Mathematicians are chronically lost and confused • Hacker News Board

Re: Mathematicians are chronically lost and confused

#81

Post by jseliger » Wed, Mar 05, 2014, 4:20 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…

The purpose of most academic papers is not to explain (let alone teach!) ideas in an intuitive manner, but rather to express them in formal, correct, unambiguous terms -- that is, to make them as accurate and critique-proof as possible for publication in some journal.

In other words, the purpose is primarily tenure and advancement.

This may be a problem.

Re: Mathematicians are chronically lost and confused

#82

Post by polymatter » Wed, Mar 05, 2014, 4:30 PM UTC

simias wrote: post #61

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…

For me the problem is that math papers/articles lack a proper API documentation. By that I mean that it's very hard for me to understand what a lot of symbols mean because mathematicians (and physicians) love to use single letters to name various concepts and functions. Worse, they also like to use the same symbol to denote different things in different fields. I'm sure it's extremely convenient to have a shorthand w…

I agree. To clarify, its not that the reader gets confused between the gas constant and the set of real numbers. The issue is never actually explicitly stating "R represents the set of real numbers" or "n is a natural number".

At uni it once took me hours to work out that "." was used for function application in one particular paper. "." was also used for multiplication and (in some example code) had the usual object oriented meaning in the same paper. It was incredibly frustrating.

Re: Mathematicians are chronically lost and confused

#83

Post by michaelochurch » Wed, Mar 05, 2014, 4:30 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…

Excellent point.

Conway's surreal numbers are awesome. Combinatorial game theory seems silly at first (why are we analyzing Hackenbush?) if you expect it to be like "regular" game theory but is mind-blowing when you actually get it in all its glory.

Re: Mathematicians are chronically lost and confused

#84

Post by [deleted] » Wed, Mar 05, 2014, 4:36 PM UTC

simias wrote: post #61

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…

For me the problem is that math papers/articles lack a proper API documentation. By that I mean that it's very hard for me to understand what a lot of symbols mean because mathematicians (and physicians) love to use single letters to name various concepts and functions. Worse, they also like to use the same symbol to denote different things in different fields. I'm sure it's extremely convenient to have a shorthand w…

[deleted]

Re: Mathematicians are chronically lost and confused

#85

Post by logicallee » Wed, Mar 05, 2014, 4:40 PM UTC

jseliger wrote: post #81

cs702 wrote: post #22

Earlier quoted context omitted.

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…

The purpose of most academic papers is not to explain (let alone teach!) ideas in an intuitive manner, but rather to express them in formal, correct, unambiguous terms -- that is, to make them as accurate and critique-proof as possible for publication in some journal. In other words, the purpose is primarily tenure and advancement. This may be a problem.

The issue is that a lot of intuitive stuff is wrong. When you formalize, you remove the simple, intuitive explanation - but you make it much harder for you to remain wrong, if you are wrong - or to become wrong, if you started off right.

As a simple explanation, consider the difference between explaining the Monty Hall problem - which might seem to be philosophical, open to interpretation - and coding it up. The moment you code it up --- go ahead, code up a monte carlo simulation that compares the two alternatives of switching doors or maintaining the same choices, and spits out a running count of which is what percentage correct. I'll wait while you code. ---

the moment you do that, you see two lines in your code that make the explanation 100% irrefutable and completely obvious.

That is why papers are written this way. Intuition can go both ways.

Re: Mathematicians are chronically lost and confused

#86

Post by dalke » Wed, Mar 05, 2014, 4:41 PM UTC

j2kun wrote: post #50

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…

That's a poor comparison. I find it hard to believe that mathematicians are still trying to decide if the set {1, 2, 3} is finite or infinite.

A "startling large part" of all science fields like this.

Physicists don't even know if the gravitational mass of an object is really the same as its inertial mass. Or if there are true magnetic monopoles. And that's after over a century of trying.

Immunologists have barely scratched the surface of how that field works.

Economists make lots of conjectures, with lots of math to back it up, but it's not a perfect predictor of the human economic system.

Biological evolution still surprises us, 150 years after Darwin and nearly 100 years after the neodarwinian synthesis.

