But numbers, and other ‘objects of reason’, are real in a different way to sense objects. And that is a stumbling block for a culture in which things are said to either exist or not. There is no conceptual space for different modes of reality (leaving aside dry, academic modal metaphysics). Which is why we can only think of them as kinds of objects, which they’re actually not. They’re really closer to kinds of acts. — Wayfarer

Well said. Starting with the natural numbers, which are ways to distinguish objects and converse about quantities, mathematics has grown to virtually unimaginable proportions over the millennia. And it has changed character from a descriptive and predictive tool to an enormous game, unbounded in some aspects, with recently formulated foundational rules.

Some compare it to chess, where material pieces are moved around a board rather than the pen or pencil upon paper, or keys and screen of a computer. Where it might differ is in potential: mathematics awaiting discovery or creation versus possible strategies or moves on the chessboard. Chess players might comment on this.

Is a crossword puzzle real? Pondering how to fill in the spaces, then doing so with pencil. Sounds a little like math. Are emerging ideas real? Of course they are. Do mathematical objects exist in some exotic realm, awaiting discovery? I think of them as commonalities of minds, the way in which human brains have evolved.

Do infinitesimals exist (in the platonistic sense)? — Michael

I've always thought of these little critters as part of the metaphysics of mathematics. They now belong to a variation of the game called nonstandard analysis.

Do mathematical objects exist in some exotic realm, awaiting discovery? — jgill

As I said, I think ‘exist’ is problematical in the context. Not that they don’t exist, but the way in which they’re real is different to empirical objects. They are ‘objects of mind’ rather than ‘objects of sense’, but I don’t think the philosophical lexicon has an appropriate term. I tried this out on ChatGPT recently and it suggested ‘transcendentally objective’, although that is hardly an elegant expression.

And it has changed character from a descriptive and predictive tool to an enormous game, unbounded in some aspects, with recently formulated foundational rules. — jgill

Consider synthetic chemistry and genetic engineering. These too are grounded in traditional chemistry and biology but now have dimensions that would never be found in nature herself. It’s analogous in some ways.

I've always thought of these little critters as part of the metaphysics of mathematics — jgill

Maybe they are to natural numbers as viruses are to organisms ;-)

Hm. But how would one substantiate this idea that numbers exist in this different way?

And why would numbers be able to exist in this way, and not flying spaghetti monsters?

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.

I just want to point out that the bolded phrase is what's at stake. You can believe that numbers and other abstracta really and truly exist without being a mathematical platonist. You merely assert that they exist because we have created them, and they will cease to exist if we also cease. Whether you want to say this or not will depend on how you wish to use the word "exist." Clearly, if you are a friend of "existence = spatiotemporal objects or arrangements thereof", then you won't want to claim even a human-made existence for numbers.

I agree. I don't think mathematical platonism is supposed to be some big metaphysical statement. It's just reflecting our experience with math: that it's something we seem to discover, that it's not owned by particular people, in other words, it's not mental or physical.

That leaves the door open to trying to explain it anyway we want, kind of like gravity is a thing, but we're still working on how to explain it.

Do infinitesimals exist (in the platonistic sense)? - 3. Infinitesimals exist according to some number systems but not others. — Michael

Starting with the natural numbers, which are ways to distinguish objects and converse about quantities, mathematics has grown to virtually unimaginable proportions over the millennia — jgill

This IS the mistake we do.

We START from natural numbers as it's the natural place to start for counting. It basically a necessity for our situational awereness, hence even animals can have a rudimentary simple "math"-system. Yet simply as mathematics has objects that are not countrable, starting with infinity, infinite sequences and infinitesimals, whole math simply cannot be based on natural numbers. This is the reason why Russell's logicism faced paradoxes. Not everything was discovered. That there exist the uncountable should make it obvious to us that natural numbers and counting isn't the logical ground on which everything mathematical is based upon.

Something really big is missing here. It's up to us, perhaps, to find the answer. Or at least get closer to it

Thanks for you @Michael to start this thread.

You can believe that numbers and other abstracta really and truly exist without being a mathematical platonist. You merely assert that they exist because we have created them, and they will cease to exist if we also cease. — J

What about the laws of logic, like the law of the excluded middle? Does that cease to obtain in the absence of rational sentient beings? I’m more inclined to the understanding that it is discovered by rational sentient beings, and with it the realisation that it must be true in all possible worlds. The alternative is to subjectivize such principles, which reduces them to social conventions. Meaning whatever reality they possess is contingent - so they can’t ‘really and truly exist’.

