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I no longer understand my PhD dissertation

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Re: I no longer understand my PhD dissertation

#241

Post by cokernel »

efaref wrote: post #216
wfo wrote: post #141

Earlier quoted context omitted.

It would be really nice if all it took to understand difficult mathematics were some easy programming tricks. The problem with looking at old code is you forget what is going on or what the purpose of different components are. The problem with looking at old mathematics is that it is genuinely very difficult to understand. You work very hard to be an expert in a field and get to a level where you can read a cutting-e…

Leslie Lamport wrote a paper on this subject over 20 years ago. http://research.microsoft.com/en-us/um/people/lamport/pubs/l... As far as I can tell, nobody has really adopted any of his proposals yet, and most papers continue to be written in pseudo-prose. Remember, mathematics used to be written in full prose (e.g. "the square of the hypotenuse is equal to the sum of the squares of the other two sides") and it took…

The structure of Doron Zeilberger's proof of the Alternating Sign Matrix Conjecture is reminiscent of this, though possibly not directly inspired by it: http://www.combinatorics.org/ojs/index.php/eljc/article/view...

Re: I no longer understand my PhD dissertation

#242

Post by Hockenbrizzle »

tfgg wrote: post #34
closure wrote: post #11

This does not surprise me in the least. Math was always extremely easy for me growing up. Up through my first differential equations class I found almost everything trivial to learn (the one exception is that I always found proving things difficult). I made the mistake of minoring in math and that slowly killed my enjoyment of it. Once I got to differential geometry and advanced matrix theory it all just became too a…

During my PhD I found I had to occasionally lock myself in a conference room with a bunch of papers and all the whiteboards to do a sort of "deep dive", in order to get everything into my head and push things forward. Things that were a bit rusty quickly came back after a little bit of effort. At the end I'd emerge with a bunch of photos of whiteboards with clear intermediate steps, and then I'd write it up as a shor…

When I was studying, I went to two schools, one using semester system and one using quarter system.

With the semester system I had plenty of time to read the material and really explore the mathematics behind what I was learning. With the quarter system I was so busy running around and catching up in different classes that I never really had the time available to sit down and do that. It sucked. I think the quarter system kind of 'beat' the interest out of me because now it is hard to get back into that state of mind.

semester > quarter :(

Re: I no longer understand my PhD dissertation

#243

Post by timroy »

chipsy wrote: post #19

It speaks to how finite we are around "knowledge." At the moment we reach understanding, we experience a sophomoric feeling of confidence. But as it fades farther and farther from our working memory, we become less fluent and more hesitant. The emerging pattern becomes one of "I can understand these interesting concepts, but it takes a lot of work and they don't last, so I have to choose to understand practically and…

You might also check out spaced repetition software. I've started putting anything I want to remember for the long haul into org-drill, an SRS package for Emacs, though Anki and Mnemosyne are more well known. I even schedule articles that I want to return to.

Re: I no longer understand my PhD dissertation

#244

Post by mmarx »

Certhas wrote: post #230
xiaq wrote: post #190

Earlier quoted context omitted.

> Programming has existed for, say, 50-100 years. We have recorded mathematical history going back thousands, with contributions from most of the most brilliant human beings to ever exist. Mathematics with a solid logic foundation has also existed only for the past century, and writing programs is actually equivalent to doing a mathematical proof as some have already pointed out. The actual problem is that when progr…

> writing programs is actually equivalent to doing a mathematical proof Only in the sense that writing in brainfuck is equivalent to writing in Python...

>> writing programs is actually equivalent to doing a mathematical proof

> Only in the sense that writing in brainfuck is equivalent to writing in Python...

Actually, in the very concrete sense that a program is a proof for the forumala that is equivalent to its type[0].

[0] https://en.wikipedia.org/wiki/Curry–Howard_correspondence

Re: I no longer understand my PhD dissertation

#245

Post by randcraw »

rjzzleep wrote: post #56
gjulianm wrote: post #42

Earlier quoted context omitted.

