<the objectivity of mathematics and the undefined symbol>
In the theory of computation (for me math at its most objective), the symbol is often left formally undefined. By 'symbol' I mean a single character like 'a' or '1.' (It is also allowable to understand '1de' as a single symbol. We'd be creating a new glyph from familiar glyphs. )Once a symbol is defined, theoretical computer science can go very far with a theory of strings and sets of these strings called languages. It can comment on whether a language can be recognized with a finite amount of memory (number of states.) It can comment on whether a language can be recognized with an infinite amount of memory (which is essentially an infinite number of states). Since languages can be understood in terms of answers to objective mathematical questions (about whether still other machines will accept this or that language, for instance), I find this kind of science highly philosophically relevant. But the edifice is built on the intuitive recognition of this undefined thing, the symbol.
If I use a pencil or a typewriter to make the mark 'a' several times, then these individual marks are not strictly identical. Moreover they are in different places. But we can recognize the marks as representing or intending the same symbol. A rose is a rose is a rose, and an 'a' is an 'a' is an 'a.' If I can't recognize that two different but similar marks intend the same abstract, distinct but otherwise undetermined entity, then I cannot understand Sipser's Theory of Computation, for instance.
So the first step into objectivity that can be made explicit (creating strings and languages formally from symbols) is founded upon a step that cannot or has not been made explicit (understanding what a symbol is or how they work). To be clear, I think this inexplicit step is objective. I trust theoretical computer science more than I trust most disciplines. I just find this dark, first leap fascinating, and I also think there's an analogy in ordinary language (the dark, first step) and science/philosophy (pursuing objectivity from an inexplicit basis.) I mostly hope to start a conversation, and it would be great to meet others, also, with a passion for the theory of computation. Perhaps because it is so 'concrete' and (in my view) for that reason especially objective.
If I use a pencil or a typewriter to make the mark 'a' several times, then these individual marks are not strictly identical. Moreover they are in different places. But we can recognize the marks as representing or intending the same symbol. A rose is a rose is a rose, and an 'a' is an 'a' is an 'a.' If I can't recognize that two different but similar marks intend the same abstract, distinct but otherwise undetermined entity, then I cannot understand Sipser's Theory of Computation, for instance.
So the first step into objectivity that can be made explicit (creating strings and languages formally from symbols) is founded upon a step that cannot or has not been made explicit (understanding what a symbol is or how they work). To be clear, I think this inexplicit step is objective. I trust theoretical computer science more than I trust most disciplines. I just find this dark, first leap fascinating, and I also think there's an analogy in ordinary language (the dark, first step) and science/philosophy (pursuing objectivity from an inexplicit basis.) I mostly hope to start a conversation, and it would be great to meet others, also, with a passion for the theory of computation. Perhaps because it is so 'concrete' and (in my view) for that reason especially objective.
Comments (39)
Character recognition software is not foolproof but then neither is human character recognition
'Mavis, look at your brother's letter here! Do you think that's an ess or a zed?'
Right. In fact our character recognition is imperfect, but it's not practically a problem. I suppose I would just comment that justifying or making explicit character recognition would happen outside of theoretical computer science (which as I understand it is a kind of pure math.) Actual computers would rely on physics, induction, etc. Roughly, the idea of the deterministic computer seems to be founded on the 'given' understanding of the symbol as symbol. Turing, for instance, assumes that his machine can recognize the symbol in the scanned square. This symbol is something like a Platonic form, though we use marks as an aid to memory. I think Turing wanted his machine to be as concrete and mark-fed as possible, but categorizing marks gives the symbol. Better said, the informal [s]program[/s] ability that maps an infinite set of marks to a finite set of intended symbols is assumed. Moreover, the mark printed by the machine is really a symbol if it is to be of value.
I think you're possible over-valuing objectivity. The point I would make is that mathematics is not necessarily objective, because it's purely inferential or logical - if this then that. And this is why mathematics can be used to determine what is objective. That is what is done through quantitative analysis, after all: you create an hypothesis, and then look at the data, to see how it fits - which is, arriving at an objective value. All of that relies on mathematical analysis. So mathematical reasoning is in some sense prior to objectivity.
I suppose what I especially mean by 'objective' here is 'not a matter of opinion.' If physics is objective, this depends on the uniformity of nature, an often implicit axiom of the gut. Math would be objective, I'd argue, as long as the truth of the implications you mention is established. It would rest (ideally) on logic alone, and not on experience and the uniformity of nature. In the theory of computation, not much is used beyond mathematical induction and proof by contradiction. The implications it establishes seem timeless. It seems logically impossible that they should become false.
That is true, but I still think ‘objectivity’ has epistemological or metaphysical implications which are unwarranted in the context. I’m not just picking nits here, either. The roots of mathematical philosophy are as ancient as philosophy itself - in the Greek tradition, the two are intertwined. But ‘objectivity’ as a criterion of truth is, I contend, something that is of very recent vintage, and carries an implication which I think is quite foreign to philosophy of mathematics in the traditional understanding.
