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Platonism in the Philosophy of Mathematics (2018)
- Ericson2314 5y agoI am hoping modern intuitionism and multi-foundational math (think Andrej Bauer) can finally mend the platonism aristotlism divide.
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- adrianN 5y agoWhat definition of "existence" are they using in this article?
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- skoodge 5y agoGregory Chaitin has an interesting take on this: "The only place where medieval theology survives is pure math." https://www.youtube.com/watch?v=1RLdSvQ-OF0 https://www.youtube.com/watch?v=1RLdSvQ-OF0 (The whole interview is actually quite nuanced on the question of whether math is discovered or invented and well worth a watch. There are also interviews with other mathematicians and physicists in the series, for example with Roger Penrose, but most other researchers seem to hold a rather dogmatic and platonist view, which is why I found Chaitin's perspective refreshing.)
- thriftwy 5y agoExistence is variative homeostasis. They do not 'exist' unless they are put in different contexts which affect their properties.
- AnimalMuppet 5y agoThat's... one definition of "exist". There are others. As all such questions, it depends on what your definition of "exist" is.
- thriftwy 5y agoWhat's the other one? Let's imagine there's a natural number exists between 7 and 8, and it is particularly sneaky so that it won't show up in any arithmetic operations performed by human and their devices. How we can argue it does not exist?
- simion314 5y ago>Let's imagine there's a natural number exists between 7 and 8 8 is by definition the label for the number after 7. See https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms So maybe you are thinking that there exists very different Set for natural numbers that has this nice properties, / I think you can try an exercise to define such axioms and rules, see if actually they are consistent and not trivial and then see if they are different then the natural numbers.
- evanb 5y agoI think you're thinking of bleem, the secret integer between 3 and 4 [ according to the short story "The Secret Number" by Igor Teper https://eyeofmidas.com/scifi/Teper_TheSecretNumber.pdf https://eyeofmidas.com/scifi/Teper_TheSecretNumber.pdf ].
- AnimalMuppet 5y ago"Exist as an idea" and "exist as a physical entity", to name two off the top of my head. There are probably more.
- Koshkin 5y agoBut is being variative necessary in order to exist? (In the context of mathematics, for example, the homeo-something seems to be all that's required.)
- andrewljohnson 5y agoIf you like such ideas, the book Reason and Persons by Derek Parfit has some mindbending chapters on whether persons exist separately from the meat, as Cartesian Egos or similar.
- playdead 5y agoHappy to see analytic philosophy trending on HN
- lordleft 5y agoThis is one of my favorite questions in the philosophy of mathematics. A related question for me is the ontology of objects we associate with Software -- algorithms, data structures, etc.
- GoblinSlayer 5y agoSoftware is a level of abstraction, but I never saw an evidence that philosophy knows the concept of level of abstraction.
- mindless_solips 5y agoWhat?!? Russell literally developed Type Theory. Even discussion of Platonic Forms ultimately goes from "lesser" abstractions like the Ideal Form of a Bed to "greater" abstractions like Beauty, Justice, Good. Please, read more philosophy. Clearer, more informed thinking might lead to better software.
- GoblinSlayer 5y agoThat's emotional level, not structural. When software supervenes on data and code, data and code supervenes on bytes, bytes supervene on bits, bits supervene on transistors, transistors supervene on electrons, electrons supervene on the field.
- quux 5y agoNumberphile did a nice episode about this topic a while back: https://www.youtube.com/watch?v=1EGDCh75SpQ https://www.youtube.com/watch?v=1EGDCh75SpQ
- criddell 5y agoIs this at all related to the mathematical universe hypothesis that says the universe is mathematics and that everything that actually is exists is only a mathematical structure?
- guerrilla 5y agoNo, the debate between Platonism and nominalism has been going on for nearly two thousand years (really getting going by the time of Occam.) Max Tegmark's mathematical universe stuff originated with him. The question here is whether mathematical objects exist independent of our minds. It's the whole "Is math discovered or invented?" question that you see in pop. sci. documentaries. You may find this[1] trichotomy more useful for a quick overview even though that article is about a different topic. 1. https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_mathematics)#Varieties https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_m...
- evanb 5y agoTegmark's mathematical universe hypothesis is like Platonism taken to the extreme: only mathematical objects exist. Standard "vanilla" Platonism says: there is an abstract world of pure forms (the mathematical side) and it is imperfectly projected [or other similar kinds of words] down to the reality we experience. Tegmark asks: why distinguish the two?
- GoblinSlayer 5y agoThe gap between mathematics and physics is big, starting with the digital paradigm that dominates math, but also stuff like relativity and time don't properly exist in mathematics and mathematics routinely forgets about them. Time, law of nature, matter itself - anything disappears when you don't think about it, as if it's solipsism. But it never happens in physics.
