The marriage of math and art in
3D printing
This is
really nice. it allows those of us who do not grasp even algebra very well to
see that the beauty of the world is mathematical and that the beauty of the
world is a reflection of the beauty of mathematics in the same way that the
brightness of the moon reflects the light of the sun, the same light that
feeds, directly or indirectly, everything on earth that grows and blossoms.
I think
that statistics served this same function for me. I wondered for a long time
why I was good at statistics but weak in math. I think I know now. Statistics
looks at the patterns in nature without ever going so deeply into the reasons
these patterns are there that the explanation is anywhere nearly as remote from
everyday life as E=MC2. Nate Silver’s predictions of the outcomes of elections
and sporting events always refer at only one or two removes to something one
can see or some aspect of social life that anyone with the ability to read
other people’s minds well enough to interact with others about as skillfully as
most of us do will grasp. Examples might be how high a wide receiver can jump
or what the typical human response to someone who is obese or a bully says
about Chris Christie’s political future.
You
have to imagine, as Einstein could but most of us cannot, what it would be like
to be pushed through the universe by the edge of a beam of light tucked neatly
under your butt to really get relativity. String theory and the rest of
theoretical physics sit several removes more distant from everyday experience
than even this. Nothing I ever encountered earning graduate degrees in English
and Psychology was ever so remote from human experience that I could not give
examples. I could always tell a story about myself or someone else that
illustrated the point I needed to make.
All of
literature from Homer to Nietzsche to Michel Houellebecq is finally nothing
more than made-up stories that illustrate what the human life-world does to all
of us who, being human, all too human, have no place else to live or go. Some
of the stories describe a world that looks like the place we find ourselves. We
call that realism. Others describe what the world would look like if
appearances were not deceiving. We call those myths. Both are fascinating and
both, if we dive deep enough, are based on a few basic patterns endlessly
articulated as variations on these themes.
These themes themselves are almost certainly
based on deeper patterns which are almost certainly based, in turn, on even
deeper patterns. Yet, if we keep diving deeper and deeper, we will eventually
come to patterns that cannot be told as stories but only written in abstract
mathematics that describe patterns we cannot imagine because they exist at so
many removes from anything we could directly live. These are the patterns that
Eugene Wigner, who came of age one generation after Einstein, earned his own
Noble Prize describing. They cannot be told as story, even as a story as
fantastic as Einstein’s story about riding a light beam.
Wigner
expressed where the exploration of the deepest patterns we could describe using
the most abstract symbols available in his lifetime left us when considered as
a story about our world when he said that we had reached the point of knowing
that the truth is not just stranger than we imagine but stranger than we can
imagine. The advent of computers has made the search for truth easier by doing
math for us that we could never do ourselves but the task of telling the
deepest truth we know as story even more hopeless. Wigner also once said:”
It is nice to know that
the computer understands the problem. But I would like to understand it too.”
For more Wigner, go here:
"I
shall try again and again to show that what is considered a mathematical
discovery had much better be called a mathematical invention" (p.22 in Lectures on the Foundations of
Mathematics: Cambridge 1939 - 1976 University of Chicago Press)
"The
mathematician is not a discoverer; he is an inventor" (Remarks on the
Foundations of Mathematics, appendix II, #2.
"The
whole modern conception of the world is founded on the illusion that the
so-called laws of nature are the explanations of natural phenomena"
(Tractatus, #6.371)
"Laws
...are about the net and not about what the net describes" (Tractatus,
#6.35)
"What
a Copernicus or Darwin really achieved was not the discovery of a true theory
but a fertile new point of view" (Culture & Value, p.18).
All
these Wittgenstein quotes have in common the idea that mathematics is as little
a product of nature and as much a product of the creative imagination of human
beings as Greek mythology. This makes “-1” real in the same sense and only in
the same sense that a “wood-nymph” is real. I take that idea for granted. That
said, the most intriguing of the Wittgenstein quotes is: "The whole modern
conception of the world is founded on the illusion that the so-called laws of
nature are the explanations of natural phenomena." The thing that makes
this quote intriguing is the unanswered question that must occur to anyone who
takes the point of view assumed by all these quotations for granted as I do.
I
always use the same example when I talk about this issue because this example
is paradigmatic; answer the question about this example and you have answered
the question in all cases where the question could be meaningfully asked. The
square root of a number is the number which when multiplied by itself equals
that number. Square root as a
mathematical concept is so old that no one can say who invented it; the concept
was part of the established tradition of mathematics that would have been
taught to Euclid, Pythagoras or any other schoolboy in ancient Greece who was
being shown how to count and measure things. Draw a square made of dots. If the
number of rows is equal to the number of columns, the diagonal will be the
square root of the number of dots contained in the square. This is simple, easy
to demonstrate with a diagram and requires no leap of faith or flight of
imagination to grasp for anyone of reasonable intelligence who can count.
