You cannot escape the need for identity conditions by focusing solely on relations rather than particulars, because relations are particulars — Pneumenon
So are you arguing against Platonism? Or constructivism? — fishfry
Take, for instance, the possible fat man in the doorway; and again, the possible bald man in the doorway. Are they the same possible man, or two possible men? How de we decide? How many possible men there are in that doorway? Are there more possible thin ones than fat ones? How many of them are alike? — WVO Quine
This is the famous two spheres argument against the identity of indiscernibles. — fishfry
I don't think anyone is saying there can't be two of something in math. There's only one set of real numbers, defined axiomatically as the unique (up to isomorphism) Dedekind-complete ordered field.
Yet we have no trouble taking two "copies" of the real numbers, placing them at right angles to each other, and calling it the Cartesian plane. Or n copies to make Cartesian n-space.
For that matter we say that a line is determined by two points. But one point is exactly like any other, except for their location, yet nobody thinks there's only one point. There are lots of points. — fishfry
I want to start with this: I'm defending Platonism, bro. — Pneumenon
The objection based on identity conditions really bothers me, so I want to say find some way to get around it and still be a Platonist. — Pneumenon
The objection based on identity conditions ... — Pneumenon
I think it's analogous to Quine's observation about modality: — Pneumenon
Take, for instance, the possible fat man in the doorway; and again, the possible bald man in the doorway. Are they the same possible man, or two possible men? How de we decide? How many possible men there are in that doorway? Are there more possible thin ones than fat ones? How many of them are alike?
— WVO Quine — Pneumenon
I think this is a deep problem. — Pneumenon
Deep problems manifest in multiple places – that's how you know they're deep. The fact that something similar happens with modality as with math ought to tip us off that this is not a surface-level concern. — Pneumenon
So how does it manifest for math? Like this: — Pneumenon
Antiplatonist: "We cannot say that mathematical objects exist in a mind-independent sense. This is because there are no clear identity conditions for them."
Platonist: "Identity is structural. Two mathematical objects are numerically identical iff they are structurally identical."
Antiplatonist: "Here are two objects. They are numerically distinct and structurally identical. Therefore, numerical identity is not structural identity. Therefore, you have not answered my objection." — Pneumenon
My solution to this dilemma, per the first post, is simple: no two mathematical objects are numerically distinct. — Pneumenon
There's only one, Math. Any two apparently distinct ones are just logically valid observations of the same object. A duck and a rabbit, if you like Wittgenstein. Attributes, if you like Spinoza. Emanations, if you like Plotinus.
(I like Spinoza. ) — Pneumenon
I agree, with a caveat.
Black's argument is this: "Two things can be indiscernible, in every single respect, and still numerically distinct". He's saying that two things can be indiscernible in every possible way and still be two things.
My argument is a lot more constrained. I'm saying, "Two things can be indiscernible, in terms of mathematical structure, and still be numerically distinct". And this is proved, IMO, by the graph example. — Pneumenon
I mean, you're not wrong. — Pneumenon
Everybody agrees that there can be two of something in math. As you note, points differ by their location. Location is structural. Therefore, points are structurally distinct. And I agree with you on that. I'm not saying, "Ha! Here are two different points. Suck it, structuralists!"
What I am saying is, "Here are two objects. They are numerically distinct, but structurally identical. Therefore, structural identity is not numerical identity". — Pneumenon
If it is possible to individuate two objects without appeal to structure, then we can find two such objects that are structurally identical. Which sinks structuralism. So the burden of the structuralist is to do math in such a way that you must appeal to structure in order to individuate two objects. — Pneumenon
You did this for points, I think. If points are individuated by location, and location is structural, then numerical identity for points is structural identity. It works.
