First time I've intentionally kudos'd
Mathematicians are chronically lost and confused
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Re: Mathematicians are chronically lost and confused
#22Secondary 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 same thing happens to me regularly, and not just with "computer science" but with other technical fields, hard sciences, and mathematics. 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.
Re: Mathematicians are chronically lost and confused
#23Secondary 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 progression of math knowledge in the US is more or less the same, since our high school is the equivalent to your last two years of secondary school combined with college.
Re: Mathematicians are chronically lost and confused
#24Earlier quoted context omitted.
I'm of the mind that there is immense value in being able to figure out difficult proofs. Absolutely. However, the rabbit hole is very deep. Many papers make leaps from one sentence to the next that, if you're not familiar with the field, can take a couple days to figure out. Even then, real world proofs are informal and therefore not air-tight. They're close enough, almost always, but there's a reason why a mathemat…
> most of it can't be done that way and remain useful to humans (like assembly language, it's too low-level for most applications). Sincere question (I'm not a mathematician): why can't it be done that way?! On top of an assembly language you can create a higher level language and on top of that an even higher level one, and it is airtight , it has to be or the code won't compile or will throw a runtime exception, th…
I think it can in theory, but the reason it hasn't been done in practice is basically that mathematics is too damn big. Look at Whitehead and Russell's original attempt to do basically exactly what you're asking for: it took them a few hundred pages to build up enough machinery to prove that 1+1=2.
Re: Mathematicians are chronically lost and confused
#25Secondary 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…
Just as a specific example, I had this experience with Bayes' Theorem " rel="nofollow">http://en.wikipedia.org/wiki/Bayes%27_theorem>. As an informal paper for my computer security class, we used Bayes' theorem to implement aimbot detection in a simple FPS. It sounds like a big, complicated theorem with a special name that some genius had to come up with and has complicated notation involving probabilities and logic symbols.
And then it's basically (paraphrasing with reckless abandon) just the probably of your event divided by the total probability space. Lots of words and jargon and theory given in countless papers and articles, and it pretty much just boils down to intuitive addition, multiplication, and division.
And our aimbot detector actually worked pretty damn well! Just gather some data points to determine probabilities, plug them into the simple formula, and it was always correct in our test cases.
Re: Mathematicians are chronically lost and confused
#26Earlier quoted context omitted.
Cantor's conception of transfinite numbers is the one that I think has done most damage.
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 numbers. Incidentally, the set-theoretic cardinals are equivalent to a special case of these;
4. Definition of numbers as equivalence classes of functions under their speed of growth as they tend to infinity, e.g., Hardy's logarithmico-exponential functions. Incidentally, the computable set-theoretic ordinals are equivalent to a special case of these.
Re: Mathematicians are chronically lost and confused
#27No, the OP is giving bad advice. Reading good foundational text books carefully is darned good advice. But for solving every exercise before moving on, no, that's not a good idea. Instead, be willing to be happy solving some 90-99% of the exercises. For the rest, guess, with some evidence, that they are incorrectly stated, out of place, just too darned hard, or some such. If insist on solving 100%, then get on the In…
You misread the article. He is not recommending that students solve every exercise; he's recommending the exact opposite.
Re: Mathematicians are chronically lost and confused
#28I'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 strikes, and they freeze up.
Some people find eventually find their way around this first road block, and future discontinuities in understanding become less stressful, and eventually understood to be a completely normal part the process.
But the usual experience is that a person's math confidence is blown and as the math truck barrels on ahead, they never catch up. They understandably accept the identity of not being "good at math".
What's missing in math pedagogy at most schools is a systematic way to deal with the discontinuities when they strike, especially that first time. We can prepare students to deal with that panic. The tough part is that the math teacher probably has 90 students on roster, but the discontinuity could hit pretty much any given lesson, for some given student.
I know so many people who have come back to intermediate math later in life and breezed through it, armed with intellectual confidence gained from other fields. They look back and wonder how they came to be so intimidated by math in their younger days. We've got to give younger people the tools and knowledge for overcoming this intimidation at a younger age. We've got to kill "I'm just not good at math".
Re: Mathematicians are chronically lost and confused
#29I started off doing a combined maths and computer science degree.
With both computer science and maths you are chronically confused. The difference being with computer science it doesn't matter so much if you don't understand something, if you can get it to work you know you are on the right track. Maths is much more progressive, each proof builds on a previous one. So if you fail to understand one step you are screwed from that point on.
After the first year I realised I didn't actually enjoy being permanently confused and so I ditched the maths to focus on computers. I do regret this. It didn't take long at all before I forgot all that knowledge I had spent years sweating over.
Re: Mathematicians are chronically lost and confused
#30Secondary 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... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating.
It's like this all the time in math papers. It often seems in the end like the ideas themselves are fairly straightforward, and it shouldn't have taken that long to understand. I think, though, that if you actually sat down and tried to explain it in more intuitive terms, you'd find that you might not be able to. True, you could find a way to convey the general idea, but without the technical details, (1) while non-mathematicians may get a surface-level understanding faster, even mathematicians will not grasp the technical aspects, and (2) it will be very hard for anyone to extend your work or for anyone to apply it to another situation, so it will only be applied in the specific contexts that you explained it in.
Language isn't built to communicate math, so doing so effectively will be either difficult to understand or imprecise. Many people claim that it would be easy to explain deep math concepts with "a couple of block diagrams and a few short paragraphs"s, but I'd challenge them to write a textbook on abstract algebra, or topology, or something like that before they make that claim.