[HN Gopher] Structuralism as a Philosophy of Mathematics
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Structuralism as a Philosophy of Mathematics
Author : FillMaths
Score : 98 points
Date : 2024-04-06 19:59 UTC (1 days ago)
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| moomin wrote:
| I'm not sure what a mathematical approach _without_ structuralism
| would look like. Like, if you consider the operations created by
| combining the rotations and flips of a triangle, and the set of
| permutations of the letters A,B and C, it's pretty obvious
| they're isomorphic and also obvious that they're different. My
| question is: is there a mathematically useful way of expressing
| that difference?
|
| Or to put it a different way, I'm not sure anything interesting
| is being said here.
| mgn115 wrote:
| Mathematical structuralism was developed to explain the
| ontology of mathematical objects, and is often contrasted with
| Platonism, which is the position that numbers are real things
| like you or I.
| denton-scratch wrote:
| I'm not real. I'm not sure about you.
| 082349872349872 wrote:
| Found the co-solipsist.
| jjgreen wrote:
| _The working mathematician is a Platonist on weekdays, a
| formalist on weekends._
|
| Reuben Hersh
| zhouyisu wrote:
| A funny insight about Platonism (if it's not funny, treat
| this as a bad joke)
|
| I think aka "[?]" therefore I know I thought aka "{[?]}"
| therefore I know I knew I have thought aka "{{[?]}}"
|
| and ...
|
| boom! The entire Math system is imported. (BTW, limitations
| known as "computation theory" is also introduced)
|
| So maybe we are actually mathematical being on a manifold
| named as "real world". To me it is more concise and profound
| than "philosophy". As we are real, so do all mathematical
| objects.
| nicklecompte wrote:
| The point is that if you are interested in the structure of
| finite sets, then there are structure-preserving isomorphisms
| between the vertices of a triangle and the set {A, B, C} so
| that (for example) permutations of the set and
| reflections/rotations of the triangle are coincident.
|
| But if you're interested in the structure of plane geometry
| then there is no such structure-preserving isomorphism because
| any map into a finite set will "delete" information about side
| length and angles.
|
| The structuralist idea is that interesting mathematics can be
| "most easily" found by considering structure-preserving
| isomorphisms and not the structures themselves. In particular
| dealing with the structures directly can obscure the
| mathematics you are trying to discover, e.g. dealing with a
| full Euclidean group when the dihedral group is all the problem
| requires.
| 082349872349872 wrote:
| If you want, it's even possible to fully commit to the
| isomorphism point of view by saying that you'll represent the
| structures themselves by their identity isomorphisms, but
| then if you "delete" the little tag telling you which
| particular endo was the identity (would a physicist say "up
| to phase"?), you might discover other interesting things...
| empath-nirvana wrote:
| Well, given that structuralism as a program didn't exist until
| basically the 19th century, you can look at the entire history
| of mathematics to see what mathematics without structuralism
| looks like. It's sort of hard to argue that the isomorphism
| between symmetries and permutations is "obvious" when group
| theory wasn't invented until the 18th century, and symmetry
| groups weren't formalized until the 19th century. Your comment
| is basically this:
|
| One fish says to another fish: "The water's nice today." and
| swims off, the other fish says "What's water?". Your entire
| mathematical world view is so permeated with the language of
| structuralism that you can't see it any more.
| klysm wrote:
| I think sometimes obvious concepts like symmetries are hard
| to distill into the appropriate mathematical language. I'm
| willing to bet the isomorphism there is obvious to most
| folks, but the expression of its mathematical essence is not.
| woopsn wrote:
| This is really an important point for mathematical
| philosophy. It is a single enterprise going back to ancient
| Babylon, China, India, Greece, Egypt, and before. The
| elementary meta-theory needs to be syntonic to mathematical
| activity and knowledge predating (or otherwise practiced
| without) formalism, structuralism, categoricity, platonism,
| etc.
| hackandthink wrote:
| Representation theory should do it:
|
| https://en.wikipedia.org/wiki/Representation_theory_of_finit...
| hackandthink wrote:
| Lawvere theories should be fine as well:
|
| "The rough idea is to define an algebraic theory as a category
| with finite products and possessing a "generic algebra" (e.g.,
| a generic group), and then define a model of that theory (e.g.,
| a group) as a product-preserving functor out of that category."
|
| https://ncatlab.org/nlab/show/Lawvere+theory#the_theory_of_g...
| raincom wrote:
| There are two kinds of mathematicians, or one can say, two
| cultures of mathematics: (a) problem solving, proving
| conjectures, etc (b) theories. Alexander Grothendieck belongs
| to (b). Paul Erdos to (a).
