For most mathematicians, most of the time, the mathematics they personally do is, in relation to different mathematical foundations, like architecture is to atoms.
Architects know and care that they are building things made of atoms. And then ... pretty much don't think about atoms because the objects and relationships they are working on are abstractions well above the fine details of atoms.
And having designed many structures with architectural methods, and seeing those buildings built and stand, it doesn't worry them to hear physicists arguing that maybe atoms are different than they thought. They figure, their experience with architecture and its artifacts has proven to be reliable, so there is no realistic threat of some new quantum theory undermining their work.
On infrequent situations, where their work needs to deal with some special property of some material, they don't have any issue dipping down any number of levels of abstraction. But as a practical matter, that is infrequent for most math.
IMO it's not far off how most python or javascript devs don't care about registers or cache misses. Someone's thought deeply about those things so you don't have to.
Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed.
Another good example of mathematicians caring which 'house of cards' their results are built on is the search for an "elementary" proof of the prime number theorem (i.e. showing it doesn't rely on complex analysis).
> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed.
In a way mathematicians can afford to do this more readily than people in software development, because if something is actually proven, then you can 100% rely on that. With software not so much. Or rather: Software usually is not proven to be correct, because that's usually expensive. In mathematics they don't have to consider the runtime of an algorithm, when they "merely" need to prove correctness. The time it needs to run is irrelevant for its correctness. And so they can stack and stack and stack, provided that each piece is proven correct, and it won't have negative consequences. Well, almost. There is some negative consequence in that another human being, wanting to understand a proof, needs to know perhaps many concepts and other proofs, in order to be able to do so. But that's probably the only reason to pursue simplicity in mathematics.
> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study).
I'd be careful about generalizing that to all or most 'mathematicians'. E.g., people working in a lot of fields won't bat an eye at invoking the real numbers when the rational or algebraic numbers would do.
I'm sure some python devs care about cache misses too. I guess my point was that the big results will be picked over again and again to understand _exactly_ which conditions are needed for them to hold.
Something the computer scientists of Hackernews might not realise is that most mathematicians are by nature Platonists, even if they would not try to defend that position when pressed.
Mathematicians begrudgingly retreat to formalism and foundations when pressed because its easier to defend, but the day-to-day of contemporary mathematics is much more an explorative process of a "real" mathematical landscape. They aren't concerned with foundations because it "feels" self-evident that the mathematics they are discovering is true (because their means of discovery, rigour and proof, "guarantee" it to be so).
A lot of the comments here are making false assumptions like "but surely mathematicians all know that their field is ultimately justified as a symbol-pushing game from some axiomatic system right?" in the same way one might say "surely all computer scientists know that every language ultimately compiles down to 1s and 0s processed by a CPU" but that is not at all how most mathematicians think about doing mathematics.
Mathematicians can do mathematics without a knowledge of foundations because the human brain and organism are highly organized, but in a different manner than a formal conceptual system. That human thinking and understanding is, on the whole, not a matter of logically manipulating formalized ideas is generally accepted in philosophy today, as in the later Wittgenstein.
This seems to me to be the same as saying that mathematicians do not care about the meaning of their theorems. That they are only playing a game. They care about consistency only because inconsistency means one can cheat in their game.
I know TFA says that the purpose of foundations is to find a happy home (frame) for the mathematicians intuition. But choosing foundation has real implications on the mathematics. You can have a foundation where every total function on the real numbers is continuous. Or one where Banach–Tarski is just false. So, unless they are just playing a game, the mathematicians should care!
Newton and Gauss and Euler did just fine without such solid foundations. If you get a PhD, very likely even a undergraduate degree in mathematics you cover this stuff, then (unless you choose foundations as your field) you go about doing statistics, or algebra (the higher kind), or analysis knowing you're working on solid fundamentals. It would be crazy if every time you proved something in one of those fields you had to state which derivation of real number you were using. And I guarantee at least 90% of PhD mathematicians could do so if they really needed to.
We are not talking about having to return to foundational axioms in every argument! Just that what axioms one chooses has an impact on which arguments are valid, and thus in turn what truths there are.
I'd say that I care deeply about the meaning behind theorems, but just find results which swing widely based on foundational quirks to be less interesting from an aesthetic standpoint. I see the most interesting structures as the ones that are preserved across different reasonable foundations. This is speaking as someone who was trained as a pure mathematician, moved on to other things, but tries to keep up with pure math as a hobby.
Yes, but most mathematicians do not seem to make this distinction between sturdy and flimsy truths. Which puzzles me. Are they unaware? If so, would they care if educated? Or do they fully commit to classical logic and the axiom of choice if pushed? I can see it go either way, depending on the psychology of the individual mathematician.