Is the language that is 'printable' (as I like to think of it) by a right linear grammar a 'regular' language (recognizable by a deterministic finite acceptor)? Yes. How do we know? Because we have an algorithm for converting such a grammar into an associated DFA. I can be certain of this. I may waver forever about the metaphysical interpretation of the symbols, but a functional understanding of the symbols is enough to establish the theorem. A person might say that math deals with the abstract 'how' of symbols. I don't have to know what they metaphysically are (if this is even [s]meaningful[/s] an objective question) to obtain objective knowledge about what is possible with them in particular formal 'games' (models of computation, for instance.)
If you are only saying that 'objectivity isn't everything,' then sure, of course not. But denying the objectivity of math strikes me as an attempt to hijack the usual meaning of a word 'objective.' (I looked it up just to make sure I wasn't imagining things --to make sure that I was being objective.)
The beauty of a symbolic approach to math is its metaphysical neutrality. Can you not relate to the joy of occasionally getting beyond the field of opinion into a field of amusing/useful objective truth? This is made possible by a move from the 'what' to the 'how.' The essential 'what' of the the symbol (apart from the givenness of it), the rules that connect it to other symbols in various ways, is arguably this same 'how.' I can grant that it is not a science of marks. It is roughly a science of symbol crunching, whose objects are publicly available albeit non-physical, at least in part. (The science would be impossible without physical marks for the aid of memory and communication.)
https://en.wikipedia.org/wiki/Theory_of_computation
Yes - but then you’re in engineering, not philosophy as such. You’re still concerned with instrumental utility. I’m not trying to hijack the meaning of the word ‘objectivity’, but to draw attention to that as a criterion. I mean, almost everyone would pass by ‘objectivity’ without a murmur, but I think it betokens a deeper issue. I will leave it at that here, there are some far more learned philosophers of maths on this forum who hopefully will chip in. Consider it a footnote. :-)
1+1=2 is a human symbolic creation.
1 apple + 1 apple = 2 apples is a human observation if two humans agree on what is an apple. Hence the manner in which math is first taught in school.
Whatever is outputted by a computer (a tool created by humans) must still be interpreted by humans with different opinions. Hence the phrase garbage in garbage out.
Objectivity cannot exist in a subjective human experience.
This fundamental first step, to overlook the fact that two distinct instances of "a" are not identical in an absolute way and are therefore not actually "the same", for the sake of calling them "the same", is the basic incoherency of logic. Some will dismiss it as a difference which does not make a difference, but if we are discussing the law of identity, then it is necessary that any difference makes a difference. Otherwise we undermine the meaning of "the same", and allow the contradictory notion that two distinct things are one and the same thing.
A similar incoherency is found in the first step of mathematics, relating to the nature of "unity". The numeral "1" signifies a basic unity. The numeral "2" signifies two distinct unities, but also one unity as "two", at the same time. In performing mathematical operations we must overlook the fundament fact that "2" signifies two distinct unities, (just like we must overlook the fact that two distinct instances of "a" are not the same), and treat it as if it represents one unity.
These incoherencies are fundamental to the logical process. Nevertheless we must overlook them, ignore them, to proceed into the logical realm. However, they are significant, and these flaws indicate that mathematics and logic are less than ideal.
I see what you mean, but I also think logic is impossible without this first step. In other words, the logic you think is violated by the many-to-one map from marks to symbols is itself founded on this map, at least to the degree that it can be formalized. All abstractions seem to include this many-to-one map/function. I see a particular dog and recognize it as a dog by negating its particularity. I place it in a category. It seems to me that knowledge is impossible without this function from particulars to categories.
Quoting Metaphysician Undercover
Again, I see what you mean. But I think we'd have to get into specific philosophical visions of mathematics. In a unary representation (often used in Turing machine computations), the number that has representation '2' in base 10 has the representation '11' (or '111' if zero is encoded as '1'). I agree that conceptually the number 2 is a unity of basic unities. When numbers are constructed in set theory, they are sometimes sets that contain all preceding numbers, where 0 is the empty set. Or 0, {0}, {{0}}, ... where the 'depth' of the empty set represents the number. In these representations, the nested 'unification' is apparent.
Quoting Metaphysician Undercover
I think math and logic are pretty ideal. For them not be to ideal, in my view, would mean that we could imagine something better. If we could imagine something better, then that would already be within math and logic, since anything better would conform to mathematical and logical requirements.
Economically I've specialized in science, but I've read many philosophers. I try to avoid name-dropping and an addiction to any particular jargon and instead approach issues in ordinary language. Comprehension/assimilation and the value of the assimilated philosophy itself is manifest or not in what I do with it. You may occasionally recognize a paraphrase of a well-known idea.
