7 ms·

I'm curious. Did either of you ever notice the implicit philosophical assumptions that you have to make to come to the conclusion that one infinity can be large

by btilly 1y ago

I'm curious. Did either of you ever notice the implicit philosophical assumptions that you have to make to come to the conclusion that one infinity can be larger than another? Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way.

dragonwriter 1y ago
> Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way. That’s not at all true at the level where you are dealing with different infinities, usually, which tends to come after the (usually, fairly early) part dealing with proofs and the fact that all mathematics is dealing with “unprovable intellectual leaps” which are encoded into axioms, and everything in math which is provable is only provable based on a particular chosen set of axioms. It may be true that math beyond that basic level doesn’t make a point of going back and explicitly reviewing that point, but it is just kind of implicit in everything later.
- btilly 1y ago
  I guarantee that a naive presentation doesn't actually include the axioms, and doesn't address the philosophical questions dividing formalism from constructivism. Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.
  - godelski 1y ago
    > I guarantee that a naive presentation doesn't actually include the axioms But you said "modern math courses". Are you now talking about a casual conversation? I mean the OP's story is that his wife just liked listening to him talk about his passions. > Uncountable need not mean more. Sure. But that doesn't mean that there aren't differing categories. However you slice it, we can operate on these things in different ways. Real or not the logic isn't consistent between these things but they do fall out into differing categories. If you're trying to find mistakes in the logic does it not make sense to push it at its bounds? Look at the Banach-Tarski Paradox. Sure, normal people hear about it and go "oh wow, cool." But when it was presented in my math course it was used as a discussion of why we might want to question the Axiom of Choice, but that removing it creates new concerns. Really the "paradox" was explored to push the bounds of the axiom of choice in the first place. They asked "can this axiom be abused?" And the answer is yes. Now the question is "does this matter, since infinity is non-physical? Or does it despite infinity being non-physics?" You seem to think mathematicians, physicists, and scientists in general believe infinities are physical. As one of those people, I'm not sure why you think that. We don't. I mean math is a language. A language used because it is pedantic and precise. Much the same way we use programming languages. I'm not so sure why you're upset that people are trying to push the bounds of the language and find out what works and doesn't work. Or are you upset that non-professionals misunderstand the nuances of a field? Well... that's a whole other conversation, isn't it...
    - btilly 1y ago
      Your guesses at what I seem to think are completely off base and insulting. When I say "modern math courses", I mean like the standard courses that most future mathematicians take on their way to various degrees. For all that we mumble ZFC, it is darned easy to get a PhD in mathematics without actually learning the axioms of ZFC. And without learning anything about the historical debates in the foundations of mathematics.
      - godelski 1y ago
        Honestly it's difficult to understand exactly what you're arguing. Because I understand laymen not understanding your argument about infinities not being real (and even many HN users don't understand code is math bit a CS degree doesn't take you far in math. Some calc and maybe lin alg) but are we concerned about laymen? I too am frustrated by nonexperts having strong opinions and having difficulties updating them, but that's not a culture problem. We're on HN and we know the CS stereotypes, right? If instead you're talking about experts then I learned about what you're talking about in my Linear 2 course in a physics undergrad and have seen the topic appear many times since even outside my own reading of set theory. The axiom of choice seems to have even entered more main stream nerd knowledge. It's very hard to learn why AoC is a problem without learning about how infinities can be abused. But honestly I don't know any person that's even an amateur mathematician that thinks infinities are physical
        btilly 1y ago
        The fact that you think I'm talking about the axiom of choice, demonstrates that you didn't understand what I'm talking about. I would also be willing to bet a reasonable sum of money that this topic did not come up in your Linear 2 course in physics undergrad. The arguments between the different schools of philosophy in math are something that most professional mathematicians are unaware of. Those who know about them, generally learned them while learning about either the history of math, or the philosophy of math. I personally only became aware of them while reading https://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395929687 https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav.... I didn't learn more about the topic until I was in grad school, and that was from personal conversations. It was never covered in any course that I took on, either in undergraduate or graduate schools. Now I'm curious. Was there anything that I said that should have been said more clearly? Or was it hard to understand because you were trying to fit what I said into what you know about an entirely unrelated debate about the axiom of choice?