I don't think they usually make the distinction in a formal sense, but I think most are aware. The space of explorable mathematics is vastly larger than what the community of mathematicians is capable of collectively thinking about, so a lot of aesthetic judgment goes into deciding what is and what isn't interesting to work on. Mathematicians differ in their tastes too. A sense of sturdiness vs flimsiness is something that might inform this aesthetic judgment, but isn't really something most mathematicians would make part of the mathematics. Often, ones interest isn't the result itself, but some proof technique that brings some sense of insight and understanding, and exploring that often doesn't make much contact with foundational matters.
No one not working on foundations has any problem with axiom of choice. It has weird implications but so what? Banach Tarski just means physical shapes aren't arbitrarily subdividable.
My 2 cents is they do justify it by the interest of the consequences, as Tychonoff or Nullstellensatz. I wouldn't call that faith: Best practices is to state Tychonoff as "AC implies Tychonoff" and that last is logically valid. Sometimes the "AC implies..." is missing, buried in the proof or used unawaredly or predates ZFC, and is a bad thing. But very ofen one now see asterisks on theorems needing it.
AC makes things much easier as it allows to play God powers. Negating AC is not significantly different from constructing mathematics that avoids AC (no assumption about validity of AC). And that makes things way harder with longer proofs and only in sub-cases of classical theorems.
Simply assuming the negation of AC is boring, as negations often are. But there are stronger statements, implying the negation of AC which might be as useful. I think for instance one could assume all subsets of the plane to be measurable. Seems convenient to me.
Same with law of the excluded middle. Tossing it out we can assume all functions are computable and all total functions in the real are continuous. Seems nice and convenient too!
To be fair, in some fields I've seen arguments between "a widget should be defined as ABC" vs. "a widget should be defined as XYZ", to the point that I wonder how they're able to read papers about widgets at all. (If I had to guess, likely by focusing on the 'happy path' where the relevant properties hold, filling in arguments according to their favored viewpoint, and tacitly cutting out edge cases where the definitions differ.)
So if many mathematicians can go without fixed definitions, then they can certainly go without fixed foundations, and try to 'fix everything up' if something ever goes wrong.
In my experience those debates are usually between experts who deeply understand the difference between ABC and XYZ widgets (the example I'm thinking of in my head is whether manifolds should be paracompact). The decision between the two is usually an aesthetic one. For example, certain theorems might be streamlined if you use the ABC definition instead of the XYZ one, at the cost of generality.
But the key is that proponents of both definitions can convert freely between the two in their understandings.
The foundations have real implications on very little of the mathematics. Say I'm working in differential equations in vector spaces. I really do not care whether the axiom of choice is true or false. I'm not building up my functions of multiple real parameters out of sets.
You say you have a foundation where that is in fact what I am doing? Great, if that floats your boat. I don't care. That's several layers of abstraction away from what I'm doing. I pretty much only care about stuff at my layer, and maybe one layer above or below.
Very little of mathematics, like analysis? I am sure the analyst will care about all functions on the reals suddenly turning continuous. (Or rather losing the discontinuous ones)
Or what of commutative algebra and their beloved existence of maximal ideals!
you're kind of coming at this backwards. it's not that someone doing analysis doesn't care about whether all functions on reals is continuous, it's that if you hand them a foundation where that's true, they'll disagree with whether your foundation is correctly modeling functions/real numbers.
Having a PhD in mathematics myself, I have been surrounded by such and had this discussion a few times. Some even like the ideas suggested!
I would say the most common counter argument is cultural: Classical mathematics is the norm in the field, hence one must use it to participate in research in this field.
But that seems to me a rather intellectually unsatisfying argument, if one cares about the meaning of the work.
It is nicer to state theorems that hold for all vector spaces, so mathematicians like to invoke AoC. However, in any applications that are practically relevant, you can obtain a basis without invoking AoC.
> But choosing foundation has real implications on the mathematics. You can have a foundation where every total function on the real numbers is continuous. Or one where Banach–Tarski is just false.
I mean, mathematicians do care about the part of the foundations that affect what they do! Classical vs constructive matters, yes. But material vs structural is not something most mathematicians think about. (They don't think about classical vs constructive either, but that's because they don't really know about constructive and it's not what they're trying to do, rather than because it's irrelevant to them like material vs structural.)
Try to be charitable. Remember, research mathematicians aren't HN commenters. They're forced to live within their intellectual limitations, however narrow those may be.
For most mathematicians, most of the time, the mathematics they personally do is, in relation to different mathematical foundations, like architecture is to atoms.
Architects know and care that they are building things made of atoms. And then ... pretty much don't think about atoms because the objects and relationships they are working on are abstractions well above the fine details of atoms.