But this doesn't apply to the graph-theoretic example. You can simply say, "There exists a graph with two vertices and no edges". If any graphs exist, this one does. And now we've individuated two vertices without appeal to structure. The structuralist must show that one vertex has a structural property that the other doesn't. By definition, this cannot be done. — Pneumenon
You can't make structure itself a primitive notion, by the way. — Pneumenon
That defeats the point of structuralism. The whole point was to abstract away from particulars and deny intrinsic properties. If you make structure primitive, then you've basically made it an intrinsic property. — Pneumenon
One last point: when I say, "Therefore, the two vertices are numerically distinct", I'm speaking ex hypothesi. In fact, I do not think the vertices are numerically distinct. — Pneumenon
There's only one. Any talk of distinction is just a way of talking about how that one object relates to itself. — Pneumenon
I'm not sure how to cash that out, exactly. Perhaps I should say: math relates to itself in every logically valid way, for every coherent system of logic. Therefore, every coherent logical system expresses math. — Pneumenon
You know, you might have missed the mark in tagging me for your resurrection of this thread. I made an offhand remark about Hilbert's famous beer mug quote, and I may have mentioned the turn towards structuralism of latter 20th century math. But I am not proclaiming myself the Lord High Defender of these ideas, nor am I particularly knowledgeable about these matters, nor do I even have any particularly strong opinions about them. You don't like structuralism, that's ok by me. You are a Platonist, so was Gödel, and so are most working mathematicians.
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Why me? All I did was quote-check the beer mug. I might as well have worn Hilbert's famous hat.
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I think you are misconstruing the hell out of mathematical structuralism; flailing at a straw man; and expecting me to engage or to be an enthusiastic advocate of something or other.
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Ok. Fine. Whatever. Tell me why you think I am the right target for this conversation. I can't hold up my end. I'm defenseless. — fishfry
You are using "numerically distinct" in some kind of crazy way. 5 and 6 are numerically distinct. Past that I have no idea what you mean.
You've used that phrase [identity condition] a few times, and I should admit that I have no idea what it means.
What is an identity condition? Can you explain what you mean? Can you give me an example or two, say with regard to some familiar mathematical examples like the set of natural numbers, or the cyclic group of order 4, or the Riemann sphere, or any other example you care to name.
I just have no idea what you mean by an "identity condition," and how this idea stands in opposition to the idea of structuralism.
Do you mean perhaps that the two representations of the cyclic group of order 4 I gave earlier should be properly regarded as two separate things? But of course they are. They also happen to have the same group-theoretic structure, so they represent the same group. It's all about the context in which we use the word "same."
Nobody is arguing with this. Clark Kent and Superman are the same and they are different. It all depends on the context.
Have I ever done so?
I just wanted to bother you 'cause I thought you'd be fun to talk to. — Pneumenon
I figured you'd just ignore me if you weren't into it. I didn't think you'd start screaming "OH GOD WHY DID YOU CHOOSE ME I WANT NO PART IN THIS MADNESS" like I jumped out of the bushes and throttled you or something. — Pneumenon
It's okay. You can just ignore me if you want. But I think you'll have fun. — Pneumenon
Okay, so some philosophers use the terms "qualitative identity" and "quantitative identity" (or numerical identity). Qualitative identity is where you can't tell them apart. Quantitative is where they're the exact same thing like Superman and Clark Kent. "Quantitative" is kinda misleading, there, so I'll drop these terms. — Pneumenon
Instead, I'll use "indiscernible" and "identical". — Pneumenon
Two things are indiscernible if you cannot tell them apart. They're identical if they are one and the same thing. So two "identical" twins can be indiscernible. But they're not identical, strictly speaking, cause they're two different people. Do I still sound like raving lunatic? — Pneumenon
An identity condition is a statement that only applies to one thing. The statement "a prime number between 2 and 5" is an identity condition, because it applies only to one mathematical object, which is 3. — Pneumenon
That sounds very simple, right? But it gets complicated. See below. — Pneumenon
(Apropos of nothing, have you ever read Augustin Rayo's The Construction of Logical Space? You might enjoy it.) — Pneumenon
I think we agree that the set {0, 1, 2, 3} is not identical to {1, i, -1, -i}. — Pneumenon
And we agree that both of those are instances of the cyclic group with order 4, if they have the right operations defined on them. — Pneumenon
Now, here are some questions:
