4 ms·

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 t

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?

davidivadavid 5y ago
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.)
mensetmanusman 5y ago
Why does anything matter
Barrin92 5y ago
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.
guerrilla 5y ago
> 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 https://plato.stanford.edu/entries/pragmatism * The squirrel thing is in section 1 of the linked article.
- paganel 5y ago
  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?".
  - guerrilla 5y ago
    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 https://philpapers.org/surveys/results.pl
throw0101a 5y ago
> 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 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 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)#Providing_assumptions_required_for_science 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.
- Errancer 5y ago
  "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.
playdead 5y ago
> 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.
- threatofrain 5y ago
  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.
  - quus 5y ago
    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.
    - threatofrain 5y ago
      It's also difficult for me to believe that deep advances in music understanding would fail to translate to actual music.
- Errancer 5y ago
  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.
  - guerrilla 5y ago
    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.
dvt 5y ago
> 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.
- Errancer 5y ago
  "This is an incredibly parochial point of view.", why? I don't really see how what you said and what I said connect.
fmoralesc 5y ago
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.
- guerrilla 5y ago
  > 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.

davidivadavid 5y ago

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.)

mensetmanusman 5y ago

Why does anything matter

Barrin92 5y ago

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.

guerrilla 5y ago

> 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 https://plato.stanford.edu/entries/pragmatism * The squirrel thing is in section 1 of the linked article.

paganel 5y ago
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?".
- guerrilla 5y ago
  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 https://philpapers.org/surveys/results.pl

throw0101a 5y ago

> 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 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 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)#Providing_assumptions_required_for_science 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.

Errancer 5y ago
"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.

playdead 5y ago

> 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.

threatofrain 5y ago
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.
- quus 5y ago
  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.
  - threatofrain 5y ago
    It's also difficult for me to believe that deep advances in music understanding would fail to translate to actual music.
Errancer 5y ago
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.
- guerrilla 5y ago
  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.

dvt 5y ago

> 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.

Errancer 5y ago
"This is an incredibly parochial point of view.", why? I don't really see how what you said and what I said connect.

fmoralesc 5y ago

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.

guerrilla 5y ago
> 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.