You cannot escape the need for identity conditions by focusing solely on relations rather than particulars, because relations are particulars — Pneumenon

I have a question, and sorry for my basic knowledge on this, but I want to learn about this very interesting topic.

My question: according to your points, are you stating that mathematical objects are dependent upon identity conditions rather than structure? If I remove those conditions, the mathematical object doesn't exist?

I'm curious about the fundamental core of each mathematical object's existence.

So are you arguing against Platonism? Or constructivism? — fishfry

I want to start with this: I'm defending Platonism, bro. The objection based on identity conditions really bothers me, so I want to say find some way to get around it and still be a Platonist.

I think it's analogous to Quine's observation about modality:

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

I think this is a deep problem. Deep problems manifest in multiple places – that's how you know they're deep. The fact that something similar happens with modality as with math ought to tip us off that this is not a surface-level concern.

So how does it manifest for math? Like this:

Antiplatonist: "We cannot say that mathematical objects exist in a mind-independent sense. This is because there are no clear identity conditions for them."
Platonist: "Identity is structural. Two mathematical objects are numerically identical iff they are structurally identical."
Antiplatonist: "Here are two objects. They are numerically distinct and structurally identical. Therefore, numerical identity is not structural identity. Therefore, you have not answered my objection."

My solution to this dilemma, per the first post, is simple: no two mathematical objects are numerically distinct. There's only one, Math. Any two apparently distinct ones are just logically valid observations of the same object. A duck and a rabbit, if you like Wittgenstein. Attributes, if you like Spinoza. Emanations, if you like Plotinus.

(I like Spinoza. )

This is the famous two spheres argument against the identity of indiscernibles. — fishfry

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.

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 mean, you're not wrong. Everybody agrees that there can be two of something in math. As you note, points differ by their location. Location is structural. Therefore, points are structurally distinct. And I agree with you on that. I'm not saying, "Ha! Here are two different points. Suck it, structuralists!"

What I am saying is, "Here are two objects. They are numerically distinct, but structurally identical. Therefore, structural identity is not numerical identity".

If it is possible to individuate two objects without appeal to structure, then we can find two such objects that are structurally identical. Which sinks structuralism. So the burden of the structuralist is to do math in such a way that you must appeal to structure in order to individuate two objects.

You did this for points, I think. If points are individuated by location, and location is structural, then numerical identity for points is structural identity. It works.

But this doesn't apply to the graph-theoretic example. You can simply say, "There exists a graph with two vertices and no edges". If any graphs exist, this one does. And now we've individuated two vertices without appeal to structure. The structuralist must show that one vertex has a structural property that the other doesn't. By definition, this cannot be done.

You can't make structure itself a primitive notion, by the way. That defeats the point of structuralism. The whole point was to abstract away from particulars and deny intrinsic properties. If you make structure primitive, then you've basically made it an intrinsic property.

One last point: when I say, "Therefore, the two vertices are numerically distinct", I'm speaking ex hypothesi. In fact, I do not think the vertices are numerically distinct. I don't think any two mathematical objects are numerically distinct. There's only one. Any talk of distinction is just a way of talking about how that one object relates to itself.

I'm not sure how to cash that out, exactly. Perhaps I should say: math relates to itself in every logically valid way, for every coherent system of logic. Therefore, every coherent logical system expresses math.

I want to start with this: I'm defending Platonism, bro. — Pneumenon

Oh I see. In your OP you were stating an objection to it only to knock down the objection. That was not clear to me, and I did not read through this entire four year old thread to discern that. My most humble apologies. Bro'.

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

You know, you might have missed the mark in tagging me for your resurrection of this thread. I made an offhand remark about Hilbert's famous beer mug quote, and I may have mentioned the turn towards structuralism of latter 20th century math. But I am not proclaiming myself the Lord High Defender of these ideas, nor am I particularly knowledgeable about these matters, nor do I even have any particularly strong opinions about them. You don't like structuralism, that's ok by me. You are a Platonist, so was Gödel, and so are most working mathematicians.

So I am poorly positioned to KNOW anything about these matters, nor particularly CARE about them. That is, I am both ignorant and apathetic about the subject at hand.

That said, I did see a bit of modern math a long time ago; and in particular, I was trained in the doctrine of isomorphism; that is, that two things that have the same structure, may as well be regarded as the same thing, in a given context.

So I believe I can perhaps explain some of these ideas, and put them into mathematical context.

But I can't offer any kind of spirited rebuttal to your Platonism, since "I don't know and I don't care."

With that said, I will plunge in and stumble on.

One more thing ...

The objection based on identity conditions ... — Pneumenon

You've used that phrase a few times, and I should admit that I have no idea what it means.

What is an identity condition? Can you explain what you mean? Can you give me an example or two, say with regard to some familiar mathematical examples like the set of natural numbers, or the cyclic group of order 4, or the Riemann sphere, or any other example you care to name.

I just have no idea what you mean by an "identity condition," and how this idea stands in opposition to the idea of structuralism.

Do you mean perhaps that the two representations of the cyclic group of order 4 I gave earlier should be properly regarded as two separate things? But of course they are. They also happen to have the same group-theoretic structure, so they represent the same group. It's all about the context in which we use the word "same." Nobody is arguing with this. Clark Kent and Superman are the same and they are different. It all depends on the context.

