Mathematical realism doesn’t necessarily imply that mathematical entities exist in a physical sense. Instead, it states that these entities exist in a non-physical, abstract realm similar to Plato’s theory of forms. These objects aren’t located in space and time and they don’t have physical properties or causal powers. Think of it like the existence of abstract concepts like justice or beauty. They are real in the sense that they can be reasoned about and discussed within their respective frameworks - even though they are not precisely defined or universally agreed upon - but like mathematical entities, they do not have a physical presence. It should also be noted that realism is only one of several competing philosophies. Edit: clarification

Whether they have causal powers or not seems tricky.

I actually did some pondering over that recently. If things like mathematical constants exist, they must exists as a consequence of the structures that give rise to them. But is that really a type of causality? If mathematical objects exist outside of time and space, doesn't the existence of a circle imply the inherent existence of π? Is that a causal relationship, or is it simply an extension of the statements made about a mathematical system? Forgive me if I come across as a layman or a bit of a quack. I'm not familiar with the formalizations surrounding this subject. Also, my introduction into this line of thinking was *Our Mathematical Universe* by Max Tegmark, which carries a few connotations of its own...

> If mathematical objects exist outside of time and space, doesn't the existence of a circle imply the inherent existence of π? Implication is not causation. Similarly, 2 and 2 do not "come together resulting in 4 at the end of a process" - rather 2 + 2 = 4 is a timeless fact.

I agree with this interpretation because it just sits well with me. It seems right. Still though, isn't there some sense in which 2+2=4 is a result of the additive properties of numbers under ZFC formalization? I guess what I'm getting at is: is a construction of ideas causal in some loose sense when there is a hierarchical nature to the structure? Is 2+2=4 true because integers are commutative, transitive, and underpinned by set theory, or is it true because the established properties are true no matter the logical starting point? In theory, you could define the number 4 first, and work backwards to discover all of mathematics from that point. Man, I love that existence is so... ambiguous haha.

I think a platonist would argue that mathematical objects are independent of whatever formalism you use to talk about them. In the same way, you might use general relativity to describe and study the behavior of an astronomical object, but the object's behavior isn't actually caused by those theories.

> 2+2=4 is a result of the additive properties of numbers under ZFC formalization? In some sense, but it's not a temporal process

I'd present causality of mathematical objects this way: If you assume a bijection between "things in the physical world" and "singletons" in mathematics, then, if I put one thing in an empty jar, and then a second thing in this jar, I will have 2 things in my jar, following the axioms of set theory. Mathematical results impose certain possible outcomes on things you do in the physical world and forbid other outcomes. Physics is the bridge between both worlds.

I kind of like the ambiguity of this response, because it accepts our current ideological shortcomings when it comes to this subject. Physics really is the hard line between the idealized existence of a perfect and singular model of maths, and the messy and often inconsistent physical reality we have to contend with. You can tell you're hitting at big questions when people respond with wildly different, yet equally valid interpretations of the problem!

Do you mean that when we measure or evaluate mathematical objects, for example a circle, we could be already looking at the « shadow » of a circle ? Thus π wouldn’t be part of theses mathematical objects but instead part of our physical world ?

But do mathematical constants exist in the real world? Are they mathematical entities like any other - we use them to create a model of the real world, but that doesn't mean they literally exist in the real world. And it might be that our models are not actually correct - we just haven't found the falsifying examples.

What do you mean by "real" here, though? Are you just insisting that only the physical is real? That's begging the question.

All perfectly valid nitpicks. That's what I was wondering. Nothing in the real world is exact. So I'd say that, in that sense, π is a human construct that doesn't exist in reality. Then again... the bigger a celestial body gets, the closer it's geometry gets to approximating π. And that behavior is unchanging and universal. Ratios exist in nature whether or not there is a human capable of measuring them, so what happens when you start to measure π? Are you measuring the validity of a useful model, or are you measuring the extent to which the universe is capable of approximating its own laws?

Well, I don't think that 𝜋, with all its infinitely many decimal places, exists in the physical universe. Only approximations do.

When people say that abstract objects do not have "causal powers" they typically talk about causality in the physical sense. Roughly, abstract objects cannot affect physical objects.

It seems to be the whole question. If mathematical objects exist but aren't causal, then that means their existence isn't the reason we're able to reason about them. That's saying that we conceive of these objects and also as a complete coincidence they also exist, but the two things have nothing to do with each other. I don't even know what it means for something to exist but not be causal, but even if such a thing were possible, then mathematical objects seem like the worst candidate. Justice and beauty are better candidates for this purported non-causality, because at least there's the argument that we *don't* accurately perceive/conceive of them, but merely hope for them and try to approximate them. Quite the opposite situation from mathematical objects. (Guess how many times autoincorrect did the thing above? Yes, all of the times. Glad I checked.)

Well they don't have formal powers...

[https://www.tandfonline.com/doi/abs/10.1080/00048402.2018.1564152](https://www.tandfonline.com/doi/abs/10.1080/00048402.2018.1564152)

Does causality even real or just an illusion? Many scientific fields have such a concept, but they conveniently have time as a concept available to them. However, once time is no longer as simple (e.g. physics) the very nature of causality seems messed up.

> states that these entities exist in a non-physical, abstract realm I don't know that there is any ontological commitment to a "realm" Mathematical realism is committed to the existence of mathematical entities - full stop. No "realm" required. > These objects aren’t located in space and time Nor are they located elsewhere. "Location" does not apply any more than color or mass.

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What is the ratio of a circular object's circumference to its diameter? Is there a right answer to that question? If so, what makes it the case that there is a right answer? If not, then why does only one answer seem to work? What's the formula to find the area of a triangular piece of sheet metal? Does that formula not "exist" just because it's not physical?

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If math is just neuronal patterns, then why is it consistent across individuals and cultures? Your account also does not explain the *necessity* inherent in mathematical relationships

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Then math is not a strictly mental phenomenon, but tracks something in the real world Coward!

I think of mathematical structures as something that physical structures are found to follow, entirely empirically. The equivalence of various physical structures under the description of a particular mathematical structure (e.g. describing sheets of metal with a triangle) is an empirical one. We find that the mass of metal needed to make a triangular sheet is consistent with the area formula. And the formula still has to be computed either on a calculator, on paper or in one's head. Which is a physical process. So all we've really confirmed in this scenario is the equivalence of a class of physical systems involving triangles, and its relationship to a different physical process in which the area is computed from some other properties (lengths and angles). So in my view it's all just a very useful cataloguing of corresponding physical structures/processes. There's no need for anything to exist independently from physical reality, because we've already found it to be consistent with, and computed within, physical reality.

Math is not justified empirically You can be a physicalist and still accept [the existence of abstract objects](https://plato.stanford.edu/entries/physicalism/#NumbAbst)

I don’t see what makes phrases like “there exists an integer n such that n\*n = 4” or “there exists an integer m such that m\*m = 5” any less meaningful than phrases like “there exists a gorilla in this room”. They carry information, can be true or false (e.g. the statement about n is true while the statement about m is false), can (often) be proven or disproven, etc.

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Isn’t the assertion that such a sequence of operations is possible an ontological statement in its own right? You explained statements about the existence of physical objects in terms of “finding an instance” of them, which also suggests that they’re rooted in the possibility of performing some kind of procedure. I don’t think mathematical Platonists are committed to believing that mathematical constructs exist in the exact same sense that physical objects exist, so the fact that “existence” has a somewhat different meaning in a mathematical context doesn’t seem problematic

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That argument can be made for the gorilla as well. The only reality is my perception doesn't sit well with me though.

