Carlo Rovelli against Mathematical Platonism
Here's a fun one: Carlo Rovelli - the renowned theoretical physicist whose name seems to be everywhere these days - published a small seven page paper in 2015 arguing against mathematical Platonism. It can be found here, and I'd encourage everyone to read it, for discussion's sake. It doesn't require a great deal of mathematical knowledge. The argument is interesting in itself, but I think it's even cooler coming from a physicist, given that physicists are usually those who are most enamoured with the idea of mathematical Platonism. Anyway, here's how the argument goes -
Rovelli begins with a simple definition of Mathematical Platonism, which "is the view that mathematical reality exists by itself, independently from our own intellectual activities." Now, he asks that we imagine a world M, which contains every possible mathematical object that could ever exist, even in principle. Not only does M include every mathematical object we have currently discovered (integers, Lie Groups, game theory, etc) it also includes every mathematical object we could possibly discover. M is the Platonic world of math. The problem, though, is that this world is essentially full of junk. The vast majority of it is simply useless, and of no interest to anyone whatsoever.
As he goes on to say, mathematics, as it stands, only includes an infinitesimal subset of M, and even if we could go on to discover M, we probably wouldn't, because most of it would be totally uninteresting. This, though, opens up a new question - what is 'interesting?' Well, interest simply is in the eye of the beholder: we develop some parts of M and not others because those parts help us do stuff, alot of the time. Rovelli asks: "Why has mathematics developed at first, and for such a long time, along two parallel lines: geometry and arithmetic?" And answers: "because these two branches of mathematics are of value for creatures like us, who instinctively count friends, enemies and sheep, and who need to measure, approximately, a nearly flat earth in a nearly flat region of physical space.... From the immense vastness of M, the dull platonic space of all possible structures, we have carved out ... a couple of shapes that speak to us".
Moreover, "there is no reason to assume that the mathematics that has developed later escapes this contingency". One way that Rovelli fleshes this out is by looking at examples of just how contingent our current mathematics is. He gives a few, but I'll focus on just one, the idea of natural numbers (1, 2, 3...). Rovelli basically asks: why does anyone think natural numbers are natural at all? We certainly find it useful to count solidly individuated items, but he notes that what what actually counts as 'an object' is a very slippery affair: "How many clouds are there in the sky? How many mountains in the Alps? How many coves along the coast of England?".
In order to make the point stick, Rovelli asks us to consider a species of intelligent beings evolved on Jupiter. Because Jupiter is fluid, most natural structures found of Jupiter would be continuous complex structures - flows, vortexes, currents, and so on. What kind of math might be developed in this environment? R: "The math needed by this fluid intelligence would presumably include some sort of geometry, real numbers, field theory, differential equations..., all this could develop using only geometry, without ever considering this funny operation which is enumerating individual things one by one. The notion of "one thing", or "one object", the notions themselves of unit and identity, are useful for us living in an environment where there happen to be stones, gazelles, trees, and friends that can be counted. The fluid intelligence diffused over the Jupiter-like planet, could have developed mathematics without ever thinking about natural numbers. These would not be of interest for her."
From all this Rovelli concludes: "Far from being stable and universal, our mathematics is a
fluttering buttery, which follows the fancies of inconstant creatures. Its theorems are solid, of course; but selecting what represents an interesting theorem is a highly subjective matter.... The idea that the mathematics that we find valuable forms a Platonic world fully independent from us is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather."
Discuss!
Rovelli begins with a simple definition of Mathematical Platonism, which "is the view that mathematical reality exists by itself, independently from our own intellectual activities." Now, he asks that we imagine a world M, which contains every possible mathematical object that could ever exist, even in principle. Not only does M include every mathematical object we have currently discovered (integers, Lie Groups, game theory, etc) it also includes every mathematical object we could possibly discover. M is the Platonic world of math. The problem, though, is that this world is essentially full of junk. The vast majority of it is simply useless, and of no interest to anyone whatsoever.
As he goes on to say, mathematics, as it stands, only includes an infinitesimal subset of M, and even if we could go on to discover M, we probably wouldn't, because most of it would be totally uninteresting. This, though, opens up a new question - what is 'interesting?' Well, interest simply is in the eye of the beholder: we develop some parts of M and not others because those parts help us do stuff, alot of the time. Rovelli asks: "Why has mathematics developed at first, and for such a long time, along two parallel lines: geometry and arithmetic?" And answers: "because these two branches of mathematics are of value for creatures like us, who instinctively count friends, enemies and sheep, and who need to measure, approximately, a nearly flat earth in a nearly flat region of physical space.... From the immense vastness of M, the dull platonic space of all possible structures, we have carved out ... a couple of shapes that speak to us".
Moreover, "there is no reason to assume that the mathematics that has developed later escapes this contingency". One way that Rovelli fleshes this out is by looking at examples of just how contingent our current mathematics is. He gives a few, but I'll focus on just one, the idea of natural numbers (1, 2, 3...). Rovelli basically asks: why does anyone think natural numbers are natural at all? We certainly find it useful to count solidly individuated items, but he notes that what what actually counts as 'an object' is a very slippery affair: "How many clouds are there in the sky? How many mountains in the Alps? How many coves along the coast of England?".
In order to make the point stick, Rovelli asks us to consider a species of intelligent beings evolved on Jupiter. Because Jupiter is fluid, most natural structures found of Jupiter would be continuous complex structures - flows, vortexes, currents, and so on. What kind of math might be developed in this environment? R: "The math needed by this fluid intelligence would presumably include some sort of geometry, real numbers, field theory, differential equations..., all this could develop using only geometry, without ever considering this funny operation which is enumerating individual things one by one. The notion of "one thing", or "one object", the notions themselves of unit and identity, are useful for us living in an environment where there happen to be stones, gazelles, trees, and friends that can be counted. The fluid intelligence diffused over the Jupiter-like planet, could have developed mathematics without ever thinking about natural numbers. These would not be of interest for her."
From all this Rovelli concludes: "Far from being stable and universal, our mathematics is a
fluttering buttery, which follows the fancies of inconstant creatures. Its theorems are solid, of course; but selecting what represents an interesting theorem is a highly subjective matter.... The idea that the mathematics that we find valuable forms a Platonic world fully independent from us is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather."
Discuss!
Comments (337)
At this stage, a Platonist mathematician might insist that the concept of a natural number latches on the structural invariant shared between two such language games. But I don't think such a defense would work since the variations in the pragmatic points of structurally similar language games being played with symbolic numerals would lead to variations in the way they are structured 'around the edge' as it were. They might not be axiomatized quite in the same way nor be projectable in the same manner to extended domains. (I'd have to conjure up some example that might be more compelling than Kripke's 'quus' example).
Just quickly, on this - Do you know if this is something that is in Daniel Everett's discussion of the Piraha language? I ask because I think there's a deep and little explored connection between grammar and math and I'm looking everywhere for resources on this, even though I've yet to read Everett's book.
I have Everett's book in my digital library but I am yet to read it too. The case of the Piraha and their (lack of) counting abilities had been a topic of discussion very many years ago on the now defunct (but still archived) Yahoo 'analytic' discussion group. From what I remember, there is an interpretive disagreement between the psychologist Peter Gordon, who also studied the Piraha (lack of) numerical abilities and Everett, who is more of a Chomskyan linguist and, hence, who is less inclined to take seriously the Whorf-Sapir hypothesis on the essential link between language and cognition. I'm siding more with Gordon's analysis than with Everett's, because of this issue, regarding this specific topic. (So, you might also be interested in digging up Gordon's relevant publications)
So his argument is that the Platonic world of math doesn't exist because it is... uninteresting? :lol:
The most general definition of mathematics I know is that it is a study of structures/relations. If there is more than one object, these objects form relational structures and mathematics studies these structures. The most general relation is "similarity" (also known as "difference"), because it is a relation that holds between any two objects. It means that the two objects have some different properties and some same properties. Which gives rise to another general relation called "instantiation", which is the relation between a property and its instance. The instantiation relation is a special kind of the similarity relation but less general than similarity since it doesn't hold between arbitrary two objects. Finally, any objects can define a collection of them (for example based on their common property, as long as such a definition is consistent), which gives rise to another general relation called "composition", which is the relation between a collection and its part. The composition relation, too, is a special kind of the similarity relation but less general than similarity since it doesn't hold between arbitrary two objects.
So, the similarity relation, together with its special kinds - instantiation and composition, defines all possible relational structures. All these three relations come together in set theory, the foundation of mathematics. In other words, the world of mathematics is a world defined by set theory (more accurately, by all consistent versions of pure set theory).
Quoting StreetlightX
As long as there are any differences on Jupiter, you can start counting.
Sure, but what sorts of things are structures and relations? Do they exist in themselves rather like intelligible forms in Platonic heaven? If you assume that they are universals that exist by themselves, quite independently from the constitutive roles of our practices of reasoning and discussing about them, then, in that case, you are begging the question in favor of mathematical Platonism.
Arguably, we reason the way we do because the world is certain way for us to reason about it.
That's true, but the recognition that the world is a certain way for us to reason about it already amounts to acknowledging a productive role for the cognizing subject in constituting it and hence is a move away from Platonism.
Why would it only be a certain way for us? Do we really think that evolution or general relativity is a certain way for us, as opposed to being a certain way for the universe?
It's a certain way for us because we are co-evolved with our environment, or Umwelt (as Uexküll uses the term in the context of ethology, but as it can be extended to the context cultural evolution as well). Evolution isn't any way for the universe because the universe is quite dumb and doesn't care about anything in particular, not even its own material unfolding.
Lots of irrealists about math make this argument. "Well, it's not useful, so these abstractions aren't real." Of course, the reality of an abstraction would only depend on its utility to us if the abstraction were not independently real to begin with. Argument begs the question, everybody go home.
Relations are objects that hold between other objects (those other objects may be relations or non-relations). Relations are inseparable from the objects between which they hold.
Interestingly enough, the fact that they enabled us to exist already begins to establish a conceptual dependence between them and us. This is not to deny that they aren't causally dependent on us. They indeed aren't. But merely to establish the causal independence that past material instantiations of formal structures (such as subsumption of past events under laws) have from us falls short from securing their conceptual autonomy in Platonic heaven.
This appear close to Russell's theory of concepts, relations and (Russellian) propositions. It is a quite Platonic theory, so, relying on it would also beg the question.
There are other foundations of mathematics which are currently in use. Even one which highlights explicitly that mathematics studies relational structures; in category theory, the category Set is a subcategory of the category of relations, Rel. The paper in the OP even points this out, referring to topos and category theory among other things.
But anyway, the thrust of the argument is: if we took the results of all possible axiomatic systems, agglomerated them into one giant object, then granted that object independent existence - what would it look like? It would contain all kinds of bizarre crap, navigating through this world you'd hardly ever find an axiomatic system which resembled anything like our own. I imagine if Bertrand Russel visited this elemental plane of mathematics, he would make observations like:
Oh dear, I seem to have stumbled over a strange sort of arithmetic with countably infinitely many things which seem like 0 insofar as 0*a=a*0=0 for each one but a+0 isn't 0+a isn't a, ever!
Looking over the graphical representation of the function, we see that it satisfies f(x+y)=f(x)+f(y) but it isn't anything like our normal linear operators! It's everywhere dense in the plane, help!
Some mother fucker over there glued a cardinal between Aleph-0 and Aleph-1 copies of the real line together...
He wouldn't make a statement like:
because there's no way in hell he'd even be able to find such sensible structures in this existence turned madhouse. The most sensible explanation for why mathematics as a field looks nothing like this writhing chaos is that mathematics is sustained through the refinement of diamonds into more diamonds, and finding new ways of mining them. It is the study of fruitful relations and structures for us, not panning the sewage which is the elemental plane of mathematics for gold.
If such a realm really does instantiate into ours, it's an incredible coincidence that so little of it resembles mathematics as a topic of study, no?
Talk of the 'external reality' is quite Cartesian sounding. Cartesian materialism may be some sort of a variety of Platonism. But the relationship between representationalism in philosophical accounts of mental content, reference and truth, on the one hand, and Platonism regarding universals, on the other hand, is rather complex. I'll come back to this conversation tonight.
Mathematics is a feature of the external world, a consequence of the fact that there are differences and thus more than one object in the external world.
Yes, so far I have focused on a different part of Rovelli's argument that seems decisive to me but may be less intuitively compelling than his main point, which you are now drawing back the focus on. Thank's for highlighting it.
As I read it there are two complementary thrusts; one is showing what kind of crazy thing the elemental plane of mathematics is (would be) and that it doesn't resemble mathematics as we study it (or its objects) at all, the other is by looking at what kind of selection criteria we use for things which are part of mathematics as a field of study. I think the first one is a better intuition pump, but the second one is a more fleshed out argument.
Uninteresting in the sense that it does not even count as mathematics; not 'uninteresting, but still mathematics'. Rovelli's other example, of linear algebra, makes this clearer:
"When Heisenberg wrote his famous paper he did not know linear algebra. He had no idea of what a matrix is, and had never previously learned the algorithm for multiplying matrices. He made it up in his effort to understand a puzzling aspect of the physical world. This is pretty evident from his paper. Dirac, in his book, is basically inventing linear algebra in the highly non-rigorous manner of a physicist. After having constructed it and tested its power to describe our world, linear algebra appears natural to us. But it didn't appear so for generations of previous mathematicians. Which tiny piece of M turns out to be interesting for us, which parts turns out to be "mathematics" is far from obvious and universal. It is largely contingent."
The fact that linear algebra even is considered mathematics is because of it's interest to us. The question is ask is about 'paths not taken' (or even, as the history of math is littered with, paths-once-taken-but-now-largely-abdondoned, like geometrical definitions of infinity, as distinct from algebraic definitions of infinity): paths that 'populate' the world of M, and which have no power at all to describe anything about our world, and thus are dismissed as not-math. As fdrake rightly points out, the counter-argument to the supposed 'unreasonable effectiveness of math' is to point out that the vast majority - in fact almost everything that could have possibly been math (all the useless junk in M), sans the tiny sliver of what we currently consider to be math - of what might be math is completely useless and is totally and utterly 'ineffective'. It's as if, having had a suit tailored and refined over more than two millennia, one is surprised to find that it fits so damn well, and then to declare: this suit must be eternal and Real! It's intelligent design masquerading as math.
Yes, there are various approaches to the study of relational structures. Maybe some are less comprehensive than others. Set theory seems to be the most popular foundation of mathematics and it seems to me that it is an exhaustive study of relational structures (of course it can never be completed even in principle, due to Godel's first incompleteness theorem).
Quoting fdrake
In set theory, all relations are defined as sets.
But no one but you is talking about 'real'. There's something very insidious about the idea that unless something is eternal and timeless it can't be real. Part of the necessity of re-evaluating Platonism is to block its usurpation of what counts as real: denying that math exists apart from humans is equally to insist that reality is far more interesting than the bland, white-padded wall picture of it painted by Platonists.
Yes, most of it might not be beautiful or useful but we are talking about metaphysics, which I don't think depends on subjective notions of beauty or usefulness.
But how mathematics looks as a category theorist is quite a lot different from how it looks like under the aspect of set theory. Say, to a category theorist, natural numbers don't look like the names of individual objects, they look like isomorphism classes of sets. Set theory was built out of intuitions about composite objects of multiple elements, category theory was built from intuitions of transformation and symmetry.
Quoting litewave
I have no idea how you took the main thrust of my post to be about beauty or truth. The main thrust is simply that most mathematical objects aren't worthy of study, and agglomerating them all together; producing the final book and the final theorem, far from the ideal vision or ultimate goal of mathematics - produces a writhing mass of irrelevant chaos. It's less Heaven, more Pandemonium.
At least as many pages in The Great Book Of Mathematics from this realm would be devoted to this picture:
as all of the research we have done and will ever do in mathematics. You shouldn't come away from this thinking about the the relationship of mathematics to the form of derp face...
But do these mathematical objects exist, or is this based on the hypothetical that they could be created if we were Jovians?
Is there a bunch of abandoned junky math that was of no value to Mathematicians but still qualifies as math? Is there a Math junkyard?
If the platonic realists are right, the name of that junkyard is the Platonic realm of forms.
Right, but I'm asking if there is a human junkyard of abandoned math, whether constructed or discovered. Because the argument turns on most of math being a junkyard. I'm asking whether this is a hypothetical, or actually historical.
Well no. There's nothing like what there would be if all the mathematical forms instantiated in the same way. Which really raises the question - when we recognise an instantiation of a mathematical object, with our platonist goggles on, are we seeing the world conforming to mathematics or mathematics conforming to the world? Sometimes we will see mathematics conforming to mathematics, but that's of no interest here.
We already have to filter out the junk in the platonic realm to obtain something resembling the collection of mathematical objects we are familiar with; what if this filter was internal to mathematics as a field of study, and roughly demarcated its contours of relevance and topics of interest?
Under that presumption, the realm of mathematical entities sure does look a lot more like its current self.
So Rovelli's argument summarized in the OP is that Platonism would be full of useless math instead of just the math we're interested in.
But what is the argument justifying this claim? Are there examples of useless math of interest to no one? What makes the case that Platonism would lead to this? Because other creatures would develop maths we wouldn't care about? Is that actually true? I'm thinking human mathematicians would actually quite interested in how much farther than us the Jovians had developed their geometry, and I'm guessing Jovian mathematicians would be quite curious about arithmetic.
Which brings about a second question. Why is utility an important criterion for math? Certainly applied math is important for various fields, but mathematicians also are interested in math for it's own sake.
Trying to give examples of 'non-mathematical' math from the canon of mathematics is a poisoned well. But I did try to give some demonstratively weird objects in my response to litewave. There are also some suggestive examples. EG, every number we're ever going to see as a digit based representation is computable, but computable numbers are a measure 0 set in the real line which is their home. We have a rich mathematical theory about continuous functions, augmented it with continuously differentiable and smooth functions, regardless generic continuous functions are nowhere differentiable. Generic functions themselves are nowhere continuous. Ways of associating elements of sets with elements of other sets generically are not functions, either.
If we have that derp face from before as a constant symbol, it satisfies the axioms of a group on a single element. We could make the same generalisation with any addition of a black pixel somewhere in the plane to the image, and we have countably infinite isomorphic copies of the group on one element that are literally just derp faces. If you were to write all of them down and establish the isomorphisms between the different representations, that's more mathematics than will ever be written solely devoted to the stupid application of an algebraic structure to the derp face.
All of these are in the elemental plane of mathematics. But they too are relatively well behaved.
Given the tiny frequency of tame objects in the vast sea of batshit lunacy that are models of some collection of axioms, it would be very hard to claim that generic mathematical objects typically are interesting or resemble/are related to the ones whose study constitutes what could be topics of mathematical study. We don't just not care about them for reasons of utility, we don't care about them because we have a standard of intelligibility which automatically excludes them from our mathematical discourse.
Yes yes, the Platonist will insist, they're still there, even though they can't be instantiated in our mathematics. But that still concedes enough for them to hang themselves with; this is our mathematics, it's mathematics for us by us, it's not an independent realm of existing objects at all, it's a conceptual web whose contours are delimited by our standards of intelligibility. All of that is mathematics, and no more.
Edit: previously I had an example of adding different multiplicative 0's to our usual arithmetic in this post, which satisfy the property of 0 like a*x = x*a=a for all other numbers x, where a is now the extra 0, but if I took another 0, b, I would have that a=a*b=b through the definition. This was based around a vague recollection from university but I obviously didn't remember it properly and I can't figure out how to set it up exactly again. In lieu of that, if we have a less restrictive 'left 0' so that a*x=a for all x, I can add any number of leading extra 0's. If you are reading this and have already read my unedited post, sorry for the flub.
Yes. This is how I understand it: more general (more abstract) mathematical objects are instantiated in more specific mathematical objects (e.g. "geometric object" is instantiated in "triangle") and ultimately in concrete mathematical objects (e.g. in concrete triangles), which are not instantiated in anything else. (Those objects that can be instantiated in other objects are also called properties.) All concrete objects are concrete collections, that is, collections of concrete objects, so all mathematical objects are ultimately instantiated in concrete collections. This fact is used in set theory, where every mathematical object is represented as a collection (set) and that's why set theory can be a foundation of mathematics.
The collections referred to in set theory are not concrete collections though but abstract collections (generalized collections), because differences between concrete collections of the same kind are not relevant for mathematical purposes. So for example, set theory does not refer to concrete empty sets but to one abstract empty set (which is instantiated in all concrete empty sets).
The approach of category theory is not to represent mathematical objects as collections but to study similarities (morphisms) directly between mathematical objects themselves. Collections, then, are treated just as one of many kinds of mathematical objects.
Quoting fdrake
Well, it depends on how you define mathematics. In the context of set theory, traditional mathematical topics can be extended to the study of all relational structures. This covers the whole relational aspect of reality and so is relevant for metaphysics/ontology.
Some categories aren't sets though.
That's not how that works. There are more categories than the category of sets, Set. Only things in Set are categories which can 'be instantiated' in sets.
Yes.
Except it isn't any more! There's more to mathematics than what can be represented through set theory.
Do some reading about it. Essentially small categories are those which have representations as sets and set relations/operations. Not all categories are small categories. So the link provides a starting point to start learning about under what conditions a category can be represented by a set!
Stanford Encyclopedia of Philosophy says:
"The axioms of set theory imply the existence of a set-theoretic universe so rich that all mathematical objects can be construed as sets. Also, the formal language of pure set theory allows one to formalize all mathematical notions and arguments. Thus, set theory has become the standard foundation for mathematics, as every mathematical object can be viewed as a set, and every theorem of mathematics can be logically deduced in the Predicate Calculus from the axioms of set theory."
https://plato.stanford.edu/entries/set-theory/
(from year 2014)
Then it's wrong. You don't actually need to go to category theory for objects which don't exist in the ZFC universe, certain large cardinals need extra axioms to model.
Yes these kinds of objections to Platonism occurred quite early on in the form of questions such as "But is there a perfect form of the turd, or the pile of vomit?".
What all versions of pure set theory have in common is the concept of set as a collection of objects that can be defined by listing those objects (set members) or by specifying their common property. As it turned out, not every definition via a common property is consistent, and since it would not be very useful to define sets only via listing of members, one must also specify with axioms what common properties can be used to define sets. Which gives rise to uncountably many axiomatic versions of pure set theory.
As Raphael Demos notes in Plato: Selections:
which is a familiar sentiment around here.
As far as the essay itself is concerned, I think a lot rides on the qualification at the beginning of the paper:
The fact that the distinction between the expressions 'to be real' and 'to exist' are brushed off says a lot. Because the assumption that platonism really amounts to saying that a mathematical domain M exists begs the question as to the nature of its existence. Are numbers only real in the mind of the individual? The species? The culture? Or, as mathematical platonists argue, are they there to be discovered in the intelligible domain?