I think it's analogous to Quine's observation about modality: — Pneumenon

Now here you are barking up the wrong tree. I know little of Quine and even less about modal logic. I'm singularly unqualified to even think about what you wrote. I simply do not have any place in my mind to hang these concepts. I plead abject ignorance.

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 simply fail to see the relevance to any ideas I may have about mathematical structuralism, modern concepts of isomorphism, or anything else. This is exactly the kind of question that, if it were to appear in a thread on this forum, I'd simply ignore. I have nothing at all to say.

I think this is a deep problem. — Pneumenon

I respect that. I just wonder why you are drawing my personal attention to it. I'm singularly unqualified to have an opinion. I have no knowledge, no interest, and I have no referents in my conceptual scaffolding. I have to just read, ignore, and move on. In some other life I was a big Quine fan and knew something of modal logic. In this life, no.

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

Perhaps you can give me a mathematical example so I can know what you are talking about.

"something similar happens with modality as with math" -- I just have no idea.

Remember, all I did was toss out the famous beer mug quote. Perhaps you think too much of me.

So how does it manifest for math? Like this: — Pneumenon

Ah! Ok. Thanks. I'll gratefully take the lifeline.

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

I can't relate that to any aspect of math I've ever encountered. You threw me an anchor, not a lifeline.

Give me a mathematical example. Something involving the number 5, say,

Again, you are arguing a thesis -- clearly one you've thought deeply about -- that I just have no knowledge of and little interest in.

Why me? All I did was quote-check the beer mug. I might as well have worn Hilbert's famous hat.

Now if you had specific questions about how the concept of isomorphism is used in math, I could definitely be of help. But I did give you a concrete example, the two representations of the cyclic group of order 4, and you didn't engage.

My solution to this dilemma, per the first post, is simple: no two mathematical objects are numerically distinct. — Pneumenon

What does "numerically distinct" mean? I'm willing to agree that 5 and 6 are numerically distinct.

Past that, I have no idea what you mean. Explain please?

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

Didn't he get excommunicated by the Jewish faith? That's all I know about the guy. I hope you see what a philosophical ignoramus you are talking to. I wish I could engage. And I wish to hell you would give me some concrete mathematical examples of what you are talking about.

You wrote this long post to the wrong person. I can't respond to any of it and I have no idea what you are referring to. But some mathematical examples would help. And when you say you're going to give a mathematical example, you don't. You just give some vague hypothetical dialog that has nothing to do with math as I understand it. I wish I could help. Maybe some of the other participants in this thread can chime in.

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

Do you mean that the two representations of the cyclic group of order 4 are "numerically distinct" yet have the same group-theoretic structure? If that's what you mean, I heartily agree.

I mean, you're not wrong. — Pneumenon

That's high praise around this place :-)

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

You are using "numerically distinct" in some kind of crazy way. 5 and 6 are numerically distinct. Past that I have no idea what you mean.

But if you are saying that two mathematical objects may have the same structure yet be presented vastly differently, well duh. That's a commonplace observation that nobody would disagree with.

I think you are misconstruing the hell out of mathematical structuralism; flailing at a straw man; and expecting me to engage or to be an enthusiastic advocate of something or other.

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

I couldn't parse that. You are talking in such vague generalities. Give me a specific mathematical example.

But the article about MacLane that I recommended to you actually explained that. It said that we can't call two objects isomorphic until we say which category we're in. So the two representations of the cyclic group of order 4 are the same as groups, and different as mathematical objects.

What is the point of making such a trivial observation that nobody disagrees with?

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

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.

You can't make structure itself a primitive notion, by the way. — Pneumenon

Have I ever done so? Are you arguing against Category theory? If structure is not a primitive notion, you must be against abstract algebra. I'm actually laughing as I write this. That is such a naive and inaccurate argument. I just don't know what you mean.

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

Who made structure primitive? Are they in the room with us right now? You're playing word games. And I gather from your discourse that you don't have much idea what structuralism in math is really about.

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

What the heck does numerically distinct mean? You mean that one is 5 and the other is 6? I don't know what you mean by that phrase.

For one thing, they're unequal as grade school numbers. For another, they're unequal as sets. And for a third reason, 6 is the Peano successor of 5, and the successor of a number is always distinct from the number.

So you're just mathematically wrong here.

There's only one. Any talk of distinction is just a way of talking about how that one object relates to itself. — Pneumenon

Only one number? What the heck are you talking about? You are making demonstrably false claims. Euclidean 4-space is mathematically distinct from the Lorentzian 4-space of general relativity. They're entirely distinct mathematical objects. They both consist of exactly the same underlying set of points; but one has the Euclidean metric and the other has the Lorentzian metric. That makes them utterly distinct; as distinct as Newton's and Einstein's conceptions of gravity.

You just made the claim that there are no distinct mathematical objects.

Do you stand by that? At least it's a claim you made that I am qualified to dispute.

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

Which tells us what?

Anyway I find this conversation interesting, but I am not really understanding what you're getting at. I could really use some mathematical examples. Do you really claim that 5 and 6 are not "numerically distinct" mathematical objects?

In any event, the observation that two mathematical objects can be

1) Very different in form and nature; and yet

2) Have the same structure with respect to some abstract properties;

is a commonplace barely worthy of note.

Superman and Clark Kent. The same in one context, utterly different in another.

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.

...

Why me? All I did was quote-check the beer mug. I might as well have worn Hilbert's famous hat.

...

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.

...

