4 ms·

As much as I like philosophy of mathematics, I never feel quite at ease with sets being some universal foundation. Lists, sets, graphs, all seem quite fundamen

by smokel 3y ago

As much as I like philosophy of mathematics, I never feel quite at ease with sets being some universal foundation. Lists, sets, graphs, all seem quite fundamental to us humans, but where in nature does one observe these weird things? A cave or a jug are highly complex things. Perhaps molecules resemble a graph, but if I understand physics correctly, atoms move like crazy and it's almost accidental that the graph structure is somewhat stable in most molecules. This led me to believe that these fundamental containers are probably a byproduct of how our brains work, more than that they are fundamental outside of those. Thanks to ChatGPT, I now know that this makes me a mathematical fictionalist, or mathematical anti-realist. Anyone care to talk me out of this? :)

2snakes 3y ago
My answer would be: when we distinguish something from something else. This is the root of all logic and cognition.
- colobas 3y ago
  I like this answer. Been reading “Gödel’s Proof” and for a demonstration where the definition of “tautology” in the main text makes use of the concepts of True and False, there is an appendix explaining that you can arrive at the same result without those concepts, just by treating things as belonging to one class vs another (there is a one to one correspondence with True and False but the meaning is arbitrary)
  - 2snakes 3y ago
    "An engineer, a physicist and a mathematician have to build a fence around a flock of sheep, using as little material as possible. The engineer forms the flock into a tight circular shape and constructs a fence around it. The physicist builds a fence with an infinite diameter and pulls it together until it fits around the flock. The mathematician thinks for a while, then builds a fence around himself and defines himself as being outside.”
    - justincredible 3y ago
      [dead]
tutfbhuf 3y ago
> I never feel quite at ease with sets being some universal foundation That's not a big issue nowadays since there is univalent foundations, which is based on ∞-groupoids instead of sets.
auggierose 3y ago
Sets being the foundation is a historical accident. It is convenient and well-studied, and if you can encode your objects and axioms in set theory, you have a sure footing (at least as sure as it gets) with respect to consistency. Otherwise, of course, lists, graphs, etc. are all objects on their own and exist independently of any encoding as sets.
zehaeva 3y ago
The ideas of sets being the universal foundation of math is arbitrary as any other bit of math. Which is to say all of it is arbitrary. Just as you ask, "where in nature does one observe these weird things?" I would to ask where in nature does one find a one? or a pi? or any other number really. Where do you find a triangle? Or a coordinate? Or any of the mathematical constructs we use day to day. Sure you can point at something triangle shaped and go "there!" but is it _really_ a triangle? Or just an approximation? Sure you can count one of something, but that's not the same as the number one. Just like you can't have pi of something. All of math is just a model that is surprisingly applicable to the real world. All of math is a byproduct of our brains. It doesn't exist, out there, in some Platonic World. Anyone who thinks so, I'm looking at you Max Tegmark, is mistaking the map for the territory.
- NotSuspicious 3y ago
  > All of math is a byproduct of our brains. It doesn't exist, out there, in some Platonic World. Anyone who thinks so, I'm looking at you Max Tegmark, is mistaking the map for the territory. I could brashly say the exact opposite and would have proven just as much as you. Fictionalists often attempt to shunt the Platonic realm into an ill-conceived emanation of the mental realm (which just so happens to itself be an accident of matter). Somehow all of this works by virtue of following a kind of mathematical logic that just so happens to not exist or something. I suspect the fundamental problem here is some kind of neurosis that psychologically compels people to reduce the quality of their thought until hard problems disappear. I hope we one day are able to build a catalog of all the ways thought goes wrong so as to prevent such nonsense from proliferating in at least some section of the world. I think a reasonably compelling way to teach yourself how to actually see The Problem (tm) is to view it from the perspective of Roger Penrose's three worlds ( https://hrstraub.ch/en/the-theory-of-the-three-worlds-penrose/ https://hrstraub.ch/en/the-theory-of-the-three-worlds-penros... ) and actually think through the implications in a contemplative, meditative way over the course of several hours. Any analysis of this issue that doesn't involve a sustained look at both logic and phenomenology is a waste of time.
  - smokel 3y ago
paulddraper 3y ago
> where in nature does one observe these weird things? That's like asking "where in nature does one observe numbers"? Literally everywhere. Your jug partitions water into the set inside the jug and the set outside it. Caves are a subset of rock formations.
- smokel 3y ago
  All that partitioning and counting can only be performed by a system that is able to abstract and differentiate between different abstractions. Brains and computers appear to be good at this, but other than that, I don't see many other structures that are capable of doing this. So, no, I don't observe numbers in the universe, if I take on the perspective of, say, a rock.
  - paulddraper 3y ago
    Um what? Are you saying you want something that can be logically reasoned by a rock?
    - smokel 3y ago
      It's obviously not logical, and possibly not even reasonable, but it seems like a viable way to get out of the frame of reference from a human being. I'm fine if this way of thinking is discarded as nonsense. I'd be happy to find more constructive ways to continue with this.
wnc3141 3y ago
I think sets are fundamental to counting. An example: how many trees are in the forest? It depends on where you draw the boundary for tree. An Aspen Grove, for example have many stems of one large interconnected tree.
- smokel 3y ago
  Try to count trees from the perspective of the sun, or from the perspective of an electron. Trees are made up of billions of atoms, in extremely differing configurations, which can only be appreciated by machinery that is able to abstract it into the concept of a tree. To the rest of the universe it's just a bunch of atoms, with no clearly defined boundary. So, counting is not necessarily a fundamental thing either. At least, according to how I like to interpret things :)
  - wnc3141 3y ago
    Excellent points. I think we are viewing "fundamental" differently. Instead of fundamental to the universe, I see counting as fundamental to the usefulness of math, which in itself is just an abstraction of the real world. (Whole Plato's cave thing)
lmm 3y ago
I prefer the HoTT/univalent foundations approach, which is about equivalences, which seem to me to capture a deep essence of what mathematics is about. Arithmetic starts with the idea that you can make some marks on the paper and manipulate them according to certain rules, and this somehow corresponds to how many sheep are in your field or what have you. Geometry starts with the idea that you can record some angles and then after your fields have flooded you can put the boundary markers back in places that are somehow the same as the places they originally were. Lists, sets and graphs are formal systems with arbitrary rules - but, crucially, they're equivalent to things we care about, and we can lift results about those formal systems (where they're easy to calculate) back along that equivalence to be facts about those real things (where they're useful).

