11 ms·

To Settle Infinity Question, a New Law of Mathematics (2013)

122 points by okket 9y ago

NotSammyHagar 9y ago
This feels like the mathematical equivalent of string theory, an untestable and apparently unprovable 'set' of ideas has shown up and captured the minds of mathematicians. The impact of these questions is that it seems to block progress?
- rocqua 9y ago
  The reason to pay attention to this stuff is that it might (just might) affect other conjectures in mathematics. A very clear example here is the continuum hypothesis. It depends on which axiom you choose whether or not the hypothesis holds. If other hypotheses (e.g. Goldbach, Riemann) turn out to have different results in differing axiom systems, that tells us something about those conjectures. Specifically, it tells us that maybe the conjectures might not be that important.
  - openasocket 9y ago
    > If other hypotheses (e.g. Goldbach, Riemann) turn out to have different results in differing axiom systems, that tells us something about those conjectures. To be clear, you shouldn't (can't ?) have something like that Riemann is true under ZFC and false under ZFC + Continuity, because that would imply that ZFC + Continuity is inconsistent. You could have that Riemann is true under ZFC + Continuity but unprovable under ZFC, though.
    - rocqua 9y ago
      Indeed, if we presume ZFC + Continuity to be consistent, then your situation cannot happen. However, it could be that ZFC + Continuity has Goldbach true, and ZFC + Forcing has goldbach false. I should say I recall reading that if the Riemand hypothesis is false, then it is provably so in ZFC, hence my inclusion of the Goldbach conjecture. [1, 2] [1] https://mathoverflow.net/questions/79685/can-the-riemann-hypothesis-be-undecidable https://mathoverflow.net/questions/79685/can-the-riemann-hyp... [2] https://www.youtube.com/watch?v=O4ndIDcDSGc&ab_channel=Numberphile https://www.youtube.com/watch?v=O4ndIDcDSGc&ab_channel=Numbe...
Sniffnoy 9y ago
This sort of thing just strikes me as so pointless. How could one possibly decide? Obviously, if we're talking about new axioms that aren't provable from the old ones, you can't use those old ones to settle the matter. The axioms under discussion don't really affect normal math so you can't appeal to their usefulness there (unlike with, say, the axiom of choice, which famously was controversial for a while). And I don't think people can really have much of a claim to have any good intuition for such things, or else we'd have had such an axiom a long time ago. My bet is neither of these will become widely accepted. That said, if I had to bet on one of the two becoming accepted, I'd go with "V = ultimate L". Why? Because mathematics already has an axiom roughly analogous, that doesn't actually affect normal mathematics but prunes away sets we don't need so people don't waste their time with them. I'm talking, of course, about the axiom of foundation. It's like, why is that even in ZFC? Ordinary math doesn't actually care about the material content of sets; it really affects nothing. The only thing it does, is that, well, without it, non-well-founded sets are something of a problem, in that they could exist, or they could not, and you can't really prove much about them either way. There are non-well-founded set theories that replace the axiom of foundation with an axiom of antifoundation, allowing you to actually prove things about non-well-founded sets -- Peter Aczel wrote a good book on the subject -- but if you just take it out and don't put anything in to replace it, then, well, you just can't really say anything about the matter. The axiom of foundation prunes these irrelevant sets away and tells you to focus on actual math rather than this mystery that's not really much of a mystery at all. So I'd bet on another axiom along those lines becoming accepted, rather than one that does the opposite. After all, if we want to take the opposite perspective that anything that can exist ought to... well, then why not non-well-founded sets, too? (Note: Not a logician, my understanding of the two axioms being discussed is based pretty much entirely on this article, so the analogy I claim may not actually work as well as I think it does.)
- KGIII 9y ago
  They speak of the Philsophy of Mathematics. I typed a lot out and chose to delete it. In short, it is considered the highest order among Mathematicians. However, us Applied Mathematicians just get on with the real work and have pretty much figured out that they are insane. It's a bit like true random and infinity. It's best not to think about it to much and only useful for post doc bar conversations or impressing stoned Econ undergrads of your preferred gender. Even then, it's best not to think about it too much.
excalibur 9y ago
Title needs a (2013) at the end. In my opinion, trying to settle the matter by adopting a new axiom is a bit like proving that blue is the best color for cars by requiring all cars to be painted blue.
- xamuel 9y ago
  A different viewpoint: "adding a new axiom" is synonymous with "further restricting the things under discussion". Imagine biologists decide that "DAG" is the correct abstraction for parent-child relationships. Then they spend years trying to prove the obviously-true fact that parents can't have infinitely many children. Of course, they never can, because DAG nodes can have infinitely many children. Finally they realize they chose the wrong abstraction, they should've chosen "DAGs where all nodes have finitely many children". To suggest adding a new axiom to ZFC is to suggest that ZFC was an over-broad abstraction for capturing what things are supposed to be "sets".
pavpanchekha 9y ago
