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I no longer understand my PhD dissertation

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Re: I no longer understand my PhD dissertation

#271

Post by stuxnet79 »

peterbraden wrote: post #156

Could part of it just be because Mathematical notation is just so bad? It's more of a shorthand than an actual tool of conveying meaning. So much context goes into establishing what a notated equation means - and that context is now gone.

I always complain about this. While I'd say I have an affinity for math, it was much more difficult to get caught up with my peers in university than let's say programming because there was just so much context / assumed knowledge required to make sense out of what the professor was stating that it took me forever to understand even the most trivial things. To be mathematically mature constant exposure is imperative. As to how to improve mathematical notation so that it is not so subjective or context dependent - I haven't a clue - seems to me like almost everyone is alright with the status quo. Not a good thing in my opinion.

Re: I no longer understand my PhD dissertation

#272

Post by stuxnet79 »

EvanPlaice wrote: post #256
xiaq wrote: post #250

Earlier quoted context omitted.

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 onl…

At least in programming it's possible to step into a function and/or work at multiple levels of abstraction. Getting a math degree just to be able to fill in the gaps created where 'it trivially follows' is equivalent to being required to memorize the API for a framework because clearly defined documentation is hidden behind a paywall. so called 'intellectual power' has little to do with it. What you're describing is…

Well said. Memorizing the API is fundamental to success in most branches of math - but this erects a barrier that a lot of able individuals are unable to cross. I've always thought this was a problem but I almost never hear of anybody complaining about this so I figured it was just me. Languishing in the established system is usually interpreted as "sour grapes".

Re: I no longer understand my PhD dissertation

#273

Post by JabavuAdams »

joshvm wrote: post #139
JabavuAdams wrote: post #123

Earlier quoted context omitted.

1) Vector3 computeNewtonianGravity(float massA, Vector3 positionA, float massB, Vector3 positionB); Which way does the returned force vector point? Towards mass A, or towards mass B? 2) Vector3 UnitVectorWithDirection(Vector3 originalVector); What does this function do when the magnitude (length) of originalVector is zero? 3) float ArcTan(float x); What is the allowable range of values for x? ...

The sibling comment about renaming is good - particularly with regard to the first function. However I think the other two are better suited to documentation rather than trying to make the function signature explain itself. Sooner or later you'll just remember that it takes a float, not the variable name. Certainly calling things like "x_returnsNaNIfNot0to1" works, but I find it a bit ugly and it gets complicated if…

Was just looking at the Oculus SDK math library, and they have some really neat ideas...

For one, handling rotation sign (is positive clockwise or counter-clockwise), left-handed vs. right-handed coordinates, etc. through c++ template parameters.

Re: I no longer understand my PhD dissertation

#274

Post by wfo »

xiaq wrote: post #190
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…

> 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…

Mathematical papers will not come with a formal proof until formal proof systems "auto-prove" button is strong enough to do everything for them and it can come as an afterthought that takes 5 minutes.

Formal proofs do nothing to help understand the mathematics, all they do is help to check for errors in logic -- have you ever read a formal proof? They are leagues more obtuse and difficult to understand than any piece of mathematics from any field.

A program is technically truth-value equivalent to doing a proof, but the resulting document/piece of text are MILES apart. The proof is an explanation in plain language to others who understand the topic why something is true. The formal proof is a sequence of opaque inference rules that the computer promises is equivalent to a real proof.

It's the difference between understanding a formula and plugging it into your calculator. The calculator is more likely to compute correctly, but it gives you no help when it comes to actually understanding mathematics.

Mathematicians as a community completely reject the notion of using formal proofs (while sometimes begrudgingly accepting they may be useful to check our work) because they completely miss the point of what we are actually doing.

Re: I no longer understand my PhD dissertation

#275

Post by fiatjaf »

reikonomusha wrote: post #249
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.

How many have read it?

Re: I no longer understand my PhD dissertation

#276

Post by fiatjaf »

argonaut wrote: post #262
fiatjaf wrote: post #206

Earlier quoted context omitted.

Nothing would be better, considering this article.

