btilly

28 ms·

481.
by btilly 1y ago
I was responding to this statement of yours, "I don't think that humans philosophically reject non-constructive results." Some of the humans who have thought about it do reject them. Some of the humans who have thought abou
482.
by btilly 1y ago
It does not matter what your best understanding of Cantor's diagonalization argument is. In some mathematical systems it means, "there are more reals than natural numbers", and in others it means, "the reals encode self-
483.
by btilly 1y ago
We can create a countable list that contains every possible description that can ever be created. For example just write down numbers in base ASCII, using a programmable markup language (like TeX) that lets us represent anything that we wan
484.
by btilly 1y ago
Please see my comment at https://news.ycombinator.com/item?id=44271589 for my explanation.
485.
by btilly 1y ago
As I pointed out at https://news.ycombinator.com/item?id=44271589 , there are systems that can accept Cantor's argument, without concluding that there are more reals than rational numbers. As you point out, there are ma
486.
by btilly 1y ago
My understanding of how Hilbert meant it is summed up in this quote from him: "Mathematics is a game played according to certain simple rules with meaningless marks on paper." I think that in part because I read Constance Reid&#
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by btilly 1y ago
I believe that you're fine with it, simply because that is what you're familiar with. And if you'd grown up with a different way of thinking about these problems, then you wouldn't be fine with it. Personally, I can work
488.
by btilly 1y ago
It is certainly easier to prove interesting theorems with formalism. You don't get caught up with such basic things like whether or not it is always possible to tell that one real number is bigger than another. But formalism leads to h
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by btilly 1y ago
That does seem extremely relevant. And is a mirror of the fundamental insight behind nonstandard analysis. Which is that that any set containing the integers that follows some set of axioms, has a nonstandard model that follows a nonstandar
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by btilly 1y ago
Thank you, fixed. And that is why I did think that. I only play with the ideas of Lisp. I've never really had to use it. So I looked at a Lisp example, saw something that looked like it was functioning as a comment, then used that comm
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by btilly 1y ago
If you enjoy such questions, I highly recommend https://www.amazon.com/Indistinguishable-Magic-Robert-L-Forw... . The best speed for interstellar travel with technologies that current theory says should be within our reach c
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by btilly 1y ago
Spoken like a true formalist. It doesn't really have to mean anything when we say that the reals are a larger set than the natural numbers - that's just the conclusion of the game that we are playing. What fraction of people who &
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by btilly 1y ago
I wish that I had specific suggestions. My overall wish that more people understood why, in intuitionist mathematics, uncountable means "self-referential" and not "more". No infinite set can have "more" element
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Peano arithmetic is enough, because Peano arithmetic encodes computation(math.stackexchange.com)
246 points by btilly 1y ago|141 comments
495.
by btilly 1y ago
This is a stack overflow question that I turned into a blog post. It covers both the limits of what can be proven in the Peano Axioms, and how one would begin bootstrapping Lisp in the Peano Axioms. All of the bad jokes are in the second se
496.
by btilly 1y ago
The current one big bill does a 5 year reversion back to the old rules, and then goes back to the problem existing. So it is the same idea as the 2017 bill. See https://exactera.com/resources/what-one-big-beautiful-bill
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by btilly 1y ago
I think it would be very good if they can make this work. I suspect that we do something not entirely unlike this, and that is why spaced repetition is so good for stuffing things into our long term memories.
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by btilly 1y ago
In order to see it, you must recognize the ways in which he fooled you. People would generally rather be fooled again than face the thought that they were fooled at some point in the past. And the more that they have been fooled, the strong
499.
by btilly 1y ago
The idea that information can be hyperlinked together predated HyperCard by decades. It goes back to https://www.theatlantic.com/magazine/archive/1945/07/as-we-m... , which was written in 1945. The same e
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by btilly 1y ago
I've always thought of "transcendental" as number theory language, though I can see how someone could argue that it is field language. But the Galois group of a field extension definitely is field language.
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by btilly 1y ago
That we can't square the circle comes from pi being transcendental. The result that you're thinking of is Galois' proof that there is no algebraic formula forroots of 5th degree polynomials.
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by btilly 1y ago
I found https://x.com/gm8xx8/status/1925768687618773079 to be a little more understandable summary of what was actually shown. Any Diophantine equation can be reduced to one of at most 11 variables and degree at m
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by btilly 1y ago
If you like the phrase, you should read https://fs.blog/chestertons-fence/ to get the full context on what it means.
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by btilly 1y ago
Can we at this distance tell the difference between bipolar, mercury poisoning, and repressed homosexual? He was also responsible for the execution of a couple of dozen people. These executions were connected to his position as master of th
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by btilly 1y ago
It makes race conditions easier to hit, and that will require multi-threaded Python to be written with more care to achieve the same level of reliability.
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by btilly 1y ago
Process creation should be fast on all Unices. If it isn't, then the lowly shell script (heavily used in Unix) is going to perform very poorly.
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by btilly 1y ago
According to https://www.theverge.com/24066646/ai-electricity-energy-watt... a reasonable estimate is half a percent of global energy usage by 2027. On the one hand, this is a lot of energy. On the other, that's i
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by btilly 1y ago
Hilbert's hopes, of course, were dashed by Gödel. However Hilbert's philosophy, named Formalism, became dominant in mathematics. In this philosophy, mathematics is just a formal game played by formal rules. When we say that someth
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by btilly 1y ago
Ah. Good point.
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by btilly 1y ago
You may have missed the same subtlety that I did. Because pi is irrational, the functions are different at all integers. Therefore, in the total order, these two functions cannot have the same order. That still doesn't resolve which on

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