- Errancer 5y agoI'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 agoI 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 agoWhy does anything matter
- Barrin92 5y agoi 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.
- cheeselip420 5y agoI highly recommend the book Anathem as it goes into this quite a bit.
- mensetmanusman 5y agoI wish my kitchen had the tile pattern that the 1000er completed:)
- jeffrallen 5y agoThis topic, slightly more popularized, appeared in the Science Diction podcast: https://www.wnycstudios.org/podcasts/science-diction/articles/algebra-from-broken-bones-to-twitter-feuds https://www.wnycstudios.org/podcasts/science-diction/article...
- AlexCoventry 5y agoI don't care whether it exists, but I care whether it helps me to reason about things I care about. What changes about life, depending on whether it exists or not? If nothing changes, what's it matter? There's probably a philosophical term for this attitude. That doesn't really answer my questions, though.
- Koshkin 5y ago> What changes about life If only that you felt like you have to spend some of your time and energy on writing your post.
- mensetmanusman 5y agoWe are all back into star dust in 100 years, what’s it matter :)
- guerrilla 5y ago> There's probably a philosophical term for this attitude. It's usually referred to as quietism[1]. Quietism on this particular topic (and it's generalization) is probably the most popular kind of quietism actually. 1. https://en.wikipedia.org/wiki/Quietism_(philosophy) https://en.wikipedia.org/wiki/Quietism_(philosophy)
- medo-bear 5y agofor me maths is just measurement, including of itself
- Koshkin 5y agoWell, measurement is physics; mathematics is calculation (in a rather general sense). Incidentally, I think the word "calculus" happens to be more reflective of the nature of the subject than the word "mathematics" which seems to hide a simple notion behind an aura of mystery. "Calculus of Infinitesimals," methods of doing calculations using the notion of the infinitely small, is easier to guess what it is all about than when you first hear the mysterious "Mathematical Analysis."
- medo-bear 5y agoi think measurement is more general than calculation. when i say mathematics also measures itself, i mean this in a pure-maths sense
- im3w1l 5y agoWhen we ask "do they exist", we invite a certain type of reasoning. "Well consider chairs. We can all agree they exist. In what ways are mathematical objects similar and different from chairs?" But I have an idea. What if we turn it upside down? "Do mathematical objects *not exist*?" Now we invite the opposite metaphor. "Well consider ghosts. We can all agree that ghosts don't exist. In what ways are mathematical objects similar and different from ghosts?" I think that fundamentally if you say that mathematical objects do exist, then you have to admit basically everything in - including ghosts. Fundamentally, definitions are a matter of taste and aesthetics, and I dislike a definition that is so broad as to contain everything. That makes it useless for reasoning and discussing.
- Igelau 5y ago> We can all agree that ghosts don't exist. I don't agree that the nonexistence of ghosts is definitively settled, and I doubt I'm alone in that here.
- Koshkin 5y agoWould you say that there exists something in common between the 21 sheep and the 21 dry peas in the shepherd's pocket which he uses every evening to see if any of the sheep have been lost? That common which exists, in a very clear and practical sense, is called a Number.
- tremon 5y agoNo, I would not say that there exists something in common between those. Cardinality is an emergent property of a set, but that doesn't mean that cardinality exists on its own. Did Plato ever make the argument that because his Shapes have corners, edges and angles, that therefore corners, edges and angles exist in the same way as the Shapes? I don't recall if he ever described Shapes in those terms.
- kdtsh 5y agoWhat does it mean for something to exist then? Does an apple exist in the same sense that a number exists? There is something in common between two chairs; there is something in common between a red ball and a red book; there is something in common between 21 sheep and 21 dry peas. It is _emphatically_ unclear that each of these three statements of 'existence' have anything at all to do with each other except in the most intuitive sense - that they refer to things which are as opposed to are not. At some point the bar for existence becomes so low that Harry Potter exists - really exists - because we can conceive of the idea of him. The question of whether numbers 'exist' is more than that.
- sohdas 5y agoI don't think there is a real correspondence between math and some kind of 'mathematical reality' or Platonic heaven. Mathematical notions of truth are just mathematical. If you say something like "2+2=4 is true", you are making the purely mathematical statement "2+2=4", nothing more and nothing less. It's like saying "Sherlock Holmes lives at 221B Baker Street".
- FredPret 5y agoIt’s more complicated than that. For a number like 2 to exist, the implication is that there are two things in the universe that can be exactly equal. Even if there are two or more of something (not a settled question in physics), the idea that you can add two such groups of two and have it be equivalent to some other group of four of that thing is only an abstraction in our heads. In reality, if you have two apples in two pockets and I have four in mine, all we can say is that there are eight regions of the universe we like to call “apples”. 2+2=4 isn’t true in the same way that Sherlock Holmes’s address is because the former depends on a shared illusion/set of abstractions.
