So, why math?
The short version of the answer is remarkably simple: math provides
a tool where you can, without ambiguity, prove that something is true or false.
I’ll get back to that – but first, I’m going to make a quick diversion, to help you understand my basic viewpoint on things.
This blog actually started in response to something specific. I was reading
Orac’s blog “Respectful Insolence”, and
he was fisking a study published by the Geiers, purporting to show a change in the trend in autism diagnoses. Orac was attacking it on multiple bases, but it struck me
that the most obvious problem with it was that it was, basically, a mathematical argument, but the math was blatantly wrong. It was making a classic statistical analysis mistake which is covered in first-year statistics courses. (And I mean
that very literally: when I was in college, I lazily satisfied some course requirements by taking a statistics course given by the Poly Sci department, and
in statistics for political scientists, they covered exactly the error made by the Geiers in November of the fall semester.) It struck me that while there were a lot of really great science bloggers – people like Orac, PZ Myers, Tara Smith, and so on – that I didn’t know of anyone doing the same thing with math.
So I started this blog on Blogger. And my goals for the blog have never changed. What
I’ve wanted to do all along is:
- To show people the beauty of math. Math is really wonderful. It’s
fun, it’s beautiful, it’s useful. But people are taught from
an early age that it’s useless, hard, and miserable. I want to show
otherwise, by describing the beauty of math in ways that are approachable
and understandable by non-mathematicians.
- To help people recognize when someone is trying to put something past
them by abusing math – what I call obfuscatory mathematics. Because so many people don’t know math, hate it,
think it’s incomprehensible, that makes it easy for dishonest people
to fool them. People throw together garbage in the context of a mathematical
argument, and use it to lend credibility to their arguments. By pointing
out the basic errors in these things, I try to help show people how to
recognize when someone is try to use math to confuse them or trick them.
- To show people that they use and rely on math far more than they think.
This relates back to the first point, but it’s important enough to
justify its own discussion. Lots of people believe that they can’t
understand math, and avoid it like the plague. But at the same time, they’re
using it every day – they just don’t know it. My favorite example
of this is from my own family. My older brother had a string of truly horrible
math teachers, and was convinced that he was horrible at math, couldn’t
understand it, couldn’t do it. You couldn’t even try to teach it to him,
because he was so sure that he couldn’t do it that he’d psych himself out
before he even started. But he’s a really smart guy. When he went to college,
he studied music. I visited him at one point, and was watching him do an
assignment for his music theory course, where they were studying something
called serial composition. He was analyzing a musical score – and what
he was doing to analyze it was taking determinants of matrices in mod-12
arithmetic! Of course, he didn’t know that that was what he was
doing; instead of the numbers 0 through 11, he was using the notes of the
musical scale. But it was taking a determinant, just using a different
symbol set. He had no trouble doing that; but try to teach him to compute
a percentage, and he’ll insist not just that he can’t do it, but that
he’s incapable of learning to do it. That kind of thing is
all too common – people do math every day, without knowing it. If they
understood whata they were doing, they might be open to learning more,
to being able to do more themselves – but because they’ve been taught
that they can’t do it, they don’t see that they do.
This will come around back to my basic point; keep reading below the fold.
I really do honestly believe that
math is absolutely fundamental to how we understand the world, and that
when we’re confronted with figuring out whether or not something is possible,
the process that we use to determine it is, at its core, mathematical. Even
more than that, I believe that it’s impossible to really understand
things like machines and how they work without math. Math isn’t just important – it’s absolutely essential.
When we look at something like the wind-powered vehicle that started this
whole discussion, the way that we try to determine whether or not it works is,
fundamentally, a mathematical process. We build an abstract model of it in our mind, and use that model to analyze it and figure out if/how it works. The only
conclusive way to figure out how/if it works is, ultimately, to build
a quantitative mathematical model, and figure out how the numbers add up.
The reason that it’s always ultimately mathematical is because math – numbers and logic – is the only way to formulate a description of the thing that can actually
be proven to be either correct or incorrect. Math provides the formalism that makes
it possible to say: “Yes, this works” in a way that can be checked by other people.
Of course, just like it’s possible to screw up an informal mechanical description of something, it’s possible to screw up the mathematical model describing something, and get an incorrect result. But the critical difference is, in math, you can show
that the model is wrong – and you can do it in an unmistakeable, undeniable way. That’s how I got convinced in this case – someone posted a mathematical argument
that demonstrated how you could get power from the ground. And even though it went
strongly against my own intuition – it was incontrovertible. I couldn’t find a problem with the math; and if there was no problem with the math, that meant that I had to be wrong. There’s no room for argument there – I had to either show where the math was wrong, or accept that it was right. Math allows you to form unambiguous
analyses like that.
To illustrate why I say math is really essential, it’s useful to think about some
examples. As I’ve said, I’ve been around the net for a while, and I’ve been absolutely
bombarded by crackpots of all kinds. My favorites are the perpetual motion folks; it
just never ceases to amaze me how many of them there are, or how committed they are to
something that can’t possibly work. Ignoring the more sophisticated ones for the
moment, there are still tons of people out there pushing variations on some of the
oldest perpetual motion machines, like the old overbalanced wheel. Those are
based on pure ignorance; we know that those things don’t know, and we know why. But there are people pushing much more sophisticated versions of the same basic idea – ranging from Brown’s Gas to magnetic motors.
Perpetual motion is impossible. We know that (or at least the sane among us know
that). So obviously, none of the perpetual motion machines work. But that doesn’t stop
their inventors from believing that they work. There are plenty of scammers out there (Steorn comes to mind) who know full well that they’re full of shit. But there are also a lot of genuinely honestly deluded folks who really, honestly believe that they’ve found something amazing, and really don’t understand that they’ve got it wrong.
