• StreetlightX
    3.1k
    Here's a fun one: Carlo Rovelli - the renowned theoretical physicist whose name seems to be everywhere these days - published a small seven page paper in 2015 arguing against mathematical Platonism. It can be found here, and I'd encourage everyone to read it, for discussion's sake. It doesn't require a great deal of mathematical knowledge. The argument is interesting in itself, but I think it's even cooler coming from a physicist, given that physicists are usually those who are most enamoured with the idea of mathematical Platonism. Anyway, here's how the argument goes -

    Rovelli begins with a simple definition of Mathematical Platonism, which "is the view that mathematical reality exists by itself, independently from our own intellectual activities." Now, he asks that we imagine a world M, which contains every possible mathematical object that could ever exist, even in principle. Not only does M include every mathematical object we have currently discovered (integers, Lie Groups, game theory, etc) it also includes every mathematical object we could possibly discover. M is the Platonic world of math. The problem, though, is that this world is essentially full of junk. The vast majority of it is simply useless, and of no interest to anyone whatsoever.

    As he goes on to say, mathematics, as it stands, only includes an infinitesimal subset of M, and even if we could go on to discover M, we probably wouldn't, because most of it would be totally uninteresting. This, though, opens up a new question - what is 'interesting?' Well, interest simply is in the eye of the beholder: we develop some parts of M and not others because those parts help us do stuff, alot of the time. Rovelli asks: "Why has mathematics developed at first, and for such a long time, along two parallel lines: geometry and arithmetic?" And answers: "because these two branches of mathematics are of value for creatures like us, who instinctively count friends, enemies and sheep, and who need to measure, approximately, a nearly flat earth in a nearly flat region of physical space.... From the immense vastness of M, the dull platonic space of all possible structures, we have carved out ... a couple of shapes that speak to us".

    Moreover, "there is no reason to assume that the mathematics that has developed later escapes this contingency". One way that Rovelli fleshes this out is by looking at examples of just how contingent our current mathematics is. He gives a few, but I'll focus on just one, the idea of natural numbers (1, 2, 3...). Rovelli basically asks: why does anyone think natural numbers are natural at all? We certainly find it useful to count solidly individuated items, but he notes that what what actually counts as 'an object' is a very slippery affair: "How many clouds are there in the sky? How many mountains in the Alps? How many coves along the coast of England?".

    In order to make the point stick, Rovelli asks us to consider a species of intelligent beings evolved on Jupiter. Because Jupiter is fluid, most natural structures found of Jupiter would be continuous complex structures - flows, vortexes, currents, and so on. What kind of math might be developed in this environment? R: "The math needed by this fluid intelligence would presumably include some sort of geometry, real numbers, field theory, differential equations..., all this could develop using only geometry, without ever considering this funny operation which is enumerating individual things one by one. The notion of "one thing", or "one object", the notions themselves of unit and identity, are useful for us living in an environment where there happen to be stones, gazelles, trees, and friends that can be counted. The fluid intelligence diffused over the Jupiter-like planet, could have developed mathematics without ever thinking about natural numbers. These would not be of interest for her."

    From all this Rovelli concludes: "Far from being stable and universal, our mathematics is a
    fluttering buttery, which follows the fancies of inconstant creatures. Its theorems are solid, of course; but selecting what represents an interesting theorem is a highly subjective matter.... The idea that the mathematics that we find valuable forms a Platonic world fully independent from us is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather."

    Discuss!
  • Pierre-Normand
    1.4k
    Rovelli's argument is compelling to me. His view on natural numbers, in particular, meshes rather well with Frege's construal of them as second order functors that are predicated of first order functors (e.g. sortal concepts) where those first order concepts do the prior work of individuating the objects to be counted. If Jupiterians don't have a need for the sorts of first order functors that we make use of to individuate discrete persisting objects, then they wouldn't have much use for our concept of a natural number either. They maybe would have a use for a somewhat isomorphic concept, however, for purposes others than counting discrete persisting material entities. Some language games are structurally (almost) identical to other language games that have very different uses or pragmatic points.