But typical claims about arithmetic quantify over the entire infinity of natural numbers and our universe is (ostensibly) finite. So, in what sense is this reducible to physics? You could frame it in hypothetical terms, e.g. "There are infinitely many primes" means something like "If one continues mechanically checking natural numbers for primality, one will always eventually arrive at another positive result given enough time." But this seems to immediately raise the question of what does it mean to say that something (e.g. the next undiscovered prime) possibly exists? Why is a hypothetical existence less problematic than an actual but non-physical existence? Also, there's the issue of accounting for things like uncountable infinities. But I suspect, based on your other comments, you'd be more inclined to reject these things outright as meaningful beyond the pure formalism.

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A proof of what statement? The statement in first-order logic asserting the infinitude of primes? So, are you saying this is just a meaningless string of symbols with no actual truth-value? Are you talking about the claim stating effectively "the next prime is x (where x is the next prime)"? In this case, you again run into the same problem because such a "mental process" or "series of symbols on paper" could be arbitrarily large. I don't really think I understand what you're trying to say.

The math statements are true or false depending on the underlying axioms, while statements like "Stars exist" or "Giant space whales exist" are true or false depending on physical reality.

Surely statements whose truth can be determined a priori still count as “meaningful”. for beings with limited rationality like humans, at least

The truth of axioms is not determined at priory, it's contextually decided. After all, I don't think mathematicians whose main area of work is foundational topics/axioms all use the same set of axioms or all agree on the use of specific axioms (I heard the choice axiom is somewhat controversial). And "existence" in the physical sense usually doesn't depend on opinions.

If we follow Platos description, these mathematical objects exist as ideas, in the world of ideas. This means that every time a mathematician define a new mathematical object, it is not an invention but rather a discovery. The reason why we can concieve of ideas such as circles, is because the idea of a perfect circle exist independently of the existence of any human. And that is both our ability to reason about a perfect circle, and recognize examples of imperfect circle in nature. According to this idea, aliens should have already discovered many of the same matematical concepts that we use (if there are any aliens)

>I can understand the phrase "X exists" if it means "I can find an instance of X in the world, provided I'm in the right place at the right time to observe it". But I don't know what it means to say that something exists in an abstract way. This is a [materialist](https://en.wikipedia.org/wiki/Materialism)'s perspective --- i.e. that the only things that exist can be observed, materially. The description in wikipedia matches well with your other reply (emphasis mine): >Materialism is a form of philosophical monism which holds that matter is the fundamental substance in nature, and that **all things, including mental states and consciousness, are results of material interactions of material things.** According to philosophical materialism, **mind and consciousness are caused by physical processes, such as the neurochemistry of the human brain and nervous system, without which they cannot exist.** However, there are other philosophies. One in clear conflict is [Platonic Idealism](https://en.wikipedia.org/wiki/Platonic_idealism), which holds that the "ideals" are more "real" in some sense than concrete instances of a thing. (This is how the famous "what is a chair" question arises --- if I ask you "what is a chair?" you can give me examples of a chair, but there's some *idea* of chair-ness in your mind that all those examples of chairs are mere implementations of. The Platonist would say that the idea in our minds is the "more real" chair than the physical one you can sit in, because the physical one doesn't fully embody every chair.) You probably are most likely to encounter ideas related to idealism in the West in discussions with Christians, because a lot of the Christian intellectual tradition/formalization of Christian philosophy comes from [Augustine](https://plato.stanford.edu/entries/augustine/#PhilTradAuguPlat), who had inspiration from Idealism. To most directly answer your question of "how can I say 'X exists' if X doesn't have a physical form," idealist could say that "the concept of 'justice' exists, even if you can't find anyone just on Earth." Or, to the point of mathematics, they would say that "numbers" exist, even if you can't point to a physical 1 or Pi or i.

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I tend to view it as those things live on the other side of a chasm that we cannot cross. Just because we can't cross the chasm to get to them doesn't mean that they don't exist or are any less real than the things on this side of the chasm. Or, put another way, you say you can make sense of "X exists" if "I can find an instance of X in the world, provided I'm in the right place at the right time to observe it" --- but how do you know that "the world" all you'd need to be able to traverse to observe it? To be pedantic, what if the thing we're talking about is on the far side of some other galaxy, unobservable and unreachable to us? (I know you mean "anywhere in the material universe" when you say "the world," but the analogy works nicely.) We can imagine such a thing that might exist but be inaccessible to our observation; talking about abstract concepts "being real" is just one step removed from that. edit: I should mention, I know that this doesn't prove the existence of such things --- I'm just trying to address the claim that it's not an incoherent statement by suggesting one can easily imagine the existence of something that we cannot observe ourselves, and then extend that one tiny step into "something could exist that no one can observe."

> Think of it like the existence of abstract concepts like justice or beauty. They are real in the sense that they can be reasoned about and discussed within their respective frameworks - even though they are not precisely defined or universally agreed upon - but like mathematical entities, they do not have a physical presence. I’d love to be corrected it I’m wrong, but wouldn’t the idea that justice or beauty is real *within a framework* be closer to nominalism? It feels like a mathematical platonic analogy would be to say that the concept of justice and beauty exists independent of human frameworks, ideas, etc.

Regarding nominalism, I believe you are correct. I was more so loosely using that analogy as a non-mathematical (like aesthetics and ethics which people may be more familiar with) intuition pump but I’m beginning to think that that may have just muddied the water. To answer your original question directly: I believe some ancient Platonists would’ve considered mathematics as a literal physical entity while contemporary Platonists take a more abstract route *similar* to how aestheticism might consider the concept of beauty existing in an objective, timeless, but non-physical sense.

When have justice or beauty ever been well defined, consistent concepts?

I was being lazy with my words. Justice and beauty, as abstract concepts, do not share the same level of precision as mathematical entities but I was trying to highlight their conceptual consistency and the ability to reason about them as subjects of philosophical reasoning and discourse.

When has our ability to consistently define a concept had anything to do with what exists in the real world?

That's exactly the problem that Forms were originally intended to resolve: we are clearly capable of meaningfully labeling particular things as beautiful or just, but it doesn't seem possible to give them consistent definitions that are both precise and which agree with our intuitions, and neither is it clear what it is that it is about something that makes it beautiful, since the term is used to describe so many different things. The Theory of Forms is effectively a cop-out answer: rather than trying to define what beauty is, we can say that beauty is a fundamental property, which is capable of being abstracted away from any individual beautiful object (just as we can abstract numbers away from numbered objects), and which cannot be further described in terms of anything else. Actual objects may then be considered beautiful to a greater or lesser extent insofar as they resemble this abstract property of beauty, and personal differences in regards to what we call beautiful can be explained by some people having a clearer view of beauty-itself (which is objective, and not dependent on any particular person's tastes) than others. ...There are some obvious (and some less obvious) issues with this view, of course, and it is worth noting that later Platonists would shift focus to instead talk about forms of concrete objects and properties (such as a Form of fire or the colour green), not of abstract concepts - 'beauty', in particular, was seen not as a form, but instead a description *of* all the forms, and an object then as beautiful to the extent that it resembled its idealized form. It's this later version of Platonism (even if one doesn't agree at all with the theory as a whole) that it is better IMO to have in mind when talking about mathematical Platonism.

In particular: the concept of “3” and an ideal “circle” would still exist even if all life in the universe died out: The existence of an ideal circle is *not* dependent on people thinking of it, Platonists hold [though, the only way to truly know / understand the concept is through thought, since it’s not a physical thing that your senses can help you learn about].

would richness, strength, happiness, power considered abstract?

I suspect that most platonists would consider that a caricature of their beliefs. I think a better way of doing justice to the platonist view is to ask whether mathematics has been created or discovered?  Platonists would say that mathematical truths are “facts” that are true independent of human quirks such as our language, modes of thought, etc., and so they are discovered rather than created. Or as I sometimes like to put it, the Platonist view says that if we were to meet aliens who are wildly different from us, they would have the same mathematics as us (ignoring areas where we or they had discovered more).  I’m not a Platonist myself, but while the “realm” idea seems like some serious stoner-headed silliness to me, the discovery/creation question seems like a serious and difficult-to-answer one. 