The article is fine as long as it goes but thinking of set theory as the "foundational laws" of mathematics is sort of a mid-20th century understanding of math. We have a better understanding of what axioms are for and how foundations should be structured. I am confident the author understands the more subtle framing I'd like to discuss, but I think it's worth describing here for readers who were confused. The main difference: this is not a new "law of mathematics" so much as a new law of set theory. No one but set theorists will be affected, and no other mathematicians care much. Foundations are not nearly as set as undergrad math teaches you. I don't mean the Axiom of Choice (which pretty much no one has any problems with anymore). However, various large cardinal axioms are pretty standard in advanced Algebraic Geometry (for example, the existence of Grothendiek Universes). Set theorists work with all sorts of large cardinal axioms all the time, and for a while it was fashionable to assume various generalized continuum hypotheses. Graph theorists, for some applications, assume "V=L", because that makes things tidy. It's fine, and no one really has a problem with it, nor do these disagreements really cause problems, even though some of these axioms are contradictory. (And even when various large cardinal axioms are assumed, often people work with an intuitive ill-founded set theory with the assumption that it can be made well-founded by ramifying using the large cardinals instead.) The 21-century understanding is something like: there's no need for there to be "ultimate" foundations. As mentioned, there is little impact on "ordinary" mathematics from different axioms of set theory, so maybe the set theorists should choose axioms of set theory, but that needn't affect other mathematicians. Obviously someone should study set theory, it's cool and there are interesting theorems and so on; and if they find that "V = ultimate L" is a useful axiom for their field of study, nothing wrong with that, much like commutative algebra is a fine field of study though of course non-commutative algebra is interesting too. But there's no realistic sense in which group theory is "defined in terms of" set theory—you can tell, because group theorists tend not to care about set theoretic axioms, and have no dog in choosing between "V=ultimate L" and forcing. The argument between "V = ultimate L" and the forcing axioms are a debate over what "set theory" should study, and each mathematician involved thinks that it should study the sort of things that they study—whether that means forcing or large cardinal axioms. Large cardinal axiom people have a well-developed kind of work they want to do: defining various forms of large cardinals, proving implications between them, and then discovering that all large cardinal axioms are totally ordered by proof strength. It's good, interesting work, and "V = ultimate-L" will really complete the research program they've been pursuing; or, proving independence results for various theorems, and a forcing axiom would really put the machinery they work with deep into the foundations of set theory, making that work much simpler and at the same time deeper. That, too, is a valuable field of research, in fact it has led to the only Fields medal in logic. I assure you that your personal understanding of sets will not be affected much by which axiom which mathematicians choose to adopt.
DonbunEf7 9y ago
Here's Woodin talking about his work in this area on a more technical level: https://www.youtube.com/watch?v=nVF4N1Ix5WI https://www.youtube.com/watch?v=nVF4N1Ix5WI
zaptheimpaler 9y ago
This is crazy. How about just doing what maths does anyways - state your assumptions? Prove stuff with V=ultimate L? Great, slap a "V = ultimate L => ..." on it. Prove stuff with forcing axiom? Cool, "forcing axiom implies ..." The entire premise of asking "whats the one true axiom?" makes no sense. Its not like the real world places any restrictions on math - its a self-contained universe. So obviously you are free to pick whatever imaginary universe you would like to study. If we ever get to a point where we can directly test which of the 2 axioms leads to more useful theories in the real world, great, but that seems pretty unlikely given they come up when talking about infinity. The obvious truth seems to be that infinity could have a few subtly different meanings depending on how you define it - and thats totally fine because the real world does not contain infinities, its a purely mental construct.
- placebo 9y ago
  Amen. Not a mathematician, but this perfectly articulates my thoughts on the subject. >the real world does not contain infinities I have no idea what "real world" even means, but I'd say that infinity does not exist within anything that can be conceptualized.
  - thomasahle 9y ago
    > infinity does not exist within anything that can be conceptualized. Yet math is consistently being conceptualized? Enabling clear thinking about concepts outside of the real world is the while point of math.
    - Chris2048 9y ago
      > math is consistently being conceptualized What does this mean?
    - placebo 9y ago
      >Yet math is consistently being conceptualized? Sure, how else could we capture it or convey it to others? The thing is that I don't see it as that different than conceptualizing things in the "real world". In the latter, conceptualization is done on input of the senses, while in math conceptualization is done on inputs from a higher level of abstraction. In both cases, something is defined and then manipulated for something useful. It's just that I believe that whatever is defined automatically becomes a finite concept otherwise it could not be grasped and manipulated, and a finite infinity seems problematic...
xelxebar 9y ago
“What truly infinite objects exist in the real world?” I'm not sure what "infinite object" means, but I do find it interesting that we can do the math anyway using only a finite number of symbols.
cooltechguy 9y ago
super crazy stuff
enturn 9y ago
If infinity is being boundless then to make anything infinity have the bounds as unknown. A practical use is in Haskell's lazy evaluation. Although maybe I'm missing the point completely.
m3kw9 9y ago
My dad would answer when I was a kid, at the end, there is just a wall, but then I’d ask what what’s beyond the wall and that was the end of the conversation
Chris2048 9y ago
I don't really understand this. "Infinity" just means without finite quantity, doesn't it? I can, theoretically, count for infinity - there are no end to the theoretical numbers. If I can prove a value can be reduced arbitrarily close to a limit, we can say something is equal to that limit for any finite resolution (for any resolution of "closeness to the limit" there is a precision in the calculation capable of delivering within that resolution) Hence, I can say "F(x) = 3 for any finite resolution", so if no infinities exist, then none are needed for mathematics. They are a calculation artifact that represents a single function over arbitrary levels of resolution.