I tend towards objective idealism - that logical and arithmetical fundamentals are real independently of any particular mind, but can only be grasped by an act of rational thought. I believe that’s more in line with classical metaphysics.

See Frege on Knowing the Third Realm, Tyler Burge.

You can believe that numbers and other abstracta really and truly exist without being a mathematical platonist. You merely assert that they exist because we have created them, and they will cease to exist if we also cease.
— J

What about the laws of logic, like the law of the excluded middle? Does that cease to obtain in the absence of rational sentient beings? — Wayfarer

Right, what I was describing as a possible position about numbers was meant to sharpen the question: Are we disputing whether abstracta as such can be said to exist, or is the dispute about whether they can exist independently of us? Like you, I find the "existing, but not independently" position re numbers to be unconvincing. Some abstracta probably have that characteristic -- the rules of chess, perhaps? -- but logic and math do not seem arbitrary in that same way. If personal testimony counts, the two mathematicians I have known well are both committed platonists, and speak fervently about the experience of math as one of discovery, not invention. But that's hardly decisive.

Meaning whatever reality they possess is contingent - so they can’t ‘really and truly exist’. — Wayfarer

It's hard to talk about existence without presupposing a certain use of the term. So I'll just point out that you're wanting "exist" to mean "not depend on something else". Or perhaps it's "really and truly exist" that has the characteristic of non-contingency? I'm not making fun; these are perfectly legitimate lines to draw, it's just that there's no agreement about which terms to assign to the resulting map.

I tend towards objective idealism - that logical and arithmetical fundamentals are real independently of any particular mind, but can only be grasped by an act of rational thought. — Wayfarer

I like this too. It suggests a useful map, one which shows some existing things as graspable by reason, others by perception (or however you want to characterize what we do with stuff in space/time). We might also want a third location on the map for imaginary things -- maybe this would be a region of non-existence. Now of course someone is going to come along and say, "Yes but what is existence really? You can't just reduce it to a dispute about terminological conventions!" To which the only reply I know is -- all together now! -- "To be is to be the value of a bound variable." In other words, it all depends what you're talking about. But how you talk about it is not arbitrary at all. There really is privileged metaphysical structure; we're just not sure about the terms to use.

What about the laws of logic, like the law of the excluded middle? Does that cease to obtain in the absence of rational sentient beings? — Wayfarer

Classical logic uses the law of excluded middle but intuitionistic logic doesn't, allowing for sentences that are neither true nor false.

You seem to be suggesting that one of these logics is correct. Which may be so if platonism is correct, but not if it isn't.

You seem to be suggesting that one of these logics is correct. — Michael

If you mean, I believe that there is a truth to logical laws that is not dependent on one or another philosophical doctrine, then yes, I do believe that. I think the law of the excluded middle, for instance, describes something inherent in the structure of reality—not something contingent on whether anyone happens to conceive of it. It is a 'metaphysical primitive,' i.e., something that can't be reduced further.

There's a subtle point at issue here—the ontological status of such principles that are not created by the human mind but can only be grasped by a rational intellect. These principles, while independent of any particular mind, require the rational intellect to apprehend them—highlighting the unique role of reason in discerning universal truths. Whereas in today's culture there is an inherent tendency to try and account for those principles naturalistically, as a result of evolutionary neurology, etc (i.e. 'naturalised epistemology'). But this again relativizes them or makes them contingent facts. Would you agree with that?

How does the issue of correctness arise? As I understand it, intuitionistic logic doesn't contradict classical logic, it only uses different semantics. Couldn't both types of logic exist platonically -- awaiting discovery by sentient beings? To put it another way, if you believe that any abstracta can exist platonically, why draw the line at a single, putatively correct logic?

There really is privileged metaphysical structure; we're just not sure about the terms to use. — J

That's why I suggested that essay about Frege. I'm no expert in Frege - in fact that essay is about the sum total of my knowledge - but it explores the idea of a 'third realm', somewhat similar to Popper's idea with the same name. Those kinds of ideas are all generally Platonistic.

Thanks, I'll read it. I too find Popper's "Three Worlds" concept helpful. It's an interesting question, whether a commitment to World 3 items necessarily involves a commitment to some form of platonism.

, Popper is unconvincing on the ontology of world 3. Searle begins to answer this with his account of institutional facts and collective intentionality.

Maths as an act of collective intent. Of course there are infinitesimals.