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

I just wanted to bother you 'cause I thought you'd be fun to talk to. I figured you'd just ignore me if you weren't into it. I didn't think you'd start screaming "OH GOD WHY DID YOU CHOOSE ME I WANT NO PART IN THIS MADNESS" like I jumped out of the bushes and throttled you or something.

It's okay. You can just ignore me if you want. But I think you'll have fun.

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.

Okay, so some philosophers use the terms "qualitative identity" and "quantitative identity" (or numerical identity). Qualitative identity is where you can't tell them apart. Quantitative is where they're the exact same thing like Superman and Clark Kent. "Quantitative" is kinda misleading, there, so I'll drop these terms.

Instead, I'll use "indiscernible" and "identical". Two things are indiscernible if you cannot tell them apart. They're identical if they are one and the same thing. So two "identical" twins can be indiscernible. But they're not identical, strictly speaking, cause they're two different people. Do I still sound like raving lunatic?

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.

An identity condition is a statement that only applies to one thing. The statement "a prime number between 2 and 5" is an identity condition, because it applies only to one mathematical object, which is 3.

That sounds very simple, right? But it gets complicated. See below.

(Apropos of nothing, have you ever read Augustin Rayo's The Construction of Logical Space? You might enjoy it.)

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."

I think we agree that the set {0, 1, 2, 3} is not identical to {1, i, -1, -i}. And we agree that both of those are instances of the cyclic group with order 4, if they have the right operations defined on them.

Now, here are some questions:

1. Is "{0, 1, 2, 3} under addition mod 4" identical to the cyclic group with order 4?
2. Is "{0, 1, 2, 3} under addition mod 4" an instance of said group?

Nobody is arguing with this. Clark Kent and Superman are the same and they are different. It all depends on the context.

They're only "different" in a figurative sense. I think Clark Kent is identical to Superman at all times. They're both one dude. "Clark Kent" shares a referent with "Superman".

I don't think that identity is dependent on context like that. There's nothing you can do to make something stop being identical to itself.

Have I ever done so?

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.

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

Yes ok all good. Having fun. Late now and have some things to do tomorrow morning so I'll get to this in the afternoon.

I just wanted to bother you 'cause I thought you'd be fun to talk to. — Pneumenon

Oh gosh I hope I haven't disappointed you ... now I feel pressure to be entertaining!

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

Not at all. Just on the one hand, feeling totally inadequate to discuss the philosophical side of this ... Quine and Spinoza for example. And on the other hand, feeling that you are mischaracterizing mathematical structuralism and attacking a strawman.

But if it's good for you it's good for me.

It's okay. You can just ignore me if you want. But I think you'll have fun. — Pneumenon

I'm having fun.

Okay, so some philosophers use the terms "qualitative identity" and "quantitative identity" (or numerical identity). Qualitative identity is where you can't tell them apart. Quantitative is where they're the exact same thing like Superman and Clark Kent. "Quantitative" is kinda misleading, there, so I'll drop these terms. — Pneumenon

I'm more confused than ever now regarding those terms. As modern math shows us, concepts like identical and same and "can't tell them apart" depend on the context. They are not absolute. I'll give examples in what follows.

Instead, I'll use "indiscernible" and "identical". — Pneumenon

Big trouble, as the two sphere example shows.

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

No, but the counterexamples come up fast. Especially the twins. Identical twins do not have the same fingerprints. Interestingly when I typed "Do identical twins ..." into my Chrome search bar, it autocompleted "have the same fingerprints," showing that tens or hundreds of thousands of people have asked the same question. So identical twins are not identical and we CAN tell them apart. Forget the twins.

But just take some mathematical examples. Is the natural number 3 identical to the real number 3? They are quite different as sets, but they are exactly the same via the usual embedding maps; which is to say that the reals contain a copy of the naturals that respects all the order and arithmetic information, so we regard the naturals as a subset of the reals, even though set-theoretically, this is false.

Consider the x-axis of the Cartesian plane in high school analytic geometry class (or Algebra II in the US). Consider the y-axis. Are they the same? Clearly not, they're different lines in the plane with only one point of intersection. Yet they are both "copies," whatever that means, of the real numbers. But in set theory there's only one copy of any given set. Where'd we get another copy? Are the x-axis and the y-axis the same?

I don't accept the distinction you're trying to make. Too many corner cases.

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

You sure? Does it apply to the real number 3, which has an entirely different set-theoretic representation?

But ok. An identity condition is a statement that uniquely characterizes a mathematical object. Like "the unique Dedekind-complete linearly ordered set" uniquely characterizes the real numbers.

Ok I'll accept that definition. But note that it's a structural condition, not a Platonic one.

I better call this out to make sure you note it.

By your definition, an identity condition is a structural condition, not a Platonic one.

I can construct the real numbers many different ways, but they are all the same real numbers, because they all satisfy the identity condition of being a Dedekind-complete, linearly ordered set.

Am I understanding your meaning of identity condition?

And am I making a valid point that this is a structural condition, which seems to negate your point that identity conditions are the opposite of structuralism?

That sounds very simple, right? But it gets complicated. See below. — Pneumenon

No, not simple. Already complicated, and undermines your point that identity conditions are the opposite of structuralism. If I'm understanding you.