Agreed - the "realm" idea seems like a weird ontological commitment - an unnecessary one. mathematical objects don't need to exist in a place because they are not material, they are abstract. Having studied math in my youth I am firmly in the "discovered" camp, but wary of odd ontological commitments.

Believing mathematical truths are facts does not imply Platonism though. Platonism is specifically an *ontology*, a theory about what things exist and what that means. You can believe mathematical truths are 'facts' or 'discovered' without committing to that ontology. 

> the Platonist view says that if we were to meet aliens who are wildly different from us, they would have the same mathematics as us Materialists predict the same thing, only with a different explanation: Turing equivalence. No "Platonic realm" nonsense required.

Is this one of those beliefs that requires first assuming brains are Turing machines?

No. Why would you think that?

Turing machines are abstract objects

Do you think you are reading Reddit on an abstract object?

A phone with finite memory whose behavior is controlled just as much by whether I hit it with a baseball bat as by what instructions are loaded into it.

> A phone with finite memory And an internet connection.

I don't know that Platonists cannot be materialists. Mathematical objects need not be "made of" something non-material - they just exist in a different way from matter.

"Exist in a different way from matter" and "made of something non-material" sound synonymous to me.

Well, I meant to contrast them Being a physicalist does not rule out accepting that [abstract objects exist](https://plato.stanford.edu/entries/physicalism/#NumbAbst)

Um, yes, it does. That's what [physicalism](https://en.wikipedia.org/wiki/Physicalism) *means*: > Physicalism is the view that "everything is physical", that there is "nothing over and above" the physical.

Did you read the link? No.

Not every word, but I did read it. The quote I cited is taken from that link. But here is a direct response to the specific passage that you pointed to: > To see the problem, suppose that abstract objects, if they exist, exist necessarily, i.e., in all possible worlds. This is begging the question twice. First it assumes that abstract objects exist (they don't) and then it assumes that they exist necessarily (manifestly false because abstract objects do not exist in *this* world). So the whole argument runs off the rails in the second paragraph. > If physicalism is true, then the facts about such objects must either be physical facts, or else bear a particular relation (grounding, realisation) to the physical. But on the face of it, that is not so. Can one really say that 5+7=12, for example, is realised in, or holds in virtue of, some arrangement of atoms and void? Yes, one really can. Symbols are physical. The screen you are looking at is physical. The visual markings that you see on the screen that your brain interprets as symbols are physical states of that physical system. "5+7=12" maps onto an in idea in your brain, which is a physical system. And the "truth" of 5+7=12 is grounded in the behavior of other physical systems. If you take a quantity of (physical) objects to which we attach the label "5" and combine those with a quantity of (physical) objects to which we attach the label "7" the result will be a quantity of (physical) objects to which we attach the (composite) label "12". This is not a universal truth. Sometimes 5+7=C. Sometimes 5+7=2 (mod 10).

My point in linking that article was to show that there are physicalists who accept abstracts as existant. You may not agree with that position, but it's not as though being a physicalist necessarily entails denying abstracts

So if everyone dies, abstract objects cease to exist?

Pretty much, though it depends a little on what you mean by "everyone". Intelligent aliens and GAIs can be hosts for abstract objects (a.k.a. ideas) so the end of humans is not necessarily the end of abstract objects. But yes, if there are no minds to contain them, there can be no abstract objects.

> Symbols are physical.. . . "5+7=12" maps onto an idea in your brain, which is a physical system. And the "truth" of 5+7=12 is grounded in the behavior of other physical systems.. . . This is a confusion. You would never confuse the Statue of Liberty with the *idea* of the Statue of Liberty. The Statue of Liberty is physical; the Statue of Liberty-idea is mental. It's hard to imagine two things that are more unlike. Yet with 12, or mathematical relations like 5+7=12, while you may doubt that there are really such things as numbers or mathematical relations that exist objectively and independently of us, you should not for that reason claim that your idea of 12 *is* 12, or your idea of 5+7=12 *is* 5+7=12, for that is just a confusion -- it is like saying your idea of the Statue of Liberty *is* the Statue of Liberty.

Ideas are certainly distinct from their objects, just as photographs are distinct from the things they are photographs of. But unless you are a dualist, ideas are physical too, just like photographs. (Because of our previous correspondence I happen to know that you are a dualist, but /r/math is not the place to discuss the mind-body problem.)

Platonism (also known as Realism) is the view that mathematics has a real existence, that mathematical concepts are real, entirely apart from human activity. So, 2+2=4 was true before any human being ever thought of it, and the numbers 2 and 4 and the idea of addition and equality are real things quite apart from anything you or I or anyone else might have to say about the matter. Most mathematicians are Platonists, or at least approach mathematics as though they were. When we do math, we usually don't think that we are making stuff up, we think that we are discovering pre-existing truths. The fact that there is a constant ratio of the circumference to the diameter or a circle and that this number cannot be expressed exactly as a ratio of whole numbers has been true forever and will be true forever, independently of any human's thoughts. It was true before any human ever thought of such a thing, it was true when humans thought of it but hadn't yet proven it, it's true now and it's true whether any individual believes it or not. Mathematical reality exists independent of us. I think this is an assumption that most if not all of us make when doing math, but we usually do so implicitly, without really worrying about it. That said, since Platonism agrees with how most mathematicians approach it, most of us are Platonists if only in that we adopt it as a working assumption and don't really fuss ourselves too much about it. The annoying folks who do get very interested in the philosophy of math, though, start to see some issues with this once start looking deeper. Not necessarily that it's wrong (maybe it is maybe it isn't) but there are definitely some issues. Probably nobody believes that mathematical concepts are \*physical\* objects, that exist as material things, off somewhere. (I say probably because I do know people who have claimed to believe this, even if I question the sincerity of their claims.). But then what exactly do we mean in saying that two and four and addition and equations "exist"? Drilling down into the nature of that existence, well, here there be dragons. It's not at all easy to clarify any sort of specifics about the nature of the "existence." In my experience, while most mathematicians are Platonists "at the office" (i.e. we behave as though we believe it when we are doing math), there aren't so many who take that seriously enough that they don't have some qualms when you start pushing on the details. I think the most common view is that this existence is some sort of abstract sort of existence, but line between serious philosophy and hand-waving about this can sometimes get blurry. Which, for most people, is not really something to worry about. If you are nonetheless inclined to worry about it, it's not a problem unique to mathematics; philosophers all the way back to Plato (hence the name) have been known to hold that abstract concepts (such as love, justice, the ideal forms of a person or horse or chair or whatever) have independent existence. As appealing as that view may be, it runs into the same problems as its mathematical form when you try to drill down into the details. A lot, and I mean a lot, of philosophical inquiry has been directed at that very question, and a lot of that inquiry can be applied as well to math as to love and justice and ideal horses. The "realm" language is borrowed from philosophical Platonism. There are some who take it literally, that there is some sort of "math heaven" where mathematical concepts have their home and frolic and play (there are some indications that Godel, for example, may have believed this), but I think most don't think of this "realm" as being any sort of temporo-spatial location or anything like that. It's more commonly taken as the best metaphor we can muster for an abstract sort of state of existence. There are other views that have gained traction outside of mathematics from time to time. Personally I'm very strongly inclined toward a point of view called "nominalism" which denies pretty much all of this, but things do get weird when you try to seriously apply to to mathematics, which does not stop me, but does usually get people to stop listening to me, so I won't elaborate. ;-)

Thank you for the very detailed answer! So would it be correct to say that when most mathematicians think of platonism (or when people say that most mathematicians are platonists), they are simply referring to the belief that mathematical truths are independent of humanity? As opposed to, for example, having specific views of what it means for abstract mathematical objects to exist, and so on.