    At this stage, a Platonist mathematician might insist that the concept of a natural number latches on the structural invariant shared between two such language games. But I don't think such a defense would work since the variations in the pragmatic points of structurally similar language games being played with symbolic numerals would lead to variations in the way they are structured 'around the edge' as it were. They might not be axiomatized quite in the same way nor be projectable in the same manner to extended domains. (I'd have to conjure up some example that might be more compelling than Kripke's 'quus' example).
  • Pierre-Normand
    1.4k
    Let me add that, here on Earth, we have the Pirahas of the Amazon rainforest who don't have any use for natural numbers, not even the number 1, nor of the existential or universal quantifiers (which Hume, I think, argued were required for grounding the practice of counting).
  • StreetlightX
    3.1k
    Let me add that, here on Earth, we have the Pirahas of the Amazon rainforest who don't have any use for natural numbers, not even the number 1, nor of the existential or universal quantifiers (which Hume, I think, argued were required for grounding the practice of counting).Pierre-Normand

    Just quickly, on this - Do you know if this is something that is in Daniel Everett's discussion of the Piraha language? I ask because I think there's a deep and little explored connection between grammar and math and I'm looking everywhere for resources on this, even though I've yet to read Everett's book.
  • Pierre-Normand
    1.4k
    Breifly, on this - Do you know if this is something that is in Daniel Everett's discussion of the Piraha language?StreetlightX

    I have Everett's book in my digital library but I am yet to read it too. The case of the Piraha and their (lack of) counting abilities had been a topic of discussion very many years ago on the now defunct (but still archived) Yahoo 'analytic' discussion group. From what I remember, there is an interpretive disagreement between the psychologist Peter Gordon, who also studied the Piraha (lack of) numerical abilities and Everett, who is more of a Chomskyan linguist and, hence, who is less inclined to take seriously the Whorf-Sapir hypothesis on the essential link between language and cognition. I'm siding more with Gordon's analysis than with Everett's, because of this issue, regarding this specific topic. (So, you might also be interested in digging up Gordon's relevant publications)
  • StreetlightX
    3.1k
    Very cool. I'll definitely check out his stuff.
  • litewave
    408
    M is the Platonic world of math. The problem, though, is that this world is essentially full of junk. The vast majority of it is simply useless, and of no interest to anyone whatsoever.StreetlightX

    So his argument is that the Platonic world of math doesn't exist because it is... uninteresting? :lol:

    The most general definition of mathematics I know is that it is a study of structures/relations. If there is more than one object, these objects form relational structures and mathematics studies these structures. The most general relation is "similarity" (also known as "difference"), because it is a relation that holds between any two objects. It means that the two objects have some different properties and some same properties. Which gives rise to another general relation called "instantiation", which is the relation between a property and its instance. The instantiation relation is a special kind of the similarity relation but less general than similarity since it doesn't hold between arbitrary two objects. Finally, any objects can define a collection of them (for example based on their common property, as long as such a definition is consistent), which gives rise to another general relation called "composition", which is the relation between a collection and its part. The composition relation, too, is a special kind of the similarity relation but less general than similarity since it doesn't hold between arbitrary two objects.

    So, the similarity relation, together with its special kinds - instantiation and composition, defines all possible relational structures. All these three relations come together in set theory, the foundation of mathematics. In other words, the world of mathematics is a world defined by set theory (more accurately, by all consistent versions of pure set theory).

    The fluid intelligence diffused over the Jupiter-like planet, could have developed mathematics without ever thinking about natural numbers.StreetlightX

    As long as there are any differences on Jupiter, you can start counting.
  • Pierre-Normand
    1.4k
    The most general definition of mathematics I know is that it is a study of structures/relations.litewave

    Sure, but what sorts of things are structures and relations? Do they exist in themselves rather like intelligible forms in Platonic heaven? If you assume that they are universals that exist by themselves, quite independently from the constitutive roles of our practices of reasoning and discussing about them, then, in that case, you are begging the question in favor of mathematical Platonism.
  • Marchesk
    2.2k
    Or you could take the Aristotelian approach and say the structures and relations exist in the world. After all, neither the Jovians nor humans create the environments they find themselves in.