(Apropos of nothing, have you ever read Augustin Rayo's The Construction of Logical Space? You might enjoy it.) — Pneumenon

No, but I'm a huge fan of professor Rayo. For a long time he put on a MOOC called Paradox and Infinity, which I took several times because I love the material so much. He went over omega paradoxes (what we'd call supertasks around here), ordinals, the nonmeasurable set, even gave an accessible proof of the Banach-Tarski theorem. This was in a course aimed at total mathematical novices. Fantastic course. He doesn't do it anymore. Last time the attendance was very low. I think everyone is into machine learning and AI these days, if you can judge by the MOOC offerings.

I think we agree that the set {0, 1, 2, 3} is not identical to {1, i, -1, -i}. — Pneumenon

But we can NOT make that agreement. That statement has no truth value till we declare what category we're in; or less technically, till we specify a context.

If we are group theorists, they ARE identical, because there is only one cyclic group of order 4. This point is even more strict if we are in the category of groups. Then there is literally only one such group, and there is no such thing as a representation or presentation or instance of it.

If we aren't in the category of groups, or if we are just beginning to learn group theory, these are two distinct groups that happen to be isomorphic.

The claim that "isomorphism is identity" is part of homotopy type theory. I'm probably lying through ignorance here but that's my sense of the matter.

So no, we do NOT agree that these two isomorphic groups are identical. In beginning abstract algebra class, they are NOT identical, even as groups. They are "merely" isomorphic.

It's only when we take the more abstract point of view of Category theory that we can't tell them apart.

So no. No agreement on this point until you tell me the context.

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

In beginning abstract algebra class, yes. In the category of groups, no, because there is exactly one cyclic group of order 4, and I have no idea what it's "made of." I have no concept of its "elements" or its "operations." It has no internal structure at all. Objects in categories do not have any internal structure. All they have is maps to other objects. This is categorical structuralism.

So, no, we do not agree that those are "instances" of anything except in the most casual sense of perhaps beginning group theory. After all, they are the exact same structure.

Would you say that 4 and 2 + 2 are two "instances" of the same number? Good question.

In fact now that I think of it, I have no idea if I've got any of this right. In the category of groups are these two separate objects that have an isomorphism between them? Or is there only one object that has no instances and no internal structure? I'm not entirely sure.

I should mention in passing that earlier today I ran my eyeballs over the SEP article on mathematical structuralism. They made a clear distinction between the philosophical development of mathematical structuralism via Benacerraf and Putnam, on the one hand; and categorical structuralism via Mac Lane and Grothendieck. So categorical structuralism is a branch or aspect of mathematical structuralism, and shouldn't be identified with it as I've been doing.

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

In the category of groups I don't think the question makes sense. Any more than if you pressed me on whether 4 and 2 + 2 are each instances of the same thing. Is instance even the right word? Maybe representation, or presentation.

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

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.

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

But of course a thing is identical to itself. That's the law of identity. But recognizing when two things are the same is one of the fundamental problems in mathematics!

Now I'm reminded of Barry Mazur's famous essay (pdf link)

When is one thing equal to some other thing, which delves into some of these categorical issues.

Have I ever done so? — Pneumenon

This was in reference to your straw man argument that structuralists make structure a primitive, or some such. I don't think the question was on point but I could be wrong.

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

All good, but I'm at the limit of my knowledge and not sure I'm even telling the truth about a few things.

ps - Poincaré said that "mathematics is the art of giving the same name to different things." Another early instance of pre-structuralism.

In fact see the entire context of Poincaré's quote here ... good reading.

https://ncatlab.org/nlab/show/isomorphism

He even extends the idea to physics.

"The physicists also do it just the same way. They invented the term ‘energy’, a word of very great fertility, because through the elimination of exceptions it established a law; because it gave the same name to things different in substance, but alike in form."

pps -- I'm wrong about there not being separate isomorphic instances of groups in the category of groups. That implies that I'm wrong about a few other things in this post. So in the end I'll agree that the two representations of the cyclic group of order 4 are indeed distinct objects, even in the category of groups.

https://math.stackexchange.com/questions/2041417/are-objects-in-the-category-grp-actually-groups-or-isomorphism-classes-of-groups

ppps -- I have resolved my confusion.

So there's a concept called the skeleton of a category, which contains exactly one copy of each isomorphism class of objects. So in the skeleton of the category of groups, there is only one cyclic group of order 4; and there aren't any variations on its representation.

So if we're in the category of groups, ({0, 1, 2, 3}, +) and ({1, i, -1, -i}, *), both exist as distinct, but isomorphic, objects.

But in the skeleton of the category of groups, they do not exist as separate objects; there's only one such group.

So once again, when asking when two mathematical objects are the "same," the answer is always that it depends on the category. And also -- the structuralist view -- we should not use the word same, but only isomorphic, because isomorphism is the only thing that matters.

Perhaps a very stupid question: why isn't Math referred simply to being a system?

Perhaps a very stupid question: why isn't Math referred simply to being a system? — ssu

Math itself is full of systems, like the system of the natural numbers, or the system of addition. So, in a sense, you might say it is a system itself, but a very complex system. @fishfry will give a much more sophisticated answer to your question.

fishfry will give a much more sophisticated answer to your question. — jgill

I'll just give the laziest answer possible. I typed "is math a system" into Google. The Google AI responded:

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

Google AI thinks so. Pretty much everything is a system, from indoor plumbing to the National Football League.

There's a discipline called general systems theory, which is ...

... 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

To each his own, but I don't feel a much difficulty in adjusting to contexts, so that in some contexts I pay attention to the formal implications of the definitions but other contexts I go with a looser flow.

