So, more than half a century ago, Eugene Wigner wrote
about, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
>> Yeah.
>> And, and, indeed one man can sit in one's living room and
do Duncan experiments and, maybe you know what that word means? I don't.
>> Duncan means thought.
>> Okay, thought experiments.
>> Thought experiments.
>> You can do these thought experiments and come up with something like the
gene-, General Theory of Relativity. You discover
black holes and the curvature of space-time.
And why is that so? Could it be that we only perceive what we
can describe, without mathematics. And vice versa?
Because mathematics is co-evolved with our brain.
>> Look, Eugene Wigman made a big problem for every generation
that came after him. To answer that question, why mathematics works.
Why does mathematics? And sometimes I think
to myself, what, how could it not work?
The world has some coherence to it.
Things don't just randomly happen. How do you
describe things that don't randomly happen? If
they don't randomly happen, you have to have
some kind of quantitative framework for explaining what
happens. So I can't imagine, on the one
hand I can't imagine a world that didn't work according to some kind of laws
and those laws being written mathematically. On the
other hand, I also can't quite understand why
mathematics works. So Eugene asked a very difficult
question. And he didn't know the answer. And
I think I, I, I don't know the answer, either. But here's what I would say.
Here's what I would say. Not, why does mathematics work, but why do we need
mathematics to explain physics, why is it
so hard to explain physics in the English language? And the reason
is, because, yeah, we may have evolved as a species
on mathematical abilities, but we evolved them, in the context,
of rather ordinary things. The ordinary things would be,
how stones move when you throw them. When it comes
to waves, we have some perspective about what a
wave is because we see water making waves and so
forth. We have a good concept of what a
force is because if I give you a good shove,
you'll say, oh, don't force me. We know what
force means. But every time we enter into a new
range of parameters We make things smaller than anything that
we were, than we grew up with. Other words, we go to the quantum world. Or we go
to the world of very, very heavy things and dense things. In other worlds,
we go to the world of black holes. We suddenly find that the intuitions and
the concepts that we evolved with are not sufficient
to understand. An example, we use higher dimensional spaces all
the time. We think we sometimes think for example, that
the that space itself may be more than three dimensions.
You evolve. You and I evolved in a world of three dimensions. We can't visualize
more dimensions. And we can't because our
it, it's not that we're not smart enough.
It's that the neural architecture, the architecture
of the brain itself. Was purposefully built, well
it wasn't purposely built, but it evolved
in the world of three dimensions. The architecture
is appropriate for three dimensions and it's
not appropriate for four, five, six, seven
dimensions. If we think that we're interested
in a space with seven dimensions or whatever,
how would we describe it if we can't visualize it. We describe it by pure
mathematics. We say a three dimensional world is described by a bunch of points
that are labeled by X, Y, and Z
that's an abstraction. I can visualize three-dimensions on
my head, but I use x, y, an z to describe them mathematically. Now I go
to four dimensions, or five dimensions. I can't
visualize it anymore. I can't visualize it anymore.
I can't close my eyes an see a, a five-dimensional world. But I can add two
more letters to the alphabet. I can add the x, y, and z, w, and
v. And now I have a five dimensional space. How do I work with it? I work with
it using algebra. I work with it using
using abstract mathematics. And I don't try to see
it in my head. In trying to explain
it to people who don't have the mathematical background,
we get stuck. I can't say, now close your eyes and view in your head five
dimensions, because I can't do it either. What I, all I can do is say, take
x, y, and z and add v and w. Eventually if, eventually a
person with this kind of mathematical background begins
to develop mathematical intuitions for things.
They don't need the same kind of visualization. So, we are stuck. We're stuck
needing mathematics because evolution didn't equip us to be able to
visualize quantum mechanics. It didn't visualize us to be able to, there
sorry, it didn't equip us to be able to visualize 11 dimensional space-time.
And tha, that's the reason. Evolution did it to us.