I think the significant thing about the Platonist view is that the nature of number is ontologically distinguishable from that of sensory objects; recall that this was the fundamental platonist distinction between doxa and pistis, on the one hand, and dianoia, on the other, given in the Analogy of the Divided Line, which is the central doctrine of Platonist epistemology. So, intelligible objects are real, in the sense of being 'the same for all who can count', but they're not sensory objects, because they're only perceptible by the rational intellect [sup]1 [/sup]. They are known, and knowable, in a way that sensible objects cannot be; and at the same time, they enable the rational intellect to arrive at a foundational level of understanding, such as hypotheses expressed in mathematical terms, which could never be elicited by sensory experience. That is the source of the kind of reverence that the ancients had for rationality and logic, which is actually what gave rise to science itself.
So perhaps 'mathematical world, M', is really just a metaphorical depiction of the Platonist intuition of the nature of numbers. But then, it is 'the existence of M' that is thrown into doubt. But maybe this doesn't do anything more than show that this particular way of allegorising Platonism is what is at fault.
There's another thing. It there is indeed a 'perspectival error of mistaking ourselves as universal', then so much for the universalising claims of science itself. Reminds me of a remark by Tom Wolfe in his celebrated essay, Sorry, but your Soul just Died[sup] 2 [/sup]:
Welcome to post-modernity. I guess.
The problem for Platonists is that they have failed to, and apparently cannot, explain in what sense the purported Platonic objects exist, or are real, in some way other than the familiar concrete (human mind independent) existence or reality of sense objects (or at least of whatever gives rise to them), and the familiar ideal (human mind dependent) existence or reality of the contents of thought, emotion and perceptual experience,
It's not difficult to explain, although it might be difficult to accept, or to understand. From Platonism in the Philosophy of Mathematics.
There's another argument, The Indispensability Argument for Mathematics, which I admit I haven't studied in depth. However, I would argue that the fact that the argument is necessary is significant.
Why do you think it might be necessary to state such an argument?
Although I don't believe in this, the indispensability argument suffers from the critical flaw of question begging.
As to how my cognitive dissonance can come to be is another question.
Not only is this mathematical realm full of junk, but it's also full of contradictions. Go figure. Because of such contradictions, mathematics is clearly not logical. So, which is more reliable, mathematics or logic?
I think that's a point Plato would readily have acknowledged; and a reason why Plato may never have been a Platonist in the modern sense of the term. Modern mathematical Platonism likely is a distortion of Plato's thought, which distortion arose from taking his metaphors literally and misconstruing acts of the intellect -- themselves always portrayed by Plato as outcomes of strenuous and protracted dialectical effort -- as passive acts of contemplation of an independently constituted domain.
Quoting Wayfarer
That's my suspicion too!
This makes sense. I don't have the knowledge to bring out how Plato became distorted, though. What history are you tracing in this idea?
I'm assuming self-awareness is necessary for any bit of math a being can do. Am I right in this assumption? I heard bees can count and I don't see them asking existential questions.
I think this is a great question, and I think it's important to show that, if most of what could-be-math is junk, then we've come across that junk before. One example that I think fits the bill is John Wallis' proof that all negative numbers are - or rather can be construed as - greater than infinity (the reasoning is simple, and I summarized it in a previous post). The upshot of Wallis' proof is that the number line (which Wallis invented), which normally looks like this:
-? < ... < -1 < 0 < 1 < ... ?
can look like this:
0 < 1 < 2 < ... < ? < ... < -2 < -1
The thing about this is that there's nothing particularly 'wrong' with this way of ordering the integers (here's a paper that fleshes it out in modern terms). The reason math doesn't opt for Wallis' construal of the number line - and his conception of infinity - is because Cantors' constural of it (now the canonial treatment of infinity) is much more productive. Wallis' number line is 'junk math'.
In a physicsforum post that discusses the paper, a commentator makes a point that's almost identical to Rovelli's with respect to Dirac that I quoted earlier: "when Dirac wrote in his book: 'principles of quantum mechanics' that the derivative of Log(x) should contain a term proportional to a so-called 'delta function' that he had just invented out of thin air a few pages back, was complete nonsense too."
(source); Compare Rovelli: "Dirac, in his book, is basically inventing linear algebra in the highly non-rigorous manner of a physicist. After having constructed it and tested its power to describe our world, linear algebra appears natural to us. But it didn't appear so for generations of previous mathematicians".
The reason it's not very easy to come up with examples of junk-math is precisely because it's... junk math. No one cares for it, and no ones cares to pursue it because its largely useless.
That said, another, perhaps less pertinent example might be geometrical definitions of infinity (i.e. definitions that rely on intuitions about physical space), which in turn relied on the fuzzy concept of the infinitesimal. Until the invention of non-standard analysis in the 60s (which provided a rigorous way of understanding the infinitesimals) mathematicians made a huge effort to understand infinity on a strictly arithmetic basis (i.e. without reference to physical space), because the logical foundations of 'geometrical infinity' were not considered to be secure. In that time before those foundations were secured, one could say that the concept of infinity teetered on the edge of 'junk math' - being 'saved', ultimately, because it's so damn useful.
Yep. All you end up getting are these miserable negative non-specifications that are more than happy to specify what Platonic entities are not, all the while dodging the question by saying that 'oohh its so hard to explain'. It's a cheap rhetorical go-to that feeds right into the elitist, exclusionary, and cultish tendencies that span all of Plato's thought. No wonder that it ends up, in its later, Christian incarnations as negative theology and cultic mysticism: making a virtue out of intellectual failure (which also helps explains Neitzsche's diagnosis of nihilism which runs an intellectual line from Plato right through to Christianity: "man would rather will nothingness than not will.").
Quoting Janus
This is a really nice historical point! Totally didn't think of that.
I mentioned in a thread the other day, that Einstein said in dialogue with Tagore that 'I cannot prove scientifically that Truth must be conceived as a Truth that is valid independent of humanity; but I believe it firmly. I believe, for instance, that the Pythagorean theorem in geometry states something that is approximately true, independent of the existence of man.' However, I argue, what this overlooks is that, to our knowledge, that is something that can only be known to h. sapiens. In fact, arguably, it is what makes humans sapient in the first place.
Now this insight has been preserved almost nowhere outside neo-thomism, but that may only be because Thomism itself was the last vestige of a real metaphysics in the Western tradition. In other words, it's not due to any particular intellectual superiority of Catholicism per se, but just that this is one of the only intellectual milieu in which the insights of Greek philosophy were kept alive (the other being Orthodoxy). Elsewhere, the effects of enlightenment rationalism and positivism were such that a 'presumptive realism' became ascendant, along with the 'reign of quantity' and neo-Darwinian materialism, which is fundamentally self-defeating in its outlook due to its denigration of reason.
Only consistently defined objects can be part of the mathematical world.
Axioms are properties of an object (also called axiomatic system). Axioms like "The continuum hypothesis is true" and "The continuum hypothesis is not true" would be contradictory if they were properties of the same object but they are not contradictory if they are properties of different objects.
If they are only perceptible by a rational mind it doesn't necessarily mean that they are inside the rational mind. They may be perceptible only by a rational mind and still be outside the rational mind - that's what Platonists/realists are saying. They are saying it because it seems inconceivable that truths about numbers didn't hold before someone perceived them or will stop holding when there is no one around to perceive them. Maybe a more accurate word would be "to infer" than "to perceive", since abstract objects like numbers don't seem amenable to sensory interaction. But to infer the existence of an object doesn't mean that the object only exists in the mind that is doing the inference. Scientists make inferences (predictions) about the external world and test them.
Indeed - that's the point. And you're right in saying they're not 'perceived' in a literal sense, but are 'grasped by reason' - seen by the mind's eye, so to speak.
Quoting litewave
But inference relies on reason which is, on the one hand, internal to the operations of thought, but, on the other hand, makes real predictions. Reason is strictly the relationship of ideas, but at the same time, rational inference can be used to predict novel discoveries about the world - which is pretty close to the meaning of Kant's synthetic a priori.
And? The unarticulated premise here is that this mutual recognition entails 'independence'. But this is clearly nonsense. That people recognize words - and especially non-referential words like 'and' and 'is' - does not entail any ostentatious, overblown metaphysics of Platonic Is's and And's- it simply entails that we've learnt how to use a language, like the slightly-less-than-dim animals that we are.
As for neo-Thomism being the last refuge of Greek philosophy - all the better that it's drawing its last, dying breaths. The sooner we forget about hot mess that is Greek philosophy and it's intellectual spawn, the better.
Doesn't the fact that basic mathematical truths are the same for anyone who can count demonstrate that they're independent of particular thinkers? What is nonsensical about that?
It's like saying: wow, look at all these various languages that have nouns! Guess Nouns must be Platonic Entities. It's reasoning made for and by idiots.
There is an undeniable phenomenon of convergence. But there are two mistakes one could make regarding such phenomena.
The mistake that animates modern naive empiricism is to explain the phenomenon of convergence -- such as the discovery of laws of nature, or of general logico-grammatical features shared by (most) natural languages -- as a result of the faithful (or approximate) reproduction, as contents of our mental representations, of the structure of an independently existing empirical ('external') reality. This mistake is almost indistinguishable from the mistake of reifying intelligible ideas, where such ideas are being ascribed a role similar to the role being played by raw 'sense data' for the empiricist.
Another mistake would be to recognize the root of the convergence as a product of shared social practices but to view the fact of this convergence as a matter entirely of socially enforced conventions.
One might avoid both mistakes though recognizing that the phenomenon of convergence is a dynamical product of the enactment of the social practice of arguing for or against doing and/or believing things. The convergent phenomena are being constituted from within the enactment of dialectical reason, by social beings, rather than from without their historically situated lots of shared discursive practices.
Totally agree. Insofar as we are (mostly) beings that count, and employ counting to engage in certain behaviors, it's simply unsurprising that our counting systems will tend to converge around certain invariants. And as Rovelli remarks, the fact that we are beings that count has itself a strong element of sheer contingency: "The development of the ability to count may be connected to the fact that life evolved on Earth in a peculiar form characterized by the existence of “individuals”. There is no reason an intelligence capable to do math should take this form. In fact, the reason counting appears so natural to us may be that we are a species formed by interacting individuals, each realizing a notion of identity, or unit. What is clearly made by units is a group of interacting primates, not the world".
I don't think it's a matter of coincidence as well that in the sphere of evolution, convergence is sometimes also used as an argument for theistic ends - the usual "wow, look at all these eyes across so many different species, must be God at work!"; well, no, eyes are the product of the dynamics involved with the properties of light and animals that move, many of whom share similar environments (with animals in the depths of the ocean evolving no eyes - what a surprise!). There's a very real parallel with those who think that convergent aspects of math similarly reflect some grand transcendent design decoupled from the material universe, rather than behavior and environment: it's nothing less than intelligent design shunted into the field of mathematics. And it's all just as bogus. Math is the last, shrinking, refuge of a God deservedly driven out of everywhere else.
Also totally agree with you about the critique of 'naive empiricism': as you said it's just the mirror image of Platonic Idealism, both of which are totally utterly blind to the evolutionary and historical dynamics of shared practices - in the case of math - and shared ecological and developmental dynamics - in the case of evolution.
I was alerted to the possibility of the distortion by a handful of scattered remarks on Plato versus Platonism by John McDowell. But I haven't pondered much on the historical roots of the distortion, nor do I feel equipped for tracing such roots anywhere earlier than the modern period.
I just did some literature search and found this paper, which makes similar remarks to McDowell's: Konrad Rokstad, Was Plato a Platonist? Analecta Husserlania. The Yearbook of Phenomenological Research, Volume CX, 2011.
Regarding what might be seen as a modern recovery of Plato's valuable insight about the dialectical nature of the constraining function of a priori 'forms' upon reason, I made this recent comment.
But money has no meaning pulled away from the world that embraces it. And in turn, that world will become nonsense if we try to understand it without the abstractions that organize it.
Mental codes require mental representations, and facilitate mental processing.
Numeracy (the ability to understand number arrangements, and perform numerical operations) and literacy (the ability to understand word arrangements, and perform verbal operations) are human faculties, hence; human universals.
Both faculties develop in parallel with mental maturation, personal experience, and social influences. Mathematical and verbal abilities develop subjectively, while mathematics and language develop intersubjectively.
"I don't care if it's real or not; I'm not interested in reality." Well, okay, then.
Okay, so how does my picture have little to do with reality? We need something better than question-begging here.
I pointed out that your argument is circular. Do you actually have a response?
Ah! I see the problem here. Platonism is a stance toward the reality of abstractions, so argument against Platonism is argument against the reality of abstractions. Thus the reason for bringing up the reality of abstractions.
Terminology is not terribly important. Instead of complaining that I'm being mean to the paper, why don't you respond to the argument?
"I want to have my own debate".
Have fun.
Do you have an argument?
I'll take that as a "no."
Ah, so you do have an argument! What is it?
I'm afraid that wasn't an argument.
In defense of a dogma seems like a really fun article. I just started it and I'm impressed by the style, precision and generosity of the argument!
Well you have to make allowances for the reading age of the audience, who are, after all, only ‘apes on a rock’.
I think this is already starting on shaky grounds. What does "exist" mean here? Mathematically it would mean something like "provable" but if we're dealing with a possible world in which everything is provable, we're dealing with the Trivial World. A world without any coherent structure at all. You later ask'
Quoting StreetlightX
And often what mathematicians mean by "mathematical interesting"ness is some set of results that do not entail triviality (every sentence becoming a theorem). Because if some math explodes into everything it loses coherency and thus can't really be analyzed at all. Non-triviality is the baseline for what mathematicians consider a theory worth investigating.
Quoting StreetlightX
I'm not really sure what his argument is supposed to be. Math platonism doesn't say math is universal in the sense he is assuming. I'd say I'm a mathematical pluralist, so I don't find myself committed to any particular math system as a matter of principle, but this doesn't preclude being a math platonist. But the preference of real numbers in the scenario he refers doesn't seem to contradict math platonism unless I'm missing something. It's just a scenario where it's more useful to apply real numbers.
That would naturally flow from representative realism, in that there is a tacit assumption that the mind 'mirrors' or 'represents' the world, which ultimately stems from Locke, and is ubiquitous in empiricism.
Quoting Pierre-Normand
However, the cardinal difference between modern and ancient philosophy in this regard, is that modern philosophy is largely underwritten by biological science, which situates h. sapiens along a continuum with other species, and which brackets out any sense of there being either a first cause or final end.
Whereas
Lloyd Gerson, What is Platonism?
Quoting Marchesk
Yep. The issue is that reality and maths can seem very far apart when one is being understood in terms of the physics of constraints and the other is being viewed as a free grammatical construction. So junk can spew out of a syntax freely and meaninglessly - as in the examples like Borges’ library which Rovelli uses. But then reality and maths can seem fundamentally connected - as is the case of mathematical physics, where Rovelli is one of the leading players. When some kind of intelligible constraint - a suitable selection principle - is applied to the random junk spewing, then that gets closer to making sense of the relation.
To be fair to Plato, he did suggest such a finality. He argued for the Good as an optimisation principle that revealed the truth, beauty and justice of a subsidiary realm of mathematical (and other) forms. So that was of course still rather mystical. But it was at least recognising that something higher - a global constraint - was needed to pick out the most meaningful or fundamental patterns and structures.
Several points. It should be noted that if there turn out to be multiple descriptions of the same thing, then that thing seems to be something beyond the mere descriptions. Rovelli gives several examples. The choice between Euclidean and spherical geometry. Heisenberg's matrix mechanics and Schrodinger's wave mechanics. And as Cartesian geometry showed, in general, algebraic maths and geometric maths offer dual descriptions of everything that seems of mathematical interest. So Rovelli seems to want to use this multiplicity of descriptions to argue for social construction. Yet it also argues strongly that everyone is feeling the same elephant. Like any use of language, it is being constrained by its encounters with a recalcitrant reality beyond.
Then if we consider how maths actually advances, it is largely by the relaxation of constraints. Maths moves up to a higher level by generalising and abstracting. Euclidean geometry gives way to non-Euclidean geometry. Then geometry gives way to topology. So the map of maths - M - is not some flat plane of existence, cluttered mostly with junk with a few bright spots of interest. Instead, maths is itself a hierarchical structure. We rise up out of the clutter of the particular detail by finding some key constraint we can relax. And having gained control over that parameter, we can then add back constraints to toy with different worlds.
Give up Euclid's parallel postulate and the geometry of space can be curved. That curvature can be added back as some positive, negative, or neutral number. So what makes maths interesting is where we find the constraints that are holding things in place - imposing a particular structure or form - and can then twiddle the knobs to discover the "world" the more abstracted description lives in. Keep abstracting and you go past set theory to arrive at category theory - at least according to current wisdom. There is an ur-form at the top of the mathematical hierarchy that speaks to what is Platonically the Good - a basic idea of a relation which is the most generalised possible constraint we can imagine.
So Plato was certainly on to something. Our physical world is not some random junk of accidents. It has an intelligible structure. But the problem with Platonism is the way it suggests a flat plane of forms - where the perfect triangle exists alongside the perfect turd. That problem can be addressed by the addition of a selection principle - a hierarchical story - which does then separate reality into its accidents and its necessities. And in Aristotelian fashion, this is what you get when the material realm becomes the source of accidents or fluctuations - the blind atomistic construction - and the formal realm supplies the downward-acting constraints which are a system's regularities and inevitabilities, its essentials, or universals, or necessities.
So we recognise a triangle as speaking purely of nature's necessities. A three sided polygon is going to have internal angles of pi, or 180 degrees. This is a truth of physical space - at least in a Euclidean setting. But a turd seems mostly a set of physical accidents. A very material construction. A mathematician might find the structuring formal principles that do in fact regulate the shape of any given turd - why a liquid one might behave differently from a more solid one. But Plato's realm of forms is really a space of abstractions. Maths explores the constraints that nature imposes on material accidents - and mostly seeks to abstract them away, because what is interesting is to gain control over them.
Most of the apparent physical constraints we encounter turn out to be accidents of history. The Universe seems Euclidean to us because it has grown so large, flat and cold. So we can generalise away the particular in the world as it seems structured right now, to work back towards to the way a world could possibly be - if we were to add other general material conditions, like it being as small and hot as possible.
The mathematical enterprise is thus about trying to discover the rational forms that structure some set of material accidents. And that dualism has to give way to the triadicism of a hierarchical metaphysics as that exercise gains scientific sophistication. Science speaks of global laws and material constants. Actual substantial existence is then what arises inbetween.
So the turd becomes an entity that arises in some sense because of the shaping laws of dynamical flows - a general model that would be paired to specific material parameters, like a measure of the viscosity of the turd in question. It is this triadic complexity, this hierarchical or systems story, which the debate over Platonism is so insensitive to, but which should already be evident from the Aristotelian version.
Then a final point. The usual view is that either maths is Platonic, or it is purely a social construction. Either it speaks truly of necessity, or it is merely always a cultural accident - reality as described from a basically freely creative human subjective point of view.
But I see nothing wrong with the inbetween position. What we are getting at with mathematical physics at least is the objective point of view - the one from the perspective which would be the Cosmos contemplating its own rational structure.
So that does apply a pragmatic constraint to the enterprise - a view that has an embodied interest. The realm of junk maths is being limited by a viewpoint which speaks to the basic finality of wanting to be embodied. It is maths of that type, structure that can produce that result, which is "true, just and beautiful". We can see the universality in it as we stumble across it.
The maths of symmetry and symmetry-breaking are a good example of that. Likewise statistical mechanics and dissipative structure theory. There are areas of maths that look very organic - because they marry the accidental and the necessary in a way in which global regularity must emerge from local randomness. Structure - as the stabilisation of instability - can develop.
So sure, when maths is understood as just a realm of everything that unconstrained syntax will produce - a Borges library - then it seems to bear no real relation to a reality in which limitation or finitude is apparent everywhere. But when maths is viewed organically - as a language to capture the emergent regularities of pure possibilities - then that is a strong selection principle to sift the wheat from the chaff. We arrive at the structures that matter because they are the most irresistible. Randomness can't erode them, because randomness is in fact constructing them.
I'm sympathetic to that view. Constrained math has a relationship with reality. Aristotle's view was more correct than Plato's. The in-between position seems more reasonable.
I like this:
Quoting apokrisis
Platonism - and Aristotle - both assume an hierarchy, but it's from a top-down, not bottom up, perspective.
'The Platonic view of the world – the key to the system – is that the universe is to be seen in hierarchical manner. It is to be understood uncompromisingly from the ‘top-down’. The hierarchy is ordered basically according to two criteria. First, the simple precedes the complex and second, the intelligible precedes the sensible. The precedence in both cases is not temporal, but ontological and conceptual. That is, understanding the complex and the sensible depends on understanding the simple and the intelligible because the latter are explanatory of the former. The ultimate explanatory principle in the universe, therefore, must be unqualifiedly simple. For this reason, Platonism is in a sense reductivist, though not in the way that a 'bottom-up' philosophy is. It is conceptually reductivist, not materially reductivist. The simplicity of the first principle is contrasted with the simplicity of elements out of which things are composed according to a 'bottom-up' approach. Whether or to what extent the unqualifiedly simple can also be intelligible or in some sense transcends intelligibility is a deep question within Platonism.
....
The hypothesis that a true systematic philosophy is possible at all rests upon an assumption of cosmic unity. This is Platonism's most profound legacy from the Pre-Socratics philosophers. These philosophers held that the world is a unity in the sense that its constituents and the laws according to which it operates are really and intelligibly interrelated. Because the world is a unity, a systematic understanding of it is possible' (Lloyd Gerson, 'What is Platonism')
(Incidentally, whether there is indeed 'cosmic unity' is very much in question in current physics, is it not?)
Quoting apokrisis
That also has a precedent in Aristotle - 'The Prime Mover is simply the formal-noetic structure of the cosmos as conscious of itself' [ibid].
Yep. Constraints act top-down.
But the tricky part - which the maths of hierarchy theory realises! - is that the causality has to go both ways. There is also the upward construction kind of hierarchy. So the story becomes about the synergy between parts and wholes. You have compositional hierarchies that are the bottom-up view, and subsumption hierarchies that are the top-down view. And a stable reality can only emerge when the two are reinforcing each other's existence.
This is the kind of balance of causality that is described by dissipative structure theory - such as Bejan's work on material flows and "constructal law" - https://www.forbes.com/sites/anthonykosner/2012/02/29/theres-a-new-law-in-physics-and-it-changes-everything/#33e609ee618d
Quoting Wayfarer
I agree with the quote. But of course calling it "conceptual" is dangerous as it does suggest the mental.
For me, calling it structurally reductive would be better. My goal would be to avoid lapsing into actual mind~matter dualism on this one.
It's funny that you would mention Gerson's book, since I added it to the Platonism folder in my digital library a few hours ago, having found it thanks to the title of the first chapter: "Was Plato a Platonist?" (which was also the title of the paper by Konrad Rokstad that I referenced earlier).
Nothing in the quoted passage explains what kind of existence (apart from the 'ideal' existence they have insofar as they are thought by humans) these abstract so-called "objects" enjoy.
Does it follow from the fact that the world is numerable, and of course numerable in the same conventional way for all (a fact that reflects the common culture of human practices), that numbers have some kind of abstract ontological existence independent of human life? This would be an objective abstract existence, (if any sense could be made of that).