The formal context I mention here is formal Z set theory and its Z based variants.

Is the natural number 3 identical to the real number 3? — fishfry

Formally, no. As you mention, they are different kinds of sets but there is an embedding of the naturals in the reals.

Informally, yes.

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

"whatever that means" indeed. I'd have to hear someone's definition of "copy".

Formally, the x-axis is {<p 0> p in R} and the y-axis is {<0 p> | p in R}. So the domain of the x-axis = R = the range of y-axis. And we can define less than relations: for all p and q in R , <p 0> less than <q 0> iff p < q, and <0 p> less than <0 q> iff p<q.

Informally, pretty much, nothing at odds with the formal notion, as far as I can think of.

"the unique Dedekind-complete linearly ordered set" uniquely characterizes the real numbers. — fishfry

Do you mean just a "strict linear ordering with the least upper bound property" or do you mean a "complete ordered field"?

Of course, we know that any two complete ordered fields are isomorphic, but that involves more than a strict linear ordering with the least upper bound property. It also involves having + and * and the needed statements about them. So, from "any two complete ordered fields are isomorphic" can we infer that "any two strict linear orderings with the least upper bound are isomorphic"? (Or maybe it's tacitly understood that there are fields involved.) [EDIT: See next post.]

Would you say that 4 and 2 + 2 are two "instances" of the same number? — fishfry

I don't know what 'instances' means, but in context of the standard interpretation of the symbols '4', '2' and '+, it is the case that 4 and 2+2 are the same. '4' and '2+2' are not the same, but they name the same thing. 4 = 2+2. 4 is 2+2.

One sense of 'instance' or 'occurrence' I do understand is that of occurrences of a symbol or string of symbols.

In '4 = 2+2' there are two occurrences of '2'. The first occurrence of '2' is as the third entry in the string and the second occurrence of '2' is the fifth entry in the string.

One formal approach is to take such strings as finite sequences, which are functions whose domain is a natural number. The string is displayed just as one character after another, but formally, it is this sequence:

{<0 '4'> <1 '='> <2 '2'> <3 '+'> <4 '2'>}

Let f = {<0 '4'> <1 '='> <2 '2'> <3 '+'> <4 '2'>}

And we know 5 = {0 1 2 3 4}.

So the domain of f is 5. The range of f is {'4', '=',' 2', '+'}. And:

f(0) is '4'
f(1) is '='
f(2) is '2'
f(3) is '+'
f(4) is '2'

So '2' is both the third entry and fifth entry in the sequence.

Or we can use tuples instead of functions. So '2+2 = 4' would be <'2', '+', '2', '=', '4'>.

But it seems to me that for additional handling, such as concatenation, functions are easier to work with mathematically even if lengthier to display.

And, for example, the string '= 2+' occurs as a substring twice in '4 = 2+2 -> (4 = 2+2 & 5 = 2+2+1)'.

Etc.

So expressions themselves are mathematical objects. But, wait, not so fast, because the symbols are not themselves mathematical objects. Ah, but we can take even the symbols to be mathematical objects. For example, we can take them to be natural numbers.* And not just mapped to natural numbers as with Godelization, but taken to actually be natural numbers.

So, for example, we regard any display of the typographic shape, that is two lines crossing as with

+

to not actually be the symbol but rather to be a visual aid for marking an occurrence of the symbol which is a natural number. For example, when we specify the symbols of a language, '+' could be, say, the natural number 16. Then, by the way, we when we Godelize a language, we don't have to map the symbols to numbers; they already are numbers.

* For an uncountable language (which is not a formal language but is a language, and is used for proving the existence of a model that provides for nonstandard analysis) we could take the symbols to be real numbers. Or we could take the symbols to be members of an uncountable ordinal.)

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:

Proof: Take any two ordinals with different cardinalities. The standard ordering on an ordinal is a strict linear ordering with the least upper bound property.

Even ridiculously trivial: The membership relation e_0 on 0 is a strict linear ordering with the least upper bound property. The membership relation e_1 on 1 is a strict linear ordering with the least upper bound property. But <0 e_0> and <1 e_1> are not isomorphic since there is no 1-1 function from 0 onto 1.

So, unless I've overlooked something, I think we need to mention that there are fields involved. (Or maybe it's tacitly understood that there are fields involved.)

This bears on another thread lately.

Two definitions of 'the continuum'"

the continuum = R

the continuum = <R L> where L is the standard less than relation on R

Definition of 'is a continuum':

A topological space S is a continuum if and only if S is connected, compact and Hausdorff

Let z be the standard ordinal ordering on 2^aleph_0.

z is a strict linear ordering on 2^aleph_0 and with the least upper bound property.

But it is not the case that <R L> and <2^aleph_0 z> are isomorphic, even though R and 2^aleph_0 are equinumerous.

So, it seems to me that the claim that R is unique to within isomorphism, or that <R L> is unique to within isomorphism, i.e. that the continuum is unique within isomorphism, should be regarded as false or it needs to be taken as tacit that what are isomorphic are the fields (fields with strict linear ordering with the least upper bound property). What are isomorphic are certain systems (S p t h) with (R + * <).

And that serendipitously ties in with the topic of "systems", as a reminder that another sense of 'system' is that of a tuple with a carrier set and operations and relations on that carrier set.

(Another sense I'd mention is 'logistic system'.)

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

Hence the question was stupid, as I assumed.

