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Huh? Mathemathical proof is not a social construct. This makes no sense.

by rapsey 4y ago

hugh-avherald 4y ago
It's certainly a social construct, but it is not merely a social construct.
karmakurtisaani 4y ago
I suppose they mean what's commonly accepted as true and can be referred to as truths. No one can read all the proofs, so they have to trust others who have. There was that one example where a mathematician "proved" something terribly complicated using his own methods and terminology developed over several years. The truth value of that kind of proof is very much a social construct.
- Agingcoder 4y ago
  Yes, mochizuki and the abc conjecture. It was an interesting conundrum : he was a really good mathematician, not a crank so his funky proof couldn't be dismissed. However, people were wary of approaching the proof since it was risky career wise (it takes time, etc). You end up with a weird situation where something is probably true, but you won't know that until a trusted group of mathematicians have read it and said so.
uKVZe85V 4y ago
The mathematical truths are the fruit of the work. The social consensus is an implementation "detail". Yet it's the implementation we have. Does this make better sense now?
kingkawn 4y ago
And yet nonetheless you feel the need to object to this formulation publicly and have it considered by others
mpweiher 4y ago
The proof is not a social construct. The truth is. The proof is a mechanism to reach that consensus, by convincing other mathematicians of a specific truth. That is all it is. There is a naive idea that a proof is a purely mechanical series of steps that provides access to truth. Last I checked, this isn't so for the vast majority of proofs in math. Such a proof would be way too tedious to construct or check by mathematicians. And if it isn't checkable, how do we know it is actually true? Automated proofs are a subfield, and (again, last I checked) controversial because they can often not be checked by humans. So for example, if the proof doesn't convince other mathematicians, then it's not a a proof. Or it might convince other mathematicians and later turn out to be wrong after all. For more on the practical aspects of math, I highly recommend The Mathematical Experience. https://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395929687 https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav... I read it in German: https://www.amazon.com/Erfahrung-Mathematik-German-P-J-Davis/dp/3764329963 https://www.amazon.com/Erfahrung-Mathematik-German-P-J-Davis...
- AnonCoward42 4y ago
  > The proof is not a social construct. > The truth is. Yeah, that is how it feels like nowadays, however the truth is bound in a narrow set of assumptions. These assumptions are bound in reality even in mathematics (One apple is one apple, you add another one, you have two). And while there is an epistemologic level to reality, you would dismiss reality entirely by calling it a social construct. The details of how a truth is communicated is in a sense a social contruct, because communication as a whole is, however nobody would call it like that. It is maybe a small reminder that meddling with language for no apparent reason is a warning sign, but this is going a bit off-topic.
  - TheOtherHobbes 4y ago
    An apple is not an apple. An apple is a subjective construct that summarises the distinguishing features of a certain kind of object as it appears to our sense. To a non-human consciousness those features may be uninteresting, irrelevant, or incomprehensible, so they might not see apples at all. But they could see " "s, which we don't even have a concept for, never mind a word. And which we either ignore or possibly don't see at all. (Imagine perceiving complex networked relationships directly instead of having to access them through symbolic models.) There's no reason why math wouldn't be the same. From experiments we know that cats can't count, but they can distinguish sizes. So cat math likely wouldn't have integers as we know them, but would have some kind of size-based analogue. I have a theory this is why Hilbert's Project failed and you always end up with an incompleteness theorem. You cannot create an absolute internally consistent mathematics, because foundational axioms depend on subjective experience, not on objective logic. So you can define integers in various more and more obscure ways. But fundamentally you have to start with the subjective experience of "integer" as a concept that matters to you. And you can't prove a subjective experience objectively.
magicalhippo 4y ago
There's a difference between a mathematical proof and accepted truths. When you learn math at school, at first you're just told that these things are true, and so it becomes an accepted truth how addition works. Only much later can you go and verify the proof from the axioms. Similarly, if someone today relies on Fermat's last theorem to hold for some of their own work, they're unlikely to have verified the entirety of Wile's proof. Rather they lean upon the experts who have, and thus have accepted the truth that the proof holds.
ookdatnog 4y ago
I think you can make a philosophical argument that a fully formal proof, where every claim is traced all the way back to the axioms, is not a social construct. But when we say "proof", we usually don't mean "fully formal proof", and there are good reasons why: 1. Fully formal proofs didn't exist for most of the history of mathematics. 2. School by and large don't teach formal proofs, and most students are probably not aware of the existence of formal proofs. 3. Even today, most professional mathematicians never write formal proofs, except perhaps as an exercise during their education. So what do we mean by "proof" if it is not an argument that is exhaustively traced back to axioms? It's really no more than "that which the teacher/other mathematicians will accept as a convincing argument". When you learned a proof of the Theorem of Pythagoras in high school, you almost certainly didn't learn a fully formal proof. You learned a proof that, at some point, just implicitly got "cut off": the proof tree didn't work all the way back to the axioms but stopped at some point where the argument would become too tedious to continue laying out in full (without even being told that the proof got cut off: your teachers perhaps even told you that this was a rigorous argument). To write such a proof, you need to judge where the acceptable cut-off point is, which is entirely based on what other people will accept as good work. Hence, a social construct. edit: if you're not convinced that the proofs you learn in school/university aren't fully rigorous, I warmly recommend trying out a proof assistant like Coq, Agda, or Lean. Try to encode some well-known piece of mathematics. Euclid's Elements is a good candidate: working through it fully formally, you'll find huge omissions in the Elements immediately.
- dorchadas 4y ago
  > I think you can make a philosophical argument that a fully formal proof, where every claim is traced all the way back to the axioms, is not a social construct. I don't think you can make this claim really, precisely because the logic we accept as, well, logical, is a social construct. Different cultures across different places have had different ways of accepting what is valid in an argument. The methods of logic we consider valid are themselves social constructs, basically.