|
| You can read Gower's paper on "The two cultures of Mathematics"
| at: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
| ectopasm83 wrote:
| https://ncatlab.org/nlab/show/structuralism
|
| >In the humanities
|
| >In the 20th century, structuralism in the humanities is
| associated with Emile Durkheim and Georg Simmel in sociology,
| Ferdinand de Saussure (and later Roman Jakobson) in linguistics,
| and Claude Levi-Strauss in anthropology.
|
| Ferdinand de Saussure, Ecrits de linguistique generale:
|
| >The notion of identity will be, in all these orders, the
| necessary basis, the one that serves as an absolute basis: it is
| only through it and in relation to it that we can then determine
| the entities of each order, the primary terms that the linguist
| can legitimately believe to have before them.
|
| >(Vocal Order) Flow of ideas: Everything that is declared
| identical by form, in opposition to what is not identical, is a
| finite term, which is not yet defined and can be arbitrary but
| represents for the first time a knowable object, while the
| observation of specific vocal facts outside the consideration of
| identity represents no object. A certain vocal being is thus
| constituted and recognized in the name of an identity that we
| establish, and then thousands of others are obtained using the
| same principle, we can begin to classify these identity patterns
| of all sorts that we take, and are obliged to take, for the
| primary and specific and concrete facts, although they are each
| in their infinite diversity only the result of a vast prior
| operation of generalization.
|
| >Couldn't we limit ourselves to implying this great fundamental
| operation? Isn't it obvious from the outset that as soon as we
| talk about a group, for example, we mean the generality of cases
| where a group exists, so there is little subtle interest in
| recalling that this entity is fundamentally and primarily based
| on an identity?
|
| >We will immediately see that it is not allowed to substitute
| abstract entities for the fact of the identity of certain
| concrete facts with impunity because we will deal with other
| abstract entities, and the only pole in the middle of this will
| be identity or non-identity.
| 082349872349872 wrote:
| The basic problem with computation is that identity among the
| inputs is not necessarily identity among the outputs, which
| explains why we often bring in bigger guns than Boolean logic.
| naasking wrote:
| From his comments:
|
| > In my view, there is a way of viewing structuralism as
| undermining the success of the indispensibility argument, because
| no particular mathematical structure is ever indispensible, since
| it can be interpreted via alternative structure
|
| I can see that, but on the other hand that alternative structure
| is equivalent up to isomorphism so you haven't really eliminated
| the structure. At best, you've probably shown that alternate
| theories that explain the same observations necessarily exist
| because our knowledge of reality's structure is incomplete and so
| we can't eliminate incorrect theories by the extra structure
| beyond the isomorphism. It does not at all undermine the idea
| that reality itself has some kind of structure.
| 082349872349872 wrote:
| Could you please briefly explain the indispensibility argument?
| (this being the first time I've run across the term; although
| if it's some philosophical thing I'm afraid I'm only interested
| in it as far as I'd be in the theological creation of the
| universe*)
|
| My view of the relation between maths and physics is set out in
| https://news.ycombinator.com/item?id=39220159 , and because we
| go from physics to maths (f) and then back from maths to
| physics (f^-1), we can conjugate with any g,g^-1 pair, meaning
| that on the maths side we might as well mod out by isomorphics
| (only care about the partial order of equivalence classes of
| the preorder of structures)
|
| (Similarly, we should find that we only care about the
| equivalence class [singular] of isomorphic realities [plural]?)
|
| ---
|
| * this is not to say that I'm not interested; see
| https://news.ycombinator.com/item?id=39886966
|
| Assuming a God who creates Creations, do we wind up with a best
| possible (principal) Creation? all possible (perhaps trivially
| so) Creations? if there is a set of Creations created, are they
| directed? Theologically, what happens if we have an infinite
| number of finite Creations, each an appropriate approximation,
| and we pass to their supremum?
| cjk2 wrote:
| I think we need a philosophy of the philosophy of mathematics to
| build an understanding around this...
| 082349872349872 wrote:
| see Pierre Bordintello, "La Demonstration" (1984):
|
| > _Mathematics entails, and it entails the entailer._
| cjk2 wrote:
| Urgh another rabbit hole. I should thank you but I'm not sure
| I will :)
| 082349872349872 wrote:
| Unfortunately this rabbit hole's reality is in the
| equivalence class of the holes belonging to Hazel and
| Fiver.
| cjk2 wrote:
| Haven't read that for years!
| 77pt77 wrote:
| What does Watership down have to do with this?
| 082349872349872 wrote:
| For a Platonist, "La Demonstration", like El-ahrairah,
| exists. (I prefer Borges style to AMS style citations)
| Biganon wrote:
| I mean you cannot quote him without also quoting Robert
| Feuerstern's 1985 essay "Logique de la Tangence" in which he
| famously wrote :
|
| "Pierre Bordintello is a fucking moron."