Perhaps you'll agree that philosophers can be the silliest kind of people as well as the most brilliant. The genre itself therefore has a mixed reputation. If I am to meet a self-proclaimed philosopher, I don't know whether I'm to meet a windbag who dresses up platitudes in mystification or a forger of new paradigms at the top of intellectual culture.
Quoting Wayfarer
As is every human being. But I am more of a theorist/artist.
Quoting Wayfarer
I don't think you actually said anything though. It's as if you pointed at something and then mention that you pointed at something. And now that's a footnote? I'm not trying to be rude, and I'm happy to drop it.
Fair enough. To try and re-state it - my point is that the criterion of objectivity is not necessarily applicable to mathematics (or logic for that matter), as the ‘objects’ of mathematics are only ‘objects’ in a metaphorical sense. They comprise what used to be called the domain of a priori truths. Objectivity introduces another element, which is derived ultimately from empiricism. But I do understand your objection, and acknowledge that I might be drawing a bit of a long bow.
The meaninglessness of the symbols is one of the reasons that it is objective. There are truths about the game of chess that follow from the rules of chess. For instance, if I have only a king left on the board, then there is no way that I can checkmate my opponent. Proof: I would have to move into check to put the enemy king in check and mate, but I'm not allowed to move into check. If the rules are unambiguous, we can obtain objective knowledge about the game using logic. Or is it just my opinion that a lone king can't achieve checkmate?
Quoting Rich
It seems to me that you have to deny objectivity altogether to bring this home. You (pretend to) doubt 1+1 =2 and yet present your doubt itself as a binding, objective truth?
The meaning of '1+ 1 = 2' is philosophically ambiguous. If, however, we specify a particular convention or 'game' of dealing with symbols as symbols, then all of this mystery vanishes --within that context.
I'm inclined to think that some philosophy types relish a corona of ambiguity/mystery for its own sake and don't like even limited, clarifying, purpose-specific formalizations. For me, on the other hand, it's beautiful that we can invent 'spaces' in which strong objectivity is possible. In such spaces, a discovered truth stays discovered (and true) and can be more or less exactly communicated. Socially it's a utopia to that extent. The 'tax' on citizenship in this realm is an intellectual hygiene.
A notion of Strong Objectivity dissolves when one turns the microscope onto it. We can fall back on a notion of Pragmatic Objectivity, which is something like 'most people that have studied the topic would agree'.
Sure, I won't argue with that. For me there is the 'false' doubt of [metaphysics] and then what I'd call genuine doubt. The notion of (infinitely) strong objectivity is arguably bankrupt if not even the most concrete, formal kind of math makes the cut. Because where is an example of such perfect objectivity? To me it 'smells' like something that comes from theology and Euclid in the first place. It's like the white smoke of an evaporated deity.
As you say, we can 'fall back.' But I'd contend that this pragmatic objectivity is what is usually meant in the first place. So perhaps we don't 'fall' anywhere. Instead it's perhaps a vice of some philosophers to misread ordinary language. (Not aimed at you or anyone in particular, but rather especially at my past self before exposure to OLP and other civilizing influences.)
Speaking of OLP, I like to call our inexplicit linguistic knowhow 'ordlang.' Just as ordlang makes philosophy possible, so an in-explicit grasp of the undefined symbol (along with ordlang, of course) seems to make a science of formal systems possible. This is an analogy I mentioned but didn't stress.
I mention this to emphasize that my claim that formal systems are objective presupposed ordlang, but then so does any pursuit of explicit knowledge, it seems to me.
Thanks for the polite response. I think you might be reading too much into the symbolic similarity between 'object' and 'objectivity.' If you look up 'objectivity,' you'll probably found what I found --no mention of objects but rather an emphasis on lack of bias. Here, also, is the top of the Wiki page on objectivity in the philosophical context:
[quote=Wiki]
Objectivity is a central philosophical concept, related to reality and truth, which has been variously defined by sources. Generally, objectivity means the state or quality of being true even outside a subject's individual biases, interpretations, feelings, and imaginings.
[/quote]
https://en.wikipedia.org/wiki/Objectivity_(philosophy)
This is what I had in mind, and this seems to be the standard definition.
Objective about what if they are meaningless? What is objective about this: @_-++&__?
Quoting mrcoffee
Rules are meaningless without a human interpreting them and applying then.
Quoting mrcoffee
Absolutely meaningless until a human applies some subjective interpretation. Often there is all kinds of differences opinion about what math means because of some ambiguity and it ends up in court. I know this from working on insurance policies.
Quoting mrcoffee
Exactly, humans give the symbols meaning and not everyone will agree. Often the consensus is within a highly biased group. Such is the nature of humanity.
You may desire objectivity, but where humans are involved, there never is. What you have possibly it's some concensus within a consenting group - maybe.
Are you familiar with formal systems? Or chess? We can discover objective truths about formal systems. I gave you a simple one.