You seem to contradict yourself insofar as it is you that is always saying that there is no (concrete) objective reality apart from subjective human experience. If there is no objective concrete reality ( we can at least make sense of the idea of concrete objective existence, as the default tendency to naive realism attests) how much less would there be an objective abstract reality (an idea that we seem to be able to make no sense of at all)?
You often seem to want to claim that the distinction between reality (or being) and existence can explain platonist claims. I think this is untrue, because the terms and the notions they are associated with are more or less interchangeable, even though it is also true that under certain restricted interpretations of them, distinctions can be made. The bottom line, in any question of being, existence or reality, is whether something is concretely real or merely imaginary. That is a real distinction even though the imaginary can in one sense be said to be real, but only insofar as it participates as idea in the concrete act of imagining.
So, for example, the idea of apophatically attempting to dodge the question of God's existence by claiming that he is real but does not exist is a conceptually fraught, perhaps I should go as far as to say intellectually dishonest, strategy. The proper question is whether God's existence is really real (i.e. completely independent of humanity) or really imaginary (completely dependent on humanity)..
What if what we understand to be "cosmic unity" is due to the fact that everything is 'at bottom' (in terms of the so-called quantum vacuum, say) irreducibly entangled with, and inseparable from, everything else. In that view nature can be it's own designer and creator; no need for any supernaturally transcendent God or abstract (as opposed to virtual) realms of form and number.
What I understand is that modern-day Platonism is more like Pythagorean idealism. Although the refutation of Pythagorean idealism is commonly attributed to Aristotle, it has been argued that Plato actually laid the grounds for this. Plato worked to expose and clarify all the principles of Pythagorean idealism, and in the process uncovered its failings. I've seen it argued that the Parmenides, though it is quite difficult to understand, serves to refute this form of idealism.
I don't understand how you can read this passage - which I agree with almost entirely - and not think to yourself: "gee this Platonism business is just as shittily reductive as the materialist reductionism that I'm always moaning about". But of course you don't care about reductionism - you just want reductionism in the 'right way'.
I believe that's exactly the point: M would be entirely trivial. This is the dilemma that the paper poses for Mathematical Platonism: either M is trivial and has no structure whatsoever (and thus largely says nothing at all about our world), or, if M is not trivial in this way, then it cannot be independent from our intellectual activity. In either case Platonism is undermined because if the former, then it has no explanatory power, and if the latter, then it simply isn't Platonism.
Quoting MindForged
The paper's index of Platonism has to do with the independence of mathematics from human intellectual activity; part of this, in turn, has to do with the modal status of our math: contingent or necessary, and to what degree? Rovelli's answer is a kind of qualified contingency: our math is contingent ("Which tiny piece of M turns out to be interesting for us, which parts turns out to be \mathematics" is far from obvious and universal. It is largely contingent"), but this contingency in turn is premised upon the kind of beings we are, and the kind of things we encounter in the world, along with what we do with them - which lends our mathematics a kind of empirical necessity (Rovelli doesn't use that term, but I think it's appropriate in this context).
I'm glad you like it. I think it has some relevance to the present topic since what is at issue, in Rovelli's polemics against the Platonic thesis that the domain of interesting mathematical objects might be identified with an ideal universe 'M' allegedly knowable a priori, is the contingency of our constitutive relation to what it is that we justifiably find interesting in mathematics (in such a way that it can so much as count as genuinely mathematical). Strawson and Grice, however, began loosening up, in their response to Quine, the false dichotomy between a prioricity (wrongly construed as a purely epistemic notion) and contingency (wrongly construed as a purely metaphysical notion), long before Kripke. They thereby recover some insight from Kant about the requirement for synthetic a priori propositions. (Much confusion arises, though, from inconsistencies in the use of the analytic/synthetic, a priori/a posteriory, and necessary/contingent pairs of predicates). This false dichotomy is something I'll have a bit more to say about in another post.
Thanks very much for those reminders. That's indeed an important thread of the history of ideas to be reminded of. Maybe Aristotle shares some part of the blame for having appropriated (while he doubtlessly improved on some of them) some features of Plato's criticism of idealism while ascribing to Plato himself theses that Plato (or Plato's Socrates) was merely expounding rhetorically. Aristotle's third man argument, if I remember, which is directed by him against Plato, is actually borrowed from Plato, if I remember.
I believe that by the time Aristotle was writing, "Platonism" was already fractured. Because Plato's ideas evolved over his lifetime there was probably never a firm "Platonist" platform. In his "Metaphysics" Aristotle directs his cosmological argument against the Pythagoreans and "some Platonists". He may have considered himself to be a Platonist, at odds with the other Platonists.
The issue which Plato exposed, which is expounded on in Aristotle's division of passive and active aspects of reality, is that Pythagorean idealism assigns to "Ideas" a passive existence, as outlined in the theory of participation. That which is participated in (the Idea) is passive (being eternal), while that which is participating is active. The cosmological argument demonstrates that the passive cannot be prior to the active in an absolute sense, so it is impossible that these passive "Ideas" are eternal. The later Neo-Platonists and Christian theologians, following more closely Plato's later work, "Timaeus", developed a metaphysics whereby "Forms" are active, in a way more consistent with Aristotle.
The relation between such an unconstrained world of math and a limited finite world is that the limited finite world is a part of the unconstrained world of math.
... Just like the material of Michelangelo's David was a smaller part of the whole block of marble, which it was carved out from. Would you say that, therefore, the statue already existed as a distinctive part of the whole block independently of Michelangelo's act of carving it out?
There being rational and transcendental numbers or Euclidean and non-Euclidean geometry doesn't mean that mathematics isn't linked together. Some have just assumed a long time ago that all numbers are rational or that Euclidean geometry is the "only geometry". Yet mathematics isn't an invented social construct that we can bend to whatever we want. All math is quite logical. The only thing is to be humble and understand that we can even now understand something wrong, just like the earlier Greeks who thought that all numbers had to be rational. It's even more easier to understand that we may have not discovered many new ways how math can be used, which will in the future opens new fields to us. To understand this doesn't in my view refute Platonism.
But there seems to be an obvious lack of a material principle in the formal realm of maths. Maths is spatial, or at best, spatiotemporal, and doesn't speak to energy or action in any basic way. It about the logical syntax of patterns and structures, and not about whatever breathes physical fire into those equations.
So the maths of the world would be the "maths" of constraining structure. The need to be constraining - constraining of material spontaneity or uncertainty or action - would itself be the big constraint on the maths that is physically relevant.
This is why statistical maths and symmetry maths does seem more real. It speaks to the naturally emergent structures of systems of constraint. There is a source of fluctuation or accident that is being limited by a global order. So that is where maths gets closest to the reality it might want to model. It incorporates the other thing of an action to be shaped.
Rovelli's realm of mathematical junk is then all the possible syntactical forms that can be generated when the forms serve no real organising purpose. Now you could say our finite material world is merely a part of that larger unconstrained universe. But I would say that Platonism - as a metaphysical position - is about the forms of nature that can do actual causal work. So there is the constraint that the forms do constrain. They must bring finitude to action, or the material aspect of nature. They must do the ontological job of stabilising accident and spontaneity and so allow a Cosmos to exist.
A lack of limits is pathological. It is the maths of constraining structure which promises to tell us the most about reality and the reasons for its existence.
My understanding exactly. And this same reasoning applies to all manner of transcendentals such as universals - they are also 'real but not existent'.
The objections to this understanding are usually based on the inability to make this distinction; hence the common objection to Platonic realism, 'where do numbers exist'? This is because we are by habit instinctively realist; we are oriented in respect of the domain of time and space, the objective realm, which for most of us defines the scope of what is real; everything that exists is 'out there somewhere' in the objective realm. Whereas logic, number, reason, and so on, transcend (or more accurately are prior to) the division of subject and object; they underlie and precede the ability to analyse the contents of the objective domain. It also includes the 'domain of possibility' which is likewise 'real but not existent' in that there are real possibilities - things that are likely to happen - but obviously this doesn't include what exists, otherwise it would be actual and not possible. (Methinks this has a lot to do with the probability wave.)
Rovelli's 'mathematical junk' is simply the all the possibilities that can be generated or conceived by a mathematically-literate mind once it exists. I don't think it says anything meaningful about the ontology of number per se. (A saying comes to mind - 'God created the integers - all else is the work of man.')
But then M isn't a possible world, it's an impossible world. Under most analyses, impossible worlds have no ontology (because then you're accepting the existence of a contradictory object). Now I don't think this makes sense since impossible worlds ought to play the same theoretical role possible worlds do to the relevant modal statements, but put that aside.
All that's needed for math platonism is for the objects referred to and quantified over in maths to be real. I think recourse to possible worlds talk is at issue here. Consider we are in a possible world where intuitionistic logic/constructive mathematics obtains. Well, the results in standard maths (in our world) are still provable. That is, if we assume the ZFC set theory and classical logic, the formal derivations will be the same if I work them out in that world, with the same formalisms, as they are when we do them here (if that's confusing, what I'm saying is that the truths of the math hold in the formalism no matter the world I do them in).
So formal truths have a sort of... transcendality? Transcendency? Whatever. They go above and beyond possible worlds, basically. Their "truth" isn't quite the same as vanilla truth statements, and so too is their "necessity" not quite the same; they hold even if the logic of the world is different because formal truths don't involve any world at all. That's kinda what I was alluding to when I mentioned provability is what maths trades in.
Quoting StreetlightX
Well, it's tricky. If we are talking about the truth (provability) of our mathematics then the answer is mathematics is necessary. But if we are talking about our mathematics's applicability to the world then that is contingent because so far as we know, there is no reason to think the structure our universe has is the only possible structure. As an example, our universe has a pseudo-Rimmenian manifold as its geometry. But it seems perfect possible that it could have had a Euclidean geometry or something else entirely. But irrespective of which one the universe does does have, the theorems about those systems are true about those systems. Maybe it's a sort of stratification of the modality. Neither geometry is made true or false based on what geometry our world happens to exhibit.
I think the "what is meant by interesting" is more about whatever math happens to hold at a world than about the platonism question. (Sorry if I'm taking this off track, I am trying to answer you, lol) I think I'll go read the paper because I'm probably mucking this up by not having done so.
The problem I see with this is that if a mathematical "object", say the number five, has no existence apart from its concrete representations, then it cannot qualify as an object at all, except in the most abstract conceptual sense ( and it is already obvious that mathematical objects are real for us in this sense), since its representations are potentially infinite in number.
(By the way, by "concrete representations" do you mean the visual or verbal symbols that represent the number, or the actual instantiations of the number in groups of objects, or sounds and so on)?
I think it is better to think of a "mathematical object" as a way of thinking or speaking, so the sameness consists in the human action. It's like, for example, traveling by train from one station to another; the journey is both always the same and yet different every time, just as each instantiation or representation of fiveness is. There is no perfect form of fiveness, just as there is no perfect form of the train journey. The sameness in both cases is the result of the human process of abstraction.
Spaces are indeed traditionally studied mathematical objects and time is treated in theory of relativity as a special kind of spatial dimension, completing a more complex mathematical object - spacetime. Now, this spacetime in which we live is not empty/uniform but it contains additional structure, like embroidery in a sheet of cloth, in the form of objects extended in space and time (which we perceive as extended in space and moving through space, changing and enduring in time), and there are also certain regularities in these extensions that we call laws of physics, laws that govern how spatially extended objects move and change in time and in relation to each other, how they interact with each other, and how these attractive and repulsive interactions define their boundaries. We may use concepts like "force" or "energy" to describe these movements, changes and interactions. Force is the product of an object's acceleration and mass, where mass is a quantity related to spacetime curvature in the place where the object is located. Energy is an object's ability to exert force over space, that is, a quantity that determines how the object accelerates another object in interaction.
So spacetime with its complex structure seems to be a specific mathematical object. One of all possible mathematical objects, and one whose structure allows the existence of what we call living conscious objects - like us.
Rather than imagining counting train journeys, what about counting holes in a sphere. There is something perfect and absolute about the distinction between a sphere and a torus. Then you can keep on adding more holes.
So sure, some objects - like train journeys - seem pretty arbitrary. But then maths does arrive at cosmically general objects when every arbitrary geometric particular has been generalised away, leaving only the necessity of a pure topological constraint.
Is it really true to say "all math is quite logical"? Within mathematics in general, there are numerous contradictions such as Euclidean vs. non-Euclidean geometry, imaginary numbers vs. traditional use of negative integers. You might argue that it is just different branches of mathematics which employ different axioms, but if one discipline (mathematics) employs contradictory premises, can it be true to say that this is logical?
If you mean truths that hold in different possible worlds, then these truths constitute a more general/more abstract/higher-order possible world.
Pardon me for barging in.
The point is, there are more than correlations between mathematics and nature; as Galileo said, and surely this is a Platonist sentiment, 'the book of nature is written in mathematics' (and as is well-known, Galileo was indebted to the revival of Platonism in the Italian Renaissance).
Nowadays there is an overwhelming urge to 'relativise' the whole matter, to say that number is something internal to or peculiar to humans - which is pretty well the impulse behind the Rovelli paper too. But thought has to conform to maths, not vice versa. I reckon the whole problem is, you can't fit this into the procrustean bed of neo-Darwinist epistemology. That's why it's such a subversive idea. It is supposed to be one of the things that died along with God. :smile:
'Concrete representations' are written symbols or representations in any material form.
Quoting Janus
The point is, it is precisely mathematics (etc) that is perfect in a way that empirical objects cannot be. X is always X (where X is a whole number); there's nothing else it can be. (Have a look at the paragraph on ontology in this Wiki article.)
BTW, Frege and Russell tried to derive mathematics from logic, and failed, for reasons later articulated by Godel (to my knowledge.)
Abstract objects and their particular representations are inseparable. There cannot be one without the other. Representations cannot exist without that which they represent, and that which is represented cannot exist without its representations. If there is no number 5 then there are no 5 objects.
Quoting Janus
The sameness is also a fact about the external world.
Sure. Relativity falls out of the greater symmetry that results from switching from a distance preserving metric to an interval preserving metric. We are now talking about a world of objects with both a location and duration to be specified. Euclidean space was just too simple to stand as a model of physical reality. Lorentzian spacetime becomes the least number of symmetries we can get away with.
But even then, with general relativity, things are still too simple. We must tack on a tensor field to specify some energy density at every point in this spacetime. We have to tell Lorentzian spacetime how it should actually curve. A literally material constraint must be glued to the floppy Lorentzian fabric to give it a gravitational structure. And even then, the quantum of action - how G scales the interaction between the energy density and the spatiotemporal curvature - remains to be accounted for. This constant could have a purely mathematical explanation, but that is the big question for frontier physics. It might also be in some sense a pure accident of nature.
So the way that maths applies to physics is a complicated story. Mathematical symmetries do tell you about the zoo of possible constraints on any physical freedoms. But maths - being traditionally founded in spatial conceptions - may then tend to canonise the symmetries that are just to simple to be real. Constraints, in themselves, may be of irreducible complexity. And so the maths that really counts remains hidden from the conventional gaze as a result.
This is the case with Rovelli’s mathematical junkyard. Once you completely deconstruct maths so that it becomes just a flat and infinite syntactical machine, then it is going to spit out endless meaningless patterns. As Einstein said, the trick is to be as simple as possible, but not too simple.
And here is where Peirce, Aristotle, and other systems or hierarchical thinkers have got it sussed. They accept the irreduible triadicity of nature, where what exists is due to the fundamental reality of the possible, the actual and the necessary. Or the potential, substantial and final.
Rovelli’s relational interpretation of quantum theory and emergent approach to quantum gravity in fact have just this triadic character. So as a systems thinker, that is why he would latch on to the way that the overly simple conventional view winds up producing a world of unconstrained junk.
Should the mechanical view of reality be called Platonism? I think not. But Plato wasn’t a hierarchy theorist like Aristotle. So while he was groping in that direction with his positioning of the idea of the Good as the top of the pile constraint, and also with his talk of the chora as a complementary material principle, a fully triadic story was not cashed out. Platonism did get stuck in a dualism of opposed existences rather than united by a trichotomy of emergence.
Not sure if you missed my reply:
https://thephilosophyforum.com/discussion/comment/219667
That's not what I mean (depending on what you mean by "hold"). Take this. Take any coherent math system and it's results (the theorems you derive) will not change no matter the possible world. I don't mean anything about those worlds remains fixed, I mean the formal system itself, no matter the system, doesn't have change depending on the world one is in or on the mathematical structure of the world one is in. E.g. the truths of Euclidean geometry are "true" even in a world that is non-Euclidean (true in the system, not about the world).
I think you can call this a sort of higher-order necessity, but recourse to possible worlds semantics is superfluous I think since these don't need those to explain their necessity the way other modal statements do.
Energy density is a quantity (number) that is related via Einstein's mathematical equation to spacetime curvature. Pure mathematics.
Quoting apokrisis
Maybe G can be derived from some general principles and maybe its value is specific just to the spacetime in which we happen to live and may have different values in other spacetimes.
Is it just that? The claim would be that it is some quantity of something. So the structuralism of the maths still leaves open the question of how to understand the material part of reality’s equation.
Quoting litewave
I would expect it can. This is strongly suggested by the fact that the Planck scale is defined by a triadic system of constants. You have an irreducible triad of dimensionless constants in c, G and h. And the whole point of a theory of quantum gravity would be to unite all three in a single theory describing a single emergent geometry.
So if hierarchical organisation is the maths of existence - the Aristotelian metaphysical picture - then mathematical physics has arrived right at that very conclusion. That is the reality that a successful combination of quantum field theory and general relativity would reveal.
Again, Rovelli is right about mathematical junk. Much of maths is the result of mere syntactical complication - a mechanistic spewing that is just too simple to model a physical reality. But where maths models actual complexity - a hierarchical view of structure and development - then ontic structural realism, as the new metaphysics, is the right way to go.
No, Rovelli's 'M' explicitly excludes contradiction: "Then the platonic world M is the ensemble of all theorems that follow from all (non contradictory) choices of axioms": It contains everything that is true under any choice of non-contradictory axioms (so yes, read the paper!).
Quoting MindForged
Again, this isn't what Rovelli's paper is about - nor do I think it ought to be about. The question is explicitly about the independence of math from our intellectual activity. Rovelli - rightly, imo - does not say anything about what is or is not 'real', partly, I suspect, because the question of 'the 'real' causes more muddles than it solves. For my part, the metaphysical prejudice that equates the real with the Platonic is, I think horribly misguided, and simply bad philosophy through and through. I'm perfectly happy to accept the reality of math, with the caveat that what counts as 'real' needs to be rethought wholesale.
Quoting MindForged
But this is just tautological: theorems are by definition true (as distinct from hypotheses). You can't milk necessity out of analyticity. Or at least, you can't milk any non-trivial necessity out of it. The question of modality turns on something like: could math be otherwise? And again, the answer is a qualified yes - given what the world is, and how we utilize math for certain purposes, no, the math looks exactly as it 'should'. But without those constraints, in principle, math could well be - again, in principle - a whole bunch of junk: pure meaningless syntax unconstrained by the necessity provided by the world in which we live (which is to say: inseparable and thus not independent from it: this is the sense of contingency Rovelli is employing).
I think 'our intellectual activity' is another abstraction. It's along the lines of a set. I'm pointing to the difficulty in finding a vantage point on abstraction.
I saw your reply. It looked too confused to be worthy of a comment. Here it is:
Quoting litewave
First, I see no definition of "object". Second, you say "axioms are properties of an object". Third, opposing axioms may describe different objects. Why this is totally confused is that you have no principle to differentiate one object from another object because you have no definition of "object". So, whenever opposing axioms are used, one might simply claim that they refer to properties of different objects. And mathematics might be composed of an endless number of inconsistent and opposing axioms each describing a different object, while each object is consistently defined by its one and only, and independent, axiom. In other words, we could make up an endless number of random axioms, each describing a different object, therefore mathematics would consist of an endless supply of random objects, each with its own axiom.
Now, let's get logical. Within logic we have subjects. You cannot attribute to the same subject, opposing predicates, without contradiction. Mathematics is a subject, so we cannot attribute to mathematics, opposing hypotheses, without contradiction.
Maybe we could say that energy density is a mapping (defined by Einstein's field equation) from spacetime curvature to real numbers at a given point in spacetime? This mapping is in turn mathematically related to acceleration that this point can impart to another point during an interaction as the energy does work.
Object is something that has properties.Quoting Metaphysician Undercover
Yes. That's the most general idea of mathematics.
Quoting Metaphysician Undercover
Is there any difference between object and subject?
Quoting Metaphysician Undercover
We don't attribute opposing axioms to the whole mathematical world, only to its parts (objects in the mathematical world). For example, zero curvature of space does not hold in the whole mathematical world but only in Euclidean spaces. And non-zero curvature of space does not hold in the whole mathematical world but only in non-Euclidean spaces.
Something in your hand or room right now differs from your names for it, and ideations about them. Those are abstract, the thing in your perceptual field is not, and they are about it. Math differs in this though, and has deeper roots. Not as easy to hold in one's hands, to see in one's perceptual field, so far harder to imagine in a mind, or idea independent way, when you have it always at hand with the things in your vicinity.
I think that it is deeper level abstraction from the names themselves, and focus on the grammar or form of language itself, abstracting a step further, and removing the content entirely, and just hijacking the pure form from language and grammar and using it to talk far more precisely, recordably, trackably, and directly about the world.
So that, to speak of mathematical Platonic forms is to say that not only the names or ideas of how they relate, repeat, and can be talked about isn't wholly invented, but the form of language mirrors the form of the world so that language already is grounded in deep mathematical order that is then abstracted from language, and then reapplied in its pure form.
The problem of find with this though, is that it isn't wrong, but it isn't truer than the content of experience, which is where all of the quality is. It is the form of everything, but the dead form of everything, so that language involves the physical likeness, and qualitative likeness, and both have a universal abstractable nature. One into mathematics, and the other aesthetics. The Platonic view is then that these features, are real and independent and not projections by us. What kind of magical dimensions within which they reside, I cannot say, but I do think that language is abstracted in both parts from reality itself, and are real and mind independent.
Well then he's not talking about the trivial world, which is the world where everything is true. He is thus, in fact, making the exact baseline assumption that all mathematicians make: No mathematics that is interesting can entail triviality. The trivial world is the quintessential impossible world, because it's the world where all the contradictions are true. I will read the paper though!
The author comes down firmly on the side of Einstein on the grounds of his scientific realism, scorning the Copenhagen Interpretation as 'obscurantism'.
The title of this book: 'What is Real?', by Adam Becker.
So maybe it's correct to say that this is a problematical question. But it remains an open question, and one that has been exacerbated, not resolved, by the very physics of which Rovelli is a foremost expert. And as this is a philosophy forum, the question of ‘what is real’ and the sense in which numbers are real, remains a valid question, and an open one, in my view. (We probably wouldn't even be able to have the debate on Physics Forum, from my experience there.)