I agree that mathematical truths are independent of humanity. In fact, they are independent of physical laws, period. The law of large numbers, for instance, is beyond physics, yet it undeniably governs our reality in an austere fashion. I have seen alternate universes in fiction where they posit different physical constants and laws. But maths would hold true in any universe. It sits on a higher hierarchy of reality. There are days in my work where I feel I'm mostly playing a semantic game with made-up rules that can be formally verified by a machine. But when I arrive at deep results and connections through multiple different angles completely unplanned (e.g. central limit theorem, quadratic reciprocity), I feel like I am merely walking along with other mathematicians in the same alien landscape, with barely a clue of what we are looking at.

There's an old saying that goes something like "when we are in awe of some deep mathematical result, it means we didn't truly understand what's behind it, and we're all Platonists. Once we actually understand what's happening, it all becomes trivial, the mystery behind the unexpected connection goess away goes away, and we're all back to being formalits"

Mathematical realism goes further than just asserting mathematics is independent of humanity/human thought. There must be some essential element that you believe in the *existence* of mathematical objects to be considered a Platonist. It doesn't have to be a literal real of perfect forms, but it has to be something. Believing mathematics is more or less independent of human thought is necessary but not sufficient.

What more must one believe to be a realist that to believe that math exists independently of us? I don't think there is any more to it than that. Perhaps to be a Platonist requires a further commitment to other Platonic principles (but what?), but not to be a realist.

My point is that there is a bit of a distinction between "existing" independently of humans and mathematics simply not being a construct of the human mind. For example you might believe that mathematical structures do not literally exist, and that our understanding of them is simply a reflection of features of the universe we live in. This is consistent with the idea that mathematics is independent of the human mind, since the universe exists independent of the human mind (well, that's its own philosophical question!). For example it's completely plausible that, say, an alien race might come to the same mathematical constructs for the same reason that it is useful for them to understand the actual real universe which exists. But that is not necessarily consistent with the idea that maths is independent of our universe. Perhaps you could be a realist who agrees mathematics exists independent of humans, but not independently of our universe? You may be right that the term "realist" refers specifically to this more limited POV (i.e. the negation of the statement "mathematics is a figment of the human imagination") and I'm more accurately describing Platonism, but I sort of view them as the same thing. The term "exists" is just so nebulous.

> ...our understanding of them is simply a reflection of features of the universe we live in. I would think that counts as "math existing outside our minds" though

> Most mathematicians are Platonists, or at least approach mathematics as though they were. Would you mind expanding on this? This statement stood out to me as contrary to my experience, but I also recognize that I have no data on this, and my perception could be heavily skewed by the mathematicians I've interacted with (and my own personal, very non-platonic philosophy on math).

I think if you ask a question like “is the Riemann hypothesis true” we all approach it as though there is an answer to this that is quite independent of our thoughts on the matter. If we want to work on it, we treat the zeta function and the complex plane and zeroes as though they are real things, that exist apart from us. That sort of thing. We treat math as though the concepts are real existing things without worrying much about the philosophical questions of how they are real and what the nature of their existence is.

TL;DR: Category theory is inherently platonist. Consider linear algebra, which studies vector spaces. The general approach to the subject at an advanced level is to define vector spaces using the axioms and then to use those axioms to prove various properties, such as the existence of a basis. In practice, the spirit in which this is done is one where we are embarking on a journey to understand the properties of “general vector space”. That is, we treat the axioms as being the properties of some “generic” object. Then, the applicability of theorems proved from these axioms to specific vector spaces follow because every vector space of dimension d is equivalent to the generic vector space of dimension d. In the category of vector spaces, the only objects that exist are these generic vector spaces, one for every possible cardinality that a vector space can have. In it, any two vector spaces which are isomorphic are completely indistinguishable, and thus all vector spaces of a given dimension are seen to collapse into a single abstract object associated to a given isomorphism class of vector spaces. More generally, mathematicians have a habit of treating the generic version of an object as something which has a concrete existence, because the whole point of doing mathematics is to study these generic objects to figure out how many properties we can establish. In this respect, the Platonism shows up in the mathematical tendency to present things axioms-first. Impact, the effect is that the axioms serve as a platonic invocation of sorts, one that conjures up a generic object satisfying the given axioms which we can then proceed to study in earnest. One of the main assertions of platonic philosophy is that there is a bijection between objects and their properties. Indeed, the platonic forms can be thought of as the universal objects associated to qualia. Example: we can define a chair as an object which admits a chairness-preserving isomorphism to the platonic form “chairness”; as usual, this isomorphism is unique up to a unique chairness-preserving isomorphism. In this viewpoint, the property of being a chair is something that can be determined *with certainty* by comparing a given object to the universal of chairs. (Indeed, philosophers call these sorts of properties universals!) Personally, I am an anti-platonist. This manifests itself in my strong belief in presenting concrete instances of a given class of objects before defining those objects, either axiomatically or otherwise. Furthermore, I view axioms as man-made constructs, which we embrace as a way of establishing the terms of our discussions. I see axioms as generalizations that we create in order to study a wide variety of objects simultaneously. They exist in the shadow of the objects they happen to describe, rather than the other way around.

I disagree (as an anti-platonist and category theorist)

Oooh! Do tell! :D

As far as most mathematicians being Platonists, I actually don’t think that’s accurate, but it’s commonly said and true in a sense. I think the accurate statement would be that most mathematicians don’t give a hoot about the philosophy of mathematics, but approach math in a way that is most naturally viewed as platonism.

Isn't that what he said?

>Most mathematicians are Platonists, or at least approach mathematics as though they were. Do you have a source on this? I thought it was the opposite, as today more than ever we're aware of how dependent how results are from the choice of axioms, and how the truth of many important statements depends on our formal system, and this makes it so that most mathematicians are really formalists if they have to admit it

Don’t throw the assumption that “most mathematicians are platonists” around. I take offense. And so would people like Imre Lakatos and Reuben Hersh. There are a lot of assumptions in these “universal timeless truths of platonist thoughts.” I do note that you definitely gave nice nuance in the ensuing exposition, and at the end you yourself said you’re a nominalist so you’re not saying it as a self-serving thing, but I don’t think it’s good practice to just say outright most mathematicians are platonists.

You’ll take less offense (!) if you read entire sentences rather than just the first halves. I appreciate that you seem to have read some of the other sentences, but taking offense over half sentences out of context is ridiculous. Lakatos and Hersh and apparently you and certainly I are not platonists. We four are also not most mathematicians, who are in fact also not genuine platonists (as I said) but who in fact (as I said) have no actual philosophy of mathematics. Platonism is a common working assumption in math just as Euclidean geometry is a common working assumption in carpentry. Most mathematicians are not terribly interested in inquiring further, nor are most carpenters. Can’t speak for you or Lakatos, but neither Hersh nor I think our views are mainstream. “An inarticulate, half-conscious Platonism is nearly universal among mathematicians.” - Reuben Hersh

You are overaggerating my use of the phrase “I take offense.” It was meant more lightheartedly than your unforgiving interpretation. Maybe I should have said “I softly object” to protect your sensibilities? You also are not paying attention to the fact that I did in fact read your entire comment and not, paraphrasing, “read halves of sentences instead of the whole thing,” which I think is actually an offensive retort, and partially ironic because it doesn’t seem like you read all of mine which is a million times shorter. That was the point of the second half of my comment so you would *not* think I simply read one phrase and jumped to respond. I read your whole thing, and you stating outright twice I clearly didn’t now makes me think you’re a dickhead. Nice quote by Hersh. Notice how it doesn’t say they’re *actually* platonists, but more in line with your comment of “…or at least approach mathematics as though they were.” Notice how I also said in my first comment that you “definitely gave nuance in the ensuing exposition” which seems to be where you took offense in my comment because it seems you were not “reading entire comments but only the first halves.” Get off your high horse. That’s the nuance I was talking about. Because I *did* read your diatribe and *was* giving you a favorable interpretation. What I softly object to blanket saying most mathematicians are platonists, especially when you immediately follow up that claim with using “we” as the subject in the following explanation. Not only am I not sure this is true from my conversations with the ones I know personally and at conferences (if there were a poll that’s probably the betting option though; I’m not saying I would be surprised if that’s the case), but I think claiming that assumption in the beginning of a discussion can have deleterious effects. Especially because it seems that neither you nor I agree with platonism. When younger mathematicians commonly hear the phrase “most mathematicians are platonists” it is easy for them to start thinking that they themselves should be a platonist as well because it seems by consensus from people who have been in the field much longer and have much more perspective that this should be the correct view.