122 points by okket 9y ago

This feels like the mathematical equivalent of string theory, an untestable and apparently unprovable 'set' of ideas has shown up and captured the minds of mathematicians. The impact of these questions is that it seems to block progress?

rocqua 9y ago
The reason to pay attention to this stuff is that it might (just might) affect other conjectures in mathematics. A very clear example here is the continuum hypothesis. It depends on which axiom you choose whether or not the hypothesis holds. If other hypotheses (e.g. Goldbach, Riemann) turn out to have different results in differing axiom systems, that tells us something about those conjectures. Specifically, it tells us that maybe the conjectures might not be that important.
- openasocket 9y ago
  > If other hypotheses (e.g. Goldbach, Riemann) turn out to have different results in differing axiom systems, that tells us something about those conjectures. To be clear, you shouldn't (can't ?) have something like that Riemann is true under ZFC and false under ZFC + Continuity, because that would imply that ZFC + Continuity is inconsistent. You could have that Riemann is true under ZFC + Continuity but unprovable under ZFC, though.
  - rocqua 9y ago
    Indeed, if we presume ZFC + Continuity to be consistent, then your situation cannot happen. However, it could be that ZFC + Continuity has Goldbach true, and ZFC + Forcing has goldbach false. I should say I recall reading that if the Riemand hypothesis is false, then it is provably so in ZFC, hence my inclusion of the Goldbach conjecture. [1, 2] [1] https://mathoverflow.net/questions/79685/can-the-riemann-hypothesis-be-undecidable https://mathoverflow.net/questions/79685/can-the-riemann-hyp... [2] https://www.youtube.com/watch?v=O4ndIDcDSGc&ab_channel=Numberphile https://www.youtube.com/watch?v=O4ndIDcDSGc&ab_channel=Numbe...