Furthermore, at the heart of the thirty-year debate between Bohr and Einstein was the argument about whether there are mind-independent objects - not numbers, but actual stuff. And, contrary to what Becker says, I think scientific realism lost that argument, although that is obviously one of the vexed questions of modern physics. But, be that as it may, one thing that everyone in the debate assumes, is that they're all dealing with the same measurements and observations; and all of those are measured and described in the common language of mathematics. It's precisely because of its independence that math is the language of physics - whether it is utilised by the inhabitants of Jupiter or Earth, I would hope.
So, I'm not convinced by Rovelli's argument; in any case, his conclusion that mathematical Platonism says that mathematics is 'fully independent' is not at all the case. Here, you're seeing the assumption that 'what is real' must, by definition, be mind-independent being smuggled into the argument. The whole question of what 'independence' means, and the relationship between the knower and the known, is a very deep one, and one that has been by no means resolved in either philosophy or science, as far as I'm concerned.
And besides - just what does ‘independent’ mean in this context? Frege, a mathematical Platonist, said that:
’Frege on Knowing the Third Realm', Tyler Burge.
I personally believe that is actually a very modest claim. The only problem with it is, that the objects it is talking about, namely, numbers and logical rules, are not actually physical; so that poses an obvious problem for physicalism, which insists that only what is physical can be considered real; and that is the only point at issue in all of this.
Well yes, but thats not was ever under consideration and is, if I may, an artifact of you not yet having read the paper.
Either you haven't read the paper you actually qualify as a clinical imbecile. Rovelli's take on MP is not a conclusion he reaches - it is literally the first line of the paper where it is given as a premise against which the argument unfolds. A premise, moreover, supplied with two citations to Penrose and Connes. If emblazoning the first line of your paper with an explicit definition together with citations counts as 'smuggling' its no wonder your entire post is an exercise in obfuscation, invoking irrelevant debates about QM along with blurbs of books you show no evidence of even having read. Muddy the waters elsewhere you intellectual cretin.
I can tell you that this is the problem right here. You have absolutely no restrictions on "object". This is what I referred to, any random thing may be an object, because properties are what we, as human beings determine and assign. So what exists as "an object" is completely arbitrary, and dependent only on the way that human beings assign properties. if someone assigns properties, there is an object there. There is no principle of unity here, nor is there a principle of identity, whereby "an object" might be an individual, particular thing.
Quoting litewave
The difference between an object and a subject is found in the way that the law of identity is applied. An object is something we point to, and we identify it that way. It need not have any definite properties, so long as we can identify it as something we can point to, "what it is" may remain indefinite. Therefore an object may be identified, even named, without having any properties assigned to it. A subject is identified through a description, as having specific properties, it is identified by "what it is". As a tool of logic, this allows that numerous different objects may be identified as "the same" subject, when the differences between them are deemed as accidental. So an object is identified as something individual, particular, and unique, while a subject is identified as something specific. One is a particular, the other a universal.
Quoting litewave
Yes, that's exactly the problem I referred to. Mathematics, as a subject, is allowed to have opposing and contradictory predications. You justify this by claiming that the contradictory descriptions describe properties of different objects. However, you cannot point to the objects, to say that this is the property of this object, and that is the property of that object, because your so-called "objects" exist only by specification; this axiom indicates the existence of this object, and that axiom indicates the existence of that object. So these so-called "objects" are really subjects. And, they exist as subdivisions of the original subject, mathematics. It is irrational and illogical to allow for contradiction within the subdivisions of one subject.
Here's an example. Suppose that natural science is specified as one subject, with subdivisions specified as biology and physics. We could say that biology and physics are distinct subjects within the subject of natural science, just like Euclidean space and non-Euclidean space are distinct subjects within the subject mathematics. However, we cannot allow that biology and physics proceed from contradictory axioms, because this would signify incoherency within the subject of natural science. Likewise, the use of both Euclidean and non-Euclidean geometry signifies incoherency within the subject of mathematics.
No, an object has properties even if no one assigns them to it. Planet Earth is round no matter whether someone assigns roundness to it. It was also round before anyone believed it was round, or before anyone even existed.
Quoting Metaphysician Undercover
Huh? An object is a particular and a subject is a universal? Where did you get this terminology?
Quoting Metaphysician Undercover
But biology is part of physics; properties of biological objects are physical properties. Curved space is not part of flat space and flat space is not part of curved space.
This is over the top and uncalled for. From where I stand, Wayfarer appears to have made some good and relevant points (which I was planning to comment on). But even is I'm wrong about that, your response still is uncalled for. Moderator, moderate yourself!
This to say nothing of the usual modus operandi of citing blurbs (that is, second hand, twice-removed, assertions that themselves contain no argument) that have zip all to do with the topic at hand - Einstein and Bohr?? - all the while spending the bulk of the post talking about an entirely different topic altogether - 'scientific' claims about 'objects' - and referencing nothing, not a jot, of argument from the paper under discussion. And then to conclude, after this tangle of complete irrelevancy that "So, I'm not convinced by Rovelli's argument" - its a morass of obfuscatory sophistry, and simply the latest in a long line of it. By all means comment on it; it will dilute the muck.
I read the Rovelli paper, and also the book that I mentioned. Rovelli's concluding paragraph in the paper is as follows:
I am question that the idea that mathematical Platonism insists on a 'fully independent Platonic world'. And I also think his motivation for the paper is because of the association of Platonism with Christian Platonism, which tends to oppose the current neo-Darwinian orthodoxy which dominates the secular academy.
It's impossible for me to discuss anything of any depth with StreetlightX without his flying into hysterical invective and insults. I am called an imbecile, cretin, imposter, and disseminator of intellectual poison. I will let others decide on why that might be motivating that, however I think it is driven by what Nagel describes in his essay Evolutionary Naturalism and the Fear of Religion.
Since it requires someone to determine what "round" means, and whether the object referred to as Planet Earth fulfills those conditions, it is impossible that what you say is true. I conclude that you believe the word "round" existed before anyone existed, because this is what is required for the earth to have been determined as round, before anyone existed. Do you not recognize that whether or not an object has a specific property is a judgement, and nothing else?
Quoting litewave
OED: subject: 1a. A matter, theme, etc., to be discussed, described, represented, dealt with, etc.. object: 1. a material thing that can be seen or touched.
If what I said does not make sense to you, then you could perhaps explain why.
Quoting litewave
Have you ever been in a university before? Biology is not part of physics.
What you say is nonsense. If there is something referred to as "space", which has properties, then it is an object according to your own definition of object. It cannot be both curved and flat because this is contradictory.
I'm not clear how this relates to the point about train journeys. The point was just that there is no single real (as opposed to conceptual or abstract) object, 'train journey' of which all train journey are representations or instantiations, and I am drawing an analogy between this and 'five'.
Quoting apokrisis
I agree, but then this generality is an abstraction not an ontologically robust object. To be sure, it is a conceptual object for us, but to imagine it has some existence independently of us is, to quote Whitehead to commit "a fallacy of misplaced concreteness".
I'm no expert on the subject, or even a (good) philosopher, but I tend to agree with Pneumenon here.
The author appears to argue that 'Mathematical Platonism...the view that mathematical reality exists by itself, independently from our own intellectual activities' is false, and it is false because mathematics is dependent on our own intellectual activities.
No doubt we use mathematics to help (I don't see any reason to believe that mathematics is absolutely essential to the task) decipher the "book of nature"; but it does not follow from this that nature is somehow in itself reducible to mathematics. Obviously nature carries the possibility of math, otherwise there would be no math. For math to manifest,(although apparently some animals can count in rudimentary ways) the evolution of beings like us, who have developed the capacity for symbolic thought would seem to be necessary.
Quoting Wayfarer
Honestly I think you should stick to trying to unpack the philosophical issues and refrain from indulging in tendentious and irrelevant social commentary.
Quoting Wayfarer
OK, so just to be clear you would consider five apples and the numeral '5' to be representations of an actual (something more than merely conceptual) object? For me the first is an instantiation and the second a representation of a characteristic or quality: namely fiveness. I don't believe the quality or characteristic of fiveness qualifies and an object except in the sense that it can be a conceptual object for us. I can't see any other coherent sense in which it could be said to be an object.
Perhaps, but do abstract objects exist independently of their being thought, and if so, how would that "existence" look?
A train journey is a substantial act. So it is hylomorphically intantiated. My point is that we can unpack this in a general fashion by abstracting away both the formal and material principle involved. We can separate the formal necessities or constraints from the material accidents or fluctuations.
Topology seems to demonstrate this metaphysical principle in action. A compact surface is the constraint placed on the most generalised system we can imagine. We get a sphere. But then as a degree of freedom or accident that can’t be suppressed, the sphere could be punctured by holes. And so you get a primal model of countability that allows you to see what is really going on in something complicated, like the idea of countable train trips.
Train trips are a bad example because they start with a human mechanical imposition of a mathematical framework on a natural landscape. We are making it the case that there are some countable set of trips by some definition we all agree.
The argument here is then over the reality of mathematical structures themselves. And to follow what that argument would be - from my own hylomorphic and constraints based view - I would want to start with a clear mental picture of what the maths might actually be claiming.
So my claim was that the abstract object in question would be this kind of topological constraint that then still results in localised definite accidents. Limits turn out to be limited in this essential fashion. You can break a symmetry, but that symmetry breaking can in turn be broken by the arrival at some new terminating symmetry.
This is the physics we have discovered. It is why we wind up with the translational and rotational symmetries which all a cosmos of countable localised actions to exist. Constrain action to some spacetime point and it can still move or spin with inertial freedom.
So maths speaks deeply to the reality we observe. I just thought that counting train journeys was a misleading example because it is in no way a fundamental, or even natural, notion of a countable object. It is an action we mechanically impose on a landscape and so depends on our willingness to be indifferent about the acts which in the end do count.
If the train stopped only halfway into the station, we would all sit around debating if that not quite completed trip should still count. We would still be arguing about accidents vs necessities. But the answers wouldn’t carry much cosmological weight.
Bear in mind that I was originally talking about a train journey between two specific destinations. So, the train would need to pass through both those destinations, or begin at one and pass through the other, or begin at one and terminate at the other, to count as a train journey from one destination to the other. So, the question of whether we should think the journey completed doesn't seem relevant.
The analogy between the train journey and fiveness is not perfect, to be sure, since the train journey is an actual process or activity, whereas fiveness is a quality or characteristic. But both can coherently be understood to be abstract objects for us; and the question remains as to whether they could coherently be thought to be objects in any sense other than the abstract.
Quoting apokrisis
I can't see how mathematics is, in itself and independently of any philosophical interpretation of its significance, claiming anything at all. And even if mathematics were claiming something in itself, I can't see how this would speak to "the reality of its structures". For me the very notion of "the reality of mathematical structures" beyond their being abstractions from concrete objects and processes, seems unintelligible. Mathematical "reality" seems to consist far more in possibility than it does in actuality, but then admittedly I am not much of a mathematician. I am open to explanations that I can make sense of, though.
Yes, I think they exist independently of being thought because they are properties that different objects have in common; they are ways in which different objects are similar; they are conditions that different objects satisfy if they are of a particular kind. But I don't think we can visualize abstract objects, because they are not objects in space. (We may be able to visualize their particular instances or a typical instance.) But apparently we can inductively infer their existence by noticing in what ways different things are similar.
I agree that they are properties, and that properties as conceived can be understood to be conceptual or abstract objects. The problem I have is that I can't see any coherent way to think that properties are abstract objects in any sense other than their being thought as such.
Another answer could be: in the collection of all possible objects, which are related by the relations of similarity, composition and instantiation. Some of those objects form topological or metric spaces, including the one in which we live.
I'm not trying to imagine where they exist, but rather what kind of existence they could be said to have. For example fiveness is undoubtedly a characteristic or quality of many groups of things, and it is undoubtedly an abstract object of thought. So it has concrete existence as a quality and ideal existence as both a quality and an object. But it does not have concrete existence as an object. What more can we say about its existence? I can't see the point in saying that it has some further kind of concrete or objective existence if no account or explanation of that purported existence can be given.
Sure. Two stakes were stuck in the ground. And so suddenly the landscape had your chosen metric imposed on it.
Quoting Janus
Yeah. But what I was arguing is that your notion of material concreteness is itself just a matching abstraction.
So sure, there must be a material aspect to reality. And you are now insisting to me about the reality of that abstracted notion. If the fallacy of misplaced concreteness is a thing, it would have to apply to your claims about however you picture this idea of definite local particulars.
This is why Peirce would grant reality to both the formal and material aspects of nature. As generalisations, they are each "concrete" - or just as concrete as each other in terms of both being essential aspects of the whole.
In a holistic metaphysics, substance is emergent. It becomes localise individuation. But you sound as if you want to treat emergent individuation as the concrete baseline reality of existence. You begin with a world of objects, rather than arrive at that world.
Quoting Janus
If we wind it back, I wasn't defending some kind of spooky seperate existence of Platonic structure. I was in fact arguing that forms are always instantiated - or would have to be intelligibly instantiable. So the kinds of structures that could exist are the kinds of structures that could dovetail with some kind of logically complementary material principle. They would have to be able to yield substantial being in interaction with that material principle.
This then leads to the question of how to conceive of that material principle in properly generic form. This would lead us towards Peirce's answer - vagueness of Firstness. Or more classically, Apeiron or Chora. Or in some modern physicalist sense, chaos or fluctuation or quantum foam. That is, a potential that is lacking in limits, but capable of being limited.
So essentially my point is that the maths that is powerful and useful when it comes to the metaphysics of possible cosmologies is the kind of maths which has this particular character. It can model the constraint of freedoms, the limitation of uncertainty, the emergence of stable habit or law.
And what is exciting is that maths could model both the formal constraints - by speaking to the necessity of certain such structures - and even the material accidents, the constants of nature that then ground that structure. These constants may turn out to be shapes - like the holes in a topological sphere. As I said, global symmetry-breaking is terminated by reaching local symmetries which it can't erase. That is why you have the particle zoo of the Standard Model. A quark or electron exists as fundamental - a fundamental excitation - because they can't be broken down any further. They put a stop to the symmetry-breaking cascade and now start to ground the construction of some kind of material content in the Universe.
That is what string theory is about. Topological irreducibility. If you curl up a higher dimensional space, you can't in the end get rid of all the kinks. You are left with some countable number of holes that then become the material character grounding the Universe. They are the knots that can't be undone.
So the material principle could be reducible to ontological structuralism - becoming the local kinks that can't in the end be rotated or translated out of existence. Matter would be part of Plato's realm, but exist in it apophatically, as the topological holes or features that can't be erased. The material part of being would be the inverse of the formal part of being.
So it was this organic conception of structure - the "co-arising holism" that physics is uncovering - that I'm contrasting to the mechanical conception of pattern generation which Rovelli is using to produce a landscape of mathematical junk.
Plato was speaking to that dawning metaphysical realisation that the intelligibility of reality is about a division of the substantial into the complementary things of the formal and material principle. Aristotle might have said it much more clearly, but the dim outlines of that emergent hierarchical view can be seen in Plato - as when he talks of The Good as a finality which acts to select certain forms, and the Chora as the need for some kind of material receptacle where structure could be instantiated.
Just consider the Platonic solids. In 2D, polygons can have any number of sides, as long as they have at least three. But in 3D, suddenly that adds a huge global constraint that limits local regularity to just 3x2 possibilities - the self-dual tetrahedron (4 triangular faces), the dual cube and octahedron (swapping faces for vertices), and the dual dodecahedron and icosahedron. So place a limit on dimensionality and only a limited number of perfectly exact resonances can fit that space.
The Platonic solids are examples of how local symmetry can become physically manifest if global symmetry is explicitly broken. And of course this mathematical realisation - this intelligible fact of any possible reality - was then used to give a Platonic account of material atomism. If material fluctuation was in fact bound by formal limitation, then these had to be the shapes that would emerge at the end of the trail. Atoms would be little triangles, and so be fiery, etc. (Of course, a sphere was the other emergent perfect shape - the one that then emerges at the infinite limit of "polygonicity".)
So yes, if maths abstracts and generalises, then of course it is stepping back towards the possible, and away from the actual or substantial.
But there are then two ways of stepping back towards generality. And hylomorphism would be about following both those paths - and being able to see the unity in the fact that they are a pair of reciprocally defined paths. Each is the other's inverse. And so the metaphysical formalisation of the description of the one can apophatically stand as the formalised description of its "other". Yin and Yang. Accident and necessity. Matter and spacetime.
It is that deep structural trick that would see Platonism - as understood charitably - being cashed out by a modern physical "theory of everything". If the material constants can be shown to be the irreducible holes produced at the limit of some process of constraint, some process of symmetry breaking, then reality would "pop out" of an intelligible mathematical description.
Again, this is the big prize that Rovelli himself is pursuing. So all his paper demonstrates is the paucity of a more conventional view of mathematics (and thence reality) as the infinite noodlings of mechanistic pattern generators.
If you want to call that "Platonism", I suppose you could. But Rovelli also wrote a book on Anaximander which showed him to be rather a lightweight on Ancient Greek metaphysics. I would rate him highly for his physical speculations, poorly for his history of philosophy.
I'm baffled by your reply. What else did you think I said? And where yet did you say anything useful about the nature of this "energy density" which you have to go off and measure?
Sure you can quantify it as an act of measurement. But that still leaves "reality" as a number being read off a dial.
So we face a big choice at that point. Either we go with the usual naive realist view - reality is whatever we think it is that we are measuring. The phenomenological is mistaken for the noumenal. Or we instead make a virtue out the very fact that pragmatism and semiotics lies at the core of all this.
The noumenal becomes the fundamentally arbitrary or vague in our metaphysical picture. We conceive of the material principle as a state of radical indeterminism - like a quantum foam. And then structure is that formal principle which can constrain this indeterminism so that it forms an emergent state of order - like a classical realm of deterministic objects.
Again, you are talking about GR. And we know from QFT that spacetime would be material enough to be populated by an infinity of gravitational self-interaction. So nothing self-stable is specified at the level of GR modelling. Einstein's field equations had to include a cosmological constant just to prevent even a homogenous spacetime from immediately shrinking out of existence due to the smallest material fluctuation.
So you have to glue together some model of global spacetime symmetries, some model of an actual material content, plus a generic material fudge factor to keep the whole fabric expanding into a future, to get to a GR description that still needs to be fixed by constraints on is material self-interactions.
It is all kind of Heath Robinson. And yet, each of these components is well-motivated in terms the general principles they express - the need to satisfy that happy triad of constraints, the principles of locality, least action and cosmology. Through the glass darkly, the maths is expressing a holistic causal structure now. We are arriving at a Platonistic view of that kind - if not the other kind, the one that wants to conceive of nature as mere mechanistic construction.
So?
The argument straightforwardly conflates mathematical objects with mathematical practices developed using, or developed to describe, those objects. No one doubts that the mathematical practices of organisms are fluid, but that's not relevant to the Platonist's claim.
I said that energy is mathematically related to the acceleration that the space point imparts to another space point during an interaction. It means that energy can do work.
You said some posts back that energy and dynamics cannot be explained only by math; that we also need a "material principle". I don't understand what you mean by the material principle, and I just outlined how energy and dynamics (acceleration during interaction) can be explained by mathematical relations and how they are related to spacetime curvature, a geometric property of the mathematical object called spacetime.
I'm not sure if you meant to phrase it how you did, but that... would be a perfectly valid argument ('it is false that the tree is blue because the tree is green - and here is why'). That said, that isn't the argument of the paper.
Quoting Snakes Alive
It's only a 'conflation' if one assumes from the outset the Platonic position on mathematical objects. The point of the paper is to ask how tenable just such a distinction is, by setting out a disjunction ('dilemma'), the choices between which are claimed to put the Platonist in an untenable bind.
Except I found the author to be saying that the tree is not blue, and he did not tell us why. The author appears only to assert, or to assume the truth, that Mathematical Platonism is false. He could have written a shorter paper with the assertion that 'Mathematical Platonism is false'. But I'll take another look regarding the dilemma you mention above.
Mathematical Platonism is either true, or it is not. Why would utility help us answer that question?
It doesn't, anymore than utility helps us decide if mind-independent rocks and such exist.
On the contrary, the problem with such a question is that most moderns are nominalists, rather than realists; they treat reality as coextensive with existence. The "objective realm" of reality is not limited to that which exists in time and space.
Quoting Janus
In this context, an object is whatever is capable of being represented. Hence qualities are objects just as much as the things that embody them, and habits (including laws of nature) are objects just as much as the events that they govern. Some of these possibilities and (conditional) necessities are real - i.e., their characters are not dependent on what anyone thinks about them - even though they do not exist apart from their instantiations.
Quoting StreetlightX
But "independent of our intellectual activity" is precisely what "real" means, assuming that "our" refers to any individual person or finite collection of people. The muddle comes from conflating reality with actuality/existence.
Again, mathematics is the science of reasoning necessarily about hypothetical states of affairs. Euclidean and non-Euclidean geometry employ exactly the same (deductive) logic, but draw different conclusions because they begin with different premises; specifically, non-Euclidean geometry adopts one fewer postulate. Imaginary numbers are the perfectly logical result of defining "i" as the square root of -1, regardless of whether this corresponds to something actual.
Quoting Metaphysician Undercover
Mathematics in itself does not require the adoption of a particular set of hypotheses; it simply derives necessary conclusions from any set of hypotheses whatsoever - including, in some cases, the conclusion that those hypotheses are contradictory. Euclidean and non-Euclidean geometry are different subjects with different hypotheses. Algebra with imaginary numbers and algebra without imaginary numbers are different subjects with different hypotheses.
Quoting Metaphysician Undercover
Nonsense. That which the word "round" signifies - the real character of roundness - existed in everything that possessed it before any human being existed, and would continue to exist in everything that possessed it after every human being ceased to exist. Do you not recognize that some judgments are true and others are false? This entails that there is a fact of the matter, which is independent of whatever anyone thinks about it. Any argument to the contrary is self-refuting.
I think one thing that your analysis is missing is the understanding of what actually constitutes knowledge in Plato's philosophy. As is often said, Plato sets the bar for what constitutes ‘true knowledge’ very high. The gist is that empirical knowledge itself, sensory knowledge, can’t be considered knowledge of something real, because the senses lie, and because the real nature of ordinary objects of perception is such that they are a combination of being and non-being.
Whereas when logical and mathematical truths are known, they are known in a way that is not possible with respect to sensibles, almost in the sense that the mind unites with the object of knowledge. That is where Platonism verges dualism, although it is often implicit. It was the use that this understanding was put to, that gave the real power to Western science, but the original impetus behind the understanding was soteriological rather than utilitarian.
Quoting apokrisis
That thinking is Darwinian, I'm sure it has nothing to do with Plato. The Good doesn't do anything. The Good is not the demiurge.
Quoting apokrisis
Likewise, this is the modern understanding - not that it's wrong on that account. But one point that Aristotle makes, somewhere - I've never been able to find it again - is that metaphysics serves no useful purpose, it is not 'for' anything. Just to be able to get an understanding of it, is reward enough, the sole justification for understanding it. That is the intent of such sayings of his as 'thought thinking thought' (although I think 'thought' is a pretty lackluster word for what he is trying to convey here.)
Quoting aletheist
That's what I meant to say. Although I think 'transcendentals' can be distinguished from 'objective reality'.