Huh? Mathemathical proof is not a social construct. This makes no sense.

by rapsey 4y ago

Huh? Mathemathical proof is not a social construct. This makes no sense.

hugh-avherald 4y ago

It's certainly a social construct, but it is not merely a social construct.

karmakurtisaani 4y ago

I suppose they mean what's commonly accepted as true and can be referred to as truths. No one can read all the proofs, so they have to trust others who have. There was that one example where a mathematician "proved" something terribly complicated using his own methods and terminology developed over several years. The truth value of that kind of proof is very much a social construct.

Agingcoder 4y ago
Yes, mochizuki and the abc conjecture. It was an interesting conundrum : he was a really good mathematician, not a crank so his funky proof couldn't be dismissed. However, people were wary of approaching the proof since it was risky career wise (it takes time, etc). You end up with a weird situation where something is probably true, but you won't know that until a trusted group of mathematicians have read it and said so.

uKVZe85V 4y ago

The mathematical truths are the fruit of the work. The social consensus is an implementation "detail". Yet it's the implementation we have. Does this make better sense now?

kingkawn 4y ago

And yet nonetheless you feel the need to object to this formulation publicly and have it considered by others

mpweiher 4y ago

The proof is not a social construct. The truth is. The proof is a mechanism to reach that consensus, by convincing other mathematicians of a specific truth. That is all it is. There is a naive idea that a proof is a purely mechanical series of steps that provides access to truth. Last I checked, this isn't so for the vast majority of proofs in math. Such a proof would be way too tedious to construct or check by mathematicians. And if it isn't checkable, how do we know it is actually true? Automated proofs are a subfield, and (again, last I checked) controversial because they can often not be checked by humans. So for example, if the proof doesn't convince other mathematicians, then it's not a a proof. Or it might convince other mathematicians and later turn out to be wrong after all. For more on the practical aspects of math, I highly recommend The Mathematical Experience. https://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395929687 https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav... I read it in German: https://www.amazon.com/Erfahrung-Mathematik-German-P-J-Davis/dp/3764329963 https://www.amazon.com/Erfahrung-Mathematik-German-P-J-Davis...