| hackandthink wrote:
| "Categoricity is central to structuralism because it shows that
| the essence of our familiar mathematical domains, including N, Z,
| Q, R, C, and so on, are determined by structural features that we
| can identify and express."
|
| I do not buy this. I feel it is the other way around. Structural
| Features are essential. Categoricity may be nice to have but why
| should I care so much about it?
| random3 wrote:
| Categories are the main and most developed tool building around
| relations that are the building blocks of structure.
| cubefox wrote:
| The quote says why you should care about it. Without
| categoricity the axioms of a theory don't define a specific
| structure.
| drsopp wrote:
| Is there any attempt to organize mathematics kind of like the OSI
| model? On the bottom we might start with the physical layer that
| can be paper+pencil marks on it or a soundwave, the next could be
| a shape, next could be what the shape stands for and so on.
| Further up we could find things like isomorphisms.
| 082349872349872 wrote:
| Or start with the morphisms, then look for the idempotents, so
| further up the structures fall out naturally?
|
| (this program may have an advantage in that it motivates
| passing from the continuous to the discrete?)
| seeknotfind wrote:
| If you want to see an organization of mathematical ideas, I'd
| recommend digging into https://us.metamath.org/. Great intro to
| formal systems, though many more layers of definitions than the
| OSI model.
|
| Though, there may be canonical structures, any universal
| structure is illusive if not non-existent.
| woopsn wrote:
| In a sense computer and electrical engineers/scientists did
| largely map out the base layers, over ~150 years from the 19th
| century through the mid 20th. I think equipment (broadly
| speaking) is foundational for mathematics. The "stack", as far
| as I interpret it, is something like
|
| 1. Being
|
| 2. Communication
|
| 3. Equipment -- a device you can put marks on and read off of
|
| 4. Discipline -- ability to reliably and skillfully manipulate
| the device
|
| 5. Submission
|
| The stage is set at this point for some "elementary
| mathematics" -- think back to elementary school.
|
| 6. Symbolism -- the equipment is not just equipment.
| Mathematical relevance springs here.
|
| 7. Geometry -- from vision we see area and edges, objects of
| perception, they are interpreted as mathematically relevant and
| hence symbolized.
|
| 8. Algebra -- manipulating our equipment with discipline, an
| equivalence is perceived between different sequences of
| operations.
|
| 9. Proposition -- conviction the relevant facts of geometry and
| algebra can be formulated clearly in declarations of the sort
| "if ... then ..., and ... (... and etc)".
|
| Higher level mathematics
|
| 10. Refinement 11. Proof
|
| 12. Application 13. Theory
|
| 14. Computer science, engineering, and design
|
| ...
| empath-nirvana wrote:
| Yes, typically the base layer is set theory (with ZFC being the
| "standard" formulation).
|
| https://en.wikipedia.org/wiki/Foundations_of_mathematics
| drsopp wrote:
| I understand what you mean, but in my context I would say
| that you need layers below this. You for example paper, ink,
| glyphs, and other things before you can start defining ZFC.
| novaRom wrote:
| Discreteness, finitenes, causality.
| random3 wrote:
| The computer and computer science is, in a sense, the result of
| the somewhat failed attempt to put mathematics on a solid
| logical grounds, in a way that resembles a stack like OSI.
| Godel's incompleteness theorems and Turing undecidability
| results showed that it's not that easy. Briefly, the proof for
| the halting problem was the Turing machine.
| andoando wrote:
| Is there any mathematics field that views mathematical objects as
| patterns of multiple numbers and considers order as a fundamental
| property? ex, 4 isnt just "4" but is either 1,1,1,1 or 3,1 or 1,3
| or 2(a), 2(b) or 2(b), 2(a)? These all represent real life
| abstractions and I feel a lot of detail is lost when we only
| think about the structure of an object as its total count, and
| not its composition/order.
| moritzwarhier wrote:
| Numbers have properties beyond their order, right, e.g.
| factorization.
|
| Maybe the field you are looking for is Number theory? Or maybe
| algebra and group theory?
|
| But I don't quite understand your sentence:
|
| > [...] I feel a lot of detail is lost when we only think about
| the structure of an object as its total count, and not its
| composition/order.
|
| Maths is concerned a _lot_ with how numbers can be constructed
| or composed from other numbers, ir other mathematical
| structures.
|
| Still, there is a difference between the abstract notion of 4
| and objects where we assign some measurable quantity (e g.
| counting similar objects, let's say turtles, and saying in
| total it's "4 turtles").
|
| Numbers are not concerned with counting alone, but that's the
| easiest way to construct numbers.