Of course my notion of it is an abstraction, but material concreteness is experienced. If it weren't we would have no way of differentiating between the concrete and the abstract in the first place.
Well, I suppose it depends on your definition of object. I do agree that qualities have an objective (meaning mind-independent in the sense that they are not constructed by the mind) existence.
‘Exist’ - that is the whole issue here. Again - does the number seven exist? If you point to a symbol, then, sure, that exists. But the number itself is something that can only be grasped by an intelligence capable of counting. But it’s real regardless - hence the distinction between ‘real’ and ‘existent’.
Indeed, the paper stands or falls on whether the world M is a fair interpretation of mathematical realism. The process then becomes our selection fo the interesting bits of M.
If something counts as an object in the sense that @Alethiest stipulates, then it certainly exists. If qualities exist, or subsist, only in their instantiations, then they are existent. My argument is only that there is no coherent sense in which we can say that they have an existence, or being or that they are real, whatever locution you prefer, beyond their instantiations and representations. To say they have is to commit the dreaded "fallacy of misplaced concreteness" and to descend into obscurantism and incoherency.
Sure. That was the surprising new thing. Human thought could be organised by this new kind of logical structure. Maths cashed out a fully constrained, very mechanical, form of pattern generation. And that machine-like deductive and state-mapping approach to causality proved to be "unreasonably effective" at delivering technological control over nature.
So this was a big intellectual shock. The idea of the Machine was revolutionary. But then physical reality isn't in fact mechanical. So the "truth" of this mechanistic ontology is not the truth of the actual world.
Machinery - mathematical machinery - can be "absolutely true" because it is based on deductive proof. You assume some axiom. You derive some consequence. It seems perfectly water-tight. That is, it all degrees of freedom of rigidly suppressed. Nothing surprising can happen to derail the sequence of events. A perfect state of constraint is in effect. The only causality operating is that of linear cause and effect sequences - blind step by step deterministic construction.
So yes, this new machine mode of thinking seemed marvellous to the first logicists and mathematicians of Ancient Greece. For them, it seemed even "divine". It was thought and reason perfected. But it is ironic that you - given the way you rail against Scientism and other continuing cultural expressions of mechanical thinking - should still seem so in awe of mathematical forms. Reality ain't a computer, is it? The Cosmos is better understood as organic. Or better yet semiotic - because semiosis is the ontology which both accepts a mechanical twist to nature, but puts in its rightful place.
What I am saying is that you are still presenting an utterly confused history of the relevance of mathematics. You want it to be some kind of door to a transcendent divine aspect of existence. You respond to the reverential view taken by Pythagoreanism. And yet that exact path - that belief that maths is the royal road to Truth - is what winds up in modern technocratic reductionism. The belief that life, mind and physical reality in general can be accounted for fully and truthfully as mechanism.
Quoting Wayfarer
Bullshit ontology is bullshit ontology, regardless of whether you are claiming god is a geometer or reality is a machine.
I agree with your analysis here. In fact to say that nature is constructed from number, or some such kind of metaphysical claim, as for example Tegmark makes, is a form of reductionism; reduction of the organic to the mechanistic.
Right, so if the "different premises" are contradictory, then it is impossible that they are each true. I think that this is a problem for those who claim Platonic realism concerning mathematics. If mathematical forms are so variable that they are contradictory, then what good are they? You use your forms for your purpose and I use contradictory ones for my purpose, and we each come up with contradictory understandings of the reality which we apply them to.
Quoting aletheist
They are different subjects, as sub-classifications within the subject of mathematics, just like my example, biology and physics are different subjects as sub-classes within the subject of natural science. It is illogical to allow that sub-classes proceed with contradictory premises, as if each of the contradictory premises is true. Simply put, having different subjects which treat contradictory premises as if they are each true, is illogical.
Quoting aletheist
This is very clearly false. Before there was the word "round", there was obviously nothing which the word "round" signifies, because there was no word "round" to signify anything . Therefore it is absolutely impossible that there was "the real character of roundness" before there was the word "round". That there is something which the word "round" signifies is very clearly dependent on the existence of the word "round".
Quoting aletheist
I do not see how this is relevant. That the world is round is a judgement. Whether any such judgement is true or false is irrelevant to the fact that such predications are judgements.
Same old, same old. Semiosis says what you "experience" is your Umwelt. Sure, the "world" must stand in back of that as a noumenal constraint, some kind of recalcitrant actuality that limits the freedom of your interpretation. But then you need to pay closer attention to how conceptions are actually formed as logical dichotomies.
If you rely on a distinction like concrete~abstract, then it is the pairing that is itself the whole of the conception. Again, if you rely on experience~conception as the distinction, it is the whole of that differentiation which is the conception.
So you talk about directly experiencing the substantial material concreteness of the actual real world. But that is still just a conception. The more strongly you believe in that dualised "othering", the more deeply you are actually embedded in the Umwelt you are creating.
So the irony here is that you are strengthening your conceptualised distance from the noumenal thing-in-itself by insisting on the absoluteness of this concrete other. Your belief in the "material" - as you conceive it, conceiving it as absolutely "other" - is what makes it as abstract as it could be in terms of you "experiencing the world".
Now this is not to deny that the world is not out there in recalcitrant fashion. Nor that a conceptual division into the concrete and the abstract is not a pragmatically useful way to structure our understanding of this world. Umwelts have to achieve our lived purposes.
But to say the rock is hard, sometimes painfully hard, is no more "real" than to say a rose smells sweet or the sky is blue.
Science is at least honest in this regard. Material properties become the numbers we can read off dials. The formal aspect of existence becomes a mathematically-expressed theory. The material aspect of existence becomes an appropriately matched act of measurement. Reality is then whatever this pragmatic system of conception tells us it to be.
Quoting Janus
Nope. You want to apply a rigid if/then cause and effect logic to the situation. So for you, it has to be the case that one thing comes before the other thing. That is the habitual materialist Umwelt you are seeking to impose on your experiences of the world.
But I am arguing for an organicist or semiotic causality where the complementary aspects of any fundamental division must co-arise ... as each is the cause of its "other".
So before we could say our minds were divided by their abstract conceptions and their concrete perceptions, our state of experience would just have been an undifferentiated vagueness - the blooming, buzzing confusion of the newborn babe. A distinction between the concrete and the abstract is a structuring that grows. And no surprise really. As time passes, the generalities of the world will make themselves known as seperate from their specificities.
But in nature, the distinction is not some absolute or dualised difference. That is itself a further twist given to it by a modelling human mind. It becomes convenient to conceive of the world in the simplest fashion where the concrete and the abstract exists as actual absolutes, rather than merely as complementary limits to a useful metaphysical distinction.
I think you're confusing yourself by over-thinking this. You're conflating talk about the experience with the actual experience. Try feeling some object in your vicinity right now. You can directly feel its material concreteness, its tangibility; it is from that basic experience that the idea of substantial material concreteness originates.
Quoting apokrisis
This seems totally wrongheaded. The experience comes before the idea about it. Animals experience the tangibility of the world as much as we do; the wind on their faces, the soil under their paws, the physicality of the struggle with the prey. This basic animal experience is what underpins all our notions about the nature of things, the distinctions between abstract and concrete. Animals cannot draw such distinctions since they lack the ability to grasp symbols.
Quoting apokrisis
No, what you experience is the material world; your "Umwelt" is not experienced except upon reflection, it is your experience of the material world.
I'm trying to say something more subtle. I am saying that number - the general thing of construction or composition - is emergent rather than basic. But having emerged as a habitual possibility of nature, it then does become basic to the greater complexity that ensues. Atomism and classicality are pretty much the truth of reality so far as we living forms are concerned, at the physical scale at which we arise.
A machine has a simple causality because it is just the sum of its parts. And nature isn't a machine - but it develops structural complexity by imposing an increasingly mechanistic order on itself.
The early universe was a relativistic gas, a hot featureless bath of radiation doing nothing but cooling and expanding. It was only because constraints emerged to cause mass particles to condense out of this spreading flow that more interesting stuff could happen. You had crunchy little electrons and protons bashing about and interacting at sub-light speed. There was now a localised form of time and action. A history of concrete or discrete events could get going against a backdrop of generalised continuity. It took a while, but a machine-like order clicked into place as the new normal. A bunch of identikit parts with constructive possibilities - individuated properties - could begin to produce a more complex world.
So physical reality as we know it is not founded in the mechanistic. But it wouldn't exist as we know it if it weren't also imbued with the propensity for a mechanistic and hierarchical form of organisation.
Numbers are then a pretty good representation of this developmental view as the very possibility of a number is emergent from the notion of an identity function. If you take a general action to the limit, like addition or multiplication, then a basic unit will emerge as the difference that doesn't make a difference, so terminating the symmetry breaking with a local symmetry. One times anything is still one. Anything plus zero is still zero. So - emergently - a basic limit, an identity function, will wind up grounding some space of functions. The units you need to justify a constructive or compositional (mathematical) reality just pops out as being eventually an atomistic regularity that can't be rotated or translated out of a freely dynamical existence.
So there is a reason why we do see a deep connection between our notions of mechanical construction and reality as it exists. The organicism of symmetry breaking or dichotomisation taken to the limit results in the emergence of fundamental units of action.
So the conception of nature as fundamentally mathematical - an atomistic construction - isn't wrong. It is a pretty good description of how reality is for us as complex creatures living in an era when the Cosmos is so extremely cold and large. But also we now know that this degree of classicality is an emergent fact. It isn't actually fundamental - unless we then go the next step that a complete holism would require, which is to realise that we are still seeking to impose a strict temporal order - a classical before and after - on a process of realisation.
So the mechanistic aspect of nature can be seen as now a finality - a cause acting from an organism's own future. Reality was being called towards the structuration that then did emerge. It was always inevitable that things would arrive there. The future becomes as real as the beginning - although now neither are real in the old privileged sense we want to give them. They have to share that foundational glory rather more equally in our conceptions of nature.
Naive realists are always saying that. They like to under-think the metaphysical complexities.
Quoting Janus
Oh please. I press hard on the desk with my finger. I poke my finger with no sense of resistance through the surrounding air. I then pick up the physics textbook that tells me the solid matter is really a void of excitations, while the airy space is crammed with Newtonian particles exerting a collective pressure and resistance on my being.
Sensations are one level of semiosis - a biological level of meaning making. But an Umwelt is an Umwelt. Psychological science gives us all the evidence we need on that.
So sure, I make a perceptual distinction between what is substantial, what is void. Or what is actual, what is imagined. My psychological modelling of reality is organised by a bunch of useful conceptual dichotomies.
Thus yes, the dichotomies originate in the world in some sense. A characteristic of this era in Cosmic development is that you have solids, liquids, gases and plasmas. Concrete is a term that applies to one of the four "material state" in some everyday, not very philosophical or scientific, fashion.
But when you tell me to demonstrate the reality of "material concreteness" by poking something ... solid ... not liquid, or gas, or plasma ...
Machinery is a metaphor here. Surely the point about the axioms of arithmetic etc is that they're true for no further reason, they are apodictic, not dependent on some other truth. That's one of the things that differentiates them from the contingent domain of phenomena. But then the discovery was that maths and geometry reflected something real and constant in the constant flux of the phenomenal domain. Sure, one of the things that this enabled was machinery - but only one. But reason itself is strictly the relationship between ideas. It doesn't need to be validated with respect to any particular state of affairs - that is why it is associated with a priori truths.
Quoting Janus
So do you think this paragraph in the SEP article on Platonism in the Philosophy of Mathematics means anything?
[quote=SEP]Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects which aren’t part of the causal and spatiotemporal order studied by the physical sciences.[/quote]
and also this:
Quoting aletheist
That is the distinction that interests me.
I do agree that 'object' is a metaphorical description for number (etc). Likewise that 'realms' and 'domains' are metaphorical descriptions, in, for example, 'the domain of natural numbers'. Hence the equivocation between the concrete domain of sensory experience, and the metaphorical domain of meaning and reference. But nevertheless the point remains clear, I hope.
The textbook explanation is parasitic upon the experience of a world of tangible objects. Only a fool would deny that. I'm making no claim about the "ultimacy" of the material world, only about the foundational character of the experience of tangible things.
You love to resort to throwing around labels like "naive realist' when you cannot come up with cogent counterarguments. This is a fairly typical ploy around here. If you paid any attention at all to what I write you should know very well I am no naive realist, so don't insult me with that sub-standard shit.
That quote refers to "objects", but it does not explain how those purported objects qualify as objects beyond their conceptual, abstract dimension. So, it remains vaguely allusory, which is fine for poetry or mysticism, but not for philosophical analysis.
In reference to the second quote; I have already acknowledged that I think number is a quality or character that is real in its instantiations and representations; concretely in the former and abstractly in the latter. To say it is real is the same as to say that the quality or character exists, as far as i am concerned, since I can't see any significant distinction between saying they are real or saying that they exist. The point I have been making is that they are not real (they don't exist) apart from their instantiations and representations (including of course the conceptual), so I am not disagreeing with that passage @alethiest wrote, except perhaps in regard to what I think about the senses of terms.
What I questioned was your notion of "experience of a world". If you accept that experience is an Umwelt - conceptually structured from the get-go, then no problems. But if you can't see that speaking of tangibility is itself an abstraction, then there is a big problem.
Quoting Janus
Like you love to throw around labels like "over-thinking".
Quoting Janus
So in claiming to escape naive realism, how are you managing to escape naive idealism here?
You are asking me to go out and feel the tangible quality of material things. And somehow that proves that materiality exists - exactly as conceived!?!
My response is that tangibility is a general conception I would apply as a contrast to some useful sense of its "other" - the intangible. And indeed, given the murkiness of what would be that dialectical other, tangibility already seems rather a confused term. Air is not tangible - until the wind blows or the plane cabin depressurises. Is the magnetic field between two magnets tangible? As a kid I spent a lot of time pushing two opposing magnets together and feeling the "bubble" or resistance that wanted to form in-between.
So you are now using "tangible" as another reified way of speaking about concrete reality - a mind-independent metaphysical framing. It sounds a little better because it also has its mind-dependent meanings. It can mean palpable or tactile - acts of mind - as well as physical, real, substantial, corporeal, or solid. Facts of the world.
I just want to draw attention to how language is being used here. And what the ontological commitments might actually be. Tangibility is a tad slippery and ambiguous about the very issue that needs to be made clear.
To again restate my own position, I am arguing that "ultimately" the material principle reduces to some kind of uncertainty or fluctuation. Naked instability. This is of course a highly abstract and not very tangible notion. And then what you are talking about is really substantiality. Formed matter. Constrained instability. Tangibility would become another word for individuation that persists due to some contextual strength of constraints.
That would be a mental picture of a concrete material reality that is the polar opposite of the usual atomistic one. So the two of us - to the degree we are immersed in these contrasting umwelts - would be taking home very different views about what we just learnt by reaching out and touching a "tangible object".
That was one of the points Rovelli argued in asking what kind of maths a Jovian would arrive at.
So again, experience turns out to be conception all the way down. Nothing enters phenomenology that hasn't already been shaped by some "fundamental" conceptual dichotomies. Sensory receptors are designed with the logic of switches that are in the business of saying yes or no to a question already posed.
This is an irreducible complexity that sits at the heart of all epistemology. It fundamentally screws up any simplicity about the relation between minds and worlds.
And I still believe this is an essential difference in our outlooks. I say that this irreducible complexity is what we must eventually embrace the best way we can. You reply by calling that over-thinking. And I still say no. That triadic irreducibility is the feature and not the bug here. It is how anything could even exist. It is what stops everything collapsing back into its own entangled confusion.
There are physical objects like rocks, mental objects like whatever you happen to be sensing now, and abstract objects like mathematical entities. I think "object" here just sort of means thing.
The objects that appear to be dependent on the minds of individuals are mental objects. We know mathematical objects aren't in that category, and if intuition informs you as to why, then you know the basis of mathematical platonism.
They are formed apophatically. They arise as the opposed choices of some dichotomy. And then they are proved "true" retrospectively because the system of thought that follows turns out to be useful. Even "unreasonably effective".
So the discrete and the continuous arise as an obvious opposition. We can take one as axiomatic, the other as its emergent construction. In maths, a line could be built up as an infinite succession of points, so close together as to leave no gaps. Or we could just as well choose the opposite story that a line can be infinitely divided until it is just a bunch of discrete points - continuous intervals with no actual length.
So axioms might seem to be basic truths too self-evident to deny. But history usually shows that a dichotomising discussion first took place. And that resulted in a pair of contrasting limits. Which in turn offered two complementary descriptions of nature. We could have picked either one - as the "fundamental" from which its "other" was the mechanically emergent construction - and arrived at a useful description of the actual world.
Quoting Wayfarer
But what do you mean by reason?
Mathematical reasoning is pure logical deduction. It is mechanistic. And so it is both powerful, but also only exists as the limit case. It invents its own reality - one where constraints rule absolutely. So we know it is only a powerful fiction. To the extent that it locks reality down to a picture of efficient causation, it is "othering" the very finality that you so want to be part of your reasonable world.
So you ought to be much more comfortable with a full Peircean model of reason - one that is based on the irreducible triad of abduction, deduction and inductive confirmation. But that is then the scientific story on reason. You have to accept the validation by empirical particulars along with the "inspired leap" of the hypothesis/axiom forming, and the logical deduction of some formal description or mathematically-articulated theory.
So you put yourself in a weird position by celebrating the inspiration and the logic - seeing in them a transcendent step from the mind to the divine - and then rejecting the third leg of this triad, the empiricism that roots reasoning back in the world with which it engages.
You pick transcendence when reason actually functions - in a way we see all the time - as a pragmatic and immanent activity.
To connect back to the OP, Rovelli's paper illustrates the logical junk that can be generated if we imagine "reason" becoming just every possible output state of every possible bit of machinery or algorithm. It is just another sad multiverse tale.
But a Peircean understanding of reasoning - the scientific model you want to reject - book-ends that ultimately irrational fecundity with rational constraints. First, hypotheses are meant to be reasonable too. Inferentially, we can see why they would be a jump in the right direction. They feel like the right kind of grounding generalisation. And then second, the correctness of those inferential guesses is validated by their empirical outcomes. Judgement of truth is passed pragmatically. Certain mathematical models would be considered Platonically true because they worked. They wound up limiting unwanted surprises.
So we don't even start on generating maths unless we have a good reason to expect successful outcomes. And we don't take much notice of that maths unless it empirically delivers such an outcome.
Once Rovelli's infinite M has been shrunk in both fashions, then really the space of essential mathematical structures becomes very small. Some folk would claim it can all be shrunk down to set theory, or even category theory.
But anyway, modelling is triadic. Reasoning is a three-legged process. And your own defence of dualism is confused as it wants to conflate mathematical abduction with mathematical deduction - the creative spirit and its divine machinery - placing this unholy transcendental pairing in opposition to the dull inductive materiality of empirical measurement.
But an immanent metaphysics sees the circle being completed by these three elements working collectively. Nature is irreducibly hierarchical in its organisation. That view gives each element its proper place as aspect of a whole.
This is a tricky point, it seems. Would you say the experience of animals conceptually structured?
It's not weird. I don't think the Western tradition of philosophy, and science, for that matter, is basically materialist in orientation. My view is that materialism kind of hijacked the Western tradition from within, although the times they are a'changing. It's great that Platonic and Aristotelian philosophy is being appreciated again, but its ultimate concern is not utilitarian in nature.
One thing I did notice:
[quote=Rovelli]Why has mathematics developed at first, and for such a long time, along two parallel lines: geometry and arithmetic? The answer begins to clarify: because these two branches of mathematics are of value for creatures like us, who instinctively count friends, enemies and sheep, and who need to measure, approximately, a nearly flat earth in a nearly flat region of physical space. In other words, this mathematics is of interest to us because it reflects very contingent interests of ours.[/quote]
Everything is, after all, a function of biological adaption; we have the kinds of maths we have, because of the kinds of creatures we are. Remember our discussion a year or so back about Donald Hoffman? Don't you think that Rovelli's statement vindicates such an analysis?
Of course. Animals don't have a linguistically-structured conception of the world - so one that reflects a higher level social organisation. But they do have one that is structured by its ecologically useful generalisations, or habits of interpretance.
So who is defending a reductionist notion of utilitarianism? I was arguing in favour of the irreducible holism of pragmatism.
Quoting Wayfarer
Sure. Atomism reigns. But it can't be defeated by transcendent dualism. It must find its way to the immanence of holism. And that has a triadic logic at its heart.
Quoting Wayfarer
Sure, things start there. All animals with large brains can count - at least they can count 1, 2, er many.
Linguistic humans then add on a level of social conceptualisation. And eventually a level of mathematical semiosis too. It became useful to count sheep so as not to be cheated at the market, or measure flat ground so buildings started off upright.
But I don't see the connection to Hoffman. Rovelli is not arguing idealism, only social constructionism surely.
And you would count that as "conceptual"? Maybe 'proto-conceptual'; I would have no argument with that.
In any case there must still be a distinction between the concept we have of a thing and the shear appearance of a thing, and also the thing we have a concept of. The further point is that the fact ( if it is a fact) that a perception must be (to at least some minimal degree) conceptually mediated, does not entail that a perceptual experience is a concept. Similarly then, an experience of material concreteness is not a concept of material concreteness.
No, this completely ignores the accompanying definitions by which existence and reality are distinct. While qualities and habits only exist in their instantiations - as characters embodied in reacting things and laws governing such events - their reality does not depend on those instantiations; again, they are what they are regardless of what anyone thinks about them. Their mode of being is that of a conditional proposition; under certain circumstances, they would be instantiated.
Why wouldn't the ability to do mathematics be an evolved ability, underpinned of course by the evolution of animals we are descended from, and the ability they would probably have had to perform basic counting?
This is very clearly false. It conflates the object of a sign with the sign itself. The reality of a character, and the existence of things that possess it, is very clearly independent of any particular system of signs that represent that character and those things. Otherwise, the same claim would apply to the world - i.e., it is absolutely impossible that there was a world before there was the word "world" - which is obviously absurd.
Quoting Metaphysician Undercover
That the world is round(ish) is a fact, whether anyone ever judged it to be so or not; i.e., the world is really round(ish), regardless of what anyone thinks about it.
It means mathematical truths have the character of the discovered. One can be wrong about math.
There are options for explaining this, or you can leave it unexplained. It's the prevailing view among mathematicians. I don't think they bother much with trying to explain it.
Then as to algebra and geometry, I don't buy evolutionary contingency as the explanation. Rather it would be the necessity of such a structural conception of nature. The two ways of seeing the world are formally complementary. For instance, as Atiyah argues....
So first up, evolutionary biology did not make us mathematical creatures. Although it did leave us - as the smartest animals, our brains already being reshaped by our language and tool-use - with highly lateralised neurobiology. Large brain animals already have that "algebra vs geometry" dichotomy wired in as a division between object recognition processing and spatial relations processing. Things vs their relations. And then humans continued on to have a strong left vs right brain dichotomy in terms of attentional style - the left specialised for focal differentiation and the right specialised for global integration (both always working together to produce the third thing of a task-appropriate balance).
