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For my first attempt at making an explanatory video I thought I should
start small, and get some experience with simple topics before really diving in
to tackle the big questions.
And then I thought, nah.
Most people probably think of math as being about numbers and equations and
and, I don't know, triangles. But where do numbers and triangles come from? Like, why are these
things that anyone ever needs to talk about?
You know, every math student ever has probably at some point thought, math class
is boring and stupid and when am I ever going to use this?
Besides which, how do we know that math is true?
Why does four plus three turn out to be seven instead of twelve, and
how does that kind of truth relate to scientific truth, which is based on evidence rather than
...I guess, pure Vulcan logic?
I think most mathematicians are probably with Plato on this.
They'd tell you the mathematical truth is more pure and essential and fully
known than anything in science.
you've probably heard that evolution is just a theory, even gravity is just a
theory, because science can't really ever fully prove anything. But math has
proofs and we know for sure they're right.
But why is science stuck defending theories about the imperfect real world while math
has these perfect truths?
Plato's idea was that there are these ideal forms, like perfect circles, which
really exist in a kind of math heaven somewhere.
And the things in math heaven work like math does and we keep realizing they
exist when we see the regular world struggling to imitate them with its
imperfect triangles and circles.
i'm not being very fair to this idea, because to me it just sounds crazy.
I mean if ideal forms don't really exist somewhere in the physical universe then
isn't the notion of their reality a metaphor?
And if so, what's it a metaphor for?
Are ideal forms of property of nature, or do they exist in our minds, or what?
There are a bunch of other ideas.
The intuitionist viewpoint says that the source of mathematical truth is human
intuition; that we verify the truth of mathematical propositions subjectively
by thinking through them and directly experiencing their truth.
Well, I don't know about you, but I have an annoying tendency to directly
experience the truth of things that turn out to be false! So,
how do we deal with that?
And then there's the social constructivist view that mathematics is
just a kind of language game that mathematicians play with one another,
making up the rules as they go along like some super nerdy version of Calvinball.
Like, the only reason four plus three is seven is that everybody says so.
As a kid, did you ever just keep asking why, why, why, and eventually your parents
or whoever would break down and say "because I said so"?
...maybe that was really the answer!
I love to mock social constructivism but there's a valid, important point here
that the consensus of mathematicians, like any human agreement, is
to some extent vulnerable to mass allusions and groupthink and all the
usual
in-group/out-group drama that pervades human social life.
Still, if it's strictly true that there's little or no connection to any
underlying reality, then it's really weird
that the rules mathematicians have happened to agree on are so great for describing
the world around us.
And I think that's pretty much the average scientist's view of mathematics:
It some kind of language of nature. You need it to describe how the universe acts
and who cares why it works? We've found that it does.
At least here we're on familiar territory for scientists. Mathematical truth if
nothing else is backed up by empirical observations of the world just like any
scientific theory.
But does that mean that mathematical proof isn't really proof?
When we want to know whether a mathematical idea is true, do we have to go
out and find a relevant example triangle, or uncountably infinite set, and
confirm them theory actually describe something in nature?
Okay, so obviously I don't really buy any of these arguments, so here's what I think:
I got this idea from Haskell Curry, an all-around awesome mathematician who
outlined it in a little paper in 1951.
I'm not sure Curry did a great job defending his idea from hostile
philosophers, but he characterized math in a way that, at least for me,
finally clarifies this vague hunch humanity has had for so long that mathematical
reasoning is somehow valid.
Besides observing perfect circles, there's another very physical and
empirical way that we interact with math.
We perform calculations and we observe their results.
What Curry thought, and what I think, is that math is a science on the exact same
footing as physics or chemistry.
Mathematical rules themselves are a phenomenon that exists in the real world
and can be observed and studied.
There may not be any perfect circles, but there certainly are real
definitions of them in the rules of geometry.
Wnd following the rules of geometry to reach conclusions about their properties
is a real phenomenon with observable results.
Math, in other words, is the science that studies the implications of systems of
rules.
And any rules count: math is not just the stuff taught in grade school, like the
straight edge and compass rules of geometry, or the equation-balancing rules
of algebra,
but also non-euclidean geometries and exotic algebras with arbitrary numbers
of operations that our commutative or not or distributive whichever way,
and the rules of games like tic tac toe and chess,
even the abstracted rules of grocery checkout lines or stock options or
gravity or quantum mechanics.
Even straightforward rules can have surprising implications, so it's not like
there's nothing to discover just because you know the rules.
The pythagoreans, for example, were scandalized to learn that the rules of
geometry led undeniably to irrational numbers like the square root of two.
But there it is, and we have a proof.
A proof that is, if you look at it another way, a reproducible experiment.
Maybe we can't prove things in math anymore than in science, but by following the
rules of geometry, we can show ourselves again and again that the hypotenuse of a
right triangle with two sides of equal length can't be the ratio between any two
whole numbers.
So in the science of mathematics, the hypotheses are ideas we have about the
implications of rules, like
"you can guarantee a draw against any opponent in tic tac toe just by playing
well", and then experiments are individual applications of the rules, like checking
a proof,
to verify that they play out in a certain way.
The experiments in math, in other words, are computations
Like the intuitionist idea that math is founded in the activity of human brains
computing an answer, Curry's view focuses on the empirical observation of
computations. But the idea isn't to directly experience their intuitive
truth, so much is to see that the faithful application of rules does
indeed lead to a particular conclusion.
Putting computation at the center of mathematics like this makes me wonder
about the connection between math and computer science.
I think math literally is the science of computation. After all, the deep
connection between mathematical proof and computability, which was discovered
as part of the investigation into the foundations of math in the early
twentieth century, is what kicked off the field of theoretical computer science in
the first place.
Somebody in computer science, maybe Edsgar Djikstra, is supposed to have said
that computer sciences is no more about computers than astronomy is about
telescopes.
If throughout history mathematicians used their brains to conduct mathematical
experiments in the same way that astronomers observed the night sky with
their naked eyes, then
computers are like telescopes through which we can better investigate the
computational nature of reality.
But what about that kid asking "when am i ever gonna use this?"
If nature is all computational and mathematical,
shouldn't math be obviously useful?
Well, of course it is. Math is everywhere. When kids ask "when am I ever going to
use this," what they mean is
"you're explaining this so badly that I don't even know what it is. You've taught
me how to swim without ever showing me a body of water big enough to swim in."
We give them little buckets, like contrived "a train leave Chicago" problems,
and then drill them endlessly on techniques for solving these non-
problems. Traditional math classes are so eager to cover every clever technique in
a subject that we end up covering them all using boring little problems that
nobody needs to solve,
and we leave people confused about why they learned to solve them in the first place,
which they promptly forget how to do anyway.
If we covered fewer techniques,
we'd have more time to tackle real problems, and kids would have a chance to get the
idea of what math is. Or, at least, what it's good for.