It being a logical system would be perhaps more fruitful, but the notion would still be in the set of self-evident "So what?" truths about mathematics. Just what kind of logical systems math has inside it would be the more interesting question. Now when mathematics has in it's system non-computable, non-provable but true parts (as it seems to have), this would be a question of current importance (comes mind the Math truths aren't orderly but chaotic -thread).

If so, perhaps the old idea of math being a tautology comes to mind: something being random and non-provable but true is... random and non-provable but true. Yet how do we then stop indoor plumbing and the National Football League being math? That the two aren't tautologies, even if indoor plumbing ought to be designed logically(?) Would it be so simple?

I think Clark Kent is identical to Superman at all times. They're both one dude. "Clark Kent" shares a referent with "Superman". — Pneumenon

That seems right to me.

Lois Lane doesn't know that they are the same, but that doesn't entail that they're not the same. What are not the same is Lois Lane's notion of Kal-El as he is in the guise of a person named 'Clark Kent' and Lois Lane's notion of Kal-El as he is the super-person called 'Superman'.

This reminds me somewhat of:

the set of natural numbers = N = the ordinal w = the cardinal aleph_0.

N, w and aleph_0 are the same thing, even though we think of it differently depending on context.

We think of N with its property of being the carrier set for the system of natural numbers.

We think of w with its property as being the first infinite ordinal.

We think of aleph_0 with its property as being the first infinite cardinal.

But still they are the same thing.

Meanwhile, Lois Lane thinks of Clark Kent with his property of being a mild mannered newspaper reporter, and she thinks of Superman with his property of being a possessor of superpowers dressed in tights and a cape. She doesn't know that Kal-El = Clark Kent = Superman, but still it is the case that Kal-El = Clark Kent = Superman. She doesn't know that the mild mannered newspaper reporter is the same being as the possessor of superpowers dressed in tights and a cape, but still there is only one being involved. Suppose she learns that Clark Kent is Superman. So it had to have been true for her to learn it. It's not the case that the sameness of Clark Kent and Superman happened only upon Lois Lane learning it. Meanwhile, Kal-El very well knows that he is Clark Kent is Superman. What happens in the phone booth is a change of apparel not a change of being.

perhaps the old idea of math being a tautology comes to mind — ssu

Statements devoid of content? (Frege)
I think not.

Not at all. Just on the one hand, feeling totally inadequate to discuss the philosophical side of this ... Quine and Spinoza for example. — 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!

Although I am wearing this shirt right now:



If that counts for anything.

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

Let me make sure I understand you. You're saying this:

"There's this thing called identity. Math tells us that identity is contextual. So we should also say that the identity of people is contextual, since math tells us about identity. For example, Clark Kent and Superman are contextual identities."

Is that right?

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

I don't understand. "Lois is unaware of the fact that Clark Kent is Superman. Therefore, Clark Kent is not Superman". That's not what you're saying, is it? If I put on clown makeup and then kill someone, I'm still guilty of a crime, even after I wash off the makeup.

Reading over your other points, I have a hunch about your position. You have used the term "representation" a few times. And you seem to think that, if A and B both represent C, then A and B are identical insofar as they represent C. You seem to identify how something is represented with what it is.

Two things.

1. Do Clark Kent and Superman represent one thing? If so, what is that one thing?
2. Would horses have four legs if nobody counted them? If so, which four would it be? The natural number 4? The rational number 4? The real number 4?

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

ok ...

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

You opened by saying you wanted to discuss mathematical structuralism; specifically, that you wanted to oppose it.

I'd be happy if we could focus on your ideas. You clarified what you meant by an identity condition, and I pointed out that it was structural, and seemed to undermine your own thesis. You didn't engage and didn't agree or disagree. I'd find it helpful if we could focus in.

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

I made the point that Clark Kent and Superman are two representations, or guises if you will, of the same entity. Just as two isomorphic groups are really the same group. I don't think the Superman analogy bears too much weight. Better to come back to your ideas about mathematical structuralism; since I had in the past some training in the fundamentals of contemporary math, including set theory and a little category theory. Then again I have a pretty good background in Superman comics.

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

They're not identical. They're isomorphic. Identity is a tricky thing in this context. Structuralism is all about isomorphism. "Sameness" and "identity" don't really come up.

If you would clarify your thesis -- perhaps recall why you initially started this thread and resurrected it -- we could be more focussed.

In modern math, isomorphism is important. Identity and sameness aren't as important; as as we're seeing, they're kind of slippery.

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

I wish we could get back to your ideas about Platonism versus mathematical structuralism. The Superman analogy does not bear too much scrutiny in this context.

In any event, if it wasn't clear, in my previous long post I eventually talked myself out of my own point. The additive group of integers mod 4 and the multiplicative group of the integer powers of the complex number i are indeed distinct groups, and distinct objects in the category of groups; although they are isomorphic. I believe that actually supports a point you were making, so if you like we could go back to that before I confused myself on that point.

Hence the question was stupid, as I assumed. — ssu

I didn't say that at all. Only that calling math a system is far too general. It doesn't tell us anything about math.

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

Yes ok.

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

I suppose you'd need a more specific definition if you wanted to uniquely characterize math. It's tricky, since the nature of math is historically contingent. In the old days they didn't even have negative numbers or fractions, or algebra; let alone the wild abstractions we have today.

I'm not sure how important it is to nail down a particular definition. Math is what mathematicians do.