And having designed many structures with architectural methods, and seeing those buildings built and stand, it doesn't worry them to hear physicists arguing that maybe atoms are different than they thought. They figure, their experience with architecture and its artifacts has proven to be reliable, so there is no realistic threat of some new quantum theory undermining their work.
On infrequent situations, where their work needs to deal with some special property of some material, they don't have any issue dipping down any number of levels of abstraction. But as a practical matter, that is infrequent for most math.
IMO it's not far off how most python or javascript devs don't care about registers or cache misses. Someone's thought deeply about those things so you don't have to.
Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed.
Another good example of mathematicians caring which 'house of cards' their results are built on is the search for an "elementary" proof of the prime number theorem (i.e. showing it doesn't rely on complex analysis).
Edit: here's a great related discussion on MathOverflow, bringing in analogies from CS: https://mathoverflow.net/questions/90820/set-theories-withou...
> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed.
In a way mathematicians can afford to do this more readily than people in software development, because if something is actually proven, then you can 100% rely on that. With software not so much. Or rather: Software usually is not proven to be correct, because that's usually expensive. In mathematics they don't have to consider the runtime of an algorithm, when they "merely" need to prove correctness. The time it needs to run is irrelevant for its correctness. And so they can stack and stack and stack, provided that each piece is proven correct, and it won't have negative consequences. Well, almost. There is some negative consequence in that another human being, wanting to understand a proof, needs to know perhaps many concepts and other proofs, in order to be able to do so. But that's probably the only reason to pursue simplicity in mathematics.
> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study).
I'd be careful about generalizing that to all or most 'mathematicians'. E.g., people working in a lot of fields won't bat an eye at invoking the real numbers when the rational or algebraic numbers would do.
I'm sure some python devs care about cache misses too. I guess my point was that the big results will be picked over again and again to understand _exactly_ which conditions are needed for them to hold.
Something the computer scientists of Hackernews might not realise is that most mathematicians are by nature Platonists, even if they would not try to defend that position when pressed.
most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism https://en.wikipedia.org/wiki/Mathematical_Platonism
Mathematicians begrudgingly retreat to formalism and foundations when pressed because its easier to defend, but the day-to-day of contemporary mathematics is much more an explorative process of a "real" mathematical landscape. They aren't concerned with foundations because it "feels" self-evident that the mathematics they are discovering is true (because their means of discovery, rigour and proof, "guarantee" it to be so).
A lot of the comments here are making false assumptions like "but surely mathematicians all know that their field is ultimately justified as a symbol-pushing game from some axiomatic system right?" in the same way one might say "surely all computer scientists know that every language ultimately compiles down to 1s and 0s processed by a CPU" but that is not at all how most mathematicians think about doing mathematics.
Mathematicians can do mathematics without a knowledge of foundations because the human brain and organism are highly organized, but in a different manner than a formal conceptual system. That human thinking and understanding is, on the whole, not a matter of logically manipulating formalized ideas is generally accepted in philosophy today, as in the later Wittgenstein.
This seems to me to be the same as saying that mathematicians do not care about the meaning of their theorems. That they are only playing a game. They care about consistency only because inconsistency means one can cheat in their game.
I know TFA says that the purpose of foundations is to find a happy home (frame) for the mathematicians intuition. But choosing foundation has real implications on the mathematics. You can have a foundation where every total function on the real numbers is continuous. Or one where Banach–Tarski is just false. So, unless they are just playing a game, the mathematicians should care!
Newton and Gauss and Euler did just fine without such solid foundations. If you get a PhD, very likely even a undergraduate degree in mathematics you cover this stuff, then (unless you choose foundations as your field) you go about doing statistics, or algebra (the higher kind), or analysis knowing you're working on solid fundamentals. It would be crazy if every time you proved something in one of those fields you had to state which derivation of real number you were using. And I guarantee at least 90% of PhD mathematicians could do so if they really needed to.
We are not talking about having to return to foundational axioms in every argument! Just that what axioms one chooses has an impact on which arguments are valid, and thus in turn what truths there are.
I'd say that I care deeply about the meaning behind theorems, but just find results which swing widely based on foundational quirks to be less interesting from an aesthetic standpoint. I see the most interesting structures as the ones that are preserved across different reasonable foundations. This is speaking as someone who was trained as a pure mathematician, moved on to other things, but tries to keep up with pure math as a hobby.
Yes, but most mathematicians do not seem to make this distinction between sturdy and flimsy truths. Which puzzles me. Are they unaware? If so, would they care if educated? Or do they fully commit to classical logic and the axiom of choice if pushed? I can see it go either way, depending on the psychology of the individual mathematician.