    Arguably, we reason the way we do because the world is certain way for us to reason about it.
  • Pierre-Normand
    1.4k
    Arguably, we reason the way we do because the world is certain way for us to reason about it.Marchesk

    That's true, but the recognition that the world is a certain way for us to reason about it already amounts to acknowledging a productive role for the cognizing subject in constituting it and hence is a move away from Platonism.
  • Marchesk
    2.2k
    but the recognition that the world is a certain way for us to reason aboutPierre-Normand

    Why would it only be a certain way for us? Do we really think that evolution or general relativity is a certain way for us, as opposed to being a certain way for the universe?
  • Pierre-Normand
    1.4k
    Why would it only be a certain way for us? Do we really think that evolution or general relativity is a certain way for us, as opposed to being a certain way for the universe?Marchesk

    It's a certain way for us because we are co-evolved with our environment, or Umwelt (as Uexküll uses the term in the context of ethology, but as it can be extended to the context cultural evolution as well). Evolution isn't any way for the universe because the universe is quite dumb and doesn't care about anything in particular, not even its own material unfolding.
  • Pneumenon
    374
    So his argument is that the Platonic world of math doesn't exist because it is... uninteresting? :lol:litewave

    Lots of irrealists about math make this argument. "Well, it's not useful, so these abstractions aren't real." Of course, the reality of an abstraction would only depend on its utility to us if the abstraction were not independently real to begin with. Argument begs the question, everybody go home.
  • Marchesk
    2.2k
    My point was there are reasons to think the structures and relations we use math to model exist in the world independent of us, since they led to us existing.
  • litewave
    408
    Sure, but what sorts of things are structures and relations? Do they exist in themselves rather like intelligible forms in Platonic heaven? If you assume that they are universals that exist by themselves, quite independently from the constitutive roles of our practices of reasoning and discussing about them, then, in that case, you are begging the question in favor of mathematical Platonism.Pierre-Normand

    Relations are objects that hold between other objects (those other objects may be relations or non-relations). Relations are inseparable from the objects between which they hold.
  • Pierre-Normand
    1.4k
    My point was there are reasons to think the structures and relations we use math to model exist in the world independent of us, since they led to us existing.Marchesk

    Interestingly enough, the fact that they enabled us to exist already begins to establish a conceptual dependence between them and us. This is not to deny that they aren't causally dependent on us. They indeed aren't. But merely to establish the causal independence that past material instantiations of formal structures (such as subsumption of past events under laws) have from us falls short from securing their conceptual autonomy in Platonic heaven.
  • Pierre-Normand
    1.4k
    Relations are objects that hold between other objects (those other objects may be relations or non-relations). Relations are inseparable from the objects between which they hold.litewave

    This appear close to Russell's theory of concepts, relations and (Russellian) propositions. It is a quite Platonic theory, so, relying on it would also beg the question.
  • litewave
    408
    As long as there are any objects in the external reality, there are also relations between them, in the external reality. Relations and the objects between which they hold are inseparable.
  • fdrake
    1.4k


    There are other foundations of mathematics which are currently in use. Even one which highlights explicitly that mathematics studies relational structures; in category theory, the category Set is a subcategory of the category of relations, Rel. The paper in the OP even points this out, referring to topos and category theory among other things.

    But anyway, the thrust of the argument is: if we took the results of all possible axiomatic systems, agglomerated them into one giant object, then granted that object independent existence - what would it look like? It would contain all kinds of bizarre crap, navigating through this world you'd hardly ever find an axiomatic system which resembled anything like our own. I imagine if Bertrand Russel visited this elemental plane of mathematics, he would make observations like:

    Oh dear, I seem to have stumbled over a strange sort of arithmetic with countably infinitely many things which seem like 0 insofar as 0*a=a*0=0 for each one but a+0 isn't 0+a isn't a, ever!

