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I'm getting quite frustrated with this kind of ontological debate. Exist or not exist, why does it matter? On top of that are so many subtle distinctions that the statement "There are mathematical objects." might or might not mean "Mathematical objects exist" – depending of the frame of reference (Meinong vs Quine vs So Many Options) – but no matter the definition it changes nothing. We are still ontologically committed to mathematical objects and treating them as truly real or abstraction which is ontology dependent on other stuff don't really make us use math differently. So what is the point of such debate?

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> Exist or not exist, why does it matter?

Yes, that's exactly how pragmatists[1] answer the question. It depends entirely on what you mean by exists and what you mean by exists always depends on what you're trying to do, i.e. your goal. That's what the whole squirrel* anectdote that William James came up with was about.

1. https://plato.stanford.edu/entries/pragmatism

* The squirrel thing is in section 1 of the linked article.

I've generally found Pragmatism to be pretty parochial and limited, I mean, looking at general statements like this one:

> that a claim is true if and only if it is useful

one cannot be judged for thinking "what good is there for any quest for true/false if all that matters is the usefulness of the thing/action itself?".

I find pragmatism to be too anchored in this socio-material reality of ours, one in which things/actions have to be "useful" (the fact that many of the pragmatists come from the Anglo world, so obsessed with usefulness and utility generally speaking, also helps accentuate this).

Back to the post per se, I think "the exist or not exist" issue/problem when it comes to mathematics is a subconscious sociologic-related plot (I'm only half-joking) of the people studying maths and believing in the power of maths so that they would impose on the rest of us, the non-maths people, the statement which says something like "what we, mathematicians, are studying does indeed exist and stands at the roots of the (scientific) world as we know it! Give us symbolic power in return!".

Tell some lay-person that what a mathematician studies is in effect just a figment of his/her imagination, and one of the questions posed in return by that lay-person might be: "why should we believe what mathematicians say more than what poets say?".

It sounds like you could have a narrower idea of what utility is than I do. For me, this is really about definitions relative to systems and those systems serving out goals. That's where utility comes in: Whether something is true or not depends on the system in which we judge it (i.e. relativism) and what system we use will depend on our goal. Our goals obviously depend on what we like and want, so axiology is clearly the root here.

You'll run into this immediately when you study logics and find out that different ones can prove different theorems thus they have varying use cases, similar to the way programming languages do. If you want to predict the future, science is the tool you want but if you want to court someone then maybe poetry is your tool.

It's not that anything has to be useful, it's just that analyzing concepts in terms of utility can clarify what we mean and illuminate the criteria necessary to decide certain questions.

I agree with your second-to-last paragraph. Everyone wants what they study to be proven to be "real." It protects the prestige of the field. I always figure there's a conflict of interest going on when you find most mathematicians are Platonists but this unanimity doesn't carry over into philosophy where careers and social lives don't ride on that result [1]. I hope my previous paragraphs clarified when I think the layperson should trust the mathematician.

1. https://philpapers.org/surveys/results.pl

> If you want to predict the future, science is the tool you want but if you want to court someone then maybe poetry is your tool.

I agree and I definitely see your point, what I'm saying (or what I'm trying to say, at least, English not being my mother-tongue and me not being a mathematician nor a philosopher) is that going one (or two steps) back one may ask him/herself about the very concept/idea of utility, more exactly on what "premise" (so to speak) we assign "goals" to the "reality" surrounding us, why do we assume there is an "utility", why do we assume there should be "goals"? Why should we instrumentalize said "reality" by seeing it through the prism of "utility" and "goals"?

I know that taking that many steps back from "natural" concepts like "utility" and achieving "goals" risks turning everything into empty metaphysics and, worse (from the pov of a mathematician or of a scientist more generally speaking) into mysticism, but imo at the higher level the mysticism of the neoplatonists (i.e. Plato's ideas taken several steps further with some pre-socratic elements added in) plus the ideas of some pre-socratics themselves are more closer to the "reality" of it all than what we ended up adopting (basically Aristotle whom we refined/corrected in some specific points). Basically I think a guy like Heraclitus was closer to grasping the "reality" around us than Aristotle ever was, but had we gone the way of Heraclitus we wouldn't have had computers and rockets and all, computers and rockets that were made possible by Aristotle's logic (and by the way how he perceived the logic process per se).