Those kinds of ideas are all generally Platonistic. — Wayfarer

Popper's "Third world" differs from Plato's world of forms in that it is entirely an artefact of language and culture and is thus constantly changing. This is in contrast to the changeless world of Plato's forms. Also, for Popper the first world (the world of lifeless physical matter and energy) is real in its own right. The second world (the world of sensations, perceptions and volitions) is the world of pre-linguistic animals. And the third world is the symbolically mediated world of abstracta— of concepts and theories.

Popper's "Third world" differs from Plato's world of forms in that it is entirely an artefact of language and culture and is thus constantly changing. This is in contrast to the changeless world of Plato's forms. — Janus

True. Although there is considerable debate about what 'Plato's world of forms' actually is or means. In any case, the reason I mentioned it, is because Popper grants a kind of irreducibility to those things that constitute the third world.

Yes, the world of abstracta is conceptually irreducible to the world of sensations, perceptions and feelings and the world of energy and matter. Each world is, conceptually irreducible to the other two. If that were not so, we would not have the three worlds

Maths as an act of collective intent. — Banno

It would have to be the way collective intent interacts with the world, right?

Yes, because that's what we do. Presumably the sort that don't interact with the world are pure maths, the ones that do, applied.

This IS the mistake we do.

We START from natural numbers as it's the natural place to start for counting. It basically a necessity for our situational awereness, hence even animals can have a rudimentary simple "math"-system. Yet simply as mathematics has objects that are not countrable, starting with infinity, infinite sequences and infinitesimals, whole math simply cannot be based on natural numbers. This is the reason why Russell's logicism faced paradoxes. Not everything was discovered. That there exist the uncountable should make it obvious to us that natural numbers and counting isn't the logical ground on which everything mathematical is based upon. — ssu

Of course, "whole math" is not "based" on natural numbers. But they did come first. It was a start, like a path of a thousand miles, one step at a time. Those simple initial steps may culminate with climbing a thousand meter peak. Get a grip, man.

How does the issue of correctness arise? — J

If platonism is correct then I suppose a "correct" logic is one that includes these mind-independent logical facts and doesn't include any logical "fictions".

For example, if the law of excluded middle is a mind-independent fact then classical logic is more correct that intuitionistic logic, and if the law of noncontradiction is a mind-independent fact then classical logic is more correct than dialetheism, and if truth is mind-independently bivalent then classical logic is more correct than fuzzy logic.

But if platonism isn't correct then no logic is "correct". They can be consistent or useful, but nothing more substantial.

Couldn't both types of logic exist platonically -- awaiting discovery by sentient beings? To put it another way, if you believe that any abstracta can exist platonically, why draw the line at a single, putatively correct logic? — J

I addressed that in the OP with respect to incompatible mathematical entities, e.g. the Quine atom which is a set that contains itself. New Foundations allows for such a thing but ZFC doesn't.

If platonism is correct then either Quine atoms are mind-independent mathematical entities or they're not.

I don't think it makes any sense to say that they platonistically exist in New Foundations but don't platonistically exist in ZFC. We can only take the approach of mathematical fictionalism and say that they exist according to New Foundations but not according to ZFC.

I believe that there is a truth to logical laws that is not dependent on one or another philosophical doctrine — Wayfarer

Which logical laws, and why those? There's classical logic, intuitionistic logic, dialetheism, three-valued logic, fuzzy logic, free logic, and so on.

Why then something as obvious and useful then "now belong to a variation of the game called nonstandard analysis", as you said? "Standard" analysis I guess goes with limits.

Especially if we look at this from the viewpoint of Platonism, saying that we have these "games" in mathematics, pick what you want and look how the game goes then, doesn't seem in line with Platonism at all. Either infinitesimals exist or they don't. If they exist, there shouldn't be any problem with something else in mathematics. And why aren't infinitesimals accepted and only belong to "nonstandard analysis"? Because we still have the puzzling problem that Newton and Leibniz faced when giving an explanation for something that is and cannot be devised into anything smaller.

Yes, because that's what we do. Presumably the sort that don't interact with the world are pure maths, the ones that do, applied. — Banno

I was thinking about things like the Fibonacci sequence. It shows up in a lot of places that have nothing to do with human consensus. There's something about the structure of math that matches up to the structure of the universe in some ways.

@SophistiCat Could you explain the thing about the number 1/137 in physics?

