http://mdzlog.alcor.net/2010/07/12/read-listen-or-comprehend...
> What’s interesting is that the pattern varies from person to person. Feynman shared his discovery with a group of people, one of whom (John Tukey) had a curiously different experience: while counting steadily, he could easily speak aloud, but could not read. Through experimenting and comparing their experiences, it seemed to them that they were using different cognitive processes to accomplish the task of counting time. Feynman was “hearing” the numbers in his head, while Tukey was “seeing” the numbers go by.
So while pushing around abstract symbols might be the way that you really grok things, I don't think this applies to everyone. I have an undergrad math minor, and the only math class that I didn't get an A in and that I really struggled with was Linear Algebra.
I think it mostly had to due with excessive symbolic abstraction. Matrix notation still really troubles me. About 5 years after that Linear Algebra class that I struggled with, I was took a computer graphics class for a masters degree. We were talking about the non-invertibility of some operation (I think it was about figuring out how far something was away given a 2d rendering of a 3d scene, where you have a degree of freedom in the scale, everything could be twice as large and twice as far away). And that gave me a really intuitive understanding between matrix invertibility and full rank matrices. I am sure that I would have done much better in a combined Linear Algebra/Computer Graphics class than I did in just the mostly symbolic and proof based Linear Algebra class.
So be careful in believing that the thought processes that obviously work well for you will also work well for others.
I would suggest that someone who has trouble (serious trouble that is) with linear algebra might have also trouble with other abstract things in computer science, like automata theory or grammars and compilers.
public interface List<T> { ... }
Tree a = Empty | Branch a (Tree a) (Tree a)
This question at math.stackexchange.com goes into a little more detail: http://math.stackexchange.com/questions/24241/why-do-mathema...
Plus, the math tradition of one-character variable names means that they've had to adopt several different alphabets just to get enough identifier uniqueness (greek, hebrew, etc., plus specialized symbols like the real, natural, and integer number set symbols). Which makes all that stuff a pain to type. And even then, there are still identifier collisions where different sub-disciplines have different conventions for the meaning of a particular character.
It's also annoying because single-character names are impossible to google for.
t a = e | b a (t a) (t a)
Tree someData = Empty | Branch someData (Tree someData) (Tree someData)
One difference between programming and mathematics is that programming is mostly interpreted in context, when it matters that this double represents elapsed time, and this double represents dollars in my checking account. Mathematics, on the other hand, is mostly interpreted out of context. I don't care what a represents, all I care about is that it enjoys the relationship ab = ba with some other arbitrary object b.
If the mantra of programming is "names are important" then the mantra of mathematics might be "names aren't important".
So if you really don't care what a variable represents, then I'd agree that a single-letter name is fine. Given that math is almost universally done with single-letter variable names, are you suggesting that in math you almost never care what a variable represents? This wikipedia article makes me think otherwise; clearly, variables often have a fairly specific meaning.
http://en.wikipedia.org/wiki/Greek_letters_used_in_mathemati...
Thanks for the link, there are some good thoughts there.
The way most mathematicians ignore this problem is by throwing two variable names next to each other, so xy is x times y. Obviously, this system is incompatible with multiple-letter variable names. I have unfortunately seen the two concepts mixed, which makes reading an equation like reading ancient Greek or Latin (which lacked spaces and punctuation).
Until there is a commonly accepted, easily distinguishable symbol for multiplication, I don't see single-letter variables going anywhere, or multiple-letter variables being used in general mathematics.
We could obfuscate further by XML replacement parentheses: <sum> <term>1</term> <term> <quotient>...</quotient> </term> </sum> But the structure is more important, right?
This shouldn’t be “memorized” as a “fact” though. As you point out, it’s the very definition of e. Which is to say, to really understand what the exponential function means implies building up a mental model about lim [n→∞] (1 + x/n)^n and its behavior, interacting with it, connecting it to other functions, seeing what happens when you combine it with other ideas: trying to integrate/differentiate it; noticing how it reacts to fourier transform; relating it to rotations, areas, imaginary numbers; writing it as a taylor expansion or continued fraction, or with a functional equation, or as the solution to a differential equation. Connecting it to derived functions such as e.g. the normal distribution, or sine, or hyperbolic sine. Generalizing it to an operation on complex numbers, or quaternions, or matrices. Thinking about what exponentiation in general really means. Coming up with algorithms for computing e’s decimal expansion or proving that e is irrational and transcendental. Solving problems with it, like to start with, continuously compounded interest (&c. &c. &c.).
No one who had really learned about exponentials in a deep way would easily forget that this is the definition of e, and that has nothing to do with lists of facts or rote memorization.
> Single-character symbols (and multiple character symbols) that not only hide tons of meaning through a dereference, which is naturally how language works, that also suggest a meaning themselves, are stupid and promote natural diseased human thinking.
I think you should try writing out some very difficult complex math proofs before you make this assertion (say, for instance, a proof . Things are bad enough when we pack ideas down. Start expanding them in the middle of the computation, and it the steps become almost impossible to see or reason about.
The whole point of assigning things short simple names is that it gives a clear analogy (to past experience w/ the conventions) that provide a shortcut to anticipating how particular types of structures and operations behave. Cramming a matrix or a multivariate function or an operator down into a single symbol helps us to treat that symbol as a black box for the purpose of the proof or problem, which helps us bring our existing mathematical tools and understandings to bear. Sometimes, we run afoul of false impressions, as when we apply intuition about the real numbers to matrices in ways that don’t quite apply, or intuition about metric spaces to general topological spaces, &c. But this is I think an unavoidable cost, and it’s why we strive to be careful and precise in making mathematical arguments.