Again, imo, Plato stands between the pre-socratics and Aristotle, but because we chose Aristotle's way and we basically built our world on Aristotle's way of thinking when we read Plato we tend to focus only on the part of his philosophy that in a way confirms this "world" that we built, so to speak (with this correlation between Plato's ideas and mathematics being just an example for that).

Anyway, these are just some ramblings :)

> more exactly on what "premise" (so to speak) we assign "goals" to the "reality" surrounding us, why do we assume there is an "utility", why do we assume there should be "goals"? Why should we instrumentalize said "reality" by seeing it through the prism of "utility" and "goals"?

We don't assume. It's just an empirical fact of our life that we have desires and goals. Don't you have desires and goals? I do.

> Basically I think a guy like Heraclitus was closer to grasping the "reality" around us than Aristotle ever was, but had we gone the way of Heraclitus we wouldn't have had computers and rockets and all, computers and rockets

This is an interesting idea. It reminds me of the Two-Truths Doctrine of Buddhism as interpreted by the Madhyamaka school. They believed that there was a provisional truth useful for doing doing daily stuff (like how you see Aristotle) and then an ultimate truth (like how you see the pre-Socatics.) I think there are monists in Western traditions who have similar views but I couldn't name any.

During my uni days I had a math lecturer who stated that it was a matter of great pride to him that all the math he created was pure and had absolutely no practical applications.

With this in mind what does usefulness has to do with truth?

Alternatively, his results where doubtlessly useful to other mathematitians in their research, does that make them more true?

If one thinks that certain mathematical are mere constructions, one might also reject certain mathematical methods. Take intuitionism as a case: Brouwer thought that mathematics was the product of the human mind, and restricted maths to whatever is constructible in a very strict sense. Finitism adds even further constraints, disallowing infinitary methods altogether. In these cases, the position seems to be that we cannot hold on to the usual commitment to mathematical entities.

> Brouwer

I want to also point out that his philosophical attitude turned out to also be very productive in the end and lead to significant contributions in our understanding of type theory which are applicable today in theorem provers and functional programming languages. He also gave new proofs of various theorems independent of various other results and made room for a lot of advances in logic. So, I think even the side-effects of pursuing this kind of project can be rather valuable and have obvious practical impact.

I would also characterize myself as more of a pragmatist, and to me, the pragmatic version of that debate is around a reframing of the question, which becomes the classic: "Why is mathematics so (unreasonably) useful?" I.e. why does math work at all?

Certainly looking at what mathematics is or could be in relation to the rest of the universe looks like a legitimate research area to then potentially guide the evolution of our mathematical tools.

I'm particularly interested in how one can reconcile naturalism/physicalism/materialism/monism with non-platonism/nominalism, i.e. if there is no such thing as Platonic ideals, what's the "physical" nature of math?

(Using lots of quotes here because I'm being lazy and not super careful with the terms I'm using — I'm aware.)

> don't really make us use math differently

Even is we suppose what you're saying is correct — the goal isn't really to change how we use mathematics, but to understand what math is in a deeper explanatory sense. If that's not a project that interests you, that's totally fine.

It would be odd if we made deep advances in understanding math at an explanatory level without having some kind of translation to how people do math.

Maybe, but it doesn’t seem like a requirement. Like having an understanding of jazz at the level of music theory vs being a jazz musician.

It's also difficult for me to believe that deep advances in music understanding would fail to translate to actual music.

Could you give me an example of meaningful insight of what math is that relies on such ontological discussions? Since my issue is not with ontological debates as such but rather ontological questions formulated in a manner which is impossible to meaningfully answer. I think we can learn great things about mathematics (and world) by looking how people do math, or how people speak about math, but asking what math is seems to me to be a way of framing that instead of providing insights is dividing people into camps based on aesthetic preferences.

We often investigate things without knowing what value, if any, doing so will produce. It seems odd to require a business plan up front here. People do have them, entire metaphysical projects rest on this question and they can definitely tell you why it matters to them, but I question whether they should have to.