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

Isn't this the definition you gave earlier?

Was this post for me?

If I mis-stated the definition, my bad.

Do you mean just a "strict linear ordering with the least upper bound property" or do you mean a "complete ordered field"? — TonesInDeepFreeze

My error if I gave the wrong definition. Which I always have to look up since I always forget it. Sometimes the Archimedean property is mentioned, other times not.

The OP is making some point about mathematical structuralism, which was the context for the discussion of instances. I wasn't entirely sure how your remarks related to that.

Isn't this the definition you gave earlier? — fishfry

It's not a definition. What definiendum do you have in mind?

Was this post for me? — fishfry

It's for whomever wishes to read it.

If I mis-stated the definition, my bad. — fishfry

What definition? I didn't take issue with a definition.

I didn't intend my posts to comment on structuralism.

What definition? I didn't take issue with an definition. — TonesInDeepFreeze

I'm gonna quit while I'm behind here.

I didn't intend my posts to comment on structuralism. — TonesInDeepFreeze

My point. That was the subject of the conversation. So I wasn't sure how to respond to your remarks without going off on extraneous tangents.

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

People may view this in different ways. But, for me, as far as the bare bones context of extensional set theoretic mathematics: Two different, but isomorphic, groups are not the same object, not the same set. They may be regarded as interchangeable as far as their common aspects are concerned, but still they are different objects. Informally they may be regarded as the same, but formally they are different.

/

Clark Kent is Superman. Clark Kent is not a representation. Superman is not a representation. He is a being. But, of course, 'Clark Kent' and 'Superman' are different things. And Lois Lane has two different notions, one of a human named 'Clark Kent' and one of a superhuman named 'Superman', but Lois Lane's notions don't negate that Clark Kent is Superman.

The proverbial example: One might have a notion of the Morning star and a notion of the Evening star and believe that they are two different celestial objects. But believing that they are different doesn't make them different. There is only Venus, which is seen at different times of the day. It's not a different object when seen in the morning than it is when seen at night.

Kar-El is seen sometimes wearing a suit and glasses and with people calling him 'Clark Kent' and seen sometimes wearing tights and a cape with people calling him 'Superman'. He's not a different being when seen wearing a suit and glasses and being referred to as 'Clark Kent' than when he is seen wearing tights and a cape and being referred to as 'Superman'.

Statements devoid of content? (Frege)
I think not. — jgill

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?

I'm not sure how important it is to nail down a particular definition. — fishfry

For example to the question in the OP it is something. Also do note that it does influence on how people see mathematics and how the field is understood and portrayed. Do people see it from the viewpoint of Platonism (numbers are real), logicism (it's logic) or from formalism (it's a game) or something else?

An apt description works wonders here.

Math is what mathematicians do. — fishfry

Oh, and this is a perfect example why it is important: with that you'll give here not only your little finger, but your left leg to the worst kind of post-modernists and sociologists that then can proclaim that math is only a societal phenomenon and a power play that a group of people (read men) do. That all this bull of math being something different, having it's own logic or being something special that tells something about reality is nonsense (or in their discourse, an act in that power play) when it's just what mathematicians declare doing. That's obviously not what you meant, but how can your statements be used is important.

These kinds of ideas, which I myself oppose, is like taking Thomas Kuhn totally out of context and using him (or the study of science as a human enterprise and interaction) as a way to claim that in this case mathematics is nothing special (perhaps from plumbing or playing a sport).

I didn't intend my posts to comment on structuralism.
— TonesInDeepFreeze

My point. That was the subject of the conversation. — fishfry

It's not clear to me whether you're suggesting that my remarks were not pertinent. But in case you are:

The conversation has had many subjects. You mentioned certain isomorphisms. I was interested in that. My remarks about that don't have to comment on structuralism.

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.

...

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

You said you wanted to focus on my ideas. Okay, then. Let me explain my motivation for the thread. Please read the following carefully. If I lose you, say where and why. I promise, it's relevant.

So, I am really really interested in whether or not mathematical objects exist in a mind-independent way. Would there be numbers even if we weren't here? I want to say yes. I think that numbers would be here even if we weren't here. I think that brontosaurus had 4 legs long before we counted them. I can't fathom what it would mean to say that it didn't. So that makes me a Platonist, because I don't think that numbers depend on our minds. At least, not in that way.

But there's a problem with Platonism. If I say that something exists, I need to identify it. If I say, "the Blarb exists", then I need to say what the Blarb is. I need an identity condition that picks out the Blarb and only the Blarb.

This is a counter to Platonism, because it confounds the motivation. Would horses have 4 legs if nobody counted them? I, the Platonist, say yes. But if you ask the me for an identity condition, I'm in trouble. See, if say that the legs of a horse are the set {0, 1, 2, 3}, then I've said that the number of horse legs is the natural number four. But is that even true? What if the number of horse legs is the set of all rationals smaller than 4, i.e. the real number 4? How would we know which one it is?

So it comes out to this:

1. To say that a certain thing exists, you need an identity condition for it.
2. You can't always get those identity conditions for mathematical objects.
3. Therefore, we can't say that mathematical objects exist.

Uh-oh!

My hare-brained solution to that was "maybe there's only one mathematical entity". Math is just a single thing. Because then math can be Platonically real without needing identity conditions. You just say that different mathematical objects are what happens when you analyze that one single object in different ways.

I know that sounds weird. Maybe an analogy will help.

