[HN Gopher] How to read mathematics (2015)
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How to read mathematics (2015)
Author : penguin_booze
Score : 38 points
Date : 2022-05-27 18:35 UTC (4 hours ago)
(HTM) web link (www.people.vcu.edu)
(TXT) w3m dump (www.people.vcu.edu)
| Koshkin wrote:
| Good advice. Also applies to reading code.
| adamnemecek wrote:
| Math became easier to read when I realized that there are
| relatively few ideas in math and most things are just
| permutations thereof.
|
| The main ones are adjoint, norm, and fixed points. I wrote
| something about this https://github.com/adamnemecek/adjoint/
| atq2119 wrote:
| That's just one subfield of mathematics. During my PhD, I
| barely touched adjoints and fixed points. Though, many fields
| of math so indeed have this sort of repeating pattern. It's
| just not the same for all of them.
| adamnemecek wrote:
| What subfield do you think I'm talking about? Which subfield
| doesn't involve adjoints and fixed points?
| bollu wrote:
| Most "hard results" in any subfield of math cannot be
| derived from pure categorical nonsense like adjunctions. At
| some point, you have to roll up your sleeve and do the
| work.
|
| How would you prove the tychonoff theorem purely with
| adjunctions? How about the representation theorem of
| finitely presented abelian groups? And on and on and on...
| gnulinux wrote:
| Interesting post. In "Linear Algebra" section what do you mean
| by `a'`? Do you mean `inv(a)`? It seems like this was never
| explained in your own formalism.
| adamnemecek wrote:
| It's some Julia pseudo code. inv is inverse. Tick is adjoint.
| OccamsRazr wrote:
| As an experienced mathematician once told me:
|
| "The way I read papers is by first reading the abstract. Then I
| try to state the results and prove them myself. When I get stuck
| I go to the paper to see what I got wrong."
|
| Mileage may vary.
| anyfoo wrote:
| "Mathematics is not a spectator sport". No quote was more useful
| to get me through Uni, and enjoying it in the process. I'm a
| software engineer, not a mathematician.
|
| That being said, a decade later I still catch myself often
| "glancing" over equations in papers and textbooks, and have to
| force myself to really look at them and check that I indeed "got
| them". I don't know what it is, maybe I just need more
| training/habit around it. There's a tendency for me to half-
| consciously say to myself "yeahyeah I'll get it from reading the
| text" or "I'll get to it later", which usually does not work.
|
| For more important equations (Taylor, sinc function, all the
| variations of Fourier Series, Fourier Transform, DFT, DTFT) I
| actually write them down as flash cards in Anki and learn them
| verbatim. Yes, I have to understand them otherwise it's useless,
| but being able to just "make the equations appear" in my head to
| look at and work with them is invaluable.
|
| Even after understanding, I won't derive the Taylor Series myself
| (and even if I did, I would not always want to repeat that), so
| the old adage that understanding is better than rote memorization
| is useless here.
| tunesmith wrote:
| I actually just skimmed this and told my friend, "Hey, check this
| out! The sum of consecutive integers starting at 1 is the product
| of the final number and the number that is two before it!" So I
| totally ignored the lesson of the essay.
| alpple wrote:
| I think it's the product of the final number and the middle
| number. (given an odd length sequence)
| SOTGO wrote:
| I find that reading math at all is sometimes not the best
| approach. When working out of a textbook I often find it more
| constructive to attempt problems first and then use the text as a
| guide to help me solve the problem, particularly when the
| textbook is quite dense. For example, even after taking years of
| analysis I still find Rudin impossible to simply "read" because
| the mathematics is so condensed and difficult to follow.
| codethief wrote:
| When you say "Rudin" which one do you mean? Big Rudin? Baby
| Rudin?
| vector_spaces wrote:
| I know he gets a lot of hate, but I personally love Rudin's
| writing style. There are more chatty analysis textbooks like
| Abbott and Carothers, but the conciseness of Rudin plays nicest
| with the way I think, and the exercises (in Baby Rudin) are
| really wonderful. They're hard but I usually felt a genuine
| sense of accomplishment when I finished one
|
| Chattier authors are nice for providing context and intuition
| and sometimes details about the historical context, but I
| personally find them to be very distracting and a bit
| overwhelming. I don't like using them for much other than a
| reference or more casual reading. On the other hand, I loved
| reading a few sentences from Rudin that I didn't quite follow,
| then pulling out some pen & paper and doing a quick validation,
| or even going on a drive and munching on them until I
| understood
|
| That's me though. I'm glad that there seems to be no shortage
| of introductory analysis texts written in all kinds of styles
| so that folks can find the ones that work best for them.
|
| As an aside, I think it's a bummer that analysis classes often
| feel like hazing courses in the math curriculum, leading many
| mathematicians to despise it. I've been very lucky to have
| great analysis teachers, or at least ones that care very much
| about pedagogy over ruthless elitism, and conveying the beauty
| and fun that lies amid the ugly bit :)
|
| Also, re Rudin: his autobiography is certainly worth reading,
| if for no other reason than for his account of surviving the
| Anschluss (the Nazi annexation of Austria during WWII) as a
| young Jew. One of my favorite bits:
|
| "On the first day of school after the Anschluss several of our
| teachers and even some students strutted around in their shit-
| colored storm trooper uniforms. (The Nazi party had been
| illegal, but had obviously existed.) One of those was the gym
| teacher whom I had always disliked. He even had a pistol
| strapped to his belt. A few days later I heard that he had shot
| himself in the foot. This was one of the very few bits of
| cheerful news at the time."
|
| It's a fairly harrowing read, and perhaps some of his best
| writing overall
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