Comment by ezyang | Hacker News Reader

Comment by ezyang | Hacker News Reader

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Brett Victor writes:

When most people speak of Math, what they have in mind is more its mechanism than its essence. This "Math" consists of assigning meaning to a set of symbols, blindly shuffling around these symbols according to arcane rules, and then interpreting a meaning from the shuffled result. The process is not unlike casting lots.

This mechanism of math evolved for a reason: it was the most efficient means of modeling quantitative systems given the constraints of pencil and paper. Unfortunately, most people are not comfortable with bundling up meaning into abstract symbols and making them dance. Thus, the power of math beyond arithmetic is generally reserved for a clergy of scientists and engineers (many of whom struggle with symbolic abstractions more than they'll actually admit).

I think gamification is a great way to teach symbol manipulation, and I think (contrary to Bret) that symbol manipulation is a prerequisite for deeper STEM ideas.

I do also believe that harder mathematical problems can also be gamified, but this process is much less well understood, and you'll probably want a few theorem prover experts around if you attempt a system like that.

I 100% agree that having symbol manipulation is required for deeper understanding and learning.

In my experience, being able to quickly and (fairly) reliably move symbols around in one's head makes futher learning much easier and faster, as one can follow proofs and explanations intuitively without feeling the need to work through each step laboriously to justify it.

As a simple example, being able to see the steps that were used to go between

    (x - 1)^2 - 2 = 0
    x = 1 +- sqrt(2)

at a glance gives one a higher "maths-learning" bandwidth, since one spends just the 3 seconds it takes to read those lines, instead of the extra 30 seconds physically doing the working.

(And, on a slightly different note, having some basic facts memorised like (for example) "d/dx sin(x) = cos(x)" is a little like the difference between data in L1 cache and that just in RAM. However, this doesn't mean that one should not understand why, or not be able to rederive it in a flash by drawing a diagram or whatever.)

> I think (contrary to Bret) that symbol manipulation is a prerequisite for deeper STEM ideas.

Can you elaborate on this further? I tend to disagree, but I don't know if I have all that much to say to back it up.

I mean, sure, there's just no good way to understand shear forces without being able to manipulate matrices (so I agree with dbaupp's sibling comment), but that doesn't mean you want to learn the rules of matrix algebra as if they were arbitrary rules enforced by the stick and carrots in a game.

In the case of mathematics, there is a fundamental sense in which the symbols and their manipulations are what you're studying. Think of it as if you were an archeologist: the symbols are the artifact of study. It's only half of the picture: you must(!) come up with a different way of intuitively understanding it--but at the end of the day if you really want to precisely say what it is you're talking about, it's symbols.

OK, I think I see what you're getting at. But we could devise a AltDragonBox game with a completely different set of arbitrary rules. (Possibly even rules that are inconsistent.) And as far as game play goes, AltDragonBox would work just as well. But there's a reason that the rules of algebra are what they are. And if you can't distinguish between DragonBox and AltDragonBox, I don't think you're learning what you need to learn.

Here's another way to look at it: we all know people who could ace their high school math tests because they had memorized the rules of manipulation but weren't good at math--as evidenced by their poor performance in college and failure to succeed in future STEM classes. If "the symbols and their manipulations are what you're studying" (which, I agree, there is some truth to), then what is it exactly that these people were lacking?

I agree that it is important to recognize the limitations of this approach. As a high schooler (and even as an undergraduate in mathematics), you learn all sorts of formal systems. As a mathematics researcher, one of your charges is inventing interesting mathematical concepts to play around with. Obviously this is a bit much to ask out of DragonBox! And even more modestly, the ability to sit down with a problem and say, "Ah, but what really is going on here?" is a deep and difficult skill to impart. I don't know what these people were lacking, but if they didn't know how to move symbols around, I might have started there...

(BTW, an inconsistent set of rules would correspond to a version of DragonBox where there was a cheat code you could enter, and then easily solve every problem. So it would not "work just as well", and this would be pretty clear to a cheating gamer.)

