Mathematicians are chronically lost and confused • Hacker News Board

Re: Mathematicians are chronically lost and confused

#41

Post by japhyr » Wed, Mar 05, 2014, 2:37 PM UTC

I currently teach math to at-risk students. I don't read all of these submissions about math education, but I skim the comments on most of them. The comments people make change the way I teach math.

I have always done a decent job of teaching math. I focus on helping students understand concepts, even when they are focusing on mechanics. I use words like "shortcut" and "more efficient method" rather than "trick" when showing students more efficient ways to solve problems. I have students do problems and projects that relate to their post-high-school goals.

But with the routines of school life, I get away from the fun of math from time to time. The comments on these submissions often remind me to go in and just tell stories about math:

- "Hey everyone, did you know that some infinities are bigger than other infinities?"

- "Hey everyone, do you have any idea how your passwords are actually stored on facebook/ twitter/ etc.?"

- "Have any of you heard the story about the elementary teacher who got mad at their class, and told everyone to add up all the numbers from 1 to 100? One kid did it in less than a minute, do you want to see how he did it?"

Thanks everyone, for sharing your perspective on your own math education, and about how you use math in your professional lives as well. Your stories help.

Re: Mathematicians are chronically lost and confused

#42

Post by 11001 » Wed, Mar 05, 2014, 2:40 PM UTC

egdelwonk wrote: post #32

What's the best way to relearn math?

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

Re: Mathematicians are chronically lost and confused

#43

Post by chalst » Wed, Mar 05, 2014, 2:40 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…

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 natural numbers exist, which does not follow: it is perfectly coherent to say that constructions such as the power set of natural numbers do not exist as a definite whole, and so do not have a cardinality. These kinds of doubt have driven constructivism which has led to interesting work in topology, measure theory, and type theory, and led to such useful applications as calculators for exact real arithmetic.

Cantor's paradise seems to be coherent (likewise I would be deeply surprised if large parts of mathematics turned out to be misconstrued) but the assumptions of large cardinal set theory are grandiose and poorly justified, and yet for a long time those people who wondered if it was wise to embrace the whole edifice were marginalised. It seems that now there are many more mathematicians who are interested in revisiting this perspective [2].

To put Wiles' metaphor in perspective, it is good if some mathematicians step outside the mansion from time to time, to see if the superstructure is up to all the crashing about that happens in the dark rooms.

[1]: https://www.math.ucla.edu/~asl/bsl/0401/0401-001.ps (Postscript file)

[2]: http://homotopytypetheory.org/book/ has been very successful

Re: Mathematicians are chronically lost and confused

#44

Post by chwolfe » Wed, Mar 05, 2014, 2:42 PM UTC

The entire post was enjoyable but I found the last paragraph to have the most actionable advice:

What’s much more useful is recording what the deep insights are, and storing them for recollection later. Because every important mathematical idea has a deep insight, and these insights are your best friends. They’re your mathematical “nose,” and they’ll help guide you through the mansion.

Re: Mathematicians are chronically lost and confused

#45

Post by AimHere » Wed, Mar 05, 2014, 2:43 PM UTC

pflats wrote: post #11

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…

I'm going to be rather dismissive in my reply, and for that, I apologize, because I'm not quite sure how else to respond. This is more or less a non-issue. Thanks to mathematicians building on Euclid for the last 2300 years, we have a system of mathematics built on a few basic principles (that you would not disagree with) and deductive reasoning. If you take a theorem that is accepted as proven, you can almost defini…

>we have a system of mathematics built on a few basic principles (that you would not disagree with) and deductive reasoning

I think it's even better than that; mathematicians don't necessarily care whether the reader 'agrees' with the axioms, or whether they're in any sense 'true' or 'false'. Mathematics is always of the form 'if these axioms are true, this theorem follows from it'.

The real world and the notions which people consider to be self-evidently true is just some messy slimy gooey gunk best left to psychoanalysts and theoretical physicists and sewer workers and the like.