  - zozbot234 1y ago
    The "philosophical questions" dividing formalism from constructivism are greatly overstated. The point of having those degrees of undecidability or uncountability is precisely to be able to say things like "even if you happen to be operating under strong additional assumptions that let you decide/count X, that still doesn't let you decide/count Y in general." That's what formalism is: a handy way of making statements about what you can't do constructively in the general case. To be fair, constructivists tend to prefer talk about different "universes" as opposed to different "sizes" of sets, but that's all it is: little more than a mere difference in terminology! You can show equiconsistency statements across these different points of view.
    - btilly 1y ago
      Yes, you can show such equiconsistency statements. As Gödel proved, for any set of classical axioms, there is a corresponding set of intuitionistic axioms. And if the classical axioms are inconsistent, then so is the intuitionistic equivalent. (Given that intuitionistic reasoning is classically valid, an inconsistency in the intuitionistic axioms trivially gives you one in the classical axioms.) So the care that intuitionists take does not lead to any improvement in consistency. However the two approaches lead to very different notions of what it means for something to mathematically exist. Despite the formal correspondences, they lead to very different concepts of mathematics. I'm firmly of the belief that constructivism leads to concepts of existence that better fit the lay public than formalism does.
Etherlord87 1y ago
You don't need an implicit philosophical assumption, you just need to define what an infinity is and the comparison method.
- btilly 1y ago
  Here's a hint. When someone makes a reference to something that was actively debated for decades, and you're not familiar with said debates, you should probably assume that you're missing some piece of relevant knowledge. https://plato.stanford.edu/entries/mathematics-constructive/ https://plato.stanford.edu/entries/mathematics-constructive/ is one place that you could start filling in that gap.
- nextaccountic 1y ago
  This looks like a philosophical stance in the philosophy of mathematics actually, and it's called formalism
edanm 1y ago
What leaps are "unprovable"? I'm curious, that doesn't sound right. For sure there are valid arguments on whether or not to use certain axioms which allow or disallow some set theoretical constructions, but given ZFC, is there anything that follows that is unprovable?
- btilly 1y ago
  When you say "given ZFC", you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean. In particular, you have made sufficient assumptions to prove that almost all real numbers that exist can never be specified in any possible finite description. In what sense do they exist? You also wind up with weirder things. Such as well-specified finite problems that provably have a polynomial time algorithm to solve...but for which it is impossible to find or verify that algorithm, or put an upper bound on the constants in the algorithm. In what sense does that algorithm exist, and is finite? Does that sound impossible? An example of an open problem whose algorithm may have those characteristics is an algorithm to decide which graphs can be drawn on a torus without any self-crossings. If our notion of "exists" is "constructable", all possible mathematical things can fit inside of a countable universe. No set can have more than that.
  - tempfile 1y ago
    You are being very cryptic. Are you trying to say that the existence of uncountable sets requires the axiom of choice? If you are, that's false. If you aren't, I'm not sure what you are trying to say.
    - ogogmad 1y ago
      He never mentioned the Axiom of Choice. I think he articulated his opinion clearly enough. It's his own subjective value judgement.
      - tempfile 1y ago
thadt 1y ago
Probably not. But this one time we had an argument and I made a statement along the lines of "I'm right, naturally." She went irrational. I lost the argument. QED
- btilly 1y ago
  LOL If she laughs at that kind of thing, I can see why you married her.
N0isRESFe8GXmqR 1y ago
[dead]
ogogmad 1y ago
It might be relevant to look at this: https://home.sandiego.edu/~shulman/papers/jmm2022-complementary.pdf https://home.sandiego.edu/~shulman/papers/jmm2022-complement... Also this: https://arxiv.org/pdf/1212.6543 https://arxiv.org/pdf/1212.6543 Assuming you haven't looked at these already, of course.
- btilly 1y ago
  I had already read the second. I'm not so enthused about the first.
mensetmanusman 1y ago
Don’t worry, we have only decided that there are two sizes of Infinitis- normal ones and really big ones.

