I hope to use this blog to find new things to see about mathematics. And this might even have some effect on how we see ourselves. I want to start with something Henri Poincare said. Poincare was an intellectual heavyweight, a mathematician, theoretical physicist, and philosopher. He was the last person to be able to know all of the mathematics that existed in his lifetime. And he made some nice remarks about mathematics and science, a bit unusual in their use of nature images. I’ve chosen to begin with this one because it led me to thoughts that contribute to this blog’s perspective:
“Though the source be obscure, still the stream flows on.”
There are no easy answers to questions about what mathematics is or where it’s coming from. But it may be that its source is obscure because we’re so close to it or, more to the point, because we’re embedded in it. Looking for it would be like looking for the source of all that we see when we open our eyes, which is very close to trying to find the source of our awareness.
I have come to think of mathematics as akin to vision. For us, seeing begins when light hits all of the material around us and makes some impression on the retina, which is built precisely to receive it. The brain then needs to piece bits of stimuli together, like movement, color, and form. Some visual theorists will go so far as to say that the brain invents the image we see. It searches out the essence of things so that, for instance, no matter how a speaker might be moving his hands in gesture, a hand is a hand is a hand. So vision has no single source. The source of what we see is, all at once, the light, the material it hits, and eye and brain tissue.
In mathematics, some of what the mind is piecing together may be the things we’ve seen, the ways we’ve reasoned, our experiences of time and distance, and pure products of our imagination. We take the contour of things we see, like the sun and the moon, and in our memory and imagination we find the circle. From two trees, two fingers, and two dogs…we form quantity and the number two. Some fundamental aspect of reason builds ‘if this then not that’ statements.
Perhaps mathematics inherited from vision the purpose of finding the essence of things. But it defines essence or equivalence and exploits it, creating extraordinary generalizations. Although it can be completely removed from the physical world, mathematics is somehow stretching what the body is made to do. And the brain keeps building structure, interlocking different parts of our experience and our thought, and finding things like analytic geometry or the derivative.
Poincare was an intuitionist meaning that he believed mathematics was grounded in intuition rather than logic. And he must be right. Logic just wouldn’t be enough. Why nature provides us the abilities that our mathematics reflects is the real mystery. Just like the question, why it has provided us eyes.
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I don’t think I said that directly but it can be pulled from the combination of what I say about visual processes (looking for the essence of things) and what I say about the circle being in the sun and the moon. I don’t think there have been observations of the kind of deficiency you are describing, but in a recent post – Seeing, Touching and Doing Mathematics – there is some observation of the kind of learning required to create the kind of object consistency you’re talking about.
Hello,
I’m looking over your old posts with a new eye and noticing some things. In this post, are you saying, essentially, that the circle is the mental pattern or construct that helps us to see the sun as the sun, no matter if it’s hiding partly by tree branches, or squashed a bit by refraction when it’s near the horizon? It’s an interesting assertion if so.
But is there any research that shows this? For instance, is there any research that indicates that, for those that somehow have a problem with shapes like circles, they get confused recognizing different manifestations of the sun as the same object.
Very interesting! It’s unusual to hear math discussed in such a cross-category way. Would you agree that vision is producing mental constructions which are then permitting us to do mathematics? I hope you follow-up soon.
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Nice post. Did you hear the story on NPR about the switch from analog signal to digital television, and end of “snow” at the end of the broadcast day? The wonderful part of the story was what TV snow actually was. Like our brains, TV tubes are built to try and put radio waves together into a visual picture, and when TV stations stop broadcasting a signal there is just random background radiation. Some of that is signals from the immediate environment, but it turns out that a lot of it is residue from the Big Bang. So the TVs are industriously putting together a little gray snapshot of our distant origins. How cool is that???