Chemists still don't come close to handling some of the reactions that natural systems have figured out.

And so on.

Re: Mathematicians are chronically lost and confused

#87

Post by pflats » Wed, Mar 05, 2014, 5:04 PM UTC

AimHere wrote: post #45

pflats wrote: post #11

Earlier quoted context omitted.

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 whi…

Oh, I agree, that's the best part. Pick your rules: oh, you picked those seven? You've got a ring; here are your math rules!

In that specific case, though, I figured I'd point out that the basic rules of math aren't usually things people squabble over. (Although I do enjoy a little mathematical philosophy from time to time.)

Re: Mathematicians are chronically lost and confused

#88

Post by j2kun » Wed, Mar 05, 2014, 5:07 PM UTC

dalke wrote: post #86

j2kun wrote: post #50

Earlier quoted context omitted.

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…

That's a poor comparison. I find it hard to believe that mathematicians are still trying to decide if the set {1, 2, 3} is finite or infinite. A "startling large part" of all science fields like this. Physicists don't even know if the gravitational mass of an object is really the same as its inertial mass. Or if there are true magnetic monopoles. And that's after over a century of trying. Immunologists have barely sc…

Good point.

Re: Mathematicians are chronically lost and confused

#89

Post by pflats » Wed, Mar 05, 2014, 5:15 PM UTC

jackmaney wrote: post #79

japhyr wrote: post #35

Earlier quoted context omitted.

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 ma…

There are more than you think, I assure you. Consider the people who write elementary curricula, who implement it in large cities, who teach middle school and high school mathematics but might prefer to teach just mathematics at an elementary level. Consider NCTM[1], TERC[2], EDC[3], and UChicago[4], and their programs and work. Consider the math coaches, who instruct their peer elementary teachers on teaching mathematics.

Until these positions exist, are respected, and are not first on the chopping block the next time budget cuts roll around, these people will continue to exist under the radar. (I'd gladly transfer into a Elementary Math Specialist position, if I was sure it wouldn't threaten my family's livelihood.)

[1]: http://www.nctm.org/resources/elementary.aspx

[2]: https://www.terc.edu/display/About/Mission+and+Vision

[3]: http://ltd.edc.org/mathematics

[4]: http://everydaymath.uchicago.edu

Re: Mathematicians are chronically lost and confused

#90

Post by api » Wed, Mar 05, 2014, 5:23 PM UTC

I completely agree about the power of math, and why programmers should learn it. There are two problems with math:

(1) Math is IMHO the worst taught of all academic subjects.

It's taught as if it were not a language. Math profs and books on mathematics never explain what the symbols mean. They just throw symbols at you and then do tricks with them and expect you to figure out that this symbol means "derivative" in this context. I have literally seen math texts that never explain the language itself, introducing reams of new math with no definitions for mathematical notation used.

I've looked for a good "dictionary of math" -- a book that explains every mathematical notation in existence and what it means conceptually -- and have never found such a thing. It's like some medieval guild craft that is passed down only by direct lineage among mathematicians.

Concepts are often never explained either. I remember struggling in calculus. The professor showed us how to do a derivative, so I mechanically followed but had no idea why I was doing what I was doing. I called up my father and he said one single sentence to me: "A derivative is a rate of change."

A derivative is a rate of change.

I completed his thought: so an integral is its inverse. Bingo. From then on I understood calculus. The professor never explained this, and the textbook did in such an unclear and oblique way that the concept was never adequately communicated. It's one g'damn sentence! The whole of calculus! Just f'ing say it! "A derivative is a rate of change!"

(2) The notation is horrible.

If math were a programming language it would be C++, maybe even Perl. There are many symbols to do the same thing. Every sub-discipline or application-area of mathematics seems to have its own quirky style of notation and sometimes these styles even conflict with each other.

Yet baroque languages like C++ and Perl at least document their syntax. If you read an intro to C++ book it begins its chapter on templates by explaining both what templates are for and the fact that type means "type is templated on int."

Math doesn't do this. It doesn't explain its syntax. See point #1 above.