> I think that's because mathematical papers place too much value on terseness and abstraction over exposition and intuition. If it only happened on mathematical papers... It is all over the place, in books, classes, etc. Few professors tell you what are they doing and why; they just throw formulas, theorems and symbols at you. Intuition is not given its necessary importance, first because it's hard to grasp some con…

In Germany you get to your advanced courses for the last two years of High school. I picked Math and Physics. We did a bunch of linear regression(with maple?) i believe. And some manual differential equations. I was one of the best in that class, but when I asked what we need this stuff for the other people in the class looked at me and said the following: "if you ask this question you're in the wrong class" I went o…

> Why can't we just provide a simple real life example first and then go on explaining the details?

But sometimes there is no real life example. Fundamentally, math is abstract. Yes, mercifully, its models often have analogues in nature, making its purpose utilitarian and intuitive. But sometimes no analogue exists. As in much of modern physics, in math, often you have only abstraction.

I think that's why math is difficult to learn. Without compelling illustrations based in the physical world the student must follow the concepts and proofs using the rigor of math's legal transformations, fortified only by the faith that these formalisms will sustain truth. But too often the practitioner must remain oblivious to the utility and implications of both the end and the means.

Re: I no longer understand my PhD dissertation

#247

Post by true_religion »

xiaq wrote: post #232
Certhas wrote: post #230

Earlier quoted context omitted.

> writing programs is actually equivalent to doing a mathematical proof Only in the sense that writing in brainfuck is equivalent to writing in Python...

True, but the point is that in principle you can write very detailed, fully formal proofs, in exactly the same manner programs are written. The rest of my comment then discusses why it is not written that way. Doing formal proofs manually is like writing turing machine instructions. Computer-assisted proving gives you higher level tools, but apparently it is still not powerful enough to be accepted in mainstream. And…

That's a very illuminating analogy.

Users of high-level languages, typically do not think about what the compiler is doing, and if in fact they do not know the compiler action--often the only way to find out is by reading the compilers source code.

This is akin to looking a math paper seeing "it trivially follows..." and your only recourse to find out why it so trivially follows is to get a mathematics degree.

Re: I no longer understand my PhD dissertation

#248

Post by reikonomusha »

vidarh wrote: post #169
nanis wrote: post #9

First, I am not sure Functional Analysis is as obscure as some other areas. But, second, this just shows, once again, that one ought never to use "clearly," "obviously" etc in proofs. It is the same principle as writing programs so they are easier for the next programmer to read. That person may be you.

I don't read maths papers, but in CS papers too, words like "clearly" and "obviously" etc. serve as red flags to me that screams "hand-waving or big gaps coming up". Often it's understandable short-cuts, but often it also turns out that the author has left out very substantial chunks of knowledge, or sometimes clearly don't understand why they got the results they did. In CS papers there's an additional red flag: Mat…

Wait, let me get this straight. It's a red flag when a computer science paper has math in it? Computer science (in the asymptotic limit) is math. And the papers ideally should read like math papers. Otherwise it's not CS.

Re: I no longer understand my PhD dissertation

#249

Post by reikonomusha »

fiatjaf wrote: post #109

The author almost realized the much more important conclusion of the fact he lived. He shouldn't conclude the article by asking "what is the purpose of studying maths?" and then giving an three stupid answers. He should have asked: is this actually "knowledge" as they say academia brings to society? Is the money researchers earn being well spent? Did I actually deserve to be remunerated by this piece of work no one u…

It's not that no one can understand it. It's that someone who has been out of math research for half a decade can't understand it.

Re: I no longer understand my PhD dissertation

#250

Post by xiaq »

true_religion wrote: post #247
xiaq wrote: post #232

Earlier quoted context omitted.

True, but the point is that in principle you can write very detailed, fully formal proofs, in exactly the same manner programs are written. The rest of my comment then discusses why it is not written that way. Doing formal proofs manually is like writing turing machine instructions. Computer-assisted proving gives you higher level tools, but apparently it is still not powerful enough to be accepted in mainstream. And…

That's a very illuminating analogy. Users of high-level languages, typically do not think about what the compiler is doing, and if in fact they do not know the compiler action--often the only way to find out is by reading the compilers source code. This is akin to looking a math paper seeing "it trivially follows..." and your only recourse to find out why it so trivially follows is to get a mathematics degree.

There is still a key difference.

With programming the information is always there. If you don't understand a higher level language but do understand the language in which the compiler is implemented, you can always read the compiler.

The "it trivially follows", the information is simply missing. Gaining a mathematics degree will often give you the intellectual power of finding the missing pieces yourself, but the only sure way is to ask the authors themselves.

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