I don't think it makes any sense to say that they platonistically exist in New Foundations but don't platonistically exist in ZFC. We can only take the approach of mathematical fictionalism and say that they exist according to New Foundations but not according to ZFC. — Michael

I see where you're going with this. But I don't think that what you're calling the "only approach" is quite so straightforward.

Suppose I say, "x exists according to Harry." You say, "x does not exist according to Sally." What is the subject of the dispute between Harry and Sally? Are they in disagreement about x, or about what 'exists' means?

Tell me how you'd be inclined to answer that, and I'll develop the thought further.

I was thinking about things like the Fibonacci sequence. It shows up in a lot of places that have nothing to do with human consensus. There's something about the structure of math that matches up to the structure of the universe in some ways — frank

And the structure of the universe isnt the product of imaginative construction? Wittgenstein would say you’re being tricked by your own grammar, that is, by hidden suppositions that project themselves onto the ‘real’ world and then seem to arise from that outside.

And the structure of the universe isnt the product of imaginative construction? Wittgenstein would say you’re being tricked by your own grammar, that is, by hidden suppositions that project themselves onto the ‘real’ world and then seem to arise from that outside. — Joshs

So you're saying that math can be a community construction without necessarily arising from any activity involving the world. It's that what we call the world conforms to thought a la the Tractatus, so it's no surprise that we find an affinity between our math and the world's shenanigans.

Do you believe that we are also products of analysis? That your individuality arises from reflection on events?

Suppose I say, "x exists according to Harry." You say, "x does not exist according to Sally." What is the subject of the dispute between Harry and Sally? Are they in disagreement about x, or about what 'exists' means? — J

The question is ambiguous. Consider these two claims:

1. According to the Lord of the Rings canon, orcs exist
2. According to crazy folk, fairies exist

Mathematical fictionalists are saying something like (1), not (2).

I don't know what Harry and Sally are saying.

So rather I am expressing skepticism towards those who would claim mathematics is 'objectively real', and also pointing out the contradiction in the term 'mathematical platonism'.

What contradiction? The only one I've seen is that "since math is a sui generis human creation that doesn't exist "objectively " then it doesn't exist objectively." Yet this is just assuming the conclusion. At best you've argued for a sort of nescience on this question, but skepticism and agnosticism are not the same thing as rejecting a thesis.

Note that I am not saying that science shows us what is real, rather it seems to heavily suggest the existence of an underlying reality because it is able to make models of how that reality works to a degree that is at least accurate enough for our human endeavors.

Ok, why can't this involve numbers, which are essential to modern science? Can we infer what biology and evolution tells us about how our sense organs work in some way corresponds to reality, but not that the math that underpins these finding does? Why is that?

The core of what I'm saying is that, as Plato argued, it is very difficult to even access the reality that underlies our world of sense experience, let alone make statements about this reality.

Your position seems far more similar to Locke, Hume, Kant, etc. To be sure, Plato acknowledges a distinction between reality and appearances, but he does not suppose that reality is some sort of noumenal "reality as divorced from all appearances." Indeed, his supposition is that threeness, circles, etc. are more real than the world of sensible appearances because they are more intelligible/necessary/what-they-are. This is, in an important sense, the exact opposite of supposing that reality is the world with all appearances (including intelligibility) somehow pumped out of it or abstracted away.

Yet this is just assuming the conclusion. At best you've argued for a sort of nescience on this question, but skepticism and agnosticism are not the same thing as rejecting a thesis. — Count Timothy von Icarus

Didn't I just tell you that what I am doing is expressing skepticism, and not making claims about what does and doesn't objectively exist?

Ok, why can't this involve numbers, which are essential to modern science? Can we infer what biology and evolution tells us about how our sense organs work in some way corresponds to reality, but not that the math that underpins these finding does? Why is that? — Count Timothy von Icarus

Our sense organs do not show us the whole picture, and the same thing appears to be true for math and science.

They're tools that help us model reality.

Your position seems far more similar to Locke, Hume, Kant, etc. To be sure, Plato acknowledges a distinction between reality and appearances, but he does not suppose that reality is some sort of noumenal "reality as divorced from all appearances." Indeed, his supposition is that threeness, circles, etc. are more real than the world of sensible appearances because they are more intelligible/necessary/what-they-are. This is, in an important sense, the exact opposite of supposing that reality is the world with all appearances (including intelligibility) somehow pumped out of it or abstracted away. — Count Timothy von Icarus

Plato's objective reality is 'the One' - an indivisible, all-encompassing unity.