[HN Gopher] A mathematical thought experiment for accepting the ...
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A mathematical thought experiment for accepting the continuum
hypothesis
Author : FillMaths
Score : 41 points
Date : 2024-07-03 15:25 UTC (7 hours ago)
(HTM) web link (jdh.hamkins.org)
(TXT) w3m dump (jdh.hamkins.org)
| FillMaths wrote:
| The pdf file is available at: https://arxiv.org/pdf/2407.02463
| drpossum wrote:
| I argue this should have been the link posted rather than to
| some blog that says very little other than to host the link
| Maxatar wrote:
| To be fair the blog is from the author of the paper.
| Maxatar wrote:
| I went into this hoping for a mathematical thought experiment,
| but rather this is merely a historical thought experiment in the
| sense of "Wouldn't it be nice of mathematicians accepted CH early
| on?". It seems the big selling point of accepting CH is that
| mathematicians would be less hesitant to use nonstandard
| analysis.
|
| For an actual thought experiment that rejects the continuum
| hypothesis, I rather enjoy the explanation found at:
|
| https://risingentropy.com/the-continuum-hypothesis-is-false/
| jerf wrote:
| That sort of argument makes me a nervous. One of my favorite
| mathematical quotes is a sort of related one about the Axiom of
| Choice, referenced and explained at
| https://math.stackexchange.com/a/787648: "The Axiom of Choice
| is obviously true, the well-ordering principle obviously false,
| and who can tell about Zorn's lemma?" That sounds like the
| "obviously false" branch of a similar debate about the
| continuum hypothesis.
| ithinkso wrote:
| I found that the sooner you ditch 'human intuition' in
| learning maths/physiscs the better you will be at it
| staunton wrote:
| I've generally found the opposite. Polemically, a true
| mathematician can write theorems where every single proof
| is riddled with errors (actual errors, not just "typos")
| but all results, building upon each other, are still true;
| a true physicist can tell you what the result of a
| calculation will be even if they are unable to actually do
| the calculation.
|
| Maybe you would call that "a mathematician's/physicist's
| intuition", rather than "human"?
| BalinKing wrote:
| Related, Terry Tao's blogpost on the development of
| mathematical intuition:
| https://terrytao.wordpress.com/career-advice/theres-more-
| to-...
| yantrams wrote:
| Thanks for sharing this blog. Tons of interesting stuff.
| codeflo wrote:
| I don't think this is a slam dunk. For this argument to work,
| the dart probability must be 100% for any function. This is
| supposed to be clear "intuitively", and then, by constructing a
| counterexample using the CH, it's concluded that the CH is
| false.
|
| But the space of functions from R to countable subsets of R is
| so vast (and so far removed from the physical world) that I
| don't think it's possible to have any "intuition" of what's
| possible in that space. And indeed, we see that there's a
| construction of a function f that doesn't conform to the
| "intuition". If there's an "intuitive" line of reasoning and a
| formal one, and they disagree, shouldn't we just conclude that
| our intuition is flawed?
| staunton wrote:
| > shouldn't we just conclude that our intuition is flawed?
|
| Alternatively, we might conclude that our intuition is right
| and instead our definition of real numbers isn't exactly what
| we want for some cases/questions.
| ajkjk wrote:
| Mostly I just find these arguments to be evidence that 'measure
| theory is not very interesting', that is, it's concerned with
| proving things about mathematical objects that you won't find
| in reality and therefore I don't care about.
|
| I wonder sometimes if there is a concrete version of the
| statement: 'there is an infinite number of interesting
| theorems', which would suggest that perhaps doing 'all the
| math' is not a good idea and we should only do the math which
| we find important.
|
| (of course, others would disagree that measure theory is
| unimportant, anyway. Shrug.)
| monadINtop wrote:
| You need measure theory for probability, economics, QFT and
| Physics, etc. And who is doing "all the math"? The vast
| majority of resarchers who "do math" are largely in PDEs and
| other fields that simply use the technology of math for
| "things that you find in reality" like engineering problems
| or machine learning and so forth. And most mathmaticians
| would agree that it is some of the most uninteresting and
| ugly kind of math.
|
| Whereas the relative minority of people who study really
| abstract things like say k-theory or large cardinals in set
| theory are largely doing it out of interest in it's intrinsic
| beauty. And this is especially true for idk, some esoteric
| subfield of tropical geometry or modal logic or something,
| who's relevance to "things you find in reality" are
| completely orthogonal as to the motivations of those people
| who chose to spend their lives uncovering the truths within
| them.
|
| Math research isn't about blindly marching from proof to
| proof by mechanical deduction with no conception of the
| larger picture like a uniform bubble spreading outwards, it
| is done by small communities of scholars who hack away at a
| specific nexus of interesting problems and structures for
| their own sake.
|
| Sometimes, like with spin bundles or lie algebras or non-
| abelian geometry, yeah you can apply it to "real" problems,
| but that's not how the theory was developed, and as a
| theoretical physicist I will tell you that you will find no
| greater blindness to the underlying structure or ugliness in
| the use of the technology than those people that exclusively
| wield the technology against "real" problems, instead of
| appreciating it for its own sake.
| ajkjk wrote:
| Well it is the theory that underpins those at the moment,
| but that doesn't say much about the counterfactual where it
| isn't.
|
| But I think I can say with confidence that none of those
| fields care about the fact that hitting a rational number
| out of the reals has probability 0. If they do something's
| wrong.
|
| Edit: oh wow your reply got a lot longer after I responded
| vitus wrote:
| I am not remotely convinced by this argument.
|
| The first flaw I see is that the author is imprecise by
| commingling probabilities (0%, 100%) with absolutes (possible,
| impossible, none, never, etc).
|
| > After all, probability-zero events do happen. Not a problem!
| Just pick two new real numbers! And if this fails, pick again!
|
| Probability-zero events happen all the time. The probability of
| getting any specific value selected uniformly at random from
| the unit interval (say, 0.232829) is zero.
|
| Probability-zero events should not be conflated with properties
| that exist nowhere.
|
| > We can now state that for any such mapping, none of the three
| reals is in the countable set assigned to the others. And this
| entails that we can prove that |(o)| > |o2|! In other words, we
| can prove that there are at least TWO cardinalities in between
| the reals and the naturals!
|
| That's... not how cardinalities work. Just because you have two
| sets with different elements does not mean they have different
| cardinalities. For instance, consider the set of integers {...,
| -1, 0, 1, 2, ...} vs the set of half-integers {..., -1/2, 1/2,
| 3/2, 5/2, ...}. These clearly have different elements, but you
| can easily construct a bijection between the two (just add 1/2
| to each element in your set of half-integers), so you can
| demonstrate that they have the same cardinality.
|
| > We define f(x) to be {y | y <= x}
|
| Um, no. This demonstrates the existence of one such mapping. It
| does not demonstrate that the set of such mappings covers any
| substantial portion of the entire space of possible mappings.
| drewcoo wrote:
| HN title is misleading.
| FillMaths wrote:
| Sorry, I tried my best. I wanted to mention the thought
| experiment part, since that is the most interesting bit. (But
| I'm not sure why it was misleading?)
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