by smokel 3y ago

2snakes 3y ago

My answer would be: when we distinguish something from something else. This is the root of all logic and cognition.

colobas 3y ago
I like this answer. Been reading “Gödel’s Proof” and for a demonstration where the definition of “tautology” in the main text makes use of the concepts of True and False, there is an appendix explaining that you can arrive at the same result without those concepts, just by treating things as belonging to one class vs another (there is a one to one correspondence with True and False but the meaning is arbitrary)
- 2snakes 3y ago
  "An engineer, a physicist and a mathematician have to build a fence around a flock of sheep, using as little material as possible. The engineer forms the flock into a tight circular shape and constructs a fence around it. The physicist builds a fence with an infinite diameter and pulls it together until it fits around the flock. The mathematician thinks for a while, then builds a fence around himself and defines himself as being outside.”
  - justincredible 3y ago
    [dead]

tutfbhuf 3y ago

> I never feel quite at ease with sets being some universal foundation That's not a big issue nowadays since there is univalent foundations, which is based on ∞-groupoids instead of sets.

auggierose 3y ago

Sets being the foundation is a historical accident. It is convenient and well-studied, and if you can encode your objects and axioms in set theory, you have a sure footing (at least as sure as it gets) with respect to consistency. Otherwise, of course, lists, graphs, etc. are all objects on their own and exist independently of any encoding as sets.

zehaeva 3y ago

The ideas of sets being the universal foundation of math is arbitrary as any other bit of math. Which is to say all of it is arbitrary. Just as you ask, "where in nature does one observe these weird things?" I would to ask where in nature does one find a one? or a pi? or any other number really. Where do you find a triangle? Or a coordinate? Or any of the mathematical constructs we use day to day. Sure you can point at something triangle shaped and go "there!" but is it _really_ a triangle? Or just an approximation? Sure you can count one of something, but that's not the same as the number one. Just like you can't have pi of something. All of math is just a model that is surprisingly applicable to the real world. All of math is a byproduct of our brains. It doesn't exist, out there, in some Platonic World. Anyone who thinks so, I'm looking at you Max Tegmark, is mistaking the map for the territory.

