logically consistent set of axioms from which one derives a mathematics is a valid derivation from Math — Pneumenon
But what happens when one modifies the logic? — Marchesk
I'm in unfamiliar territory but does the type-token distinction seem relevant to the OP? Speaking in terms of the geometrical object triangle, there's a type triangle and all other triangles are its tokens. Quine's notion of Plato's beard seems to ignore/overlook this. — TheMadFool
A challenge to Platonism, which is IMO one of the more serious ones, is that mathematical objects lack clear identity conditions. — Pneumenon
Tegmark's mathematical universe — jgill
I would have thought that the identity conditions of integers was abundantly obvious. I mean, any integer is distinct from all other integers - how does that not constitute an 'identity condition'? — Wayfarer
As for the triangle, it's 'a flat plane bounded by three intersecting straight lines'. That applies to any triangle. The 'form' is not the shape. — Wayfarer
:100:The concrete world is an abstract object: it's just the one that we're a part of.
Because you can make abstract objects from collections of other abstract objects, then yes OP, you can say that there is just one abstract object of which all other objects are parts, and that one all-encompassing abstract object is the entirety of existence in the broadest possible sense. — Pfhorrest
How can we say that 2 represents a unique and particular object? That is the identity condition. — Metaphysician Undercover
We can attempt to provide identity through the means of a definition, but the definition always allows that more than one thing can be identified as fulfilling the identity conditions of the definition. — Metaphysician Undercover
Other scholars—especially those working in other branches of science—view Platonism with skepticism. Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.
Platonic objects lack clear identity conditions — Pneumenon
Right. So 'identity condition' pertains to individual identity, something unique and particular. What is the source or definition of 'identity condition'? — Wayfarer
Be that as it may, a triangle will never have other than three sides. — Wayfarer
I know the identity condition as the "law of identity", which states that a thing is the same as itself. — Metaphysician Undercover
"triangle" does not suffice as a thing's identity because many things are said to be triangles. — Metaphysician Undercover
To those throwing rocks at mathematical Platonism (as I do when I'm taking the other side of this debate), was 3 prime before there were intelligent life forms in the universe? — fishfry
[Rationalist philosophers] claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought. But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies. ...The indispensability argument in the philosophy of mathematics is an attempt to justify our mathematical beliefs about abstract objects, while avoiding any appeal to rational insight. Its most significant proponent was Willard van Orman Quine.
But the most succinct formulation of 'the law of identity' is 'a=a'. So are you saying that 'a' doesn't have an identity? — Wayfarer
Many things can be triangles, but that is only insofar as those things assume that form. The form itself is not a thing. Only three-sided flat planes bounded by lines constitute a triangle but that covers an endless variety of things. That's the 'thing' about universals. — Wayfarer
To those throwing rocks at mathematical Platonism (as I do when I'm taking the other side of this debate), was 3 prime before there were intelligent life forms in the universe? If that's too easy (I don't think it is), were there infinitely many primes? Or at least no largest prime? — fishfry
It does not make any sense at all to ask about the meaning of the word "prime", or any word for that matter, at a time before the word existed. — Metaphysician Undercover
Many mathematical axioms such as those of set theory rely on the assumption of mathematical objects — Metaphysician Undercover
whether there was something which we refer to with "3", and something which we refer to with "prime", which existed prior to the existence of these words, and that is a difficult metaphysical question without a straight forward answer. — Metaphysician Undercover
So, you're a relativist after all? — Wayfarer
Can you name one such? — fishfry
It doesn't matter if you call sets "beer mugs" as Hilbert pointed out. It's the properties and relations that matter, not the nature of individual things. — fishfry
I think we've been through this before. — Metaphysician Undercover
You insisted on an unreasonable separation between "objects" and "mathematical objects", such that mathematical objects are not a type of object. — Metaphysician Undercover
We could start with the axiom of extensionality. Any axiom which treats numbers as elements of a set, treats the numbers as objects. — Metaphysician Undercover
The issue though, is that set theory treats them as "individual things", therefore Platonism is implied. Set theory relies on Platonism because it cannot proceed unless what 2, 3, 4, refer to are objects, which can be members of a set. — Metaphysician Undercover
I just recognize that it is impossible for a word to have a meaning before that word exists. — Metaphysician Undercover
Your question therefore asks, whether there was something which we refer to with "3", and something which we refer to with "prime", which existed prior to the existence of these words, and that is a difficult metaphysical question without a straight forward answer. — Metaphysician Undercover
Well, mathematical objects are abstract objects. But I agree that numbers aren't like rocks. That doesn't mean that numbers don't exist. It only means that numbers are abstract. And, per structuralism and Benacerraf's famous essay, What Numbers Cannot Be, numbers are not any particular thing. They're not actually sets, even though they are typically represented as sets. Numbers are the abstract things represented by sets. I suspect you and I might be in violent agreement on this point, but I'm not sure. — fishfry
We're just saying that the membership relation holds between the abstract things represented by the symbols. — fishfry
You can, if you like, view the entire enterprise as an exercise in formal symbol manipulation that could be carried out by computer, entirely devoid of meaning. It would not make any difference to the math. — fishfry
We could go down a rabbit hole here but just tell me this. Do you believe that if E is the set of even positive integers, then E = {2, 3, 4, 6, ...}. Do you agree with that statement? Or do you deny the entire enterprise? I'm trying to put a metric on your mathematical nihilism. — fishfry
