81–90 of 216 posts
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And Mathematics is Computer Science. Took me most of a lifetime to realize this.
I'd argue for the converse: Computer Science is Mathematics
I remember when I was a student a friend of mine telling me this guy was giving a seminar and he'd just completed day one and everyone was really excited he was going to prove FLT. Of course, the guy in question was Andrew Wiles. He then spent months patching up problems they found prior to publication and finally the whole thing got published. It was a hugely exciting thing when you were studying mathematics. All of…
As a CS undergrad at Berkeley in the 90's I took an upper division math class in which we worked through the "old school" proof which was brand new and exciting then. Pretty much everyone else in the class was a math grad student. I don't think I understood more than 20% of the material! :)
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> no house of cards As I understand TFA, from a formalist’s perspective, this is not necessarily the case. People were building on swathes of mathematics that seem proven and make intuitive sense, but needed formal buttressing. > _actually experts_ at a deep and rigorous technical field Seeing as the person you’re addressing was a mathematics graduate student, I’m sure they know this.
> Seeing as the person you’re addressing was a mathematics graduate student, I’m sure they know this. The OP (u/boothby) was not the person I was addressing (u/bell-cot).
> If I could give my many-decades-ago younger self some advice for math grad school
> The experts are in agreement that the important ideas are robust enough to withstand knocks like this, but the details of what is actually going on might not actually be where you expect them to be. Past researcher in pure math here. The big problem is that mathematicians are notorious for not providing self-contained proofs of anything because there is no incentive to do so and authors sometimes even seem proud to…
Speaking as a current researcher in pure math -- you're right, but I don't think this is easily resolved. Math research papers are written for other specialists in the field. Sometimes too few details are provided; indeed I commonly gripe about this when asked to do peer review; but to truly provide all the details would make papers far, far longer. Here is an elementary example, which could be worked out by anyone w…
WTF
>i.e. the left-side is big-O of the right.
Oh.
This makes me wonder if there'll ever be a significant bug discovered in Lean itself, breaking past formalisation work.
So I have wondered about PdF's possible thinking.
Now, the degenerate case of n=2 is just the Pythagorean Theorem
c^2 = a^2 + b^2
Now, the PT is successful because the square b^2 can be "squished" over the corner of a^2, leaving a perfect square c^2. [Let's let the a^n part be the bigger of the two.]
5^2 = 4^2 + 3^2
25 = 16 + 9
25 = 16 + (4 + 4 + 1)
Now, for cubes, we now know that a cube cannot be "squished" onto another cube's corner in such a way that makes a bigger cube.
I'm not up for breaking out my (diminishing) algebra right now (as it's a brain off day), but that b^3 cube would need to break down into three flat sides, three linear edges, and the corner.
This fits my physical intuition of the problem and seems to me to be a possible way that PdF might have visualized the cubic form. Now, I have zero mathematical intuition about such things (or necessary math skills either), but the physical nature of the problem, plus the fact that we now know that the cubic and beyond don't work, leads me to wonder if this is an interesting approach. It also makes me curious to know why it isn't, if that is indeed the case, (which I assume is probable).
As to the n=4 and beyond, I would just hand-wave them away and say something like, "Well, of course they are more fractally impossible", which by that I mean that a real mathematician would have to say exactly why the n=3 case failing means that the n>=4 case would also not work. (My guess here is that there just becomes less and less possibility of matching up higher-dimensional versions of a two-term addition.)
Anyway, just a thought I had some years ago. I even built a lego model of the bits of cube spreading out along the corner of the a^3 cube.
I would enjoy learning why I'm so utterly wrong about this, but I found it a fun mental exercise some years ago that's been swimming around in my brain.
Thanks in advance if there are any takers around here. :-)
[And, my favorite Horizon episode is 'Freak Wave'.]
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Speaking as a current researcher in pure math -- you're right, but I don't think this is easily resolved. Math research papers are written for other specialists in the field. Sometimes too few details are provided; indeed I commonly gripe about this when asked to do peer review; but to truly provide all the details would make papers far, far longer. Here is an elementary example, which could be worked out by anyone w…
>Here is an elementary example, which could be worked out by anyone with a good background in high school math WTF >i.e. the left-side is big-O of the right. Oh.
If you're interested in this stuff at all, check out some code. Example: https://github.com/ImperialCollegeLondon/FLT/blob/main/FLT/M... Also check out the blueprint, which describes the overall structure of the code: https://imperialcollegelondon.github.io/FLT/blueprint/ I'm very much an outside observer, but it is super interesting to see what Lean code looks like and how people contribute to it. Great thing is tha…
Most (larger) Lean projects still have "unit tests". Those might be, e.g., trivial examples and counter examples to some definition, to make sure it isn't vacuous.
I remember when I was a student a friend of mine telling me this guy was giving a seminar and he'd just completed day one and everyone was really excited he was going to prove FLT. Of course, the guy in question was Andrew Wiles. He then spent months patching up problems they found prior to publication and finally the whole thing got published. It was a hugely exciting thing when you were studying mathematics. All of…
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[flagged]
I just meant in the context of math skills, but I definitely acknowledge that it was poor phrasing on my part.