- talaketu 5y agoYou are unsure whether "2 + 2 = 4" is real, but certain that "2 + 2 + 4 = 8" is.
- tremon 5y agoFor a number like 2 to exist, the implication is that there are two things in the universe that can be exactly equal. I'm not convinced. You are defining the number 2 as a material cardinal, and then coupling the existence of the number 2 to the existence of two completely identical material objects. But why would the existence of those objects give rise to the existence of the number 2? Can't I similarly define that the number 2 is a materialized ordinal, and simply count the revolutions of a moving object, and state that the number 2 is instantiated by observing the periodicity of a single material object?
- adrianN 5y agoNo, I don't think there is any such implication. Mathematics works by defining some axioms and some rules on those axioms and then people think hard to see what theorems are true in this system. Being able to define something doesn't seem to imply that such a something must exist.
- JabavuAdams 5y agoMathematical platonism is ridiculous quasi-religious mumbo-jumbo that keeps philosophers employed. You've only ever explained a circle to other humans. You've only ever learned about circles from other humans, or artifacts (not circles!) authored by humans. No human has seen a perfect mathematical circle with their senses, because they don't exist as physical objects outside of our skulls. They exist as physical objects inside our skulls in the form of neural correlates and latent representations. A circle is like a meme. It's compelling because it's so simple (to us anyway). But, try to really explain a mathematical circle to a six year old and especially how it is different from the representation of a circle that you've just drawn. Watch even a smart kid struggle a bit with this, before getting it. It's not as simple as we might think, having mastered the concept. That doesn't mean it has an independent existence in some realm that is not embodied. It just means the concept is hard, then simple due to its abstractness and otherworldly simplicity / power balance. EDIT> apostrophe
- Koshkin 5y agoOne could say that mathematics does not study (abstract) objects; rather, it studies relations that do, in fact, exist between (real) objects. Even a circle is merely a relation - that between what lies on the circumference and, for example, what's at the center (or the axis, if we speak about a cylinder); this also manifests itself in the form of rotation, ostensibly a physical phenomenon.
- davidivadavid 5y agoThat's kind of a cop out, though. Relations are presumably abstract objects, too?
- Koshkin 5y agoRelations often realize themselves as physical interactions.
- davidivadavid 5y ago
- romwell 5y agoThe correct answer is: "why give a fuck?". The only practical value of having an answer to this question is giving philosophy students something to write about. Mathematics is a human activity, so whether you think it's "exploring" or "inventing" mathematics, you would agree that we aren't pursuing the exploration or invention in all possible directions. So let's look at what we, the humans, actually do. Any mathematical object ever defined is ultimately an abstraction and a model of something the humans actually experience in the physical world. Number theory is abstract, numbers are not. You can count your sheep in a pasture; you can count your neighbor's. Natural numbers model the idea that neither of you has more than the other. This is just a model, of course; you have a sheep named Dolly which is dear to you, and your neighbor has one named Sue which gives more wool than any sheep in the county. So you really don't have the same thing, because these sheep are not interchangeable. But we have a concept — natural numbers — which models the idea of sameness if we, for a second, forget that. It's an imperfect model; the concept of something that can grow without bound is an outcome of this imperfection. When you use numbers, you might as well assume that 10^(10^100) is the largest number there is. Numbers of that scale are so large, that all computers in the world taken together aren't enough to just store this number. But modeling the concept of "not knowing what the bound is" is modeling an experience we run into all the time. 640KB isn't enough for everyone. We dispense with this abstraction with ease, though. Most calculations in practice are done with 64-bit "numbers". These numbers don't behave like the one in math do, but it's OK. Even simpler, our clocks have just 12 numbers, even though the time doesn't stop running. Euclidean geometry can be verified experimentally. Cut out a triangle out of paper; tear off the angles; put them together, and see that they make up a straight line. It's an imperfect model, and you don't need Einstein or Lobachevsky to tell you that. Long-distance sailing made people run into this model breaking. The corners of a triangle you cut out of orange peel with a straight knife won't make a straight line. But that only gives rise to other kinds of geometries being interesting to us. People do math because it's beautiful, and something that doesn't connect to any human experience is decidedly not beautiful. Take a deck playing cards, or just a dozen cards, and come up with a set of rules of how a pair of cards can be traded for a single one. You can make different rules depending on which of the two cards is on top; 2-for-1 is the only restriction. We could come up with an immense number of complicated systems that way. But most of them would be so arbitrary, they won't connect to anything. So nobody cares about systems like that. Unless they come from something else and therefore have more than an arbitrary structure. Like finite symmetry groups, e.g. rotations of a cube. You can model that with such card rules. Then it's interesting. So the question of whether mathematical objects exist is beside the point. The reality is that all mathematical objects that humans care to study are, ultimately, modeling an experience that does. And this is consistent with mathematics truly being an art, not a science. Because art that doesn't abstract human experience in some way is not something we care about. Strings of letters spit out by a random number generator aren't an exciting read. That's what Vladimir Arnold means when he says, seemingly controversially, that mathematics is a branch of physics (in which experiments are cheap)[1]. It's that mathematics that doesn't connect to the world we live in is too boring to exist - or be invented and remembered. [1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
- svat 5y agoThis is from the Stanford Encyclopedia of Philosophy; see also the corresponding article in the Internet Encyclopedia of Philosophy: https://iep.utm.edu/mathplat/ https://iep.utm.edu/mathplat/ Also, frequently quoted (by me at least): from an interview with Yuri Manin (http://www.ams.org/notices/200910/rtx091001268p.pdf http://www.ams.org/notices/200910/rtx091001268p.pdf): > I must explain to you how I imagine mathematics. I am an emotional Platonist (not a rational one: there are no rational arguments in favor of Platonism). Somehow or other, for me mathematical research is a discovery, not an invention. I imagine for myself a great castle, or something like that, and you gradually start seeing its contours through the deep mist, and begin to investigate something.