AnonCoward42 4y ago
> The proof is not a social construct. > The truth is. Yeah, that is how it feels like nowadays, however the truth is bound in a narrow set of assumptions. These assumptions are bound in reality even in mathematics (One apple is one apple, you add another one, you have two). And while there is an epistemologic level to reality, you would dismiss reality entirely by calling it a social construct. The details of how a truth is communicated is in a sense a social contruct, because communication as a whole is, however nobody would call it like that. It is maybe a small reminder that meddling with language for no apparent reason is a warning sign, but this is going a bit off-topic.
- TheOtherHobbes 4y ago
  An apple is not an apple. An apple is a subjective construct that summarises the distinguishing features of a certain kind of object as it appears to our sense. To a non-human consciousness those features may be uninteresting, irrelevant, or incomprehensible, so they might not see apples at all. But they could see " "s, which we don't even have a concept for, never mind a word. And which we either ignore or possibly don't see at all. (Imagine perceiving complex networked relationships directly instead of having to access them through symbolic models.) There's no reason why math wouldn't be the same. From experiments we know that cats can't count, but they can distinguish sizes. So cat math likely wouldn't have integers as we know them, but would have some kind of size-based analogue. I have a theory this is why Hilbert's Project failed and you always end up with an incompleteness theorem. You cannot create an absolute internally consistent mathematics, because foundational axioms depend on subjective experience, not on objective logic. So you can define integers in various more and more obscure ways. But fundamentally you have to start with the subjective experience of "integer" as a concept that matters to you. And you can't prove a subjective experience objectively.

magicalhippo 4y ago

There's a difference between a mathematical proof and accepted truths. When you learn math at school, at first you're just told that these things are true, and so it becomes an accepted truth how addition works. Only much later can you go and verify the proof from the axioms. Similarly, if someone today relies on Fermat's last theorem to hold for some of their own work, they're unlikely to have verified the entirety of Wile's proof. Rather they lean upon the experts who have, and thus have accepted the truth that the proof holds.

ookdatnog 4y ago

I think you can make a philosophical argument that a fully formal proof, where every claim is traced all the way back to the axioms, is not a social construct. But when we say "proof", we usually don't mean "fully formal proof", and there are good reasons why: 1. Fully formal proofs didn't exist for most of the history of mathematics. 2. School by and large don't teach formal proofs, and most students are probably not aware of the existence of formal proofs. 3. Even today, most professional mathematicians never write formal proofs, except perhaps as an exercise during their education. So what do we mean by "proof" if it is not an argument that is exhaustively traced back to axioms? It's really no more than "that which the teacher/other mathematicians will accept as a convincing argument". When you learned a proof of the Theorem of Pythagoras in high school, you almost certainly didn't learn a fully formal proof. You learned a proof that, at some point, just implicitly got "cut off": the proof tree didn't work all the way back to the axioms but stopped at some point where the argument would become too tedious to continue laying out in full (without even being told that the proof got cut off: your teachers perhaps even told you that this was a rigorous argument). To write such a proof, you need to judge where the acceptable cut-off point is, which is entirely based on what other people will accept as good work. Hence, a social construct. edit: if you're not convinced that the proofs you learn in school/university aren't fully rigorous, I warmly recommend trying out a proof assistant like Coq, Agda, or Lean. Try to encode some well-known piece of mathematics. Euclid's Elements is a good candidate: working through it fully formally, you'll find huge omissions in the Elements immediately.

dorchadas 4y ago
> I think you can make a philosophical argument that a fully formal proof, where every claim is traced all the way back to the axioms, is not a social construct. I don't think you can make this claim really, precisely because the logic we accept as, well, logical, is a social construct. Different cultures across different places have had different ways of accepting what is valid in an argument. The methods of logic we consider valid are themselves social constructs, basically.