But I can even answer your insincere question. _-++&__? is a finite sequence of symbols from a familiar alphabet containing one ?, one &, two +, etc. Anyone who can read the symbols as ordered from left to right (or right to left) will presumably agree, though some antisocial types might pretend not to (just as others will pretend to need a proof that they exist in another context, lost in vanity.) I count 8 symbols, so the length of that string is 8. These are objective truths in any reasonable sense of the word. On the other hand, we can tease out the meaning of your English words as words (and not strings) indefinitely.
Quoting Rich
Sure, and the sun rises in the morning. Who has or would say otherwise? Are you sure you aren't tilting your lance at windmills here?
Quoting Rich
I think you still haven't grasped the difference between a formal system and the entirely different issue of its connection to practical reality. Formal systems are about as unambiguous as it gets. That's the point. I'm not sure what kind of bold assertion on my part you think you are responding to.
I suspect that the proportion of the population that would agree that a simple mathematical proof is valid will generally be smaller than the proportion that would say that killing an innocent person is wrong, yet we tend to regard the latter as subjective and the former as objective. I find that strange, and hence tend to avoid making subjective/objective distinctions.
Why? Because you believe so? Unless you are not human, all lying are doing is saying what you believe is objective, which is generally the way it goes.
Quoting mrcoffee
People say all kinds of things about the sun, and it has changed over the eons. All you are doing is repeating a phrase taught to you in school and I guess since it was taught in school and everyone you b bump into probably agrees with you then it automatically becomes objective? No. It is simply a phrase and idea that you and most other in your population were taught. Eons from now, something else may be taught. Everything changes.
Formal systems are created by humans and are subject to change, which is what makes them subjective.
If objectivity doesn't exist, then the word is meaningless. Moreover, if objectivity does not exist, then you are just spouting opinion and bias above. I'm happy with 'reducing' objectivity to a kind of inter-subjective consensus. Others may look for holiness in this or that word, but it's not one of my vices.
To me it's old news, the wisdom of the sophist. All speech is a tool in the hand of irrational will-to-power. Yes and no. It doesn't scare me. It doesn't shatter the gold statue of Objectivity that I keep at the center of a temple of Reason in my back yard. No. It's just ambiguous. That doesn't mean it's worthless or false. It just means that it's unlikely to be objective. How can we agree on something if we don't know what we are agreeing on? I can offer you hundreds of lofty and sinister phrases like that. Having read the wicked books and the 'sentimental'/spiritual books (or enough of them), I'm rarely surprised by personalities these days. It's a special occasion when I bump into a truly original and relevant thinker.
It exists as a concept as opposed to subjective. Whether a human can be objective, I have yet to find the case.
Quoting mrcoffee
Exactly. I am saying the concept cannot be applied to anything perceived by humans. Others have their own subjective opinions about objectivity.
Quoting mrcoffee
Yes, people have their faith and their God and their idols. It just seems that it is part of the human character.
It is indeed, but it's also of historically recent provenance - notice in that article you link to, that it says '"Objectivity" is an aspect of philosophy that originated in the early nineteenth century.' And I maintain that there are historical reasons for that, coinciding with the advent of the naturalist outlook in the modern sense. But I won't persist as it's obviously a pretty abstruse idea - I will take it back to the workshop and spend a bit more time polishing it.
I have only capitalized 'objective' recently as a joke, just to make that clear. For me philosophy is largely about demystification. The demystification of naive views or understandings of objectivity (the deflation of Objectivity) is (for me) old news. I understand myself to be working from a second demystification of philosophical language itself.
For instance, we can say that math is 'just' another language game. We can say that humans are 'just' another animal. But what is this 'just'? It's an intentional ignorance of difference, useful in some contexts and questionable in others. I'm pro-choice, but calling an 8-month fetus 'just tissue' rings false for me. Similarly, putting math on the level of politics strikes me as a bit ridiculous. Yes, we almost universally want murder outlawed. You have picked, however, the least controversial issue possible, with the exception perhaps of laws that protect children. We need only look at real politics (actual controversy) to see the great gap. Or we can look at the endless metaphysical wrangling over the meaning of words. We can look at this very conversation. Educated people can and do disagree on a regular basis about politics and philosophy. But those who are educated enough to read math hardly ever disagree. A student or an expert may be temporarily confused, but generally math is an especially normal discourse. Philosophy, on the other hand, is the supremely abnormal discourse, with a permanent identity crisis --its glory and folly.
I'm not suggesting that there is an absolute gulf between the science of formal systems and other language games, but then I think the concern with absolute gulfs is artificial in the first place. As I see it, it's the philosophers I'm criticizing who need absolute difference, absolute certainty, etc. I see no point in understanding the unqualified word 'objectivity' to have an 'impossible' meaning.
EDIT: On re-reading this, the tone is a little harsher than intended. Too much coffee in Mr. Coffee, perhaps...