Sniffnoy 9y ago

This sort of thing just strikes me as so pointless. How could one possibly decide? Obviously, if we're talking about new axioms that aren't provable from the old ones, you can't use those old ones to settle the matter. The axioms under discussion don't really affect normal math so you can't appeal to their usefulness there (unlike with, say, the axiom of choice, which famously was controversial for a while). And I don't think people can really have much of a claim to have any good intuition for such things, or else we'd have had such an axiom a long time ago. My bet is neither of these will become widely accepted. That said, if I had to bet on one of the two becoming accepted, I'd go with "V = ultimate L". Why? Because mathematics already has an axiom roughly analogous, that doesn't actually affect normal mathematics but prunes away sets we don't need so people don't waste their time with them. I'm talking, of course, about the axiom of foundation. It's like, why is that even in ZFC? Ordinary math doesn't actually care about the material content of sets; it really affects nothing. The only thing it does, is that, well, without it, non-well-founded sets are something of a problem, in that they could exist, or they could not, and you can't really prove much about them either way. There are non-well-founded set theories that replace the axiom of foundation with an axiom of antifoundation, allowing you to actually prove things about non-well-founded sets -- Peter Aczel wrote a good book on the subject -- but if you just take it out and don't put anything in to replace it, then, well, you just can't really say anything about the matter. The axiom of foundation prunes these irrelevant sets away and tells you to focus on actual math rather than this mystery that's not really much of a mystery at all. So I'd bet on another axiom along those lines becoming accepted, rather than one that does the opposite. After all, if we want to take the opposite perspective that anything that can exist ought to... well, then why not non-well-founded sets, too? (Note: Not a logician, my understanding of the two axioms being discussed is based pretty much entirely on this article, so the analogy I claim may not actually work as well as I think it does.)

KGIII 9y ago
They speak of the Philsophy of Mathematics. I typed a lot out and chose to delete it. In short, it is considered the highest order among Mathematicians. However, us Applied Mathematicians just get on with the real work and have pretty much figured out that they are insane. It's a bit like true random and infinity. It's best not to think about it to much and only useful for post doc bar conversations or impressing stoned Econ undergrads of your preferred gender. Even then, it's best not to think about it too much.

excalibur 9y ago

Title needs a (2013) at the end. In my opinion, trying to settle the matter by adopting a new axiom is a bit like proving that blue is the best color for cars by requiring all cars to be painted blue.

xamuel 9y ago
A different viewpoint: "adding a new axiom" is synonymous with "further restricting the things under discussion". Imagine biologists decide that "DAG" is the correct abstraction for parent-child relationships. Then they spend years trying to prove the obviously-true fact that parents can't have infinitely many children. Of course, they never can, because DAG nodes can have infinitely many children. Finally they realize they chose the wrong abstraction, they should've chosen "DAGs where all nodes have finitely many children". To suggest adding a new axiom to ZFC is to suggest that ZFC was an over-broad abstraction for capturing what things are supposed to be "sets".

pavpanchekha 9y ago

The article is fine as long as it goes but thinking of set theory as the "foundational laws" of mathematics is sort of a mid-20th century understanding of math. We have a better understanding of what axioms are for and how foundations should be structured. I am confident the author understands the more subtle framing I'd like to discuss, but I think it's worth describing here for readers who were confused. The main difference: this is not a new "law of mathematics" so much as a new law of set theory. No one but set theorists will be affected, and no other mathematicians care much. Foundations are not nearly as set as undergrad math teaches you. I don't mean the Axiom of Choice (which pretty much no one has any problems with anymore). However, various large cardinal axioms are pretty standard in advanced Algebraic Geometry (for example, the existence of Grothendiek Universes). Set theorists work with all sorts of large cardinal axioms all the time, and for a while it was fashionable to assume various generalized continuum hypotheses. Graph theorists, for some applications, assume "V=L", because that makes things tidy. It's fine, and no one really has a problem with it, nor do these disagreements really cause problems, even though some of these axioms are contradictory. (And even when various large cardinal axioms are assumed, often people work with an intuitive ill-founded set theory with the assumption that it can be made well-founded by ramifying using the large cardinals instead.) The 21-century understanding is something like: there's no need for there to be "ultimate" foundations. As mentioned, there is little impact on "ordinary" mathematics from different axioms of set theory, so maybe the set theorists should choose axioms of set theory, but that needn't affect other mathematicians. Obviously someone should study set theory, it's cool and there are interesting theorems and so on; and if they find that "V = ultimate L" is a useful axiom for their field of study, nothing wrong with that, much like commutative algebra is a fine field of study though of course non-commutative algebra is interesting too. But there's no realistic sense in which group theory is "defined in terms of" set theory—you can tell, because group theorists tend not to care about set theoretic axioms, and have no dog in choosing between "V=ultimate L" and forcing. The argument between "V = ultimate L" and the forcing axioms are a debate over what "set theory" should study, and each mathematician involved thinks that it should study the sort of things that they study—whether that means forcing or large cardinal axioms. Large cardinal axiom people have a well-developed kind of work they want to do: defining various forms of large cardinals, proving implications between them, and then discovering that all large cardinal axioms are totally ordered by proof strength. It's good, interesting work, and "V = ultimate-L" will really complete the research program they've been pursuing; or, proving independence results for various theorems, and a forcing axiom would really put the machinery they work with deep into the foundations of set theory, making that work much simpler and at the same time deeper. That, too, is a valuable field of research, in fact it has led to the only Fields medal in logic. I assure you that your personal understanding of sets will not be affected much by which axiom which mathematicians choose to adopt.