The Platonist explanation is that these 'things' - they're not actually things, which is part of the point - are discerned by the rational intellect, nous. — Wayfarer
They transcend individual minds, but they're constituents of rational thought because thought must conform to them in order to proceed truly. — Wayfarer
Wouldn't the way of thinking, which would be judged as the rational way, be itself dependent on and therefore determined by the particular situation, or context? So for example, there is some sort of way of thinking which is the correct way for a particular situation, and this transcends all individual minds, making it the correct way for any mind, in that situation. But that correctness, and the way of thinking itself which is correct, is determined by and specific to the particular situation itself. Would you agree with this? — Metaphysician Undercover
I think that modern Platonism treats the abstract as things. — Metaphysician Undercover
Well, I wouldn't go so far as to call it a violent agreement. The point of the op I believe, is that it's incorrect to call mathematical objects "objects" at all, because they do not fulfill the requirement of identity. — Metaphysician Undercover
And so, if we start talking about them as if they are objects, and believe that they have identities as objects, and treat them that way, when they do not, there is bound to be problems which arise. — Metaphysician Undercover
Where we agree is that they are "abstract", but the problem is in where we go from here. — Metaphysician Undercover
Here is where the difference between us appears to arise. You are saying that there are "abstract things represented by the symbols". That's Platonism plain and simple, the "abstract things" are nothing other than Platonic Ideas, or Forms. See, you even allow that there are relations between these things. — Metaphysician Undercover
But from my perspective, a symbol has meaning, — Metaphysician Undercover
and meaning is itself a relation between a mind and the symbol. So I see that you've jumped to the conclusion that this relation between a symbol and a mind, is itself a thing, — Metaphysician Undercover
and you then proceed to talk about relations between these supposed things which are really just relations, and not things at all, in the first place. — Metaphysician Undercover
If you can follow what I said above, then I'll explain why there's a real problem here. — Metaphysician Undercover
The relation between a symbol and a mind, which is how I characterized the abstract above, as meaning, is context dependent. When you characterize this relation, the abstract, as a thing, you characterize it as static, unchangeable. — Metaphysician Undercover
This is what allows you to say that it is the same as manipulating symbols devoid of meaning, the symbol must always represent the exact same thing. — Metaphysician Undercover
But here is where this thing represented, the abstract, fails the law of identity, the meaning, which is the relation between the mind and the symbol, is context dependent and does not always remain the exact same. — Metaphysician Undercover
I think we've discussed this enough already, for you to know that I denounce all set theory as ontologically unsound, fundamentally. — Metaphysician Undercover
It doesn't mean a whole lot though, only that I think it's bad, like if I saw a bunch of greedy people behaving in a way I thought was morally bad, I might try to convince them that what they were doing is bad. — Metaphysician Undercover
However, if it served them well, and made their lives easy, I'd have a hard time convincing them. — Metaphysician Undercover
But, the point about universals is that they're universally applicable, isn't it? They're applicable in any context. Think about scientific laws, which I think must in some sense be descended from such ideas. Water doesn't sometimes flow uphill, for instance. Think also about Kant's deontological ethics, which individuals are obliged to conform to if their actions are to be ethically sound. — Wayfarer
Furthermore, I can't arbitarily designate the rules of math or the laws of logic, I have to conform to them, as much as I'm able (which in my case, is not very much). I can adapt them to my situation, I can use them to advantage, but I can't change them. (Again, the clearest exposition of these ideas are in the Cambridge Companion to Augustine, on the passage on Intelligible Objects.) — Wayfarer
Modern thought treats everything as a thing. (Who's paper is it, 'What is a thing'? Heidegger, I think.) Anyway, the point is, the modern mentality is so immersed in the sensory domain, that it can only reckon in terms of 'things'. Things are 'what exists' - which is what throws us off about mathematical concepts, they're not things, but they seem real, so 'what kind' of reality do they have? In our world, real things can only be 'out there', the only alternative being 'in the mind'. But in reality, 'out there' and 'the mind' are not ultimately separable - hence, as I say, the logic of objective idealism. But it takes a shift in perspective to see it. — Wayfarer
But now you say, "it's incorrect to call mathematical objects "objects" at all, because they do not fulfill the requirement of identity." When a while back you disagreed that 2 + 2 and 4 represent the same mathematical object (regarding which you are totally wrong but nevermind), that was one thing. But now you seem to be saying that 4 = 4 is not valid to you because mathematical objects don't fulfill the law of identity. Am I understanding you correctly? Do you agree that 4 = 4 and that both sides represent the same mathematical object? Or are you saying that since there aren't any mathematical objects, 4 = 4 does not represent anything at all? — fishfry
4 = 4 is true by the law of identity, yes or no? — fishfry
Do you at least accept that math can be regarded as a formal game without regard to meaning? — fishfry
But, are you claiming that 4 means one thing to you today and other thing tomorrow? — fishfry
The point in enforcing a "law of identity", is to recognize and adhere to this understanding, that this gap exists, so that we do not push Platonism to its extremes, claiming that the physical universe is composed of mathematical objects. This is impossible, because mathematical objects are universals, but the universe is composed of unique particulars. — Metaphysician Undercover
if the proper knowledge of the senses is of accidents, through forms that are individualized, the proper knowledge of intellect is of essences, through forms that are universalized. Intellectual knowledge is analogous to sense knowledge inasmuch as it demands the reception of the form of the thing which is known. But it differs from sense knowledge so far forth as it consists in the apprehension of things, not in their individuality, but in their universality.