1. Is "{0, 1, 2, 3} under addition mod 4" identical to the cyclic group with order 4?
2. Is "{0, 1, 2, 3} under addition mod 4" an instance of said group? — Pneumenon
They're only "different" in a figurative sense. I think Clark Kent is identical to Superman at all times. They're both one dude. "Clark Kent" shares a referent with "Superman". — Pneumenon
I don't think that identity is dependent on context like that. There's nothing you can do to make something stop being identical to itself. — Pneumenon
Have I ever done so? — Pneumenon
Nah, man. It was just a tangential aside. It's okay. I'm not trying to throttle you. I'm just talking to you. I'm just writing this on my laptop. I'm just under your floorboards. Ominously cracking my long, gnarled fingers. And salivating. — Pneumenon
Perhaps a very stupid question: why isn't Math referred simply to being a system? — ssu
fishfry will give a much more sophisticated answer to your question. — jgill
Yes, mathematics can be considered a system, as it is a structured set of rules, axioms, and concepts that are interconnected and used to reason about and describe patterns and relationships, often through symbols and operations; essentially forming a logical framework for understanding abstract concepts — GoogleAI
Perhaps a very stupid question: why isn't Math referred simply to being a system? — ssu
... the transdisciplinary[1] study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or artificial. Every system has causal boundaries, is influenced by its context, defined by its structure, function and role, and expressed through its relations with other systems. A system is "more than the sum of its parts" when it expresses synergy or emergent behavior. — Wiki
Is the natural number 3 identical to the real number 3? — fishfry
Consider the x-axis [and the] the y-axis. Are they the same? Clearly not, they're different lines in the plane with only one point of intersection. Yet they are both "copies," whatever that means, of the real numbers. But in set theory there's only one copy of any given set. Where'd we get another copy? Are the x-axis and the y-axis the same? — fishfry
"the unique Dedekind-complete linearly ordered set" uniquely characterizes the real numbers. — fishfry
Would you say that 4 and 2 + 2 are two "instances" of the same number? — fishfry
I'll just give the laziest answer possible. I typed "is math a system" into Google. The Google AI responded: — fishfry
Pretty much everything is a system, from indoor plumbing to the National Football League. — fishfry
I think Clark Kent is identical to Superman at all times. They're both one dude. "Clark Kent" shares a referent with "Superman". — Pneumenon
Not at all. Just on the one hand, feeling totally inadequate to discuss the philosophical side of this ... Quine and Spinoza for example. — fishfry
As modern math shows us, concepts like identical and same and "can't tell them apart" depend on the context. They are not absolute. — fishfry
Easily falsified in-universe. Lois Lane has no idea. It's the glasses, apparently. A little willing suspension of disbelief on that point. How can she possibly not see that they're the same guy? But she can't tell. — fishfry
Look, I'm not educated. At all. I have neither a degree nor a high school diploma. I have what Americans call a "GED", and that's it. My job is to sit at home in my underwear and write code. If I tried to show up at an academic conference, they'd kick me out just based on smell. I can make computers go bleep bloop, though! — Pneumenon
Let me make sure I understand you. You're saying this:
"There's this thing called identity. Math tells us that identity is contextual. So we should also say that the identity of people is contextual, since math tells us about identity. For example, Clark Kent and Superman are contextual identities." — Pneumenon
I don't understand. "Lois is unaware of the fact that Clark Kent is Superman. Therefore, Clark Kent is not Superman". That's not what you're saying, is it? If I put on clown makeup and then kill someone, I'm still guilty of a crime, even after I wash off the makeup. — Pneumenon
Reading over your other points, I have a hunch about your position. You have used the term "representation" a few times. And you seem to think that, if A and B both represent C, then A and B are identical insofar as they represent C. You seem to identify how something is represented with what it is. — Pneumenon
Two things.
