trivia, Diophantus, living in the third century, seems aware of Lagrange's four-square theorem, and if he had an actual proof of it then it was effectively lost for 1,500 years (we only have some of his books).
Category theory is beyond my pay grade. It's quite popular (the Wiki page has over 600 views per day - people want to know what it's all about). So far it seems not to have included classical complex analysis. When I look at diagrams what is familiar is composition of functions, of which I am fairly proficient.
This is a very interesting paper on the subject. Thank you, @Count Timothy von Icarus! :grin: :up:
Too bad that the basics of category theory aren't taught in school. But then again, the educational system doesn't care much about the philosophy of mathematics or the foundations of mathematics.
I assume you are speaking of "school" as in "university". It is taught at some elite institutions and some not so high on the scale as well. If you mean elementary or high school the very thought is laughable.
I remember being in a discussion with several mathematicians fifty years ago when category theory came up. The consensus was that it "doesn't do anything". It may not prove any theorems other than those about itself. With set theory one can start from scratch and build a logical system, but CT requires knowledge of the various facets of the category. It's mostly an outgrowth of abstract algebra.
Computer science may be another matter.