> there's a reason that the rules of algebra are what they are

But which algebra, and which rules ? There are infinitely many algebras ( eg. Algebra over the field of reals, Banach Algebra, relational algebra, boolean algebra, sigma algebra etc. ) The "rules" are really constructs you decide that apply to the elements of the space that conform to your algebra. So for example the reals are a field that have ordering, so you can talk about less than and greater than, but the complex numbers don't have an imposed order and you'd have to first define a norm to map them onto the reals. The AltDragonBox with its own inconsistent arbitrary rules will still have some algebraic encoding. Whether that's useful to you is debatable. Like in my algebra I could overload plus to mean multiply and square root to mean divide by 7 and add -3 and then try to figure out what exponentiation works out to. It would be interesting...maybe not useful, but its still an algebra. Maybe you won't have closure...the elements may not end up in a field or even in a semigroup...its a nice make-believe algebra.

Yes, that's a very impressive display of all the math you must surely know, but its completely misses the point. The rules of elementary algebra really are special, and it has to do with their correspondance to real things in real life. There's a reason mathematicians don't just enumerate all possible algebras and study them one by one.

OT: There's a great bit in Neal Stephenson's Anathem where these monk-like scientists are made to copy out subtly wrong mathematics and scientific proofs as a punishment.

AltDragonBox could be exactly that -- day and night cancel, except for symbols when its constellation is rising, in which case they divide, except for odd numbered Fridays in a leap year. Oh, and do it with numbers instead of day/night symbols.

Yes, there are loads of different forms of algebra and geometry. Sometimes people throw out what people though were fundamental rules and come up with new geometries (e.g. non-Euclidian geometry which turns out to be very helpful to describe relativity).

A lot of the time though, the other rules for alternative algebras produce very dull and boring algebras.

>but at the end of the day if you really want to precisely say what it is you're talking about, it's symbols

Hear hear! I'd go one step further and say its ALL symbols. Any associated real-life meanings that help a human intuitively understand the equation is purely coincidental and actually a distraction. I've repeated this argument ad-nauseam : http://news.ycombinator.com/item?id=4085558

Don't know who it was ( Martin Gardner ? ) who once said three dinosaur plus two dinosaur is still five dinosaur. The implication is that symbol pushing and symbol manipulation is way more fundamental than having humans around who can associate three and two with human artifacts and then add them to satisfy their intuition. The dinos will add up to 5 regardless of the human intuition.

Interesting take. Could you comment on my sibling comment?(Here: http://news.ycombinator.com/item?id=4106554 )

If it's just all symbols then computers should be better at it than us. The reason we can prove deep theorems by choosing the right path through an impossibly large combinatorial space is because we perceive structure and meaning, and we use that to guide us. We gaim an intuition for something that's "going on underneath" and so don't just perform random searches.

Explaining simple proofs to students often leaves them feeling that they've followed the steps, but don't undertand. There is more than just symbols.

Humans have a lot more bandwidth and computational power than you might think. And the search space for most proofs is not all that large compared to say a go board once you consider how many different proofs also work.

PS: 1-10 petaflops by some estimates, just not that many significant digits per calculation.

When you say `most proofs' do you mean proofs that you would find in an intro level course, or do you something more. I strongly disagree that the search space for most proofs is as small as go. This may be true once you restrict to a suitably relevant field, but this is a nontrivial reduction which takes a great deal of insight!

Fermat's last theorem resisted the attempts of mathematicians for three hundred years because it required insights so complex they couldn't be formulated without a deep understanding of disparate subfields.

To tie this back to the go analogy, the search space of go is large because the branching factor is big (<400) and because the number of moves is quite large (<400 as well, for all but a very few bizarre situations). For real proofs, while the branching factor may be substantially smaller (given some axiomatic system), the length of the proof is much much longer. The exponent in proofs beat the branching factor of go.

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