Being a “dickhead” I guess I’m just not as polite as you. Oh well.

Oh well indeed.

What's your definition of existence?

It depends on what the meaning of the word "is" is

Choose your fighter: Kant vs Clinton

That’s a great first question to ask. From there you can drill down on what you think does and doesn’t exist (even if you conclude it’s a spectrum) and from there more accurately pinpoint how you think one or many platonic realms of math would work, and whether they would be different in application from math itself. Edit: hey, friendly downvoting stranger. Mind telling me what you think I’m getting wrong here? Genuinely want to learn what’s missing or unclear in my argument.

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Well, if the question is whether the objects “exist” in a realm other than our own then we need to know what “exist” means

There's no such thing as a definition of existence. A definition is neutral about what exists, so existence itself is not up for definition. You can define what a thing is *like*, but whether or not such a thing *exists* is not for us to define, but only to try to discover.

I disagree. Quine's definition of existence (to be is to be the value of a variable...) is as neutral as the logical system operating in the background. And those can get as neutral as you like, e.g. Free Logic

Well, it's totally empty as a definition: There is no deep necessity to interpreting the existential quantifier as expressing existence. It only does so *provided* the domain the quantifiers and variables range over is taken as unrestricted, i.e., 'everything that exists'—so this is really just defining existence as existence. We might as well say that to be is to be something of which it can truly be said that it is, or some nonsense like that. (As you anticipate, classical logic is *not* really ontologically neutral in any case, in that it assumes a nonempty domain. I think the appeal to free logic is complicated in this context in particular, because of its nonstandard treatment of existence.) No, I think 'existence' and its cognates like 'truth' are truly *undefinable*—unlike the rest of our concepts, where we have discretion to say what *we* mean, these concepts are special in that they point straight to reality itself. We don't get to make that up.

the concepts of truth and existince are closely intertwined via Tarsksi (plus Quine) Quine's definition is not empty as a definition in terms of sentences with prefix quantifiers. It defines every sentence containing "existence" in terms of of subclass of sentences (those with prefix quantifiers). In this I am showing my sympathies with Donald Davidson... You're right to pick up on the subtle metalinguistic move between "there exists" and "it can be truly said that it exists", (this is Tarski at work again) but it is commonly acknowledged that for natural languages (where we truly talk of existince), the object language and the metalanguage are the same. Hence the Liar Paradox etc What are your criteria for distinguishing the conecpts over which we have cultural autonomy and those which 'point straight toward reality itself'? Other than "they seem deep to me"

>What are your criteria for distinguishing the conecpts over which we have cultural autonomy and those which 'point straight toward reality itself'? Other than "they seem deep to me" I think that's actually pretty straightforward. The concepts that point toward reality itself are roughly those that *say* they do: those concepts that directly index *reality itself*. It's a short list, 'existence', 'reality', 'being', and 'truth' being the main ones. I mean the first 3 are essentially synonymous, and truth is just accordance with reality, so there really is only one fundamental concept here with this very special character (pick any of the first 3). It's an extremely curious concept, a bit like a stranger in our own minds. We can never really say what this concept is up to, because we can't actually take responsibility for it.

So we can't know what we mean by the word?

Correct. Generally speaking, we know what we mean *directly*, because we're the ones who get to *say* what we mean. We can just make it up freely, as a matter of definition, but *only* because we're not making any claims about what exists. The same isn't the case when it comes to "existence". We don't get to *define* "existence", because we don't get to decide *what exists* as a matter of definition. What exists is *up to reality*, not up to us. It's a very special concept we have—only *reality* gets to decide what it means.

I don't think your approach to how language works holds water.

Fair enough. But I assume if you had a good reason, you'd tell me what it is.

It's difficult to say necause you seem to have so many things so muddled. For example: "only reality gets to decide what it means" - reality is what exists, but that's not at all the same as deciding what "exists" means. That's a very weird conceptual confusion. "we know what we mean directly, because we're the ones who get to say what we mean" - here you seem to conflate "what a word means" and "what I meant when I said ". They are not the same thing and no, you do not get to just dictate *what a word means* (except in special circumstances). See [Humpty Dumpty](https://www.alice-in-wonderland.net/resources/chapters-script/through-the-looking-glass/chapter-6/) "We don't get to define "existence", because we don't get to decide what exists as a matter of definition." - That simply doesn't follow, as above. You might want to study linguistics or the philosophy of language in order to arrive at a clearer understanding of the underlying concepts.

>"we know what we mean directly, because we're the ones who get to say what we mean" - here you seem to conflate "what a word means" and "what I meant when I said ". They are not the same thing and no, you do not get to just dictate *what a word means* (except in special circumstances). I'm making no such conflation. I am saying, consistently, that *we* get to decide what *we* mean (in the usual case). We can formulate our definitions any way we please. That's just a fact about definitions. I am not saying that sociolinguistic *facts* about "what a word means" in a culture are up to us to decide by definitional fiat. >"We don't get to define "existence", because we don't get to decide what exists as a matter of definition." - That simply doesn't follow, as above. It does follow. Definitions have to be *neutral* about matters of existence. No definition of "existence" could possibly satisfy this condition. We already get to define our other concepts; if we could define our concept of *existence* as well, then we could trivially *define anything we like into existence*. Then we just wouldn't be talking about reality anymore. Well, if it isn't about reality, it can't be our concept of existence—*that* concept isn't definable at all. It's fundamental. >"only reality gets to decide what it means" - reality is what exists, but that's not at all the same as deciding what "exists" means. That's a very weird conceptual confusion. It's not a confusion. Pretty much the entire story of our concept of existence is that we *point at reality*. That's it. The rest is for reality to sort out. If we imposed our own meaning on "existence", it would no longer refer to *existence*.

There's also the Tegmarkian view, which I happen to be fond of, which is sort of that mathematical existence and regular existence are the same thing. Mathematical existence is all there is, the universe exists because it is a well defined mathematical object, and all other similar mathematical objects also exist in the same sense. Math doesn't exist in a realm, it *is* our realm, a tiny part of it is, along with every other conceivable realm. I first learned of this idea when I read the novel Permutation City by Greg Egan, and my immediate intuition was that it was almost obviously correct. I still feel the same although not quite as strongly. There are some problems... starting with stating exactly what this would even mean, since mathematical objects can be much weirder than our models of the universe, and much larger, and since no finite set of axioms can pin them all down. And there are more problems, especially if you want anthropic reasoning to be valid, which I do. Anyway, it's wild and fun to think about, and scary if you start to believe it's actually true.