If you are saying that humans are biased, then of course. Objectivity is something we pursue. In the case of formal systems there is very little room for bias. Interpreting formal systems from the outside of the system opens up the interstellar spaces for bias.
Quoting Rich
We don't have to see an impossibly perfect instantiation of the concept to apply the concept. I doubt that anyone honestly believes that some disciplines aren't more objective than others. If formal systems give up meaning entirely for almost perfect objectivity, then physical science gives up certain aspects of experience to obtain its own above average objectivity. It's a trade-off.
Quoting Rich
Hey, that statue thing was a joke. But, yes, folks have their idols. Even you and me. But objectivity isn't my idol. It just has its charms.
Some people do, others don't, most couldn't care less. It's just something some people like to do. No harm unless they try to enforce their own subjectivity upon others.
Quoting mrcoffee
My preference is recognizing it as just another human creation and then dealing with it for v what it is. U have no need to gain higher ground by claiming objectivity.
Quoting mrcoffee
If humans and money are involved, forget about objectivity. The more money, the less objectivity. Call it the inverse Rule of Objectivity and Money.
I think everyone cares when they are on that operating table. They sure hope that there is a right way to transplant a heart, for instance. And folks also tend to want their words to be binding on others. A relatively uneducated person can be quite fierce about their political reasoning. Of course I agree that not everyone is interested in science and math for their own sake.
Quoting Rich
I don't remember claiming higher ground, but of course I'm aware that this is the basic rhetorical move. I'd say that objective pursuits are about as far as we can get from the endless sophistry. Of course these pursuits can be taken up (from the outside) and used as tokens by the sophist, but that's a triviality and a different issue.
I think they only hope that the operation succeeds. Even if there is a right way to transplant a heart, doing it that way does not guarantee success, which is all the patient cares about. Further, I bet the humans of 500 years' time would say that the way the world's most esteemed heart surgeons currently transplant hearts is the wrong way compared to what they do.
I'm still struggling to see a need for the concept of objectivity. Instead we can just say that as humans we have evolved to instinctively trust what has worked in the past, so we trust surgeons, techniques and theories about how hearts function, that have worked in the past.
I'm not implacably opposed to the concept of objectivity. I'm ready to be convinced of its usefulness. Indeed I used to be a big believer in objectivity, seeing it as a crucial cleaver that cut through, dividing two fundamentally different types of thinking. But I gradually lost that certainty, and now all I see is differing degrees of confidence in beliefs, some of which are hard-wired into us by evolution.
That is of course their primary hope. But I think they associate the likelihood of success with the trustworthiness or lack thereof of the discipline. Why not drink some magic potion suggested by a random stranger and hope for success? I agree that objectivity is 'cashed' pragmatically. Esthetic reasons aside for the moment, we don't want objectivity for its own sake. We want tools we can trust, tools that won't break in our hand. But this takes us right back to objectivity.
Joe the Metaphysical Plumber tells me a purple spirit in haunting my pipes that he can exorcise for a small fee. Do I trust him? Probably not. Maybe he believes what he says, but maybe that's just his opinion (an idea false/useless for me despite its perceived use/truth for him). And that's probably because I'm familiar with other outlandish claims that didn't cash out. A child will believe perhaps in a giant bunny that brings candy once a year, that a fat man in red brings toys.
At some point I start to reflect on this critical process itself (a pattern in my trusting or not). I come up with a word for the most trustworthy kind of claims. I notice that staying close to uncontroversial facts as a foundation seems to work. These might be facts about stones of different sizes dropped from a height or games of chess. In both cases an inexplicit knowhow grounds the non-controversiality of these facts. (I can't say what exactly it is to understand how a bishop moves, or what a person is, but it's not usually a problem.) Someone claims to find underground water by walking around with little sticks. OK. Let's see if they find water better than someone wandering around guessing. If they fail, it's logically possible that our observation somehow fizzled their power. But who cares? We need their 'magic' to work as needed, and not only when we aren't asking it to.
I may even abandon an interest in explanation and focus on prediction and control, deciding that explanation often muddies the water. I can trust a black box if it gets the job done. That's how most people use their smartphones, I think. Prediction and control gets us near the pragmatic essence. What's left is morale, let's say. But even that is control of mood. Morale is tricky, though, because my mood is not a public entity. My behavior, sure, but whether I am 'happy' or 'enlightened' or 'saved' in some special 'metaphysical/religious' sense seems controversial. I may believe it, and others may shrug or spit. I can insist that it's an objective truth, that I am free of bias. But bias is almost by definition invisible to the biased person. We're back to the plumber believing in the purple spirit. I'm not denying the purple spirit. I'm saying it's going to be invisible for my criterion unless it leads to predictions and control. To be sure, these 'morale' entities obviously affect behavior, and I can imagine a black box that maps pubic words to predicted behavior. But the 'morale entities' themselves seem destined for invisibility in this sense, which is the way that many of their users like them, I think, for reasons that aren't necessarily cynical.