Greek math had its roots in practical problems
involving physical things that needed to be counted, measured or built.
Negative numbers came much later when some mathematicians began using them to
solve math problems that were purely intellectual exercises. They were driven
by their own abstract curiosity to solve problems that even the people solving
them never thought had any practical application to counting, measuring or
building anything. At first, the people using -7 or -9 to solve these
intellectual puzzles never claimed that -7 or -9 were anything other than
creations of their own imagination. No one had the audacity to claim that
negative numbers and the wild flights of mathematical fancy that they made
possible had any significance beyond the joy solving these problems or
following their solutions gave to a very small group of people whose idea of
fun was rather peculiar.
Anything
involving negative numbers is already one remove in abstraction away from
anything that can exist physically and be directly observed. Asking what the
square root of a negative number might be takes us two removes away from
anything physical. The invention of negative numbers was a flight of fancy as
pure as the creation of any myth. The invention of i to represent the score
root of a negative number was a second flight of fancy that could only start
from the Never-never land where taking the first lands us. I can now ask, from
where these two leaps of fancy put me, the question that truly intrigues me; if
"the mathematician is not a discoverer; he is an inventor “ does this not
mean that, for example, Lewis Carroll, was composing works of imagination every
bit as much when he was doing logic as when he was writing Through the Looking
Glass?
Myth is not a realistic report of human
experience. Myth is an imaginary place where you can stand to look back into
human experience and see it more clearly, clearly enough to see that a factual
report is not the truth of human experience because the facts are not the
truth; what the facts mean is the truth and that is best seen not from inside
the facts but from the perspective of myth, from the place one or more leaps of
fancy and faith taken beyond the facts might leave you. This is all
boiler-plate archetype theory straight out of Jung. Some leaps beyond the facts
are useful, even necessary to our full development as spiritual beings having a
physical experience, so we see those same leaps taken over and over again by
many different people who describe the place they land in remarkably similar
ways.
Mathematics
is not a report of physical observables. Mathematics is an imaginary place
where you can stand to look back into the physical and see it more clearly,
clearly enough to see that physical observables are not the whole truth even of
what is happening in the physical world because the observable is not the real;
what the observable means is the real and that is best seen not from inside the
physical but from the perspective of mathematics, from the place one or more
leaps of fancy and faith taken beyond the facts might leave you. Some leaps
beyond the physical are useful, even necessary to our full development as
spiritual beings having a physical experience in a hypercube where the physical
is but one slice, the slice that contains our bodies, so we see those same
leaps taken over and over again by many different people who describe the place
they land in remarkably similar ways.
I’m
seeing Myth and Mathematics as two complimentary ways of seeing the physical as
a profusion of signs and the real as what these signs indicate, the one based
on narrative and the other based on numbers. This would make -1 and the square
root of -1 necessary leaps of fancy and faith in the same sense as the dying
god and building the temple on the very stone that the master builder rejected.
The things we make up in our flights of fancy, either narrative or numerical,
turn out to be real because imagination is a sense as much as sight or hearing.
Imagination is not a physical sense but the sense that makes us aware of what
we can know of the vaster real of which the physical is only one slice, only a
profusion of signs that can mean the vaster real if properly read.
We can
know through imagination and intuition because we are not just the bodies that
exist in the physical but spirits who can see what the physical means from the
perspective of the real. Our imaginings and intuitions tend to leap beyond the
facts given by direct observation of the physical in the general direction of
the real by reading the given facts as signs. We are pulled beyond the given by
a search for larger patterns that do not fully manifest in the given, pushed
beyond the given by a need both emotional and intellectual to see more than is
there. The “more” we add feels right and true to the extent that this “more”
makes the given into clues to something much more richly interesting and
satisfactory than just what is there. They (mathematicians or, as I am
thinking of them, number poets) were driven by their own abstract curiosity to
solve problems that even the people solving them never thought had any
practical application to counting, measuring or building anything.
Yet the
world we have counted out, measured and built since, complete with nuclear bombs
and computers, only works because the things they imagined turned out to be
real enough to make possible the construction of these physical things. The
question left unanswered by knowing that "the mathematician is not a
discoverer; he is an inventor" and that "The whole modern conception
of the world is founded on the illusion that the so-called laws of nature are
the explanations of natural phenomena” is the question of why when we act as if
these “inventions” were discoveries about the real and construct nuclear bombs
and computers under the influence of this “illusion” the damn things, for
better or worse, work. I believe all the craziness I have written above about
mathematics because this is the only explanation I can come up with for how and
why this should be the case that really makes sense to me. Being able to
converse with someone half a world away because a machine based on circuitry
that can only be designed under the assumption that -1 has a square root is
only slightly less astonishing to me than being sexually propositioned by a
randy wood-nymph during a stroll through the park.
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