by btilly 1y ago

dragonwriter 1y ago

> Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way. That’s not at all true at the level where you are dealing with different infinities, usually, which tends to come after the (usually, fairly early) part dealing with proofs and the fact that all mathematics is dealing with “unprovable intellectual leaps” which are encoded into axioms, and everything in math which is provable is only provable based on a particular chosen set of axioms. It may be true that math beyond that basic level doesn’t make a point of going back and explicitly reviewing that point, but it is just kind of implicit in everything later.

btilly 1y ago
I guarantee that a naive presentation doesn't actually include the axioms, and doesn't address the philosophical questions dividing formalism from constructivism. Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.
- godelski 1y ago
  > I guarantee that a naive presentation doesn't actually include the axioms But you said "modern math courses". Are you now talking about a casual conversation? I mean the OP's story is that his wife just liked listening to him talk about his passions. > Uncountable need not mean more. Sure. But that doesn't mean that there aren't differing categories. However you slice it, we can operate on these things in different ways. Real or not the logic isn't consistent between these things but they do fall out into differing categories. If you're trying to find mistakes in the logic does it not make sense to push it at its bounds? Look at the Banach-Tarski Paradox. Sure, normal people hear about it and go "oh wow, cool." But when it was presented in my math course it was used as a discussion of why we might want to question the Axiom of Choice, but that removing it creates new concerns. Really the "paradox" was explored to push the bounds of the axiom of choice in the first place. They asked "can this axiom be abused?" And the answer is yes. Now the question is "does this matter, since infinity is non-physical? Or does it despite infinity being non-physics?" You seem to think mathematicians, physicists, and scientists in general believe infinities are physical. As one of those people, I'm not sure why you think that. We don't. I mean math is a language. A language used because it is pedantic and precise. Much the same way we use programming languages. I'm not so sure why you're upset that people are trying to push the bounds of the language and find out what works and doesn't work. Or are you upset that non-professionals misunderstand the nuances of a field? Well... that's a whole other conversation, isn't it...
  - btilly 1y ago
    Your guesses at what I seem to think are completely off base and insulting. When I say "modern math courses", I mean like the standard courses that most future mathematicians take on their way to various degrees. For all that we mumble ZFC, it is darned easy to get a PhD in mathematics without actually learning the axioms of ZFC. And without learning anything about the historical debates in the foundations of mathematics.
    - godelski 1y ago
      Honestly it's difficult to understand exactly what you're arguing. Because I understand laymen not understanding your argument about infinities not being real (and even many HN users don't understand code is math bit a CS degree doesn't take you far in math. Some calc and maybe lin alg) but are we concerned about laymen? I too am frustrated by nonexperts having strong opinions and having difficulties updating them, but that's not a culture problem. We're on HN and we know the CS stereotypes, right? If instead you're talking about experts then I learned about what you're talking about in my Linear 2 course in a physics undergrad and have seen the topic appear many times since even outside my own reading of set theory. The axiom of choice seems to have even entered more main stream nerd knowledge. It's very hard to learn why AoC is a problem without learning about how infinities can be abused. But honestly I don't know any person that's even an amateur mathematician that thinks infinities are physical
      - btilly 1y ago
        The fact that you think I'm talking about the axiom of choice, demonstrates that you didn't understand what I'm talking about. I would also be willing to bet a reasonable sum of money that this topic did not come up in your Linear 2 course in physics undergrad. The arguments between the different schools of philosophy in math are something that most professional mathematicians are unaware of. Those who know about them, generally learned them while learning about either the history of math, or the philosophy of math. I personally only became aware of them while reading https://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395929687 https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav.... I didn't learn more about the topic until I was in grad school, and that was from personal conversations. It was never covered in any course that I took on, either in undergraduate or graduate schools. Now I'm curious. Was there anything that I said that should have been said more clearly? Or was it hard to understand because you were trying to fit what I said into what you know about an entirely unrelated debate about the axiom of choice?
- zozbot234 1y ago
  The "philosophical questions" dividing formalism from constructivism are greatly overstated. The point of having those degrees of undecidability or uncountability is precisely to be able to say things like "even if you happen to be operating under strong additional assumptions that let you decide/count X, that still doesn't let you decide/count Y in general." That's what formalism is: a handy way of making statements about what you can't do constructively in the general case. To be fair, constructivists tend to prefer talk about different "universes" as opposed to different "sizes" of sets, but that's all it is: little more than a mere difference in terminology! You can show equiconsistency statements across these different points of view.
  - btilly 1y ago
    Yes, you can show such equiconsistency statements. As Gödel proved, for any set of classical axioms, there is a corresponding set of intuitionistic axioms. And if the classical axioms are inconsistent, then so is the intuitionistic equivalent. (Given that intuitionistic reasoning is classically valid, an inconsistency in the intuitionistic axioms trivially gives you one in the classical axioms.) So the care that intuitionists take does not lead to any improvement in consistency. However the two approaches lead to very different notions of what it means for something to mathematically exist. Despite the formal correspondences, they lead to very different concepts of mathematics. I'm firmly of the belief that constructivism leads to concepts of existence that better fit the lay public than formalism does.