    Looking over the graphical representation of the function, we see that it satisfies f(x+y)=f(x)+f(y) but it isn't anything like our normal linear operators! It's everywhere dense in the plane, help!

    Some mother fucker over there glued a cardinal between Aleph-0 and Aleph-1 copies of the real line together...

    He wouldn't make a statement like:

    Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

    because there's no way in hell he'd even be able to find such sensible structures in this existence turned madhouse. The most sensible explanation for why mathematics as a field looks nothing like this writhing chaos is that mathematics is sustained through the refinement of diamonds into more diamonds, and finding new ways of mining them. It is the study of fruitful relations and structures for us, not panning the sewage which is the elemental plane of mathematics for gold.

    If such a realm really does instantiate into ours, it's an incredible coincidence that so little of it resembles mathematics as a topic of study, no?
  • Pierre-Normand
    1.4k
    As long as there are any objects in the external reality, there are also relations between them, in the external reality. Relations and the objects between which they hold are inseparable.litewave

    Talk of the 'external reality' is quite Cartesian sounding. Cartesian materialism may be some sort of a variety of Platonism. But the relationship between representationalism in philosophical accounts of mental content, reference and truth, on the one hand, and Platonism regarding universals, on the other hand, is rather complex. I'll come back to this conversation tonight.
  • litewave
    408
    Of course, the reality of an abstraction would only depend on its utility to us if the abstraction were not independently real to begin with.Pneumenon

    Mathematics is a feature of the external world, a consequence of the fact that there are differences and thus more than one object in the external world.
  • Pierre-Normand
    1.4k
    But anyway, the thrust of the argument is: if we took the results of all possible axiomatic systems, agglomerated them into one giant object, then granted that object independent existence - what would it look like? It would contain all kinds of bizarre crap, navigating through this world you'd hardly ever find an axiomatic system which resembled anything like our own.fdrake

    Yes, so far I have focused on a different part of Rovelli's argument that seems decisive to me but may be less intuitively compelling than his main point, which you are now drawing back the focus on. Thank's for highlighting it.
  • fdrake
    1.4k


    As I read it there are two complementary thrusts; one is showing what kind of crazy thing the elemental plane of mathematics is (would be) and that it doesn't resemble mathematics as we study it (or its objects) at all, the other is by looking at what kind of selection criteria we use for things which are part of mathematics as a field of study. I think the first one is a better intuition pump, but the second one is a more fleshed out argument.
  • StreetlightX
    3.1k
    So his argument is that the Platonic world of math doesn't exist because it is... uninteresting?litewave

    Uninteresting in the sense that it does not even count as mathematics; not 'uninteresting, but still mathematics'. Rovelli's other example, of linear algebra, makes this clearer:

    "When Heisenberg wrote his famous paper he did not know linear algebra. He had no idea of what a matrix is, and had never previously learned the algorithm for multiplying matrices. He made it up in his effort to understand a puzzling aspect of the physical world. This is pretty evident from his paper. Dirac, in his book, is basically inventing linear algebra in the highly non-rigorous manner of a physicist. After having constructed it and tested its power to describe our world, linear algebra appears natural to us. But it didn't appear so for generations of previous mathematicians. Which tiny piece of M turns out to be interesting for us, which parts turns out to be "mathematics" is far from obvious and universal. It is largely contingent."