How do you refute one of these people?
Let me take one example – Brown’s gas. Brown’s gas is a name for a mixture
of hydrogen and oxygen, in perfect two-to-one atomic proportions. If you look around the web, you’ll find hundreds of people who really, sincerely believe that they
can produce more energy by burning Brown’s gas than they consume producing it.
In fact, they can even provide calorimetric measurements showing that
the amount of energy coming from their burner is larger than the amount of electricity
that it took to split the water into hydrogen and oxygen.
The usual argument over this follows roughly the following form:
- Look! Brown’s gas provides clean free energy!
- No, that’s impossible.
- Yes, it does!
- No, it can’t. Burning hydrogen and oxygen produces an quantity of
energy exactly equal to the amount of energy it takes to split the
water molecule; and since you can’t extract all of the energy from burning it,
you’re losing energy in the process.
- Yes, it does work! In perfect proportions, hydrogen and oxygen combust in
an implosive manner rather than an explosive one, and that gives us more
energy than it took to split the water!
- No, it can’t.
- Yes, look, here’s some measurements!
And so on. How can you possibly really refute them?
You can argue ’til you’re blue in the face, and the argument will just go around
in circles: “Yes it does”, “No it doesn’t”, forever. No informal argument will ever
produce a conclusive result.
But once you bring math into the discussion, you can force a conclusion. By
working out the energy inputs and the energy outputs, and comparing them. That’s math.
In the Brown’s gas case, in their measurements, the input energy is the amount of
electricity that it took to split the water into hydrogen and oxygen; the output is
the amount of energy they produced by burning them. But there’s a missing element in
the equation: the burning measurements come from burning a mixture of
compressed hydrogen and oxygen. The burning fuel is coming out at a high
speed even before combustion. In the “energy out” measurements, the
measurement includes energy coming from the decompression of the gases.
Brown’s gas proponents will insist that the energy it took to compress the gas
was less than the surplus they’re seeing from burning it. If you use direct
measurements, they’ll claim that it’s unfair, because of inefficiencies
in the compressor you measured, or problems with the valves on the compressed gas
cylinders, or any of a million other problems.
How can you prove them wrong?
Math. You can compute the amount of energy required to split the water
and compress the gas in an ideal system, where the compressors and valves
and electrodes and such are all perfect and lossless. And still, the amount of energy
required in that perfect lossless system to split water and compress the resulting
gases will be greater that the amount of energy extracted by burning
Without the mathematical argument, you can spend an endless amount of time trapped
in arguments about the specific compressor, the mechanics of combustion, and so on. Because for every “But that takes more energy”, they’ve got a comeback saying “No it doesn’t because …”. It isn’t until you reach the point of specific quantitative comparisons – that is, mathematical comparisons – that you get an unambiguous answer.
When I make arguments in cases like this, one of the most common responses is
something along the lines of “But that doesn’t need math. My mechanical intuition
would have convinced me of that without using any math!”, or “I don’t need math: a
working demonstration is conclusive.” My response comes in two parts:
- Mechanical intuition is a fancy word for “doing math in your head”.
It’s one of those examples I mentioned at the beginning of this
article, where I said that people constantly do math without
knowing that it’s what they’re doing. But when your mechanical
intuition tells you that the energy in a system can’t work, you’re
really intuitively saying “It doesn’t add up”; that is, you’ve got
an intuitive sense of the math of the underlying system, and using
that, you’re able to see that it doesn’t work. Even the terminology
that’s commonly used in that situation reflects the underlying math:
“It doesn’t add up”.
- Mechanical intuition can be fooled, and the only way to conclusively show
that the intuition wrong is mathematical analysis. The mechanical
intuition is based on an understanding of the mechanics and relationships
of the components of a system. If that understanding is wrong, you’ll get
a wrong answer. How can you prove to someone that their
mechanical intuition is wrong? It comes back to some kind of
analytical computation – i.e., math.
- Demonstrations are great, provided you really understand what’s being
demonstrated. Just look around the net at the thousands and thousands of
people who believe in magnetic free-energy engines! They’ve got
demonstrations that really appear to work; they’ve got explanations that sound
incredibly convincing. But they’re wrong – because they’ve left
some element out of the analysis. How can you really prove that?
Math: the inputs don’t match the outputs, and you can show that with
an analytical result.
Once again, let me try another demonstration. You can create lots of
cart-like devices that work on treadmills. You can produce
a pairing of a torque-adjusting treadmill with a device to create the appearance
of the vehicle being accelerated by something other than the treadmill. That doesn’t mean that it really is. If you don’t know the mechanics of how the treadmill works,
then putting your vehicle on the treadmill could triggering the treadmill to increase its motor output. If you don’t understand how your treadmill works, how can you be sure that it’s actually introducing energy to the system in a way you didn’t account for?
An important thing to remember here is that math doesn’t have to be
some gloriously complex system of equations. You don’t need to do
a complete navier-stokes computation to show how a wing works. You don’t
need to build molecular models of friction to describe a block sliding down
a ramp. Math is just a framework for formal analysis; it’s the formal
application of logic to provide a very precise description of something.
Putting meters at various locations on a device and comparing the results
is math. One way of working around the problem that I suggested above with
misunderstanding the operation of a treadmill would be to measure the energy being consumed by the treadmill. When you see that the energy consumed by the treadmill increases when a vehicle is placed on it, you’re observing that there’s
an unanticipated energy input to your system. To know if that’s really what’s producing your results, you need to compute how much influence that additional
power can produce. And as soon as you see the work “compute”, you’re in math-land.