I know this is a long response, but I tried to connect objectivity to a pragmatic epistemology. In short, objectivity is a concept that helps us decide which tools to trust.
Quoting andrewk
Has the concept of objectivity served us well in the past? I'm not saying that this is an easy question. I'd suggest that it has served us well, but then I think most words that remain in the language are probably pulling their weight --even those filthy 'morale entities.'
Yes I agree that logic is impossible without this fundamental first step. But if contradiction is inherent within the first step, don't you see this as a problem? The first law is the law of identity. The second law is the law of non-contradiction. The very first step, to identify, has contradiction inherent within. So we identify first, then the law of contradiction says we cannot contradict. Aren't we obliged to either forfeit the law of non-contradiction, or go back to our mode of identification and rectify this problem of contradiction inherent within identification?
Quoting mrcoffee
I'm no mathematician, but I've noticed that set theory has contradiction inherent within it as well. I do not believe that the resolution to the problem of contradiction being inherent within the first step, is to introduce other contradictions to cover it up.
Quoting mrcoffee
To demonstrate imperfections within something is to demonstrate that it is less than ideal. It is not necessary to show the ideal, in order to demonstrate that what we have is not ideal. It may be the case that there is no such thing as the ideal, and then thinking what you have to be ideal is mistaken, while it is impossible to put forward an alternative as the ideal. To demonstrate problems within a system does not require that one put forward resolutions to the problems.
Someways definitely better than others but people still die. In fact, accidental deaths in hospitals are one of the leading causes of deaths in the U.S. one study putting it at over 400,000 per year, the third leading cause of accidental death.
Indeed. We just generally try to maximize our chances for success.
I'm not sure that we have a problem. We can distinguish between the individual marks and then categorize these marks. For instance, let A = {s2e, 2rt, e42}. Let f be defined on all strings over lower case letters and decimal digits so that f(s) = 0 if s does not contain the symbol 2 and f(s) = 1 if s does contain the symbol 2. Then f(s2e) = f(2rt) = f(e42) = 1. A less formal example would be three different dogs, each recognized as belonging to the category 'dog.' I can only offer this example because we already understand this category 'dog.' My formal example suggests how math basically scrubs 'ordlang' logic of its ambiguity, which is gets from our fuzzy language, so that it's structure can be focused on and examined.
Quoting Metaphysician Undercover
I find the 'law' metaphor a little awkward here. "Each thing is the same with itself and different from another." This is a tautology or close to a tautology. We can give the same thing different names, but we wouldn't generally give different things the same name, remembering that a name would not be a fixed context-independent token in ordlang. I can talk about 'John' successfully if the context specifies which John and enriches the token 'John.' Of course a formal system avoids what could go wrong here by insisting that the token completely specify the entity. In a formal system the token is the entity. So we scan the page, see the mark 'abc' which is microscopically different from the mark 'abc' on the previous page, and map both automatically to the token or sequence of tokens 'abc.' (This is slightly tricky, because I can only talk about these marks by using tokens.
Quoting Metaphysician Undercover
This 'law' again just seems to codify or tautologize the syntax of 'ordlang.' It's indulgent of me to try and fire up a meme, but I am trying to point at an inexplicit know-how that is also invisible in its smooth functioning.
Quoting Metaphysician Undercover
I hope I addressed this above, if obliquely.
Quoting Metaphysician Undercover
Depending on what you mean by 'contradiction,' that would make you famous. I think you mean that you have philosophical reservations about set theory ( the formal system itself). I can relate to that. When we talk about systems from the outside, we leave the cosy objectivity available within the system, as I see it. If I understand correctly, a mathematical contradiction in this context would be a 'legally' generated string with certain properties. Or rather it could be put in such a form. In practice almost no one works in 'machine code,' though I understand that to be a speciality itself. For example: https://en.wikipedia.org/wiki/Automated_proof_checking
Quoting Metaphysician Undercover
If we view formal systems as pieces of technology, then it makes sense to me that we might tinker with them. Cantor's original set theory was beautiful and revolutionary, but it allowed some contradictions. Mathematicians didn't want to just give up on something so promising, so they added a few constraints in the hope that they could avoid contradictions. They at least fixed the obvious ones, and they did this without losing much. I have Suppes' book, and it follows the history closely, with some great quotes of the philosophical motivations of those who fixed set theory up. The axiom of infinity especially begs for philosophical dicussion: https://en.wikipedia.org/wiki/Axiom_of_infinity.
Quoting Metaphysician Undercover
To me this is far from obvious. How do we evaluate/demonstrate the relation 'less than' without an image of the ideal? I will agree that we can prove the impossibility of certain systems. Every finite field has order p^k where p is prime and k>0, so we can't have such a field with 6 elements. If we determine that a finite field with 6 elements would somehow be useful or beautiful, then there would be a gap between the real and the ideal. But if we consider possibility (that the system works) to be an essential feature of the ideal, then I don't currently see how we couldn't immediately institute a particular vision of the ideal.
