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Sheafification – The optimal path to mathematical mastery: The fast track (2022)

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Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#31

Post by jebarker »

falcor84 wrote: post #22
lsc4719 wrote: post #15

This is wrong. In Computer Science language, human learning works in greedy way. We make locally optimal decisions in life. We cannot learn something in globally optimal way. We learn something in locally optimal way. And by repeating that we can reach somewhere at some point.

Here's my take on this - you should try to organize your local space of affordances and rewards in such a way that locally optimal choices will over time get you closer towards your global objective.

My impression is that most people don't have a global objective.

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#33

Post by bwfan123 »

The author misunderstands knowledge and mastery.

Mastery comes from problem solving and practice - not from reading books. So, I would advise students to limit what you read, but spend a lot of time in problem solving. Get the basics in place. Start with euclid's elements and master that first.

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#35

Post by nathan_compton »

bwfan123 wrote: post #33

The author misunderstands knowledge and mastery. Mastery comes from problem solving and practice - not from reading books. So, I would advise students to limit what you read, but spend a lot of time in problem solving. Get the basics in place. Start with euclid's elements and master that first.

Weird!

I think Euclid is fine as a historical document and interesting in a broad sense, but its kind of silly to start with a document from 300 BCE when you could start with, for example, "How to Solve It" (Polya). And even that text could use a rewrite to make it much more readable.

I have completed substantial education in both mathematics and physics and I would say one of the weaknesses of the standard system of courses in physics is that it more or less recapitulates the development of physics in a historical shape, which substantially obscures mathematical structures which are shared between disciplines developed at different times. For example, unless you were lucky or very curious you might never appreciate that the bras and covectors relate to kets and vectors in fundamentally the same way. You might only have had a vague sense that physics involves making a lot of sandwiches.

Don't get me wrong, I'd be delighted if my kid's school broke out The Elements, but I just don't think its an obvious pedagogical strategy to start math instruction (self or otherwise) there.

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#36

Post by nathan_compton »

stared wrote: post #16

I am not sure if I agree with the list. I mean, one red flag is the frequent mention of Landau & Lifshitz. It is considered a "standard textbook", but I feel it stuck there by inertia. There are quite a few choices of both less boring and more insightful. (Back when I was reading such stuff, 20 years ago, the Feynman Lectures provided orders magnitude more insight. And fun.)

I dislike Feynman, frankly. So many of his weird hand-wavy metaphorical descriptions of shit obscure more than clarify the really important elements of the physics.

To name an example: Feynman is the source of the popular idea that in special relativity we can think of a particle as having a constant 4-vector with length c and that movement changes the direction of the four vector into the spatial directions, thus "slowing" the speed through time.

This is a very strange way of thinking about this stuff because the entire point of special relativity is that there is no objective state of affairs about velocity at all. It's meaningless to talk about the velocity of a single particle because velocity is a relative quantity. Also, I'm just generally suspicious of all this "hyperbolic rotation" stuff. I mean its true as far as the mathematical structure is concerned, but most of the time metaphors which try to get us to think of a minkowski space as being a lot like a normal 4d euclidean space confuse us or at least hide the real interesting structure, which is that in a minkowski space much of the 4d structure implied by a set of events is redundant. That is, spacetime is less than space and time together, not more.

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#37

Post by ceh123 »

jebarker wrote: post #30

2-3 seems wildly optimistic for reading that list and actually consolidating the knowledge. During my math PhD I ready a fraction of that number of books. It’d be more convincing if someone was provided as an example of “mastering mathematics” (whatever that means) through this curriculum and timeframe.

Even one of these topics I would say it would take most PhDs at least 2-3 years to “master”. I feel like at the end of my math PhD (5 years, 3 focused solely on my research area) I had just scratched the surface of mastery in my sub field, and that’s with 3 published papers.

I guess you’re right though, defining “mastery” is the key missing point here.

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#38

Post by srean »

nathan_compton wrote: post #36
stared wrote: post #16

I am not sure if I agree with the list. I mean, one red flag is the frequent mention of Landau & Lifshitz. It is considered a "standard textbook", but I feel it stuck there by inertia. There are quite a few choices of both less boring and more insightful. (Back when I was reading such stuff, 20 years ago, the Feynman Lectures provided orders magnitude more insight. And fun.)

I dislike Feynman, frankly. So many of his weird hand-wavy metaphorical descriptions of shit obscure more than clarify the really important elements of the physics. To name an example: Feynman is the source of the popular idea that in special relativity we can think of a particle as having a constant 4-vector with length c and that movement changes the direction of the four vector into the spatial directions, thus "s…

In a way Feynman was also the 3b1b of his generation. He was very good at coming up with and communicating with visual metaphors. To the laypeople at the receiving end, these give a strong illusion of understanding.

That's ok if you are not going to compute or design or build anything with it. But they are very inadequate when it is time to shut up and compute.

Feynman (and I am sure) Grant Sanderson could/can operate at a virtuoso level at both the visual imagery and the compute layers. But their popularity with the masses is because of the visual imagery they could conjure up.

On the other hand for those who can already compute for themselves, the metaphors can be a big help for building intuition as long they think in the same style.

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#39

Post by tomku »

marifjeren wrote: post #28

Not much attention given at all to explaining what order in which these should be read or what optimality means. This is just a list of books some guy is proud to have read

Lists like these are almost always more accurately titled "How I wish I'd been taught X" and are aimed squarely at a student who has never really existed, one who simply learns whatever is put in front of them and can therefore be steered by a heavy stack of "the best" books away from all the mistakes and dead ends and frustrations that real students face. Real pedagogy's tougher than that!

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#40

Post by math_dandy »

Not sure about these books as a self-study curriculum — their unifying theme seems to be that they require a reasonable level of mathematical maturity going in. But, they absolutely comprise an excellent “greatest hits” list of math books in the most influential subdisciplines. You’re guaranteed to learn a tonne if you study any one of these books.
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