So in a general way, biological evolution structured the animal brain with the kind of Gestalt figure~ground logic that would be needed to see the world "as it is". The world could be experienced as an intelligible whole because it was always being analysed in terms of its fundamental structuring dichotomies.
But then a mathematical turn of mind - one which actually see the world in terms of mechanistic construction, acts of counting and measuring - was a cultural evolutionary step. Ordinary language got it going. Then mathematical language crystalised the practice as an actual machinery, a teachable syntax.
So yes, a mechanistic conception of nature was of evolutionary value. But that couldn't become apparent until after language was developed.
And on the other hand, our neurobiological legacy was not absolutely devoid of logical structure. But it was the much more organic logic of dichotomisation and symmetry breaking. This is the logic that reflects the deepest structural reality of the Cosmos.
And then it is no surprise that both geometry and algebra developed hand in hand. They are simply the two complementary ways that structure itself breaks down. One is ground, the other is figure. One is global, the other local. We can choose either as the starting point from which to work back towards the other.
Quoting Banno
Exactly. The structure of the argument of the paper is that of a reductio. Those who see the paper as begging the question seem to miss this entirely. The paper takes place on the grounds of mathematical Platonism, and attempts to dismantle it internally, and not from some position outside of it.
Again, something exists iff it reacts with other things; something is real iff it is what it is regardless of how anyone thinks about it. Numbers clearly do not exist, because they do not react with anything; yet they are clearly real, because they are what they are regardless of how anyone thinks about them.
Quoting Janus
I am saying that the reality of numbers does not depend on their particular instantiations (existence), and hence that at least some "mere logical possibilities" are real - i.e., independent of how anyone thinks about them.
Mathematical proofs are justified by their consistency or lack of contradiction. They cannot be justified by anything outside of math as far as I can see. Mathematics is the elaboration of basic counting operations, an elaboration that is enabled by the ability to manipulate symbols and symbolic operations.
I can't see what counterpoint you are trying to make here.
It's true you can be wrong about math. But at the basic level this can be seen by manipulating physical objects. Say you want proof that 8 x 13 = 104. All you need to do is arrange 104 objects in groups of 8 and see if there are thirteen of them.
No. That's what it is stipulated to mean. The idea that what is real cannot refer to things that are products of our activity is a malicious piece self-serving philosophical claptrap that Platonists have traded in since day one. I agree that it's the usual, most widely employed understanding of the term, but that only attests to the fact that people are not particularly bright.
Rovelli, who is happily a bulb above the rest, rightly avoids the whole semantic debate altogether.
I think the key point here is that saying a number is what it is regardless of what anyone thinks about it is not the same as saying that a number is what it is independently of all thought whatsoever.
I agree that mathematical objects are not dependent on any individual mind; but it does not follow from that that they are real independently of their being instantiated in actual things or thought by actual minds.
Proof isn't the issue, but I gather you're saying number is a property of physical objects and nothing more? Correct?
Yes, I think the semantic debate is a red herring.
I would say that number consists in difference and similarity; which can be both perceived and conceived; the former being concrete and the latter abstract. I'm not proposing any absolute division between the concrete and the abstract; because they do partake of one another.
My argument is that conception and perception would be the two extremes of the one process. To speak of one versus the other is merely to highlight the grounding generality vs the focally individuated particular. You actually have nothing without having both together striking an appropriate balance. So you need three things. The limits of the conceived, the limits of the perceived, and the outcome of that which is then "the experiential state".
That is a holistic description of experiencing as a general psychological process. And that would apply to the business of modelling the world whether it was neurological, linguistic, or mathematical.
But then you introduce the further issue of how the different levels of semiosis might relate - given that they do seem differentiable in principle, but also again are always in action together in any brain trained to understand its world in a "modern" way.
So you want to argue something familiar - the biology does the perception, the sociology does the conception. And there is a rough truth to that. But I am saying it becomes an unacceptably rough description if we want to be thorough-going metaphysical holists. It wants to make absolute a distinction which can't absolutely exist.
My original response was a result of you both accusing me of misplaced concreteness and then refusing to recognise that you yourself were doing just that with your own talk of "materiality" or "tangible objects".
Science is always talking about matter. But it long ago dropped most of what most folk think "matter" means from their everyday linguistic talk about the kind of neurobiological perceptions they experience. Science is now talking about matter as fields, and fields as information. It is galloping along because it doesn't restrict itself to ordinary language descriptions of neurobiological conceptions.
I was pointing out how your own use of the term was still overly concrete in taking the meaning of "material" as just something one could simply point at the world and exclaim: "See, right there before your eyes, just as it looks and feels." To avoid the charge of naive realism, or simple epistemic confusion (they are usually the same thing), you would have to show how you were meaning something more sophisticated.
And are difference and similarity properties of physical objects?
No, I would say that objects have qualities or characteristics which are different from and similar to other objects. So it is the qualities or the characteristics which are the properties of objects, not difference or similarity per se. In other words I think it is better to think not in terms of qualities constituting difference and/ or similarity as such, but in terms of qualities being different and/ or similar to other qualities.
I agree and I say as much in the quote directly above.
Difference and similarity are properties of relationships then? Your presentation is stuffed full of abstractions. I pointed this out before: they're kind of indispensable.
My own opinion is that form and the object of formation are products of reflection. In very much the same way we separate ourselves out from the world on reflection, we separate universals from particulars. Then from that dismantled arrangement, we try to understand the world. Make sense?
Exactly right; according to Peirce, reality is independent of what any individual mind or finite collection of minds - including, notably, the collection of all actual minds - thinks about it; but reality is not independent of thought in general. As he once put it, "just as we say that a body is in motion and not that motion is in a body, we ought to say that we are in thought and not that thoughts are in us." In fact, another of his definitions is that reality is whatever would be included in the ultimate consensus of an infinite community after infinite inquiry. This is obviously a regulative ideal, not something that could ever actually be achieved.
To try and clarify why I remain unconvinced, consider the author's synopsis of his paper:
So, M is the "platonic world" of mathematical facts. The author observes that if M is too large then it is uninteresting to us, and if it is smaller and interesting then it is not independent of us.
Perhaps I fail to grasp the dilemma, but can't we just accept both the existence of a very large M and also that we are only interested in a small subset of it? I don't see why our interest in only a small subset of M - the part which we find interesting or useful - should falsify the existence or independence of the much larger M.
If we assume from the outset that M is the "platonic world" of mathematical facts - even those which are of no interest to us - then we cannot also say that M might contain only those mathematical facts that we find interesting (and therefore M is not independent from us). And only one of these resembles mathematical platonism.
"Mathematics may be the investigation of structures. But it is not the list of all possible structures: these are too many and their ensemble is uninteresting. If the world of mathematics was identified with the platonic world M defined above, we could program a computer to slowly unravel it entirely, by listing all possible axioms and systematically applying all possible transformation rules to derive all possible theorems. But we do not even think of doing so. Why? Because what we call mathematics is an infinitesimal subset of the huge world M defined above: it is the tiny subset which is of interest for us. Mathematics is about studying the “interesting” structures". (my emphasis)
Note the identification of what is mathematics with what is 'important' to us; or, contrapositively, the exclusion of most of M as that which is not mathematics. Or more starkly still: most of M is not mathematics. It's not that we pick out some of interesting parts of math out of a wider set of math: it is that what we don't pick out is not even considered math. This is the import of Rovelli's metaphor of the sculpture and the stone: the stone really does 'contain' every possible sculpture that could be made from it, but what it contains is a kind of sheer potential, indefinite and undifferentiated such that the stone cannot be identified with 'every possible sculpture'. The stone is not a sculpture in the same way that M is not to be identified with math ("Mathematics... is not the list of all possible structures").
Importantly, this is not an assumption that Rovelli makes: this really is how math is, how it 'works'. So the question is: why does math look the way it does (and not otherwise)? What selection principle was employed to sculpt the indefiniteness of M into what is, in fact and in reality, considered math? What Rovelli essentially points out is that Platonism can provide no such principle, because it specifically divorces math from the practice of mathematical activity, which it considers something of an epiphenomenon, and which thus cannot play any constiutitve role in defining math. This is in contrast, Rovelli points out, to how math actually proceeds, wherein 'interest' provides just the selective principle that is missing from Platonism. And if this is the case, then Platonism cannot possibly be true.
That's the argument at play here; those who think that the paper simply proceeds on the basis of begging the question simply lack any basic comprehension ability.
Phew. Natural selection saves the day, again.
So the argument goes:
1. Mathematical platonism is the view that mathematical reality exists by itself, independently from our own intellectual activities.
2. But what we call mathematics - containing only what is important to us - is but "an infinitesimal subset" of mathematical platonism.
3. Mathematical platonism cannot explain which mathematical facts are important to us.
4. Therefore, mathematical platonism is false.
?
No it does no such conflation. It states the simple fact that it is impossible, that there is the object which corresponds with a particular sign without the existence of that sign. Correspondence requires two things, the sign and the object. Without the sign there is no correspondence, therefore no object which corresponds.
Quoting aletheist
Can you not grasp the very simple fact that it is impossible to have "the character which is represented" without the sign which represent? One is clearly prior to the other as "the character which is represented" requires for its existence, the act of representation, and this act is dependent on having a sign which represents.
What you describe as "obviously absurd" is also obviously true. Sometimes truth is stranger than fiction. There is something which the word "world" refers to. Prior to the existence of this word, there was nothing that the word "world" referred to. Therefore there was no world prior to the word world.
However, many of us project, and claim that the thing which is now referred to by "world" had existence prior to the word. This requires a separation between the thing and the word. The thing must be conceived of as independent from the word. But when we conceive of the thing as separate from the word, we deny any necessary relationship between the thing and the word. So there is no logic which allows us to claim that this independent thing, which is conceived of as existing independently from the word associated with it, is actually the thing which is referred by this word. Therefore the claim involved with this projection, that the independent thing is the thing referred to by the word, is unsupported logically. The thing which is assumed to have existed before the word, cannot be proven to be the same thing as the thing referred to by the word. This inability of logic to prove that the independent thing is the thing referred to by the word, is indicative of the simple fact referred to above, that there is no such thing as the thing referred to by the word, without the word.
P1. Any account of mathematics would need to explain why mathematics is the way it is.
P2. Mathematical Platonism is the view that there is a world M, that contains all possible mathematical objects and truths.
P3. Mathematics is but "an infinitesimal subset" of any such mathematical reality.
P4. Any account of mathematics would need to explain why P3 is the case, in order to satisfy P1.
P5. Mathematical Platonism has no way to explain why P3 is the case.
C1. Mathematical Platonism cannot satisfy P1.
Ergo, Mathematical Platonism fails to have any explanatory force with respect to mathematics.
I 'excluded' the question of interest because the argument works without it. 'Interest' is Rovelli's effort to provide a positive explanation that he finds lacking in Platonism. The negative argument works without any reference to it. Rovelli weaves both the positive and negatives aspects of the argument together in the paper, but isolating the negative aspect makes the 'argument against Platonism' easier to see, I think. I'm not super confident about my construction of syllogisms (it's not something I'm trained in, and I find it hard to think with them), but I'm happy to hash this out if possible.
I don't know. It just sounds a bit like bemoaning the fact that mathematical platonism is unable to tell us which mathematical facts are interesting to us (or "why mathematics is the way it is"), despite mathematical platonism being the view that mathematical facts exist independently of our intellectual activities.
Have been away (so this is an answer to page 4)
Euclidean and non-Euclidean geometry simply starts from different premises (or should I dare to say axioms). The geometry on a blank paper and the geometry on a sphere are different, but their existence doesn't make one or the other illogical. The only mistake is if you assume that all geometry is, let's say Euclidean (and that the parallel postulate is universal). That argument is wrong, but it doesn't make either geometry illogical. Especially in set theory you can choose your axioms and have different kinds of set theories with different answers, but that in my mind don't make them illogical.
Quoting Metaphysician Undercover
Premises (axioms) can make the math to seem contradictory, but can be totally logical. Only if you prove that something that we call an axiom is actually false, then is the statement simply wrong.
Thank you for so convincingly demonstrating the patent absurdity of nominalism.
No, not really. If you begin with the neutral position, it is the one making the argument that begs the question.
It doesn't matter if you're a Platonist or not; the argument is simply bad.
Even if we assume a lot of math is useless, or, let's even assume the vast majority of it is, so what? How does it then follow from this claim that Platonism does not exist? It doesn't.
Yes! I think basically think that Wittgenstein basically hit the nail on the head with his reflections on math and that everyone else has more or less been playing catch-up ever since (and failing rather miserably, at that!). That said, I say this only having gleaned Witty's position from some selective reading of the Lectures and primarily the work of Bob Clark and Paul Livingston. I read the two papers you linked by Rodych, and while I have minor quibbles (I wouldn't call Witty a finitist - or an 'infinitist', for that matter, insofar as I think his position explodes the terms of that debate - in a productive manner), I really liked the way they tracked Witty's evolving thoughts on math across his work.
But yes, my enthusiasm for Rovelli's paper is partly coloured by the Wittgenstinian hue with which I bring to it.
It's difficult for me to improve much on @fdrake's summary of Rovelli's argument, earlier in this thread. This is a broadly negative argument, however, that consists in highlighting that the version of mathematical Platonism which Rovelli is targeting incorporates too many items into the set of what Platonists themselves intuitively feel are entitled to be counted as intelligible mathematical patterns. The argument relies on the acknowledgement that the manner in which we sort out the wheat (fruitful mathematical theories) from the chaff (unprincipled and uninteresting sets of axioms) reflects contingent features of our specific form of life. This consideration, supporting the negative argument, fails however, it seems to me, to properly account for the fact that mathematical truths appear to have a grade of necessity (and degree of generality) somehow intermediate between, and qualitatively distinct from both, logical necessity and pure contingency. (I've suggested that Kant, and neo-Kantians such as Sellars, are gesturing towards the right kind of necessity with the concept of synthetic a priori propositons). But that's not what you're asking about. Maybe I'll comment more about this in another post.
Much hangs on what "radically different" means. It could mean that two existing forms of life are incommensurate in such a way that mutual understanding is impossible in principle. Or a form of life could be radically different than our own in the mundane sense that it is difficult for us to fathom prior to having gained some acquaintance with it. Michael Thompson has argued that there is a multiplicity of practical forms of life, while Sebastian Rödl has argued that there is only one. Donald Davidson, in his paper On the Very Idea of a Conceptual Scheme, also argued that the very idea of a multiplicity of mutually incommensurable practical forms of life (or conceptual schemes) is incoherent. That's an issue that Rovelli doesn't contend with. It points to the possibility of a middle term between viewing mathematical theories in a way that make them inherit the contingency of the forms of life which they speak to, on the one hand, and viewing them to be universal in a way that makes them independent of (or unsoiled by) any embodied and situated life form whatsoever, on the other hand.
I'd be really interested in any more you have to say about that.
If you don't mind me referring you back to old posts of mine, I've sketched my understanding of the significance of synthetic a priori propositions, here and there, with reference to John Haugeland's neo-Kantian (and Sellars inspired) view of the constitution of the necessary standards (such as mathematical rules, social practices, and/or laws of nature) that make possible objective empirical judgments.
Haugeland's view of constitution, which he further elaborates in his paper Truth and Rule Following, is intermediate between the idea of (contingent) invention of rules and discovery (of 'intelligible' or 'independent' laws).
We're trying to decide between two positions, A and B.
You present an argument that A is true, because of fact X.
Your interlocutor points out that nothing about the truth of X entails A, and so this is a bad argument: the conclusion has nothing to do with the premises.
You respond by saying, well of course it's still possible that B given X, if you take for granted that B!
This is not an effective argumentation strategy.
The fact is that whether or not mathematical objects are interesting to creatures that study or make use of them has nothing to do prima facie with whether there are not such objects. This is, as it stands, a terrible argument. If you want to try to make it better by trying to provide some bridge principles (why should we think that the existence of mathematical objects is contingent on, or interestingly tracked by, the interests of creatures? We're not inclined to think such a thing for any non-human creatures: things are still countable by number, even though there are no creatures who can count!), that's fine, but you can't simply double down.
In other words, we are looking for some reason to accept a premise like:
"A mathematical object exists only if it is interesting to some creature."
This is on its face an absurd claim. Can you give us a reason to believe it?
I thought you'd be interested on Terence Tao's thoughts on the development of mathematical skill. He has three distinct stages of competence:
(1) Pre-rigorous; like manipulating apples on a table, 2 apples plus 3 apples is... count them.... yes! 5 apples. Or later: 'the derivative of a function is the slope at that point... look if we take the derivative formula for x^2 - yes class that's 2x - and line it up with the point (1,1) on the line, yes! you can see the slope is the same as 2x'.
A pre-rigorous stage mathematician has enough of a feel for a topic to perform basic computations; they will form non-systematic insights and heuristics which they cannot easily translate into formal mathematics. They can follow handle-crank rules without any intuition as to why it works.
A good example here is, probably what most people still have, the ability to times 47 by 12 through the algorithm which looks like:
47
12
-----
and by carrying tens and so on. Most of us could do this with some degree of effort, but god knows precisely how it works. It usually occurs while young and inexperienced with a topic; spanning pre-university and some undergrad education.
(2) Rigorous: like learning how to prove 1+1=2 through a set theory or the Peano Axioms, or how to define fractions and real numbers consistently; epsilon-delta and epsilon-N 'turn the handle' proofs for convergence and continuity. Manually computing derivatives from their definition. This usually occurs mid-undergrad until the masters thesis on a topic, or advanced modules on topics.
A rigorous stage mathematician largely learns definitions and computes proofs and more advanced calculations by reducing them to simpler ones.
Developing the example; a rigorous stage mathematician would be able to prove why the above algorithm for computation works: you can do this by formally summing the stages:
2*7=10+4, remove tens to obtain the units digit, 4, store 10 to the tens computation (total sum 4)
2*40=80, now 90 while adding the 10 (total sum 94)
end stage 1
0*47=0
10*47=470, add previous total sum yielding 564
end
2*7+2*40+10*47=2*(47)+10*47=12*47
(3) Post rigorous: a post-rigorous mathematician can reason intuitively and through analogy about mathematical structures, their intuitions and analogies usually can be translated into proofs, or they will have the ability to find counter examples. The post rigorous stage can be obtained by mathematicians by updating intuitions formed in the pre-rigorous stage to respect the formalisms learned in the rigorous stage. This obtains after advanced modules or long theses are completed, typically the sphere of research or practicing mathematicians in their required competences.
A post rigorous modification of the above algorithm works to compute the product in binary:
[hide=details]
12=0*2^0+0*2+1*2^2+1+2^3=b=1100
47=1*2^5+0*2^4+1*2^3+1*2^2+1*2+1*2^0=101111
101111
1100
gives
0
0
010111100
101111000
2^0 col = 0
2^1 col = 0
2^2 col = 1
2^3 col=0 carry 1
2^4 col = 1 carry 1
2^5 col = 1 carry 1
2^6 col = 0 carry 1
2^7 col = 0 carry 1
2^8 col = 0 carry 1
2^9 col = 1
2^9=512, 2^5=32,2^4=16,2^2=4, 512+52=564
1000110100
[/hide]
I winged this off the intuition that 'carry the ten' was the same as 'carry the base' and filled out the details later.
To be sure, post-rigour is topic specific; while I'm sufficiently familiar with positional notation and base changes to intuit quite a lot about them, I'm nowhere near as familiar with formal proofs in ZFC, so I couldn't wing deductions in ZFC by 'getting the right idea' first then 'working out the details'.
The motivation for bringing this up being that how the conceptual links work between synthetic a-priori propositions, or synthetic a-priori ideas/intuitions, are competence dependent and topic specific. Which isn't to say that the fact that the algorithm above generalises to any base system is dependent upon my competence, but the expression of that fact depends entirely on the 'right links' and 'right intuitions' being in place in their expresser. So there's a historical component to it; there's no way I'd've been able to wing the above generalisation without floating point arithmetic, hex and binary manipulations being part of computing courses. Having the 'right links' and 'right intuitions' is as much a function of the intellectual milieu as creative competence.
Would you rather people address the argument in this post:
https://thephilosophyforum.com/discussion/comment/220479
Than the one in the OP (they are not the same)?
MP posits the existence of a realm of independent, abstract, mathematical objects. Why should this ontology be required to "explain why [our] math is the way it is"?
It's not that one or the other principle is illogical, because it requires two opposing principles for there to be contradiction, and the contradiction is what is illogical. That the object, space, is described in these two contradictory ways is what is illogical.
Quoting ssu
What is the case, is that this subject, mathematics, which allows that the objects which it deals with are described in contradictory ways, is illogical. That one geometry is consistent within its own system, yet inconsistent with another geometry, or one kind of set theory is consistent within its own system, yet in inconsistent with another set theory, doesn't make any of them, in themselves, illogical. But this is not the issue. The issue is that the discipline of mathematics, which allows within it, such inconsistencies, is illogical. It's not that this or that branch of mathematics is illogical because this branch is inconsistent with that branch, but that mathematics itself is illogical for allowing such inconsistencies within the discipline.
Quoting ssu
Opposing axioms cannot both be true. Therefore, one or the other, or both must be false. It is illogical to hold opposing axioms, as both true, and this is what mathematics does.
Quoting aletheist
Yes, the argument is very simple and clear, isn't it? So much so that you have no counterargument except, "that's absurd". I thought you said that the argument is "self-refuting". Is that how it is refuted, by you saying it's absurd?
I think both arguments are very bad, but the response will change depending on which is addressed.
Good post. I agree it is about finding the (pragmatist) middle path.
So in my simple-minded way, the debate is about the interaction between the formal and the material in a hylomorphic conception of nature. Maths stands for a notion of all the forms that are possible. And as such, it is unconstrained by material considerations. Practical considerations of energy and matter - whatever it is that instantiates a form as something actually physical - are left out of the story.
And that disconnection is where all the confusion arises. It in fact is the same confusion that leads to mind~body dualism. It is only once psychology is understood as something embodied and function serving that we can see why physics acts to produce the kind of "information processing" nervous system that we see.
So the maths that matters is the maths of purpose-serving structures. They do reflect the features of a form of life. But the features need not be contingent, nor the way of life just ours. Instead, we could find our centre ground in the standard structuralist claim that there are generically necessary "forms of life". For actualised substance to be the case, there has to be a systematic organisation - that interaction between downward constraining forms and upward constructing material degrees of freedom.
So tracking the genericity of structures leads you towards a thermodynamic, probabilistic, and indeed semiotic, metaphysics. Life is certainly a form of thermodynamics - a semiotic elaboration on a dissipative structure. And even the Cosmos is a "form of life" in being a dissipative structure at a universal level.
In the grand scheme then, we would seek to unite the formal and the material in the substantial. That hylomorphism must be the thread that connects existence at its every level, its every stage of complexity. So substantial reality can be said to have a general interest in the formal structures that work - the structures that can harness material flows or entropy to achieve the goal of wresting stablility from impermanence. A "way of life" boils down to just that. And the maths that matters most, the maths that has true reality, is that which describes the kinds of forms or structures which subserve that fundamental purpose.