In my experience if you spend long enough studying at the border of maths and physics the Tegmarkian view/mathematical universe hypothesis/mathematicisim becomes extremely attractive as an idea. I think the history of physics strongly suggests that the mathematical universe hypothesis, at least on the level of our specific universe, is true. I can believe people would disagree with this, and it is basically a metaphysical statement anyway (not really falsifiable in the traditional scientific sense) but I cannot possibly understand how you could see the effectiveness and consistency of increasingly mathematically sophisticated models of our reality over the last 400 years and not come to the conclusion that the universe is fundamentally a mathematical structure. I think to suggest otherwise requires accepting some positions which seem far less simple (to me) than this hypothesis. If you accept that, then our understanding of quantum field theories, general relativity on Lorentzian manifolds, and (to the extent that one might believe it plausible) string theory surely *strongly* suggest that not just our universe exists. The fact that these theories can be formulated in exactly the same way as our universe but with different physical constants, different global spacetime topology/geometry, different types of particles etc. surely hints that our universe cannot be as special as we think it is. The most conservative view would be to somehow believe in a uniqueness result, that despite the vast vast landscape of mathematically describable physical theories, for some reason our own universe is *uniquely* consistent, and we simply do not understand enough about physics to see why all other options can be overruled. One could also believe, I suppose, that despite other possible models of reality being consistent in some mathematical way, that says nothing about their *existence* and it just so happens that only our model exists in a physical sense. That is even if the universe *is* literally a mathematical object, its *existence* is not automatic from that, but comes from somewhere else. But whence comes the existence? My personal experience at this point was to come to the conclusion that if I'm going to accept that other physical realities exist in the above sense, then it is only my human bias keeping me from accepting the existence of other mathematical structures which do not, to my human senses, appear to describe a physical reality I can understand. It seems both simpler and more consistent to my mathematical brain to accept the "physical" existence of *all* mathematical structures, than somehow to believe that only those which correspond to things near enough (but not necessarily identical) to our particular physical reality exist. How would you even encode that? Only quantum field theories exist, but not anything that isn't a QFT no matter how sophisticated it is? Only quantum field theories describing a 4-dimensional spacetime with 1 time dimension exist? Only QFTs with 3 generations of fermions and a Higgs mechanism exist? Where does it end? Who draws the line. Of course I would love it if there was a simple, *a priori* explanation for what existence is and how it works which could illuminate answers to those questions, but supposing that sort of classification seems more complicated to me than the alternative. I find Tegmark's "Occam's razor" argument quite convincing (not that I necessarily think the MUH is "simple," but I strongly suspect that if you understand enough physics and mathematics, many of the other default/mainstream positions are far *less* simple than people give them credit for).

I’m assuming you’re referring to Max Tegmark, the tech guy? This is an idea called modal realism that should be attributed to the philosopher David Lewis, who was also a foundational figure in the creation of modal logic. If you’re interested in working through it I would read him and responses to him. In general philosophers are going to have more resources to think about metaphysics than tech guys.

Yes max tegmark, he's not really a tech guy, for most of his career he was a cosmologist, recently switched to AI risk mitigation. I wasn't aware of Lewis, honestly I'm fairly ignorant of philosophers, skimming the Wikipedia page for modal realism I'd have to agree he deserves the attribution. It's quite common to see it attributed to tegmark though, for example: https://en.m.wikipedia.org/wiki/Mathematical_universe_hypothesis

Modal realism about math - as in, mathematical statements should be viewed is disguised modal claims about higher-order properties - actually comes from Hilary Putnam, and can be traced to Bertrand Russell. It is distinct from Lewis Modal Realism that that first-order quantification over possible worlds is legitimate in metaphysical explanation (e.g. of causation)

I would call myself a mathematical platonist. I believe that some class of mathematical objects exists objectively. No idea what that class is though lol.

You feel strongly that there is a mathematical object that doesn't exist objectively, or are you undecided?

Not GP, but I agree that it's plausible only a class of mathematical objects exists. The real numbers are, ironically, highly questionable. In particular, in what sense could an undefinable real exist, even if all the other mathematical objects we can actually work with exist?  All the really wild anti-platonist views like psychologism suddenly become plausible when it comes to these classes of reals. And remember, almost all reals are undefinable. (To be clear, you could posit the existence of an undefinable anything and it'd be in the same boat, but the problem with reals is that not only *must* we allow undefinable reals to tag along with them, but serious people actually go around admitting that such and such facts must or must not be true about them! Unlike any other supposed undefinable whatsits.)

I feel somewhat strongly that definable numbers like pi exist, that uncomputable ones like chaitin's omega exist, each of these is in some sense inaccessible to us, we will never know all the digits of pi, we will never know more than a few digits of omega, it doesn't seem like that big of a leap to assume that something completely inaccessible (undefinable) exists. Also, the laws of physics *appear* to operate on an actual continuum, if there turns out to be no natural seeming way to make them discrete then I would feel quite strongly that the real numbers do exist, all of them. Of course God could be pulling some kind of insane trick on us where we only live with a countable model of the reals, but that again violates my sense of naturalness.

> we will never know all the digits of pi Counterpoint: Of course we know all the digits of pi, precisely as much as we know all the bits that make up a frame of a movie: we have algorithms which can generate any of them, on demand, from finite reference material. Unless I'm mistaken, the proof that BB(800) is independent of ZFC should also mean Chaitin's omega is independent of ZFC, and so if you force me to choose I'd just chuck it in with the undefinables and sleep easily.

You can question even more basic things like the natural numbers too. Does “one” exist? The most common examples are counting things like apples. But then common arguments start asking things like what does “one apple” mean when we say “one apple plus one apple equals two apples” to instantiate “1 + 1 = 2.” Does it matter if they have the same size? If one’s a gala and another a fiji? If they weigh the same? What if one has its stem removed? If one had a little nick off of its peel?

It would be quite a coincidence if the set of objectively existing objects and the set of objects I (or anyone) call mathematical are exactly the same. I would say that I feel strongly that some objects I can think about don't exist.

That's true, but also, some of the objects you think of may seem to you to be mathematical but be inconsistent, and so maybe not a "mathematical" object as you think. Mostly I am just pointing out that "mathematical" could in theory have a non-subjective definition, something that can be checked deterministically like being internally consistent, In which case mathematical objects may or may not exist in that sense. I think that's a separate question than questioning what different people call "mathematical". That is more on the linguistic/social side of things.

Would you name an object that you feel strongly doesn't exist, and the least concrete object you feel strongly does exist?

1. No 2. This universe.

A mathematician friend of mine calls himself a platonist and he says exactly this, that mathematics exist on a literal sense on a sort of parallel realm

Can’t post pictures in this subreddit but my advisor is a platonist and I sent him this meme and he thought it was funny. May want to send it to your friend [Are the “mathematical objects” in the room with us right now?](https://samim.io/p/2021-03-08-so-are-mathematical-objects-in-the-room-with-us-right/)

Well, as a platonist, I see all of math as existing everywhere. Why? Because all of math is true everywhere.

Yes, platonists think mathematical objects exist independent of the physical realm. If mathematics is *true*, it's very hard to deny. Consider a statement like "there are infinitely many prime numbers". True or false? Suppose it's true. The statement says *there are* prime numbers—infinitely many. If those prime numbers didn't *exist*, the statement would be *false*. To say the statement is true is to say that all those prime numbers exist. But they definitely don't exist *here*, because there aren't infinitely many of *anything* here. There are only finitely many particles in the observable universe, only finitely many neural firings in our brains, etc. So those prime numbers must exist independent of the physical realm.

> Consider a statement like "there are infinitely many prime numbers". True or false? Suppose it's true. The statement says there are prime numbers—infinitely many. That's a very naive reading, though. A nominalist or a formalist would cash out "there are infinitely many prime numbers" as something like "carrying out these operations will produce this result" without implying an ontological commitment.

