Mathematicians are chronically lost and confused • Hacker News Board

Re: Mathematicians are chronically lost and confused

#71

Post by j2kun » Wed, Mar 05, 2014, 3:55 PM UTC

fidotron wrote: post #5

This misses the dangerous part, which is mathematicians in groups can confuse each other into accepting ideas which are basically nonsensical, especially if the counter argument relies on some obvious but intuitive observation of reality but cannot be easily formalised within their chosen framework of the moment. As a consequence of this it wouldn't surprise me if the overwhelming majority of maths was actually incoh…

You're right that there is a big social aspect to mathematics that has bitten people in the butt many times throughout history.

But to say that an "observation of reality" should have any effect on existing mathematics is silly. Though mathematics might take inspiration from the physical world, it is removed from it by design. When some observation of reality disagrees with mathematics, that usually means the mathematical framework in question needs to be generalized or specialized, not altered.

This has happened over the years, for example, with measure theory, Fourier analysis, Lie theory, computational complexity theory, and many others.

Re: Mathematicians are chronically lost and confused

#72

Post by ChristianMarks » Wed, Mar 05, 2014, 3:57 PM UTC

Fair enough. I tried in my youth to solve every problem I came across. There were many I couldn't solve. It took a while before I developed the wisdom and discipline not to solve every problem no matter how long it took. By a while I mean decades. I sacrificed the possibility of family life, have stopped talking to my uncomprehending stepfather, and have kept my social interactions to an absolute minimum to pursue my consuming interest. (I mention this as a point of pride.) I find myself continually astonished by the ingenuity of solutions I probably could never have imagined after years of work. Perhaps, after a lifetime of effort that must be continually maintained, I have attained the level an entering freshman at Harvard. At this stage, I may be reduced at best to connoisseurship of some aspects of mathematics.

Now for some reflections on attitudes. Mathematicians sometimes act as if they believe that expertise in mathematics transfers to expertise in mathematics education. Suppose you are a sensitive student, lacking in confidence. You open Korner's beautiful book on Fourier Analysis, and the first thing you are greeted with is "This book is meant neither as a drill book for the successful student nor as a lifebelt for the unsuccessful student." Korner does not mention other references suitable for the successful and the unsuccessful student. You take this comment to mean that Korner would let the unsuccessful student drown. There is no implication, but this is the psychological import, the implicature. Why mention the unsuccessful student at all? Why not say who the book is for, without planting this gratuitous image in the reader's mind? It would take some time to return to this book, to get past the wonder at a mind capable of such an incidental, dismissive, off-handed acknowledgement of "the unsuccessful student."

You could say this is "overthinking." Such remarks, microagressions as they are termed today, "perpetrated against those due to gender, sexual orientation, and ability status", are sometimes revealed in the asides of mathematical authors [1].

And now if only mathematics educators would evaluate their students on the state of their confusion!

[1] http://en.wikipedia.org/wiki/Microaggression

Re: Mathematicians are chronically lost and confused

#73

Post by j2kun » Wed, Mar 05, 2014, 3:58 PM UTC

egdelwonk wrote: post #52

11001 wrote: post #42

Earlier quoted context omitted.

Textbooks and exercises. Avoid shortcuts such as online tutorials or anything with "for hackers" in the title.

I've been using Khan Academy to refresh my skills and it's been really helpful, but do you have any recommendations for textbooks? I'm 10 years out of school and haven't really needed to flex my math muscles in years.

If you're past calculus, I'd say start with Sheldon Axler's Linear Algebra Done Right (it has the most immediate applications, and linear algebra is essential for almost all higher mathematics).

Re: Mathematicians are chronically lost and confused

#74

Post by j2kun » Wed, Mar 05, 2014, 4:00 PM UTC

nbouscal wrote: post #67

graycat wrote: post #37

Earlier quoted context omitted.

No, I'm correct: He set up an extreme straw man to knock it down. I clearly agreed that his extreme straw man is foolish. There is a common reason students fall for his straw man: They are concerned that if there is an exercise they can't work they are missing something important. My advice was, instead, for a very diligent student, to solve 90-99% of the exercises and just let go of the last few as illposed, stated…

This is a good example of how being correct is completely irrelevant if you can't communicate it well. That said, I still maintain that you're thoroughly misunderstanding the position the OP was arguing for.

For the record, I'm saying two things:

1. I hear about lots of people trying to do every exercise. 2. I think this is bad if it causes them to quit out of frustration. 3. I provide some tips on how to deal with frustration, and to know that you're in good company.

Re: Mathematicians are chronically lost and confused

#75

Post by nly » Wed, Mar 05, 2014, 4:00 PM UTC

jorleif wrote: post #58

makmanalp wrote: post #55

Earlier quoted context omitted.