| andoando wrote:
| What I mean is that there are multiple structures that are
| equivalently "4". 1,3 or 2,2, etc. 1 AND 3 depicts two
| distinct ovjects (or one object split into 2 parts) and is
| fundamentally a different idea than the structure represented
| by 4. However in all the math Ive seen this are simply
| reduced and we just care about the end result.
|
| I am saying this equivalence isnt a fundamental property, but
| one merely useful toward a purpose. I am interested in
| mathemathics where one can reason about and do operations on
| ordered sets of discrete numbers or booleans.
|
| I suppose matrices, boolean algebra or category theory is the
| closest to what I am thinking of, but I need to learn more
| about that.
|
| I am aware of vector spaces but keep in my mind by 1,1 or 1,3
| I am not talking about points in a multidimensional
| coordinate space.
| ndriscoll wrote:
| In some sense, basically every field of math makes this kind of
| distinction. e.g. it's acknowledged that for a vector space V,
| the tensor products V[?](V[?]V) and (V[?]V)[?]V are not _equal_
| , but they are _equivalent_ in the sense of e.g. a linear
| isomorphism, or maybe in the sense of an algebra isomorphism or
| something, and mathematicians do have machinery to track _in
| what sense_ they consider two non-equal things to be
| equivalent, and they take the time to prove that operations
| they want to define on _equivalent_ but non-equal things give
| _equivalent_ results. Category theory is partly concerned with
| this kind of machinery.
|
| So they do care about this, but then invented ideas like
| equivalence relations and quotients so that they can not have
| to _constantly_ care about it.
|
| With programming languages or a proof assistant like lean, by
| default non-equal things are not the same, and you need to do
| some bookkeeping to carry around proofs that things are
| equivalent in the way you need them to be. The tricky thing is
| actually making it ergonomic to be able to think of (1+1)+1 and
| 1+(1+1) as "the same" without constantly needing to think about
| it. It's hard to do math in most programming languages partly
| because they lack the machinery necessary to have contextual
| notions of equivalence (but also because they lack dependent
| types).
|
| In some fields, that extra bookkeeping is really important.
| E.g. in geometry you might be able to choose coordinate systems
| at each point of some space, but perhaps not in a way that is
| globally defined and consistent with the types of equivalence
| that you need, so you need to track everything through local
| coordinate changes as you move around.
|
| Or a simpler example is a vector space and its dual: they're
| isomorphic, so in some sense equivalent, but not canonically
| so, so you need to track the equivalence you're using, and maps
| on one are "mirrored" when applied to the other. More
| generally, any finite dimensional vector space is characterized
| (in the sense of equivalence up to linear isomorphism) by a
| single number (its dimension), so in some sense it "is" that
| number, but we also think of them as each being different,
| depending on context. So f.d. vector spaces can in some sense
| be thought of as "numbers that remember which vector space they
| are".
| hyperthesis wrote:
| Well, there's the fundamental theorem of arithmetic, that there
| is exactly one composition of prime factors for every natural
| number (excluding ordering).
| https://wikipedia.org/wiki/Fundamental_theorem_of_arithmetic
| whiterknight wrote:
| That's how integers are defined in set theory.
| joe_the_user wrote:
| OK, the confusing thing is that Dedekind proved that a two second
| order models of arithmetic are equivalent/unique but Godel
| essentially proved that there are an infinity of first order
| models of arithmetic. But it seems logical that a second order
| model of arithmetic would contain a first model and that you
| couldn't say "which" model contained.
|
| I probably phrased that wrong but I think the question is clear
| woopsn wrote:
| So for example, do programs 1 and 2 compute the same thing? In
| general this is quite difficult and unsolvable. But then, if so,
| is the complexity of 1 less than 2? What is the minimum
| complexity of any program that computes the thing? These are
| relevant problems we want to solve. They involve considering
| isomorphism, orderings, limits, etc. - structuralism - but in
| many cases the maps are not there to support it, or there is no
| way to find them. The relationship between two complexity
| classes, or even two binaries.
|
| At an elementary level what we deal with are representations. In
| CS this problem is severe, for example almost all boolean
| functions have exponential circuit complexity - but we cannot
| offer even a single example up. It's an open question how
| powerful graph isomorphism even is.
|
| I don't mean to knock the structuralist insight, it is a powerful
| "imperative" as the article says. There is just incompleteness
| everywhere, in less mature fields especially, where they work in
| spaces that can barely be classified as of yet. The knowledge is
| generated there, and then organized within a higher scheme.
| Topology sprung partly out of Euler's attack on the Konigsberg
| bridge problem (I suppose structuralism overall does to). It is a
| revealing perspective, but it seems the insight comes _after_ the
| large amount of work, not usually before.
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