Clearly, "is the same as" does not mean the same thing as "is equal to". — Metaphysician Undercover
I guess you don't remember the key points (from my perspective) of or previous discussions. — Metaphysician Undercover
What I objected to was calling things like what is represented by 4, as "objects". I made this objection based on the law of identity, similar to the op here. You insisted it's not an "object" in that sense of the word, it's a "mathematical object". And I insisted that it ought not be called an object of any sort. So you proceeded with an unacceptable interpretation of the law of identity in an attempt to validate your claim. What I believe, is that "mathematical object" is an incoherent concept. — Metaphysician Undercover
This depends on what = represents. — Metaphysician Undercover
Does it represent "is the same as", or does it represent "is equal to"? — Metaphysician Undercover
From our last discussion, you did not seem to respect a difference between the meaning of these two phrases. — Metaphysician Undercover
If you're still of the same mind, then there is no point in proceeding until we work out this little problem. — Metaphysician Undercover
This is why I say context of the symbol is important. When the law of identity is represented as a=a, = symbolizes "is the same as". But when we write 2+2=4, = symbolizes "is equal to". If we assume that a symbol always represents the very same thing in every instance of usage, we are sure to equivocate. Clearly, "is the same as" does not mean the same thing as "is equal to". — Metaphysician Undercover
No, of course not, that's clearly a false representation of what math is. That would be like saying that 2+2=4 could be considered to be valid regardless of what the symbols mean. That's nonsense, it's what the symbols mean which gives validity to math. — Metaphysician Undercover
Yes, that is exactly the case it can even mean something different in the same sentence. When some says "I reserved a table for 4 at 4", each instance of 4 means something different to me. — Metaphysician Undercover
And, as I explained to you already, when someone says 2+2=4, each instance of 2 must refer to something different or else there would not be four, only two distinct instances of the very same two, and this would not make four. — Metaphysician Undercover
The existence of two requires two copes of one. Then there are two objects which are separable and yet absolutely indistinguishable.
But I insist that this appeal to eternal abstract beings which exist in copies is illicit. If you claim the existence of two copies of one, what differentiates them? There would have to be a feature to distinguish them; yet arithmetic insists there is no such feature. As eternal abstract beings they have to be identical in every respect. Then how can they be plural? I conclude that there can’t be identical copies of the pure number one.
Clearly, "is the same as" does not mean the same thing as "is equal to".
— Metaphysician Undercover
I say it does. I think you're splitting hairs for the sake of argument. — Wayfarer
I say it does. I think you're splitting hairs for the sake of argument. — Wayfarer
My mind has blocked them out as traumatic experiences. — fishfry
What's incoherent is you objecting to 4 = 4 as an instance of the law of identity — fishfry
In a given mathematical context, a given symbol holds the exact same meaning throughout. — fishfry
Hard to believe there are two people who assert this nonsense, not just you alone. — fishfry
By the way there is a standard formalism for obtaining multiple copies of the same object, you just Cartesian-product them with a distinct integer. So if you need two copies of the real line RR, you just take them as R×{1}R×{1} and R×{2}R×{2}. It's not that mathematicians haven't thought about this problem. It's that they have, and they have easily handled it. As usual you confuse mathematical ignorance with philosophical insight. — fishfry
In any event, you avoided (as you always do when presented with a point you can't defend) my question. If set theorists are not only wrong but morally bad, is Euclid equally so? You stand by your claim that set theorists are morally bad? Those are your words. Defend or retract please. — fishfry
Particulars are real insofar as they're instantiations of the idea, which is their unchanging form; that is the sense in which the ideas 'lend being' to particulars, or particulars are said to 'participate in' the form. — Wayfarer
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