9 ms·

Structuralism as a Philosophy of Mathematics

127 points by FillMaths 2y ago

moomin 2y ago
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 2y ago
  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 2y ago
    I'm not real. I'm not sure about you.
    - 082349872349872 2y ago
      Found the co-solipsist.
  - jjgreen 2y ago
    The working mathematician is a Platonist on weekdays, a formalist on weekends. Reuben Hersh
  - zhouyisu 2y ago
    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.
  - raincom 2y ago
    The problem is: do numbers exist? One group says, they don't. Another say, they do exist in the Platonic world, but this Platonic world is accessible from the world we inhabit in. Next time, look at the folks who looks for Platonic love:)
- nicklecompte 2y ago
  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 2y ago
    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 2y ago
  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 2y ago
    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 2y ago
    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 2y ago
  Representation theory should do it: https://en.wikipedia.org/wiki/Representation_theory_of_finite_groups#Permutation_representation https://en.wikipedia.org/wiki/Representation_theory_of_finit...
- hackandthink 2y ago
  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_groups https://ncatlab.org/nlab/show/Lawvere+theory#the_theory_of_g...
- raincom 2y ago
  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 https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
ectopasm83 2y ago
https://ncatlab.org/nlab/show/structuralism 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 Lévi-Strauss in anthropology. Ferdinand de Saussure, Écrits de linguistique générale: >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 2y ago
  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.
- deleted 2y ago
  [deleted]
naasking 2y ago
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 2y ago
  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 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 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?
  - nyssos 2y ago
    > Could you please briefly explain the indispensibility argument? An extremely rough caricature: - Scientific realists claim that we should believe the entities that are indispensable to our best scientific theories (electrons, for instance) exist. - Mathematical entities are indispensable to our best scientific theories - So scientific realists should be realists about mathematical entities.
cjk2 2y ago
I think we need a philosophy of the philosophy of mathematics to build an understanding around this...
- 082349872349872 2y ago
  see Pierre Bordintello, "La Démonstration" (1984): > Mathematics entails, and it entails the entailer.
  - cjk2 2y ago
    Urgh another rabbit hole. I should thank you but I'm not sure I will :)
    - 082349872349872 2y ago
      Unfortunately this rabbit hole's reality is in the equivalence class of the holes belonging to Hazel and Fiver.
      - cjk2 2y ago
        Haven't read that for years!
      - 77pt77 2y ago
        What does Watership down have to do with this?
        082349872349872 2y ago
        For a Platonist, "La Démonstration", like El-ahrairah, exists. (I prefer Borges style to AMS style citations)
  - Biganon 2y ago
    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 2y ago
"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 2y ago
  Categories are the main and most developed tool building around relations that are the building blocks of structure.
- cubefox 2y ago
  The quote says why you should care about it. Without categoricity the axioms of a theory don't define a specific structure.
  - auggierose 2y ago
    So what? We can still only prove those theorems that also hold in all of those other structures that we don't mean, but sweep up anyway.
    - cubefox 2y ago
      It's not clear what "the structure we mean" means without a way of definitely identifying it in our mind. Categorical axioms just make this explicit.
      - auggierose 2y ago
        Yes, that's true. I think it is good to have a logic with a semantics which allows you to clearly say what the "standard models" are. For those models categoricity should be achievable. But at the same time it is also clear that there will always be other models, for which categoricity doesn't hold.
drsopp 2y ago
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 2y ago
  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 2y ago
  If you want to see an organization of mathematical ideas, I'd recommend digging into https://us.metamath.org/ 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 2y ago
  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 ...
andoando 2y ago
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 2y ago
  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 2y ago
    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.
    - moritzwarhier 2y ago
joe_the_user 2y ago
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
- rwl 2y ago
  The distinction between "first order" and "second order" is in the first instance a distinction at the level of formal languages. A second order language has more complicated syntax and semantics: it allows variables in predicate position, which (in the standard semantics) take values from the entire powerset of the underlying domain. This makes second order languages, including the language of arithmetic, much more expressive: they can distinguish models that first order languages can't. Those infinitely many non-isomorphic models of arithmetic expressed in a first order language can be distinguished, and excluded, as models of arithmetic expressed in a second order language. That's why second order arithmetic is categorical: all of its models are isomorphic. Yes, a model of second order arithmetic contains a model of first order arithmetic, but within the second order language, you can say "which model it is" (up to isomorphism). It's only if you restrict yourself to a first order language that you can no longer say anything which will be true in that model, but false in any non-isomorphic one.
woopsn 2y ago
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 Königsberg 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.