Admittedly, we don't always know ahead of time whether we are desiring the impossible.
Quoting Metaphysician Undercover
I agree in general, but my concerns above address the case of formal systems.
As I said, my mathematics is not good, so I don't see how the example deals with the problem. Say object #1 and object #2 are seen to be different, but not identified as different. They are both identified as "dog". We know the objects are distinct, be we are identifying them as the same. You might say that this is just a categorization, but for the sake of the logical process which follows the identification, they are the same. So for the sake of the logical process they are said to be the same, when they are really different.
Quoting mrcoffee
The meaning of "less than" is not demonstrated, it is stipulated by definition, in reference to an order. I don't think "less than" can be judged without reference to the definition, and therefore the order. If you want to argue that a definition is an ideal, I don't think you could succeed because definitions are not perfect, due to the ambiguity of words.
And, since unities can be subtracted from and divided, as well as added to and multiplied, I don't know how you would designate the ideal order. To have an ideal order, I think would require having an indivisible first unity. Zero might serve the purpose, but it allows for the possibility of a negative as well as a positive order. They cannot both be "the ideal order" unless "zero" is defined in relation to some ideal good. Then we'd have less and more in relation to that good.
Quoting mrcoffee
I think that this is contradictory. "Possibility" implies necessarily a multitude. There cannot be just one possibility or else that possibility would be the only possibility, and therefore a necessity and not a possibility. This contradicts "a particular vision of the ideal", which implies necessarily the one and only. In other words, it contradicts the notion of "the ideal" to allow that possibility inheres within.
Right. But in the science of formal systems we discover the relationships of categories/symbols/tokens and not of the marks we need to aid memory and communication. The theorems aren't about the marks. Beyond that, there's no denial that the marks are different. There's just no interest in the mark except as the representation of a category.
If I draw the letter a in two ways, even a child can agree that the marks are different and yet the 'same' (the same letter). This is an informal computation of the many-to-one function from marks to symbols. Some might prefer to use 'symbols' for what I mean by marks, which is fine. So for clarity I can just talk about the categorization function or categorization itself, which is allowed to place 2 or more different objects in the same conceptual bin. This is just an ability we find ourselves with. Existence is ultimately mysterious, etc. But I don't think there's problem with categorization. A person would have to use categories successfully in order to argue for their failure.
Quoting Metaphysician Undercover
But surely you didn't mean the usual order on the integers or real numbers? My point is that if something is less than ideal within or about mathematics, that this would tend to involve a notion of the ideal. Anything ideal for mathematics would already be in the right form for immediate adoption. I did give an example we could want a particular formal system (a field with 6 elements) and discover that such a thing is impossible. If we want the impossible, then math (in this case 'logical reality') is not ideal.
This critical anti-realist insight renders rules and laws as having trivial epistemological significance, since the meaning of rules and signs is ultimately grounded, explained and justified in terms of our behavioural dispositions, as opposed to our behavioural dispositions being grounded or justified in terms of mind-independent rules and laws. This in turn ought to lead to a rejection of the free-will-determinism dichotomy.
This is not "the science of formal systems", this is philosophy. In philosophy we are concerned with understanding reality as a whole, so we cannot dismiss certain contradictions and inconsistencies as irrelevant to the field of study.
If it is necessary that we take two distinct things, which have a very similar physical appearance (two distinct instances of a symbol), and assume that they are "the same", despite the fact that they are clearly not the same, in order to understand some aspect of reality, then as philosophers we ought to recognize and take interest in this, to determine what the implications of such a contradiction might be.
Quoting mrcoffee
The point is to recognize the difference between "the same" and "similar". We all agree that there is a fundamental difference between these two. I would say that it is a categorical difference, "the same" always indicates one and only one, the one and only, while "similar" always indicates a multiplicity. The law of identity, as expressed by Aristotle claims that "the same" refers to the one and only. This is supported by the Leibniz principle of indiscernibles, if it appears as two distinct things, but the two are absolutely identical, then they are in fact the same, one not two.
I agree that we can categorize, and place similar things in the same category. Where I disagree is that it is proper to call these similar things "the same". The category itself is "the same", as the one and only, but the individual items are not the same, they are similar. Therefore we have a categorical separation between the category and the members of the category. The category is "one", the members are "many". If I remember correctly, set theory violates this categorical separation, being based in a fundamental category error.
Quoting mrcoffee
Yes, we refer to the notion of "the ideal". Let's say that it must be perfect, cannot be otherwise without loosing the status of "the ideal", therefore it is unique, the one and only, the ideal. If it can be demonstrated that any particular principle does not fulfill this criteria, then we can conclude that the principle is not the ideal. I agree that in excluding the principle from the category of "ideal", we refer to "a notion of the ideal". Do you recognize the difference between "the ideal", and "the notion of the ideal"? We make a category, "the ideal", and have a notion of what is required of something to be put in that category. But the category may be empty, like the empty set. There may be nothing to put in that category. There is a notion of the ideal, but we haven't found the ideal. That we have a notion of the ideal doesn't mean that we have the ideal.