Etherlord87 1y ago

You don't need an implicit philosophical assumption, you just need to define what an infinity is and the comparison method.

btilly 1y ago
Here's a hint. When someone makes a reference to something that was actively debated for decades, and you're not familiar with said debates, you should probably assume that you're missing some piece of relevant knowledge. https://plato.stanford.edu/entries/mathematics-constructive/ https://plato.stanford.edu/entries/mathematics-constructive/ is one place that you could start filling in that gap.
nextaccountic 1y ago
This looks like a philosophical stance in the philosophy of mathematics actually, and it's called formalism

edanm 1y ago

What leaps are "unprovable"? I'm curious, that doesn't sound right. For sure there are valid arguments on whether or not to use certain axioms which allow or disallow some set theoretical constructions, but given ZFC, is there anything that follows that is unprovable?

btilly 1y ago
When you say "given ZFC", you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean. In particular, you have made sufficient assumptions to prove that almost all real numbers that exist can never be specified in any possible finite description. In what sense do they exist? You also wind up with weirder things. Such as well-specified finite problems that provably have a polynomial time algorithm to solve...but for which it is impossible to find or verify that algorithm, or put an upper bound on the constants in the algorithm. In what sense does that algorithm exist, and is finite? Does that sound impossible? An example of an open problem whose algorithm may have those characteristics is an algorithm to decide which graphs can be drawn on a torus without any self-crossings. If our notion of "exists" is "constructable", all possible mathematical things can fit inside of a countable universe. No set can have more than that.
- tempfile 1y ago
  You are being very cryptic. Are you trying to say that the existence of uncountable sets requires the axiom of choice? If you are, that's false. If you aren't, I'm not sure what you are trying to say.
  - ogogmad 1y ago
    He never mentioned the Axiom of Choice. I think he articulated his opinion clearly enough. It's his own subjective value judgement.
    - tempfile 1y ago

thadt 1y ago

Probably not. But this one time we had an argument and I made a statement along the lines of "I'm right, naturally." She went irrational. I lost the argument. QED

btilly 1y ago
LOL If she laughs at that kind of thing, I can see why you married her.

N0isRESFe8GXmqR 1y ago

[dead]

ogogmad 1y ago

It might be relevant to look at this: https://home.sandiego.edu/~shulman/papers/jmm2022-complementary.pdf https://home.sandiego.edu/~shulman/papers/jmm2022-complement... Also this: https://arxiv.org/pdf/1212.6543 https://arxiv.org/pdf/1212.6543 Assuming you haven't looked at these already, of course.

btilly 1y ago
I had already read the second. I'm not so enthused about the first.

mensetmanusman 1y ago

Don’t worry, we have only decided that there are two sizes of Infinitis- normal ones and really big ones.