127 points by FillMaths 2y ago

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 2y ago
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 2y ago
  I'm not real. I'm not sure about you.
  - 082349872349872 2y ago
    Found the co-solipsist.
- jjgreen 2y ago
  The working mathematician is a Platonist on weekdays, a formalist on weekends. Reuben Hersh
- zhouyisu 2y ago
  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.
- raincom 2y ago
  The problem is: do numbers exist? One group says, they don't. Another say, they do exist in the Platonic world, but this Platonic world is accessible from the world we inhabit in. Next time, look at the folks who looks for Platonic love:)
nicklecompte 2y ago
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 2y ago
  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 2y ago
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 2y ago
  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 2y ago
  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 2y ago
Representation theory should do it: https://en.wikipedia.org/wiki/Representation_theory_of_finite_groups#Permutation_representation https://en.wikipedia.org/wiki/Representation_theory_of_finit...
hackandthink 2y ago
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_groups https://ncatlab.org/nlab/show/Lawvere+theory#the_theory_of_g...
raincom 2y ago
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 https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

ectopasm83 2y ago

https://ncatlab.org/nlab/show/structuralism 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 Lévi-Strauss in anthropology. Ferdinand de Saussure, Écrits de linguistique générale: >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 2y ago
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.
deleted 2y ago
[deleted]

naasking 2y ago

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 2y ago
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 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 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?
- nyssos 2y ago
  > Could you please briefly explain the indispensibility argument? An extremely rough caricature: - Scientific realists claim that we should believe the entities that are indispensable to our best scientific theories (electrons, for instance) exist. - Mathematical entities are indispensable to our best scientific theories - So scientific realists should be realists about mathematical entities.

cjk2 2y ago

I think we need a philosophy of the philosophy of mathematics to build an understanding around this...

082349872349872 2y ago
see Pierre Bordintello, "La Démonstration" (1984): > Mathematics entails, and it entails the entailer.
- cjk2 2y ago
  Urgh another rabbit hole. I should thank you but I'm not sure I will :)
  - 082349872349872 2y ago
    Unfortunately this rabbit hole's reality is in the equivalence class of the holes belonging to Hazel and Fiver.
    - cjk2 2y ago
      Haven't read that for years!
    - 77pt77 2y ago
      What does Watership down have to do with this?
      - 082349872349872 2y ago
        For a Platonist, "La Démonstration", like El-ahrairah, exists. (I prefer Borges style to AMS style citations)
- Biganon 2y ago
  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 2y ago

"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 2y ago
Categories are the main and most developed tool building around relations that are the building blocks of structure.
cubefox 2y ago
The quote says why you should care about it. Without categoricity the axioms of a theory don't define a specific structure.
- auggierose 2y ago
  So what? We can still only prove those theorems that also hold in all of those other structures that we don't mean, but sweep up anyway.
  - cubefox 2y ago
    It's not clear what "the structure we mean" means without a way of definitely identifying it in our mind. Categorical axioms just make this explicit.
    - auggierose 2y ago
      Yes, that's true. I think it is good to have a logic with a semantics which allows you to clearly say what the "standard models" are. For those models categoricity should be achievable. But at the same time it is also clear that there will always be other models, for which categoricity doesn't hold.

drsopp 2y ago

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 2y ago
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 2y ago
If you want to see an organization of mathematical ideas, I'd recommend digging into https://us.metamath.org/ 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 2y ago
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 ...

andoando 2y ago

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 2y ago
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 2y ago
  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.
  - moritzwarhier 2y ago

joe_the_user 2y ago

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

rwl 2y ago
The distinction between "first order" and "second order" is in the first instance a distinction at the level of formal languages. A second order language has more complicated syntax and semantics: it allows variables in predicate position, which (in the standard semantics) take values from the entire powerset of the underlying domain. This makes second order languages, including the language of arithmetic, much more expressive: they can distinguish models that first order languages can't. Those infinitely many non-isomorphic models of arithmetic expressed in a first order language can be distinguished, and excluded, as models of arithmetic expressed in a second order language. That's why second order arithmetic is categorical: all of its models are isomorphic. Yes, a model of second order arithmetic contains a model of first order arithmetic, but within the second order language, you can say "which model it is" (up to isomorphism). It's only if you restrict yourself to a first order language that you can no longer say anything which will be true in that model, but false in any non-isomorphic one.

woopsn 2y ago

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 Königsberg 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.