I don't think they usually make the distinction in a formal sense, but I think most are aware. The space of explorable mathematics is vastly larger than what the community of mathematicians is capable of collectively thinking about, so a lot of aesthetic judgment goes into deciding what is and what isn't interesting to work on. Mathematicians differ in their tastes too. A sense of sturdiness vs flimsiness is something that might inform this aesthetic judgment, but isn't really something most mathematicians would make part of the mathematics. Often, ones interest isn't the result itself, but some proof technique that brings some sense of insight and understanding, and exploring that often doesn't make much contact with foundational matters.
No one not working on foundations has any problem with axiom of choice. It has weird implications but so what? Banach Tarski just means physical shapes aren't arbitrarily subdividable.
Banach Tarski is not about physical shapes.
The thing is, the foundations negating axiom of choice are just as consistent as those with. So, how do mathematicians justify their faith in AC?
My 2 cents is they do justify it by the interest of the consequences, as Tychonoff or Nullstellensatz. I wouldn't call that faith: Best practices is to state Tychonoff as "AC implies Tychonoff" and that last is logically valid. Sometimes the "AC implies..." is missing, buried in the proof or used unawaredly or predates ZFC, and is a bad thing. But very ofen one now see asterisks on theorems needing it.
AC makes things much easier as it allows to play God powers. Negating AC is not significantly different from constructing mathematics that avoids AC (no assumption about validity of AC). And that makes things way harder with longer proofs and only in sub-cases of classical theorems.
Simply assuming the negation of AC is boring, as negations often are. But there are stronger statements, implying the negation of AC which might be as useful. I think for instance one could assume all subsets of the plane to be measurable. Seems convenient to me.
Same with law of the excluded middle. Tossing it out we can assume all functions are computable and all total functions in the real are continuous. Seems nice and convenient too!
To be fair, in some fields I've seen arguments between "a widget should be defined as ABC" vs. "a widget should be defined as XYZ", to the point that I wonder how they're able to read papers about widgets at all. (If I had to guess, likely by focusing on the 'happy path' where the relevant properties hold, filling in arguments according to their favored viewpoint, and tacitly cutting out edge cases where the definitions differ.)
So if many mathematicians can go without fixed definitions, then they can certainly go without fixed foundations, and try to 'fix everything up' if something ever goes wrong.
In my experience those debates are usually between experts who deeply understand the difference between ABC and XYZ widgets (the example I'm thinking of in my head is whether manifolds should be paracompact). The decision between the two is usually an aesthetic one. For example, certain theorems might be streamlined if you use the ABC definition instead of the XYZ one, at the cost of generality.
But the key is that proponents of both definitions can convert freely between the two in their understandings.
The foundations have real implications on very little of the mathematics. Say I'm working in differential equations in vector spaces. I really do not care whether the axiom of choice is true or false. I'm not building up my functions of multiple real parameters out of sets.
You say you have a foundation where that is in fact what I am doing? Great, if that floats your boat. I don't care. That's several layers of abstraction away from what I'm doing. I pretty much only care about stuff at my layer, and maybe one layer above or below.
Very little of mathematics, like analysis? I am sure the analyst will care about all functions on the reals suddenly turning continuous. (Or rather losing the discontinuous ones)
Or what of commutative algebra and their beloved existence of maximal ideals!
you're kind of coming at this backwards. it's not that someone doing analysis doesn't care about whether all functions on reals is continuous, it's that if you hand them a foundation where that's true, they'll disagree with whether your foundation is correctly modeling functions/real numbers.
At which point we would have an interesting debate! I could tell them all about how this foundation will give them a more nuanced view on continuity!
I suggest you go meet some PhD mathematicians and have that discussion.
Having a PhD in mathematics myself, I have been surrounded by such and had this discussion a few times. Some even like the ideas suggested!
I would say the most common counter argument is cultural: Classical mathematics is the norm in the field, hence one must use it to participate in research in this field.
But that seems to me a rather intellectually unsatisfying argument, if one cares about the meaning of the work.
Do you not care if your vector space has a basis?
It is nicer to state theorems that hold for all vector spaces, so mathematicians like to invoke AoC. However, in any applications that are practically relevant, you can obtain a basis without invoking AoC.
> But choosing foundation has real implications on the mathematics. You can have a foundation where every total function on the real numbers is continuous. Or one where Banach–Tarski is just false.
I mean, mathematicians do care about the part of the foundations that affect what they do! Classical vs constructive matters, yes. But material vs structural is not something most mathematicians think about. (They don't think about classical vs constructive either, but that's because they don't really know about constructive and it's not what they're trying to do, rather than because it's irrelevant to them like material vs structural.)
Try to be charitable. Remember, research mathematicians aren't HN commenters. They're forced to live within their intellectual limitations, however narrow those may be.