1. Do Clark Kent and Superman represent one thing? If so, what is that one thing?
2. Would horses have four legs if nobody counted them? If so, which four would it be? The natural number 4? The rational number 4? The real number 4? — Pneumenon
Hence the question was stupid, as I assumed. — ssu
It being a logical system would be perhaps more fruitful, but the notion would still be in the set of self-evident "So what?" truths about mathematics. Just what kind of logical systems math has inside it would be the more interesting question. Now when mathematics has in it's system non-computable, non-provable but true parts (as it seems to have), this would be a question of current importance (comes mind the Math truths aren't orderly but chaotic -thread). — ssu
If so, perhaps the old idea of math being a tautology comes to mind: something being random and non-provable but true is... random and non-provable but true. Yet how do we then stop indoor plumbing and the National Football League being math? That the two aren't tautologies, even if indoor plumbing ought to be designed logically(?) Would it be so simple? — ssu
Unless I've overlooked something, it seems to me that it's easy to prove that it is not the case that any two strict linear orderings with the least upper bound property are isomorphic: — TonesInDeepFreeze
Do you mean just a "strict linear ordering with the least upper bound property" or do you mean a "complete ordered field"? — TonesInDeepFreeze
What definition? I didn't take issue with an definition. — TonesInDeepFreeze
I didn't intend my posts to comment on structuralism. — TonesInDeepFreeze
I made the point that Clark Kent and Superman are two representations, or guises if you will, of the same entity. Just as two isomorphic groups are really the same group. — fishfry
More like that the truths in mathematics are tautologies: a statement that is true by necessity or by virtue of its logical form. Wouldn't that description fit to mathematics?Statements devoid of content? (Frege)
I think not. — jgill
For example to the question in the OP it is something. Also do note that it does influence on how people see mathematics and how the field is understood and portrayed. Do people see it from the viewpoint of Platonism (numbers are real), logicism (it's logic) or from formalism (it's a game) or something else?I'm not sure how important it is to nail down a particular definition. — fishfry
Oh, and this is a perfect example why it is important: with that you'll give here not only your little finger, but your left leg to the worst kind of post-modernists and sociologists that then can proclaim that math is only a societal phenomenon and a power play that a group of people (read men) do. That all this bull of math being something different, having it's own logic or being something special that tells something about reality is nonsense (or in their discourse, an act in that power play) when it's just what mathematicians declare doing. That's obviously not what you meant, but how can your statements be used is important.Math is what mathematicians do. — fishfry
I didn't intend my posts to comment on structuralism.
— TonesInDeepFreeze
My point. That was the subject of the conversation. — fishfry
I'd be happy if we could focus on your ideas. You clarified what you meant by an identity condition, and I pointed out that it was structural, and seemed to undermine your own thesis. You didn't engage and didn't agree or disagree. I'd find it helpful if we could focus in.
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But ok. An identity condition is a statement that uniquely characterizes a mathematical object. Like "the unique Dedekind-complete linearly ordered set" uniquely characterizes the real numbers.
Ok I'll accept that definition. But note that it's a structural condition, not a Platonic one. — fishfry
Along Benacerraf’s lines, mathematical objects are viewed as “positions” in corresponding patterns; and this is meant to allow us to take mathematical statements “at face value”, in the sense of seeing ‘0’, ‘1’, ‘2’, etc. as singular terms referring to such positions.
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By focusing more on metaphysical questions and leaving behind hesitations about structures as objects, Shapiro’s goal is to defend a more thoroughly realist version of mathematical structuralism, thus rejecting nominalist and constructivist views (more on that below).
[Shapiro] distinguishes two perspectives on [positions in structures]. According to the first, the positions at issue are treated as “offices”, i.e., as slots that can be filled or occupied by various objects (e.g., the position “0” in the natural number structure is occupied by ∅ in the series of finite von Neumann ordinals).
a societal phenomenon and a power play that a group of people (read men) do — ssu
More like that the truths in mathematics are tautologies: a statement that is true by necessity or by virtue of its logical form. Wouldn't that description fit to mathematics? — ssu
when it's just what mathematicians declare doing. — ssu
if say that the legs of a horse are the set {1, 2, 3} — Pneumenon
then I've said that the number of horse legs is the natural number four. But is that even true? What if the number of horse legs is the set of all rationals smaller than 4, i.e. the real number 4? How would we know which one it is? — Pneumenon
1. To say that a certain thing exists, you need an identity condition for it.
2. You can't always get those identity conditions for mathematical object — Pneumenon
So, I am really really interested in whether or not mathematical objects exist in a mind-independent way. Would there be numbers even if we weren't here? I want to say yes. I think that numbers would be here — Pneumenon
One might have a notion of the Morning star and a notion of the Evening star and believe that they are two different celestial objects. But believing that they are different doesn't make them different. There is only Venus, which is seen at different times of the day. It's not a different object when seen in the morning than it is when seen at night. — TonesInDeepFreeze
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