I've argued the same with a mathematician friend of mine. I hate academic papers because of their seemingly convoluted and backwards way of explaining things. His answer was that papers were not made to convey thoughts to laymen, they were made to communicate facts and proofs with as little ambiguity as possible, optimized for reading by other mathematicians. It's meant to be high bandwidth (hence the terse style and…

I think the process of peer review often causes papers to be optimized for being hard to critique rather than easy to understand. I have myself participated in writing papers where we decided to leave out some non-crucial but very useful detail just because it opens up too many questions and opportunities for critique.

So "let's keep the source code closed so nobody can see how awful we are and just show off our wonderful working program"

Re: Mathematicians are chronically lost and confused

#76

Post by wisty » Wed, Mar 05, 2014, 4:04 PM UTC

nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

See Feynman's rant on textbooks - http://www.textbookleague.org/103feyn.htm which is still valid.

Maybe not the corruption angle so much; but special interests. Now every book is designed to not offend the politically correct Californians, or the religious right in Texas, and to convince people that they are keeping up with the latest fads - http://www.edutopia.org/muddle-machine

Re: Mathematicians are chronically lost and confused

#77

Post by nly » Wed, Mar 05, 2014, 4:07 PM UTC

ronaldx wrote: post #64

nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

> Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. I've been a secondary math teacher in the UK and I want to defend this point a little. The job of a secondary math teacher at this level is to teach everyone math, particularly including a majority who don't have a strong interest and won't go on t…

I wasn't aiming to criticise the particular set or scope of topics taught at each stage, more so the fact there's no provision to ensure students are aware of what branches of mathematics even exist. There's no formal introduction or grounding or broad strokes.

It's like teaching science where there's no innate notion of biology, chemistry and physics as three separate disciplines... or teaching history without putting everything in its place on a timeline and your students end up not knowing whether Tudor times ended before or after the Ottoman empire fell.

It's the big picture exposure that I think maths lacks, and I personally never bothered to step back and actually look for that big picture until I was already in my 20s. The way maths is taught makes you feel like it's suddenly going to open up, but all that actually happens is you follow one branch.

Re: Mathematicians are chronically lost and confused

#78

Post by waqf » Wed, Mar 05, 2014, 4:08 PM UTC

chalst wrote: post #43

fidotron wrote: post #5

This misses the dangerous part, which is mathematicians in groups can confuse each other into accepting ideas which are basically nonsensical, especially if the counter argument relies on some obvious but intuitive observation of reality but cannot be easily formalised within their chosen framework of the moment. As a consequence of this it wouldn't surprise me if the overwhelming majority of maths was actually incoh…

Since fidotron has been piled on, let me defend the point in his/her post. Good mathematics has come out of being worried that what other mathematicians have done isn't quite right, and I think the perspective of the article doesn't acknowledge that. For example: Cantor's theorem is quite true, only cranks doubt it [1]. But many mathematicians take it to have the corollary that cardinalities greater than that of the…

Thank you, I wanted to say this but you put it much better.

Re: Mathematicians are chronically lost and confused

#79

Post by jackmaney » Wed, Mar 05, 2014, 4:12 PM UTC

japhyr wrote: post #35

acjohnson55 wrote: post #28

I've felt this is the case for a long time. A lot of people have a smooth experience in math for years until they hit their first serious discontinuity. That could happen anywhere: times tables, fraction arithmetic, two-step equations, geometric proofs, radicals, limits, or maybe even college math. The reaction is nearly universal though. The person thinks, "holy crap, I guess I'm actually not good at math", anxiety…

as the math truck barrels on ahead I've been teaching math to at-risk high school students for the last 10 years. I have spent more time helping students understand that they are not stupid, that something just got in the way of their learning at one point, and they never understood anything after that. I'm going to use your quote in some of these conversations now. What most of my students think: "I could never do m…

> I would like to see every elementary school have a math specialist, who knows advanced math, to help students with their overall understanding when they get off track.

Note: This response is US-centric.

These individuals are exceedingly rare (if they exist at all). In fact, I would be absolutely shocked if 100 such people existed. College students who go into Elementary Education are stereotypically terrified of mathematics, and they have (at most) one required math course. This course is a "general methods" course that essentially acts as a survey of the elementary school mathematics that they'll be teaching.

Re: Mathematicians are chronically lost and confused

#80

Post by vidarh » Wed, Mar 05, 2014, 4:16 PM UTC

nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

> I loath academic papers. Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper...

I systematically ignore maths in CS papers. If the concept isn't described with code or pseudo-code at least, I will look at who cites the paper, and look for someone who has replicated enough in code. 90%+ of the time I can avoid dealing with the maths entirely, or get enough details that I can glean the rest without bothering trying to actually understand the maths properly.

Usually I find the maths tends to obscure vast amounts of missing details.

Of course, how well that works depends on the specific field.