DonbunEf7 9y ago

Here's Woodin talking about his work in this area on a more technical level: https://www.youtube.com/watch?v=nVF4N1Ix5WI https://www.youtube.com/watch?v=nVF4N1Ix5WI

zaptheimpaler 9y ago

This is crazy. How about just doing what maths does anyways - state your assumptions? Prove stuff with V=ultimate L? Great, slap a "V = ultimate L => ..." on it. Prove stuff with forcing axiom? Cool, "forcing axiom implies ..." The entire premise of asking "whats the one true axiom?" makes no sense. Its not like the real world places any restrictions on math - its a self-contained universe. So obviously you are free to pick whatever imaginary universe you would like to study. If we ever get to a point where we can directly test which of the 2 axioms leads to more useful theories in the real world, great, but that seems pretty unlikely given they come up when talking about infinity. The obvious truth seems to be that infinity could have a few subtly different meanings depending on how you define it - and thats totally fine because the real world does not contain infinities, its a purely mental construct.

placebo 9y ago
Amen. Not a mathematician, but this perfectly articulates my thoughts on the subject. >the real world does not contain infinities I have no idea what "real world" even means, but I'd say that infinity does not exist within anything that can be conceptualized.
- thomasahle 9y ago
  > infinity does not exist within anything that can be conceptualized. Yet math is consistently being conceptualized? Enabling clear thinking about concepts outside of the real world is the while point of math.
  - Chris2048 9y ago
    > math is consistently being conceptualized What does this mean?
  - placebo 9y ago
    >Yet math is consistently being conceptualized? Sure, how else could we capture it or convey it to others? The thing is that I don't see it as that different than conceptualizing things in the "real world". In the latter, conceptualization is done on input of the senses, while in math conceptualization is done on inputs from a higher level of abstraction. In both cases, something is defined and then manipulated for something useful. It's just that I believe that whatever is defined automatically becomes a finite concept otherwise it could not be grasped and manipulated, and a finite infinity seems problematic...

xelxebar 9y ago

“What truly infinite objects exist in the real world?” I'm not sure what "infinite object" means, but I do find it interesting that we can do the math anyway using only a finite number of symbols.

cooltechguy 9y ago

super crazy stuff

enturn 9y ago

If infinity is being boundless then to make anything infinity have the bounds as unknown. A practical use is in Haskell's lazy evaluation. Although maybe I'm missing the point completely.

m3kw9 9y ago

My dad would answer when I was a kid, at the end, there is just a wall, but then I’d ask what what’s beyond the wall and that was the end of the conversation

Chris2048 9y ago

I don't really understand this. "Infinity" just means without finite quantity, doesn't it? I can, theoretically, count for infinity - there are no end to the theoretical numbers. If I can prove a value can be reduced arbitrarily close to a limit, we can say something is equal to that limit for any finite resolution (for any resolution of "closeness to the limit" there is a precision in the calculation capable of delivering within that resolution) Hence, I can say "F(x) = 3 for any finite resolution", so if no infinities exist, then none are needed for mathematics. They are a calculation artifact that represents a single function over arbitrary levels of resolution.