NotSuspicious 3y ago
> All of math is a byproduct of our brains. It doesn't exist, out there, in some Platonic World. Anyone who thinks so, I'm looking at you Max Tegmark, is mistaking the map for the territory. I could brashly say the exact opposite and would have proven just as much as you. Fictionalists often attempt to shunt the Platonic realm into an ill-conceived emanation of the mental realm (which just so happens to itself be an accident of matter). Somehow all of this works by virtue of following a kind of mathematical logic that just so happens to not exist or something. I suspect the fundamental problem here is some kind of neurosis that psychologically compels people to reduce the quality of their thought until hard problems disappear. I hope we one day are able to build a catalog of all the ways thought goes wrong so as to prevent such nonsense from proliferating in at least some section of the world. I think a reasonably compelling way to teach yourself how to actually see The Problem (tm) is to view it from the perspective of Roger Penrose's three worlds ( https://hrstraub.ch/en/the-theory-of-the-three-worlds-penrose/ https://hrstraub.ch/en/the-theory-of-the-three-worlds-penros... ) and actually think through the implications in a contemplative, meditative way over the course of several hours. Any analysis of this issue that doesn't involve a sustained look at both logic and phenomenology is a waste of time.
- smokel 3y ago

paulddraper 3y ago

> where in nature does one observe these weird things? That's like asking "where in nature does one observe numbers"? Literally everywhere. Your jug partitions water into the set inside the jug and the set outside it. Caves are a subset of rock formations.

smokel 3y ago
All that partitioning and counting can only be performed by a system that is able to abstract and differentiate between different abstractions. Brains and computers appear to be good at this, but other than that, I don't see many other structures that are capable of doing this. So, no, I don't observe numbers in the universe, if I take on the perspective of, say, a rock.
- paulddraper 3y ago
  Um what? Are you saying you want something that can be logically reasoned by a rock?
  - smokel 3y ago
    It's obviously not logical, and possibly not even reasonable, but it seems like a viable way to get out of the frame of reference from a human being. I'm fine if this way of thinking is discarded as nonsense. I'd be happy to find more constructive ways to continue with this.

wnc3141 3y ago

I think sets are fundamental to counting. An example: how many trees are in the forest? It depends on where you draw the boundary for tree. An Aspen Grove, for example have many stems of one large interconnected tree.

smokel 3y ago
Try to count trees from the perspective of the sun, or from the perspective of an electron. Trees are made up of billions of atoms, in extremely differing configurations, which can only be appreciated by machinery that is able to abstract it into the concept of a tree. To the rest of the universe it's just a bunch of atoms, with no clearly defined boundary. So, counting is not necessarily a fundamental thing either. At least, according to how I like to interpret things :)
- wnc3141 3y ago
  Excellent points. I think we are viewing "fundamental" differently. Instead of fundamental to the universe, I see counting as fundamental to the usefulness of math, which in itself is just an abstraction of the real world. (Whole Plato's cave thing)

lmm 3y ago

I prefer the HoTT/univalent foundations approach, which is about equivalences, which seem to me to capture a deep essence of what mathematics is about. Arithmetic starts with the idea that you can make some marks on the paper and manipulate them according to certain rules, and this somehow corresponds to how many sheep are in your field or what have you. Geometry starts with the idea that you can record some angles and then after your fields have flooded you can put the boundary markers back in places that are somehow the same as the places they originally were. Lists, sets and graphs are formal systems with arbitrary rules - but, crucially, they're equivalent to things we care about, and we can lift results about those formal systems (where they're easy to calculate) back along that equivalence to be facts about those real things (where they're useful).