So if the mathematical realm is some vast landscape of possible algorithms, then only a limited number would be highly effective at doing the job of harnessing material instabilities. The "good" maths would be that which has the emergent property of producing finitude or constraint. Junk maths just wanders off forever in open-ended fashion. Good maths - like symmetry maths or fractal maths - speaks of nature because it speaks of self-organising limitation and closure.
Of course, mathematicians are free to explore the "junk". They could just play with open-ended patterns as a matter of contingent human choice. The constructed view of maths could be true as part of the socially-accepted "mathematician's way of life". :)
But still, humans have historically valued the maths that could speak to natural structure. And natural structure is hylomorphic. What matters is how constraints can organise freedoms to produce substantial actuality.
We can kick around Plato as if he were some kind of ancient mystical fool. But from my own readings, what strikes me is how dimly we appreciate the systems-based revolution that was Greek metaphysics - especially in the writings of Anaximander and Aristotle. It was how science was born. And then the systems view got obscured because causal reductionism took over. Nominalism made formal/final cause appear mystic and uncool. And we have been living with that metaphysical confusion ever since.
:up:
[quote=J P Hochschild]Characterized by forms, reality had an intrinsic intelligibility, not just in each of its parts but as a whole. With forms as causes, there are interconnections between different parts of an intelligible world, indeed there are overlapping matrices of intelligibility in the world, making possible an ascent from the more particular, posterior, and mundane to the more universal, primary, and noble. In short, the appeal to forms or natures does not just help account for the possibility of trustworthy access to facts, it makes possible a notion of wisdom, traditionally conceived as an ordering grasp of reality. [/quote]
I said that recognizing some judgments as true and others as false entails that there is a fact of the matter, which is independent of whatever anyone thinks about it; and that any argument to the contrary is self-refuting. Why? Because disputing it requires presupposing it.
Quoting Metaphysician Undercover
You seem to be asserting that something is not real unless and until a word for it exists, which is what I find patently absurd. The reality of (what we call) roundness and the world does not depend on the existence of those names. The world was real, and was really round(ish), before humans ever existed.
I can't see your argument. If some judgements are true, and others false, then truth and falsity is a property of the judgement. Therefore it is impossible that truth and falsity are independent of the judgement.
Quoting aletheist
No I am saying that there is no such thing as the thing referred to by a word without the word. How is that absurd? Therefore it is impossible that there was a thing which the word "world" refers to before there was the word "world". To say that there was a reality of roundness before there was a word "round" is what is absurd. What would dictate what roundness is without the word and a corresponding concept? So how could there be a reality of roundness without this?
I never suggested otherwise. However, if a particular judgment is true, why is it true? And if a particular judgment is false, why is it false? In both cases, the answer is that there is a fact of the matter, and that fact is independent of whatever anyone thinks about it. A true judgment represents a fact, while a false judgment does not.
Quoting Metaphysician Undercover
Because it entails that the reality of an object somehow depends on the existence of a sign that represents it; but reality is precisely that which is as it is regardless of any representation thereof. In other words, a real thing (or quality or habit) is that thing (or quality or habit) regardless of whether there is any word that refers to it. The thing (or quality or habit) came before the name that some humans arbitrarily invented for it. There is no necessary connection between most words and most things (or qualities or habits), only a convention by which the words refer to the things (or qualities or habits) within a particular language or other system of signs.
Quoting Metaphysician Undercover
This is exactly backwards. What would prompt the creation of the word "round" if there was nothing already observable for which such a name was needed?
I find the paper's attempted refutation to be unsuccessful.
The author says if M is too large then it is uninteresting. But what the author means to say is that if M is too large then it is mostly uninteresting to us (and perhaps also: at this point in time). Obviously, there are some parts of M that we do find interesting, because "the value is in the selection, not in the totality". The author says that M "contains too much junk", but again, as judged by us (or, alternatively, by the Jovians).
The Jovians might have a different mathematics to us and they might find different parts of M more useful than we do, but presumably we and the Jovians each borrow true theorems and objects from the same universal M. Likewise, we may have discovered how interesting linear algebra is only recently, but it could always have been a part of M, waiting there for us to use it. The same can be said of 2d and 3d (and perhaps 4d) geometry, arithmetic, set theory, logic, category theory, topos theory and whatever mathematics the Jovians might use.
The author says that if M is smaller and interesting then it is not independent. But then neither is it the M of mathematical platonism, I would argue.
The author says that the complete independence of MP "is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather."
Here I agree with the author. I think he has as much chance of refuting MP as he would have of refuting the existence of God.
I think you are wrong here aletheist. True judgement implies correspondence. Why is it true? It is true because there is correspondence between the judgement and reality. You have provided no premise whereby you can claim that if there is correspondence, one of the corresponding things is necessarily independent of the other.Therefore your conclusion of an independent fact is illogical, invalid, non sequitur. "True" implies correspondence which implies a relationship, it does not imply independence.
Quoting aletheist
The reality of an object is dependent on the process whereby the object is individuated from its environment the rest of reality. This process is carried out by sentient beings. Unless you can demonstrate how this process of individuation is carried out by something other than sentient beings, or demonstrate how this is false, then you have no argument.
Consider this example. We perceive the earth as an object. But we know that the earth is not an independent object, it is part of the solar system. We individuate it, separate it from its context, and assign to it the status of "an object", when it is just as appropriate to say that it is not an object, but part of an object, the solar system. Its independence as "an object" is artificial, created by us. Its status as an object is simply a product of how we individuate things. There are many such examples in scientific theory, electrons, and protons are said to be objects. Science is full of things which we know to exist only as parts, but we treat them in theory as independent objects.
Quoting aletheist
Whether or not the thing came before the symbol which represents it, is debatable, The thing, the object, is dependent for its existence as an object, on the process of individuation carried out by the sentient being which individuates it, as described above. To determine which "came before", we must consider the temporal nature of this process. The sentient being exists at the present, and assigns the status of "object" to things observed to have temporal extension, things which remain relatively unchanged over a period of time, or demonstrate temporal continuity. Since the being exists only at the present, this, observing temporal continuity, requires memory of what has been in the past. And memory requires that what has been, is represented. Representation requires symbols. So I believe that the logic demonstrates that the symbol is prior in existence, to the object.
Quoting aletheist
Imagination! Imagination may not be something real to you, but it is the inspiration behind creation.
I have no motivation to defend MP as I'm not a platonist. I'm just pointing out that the author's supposed refutation fails, or at the very least it fails to convince me. Do you have a knock-down argument to refute the existence of God?
This is incredibly silly. A failure of X to explain Y cannot entail that Z must in turn be implicated. That's just a basic failure of logical form, let alone content.
If you are referring to whether or not your argument implies our interest, it's about our mathematics. Do you claim that we have no interest in our mathematics, or (i.e.) in our infinitesimal subset of M?
I'm not interested enough to read it either (ha ha). I think we're ultimately limited to phenomenology of such things. I am interested in reading more Haugeland, though, (as soon I discover where his essays are stashed.)
When we discussed Pattern and Being, it was available online. Truth and Rule Following (which further develops the same ideas) only has been published as the last chapter of Having Thought: Essays in the Metaphysics of Mind, HUP, 2000. I can't legally post pdf documents but I can PM you a treasure map.
Awesome! Thank you!
Thanks very much for that. I think this idea, suitably adapted, might dovetail nicely with Wiggins's account of someone's conception of a concept, construed as the Fregean sense of this concept (while the concept itself still lives at the level of Bedeutung, or of Fregean reference). Tao's idea of the development in skill/understanding lines up with Wiggins's idea of the improvement of a conception that enables, at once, a better grasp of the sense of a concept and an active participation into its constitutive practice.
Wiggins develops this idea most fully in The Sense and Reference of Predicates: A Running Repair to Frege's Doctrine and a Plea for the Copula
Just like the pragmatized (embodied and situated) neo-Kantian account of Sellars, Haugeland and Bitbol, Wiggins's account of the way in which we grasp concepts steers a middle path between anchoring them into merely contingent features of the embodied subject or making them fully 'independent' of us (in the manner of modern Platonism). The precise account of this cognitive anchoring, though, appeals to some features of modality and reference that are indebted to Frege and to Kripke (and Putnam). Those features have been highlighted by Gareth Evans, also, in The Varieties of Reference (in the chapter on proper names, which discusses reference to natural kinds, also). I've written some posts about this many years ago on a Yahoo discussion group. I'll try to locate them.(*)
(*) Here they are: see mainly this post, which was a followup on this one.
I continue to find your argumentation nonsensical, and have decided to stop wasting my time with it. Cheers.
Seconded!
I sometimes get the feeling that analytic philosophers hide that they're talking about anything interesting by talking about language. So my eyes often glaze over when they shouldn't. I will read your notes on them, though.
I sometimes get this feeling too, but have seldom gotten it while reading either Kripke, Putnam, Evans or Wiggins. They are quite adept at navigating between the formal and the material modes of speech, as Carnap might have put it. Pondering over how things can sensibly be said to be (philosophy of language), and how things can sensibly be thought to be (metaphysics and philosophy of thought), often are one and the same inquiry.
Since you've already primed me to think of this in terms of the creation of the synthetic a priori and how that creation interfaces conceptually with the world, it's interesting. Maybe my new mantra reading on similar topics should be: 'they're talking about how we interface with the world through language'. Your prose in the first note isn't making my eyes glaze, though, so thank you for that.
Yes, and when they fail to do so, then, maybe, they're just passing off linguistics as philosophy of language, or they are falling prey to psychologism in the sense Husserl and Frege warned against. (Not that there is anything wrong with pure linguistics or scientific psychology, per se, but it's not philosophy.)
Just for reference, Michael Luntley's Contemporary Philosophy of Thought: Truth, World, Content, Blackwell, 1999, is a very good introduction to the philosophy of language qua philosophy of thought.
It will be added to the list. Thank you for the reference.
Quoting Manuel
Quoting Manuel
It's a question I'm very interested in. One point I've often raised is that numbers (and logical laws and so on) are not dependent on your or my mind, but can only be grasped by a rational intelligence. So they're mind-independent in the sense that they're the same for anyone, but mind-dependent in the sense they can only be grasped by an intelligence capable of counting. This clashes with the empiricist view that numbers (and the like) must be considered a product of the mind. It implies they are real but in a manner different to empirical objects of perception. And mainstream philosophy has no way to accomodate different kinds of real - for it, something is either real or it's not.
See the essay What is Math? in the Smithsonian magazine. James Robert Brown puts the Platonist argument for the reality of number. But the objections are that if numbers are not empirical objects, then you're opening the door to all kinds of 'mystical nonsense':
These objections speak volumes, in my opinion.
I'll check that article out, thanks for sharing.
Empiricism goes out the window if empiricism is construed as implying "publicly observable phenomena". But if you include experience in empiricism, as one must, if any empiricism is going to make any sense at all, then it remains as a method of investigation.
I agree with what you say about math being independent but requiring a mind to comprehend it. Mathematics is extremely strange and may be one of the reasons why Plato required knowledge of geometry to enter his academy, aside from its timeless otherwordly nature.
What made it interesting to me was the (I thought) simple observation: that numbers (and the like) are unlike phenomenal objects, in that they're not composed of parts (strictly speaking that is only prime numbers) and they don't come into, or go out of, existence (i.e. they're not temporally delimited.) So they exist on a different level, or in a different sense, to objects, all of which are composed of parts and temporally delimited. But then, the idea that there can be different levels of existence, or different senses of existence, turns out to be a metaphysical question.
This was the subject of my first post on the predecessor forum to this one. An excerpt from that was as follows:
And no, I don't think it's anything to do with the pursuit of maths as such. There are excellent mathematicians, I have no doubt, who are inclined towards a Platonist view - Roger Penrose and Kurt Godel being two - but there are doubtless very many who don't. And you don't have to know much about maths to understand the major issue, that being the reality of intelligible objects.
//I guess in hindsight that excerpt is a bit over the top. I hadn't planned it, I just sat down to say something and that is what I came up with. I still think it's OK, though.//
Six Fools
[i]One day, six fools from a certain village set out for pilgrimage. On their way, they had to cross a river swimming. After crossing the river, one of them counted them, not counting himself. He counted five. They were very upset at losing one of them. To be sure, each one of them counted again in the same way that the first one did and counted five. They informed the matter to a passerby. The passerby was amused at their stupidity. He agreed to produce the lost man. He took a stick and gave a blow on each head until he counted six. The six fools thanked him again and again for producing the lost man ‘miraculously’.
Moral : The stupid are stock of laughter.[/i]
Long gone. I revived this thread because it was relevant to the point I was making elsewhere. Prior to that the last post was 4 years ago.
Quoting Agent Smith
"Materialism is the philosophy of the subject who forgets to take account of himself" ~ Arthur Schopenhauer.
My guess is your parable was intended to make this point.
Idealism is the philosophy of the subject who forgets to take account of being a body. :eyes:
Thus, the a priority of the material (fundamental) and a posteriority of the ideal (emergent).
A pity. Streetlight had interesting things to say but he was a bit brusque in his conduct.
Anyway, as for your comment on different kinds/levels of existence, there's Meinong and his jungle to consider.
The senses and real
You can't see nor smell nor taste air, but it is real.
You can't hear a spanner fall on the moon, but it is real.
You can't touch a radio wave but it is real.
If so, just because you can't see, smell, hear, taste or touch a number, it doesn't mean numbers are not real.
Yes, subsists is correct in Meinong's universe. Speaking for myself, I posit that are there are two universes, a) the physical universe and b) the mental universe and numbers exist in the latter while rhinos, the Eiffel tower, etc. exist primarily in the former.
Universe + Ideaverse. We're explorin' the latter here aren't we? You seem to have visited many worlds from what I can gather from your writings and thoughts. I on the other hand have just begun my voyage. My ship was damaged and my navigator died. I'm low on fuel - crash landing on the nearest world. Wish me luck. Out! Hiss ... Crackle .... Crackle :rofl:
I think philosophy is like that for me.
It's an intrinsically satisfying activity which always leads to something more, unfinished.
This is just a reminder of how good Streetlight was and how much he contributed to the forum.
And yet the concept of number would be incoherent without the prior construction of the concept of a multiplicity , which itself implies the concept of persisting self-identical empirical object.
Husserl, in Philosophy of Arithmetic, describes a scenario for the stages of development of the modern consort of number:
“Let us transport ourselves into the early stage of the development of a people. The repeated interest in sensible groups of objects the same in kind had already led to the apprehension of a certain analogy, and therewith of a shared characteristic founding it; and thus it had led to the concept of multiplicity, which at this level, of course, being much less abstract than on our own, restricted itself to multiplicities of homogeneous and sense per-ceptible contents. The drive to communicate concerning the events of practical life, in which determinate groups of such objects played a great role, led here (when circumstances were particularly favorable) more easily than in other areas to the thought of an imitation by sensible means of the things repre-sented.
This thought would be immediately suggested by the hands. These visibly prominent organs, which the individual chief-ly employed in both serious and playful activities, and which (depending upon the position of the fingers) presented varying sensible group formations (the clusters of fingers), must accord-ingly have come immediately to mind for the imitation and sym-bolization of corresponding groups of arbitrary other objects? Thus the "finger numbers" arose within sign language as the first number signs. Indeed we can very well claim still more: it is as a rule only on this path of the sense perceptible that a sharp differentiation and classification of the determinate number forms could first come about at all.
In a certain manner one of course already possessed the number concepts when the analogy of different groups equinu-merous to one another and to groups of fingers was grasped. But only through a constant back-reference from groups of the most various types to the finger groups, sharply distinct in sensible appearance, did the finger numbers rise to the level of Representatives of general concepts, of general characteristics of groups classified in terms of more and less. Without fear of paradox we can say: the concepts 1, 2, 3, ... as the species of the general concept of multiplicity, as specifications of the "how many," first came to a more determinate consciousness in the conceptual signification of number signs on the fingers.”
Of course, but I don't see any particular conflict with what I'm saying. I had the idea that arithmetic and geometry developed greatly with the establishment of the first agrarian cultures in the Nile delta and Fertile Crescent, used for calculating parcels of land and tallying grain harvests and the like.
(I'm quite interested in the basics of Husserl's philosophy of number but it appears daunting.)
[quote=What is Math;https://www.smithsonianmag.com/science-nature/what-math-180975882/]Rovelli [calls into] question the universality of the natural numbers: 1, 2, 3, 4... To most of us, and certainly to a Platonist, the natural numbers seem, well, natural. Were we to meet those intelligent aliens, they would know exactly what we meant when we said that 2 + 2 = 4 (once the statement was translated into their language). Not so fast, says Rovelli. Counting “only exists where you have stones, trees, people—individual, countable things,” he says. “Why should that be any more fundamental than, say, the mathematics of fluids?” [/quote]
It's fundamental because of the way the world is, because we are indeed embodied beings and the world is constrained to exist in certain ways. But according to this argument, there are no necessary facts, everything just happens to be the case - everything is in some basic sense arbitrary, it could just as easily be otherwise. This, I think, is ultimately a form of nihilism.
Elsewhere Rovelli appeals to the Buddhist philosopher N?g?rjuna to justify his 'relational quantum mechanics' but he neglects the ethical dimension of N?g?rjuna philosophy, without which it would indeed be merely nihilistic. Rovelli appears to interpret N?g?rjuna to be saying that 'nothing really exists', which is the common, but fallacious, charge made against N?g?rjuna and the Madhyamika generally. See Bernardo Kastrup's Here I Part Ways with Rovelli:
[quote=Bernardo Kastrup]What Rovelli seems to be now saying is that, although the physical world is constituted of no more than relationships, there is no underlying, non-physical world to ground those relationships. This is problematic for a number of reasons. For one, it immediately runs into infinite regress: if the things that are in relationship are themselves meta-relationships, then those meta-relationships must be constituted by meta-things engaging in relationship. But wait, those meta-things are themselves meta-meta-relationships... You see the point. It's turtles... err, relationships all the way down.[/quote]
That article ends in a way not at all incompatible with the article from Street's OP.
Maths is a construction, matching the way the world is for much the same reason that a glove "just happens" to match a hand - it was made to fit.
Those mathematical objects are constructed, as the statue is constructed from Michelangelo’s Stone.
I agree. That takes it to the realm of the meaning of words: reality.
Quoting Moliere
:up: :up: Spot on, my friend! It's an ongoing exploratory adventure.
Right. I was recommended a book by Fooloso4 - again, a very difficult read - but I found this snippet which exactly described my original intuition as to what the Ancients were seeking through mathematical knowledge:
[quote=Jacob Klein, Greek Mathematical Thought and the Origin of Algebra]Neoplatonic mathematics is governed by a fundamental distiction which is indeed inherent in Greek science in general, but is here most strongly formulated. According to this distinction, one branch of mathematics participates in the contemplation of that which is in no way subject to change, or to becoming and passing away. This branch contemplates that which is always such as it is and which alone is capable of being known: for that which is known in the act of knowing, being a communicable and teachable possession, must be something that is once and for all fixed. [/quote]
Bertrand Russell remarks, in the History of Western Philosophy, in his chapter on Pythagoras, that it is this combination of mathematics and mysticism which distinguishes the Western philosophical tradition from the Asiatic. It is also, one could argue, why the 'scientific revolution' occured in the West (contentious claims, I know.)
This kind of insight goes back to the Platonist distinction between the 'unreliable' testimony of sense, and the (supposedly) apodictic certainty of rational or mathematical reason. Of course there's been a lot of water under the bridge since, but I still feel there's an element of truth which has been largely forgotten. That's the thread I've been following.
Quoting Streetlight
There is no junk, there is only mathematical truth. An example is chess: Chess is a concept built on a few simple ideas and rules. Once these ideas exist chess exists, potentially, in its entirety. But this entirety contains good chess and 'bad' chess. All possible games of chess from genius to silliness. They exist because chess as a concept exists. If math exists all math exists potentially. Is there a difference between an actual Platonic realm (containing good math) and a potential Platonic realm (containing 'junk')?
Quoting Streetlight
Very woolly thinking here. Numbers exist as an abstraction, there is no need to have 2 or 3 actual things to have numbers:
/ = 1
// = 2
/// = 3 etc.
If all mathematics exists then it is natural for our experience to awaken (induction) particular aspects of mathematics. eg. If there are sheep their multiplicity might awaken numbers in our intuition. If we lived without a need for numbers their existence might not have been awakened by our experience. But numbers would exist anyhow.
As for the Platonic realm - does it mean that the number 3 exists in some concrete reality or does it mean that in the depths of mathematical reality there is a potential for '3' to exist - depending on what events bring it into existence? That is, is there a mathematical potential, above our specific forms of math, that makes these forms of math possible? If we say mathematics 'exists' we have to be very clear on what we mean by 'exist'.
The concept of chess can exist without a single game of chess being played in 'real' terms. In this situation, does chess exist if it exists as a concept but no games are ever played? What is the difference between a game of chess that is played and one that is merely possible, because the concept of chess exists? I think the Platonic realm does exist in the sense that it makes all kinds of math possible but not necessarily realized. I don't think he is getting very far by making a distinction between math that we have become aware of and math that might have been if the world had been different.
Maybe the Platonic realm is God's Mind, which contains all possible forms of math in the way the rules of chess makes all chess games possible.
Quoting EnPassant
This is an excellent point. Once a concept is defined within an existing framework of mathematics, in a sense all that logically flows from it is potential, awakened by diligent investigations and discovery. The question remains, When a new concept seems to appear out of nowhere, is that creation or discovery?
(I speak from personal experience, not philosophical conjecture)
:100: :clap: Well said. The way I put it is that numbers are real, but they're not existent in the sense that phenomenal objects are existent, but mainstream thought can't accomodate this distinction because there is no conceptual category for intelligible objects, in the Platonic sense.
Quoting Wayfarer
Yes, mathematical potential exists because our forms of mathematics exists - linear equations, set theory etc. So the question is, where is this potential? Is it merely inside our skulls or does it exist independently of the human brain? Is it universal?
You may have answered this earlier in the discussion, but my obvious question is: what do you mean by real? If you take a Kantian view of the matter, mathematical objects are universal, necessary, and objective. All good so far. But at the same time, this objectivity for Kant is possible only via the subject of experience, by means of the faculties of understanding and intuition. Rather than full-on objectivity, this might be closer to intersubjectivity, in that it's only objective within the realm of human subjects.
Doesn't this almost look more like mathematical psychologism than platonism? The latter would demand that mathematical objects are entirely independent of human minds, and Kant is not quite able to say that, no matter how much he'd like to.
So, do we bite the platonic bullet and assert that Kant underestimated the realness of mathematical objects, or do we retreat to the Kantian middle-ground?
Excellent question. And the fact that this causes us to ask 'what we mean by "real"' is central to the whole matter. As you say, Kant is usually said to adhere to conceptualism, which is a kind of middle ground regarding universals. But my objection is that the rules of logic and arithmetic are the same for all who think. The paradoxical quality which this implies is that whilst they are independent of any particular mind, they can only be grasped by the mind. So they're mind-independent, in the sense of being independent of any particular mind, but only perceptible by reason. I think that's suggestive of the not-often-discussed philosophical attitude of objective idealism.