    The fact that linear algebra even is considered mathematics is because of it's interest to us. The question is ask is about 'paths not taken' (or even, as the history of math is littered with, paths-once-taken-but-now-largely-abdondoned, like geometrical definitions of infinity, as distinct from algebraic definitions of infinity): paths that 'populate' the world of M, and which have no power at all to describe anything about our world, and thus are dismissed as not-math. As fdrake rightly points out, the counter-argument to the supposed 'unreasonable effectiveness of math' is to point out that the vast majority - in fact almost everything that could have possibly been math (all the useless junk in M), sans the tiny sliver of what we currently consider to be math - of what might be math is completely useless and is totally and utterly 'ineffective'. It's as if, having had a suit tailored and refined over more than two millennia, one is surprised to find that it fits so damn well, and then to declare: this suit must be eternal and Real! It's intelligent design masquerading as math.
  • tim wood
    1.2k
    From the paper referenced in the OP, the first sentence: "Mathematical platonism [1] is the view that mathematical
    reality exists by itself, independently from our own intellectual activities."

    Unless the meaning of this lead sentence from the paper is substantively dealt with, the entire subject, interesting as it is, becomes an exercise either in competing definitions, or one akin to how many angels fit on a the head of a pin. And it seems the problem can be simplified. There are relatively short concatenations of letters, say 50 or 100, that have never been written down, never seen. Do they exist? Are they discovered (discoverable)? Or "constructed"?
  • litewave
    408
    There are other foundations of mathematics which are currently in use.fdrake

    Yes, there are various approaches to the study of relational structures. Maybe some are less comprehensive than others. Set theory seems to be the most popular foundation of mathematics and it seems to me that it is an exhaustive study of relational structures (of course it can never be completed even in principle, due to Godel's first incompleteness theorem).

    in category theory, the category Set is a subcategory of the category of relations,fdrake

    In set theory, all relations are defined as sets.
  • StreetlightX
    3.1k
    Lots of irrealists about math make this argument. "Well, it's not useful, so these abstractions aren't real." Of course, the reality of an abstraction would only depend on its utility to us if the abstraction were not independently real to begin with.Pneumenon

    But no one but you is talking about 'real'. There's something very insidious about the idea that unless something is eternal and timeless it can't be real. Part of the necessity of re-evaluating Platonism is to block its usurpation of what counts as real: denying that math exists apart from humans is equally to insist that reality is far more interesting than the bland, white-padded wall picture of it painted by Platonists.
  • litewave
    408
    But anyway, the thrust of the argument is: if we took the results of all possible axiomatic systems, agglomerated them into one giant object, then granted that object independent existence - what would it look like? It would contain all kinds of bizarre crap, navigating through this world you'd hardly ever find an axiomatic system which resembled anything like our own.fdrake



    Yes, most of it might not be beautiful or useful but we are talking about metaphysics, which I don't think depends on subjective notions of beauty or usefulness.
  • fdrake
    1.4k
    In set theory, all relations are defined as sets.litewave

    But how mathematics looks as a category theorist is quite a lot different from how it looks like under the aspect of set theory. Say, to a category theorist, natural numbers don't look like the names of individual objects, they look like isomorphism classes of sets. Set theory was built out of intuitions about composite objects of multiple elements, category theory was built from intuitions of transformation and symmetry.


    Yes, most of it might not be beautiful or useful but we are talking about metaphysics, which I don't think depends on subjective notions of beauty or usefulness.litewave

    I have no idea how you took the main thrust of my post to be about beauty or truth. The main thrust is simply that most mathematical objects aren't worthy of study, and agglomerating them all together; producing the final book and the final theorem, far from the ideal vision or ultimate goal of mathematics - produces a writhing mass of irrelevant chaos. It's less Heaven, more Pandemonium.

    At least as many pages in The Great Book Of Mathematics from this realm would be devoted to this picture:
    latest?cb=20140504184221

    as all of the research we have done and will ever do in mathematics. You shouldn't come away from this thinking about the the relationship of mathematics to the form of derp face...
  • Marchesk
    2.2k
    th. The main thrust is simply that most mathematical objects aren't worthy of study,fdrake

    But do these mathematical objects exist, or is this based on the hypothetical that they could be created if we were Jovians?

    Is there a bunch of abandoned junky math that was of no value to Mathematicians but still qualifies as math? Is there a Math junkyard?
  • fdrake
    1.4k


    If the platonic realists are right, the name of that junkyard is the Platonic realm of forms.
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