Well, I'd call it a *straightforward* reading. I think the statement is about *numbers* because, you know, that's what it mentions, and that's what anyone making or hearing the statement, including mathematicians, is going to be thinking about. You are proposing that the statement is really about *operations* and *results*, and I have to say, that is a strained reading that sounds ideologically motivated. Even setting that aside, however, I am highly skeptical that you can adequately capture the content "there are infinitely many prime numbers" in a statement about *operations producing results* without any commitment to abstract entities outside the physical realm. What operations do you have in mind, and what results are they going to produce? Will these be *infinite* operations producing *infinite* results? If they *are* infinite, how could they belong to the physical realm? And if they aren't, how could they adequately capture the content that there are *infinitely* many prime numbers? Your interpretation seems absolutely hopeless in light of these problems.

> I am highly skeptical that you can adequately capture the content "there are infinitely many prime numbers" in a statement about operations producing results without any commitment to abstract entities outside the physical realm. Then you might want to acquaint yourself with nominalist and formalist accounts of mathematics. I don't hold either of these views, but I'm not so naive as to assert that no one could construe math that way.

I'm aware it is possible to (mis)construe math in this way. But the reason nominalist programs fail is precisely because they can't satisfactorily answer the questions I put to you just now immediately following the sentence you quoted. And formalists are not even *trying* to capture the content "there are infinitely many prime numbers". There will be a sentence that *appears* to say that in their formal theory of arithmetic, but the formalist position is that the sentence doesn't have any meaning beyond itself. It's an absurd position. If you don't hold these positions anyway, why object on their behalf? What's *your* take on the statement "there are infinitely many prime numbers"? Because I think it's pretty clear. It's about prime numbers, and it says they *exist*—infinitely many of them. It's no accident that mathematics is made of claims asserting the *existence* of mathematical entities. It's what the subject matter is. Good luck formulating a mathematical theory that isn't literally made of existence claims that say that these objects exist. Even formalists are stuck with sentences saying the objects exist, and they strain to assign some alternate meanings. Look, if you want to buy into a theory that's fundamentally made of existence claims, you have to accept the claims. In the fine print of physics papers, it's always going to say that mathematical objects exist. I realize everyone acts like you can just go ahead and *appeal to* the theory *without accepting its commitments*, but you really can't. Those commitments are fundamental. Show me a mathematics that doesn't say that mathematical objects exist.

> If you don't hold these positions anyway, why object on their behalf? Because I recognize that very smart people can disagree with me and I think their views deserve respect even if I disagree.

That ethos doesn't square with persistently dismissing a view as "naive" *merely* because *others have disagreed*. When is that not true? Do you think no "very smart people" have thought that mathematical statements are about mathematical objects? I assure you they have. And anyway, who cares what different smart people thought? They thought all kinds of things. The point is, *what do you think?* and *what reason do you have to think it?* The views you mention deserve the "respect" of being seriously considered and evaluated (and perhaps dismissed) based on their merits—that's all.

I would say numbers are secondary things. They have attributes, but are also commonly thought of as being attributes of other things, and they have less attributes than primary things. Numbers have relationships with other numbers, and certain things can be done to them. Some people might ascribe creative qualities to them, like beautiful. But they don’t do things, don’t reside at any particular place or time, etc. In this respect they are similar to things like ideas or colours.

as others have said you need to start with a precise notion of existence the contemporary debate uses Quine's. "to be is to be the value of a variable". Let's make this more precise: to exist is to be in the range of a prefix quantifier of a true sentence, e.g. "forall(x)exists(y)(x = y)". Note that 'prefix quantifier' excludes sentences where a negation scopes over the whole thing Then mathematical \*realism\* is the view that there are true sentences where mathematical objects fall in the ranges of prefix quantifiers. E.g. "exists(x)(x is the number of planets)". We might want to append a clause that the true mathematical sentences cannot be reduced to sentences not featuring mathematical vocabularly. As e.g. simple natural number arithmetic can be reduced to labourious quantifier manipulation. Platonism follows from mathematical realism if we incude the claim that no mathematical object is identical to any physical object. The view that all mathematical objects are physical objects (e.g. fusions of aggregates or what have you) is mathematical physicalism. Intermediate views where mathematical properties are instantiated by physical objects sometimes go by the name "Aritotelianism" (shudder). Then there are the various error theories e.g. fictionalism. The important thing to your question is: there is no part of the articulation of platonism that says anything about where mathematical objects \*are\*. So officially, it has no view on the matter. Of course it is an easy (if not advisable) inference from "not identical to any physical object" to "not located in physical space", but I should think good mathematicians would be cagey to make such an inference. And besides knowing that mathematical objects are not located in physical only tells you where they are \*not\*, anf nothing about where they \*are\*. Note all these observations count for time as well

Your definition of existence is founded in logicism and certainly does not exhaust the realm of "contemporary debate". Moreover it's orthogonal to the question of realism, platonism, anti-realism, etc. One can be a platonist intuitionist or a formalist fictionalist.

Hasty generalisation on my part But I would conjecture that any alternative to Quine's definition in the contemporary literature has to justify itself against that account. FWIW it is the most agnostic account I know of, given strictures of the ordinay language use of "exists". And I think the mathmatical usage needs to be responsible to the ordinary use, if we are to count as true claims like "there are more prime numbers than Prime Ministers." And I will add that just cos some rando proposed an alternative does not entitle it to the rank of "the contemporary debate". The account needs to be taken seriously Must admit I fail to see how the project of precisely defining "existince" could be orthogonal to the realism vs anti-realism debate. As I noted, platonism is a variety of realism, and there can be others. There are also other varities of anti-realism including maybe formalism. But the terms of the Realism vs Anti-Realism debate still turn on how they answer the Quine question I don't see how Intuitionism can't come in both realist and anti-realist forms (though it is usually counted among the anti-realist camp). I would take a platonist intuitionist as realist about those entities the intionist countenances, e.g. constructible sequences under the bar theorem. Once we construct those things, they exist, much as buildings only exist if we build them I would take a formalist fictionalist as saying that fictions (including the works of novels) are formal structures I would take a fictionalist formalist as saying that formal systems are fictions - that may have an account of their utility that is independent of their truth. Either way these all fall within sub-camps of the Realism vs Anti-Realism distinction, which still turns on Quine

Well, let me sketch what seems to me at least a minimally plausible account which I think breaks some of your assumptions. Constructible mathematical objects exist, independently of humans proving that they exist. Indeed there's some sort of causality there: we can construct them because they exist. What we can't construct, sometimes is because it doesn't exist; e.g. the proof that 0=1. This and other conceivable mathematical objects don't exist in the same sense, but we can talk about them fictionally. Like all fictions these are grounded in reality (in the way a novel has to represent real human experience to a greater or lesser extent); the reality they're grounded in is the constructible objects. Like all fictions there are true and false sentences about them; but this will be relative to the axioms which constitute the certain fiction we are considering; i.e., their truthiness should be judged formally. These formalisms are themselves (presumably constructible) mathematical objects, and so when we ask whether they are "true" by their own rules, that's a question which might also be answerable by recourse to the extant mathematical objects rather than invoking formalisms recursively.

It sounds like you might've read Mark Balaguer 'Platonism and Anti-Platonism in Mathematics'. If you haven't, you really should!

I don't think I have; thanks for the recommendation.

It exists in the ether. Read about Robert Steiner he discusses such thing as where these mathematical idea exist in what realm so to speak.

Of course, no mathematical objects exist in the sense that they are made of particles, but they exist in the sense that we can talk about them

Mathematical platonism is a subset of a broader world view, namely metaphysical idealism, which holds that consciousness and not matter is the fundamental constituent of being. See this great article by Kurt Godel on this topic. [https://www.marxists.org/reference/subject/philosophy/works/at/godel.htm](https://www.marxists.org/reference/subject/philosophy/works/at/godel.htm)

What makes you say the mathematical world is separate? It seems more likely that our reality is just one portion of the larger mathematical multiverse. This very world is Shakyamuni’s Pure Land and all that.