So, for example, you know the duckrabbit?



It looks like a duck if you look at it one way. It also looks like a rabbit, though not both at once. However, you can't just see it however you want; it's not a duckrabbitgorilla. It's not a duckrabbithouse. It's not a duckrabbithitler. So there are two valid ways of seeing it, but only one can be used at a time. And some ways of seeing it are invalid.

What if math is kind of like that? There's a single mathematical reality, but it looks different depending on how you analyze it. So if you bring a certain set-theoretic framework, math gives you real numbers, and if you bring a different one, math gives you rational numbers. But the rationals and reals aren't separate, self-identical objects. They're just representations, ways of representing one underlying reality.

Basically, you don't need identity conditions for a representation, because representations don't need to be self-identical in that sense. We can answer the question, "Did horses have 4 legs before anyone counted them?" with "yes" (which is what I wanted). That's because the underlying mathematical reality is Platonic and never changes. But then we have this question: "Was that the rational number 4, or the real number 4, or what?" That's the question that initially flummoxed us. But if there's only one mathematical object, that question is no longer sensible. Represent the number of its legs however you like. You can represent it as rational 4, or real 4, or natural 4. The underlying reality is the same.

I'll come back to that in a moment. Now for structuralism.

The SEP article on structuralism tells me that there's a methodological kind of structuralism, which is basically just a style of doing math. Then there's a metaphysical structuralism, which is an ontology. The former is a style of mathematical praxis and the latter is a philosophical position.

You seem to waver between methodological and metaphysical structuralism, and it confuses me. On the one hand, you take a stance like, "I'm no philosopher. I just find this to be an interesting way of doing math". On the other hand, you do seem interested in the philosophical implications of structuralism, e.g. when you said that modern mathematics tells us new things about the notion of identity. And you're on a philosophy forum discussing it, rather than a math forum.

So let's put the discussion on these questions:

1. Does methodological structuralism imply metaphysical structuralism? Or at least, enable it?
2. Is metaphysical structuralism compatible with Platonism? If so, is it compatible with my idea that there is, in a sense, only one mathematical object?

P.S. the SEP article has it,

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.

...

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).

We ask, "Is the number of horse legs the natural number 4 or the rational number 4?". Well, for Shapiro, the number of horse legs occupies a position. That same position is filled by {1, 2, 3} in the naturals and { x ∈ Q : x < 4 } in the reals. So the answer is, "The number of horse legs is both of those".

The identity condition, then, is "That unique office occupied by {1, 2 3} in the natural numbers". But there are plenty of other identity conditions that pick out the same object: "That unique office occupied by { x ∈ Q : x < 4 } in the reals" picks it out as well. At this point, the numbers themselves are identity conditions for offices!

Maybe I don't need to reduce math to one object after all...

I hope I didn't wander off too far. And I hope that this is, at least, interesting.

a societal phenomenon and a power play that a group of people (read men) do — ssu

In America women make up 25-30% of PhD students. 15-20% of math faculties. Not entirely men.

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

To me this seems like a word game. Describing a theorem, if A then B, requires specific terms, going beyond its "logical form". The idea of categorizing math as a tautology contributes nothing to its practice.

when it's just what mathematicians declare doing. — ssu

Years ago the maid for a prominent mathematician was asked what her employer did. She replied, after some thought, "He scribbles on pieces of paper, grumbles, then wads the papers up and throws them in the trashcan."

if say that the legs of a horse are the set {1, 2, 3} — Pneumenon

I wouldn't say that.

(1) the set has only three members, not four, (2) the legs of the horse is not that set; rather the cardinality of the set of legs of the horse is that set.

So, I would say: The cardinality of the set of legs of the horse is 4 = {0 1 2 3}.

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

You could say that there are different forms of "number of":

cardinality - every cardinality is an ordinal (a finite ordinal, i.e. a natural number if the set is finite).

cardinality_q - every finite cardinality is the embedded natural number in Q

cardinality-r - every finite cardinality is the embedded natural number in R

Then each appropriate statement, couched in each form, about the set of horse's legs is true,

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

If we're talking about formal mathematics:

(1) To define a constant symbol, we need first an existence_&_uniqueness theorem.

That's syntactical.

There are only countably many constant symbols, so only countably many such definitions.

(2) To define an object b in a model M (hope I've not made a mistake):

we need a formula F, whose only free variable is x, such that M satisfies F iff x is assigned to b.

Then b = the p such that M satisfies F iff x is assigned to p.

/

Perhaps, this might work (I don't know, it only occurred to me just now):

Let the mysterysaurus be a dinosaur that we inferred existed but don't know the number of its legs. To say that the mysterysaurus had four legs might not require a mathematical definition of 'four'. Rather, you could be saying "If you were an observer when the mysterysaurus was extant, then when you looked at one leg after another, and said the words, 'one', 'two', 'three', 'four' - one of those for each leg - then you would stop at 'four'". Yes, observation is mentioned, but isn't the statement independent of my knowledge? Does that hold up?

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

Was it Einstein that said something to the affect that God gave us the natural numbers, all else of mathematics are mans' ? Or something like that. When I think of one of my theorems about the indifferent fixed point of an infinite composition of LFTs that converge to a parabolic LFT I never stop and wonder if all that preexists in a timeless realm beyond the thoughts on man. I do believe it exists in a potential way.

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

I'm always amused by this common philosophical example, since Venus isn't a star at all.