Except that some % of PhDs go on to be professors (who do a real service), and every once in a while you get a PhD student whose work probably contributes hundreds of millions of value to society. And every couple years or so, you have a PhD student who contributes billions to society in value. Not to mention PhDs usually have to TA (teach students), which accounts for some of their pay.

There's no example of a PhD work that contributed hundreds of millions of value to society.

(I could say there's no way to measure "value to society", in fact this concept means nothing, but I agree on settling on "an enourmous amount of productive capital to someone, not exactly the society").

Re: I no longer understand my PhD dissertation

#277

Post by ezy »

cokernel wrote: post #241
efaref wrote: post #216

Earlier quoted context omitted.

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...

"subsubsubsublemma" kind of cracked me up. He seems very dedicated to not numbering the nested schema in this proof. :-)

Re: I no longer understand my PhD dissertation

#278

Post by argonaut »

fiatjaf wrote: post #276
argonaut wrote: post #262

Earlier quoted context omitted.

Except that some % of PhDs go on to be professors (who do a real service), and every once in a while you get a PhD student whose work probably contributes hundreds of millions of value to society. And every couple years or so, you have a PhD student who contributes billions to society in value. Not to mention PhDs usually have to TA (teach students), which accounts for some of their pay.

There's no example of a PhD work that contributed hundreds of millions of value to society. (I could say there's no way to measure "value to society", in fact this concept means nothing, but I agree on settling on "an enourmous amount of productive capital to someone, not exactly the society").

A one minute google search reveals:

Radioactivity. http://www.nobelprize.org/nobel_prizes/themes/physics/curie/

Nash equilibrium. http://rbsc.princeton.edu/sites/default/files/Non-Cooperativ...

Emergence of digital circuit design. https://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_a...

A pretty ridiculous thing to claim.

Re: I no longer understand my PhD dissertation

#279

Post by pc2g4d »

stuxnet79 wrote: post #272
EvanPlaice wrote: post #256

Earlier quoted context omitted.

At least in programming it's possible to step into a function and/or work at multiple levels of abstraction. Getting a math degree just to be able to fill in the gaps created where 'it trivially follows' is equivalent to being required to memorize the API for a framework because clearly defined documentation is hidden behind a paywall. so called 'intellectual power' has little to do with it. What you're describing is…

Well said. Memorizing the API is fundamental to success in most branches of math - but this erects a barrier that a lot of able individuals are unable to cross. I've always thought this was a problem but I almost never hear of anybody complaining about this so I figured it was just me. Languishing in the established system is usually interpreted as "sour grapes".

It strikes me as a collective means for mathematicians to reduce entry into their field and thus to increase their own salaries. Maybe you don't have to get a "math license" to practice math, but you have to somehow acquire a huge body of knowledge that is almost never spelled out fully in the mathematics literature. This will prevent many people from becoming mathematicians and thus make math a more exclusive and lucrative endeavor.

To be cynical about it.

I guess that's not much different from most fields' use of specialized vocabulary, etc. There are many informal methods of dissuading people from competing in your own little corner of the labor market.

Re: I no longer understand my PhD dissertation

#280

Post by Houshalter »

tokenadult wrote: post #39

An interesting read. But I think the author should have explicitly written out the point he is really making: you can't be too careful about making your writing clear, even to yourself. I recall reading (I'd point to the book with a link if I could remember in what book I read this) that mathematicians who occasionally write expository articles on mathematics for the general public are often told by their professiona…

This is a really common experience. See Explainers Shoot High, Aim Low!: http://lesswrong.com/lw/kh/explainers_shoot_high_aim_low/

>We miss the mark by several major grades of expertise. Aiming for outside academics gets you an article that will be popular among specialists in your field. Aiming at grade school (admittedly, naively so) will hit undergraduates. This is not because your audience is more stupid than you think, but because your words are far less helpful than you think. You're way way overshooting the target. Aim several major gradations lower, and you may hit your mark.

This phenomenon has a name, the Illusion of Transparency: http://lesswrong.com/lw/ke/illusion_of_transparency_why_no_o...

Also see some of the other posts there on the issue, it's quite interesting.

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