- missingrib 5y agoI wonder why he thinks there are no rational arguments in favor of Platonism. I think there have been a lot of convincing ones over the last few thousand years.
- goatlover 5y agoYeah, the physicist Tegmark has arguments for a mathematical multiverse. Sean Carole has a podcast interviewing Justin Clark-Doane who argues that at least arithmetic must be real if the world is to be understood as consistent. https://www.preposterousuniverse.com/podcast/2021/03/08/137-justin-clarke-doane-on-mathematics-morality-objectivity-and-reality/ https://www.preposterousuniverse.com/podcast/2021/03/08/137-...
- GoblinSlayer 5y agoFlat Earth is sort of consistent somewhere too. But platonism isn't an idea on the level "looks like that somewhere", it aspires to be a deeper fundamental view of reality.
- mellosouls 5y agoAlthough he may not be convinced by the rational arguments, I took the quote as mostly expressing the emotional/irrational draw of it; the dismissal as hyperbole to support that as a rhetorical device.
- pdonis 5y agoFor the article to say that the "Existence" claim is "tolerably clear" is rather disingenuous. The basis for this statement in the article is that you can formalize the "Existence" claim with an existential quantifier; but the whole point of an existential quantifier in mathematics is that it has no meaning unless you already know what domain is being quantified over. And in the absence of some well-defined mathematical domain, the existential quantifier basically becomes a tautology: it means "object x exists in the domain of whatever objects exist". Given this, it's no wonder that such "existence" claims have always been problematic and have always been viewed with skepticism by many non-philosophers and even many philosophers.
- joe_the_user 5y agoYeah, I think that the reality of mathematical "objects" is very relative. That you can apparently construct an approximation to a Turing machine to any degree says that things behaving like finite mathematical objects can be readily constructed. But I don't think that would satisfy a Platonist.
- mjburgess 5y agoYour observation here really, I think, solves your own problem. We shouldn't think of there being a universal quantifier (ie., a universal "Exists") -- but a variety of ones depending on the domain. We can construct these domains by enumerating objects which share (eg.,) a property. So there is the physical domain of spatio-temporal objects, fictional domain of fictional objects, etc. The question of whether something exists then becomes a "thin" one: does the relevant quantifier apply? In this sense most things exist -- and this to me feels kinda obviously correct. Debates are mostly confusions over the domain: are mathematical objects in the physical domain? NO. And no one thinks they are. '2' has no spatio-temporal properties. If we just drop the idea of a privileged quantifier, everything becomes much clearer. There is no "really real", or "really exists!". There are just enumerable domains of objects of which we can form claims.
- pdonis 5y ago> Your observation here really, I think, solves your own problem. It's not my problem; I am not a mathematical Platonist, so I have no issue at all with the idea that any claim about mathematical objects "existing" is simply in a different domain from claims about physical objects "existing". However, for someone who is a mathematical Platonist, there is a problem, because mathematical Platonism appears to be claiming that "exist" is not domain-specific. For example, the article under discussion formalizes the "existence" claim as "Ex Mx" (I don't have the backwards "E" on my keyboard), where "Mx" is the predicate "x is a mathematical object"--i.e., it separates the "existence" claim (the quantifier) from the "domain" claim (the predicate). This only makes sense if the quantifier is not domain-specific but general--i.e., if the quantifier is the same one for all possible domains, so that the existence claim is an absolute one. Which, as I noted in my previous comment, and as you appear to agree, is simply not how existential quantifiers in mathematics are supposed to work.