Since writing that blog post, I have thought a lot more about the reality of math. And one reason is that I have since written a book about a set of fractal curves that I “discovered”. Or, did I “design” them? (read on to find the answer :)
Rotman’s review is quite critical. Tegmark claims that mathematics is the very foundation of the structure of the universe. He goes as far as to say that “Our reality isn’t just described by mathematics – it is mathematics”.
Rotman would be one of a growing number of mathematician/philosophers who take issue with this kind of claim. They would counter that mathematics is a human invention. My first encounter of this idea was through Lakoff an Nunez in their book, Where Mathematics Comes From – How the Embodied Mind Brings Mathematics into Being. This book explores mathematics from the point of view of cognition and linguistics.
All very well and good. Math comes from human brains. But the more I engage in mathematical activities, the more sympathetic I become with mathematicians who believe that they are “uncovering” great truths rather than creating ideas out of thin air. While exploring mathematical ideas, the feeling that I am discovering something outside of myself is very strong. What is the reason for this feeling? What is happening in my brain that makes this feeling so strong?
Why I am Becoming a Mathematical Agnostic
I am just about finished reading The Number Sense by Stanislas Dehaene, who is working on the cutting edge of brain imaging and identifying cerebral structures associated with abstract number and calculation.
Dehaene believes that math is a human invention.
However, he acknowledges the long reality that precedes us. According to Dehaene, the notion of number is perhaps the most primal mathematical construct that our brains perceive, and we share a rudimentary number sense with other animals.
No surprise here: the universe – at least on the scale of mammals and birds, has clumps of matter and events that we perceive and that are meaningful to us. And of course the biosphere is doing its part to continue clumpifying matter and events, including me writing this blog post.
“Number” could be seen as an emergent property of the biosphere. Animals have evolved behaviors associated with number (as well as symmetry, and other attributes associated with math). In turn they impose their emergent representations back onto the environment as part of an ongoing feedback loop of complexification. Humans have simply taken this feedback loop to a conscious level.
My conclusion: to say that math is strictly a human invention is taking it too far. Math is not arbitrary. We are made out of structured matter that exists throughout the universe. Our bodies, brains, and minds are tuned-in to that structure. We evolved with it and in it. Our language, as well as our genes, are “about” the biosphere, which is “about” the universe.
On Discovering vs. Designing
In Brainfilling Curves, I chide Mandelbrot for saying he “designed” certain fractal curves. I can’t say that I blame him, considering how clever the snowflake sweep is, for instance. But I prefer to say he discovered it.
And I claim that many of the fractal curves in my book were discovered by me. But it gets a little fuzzy at times. Geometrical objects range in primality from the circle (which no one would claim to have designed) to rare, high-order self-avoiding space-filling curves, many of which I have spent years uncovering. The more rare, the more unlikely the geometrical object, and the more information required to describe it, the more like design it becomes. I see it as a continuum. At the far extreme of this continuum would be a painting by Kandinsky.
I believe that we feel the sensation of mathematical discovery because of the evolution of our brains, language, and culture, from which we cannot escape.
The evolution of our brains, language, and culture are a continuation of the evolution of the structure of the universe. Thus math, while it may not originate from the universe, is a language we invented which is finely tuned to it.
And since we are made of the stuff that our math describes, our math feels…