The
mathematicians we’ve encountered in this book are not just puncturers of
unjustified certainties, not just critics who count. They found things and they
built things. Galton uncovered the idea of regression to the mean; Condorcet
built a new paradigm for social decision making; Bolyai created an entirely
novel geometry, “a strange new universe”; Shannon and Hamming made a geometry
of their own, a space where digital signals lived instead of circles and
triangles; Wald got the armor on the right part of the plane.
Every
mathematician creates new things, some big, some small. All mathematical
writing is creative writing. And the entities we can create mathematically are
subject to no physical limits; they can be finite or infinite, they can be
realizable in our observable universe or not. This sometimes leads outsiders to
think of mathematicians as voyagers in a psychedelic realm of dangerous mental
fire, staring straight at visions that would drive lesser beings mad, sometimes
indeed being driven mad themselves.
It’s not like
that, as we’ve seen. Mathematicians aren’t crazy, and we aren’t aliens, and we
aren’t mystics.
“What’s true is
that the sensation of mathematical understanding—of suddenly knowing what’s
going on, with total certainty, all the way to the bottom—is a special thing,
attainable in few if any other places in life. You feel you’ve reached into the
universe’s guts and put your hand on the wire. It’s hard to describe to people
who haven’t experienced it.
We are not free
to say whatever we like about the wild entities we make up. They require
definition, and having been defined, they are no more psychedelic than trees
and fish; they are what they are. To do mathematics is to be, at once, touched
by fire and bound by reason. This is no contradiction. Logic forms a narrow
channel through which intuition flows with vastly augmented force.
The lessons of
mathematics are simple ones and there are no numbers in them: that there is
structure in the world; that we can hope to understand some of it and not just
gape at what our senses present to us; that our intuition is stronger with a
formal exoskeleton than without one. And that mathematical certainty is one
thing, the softer convictions we find attached to us in everyday life another,
and we should keep track of the difference if we can.
Every time you
observe that more of a good thing is not always better; or you remember that
improbable things happen a lot, given enough chances, and resist the lure of
the Baltimore stockbroker; or you make a decision based not just on the most
likely future, but on the cloud of all possible futures, with attention to
which ones are likely and which ones are not; or you let go of the idea that
the beliefs of groups should be subject to the same rules as beliefs of
individuals; or, simply, you find that cognitive sweet spot where you can let
your intuition run wild on the network of tracks formal reasoning makes for it;
without writing down an equation or drawing a graph, you are doing mathematics,
the extension of common sense by other means. When are you going to use it?
You’ve been using mathematics since you were born and you’ll probably never
stop. Use it well.
