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https://www.smbc-comics.com/comic/the-largest-number-2 Tangentially related comic
https://www.youtube.com/watch?v=RkP_OGDCLY0
Theoretical mathematics is often absurd, maybe that's why I like the sketch so much.
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I don’t like this framing because it implies any piece of pure math becoming useful is just a matter of time, and I see no reason to suspect this is the case.
> I see no reason to suspect this is the case Isn’t history enough reason?
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Others have addressed why it matters (or doesn't matter) when viewed from outside mathematics. But within mathematics, unsolved problems usually matter for two reasons: (1) When lots of really smart people spend lots of time trying to solve something, and fail to do so, it becomes even more interesting for other smart people. A well-known example is the Collatz Conjecture[0] which, by most accounts is a meaningless p…
This is an excellent answer. I have a mostly unrelated question. In the case of the Collatz conjecture, it seems all but proved: > If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. There are lots of hist…
I think the difference here is that math is much more abstract than physics. The concepts backing math are built off of very low level abstractions about the world. For math, over a long period of time more and more relationships and rules were deduced from what had already been induced from reality. And if you CAN deduce, you should likely, as it provides a proof for some idea or concept that induction never could (but only if your previous inductions and deductions were accurate).
Physics on the other hand, cannot deduce as readily. It is primarily based in the realm of gathering more and more data from the world and observing physical relationships firsthand. This cannot be done in abstract math because abstract math does not exist in reality. There is nothing to observe, it is mostly abstractions based on lower level observations in reality. For example, you can measure the effects of gravity firsthand, but you cannot measure infinity, a mathematical abstraction. Infinity does not exist in reality. It is simply a useful abstraction for things that are too large or small for us to meaningfully measure.
I'm not trying to be flippant, although it may come off that way: why does any of this matter?
Great article. Since it looks like a lot of folks are interested in this article, some extra background. First, what is forcing? The article actually has a great description of ultrapowers (a key part of the construction) but it goes by a little fast, so you might like Tim Chow's "A beginner's guide to forcing" [1] which does a good job not only laying out the mathematical details at a high level, but also really cle…
Minor quible: Just because a potential axiom is independent of ZF(C) doesn't make it necessarily "okay" to add. Potential axioms can be unsound, for example if they prove new / untrue Sigma_1 statements. As an example, even in the likely circumstance that ¬Con(ZFC) is independent of ZFC, it wouldn't be "okay" to add ¬Con(ZFC). While the resulting system would be consistent, and does have models, the resulting system is unsound (in the sense of Tarski) because it asserts the existence of natural numbers that have no "written form" (i.e. the existance of natural numbers that are larger than any term you can write to denote a natural number).
That said, Martin's axiom, like the CH (or the axiom of choice), does not have any arithmetic consequences, and thus doesn't fall into this category of problematic axioms.
I'm not trying to be flippant, although it may come off that way: why does any of this matter?
We don’t know but looking into the past hints at the future. Cantor, Hilbert and Gödel gave us Church who gave us Turing. Turing and Flowers gave us the machines as well as the theory. All of them put together gave us type systems and types are how you formally prove that your 747 software is free of, if not all bugs, then at least certain large classes of error. There is a clear line of connections from Cantor (1890…
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"Didn't you just conclude that it's impossible to have a set of all real numbers?" Cantor's diagonalization proof proves that it's impossible to list all the real numbers with a list of size aleph-0, which is the cardinality of the set of natural numbers. The forcing proof is an attempt to prove you also can't do it with the a list of the size aleph-1, which is the size of the power set of aleph-0. It purports to pro…
I think this is a typo: > aleph-0, which is the cardinality of the set of real numbers. aleph-0 is the cardinality of the set of natural numbers, and (as you say) is not the cardinality of the real numbers.
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"Didn't you just conclude that it's impossible to have a set of all real numbers?" Cantor's diagonalization proof proves that it's impossible to list all the real numbers with a list of size aleph-0, which is the cardinality of the set of natural numbers. The forcing proof is an attempt to prove you also can't do it with the a list of the size aleph-1, which is the size of the power set of aleph-0. It purports to pro…
> aleph-1, which is the size of the power set of aleph-0 In ZFC, aleph-1 is not the size of the powerset of aleph-0. Instead, aleph-1 is the next larger cardinal number after aleph-0. The size of the powerset of aleph-0 is called continuum or beth-1. In ZFC, we can show that the size of the set of real numbers equals continuum, but it is not possible to relate continuum to a specific aleph-k (though some can be ruled…
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This is an excellent answer. I have a mostly unrelated question. In the case of the Collatz conjecture, it seems all but proved: > If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. There are lots of hist…
No amount of empirical data is ever "good enough" for the mathematical standard of proof. (It may be enough for mathematicians to "believe" something in some informal sense, but not enough to consider it "proved".) You can see some examples at the answers to these questions; maybe at least one of them will be interesting to you: - https://math.stackexchange.com/questions/514/conjectures-tha... - https://math.stackexc…
I've often wondered "Why can't we do something like that, but for all instances of things like the Collatz conjecture?" Of course, it's computationally infeasible. But that raises the question: Suppose the four-color theorem was only able to prove 95% of cases rather than 100%. Isn't it at least sort of valuable to do so? Or is that last 5% all the difference?
I suppose what bugs me about math is, physicists are proved wrong all the time. Mathematicians are rarely proved mistaken, because they construct assumptions that you can't disagree with. There's no chance for empirical data to falsify one's assertions.
But that's an odd kind of debate, and not too productive. I just can't help but wonder about stuff like this.
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Others have addressed why it matters (or doesn't matter) when viewed from outside mathematics. But within mathematics, unsolved problems usually matter for two reasons: (1) When lots of really smart people spend lots of time trying to solve something, and fail to do so, it becomes even more interesting for other smart people. A well-known example is the Collatz Conjecture[0] which, by most accounts is a meaningless p…
This is an excellent answer. I have a mostly unrelated question. In the case of the Collatz conjecture, it seems all but proved: > If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. There are lots of hist…
But also, it's just the nature of mathematics, proving what is true is just as important (if not more so) than knowing what is true.
From a practical perspective, doing mathematics, you often don't really grok why something is true until you prove it.
From a philosophical perspective, there's not much in the universe we can know for sure (as you mention with the gravitational constant), but a mathematical proof we know for sure. That's the beauty of it.
if X is true and the implication (X => Y) is true, then it must be that Y is true. By definition of a logical implication.
Add in a whole bunch of axioms and suddenly you can take these simple logical atoms and build them into a field that spans the working tools of every engineer and beautiful arcane subjects like set theory. And every single thing we prove, we know to be true, as we build it from logical atoms that can be traced back to definitions and axioms.
It "could be" that for certain types of matter or at certain distances, the gravitational constant changes. It cannot be that there exists a bijection between the natural numbers and the real numbers under ZFC. Cantor's diagonalization argument *proves* it.