Re: Mathematicians are chronically lost and confused

#46

Post by j2kun » Wed, Mar 05, 2014, 2:45 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 think part of the problem is that the math track is way too linear. It doesn't need to be that way, as I've written about a few times in the past.

Re: Mathematicians are chronically lost and confused

#47

Post by [deleted] » Wed, Mar 05, 2014, 2:46 PM UTC

cs702 wrote: post #22

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…

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... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating. This is SO TRUE. The sa…

[deleted]

Re: Mathematicians are chronically lost and confused

#48

Post by tel » Wed, Mar 05, 2014, 2:47 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…

The reason is that math papers are usually tailored to another mathematician and thus abuses that to avoid a few communication pitfalls.

Firstly, it's broadly considered to be the case that mathematical ideas are not understood until you've gotten them "from a few different angles". Math builds upon itself so much that an idea may be almost useless on its own and produce true value in lying at the nexus between many convergent ideas. For instance, statistics as a field enjoys a very nice convergent point between logic, measure theory, and information theory (among others). Approaching it from all of these positions can lead to important mental breakthroughs and a paper or book author desires to appeal to these various "roads" to their topic. Without providing that context it might be said that the presentation is very sparse.

Secondly, almost conversely, each reader is likely to be more familiar with a subset of the possible roads to the author's topic of interest. By covering as many roads as possible as they approach their goal topic they provide more chances for the reader to pick an approach they find most comfortable and follow it (lightly ignoring the rest) to the goal. A simple block diagram might be the best way to present it to you, but only a private tutor could specialize their presentation so much.

---

There's an art to reading a math paper when you're an outsider to the primary topic. You want to breeze through the paper at a high level first, slowly collecting the exposition points which are most applicable to your own method of understanding. After that, iteratively deepen your reading while looking up topics which you feel you almost-but-perhaps-not-quite-enough understand. You can very easily read a paper and get enormous value while failing to connect to 60-70% of what's written.

Re: Mathematicians are chronically lost and confused

#49

Post by dborg » Wed, Mar 05, 2014, 2:48 PM UTC

chalst wrote: post #26

michaelochurch wrote: post #15

Earlier quoted context omitted.

How are transfinite numbers "nonsensical"? When you get into infinity, you have two notions of "number" that diverge. Mathematical operations on them do different things. (For example, cardinal "exponentiation" is the power set; ordinal "exponentiation" is something different and smaller.) One is size , but proper subsets can have the same size at infinity (integers, even numbers, rationals). That's where Aleph-0 (ca…

Well put; a little quibble: these are the two notions of infinity that most interest set theorists, but there are many other notions of infinity in mathematics, e.g., 1. Representation of geometric entities "at infinity" in, e.g., the point at infinity from the projective sphere that allows straight lines to be treated as circles; 2. Infinitesimals; 3. Game-theoretic constructions of infinite numbers, e.g., in Conway…

Sorry to give a minor correction to a little quibble, but it is the ordinals, not the cardinals, that are a special case of Conway numbers. The cardinals are equivalence classes of these of the form [א_a,א_(a+1))

(Also you can get infinitesimals from Conway's construction as well)

Re: Mathematicians are chronically lost and confused

#50

Post by j2kun » Wed, Mar 05, 2014, 2:49 PM UTC

dalke wrote: post #33

"If you’re going to get anywhere in learning mathematics, you need to learn to be comfortable not understanding something." This is true for all research. And I don't mean just the physical sciences either. Historians and sociologists are also chronically "lost and confused." Otherwise it wouldn't be a topic worth of study. This is why students who are "good at X", whether it be math, German, sports, or programming,…

I think it's a good point, but I still think the kind of lost and confused in mathematics is more embarrassingly extreme. Imagine a few hundred historians trying to discern when King George I died, and after 50 years of work they conclude, "All we know for sure is that it was between the day he was born and yesterday." A startlingly large part of mathematics feels like this.

And I think the reason is that "prevailing theories" mean nothing in mathematics.