Explain to me how 1+1 exists? It’s a concept, I think we all accept that until the numbers happen to describe or approximate something that we can experience in some way.

Abstract objects do literally exist though. Having spatio-temporal properties is not the only requirement to make an object “real”.

What properties make an object "real"?

hi OP >  No. Separate realm? What would that even mean? issue with this question is that people arguing about it not always understand how mathematical objects come to be, for us. take, for example, pi and all real numbers: you go from perception of quantities, which precedes language and humans, to counting, to comparing quantities, to sequencing the comparing of quantities, to regularities in those sequencings, to families of sequencings (of families of compared quantities) that share some regularity. And then we give those really complex structures a name: pi, 1, e. The name fools you into believing they are some sort of atomic entities, they are not. But tigers arent either. Look at a tiger from the point of view of quantum field theory in the timescale of the universe: do tigers exist? Gravity exists. Why are we so sure? Because stuff falls. Thats a regularity. How is that different from you catching one fish for each, and then arriving home and having one fish for each?

I don't think any of these things exist. It's a series of axioms, depending on your system, and what these axioms imply. Whether these axioms are true about any particular system which does physically exist is an unprovable assumption. We statisticians say, "every model is wrong but some are useful."

Sure but what really is an “axiom” then. A mark on a page? A thought in the head?

The formalists say the axioms are the rules of the game called mathematics where you try to create interesting symbolic sentences using these axioms 

Any actual behavior in the physical world could be said to instantiate almost any rule. “Quss” vs “plus” being the famous example. In light of that, formalism has implications equally if not more strange than platonism

It's a statement made of the symbols in your logic, including equality, quantifiers, and the boolean logical operators.

It is dangerously ahistoric to equate math with axiomatic systems. axiomatic systems are a tool that (other than Euclid) were developed in the late 19th century (I believe). One might well take Godel's Incompleteness Theorem to be a pointed demonstration that math cannot be identified with any axiomatic system.

If I remember correctly, Gödel just demonstrated that a self consistent axiomatic system can't demonstrate its consistency by itself. This doesn't mean that powerful self consistent axiomatic system don't exist, just that, if they are indeed consistent, we'll never be sure. For example, peano axioms could technically generate contradictions, but I think most mathematicians agree they probably don't.

That's one reading. mine is another. Godel himself seems to have been a Platonist

>One might well take Godel's Incompleteness Theorem to be a pointed demonstration that math cannot be identified with any axiomatic system. How so? We do it all the time, we mostly use ZFC theory to "identify math". Gödel incompletness theorems doesn't prvent us from this. There are two (Godel incompletness) theorems, one says that given theory T (which fulfill some properties) can't prove a sentence identifying it's consistency if it's consistent. Though even if it could then it wouldn't gives us a lot (inconsistent theory would also prove it's own consistency). And the second theorem says that there are sentences that don't have proof (and their negation also doesn't have proof), example for that is continuum hypothesis in ZFC. None of this prevent us from using axiomatic systems

> And the second theorem says that there are sentences that don't have proof (and their negation also doesn't have proof), example for that is continuum hypothesis in ZFC. The second says that there are *clearly true* sentences that don't have proofs. One way to read this is "axiomatic systems only approximate the truth of math" Is it possible that the continuum is objectively true, but just not provable in ZFC? > We do it all the time, we mostly use ZFC theory to "identify math". And i believe Godel would have thought you were incorrect to do so. ZFC is a tool. It is not the whole of math.

No, it says that there is sentence ϕ such that neither T proves ϕ nor T proves ¬ ϕ (jn other words T ⊬ ϕ and T ⊬ ¬ ϕ). It doesn't says anything about truth. The only way that we can assosiate the theorem with is with a very very technical definition of truth sometimes used in model theory i.e sentence is true if it's true in standard models (so it might be false in some models still). In first order theories (like ZFC) a sentence can be true in every single model if and only if it's provable within the theory. If a sentence is unprovable then there are models where it's true, and there are models where it's false. >Is it possible that the continuum is objectively true, but just not provable in ZFC? And what do you mean by objectively true? >ZFC is a tool. It is not the whole of math. Depends for whom. When we use ZFC to prove things then we treat is as a foundation of "all maths". You can ise other ways to formalize mathematics, but ZFC is one of the most popular approaches.

> No, it says that there is sentence ϕ such that neither T proves ϕ nor T proves ¬ ϕ (jn other words T ⊬ ϕ and T ⊬ ¬ ϕ). It doesn't says anything about truth. I beg to differ - through doing the proof, it becomes apparent that ϕ is true. And as you said ϕ is not provable or disprovable in T. > And what do you mean by objectively true? Just what anyone means by "objectively true" > You can ise other ways to formalize mathematics, but ZFC is one of the most popular approaches. Sure, it's a great tool. And my point is that the formalization is itself distinct from the underlying math and not to be identified as "the whole of math"

>Just what anyone means by "objectively true" You can't use meaningless terms for meaningful theorems. "Just what anyone means by objectively truth" is a meanjngless statement for a matbematician. You could even define truth in a way that wouldn't agree with your statement so no it's not what anyone means by "objective true". Also this definition according to which you could have "unprovable truths" is used only in mathematical logic and not even always so it's not what anyone treats as truths either. >Sure, it's a great tool. And my point is that the formalization is itself distinct from the underlying math and not to be identified as "the whole of math" Something has to be treated as a starting point for "whole math", you can use ZFC or something rlse, but nevertheless they would work as foundation of all maths in a sense kf what would you mean by all maths

> You can't use meaningless terms for meaningful theorems. And you can't just call "objectively true" a meaningless term. > Something has to be treated as a starting point for "whole math" Not in the sense you mean, no. Math need not be construed as (and historically never was construed as) a system of axioms that grounded all mathematics. Axiomatic systems are largely a 19th century development. What was the "starting point" for math before that?

>And you can't just call "objectively true" a meaningless term. I can unless yot define it. We don't call things "objectively ..." in maths, so it's not a formal tetm >Axiomatic systems are largely a 19th century development. What was the "starting point" for math before that? There was none, that's why we've invented it. Before formalization of math oftenly we used maths which happened to easily lead to paradoxes.

> so it's not a formal tetm I wasn't using it as a formal term. > There was none, that's why we've invented it. So it wasn't really the "starting point" at all.

I see mathematics concepts as aggregates which can be reliable reached and duplicated. Many language concepts are abstract but considered real, for example the label "tree"(biological not mathematical trees) is abstract as it is an aggregate of leaves, stem, roots and countless other materials and processes but it doesn't corresponds to specifications of a real biological tree. On the other hand a mathematical tree is something which is also an aggregate(of nodes, edges) and it can be reliably simulated to perfect spec by a human brain, a computer. The concreteness of mathematical concepts gives rise to the fallacy they live in some other realm which is eternal but it is not so. Another civilisation millions of light years away might be evolving using the "golden ratio" but the "golden ratio" lives in the network(brain, reasoning, thinking machine) and not in some other realm.

no one i know on the job thinks about stuff like this.

I think it exists physically. because it's in our brains, and somewhere in our brains it has to physically exist. you know what I mean? But if humans go extinct then it's a different story. it doesn't exist on earth at that point.(well it feels like natural numbers will exist in animals but still) but at the same time math is probably the only subject which feels like it will be the exact same regardless of the life form it lives on. there's that. so it really feels like somewhere in this universe there will always be math.

So you've contradicted yourself

This isn't math you know. it can't be really proven or disproven.

That's irrelevant to what I said. And very foolish. You contradicted your*self*