I mean philosophy, metaphysics and philosophy of mathematics have over 2k years of history and the only example of the influence that I know it had was that ancient Greeks didn't want to use 0 in accordance with the rule that being is and non-being is not, similarly with Pythagoreans hiding the existence of irrational numbers. If you know of some examples which had positive effect then I would really like to know about them and this is not a sarcasm or anything. I just fail to see the importance of the debate.

Search "Brouwer" in this thread.

Yea I saw that, I will check him out, thanks.

> So what is the point of such debate?

This is an incredibly parochial point of view.

Kant's seminal "existence is not a predicate" argument has had a foundational impact after he put the rationalist vs. empiricist debate to rest. In fact, it's extremely deep, arguing that the P in ∃xPx can't be "exists." Modal logicians have tried to come up with more clever ways of circumventing this, e.g. E(t) := ∃x(x=t)—although this isn't entirely non-problematic, either.

For that matter, the question of anything existing is profoundly human, so asking "what is the point" kind of misses the point.

"This is an incredibly parochial point of view.", why? I don't really see how what you said and what I said connect.

i mostly agree with this but I think the philosophical position might have an impact on how open one is to new methods in mathematics.

If one holds the view that mathematical objects can be understood as 'language games' rather than real ontological objects I think it creates a more pragmatic view on what is permissible, what constitutes a proof, whereas I think the more Platonic views might make someone more purist in how they approach maths.

> I'm getting quite frustrated with this kind of ontological debate. Exist or not exist, why does it matter?

One's philosophical / metaphysical worldview determines how one views reality.

As an example, some variations of Islam follow(ed) occasionalism:

> Occasionalism is a philosophical doctrine about causation which says that created substances cannot be efficient causes of events. Instead, all events are taken to be caused directly by God. […] The doctrine states that the illusion of efficient causation between mundane events arises out of God's causing of one event after another. However, there is no necessary connection between the two: it is not that the first event causes God to cause the second event: rather, God first causes one and then causes the other.

* https://en.wikipedia.org/wiki/Occasionalism

Whereas Christianity rejected it and went with secondary causation:

> Secondary causation[1][2][3] is the philosophical proposition that all material and corporeal objects, having been created by God with their own intrinsic potentialities, are subsequently empowered to evolve independently in accordance with natural law.

* https://en.wikipedia.org/wiki/Secondary_causation

So in in the first case asking "Why did X happen?" you answer "God willed it.", while the second case you would say "There is something in Object A that interacted with Object B." The latter then leads you down the path of examining objects and their relationships, as opposed to chalking events to spirits, gods, or God exclusively. Plants/crops growth because of something with-in themselves and not because of Ceres / Demeter willed it.

Without this worldview, you don't operate under (e.g.) the zeitgeist of being able to investigate Nature:

> That this objective reality is governed by natural laws;[35][36]

> That reality can be discovered by means of systematic observation and experimentation.[35][36]

> That Nature has uniformity of laws and most if not all things in nature must have at least a natural cause.[36]

* https://en.wikipedia.org/wiki/Naturalism_(philosophy)#Provid...

So on a day-to-day basis you may not deal with these axioms/beliefs, but they are there nonetheless.

"One's philosophical / metaphysical worldview determines how one views reality." I fully agree but I fail to see how it is true in the case of ontology of mathematics. All ontological frameworks will have to explain the existence of the same entities so what is left is the difference of vocabulary and maybe cultural practices embedded in each view, but the reality will be the same. You won't meaningfully deny the existence of triangle only because you changed your ontology.

No, but it might change your perspective on things and lead you to explore a different set of ideas (something that may end up being quite important when doing scientific research?).

Lots of folks are asking questions in that vein:

* https://aeon.co/essays/does-philosophy-still-need-mathematic...

Do imaginary numbers exist? They used to be seen as "as fictitious or useless" until Euler came along:

* https://en.wikipedia.org/wiki/Imaginary_number

What about octonions in the present day?

* https://news.ycombinator.com/item?id=17575585

* https://en.wikipedia.org/wiki/Octonion

It's kind of hard to tell what idea will lead where.

Why does anything matter

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