So again that raises the whole question of the nature of their reality. The usual response is
Quoting EnPassant
...because we're accustomed to thinking of what is real as being 'out there somewhere'. But notice that underlying this question is the implicit division of self-and-world - the sense that what is 'in here' (the activities of the mind) and what is 'out there' (the objective domain) are exhaustive of what is real. That is the implicit metaphysic of modern individualism.
Note 'Augustine on Intelligible Objects':
[quote=The Cambridge Companion to Augustine][i]1. Intelligible objects must be independent of particular minds because they are common to all who think. In coming to grasp them, an individual mind does not alter them in any way; it cannot convert them into its exclusive possessions or transform them into parts of itself. Moreover, the mind discovers them rather than forming or constructing them, and its grasp of them can be more or less adequate. Augustine concludes from these observations that intelligible objects cannot be part of reason's own nature or be produced by reason out of itself. They must exist independently of individual human minds.
2. Intelligible objects must be incorporeal because they are eternal and immutable. By contrast, all corporeal objects, which we perceive by means of the bodily senses, are contingent and mutable. Moreover, certain intelligible objects - for example, the indivisible mathematical unit - clearly cannot be found in the corporeal world (since all bodies are extended, and hence divisible.) These intelligible objects cannot therefore be perceived by means of the senses; they must be incorporeal and perceptible by reason alone. [/i][/quote]
This is of course strongly and adamantly rejected by Rovelli and empiricist philosophers generally. Oil and water, because of its obviously theistic heritage and implications (after all, Augustine is said to be the 'third most senior Christian' behind only the Apostle Paul.)
So where I'm coming to is that number (etc) are real as 'structures within reason'. They're concepts, but not as the product of the mind. They are real as the constituents of reason, what Frege described as the 'laws of thought' (see Frege on Knowing the Third Realm, Tyler Burge.) They're how the mind orders and organises its experience in the world, but they're not themselves part of experience (transcendental in the Kantian sense.) Hence, real, but not corporeal. Which is why it is incompatible with naturalism and empiricism.
Quoting Wayfarer
We might combine these two questions, to ask what does it mean to say that potential is real. The best way to look at this, in my opinion, is in respect to the nature of time. The reality of "potential" can be found to inhere within the way that time passes at the present. In relation to the future, there is real possibility as to what will come to be. This real possibility constitutes the reality of potential.
From the perspective of the living breathing human being, there is real possibility (therefore real potential) with respect to future acts. This real possibility is what gives human beings their power of choice, and their power to create. Mathematics is a great tool in exercising this power, therefore the reality of mathematics, in our understanding of it, is related directly to human potential.
But this opens the question of how human potential is related to real potential. We, from our human perspective, comprehend real possibility to inhere within the passing of time. The passing of time provides us with real possibility in future acts. However, it appears to us, that this real possibility requires the human mind to manifest its realness. How this could be the case is extremely difficult to grasp. How could it be that physical existence appears to progress in a completely determined manner of causation, yet somehow the human mind grasps real possibility to inhere within this determined world?
This is to say that the physicist will model the passing of time in the physical world as deterministic, and maybe even some would claim that this is a real representation of the world, yet this model excludes the reality of possibility. Then the philosopher will step in and say wait, human potential demonstrates real possibility. Now we get a sort of compromised understanding. The compromise is to say that there is real possibility, real potential within the world, but that real potential only exists as a property of the human mind, as ideas and conceptions within the human mind.
Any rigorous analysis of this compromised understanding will demonstrate that it is faulty. If the human being has real capacity to change things in the world, the potential for change must inhere within the world itself, in order that the world itself may be changed. And if the capacity to change things in the world is only a property of the human mind, it is an illusion, a falsity. The one perspective is that of free will. The other perspective is that of determinism. The compromised understanding is compatibilism.
I think it’s the ‘realm of possibility’ and that it is a real realm, in a way analogous to ‘the realm of intelligible objects’.
I agree, and I see a problem with the determinist attitude. Describing activity in the physical world in terms of efficient causation has been a very useful and practical venture. The problem is that this descriptive format has limitations which the determinist ignores or denies. We find that within human beings there is an active mind, working with immaterial ideas, to have real causal affect in the physical world. Causation from the mind, with its immaterial ideas is described in terms of final cause (goals purpose and intent), choosing from possibilities, which is completely distinct from efficient causation.
So there is a very real need to recognize the limitations of "efficient causation" as an explanation of the activities in the physical world. And we need to accept the reality of the immaterial "final cause" as having real efficacy in the material world.
According to one of the two main accounts of causality, namely the perspectival "interventionist" interpretation, a causal model is a set of conditional propositions whose inferences are conditioned upon variables that are considered to have implicative relevance but which are external to the model, such as the hypothetical actions of an agent. These models, whose use is now widespread in industry and the sciences, are thus naturally "compatibilist" in conditioning all models inferences upon hypothetical or possible values of external variables that are considered to be chosen freely. So I presume you are criticising earlier historical conceptions of causality such as Bertrand Russells', which assumed a causal model to be a complete description of a system's actual dynamics (thus making cause and effect redundant notions).
What I don't follow is the relevance of a "final cause", unless it is surreptitiously being used to refer to an initial cause, i.e. a bog standard cause. For example, if I am working to build a shed in the back garden, what is the "final cause" of the shed here? Obviously my thoughts, goals and motivation throughout the project cannot be considered a literally "final" cause, which speculation notwithstanding, leaves the resulting actual shed as the only remaining contender for the final cause. Are you insinuating that the resulting shed caused me to build it? (which incidentally isn't likely to look anything like my imagined shed due to my terrible practical skills)
I think chess is a good analogy. Once the concept of chess exists all possible chess games are given potential. Once a chess game is played (even in one's mind) that chess game becomes real.
Very illuminating, thank you. Also has relevance in quantum physics, I would think.
Quoting sime
I'll let MU answer for himself, but I had thought the Aristotelian 'final cause' was 'the reason why x exists'. For instance, the final cause of a match is fire, as that is the reason why matches are made. That's also a good example, because the efficient cause of the fire is the match, which says something about the possible relationship of 'efficient' and 'final' causes.
Quoting EnPassant
I'm a follower of an excellent chess channel on Youtube, hosted by an ebullient Serb, Agadmator. He makes a point of saying, in every game, the point at which 'this position has never previously been reached before, from here on. it's a totally new game'. Usually happens around moves 8 -11. I guess databases must be used to identify that, but it serves once again to remind one of the infinite number of possible combinations of moves in Chess.
I think 'chess is the possibility-space (i.e. actuality) of all chess games and players are the potential realizers of all chess games' is clearer.
Look at it this way. The "variable" in the model described is a freely chosen act. But from the perspective of the agent, the chosen act is not a variable. It is known by the agent, chosen, and in that way determined. So to model the chosen act as a variable does not provide a good description of what a chosen act really is.
Quoting sime
"Final cause" is the intent, the purpose. So it is exactly the case that your thoughts, goals, and motivation are literally the final cause of the shed. Whatever reason you had, whatever purpose you had in your mind, this is the reason why the shed was built. Therefore these ideas, as intent, are the cause of your actions, and by extension the cause of existence of the shed. This is the basis of the concept of "intent" in law, the decision to bring about consequences.
That is why "variable" does not serve as an adequate representation. The fact that you wanted a shed, and this motivated you to go out and built a shed, is the cause of the shed. And you could further specify the particular purpose you had in mind for the shed when you built it. The intent, purpose in mind, or "final cause", is not a "variable" in the coming into existence of the shed, it is the cause of existence of the shed
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:up: I think the whole idea of final causation was a casuality of the Scientific Revolution and the rejection of scholastic/Aristotelian ideas of causality. Note however Aristotle's Revenge by Edward Feser
Yes, I have looked at some of his videos. V. good.
I think that's right, that "Revolution" came along with the rejection of Aristotelian philosophy. It started when Galileo and others demonstrated faults in his physics. Then there was no need to teach his physics, and this attitude progressed through his other topics, and eventually even his logic was removed from the standard curriculum.
As a part of his physics, "final cause" was an early casualty. It was completely removed from the field of physics, as irrelevant. The social sciences, such as law, replaced "final cause" with "intention". Now, "intention" retains the status of "final cause", as causal. But in as much as intention is seen as causal in law, this is far removed from physics, so the relationship between these two types of causes, efficient cause and intention, is not very well upheld in any discipline.
Because of this, we have no accurate representation of intention as a cause in the physical world, physics using "efficient cause". There is intention in law, where it is implied that intention is a cause, and there is efficient cause in physics, where it is presumed that there are no other forms of causation. As a result, intention is commonly comprehended as a form of efficient causation. Then there is no understanding of "final cause" at all, in any scientific discipline.
You can see this from sime's reply. The idea that thoughts and goals caused the existence of the shed is off-handedly rejected, because it is inconsistent with the understanding of "cause", as efficient cause. After this off-handed rejection, sime is left with the incoherent proposition 'the shed caused me to build it', as a representation of final cause.
I understand Aristotle's definition of a 'final cause', but it makes no sense to me to muddle such "final causes" with the "causes" meant by the modern scientific definition of "causes" that refer to experimental inventions that go on to produce measurable effects. Especially considering the fact that 'Final Causes' are reducible to iterative evolutionary or adaptive feedback loops between an agent or population and their environment that are understandable in the bog-standard "initial cause" sense.
'Final causes' might be reasons with cognitive significance but imo reasons and causes are best kept apart, for they don't obey the same logic.
You don't seem to understand causation sime. There is no scientific definition of cause. Cause is a philosophical concept.
I think that's misleading and somewhat inaccurate. It is true that causation didn't undergo strict formalisation until the twenty first century, economists being among the earlier pioneers of causal modelling in the twentieth century, and that causation is still undergoing formalization in tandem with the brother concepts of probability and temporal logic. Nevertheless, there has been a rapidly converging consensus in both the scientific community and industry in recent decades to the formal identification of causes with particular variables of a probability model, that if intervened upon by the actions of an experimenter, are expected to produce observable changes in the correlations among variables that lie "downstream" of the intervention. See Judea Pearl for an authoritative account.
Causal models merely express the concept that doing something leads to observations that otherwise wouldn't occur. Unlike Russell's conception, the modern meaning of causality is counterfactual. Causal models essentially define causes as being 'initial' with respect to the causal orders they define or describe, making "final causes" an oxymoron in the sense of the causal order.
Nevertheless, causal models have nothing to say regarding the order and linearity of time itself unless their variables are given additional temporal parameterization. All that they demand is that causes are considered to be controllable preconditions of their effects, not that causes are necessarily temporally prior to their effects in some absolute sense, which might well be considered a matter of perspective.
If this is true, it's proof that there is no formalized definition of "cause".
And, since there are two distinct principal types of causation, efficient and final, there will never be an acceptable formalization of causation until the relationship between the two is represented properly. Formalization of one principal type of causation while excluding the other principal type of causation does not give a true formalization.
Quoting sime
This is exactly why a formalization is impossible, and causation will always be philosophical rather than scientific. This provides no basis toward understanding the cause of "doing something". So, a person does something and this causes something which otherwise wouldn't occur. If we want to know whether the thing which otherwise wouldn't have occurred is intentional, or accidental, we need a much better principle than this. And if you claim that this is irrelevant to "causation", all that matters is whether the thing otherwise wouldn't occur, you fail to properly represent "final cause" in your formalization, and you provide no principles for excluding accidents from our actions. However, it's quite obvious that the effort to exclude accidents is very important.
Quoting sime
The use of "final" in "final cause" seems to be misleading you. "Final" is used in the sense of "the end", and "end" is used in the sense of "the goal" or "objective". The terms "end", and "final" are used when referring to the goal or objective because the intentional cause is what puts an end to a chain of efficient causes when looking backward in time. So if D caused E, and C caused D, B caused C, and A caused B, we can put an end to that causal chain by determining the intentional act which caused A. It is called "the end", or "final" cause because it puts an end to the causal chain, finality.
Take a chain of dominoes for example. We look at the last fallen domino and see that the one falling prior to it caused it to fall. Then the one prior to that one caused it to fall. When we continue to follow this chain of causation, we find the intentional act which started the process, and say that this is "the final cause", because it puts an end to that causal chain. The terminology is derived from our habit of ordering things from the present, and looking backward in time, so that the causes nearest to us at present appear first, and the furthest are last.
Quoting sime
This I do not understand at all. The fact that accidents are still considered to be caused, demonstrates that causes are not necessarily "considered to be controllable preconditions". Furthermore, I've never heard of a causal model which allows for a cause to be after its effect. You simply create ambiguity here by saying "in some absolute sense" because the principle of relativity of simultaneity allows that from the perspective of different frames of reference, the temporal order of two events may be reversed.
The fact that you say the cause is a "precondition" of the effect, implies a temporal order in itself. So to say that causes are not necessarily temporally prior to their effects is blatant contradiction whether or not you qualify this with "in some absolute sense".
But "Final causes" are representable in terms of bog standard causation without invoking teleological purposes, as demonstrated by reinforcement-learning algorithms that train a robot to implement "goal seeking" behaviour via iterative exploration and feedback . In this case, one might say that the "final cause" of the trained agent's behaviour is the trained evaluation function in the agent's brain that maps representations of possible world states to their estimated desirability. In other words, the final cause refers not to the actual goal-state in the real world that observers might colloquially say the learning agent "strives towards", but to the agent's behavioural policy and reward function that drive the agents behaviour in a mechanistic forward-chain of causation from an initial cause in a manner that is teleologically blind.
The agent's actions are not being "pulled" by the goal in any literal sense, so I am at a loss as to the incentive for mixing up purposes which refer to behaviour that converges towards a goal state, and causation which makes no reference to goal states.
Quoting Metaphysician Undercover
If you accept the distinction between purposes and causes, then there is no case for the concept of causation to answer to regarding the distinction between intentions and accidents. For that's purely a matter of teleology and not causation.
Quoting Metaphysician Undercover
A is at the beginning :) Either a "final cause" is used to refer to a bog-standard initial cause that implies none of the teleological controversy commonly associated with aristotolean "final causes", else "final cause" refers to a teleological concept such as a purpose that is defined in relation to a goal state that is external to an agent's brain and that plays no causal role in the agent's behaviour, despite the fact the agent's behaviour converges towards the goal state.
Quoting Metaphysician Undercover
I suspect you are deviating from the commonly accepted notion of "final cause". The whole point of the "finality" in "final cause" is to imply that teleological concepts are necessary for explaining the effects of causation, which isn't the case in the dominoes example; teleology is explainable in terms of purposeless causation, as AI programmers demonstrate. But causation isn't explainable in terms of teleology. To mix up the concepts leads to confusion.
Quoting Metaphysician Undercover
Which demonstrates the point i was trying to make, that what we call the "temporal order" has to be distinguished from the "causal order". That A causes B but not vice versa, doesn't necessitate that A occurs before B in every frame of reference. Also recall the time-symmetry of microphysical laws, models of backward causation etc.
As I explained already, this does not give a true representation of "final cause" because it provides no real basis for a distinction between consequences which are intended, and consequences which are accidental. In other words, if final cause was truly determinable from an agent's behaviour, all accidental acts by the agent would necessarily be intentional acts.
Quoting sime
Exactly, and that's why the model fails. Final cause is teleological purpose, by definition. You give me a model without purpose and teleology therefore your model models something other than final cause.
Quoting sime
Why do you think that "purpose" ought to be defined in "a goal state that is external to an agent's brain"? Obviously, the goal which motivates (causes) one to act is within the agent's mind, and nowhere else. Furthermore, the truth and reality of acting toward goals is that such actions are not always successful. So the external state which is brought into being (caused) by the agent's actions is not necessarily consistent with the goal which motivated the action. Therefore the only true representation of the motivating factors (causes), must be to represent what is in the agent's mind.
Quoting sime
You "suspect" something, but according to what I've stated above, you are obviously quite wrong in your suspicious mind. "Final cause" was proposed as a means toward understanding the purpose behind intentional actions, as the cause of these acts. It's obviously not intended as a means toward understanding the effects, because the effects are plainly observable and do not require teleology.
Take Aristotle's example. Why is the man walking? To be healthy. The action is walking, the cause is the man's desire to be healthy. Whether or not the man actually is healthy or becomes healthy from walking doesn't even enter the scenario. We see him walking, we ask for the cause of him walking, and it is his idea (goal), to be healthy, which is the cause. We cannot judge teleology from the effects because often the person's ideas and beliefs are incorrect. Therefore the effects do not properly reflect the cause in a logical way. The man might become ill from walking, and we would never know that the cause of him walking was to be healthy, unless he told someone this.
Quoting sime
Since temporal order is what defines causation, separating the two only renders causation as unintelligible.
Firstly, what makes you think that there is an objective matter of fact as to whether an effect was intended or accidental? Secondly, if there are such facts, then what do those facts consist of?
If we narrowly interpret the meaning of an "intention" as referring only to the agent's internal state, , then intentions as such cannot be teleological, for the agent's actions are explainable without final causes.
So in order for intentions to be considered teleological, one must consider both what is going on inside the agent as well as the environmental effects that the agent's behaviour produces, - effects which play no causal role in the agent's history of decision-making. Yet this understanding of 'intentionality' as a type of relationship between the agent's behaviour and the environmental biproducts of his actions, in turn implies that the agent is fallible with regards to knowing what his intentions are. For who now gets to decide what the agent truly intended?
Note that the problem of "Inverse Reinforcement Learning" is the problem of inferring an agent's overall goals from a history of the agent's behaviour, including the environmental consequences it's actions. It is a chicken-and-egg paradox; In order for observers to estimate an agent's overall goals given a history of it's behaviour, they must assume that the effects of the agent's actions were in accordance with it's intentions, that is to say, they must assume that the agent is an expert who understands his environment. But how can it be known whether the agent is an expert? Only by assuming what the agent's goals are :)
This implies that teleological concepts are either semantically or epistemically under-determined.
Quoting Metaphysician Undercover
Yes, if taken in the very hard sense of "separation". I'm referring to the fact that different observers from different perspectives, each of whom controls different variables, might have conflicting views as to what was the cause/intervention and what was the effect in a given situation.
Suppose Alice believes that if she presses button A, then a distant observer Bob will press Button B, otherwise Bob won't press button B. No other information is assumed.
Her causal belief might be represented by A => B.
Logically, this is equivalent to asserting NOT B => NOT A.
Therefore, in the event that Alice decides not to press the button, i.e. that event NOT A occurs, shouldn't Alice be open to the possibility that her decision not to press A was the effect of Bob deciding on NOT B 'before' Alice made her decision?
Posited examples of backward causation look a bit like teleology, but are categorically different. ,
Just saw this, but have to say excellent post. Clear and interesting.
Would be nice to have decent mathematical skills to delve into this topic with more detail, but, I suppose basic arithmetic already offers plenty of food for thought.
Have a glance at the wikipedia entry on the word teleonomy. It is a neologism coined in 1958 by a biologist to accomodate the awkward fact that virtually everything in biology is goal-directed, while trying to differentiate it from the Aristotelian 'teleology', a boo-word for modern science.
Furthermore, there's been an increasing recognition of the significance of telos and teleology in biology, with Aristotelian ideas being re-considered. An idle search of Aristotle and DNA will return some interesting papers on that subject.
Thanks! I was terrible at school maths, much to my later regret in life, but the point is philosophical rather than arithmetical - as jgill says, many maths educators are not the least interested in the philosophical question.
I must admit that in my old age one philosophical issue does interest me: where does the set of potentials created by a "new" concept or discovery reside? And what triggers actualization in the arena of mathematical knowledge? My mathematical interests have always been in infinite compositions, and I see a structure therein that might model this process. :cool:
I'm not conversant with his areas of expertise and that is usually a substantial impediment in mathematics. But I appreciate you pointing him out. :smile:
It doesn't matter as to whether there is such an objective matter of fact. What matters is that it's a useful distinction which demonstrates your model as faulty. The example demonstrates this. The man walks for the purpose of health. Health is the man's purpose for walking. If the man then proceeds to get an injury and dies from walking, we cannot conclude that getting sick and dying was the purpose of his walking, because this is contrary to his true purpose.
That it is a "subjective fact" that he was walking for his health, rather than an objective fact, is irrelevant to the reality of the situation. And as philosophers, what we are trying to understand is the reality of the situation. We are not attempting to constrain "reality" to objective fact, when reality also consists of subjective facts.
Quoting sime
No the actions are not explainable without final cause. That's the point to the example of accidents. Such an explanation would be wrong, like in a court of law when they demonstrate from the physical evidence that the perpetrator's intentions were X, when in reality the intentions were Y. The explanation is wrong, plain and simple.
Quoting sime
Again, this is wrong. The environmental effects produced ny the intentional action cannot enter into a true understanding of the agent's intentions, because they mislead, as I just explained. The only things which can enter into such a determination are the precedent conditions. This is the only way to give a true representation of the position which the agent is in at the time. The agent, at the time does not have access to the outcome of the actions being deliberated on, therefore in understanding the agent's mind-set (intentions) at that time we cannot allow the outcome of the agent's actions to influence our judgement, because the outcome might be totally inconsistent with the intention, as explained. This becomes extremely relevant when the agent's intent is to deceive. In this case, the actions are intended to mislead.
Quoting sime
Yes, this is exactly the problem. That teleological concepts are "under-determined" is very obvious to me, because of the subjective nature. Is it not obvious to you?
Quoting sime
No, if Alice believes that pushing A will cause Bob to push B, as your premise states, then there is no stated premise which denies Bob from pushing B even without Alice pushing A. You'd need to state that B occurs if and only if A. But then B is completely dependent (causally) on A, and there is no indication that not B could cause not A, as this would require a reversal of the dependence, and there is no statement of A if and only if B. Therefore Alice is continually free to push A at any moment of time, and the fact that Bob has not yet pushed B has no relevance because Alice's choice is dependent on something else.
This is not true in my view. Not all empirical phenomena are corporeal. A rainbow is not corporeal, for example. An atom, an electron, a photon, a quark—corporeal?
And also relations and functions are not corporeal, in the sense of being being embodied or objects of the senses, even though they may be instantiated as or in a series of discernible material states.
Corporeal definition - of the nature of the physical body; bodily.
material; tangible:
corporeal property.
Rainbows comprise light refracted through water droplets. Nothing incorporeal about that.
Quoting Janus
Part of my point.
The point was that there is no body of the rainbow: it is not tangible, cannot be bodily felt, even in the subtle way that clouds can be felt.. It looks like it is a corporeal object, but it is a purely optical phenomenon.
As I said about relations and functions or processes, they are entirely comprised of series of observable physical states, yet they are not, taken as a whole, corporeal objects, yet they are entirely physical. If you want to say that a relation, process or function is not physical then you should be able to identify their non-physical components.
As for whether relations and the like are ‘entirely physical’ - I would call that into question also. Consider the models of mathematical physics - insofar as they are mathematical models, then they synthesise physical observations into a mathematical and rational framework. And the question of the nature and unreasonable efficacy of mathematics in the natural sciences is the point at issue.