This is why I used zero as the principle for ordering. Let's say someone claims that zero fulfills our notion of the ideal. The argument is that we haven't found any ideal, the category is an empty set, therefore zero is the ideal. However, zero allows for the possibility of ordering toward the negative or the positive, two distinct possibilities. So there is inherent within "zero" two distinct possibilities. Therefore it cannot be the ideal because the ideal must be one unique perfection. The ideal is like the empty set, but it cannot even be represented as zero, because we cannot put zero into that set, because this leaves it not empty.
And it's dark at night.
Quoting Metaphysician Undercover
Unless we dismiss them at dead ends or as not really being contradictions. We decide all the time (implicitly at least) what is and is not worth talking about.
Quoting Metaphysician Undercover
I don't see any mystery in the process. I know very well that two marks are different as marks. I just want these marks to function as symbols. The formal system example just shows this categorization at its nakedest. As you read this sentence, you see words and not marks. As you write your replies to me, you think in terms of words/symbols. This know-how is ultimately mysterious and elusive. I can't say what it is to mean in some conclusive end-of-conversation way. I think it's interesting to try, at least for awhile. But then I may want to actually look into something that I can be far more conclusive about. I think this touches on how virtuous an individual finds philosophical hand-wringing.
Now my use of the phrase is already taking a certain side. I'm an ex-philospoher being a smart-ass, one might say. But I think it's good 'philosophy' to demystify certain classic poses. My 'complaint' would be perhaps that this 'handwringing' is too easy. I think that's why the genre isn't admired much by its non-participants. As a reader of philosophy, I certainly don't side with 'anti-intellectualism,' but sometimes 'anti-intellectualism' is just the flip side of handwringing --a word the handwringers have for those unimpressed by their sweaty palms. To be clear, I'm interested in a synthesis of both sides, and I try to sort effective handwringing (which is perhaps the most powerful kind of talk and thought) from running around in the same old circles compulsively. I'm well aware that I am far from the objectivity of formal systems in presenting my views and preferences.
Quoting Metaphysician Undercover
That's tricky in math, though. We can have whatever we dream up, with a certain constraint on the dream that keeps it mathematical. I mentioned a field with 6 elements because I can imagine a real world application that might be possible if such a field were possible. Maybe I'm designing a code and such a thing would be convenient, the perfect size. The practical ideal exists, and yet the logical structure of human cognition makes such a thing impossible. Abstract algebra can be read as implicitly psychological. While proofs can be formally true and meaningless, they tend to be written from and for an intuition of necessity.
Quoting Metaphysician Undercover
I can't understand what you are trying to say here. Since '0' is just one part of a system (or of many systems), I can't imagine anyone saying that it itself is or is not ideal. I can only guess where you are coming from, but I can say that I found math far less metaphysical upon studying it than I first understood it to be. Or rather it's metaphysical in the driest and most desirable of ways. It works with basic structural intuitions.
https://en.wikipedia.org/wiki/Peano_axioms
This is a difference of opinion then. You dismiss these inherent contradictions as "dead ends", "not worth talking about". I consider them as having important ontological significance.
Quoting mrcoffee
The point is that "0" does not work within a basic structural intuition, as you claim, it works only by arbitrary designations, different intuitions which vary. One cannot say what "0" symbolizes because depending on how it is used, this varies.
For example, put "0" on a number line. You can count down, 3,2,1,0,-1,-2, etc.. Here, "0" is just an equal integer. You count two equal intervals between 3 and 1, and likewise, two equal intervals between 1 and -1. The intervals are all equal. This clearly does not represent what "0" really does in mathematics, it has a special place, of greater significance than any other integer.
In other applications, "0" occupies a special place, unlike any other integer. This is because when we count down the positive integers we are proceeding toward "less than", but as we pass zero we cross a categorical separation, into the negative integers, so the negative numbers increase and we actually count "more than" of a different category, the negative integers, as we continue to count down. Counting down from zero, through the negative integers is actually counting up, increasing the number, of a categorically different thing, the negative rather than the positive.
The different, arbitrary, functions of "0" become evident when you try to multiply negative numbers. One convention says that when you multiply negative numbers, a double negative makes a positive, so negative numbers multiplied together makes a positive number. But this is not a very sensible convention because it doesn't allow that there is a square root of a negative number, and it doesn't reflect the fact that when we count into higher and higher negative numbers, we are actually increasing the quantity of a different category, the negatives, not simply descending by integer. So another convention allows for imaginary numbers. But no existing convention really represents "0" properly, as making a categorical division, such that we increase from zero into two distinct categories, the positive, and the negative.