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Comment on this entry

Beautiful, Unreasonable Mathematics


Gregory Benford


Ethical Technology

January 17, 2013

I find most beautiful not a particular equation or explanation, but the astounding fact that we have beauty and precision in science at all. That exactness comes from using mathematics to measure, check and even predict events. The deepest question is, why does this splendor work?


...

Complete entry


COMMENTS



Posted by b.  on  01/17  at  02:15 PM

Thank you. I really appreciate this text.

I’m thinking through this very topic, and I’m curious what your response would be. You don’t take into account the very human ability to abstract, to throw away details that refute the theory, in order to concentrate on the details that reinforce it. (Think of false memories for an example of to the extreme degree we can replace reality with theory.) This is not something we do consciously, but I would ague is inherent in how our minds/brains work.

When it comes to measurement, we must actively choose something to measure. In short, we must have a theory in place a priori that expects a degree of importance of that parameter. Because its a priori, it already exists in our minds, is already situated in a social and cultural context of ideas and values. We can’t measure everything at the same time. Of course these theories can be correct, but more than one theory can explain the same measurements, and may even be incompatible: Perhaps for example particle-wave duality.

How can we know that the regularities we see are not due to a particular interaction of scale, perspective and existing theory? Take any example of symmetry, does that symmetry hold in all contexts? If we zoom in close enough (with a microscope) is it still symmetrical? You say the world does not perfectly manifest the math, but “approximately realizes them”, this approximation sounds like the kind of abstraction I began with. To reinforce the theory, we emphasize those measurements at particular scales through particular points of view that indicate symmetry and ignore the scales and points of view that refute symmetry.

Our cognitive and perceptual abilities are directly tuned to this ability to see a pattern despite the refuting evidence, which we just call “noise”.

The difficulty in determining whether it was us or math that came first is that we have all learned a set of concepts and theories as we are brought up. Our very acquisition of language involves the refinement of arbitrary categories of things out in the world, binary distinctions between dogs and cats that are not hard, but soft gradients. Anything you say about a dog, can be said about a cat (to some degree, at some scale, from some perspective). Any single line that separates their properties, will likely fall apart under closer inspection. I would argue this is because the world is dense and continuous, no matter how deep we get we will continue to find more density and continuity.

I’m not saying math is not real, I think it is real the same sense as Dennet’s weak realism. I am saying that the reality represented in math is a function of both our human biases, expectations and desires, and the real physical processes that occur in the material world. I don’t think these two parts are so easily separated. Math can be both real and socially constructed at the same time.





Posted by SHaGGGz  on  01/17  at  07:51 PM

@b.: I think you’re right. Something vaguely analogous would be how culture and language can shape human perception of color, separating it into some categories and not others. Despite the fact that the EM spectrum is continuous and exists independently of humans, the nature of human physiology shapes what is being perceived; despite the fact that the logical possibility space described by mathematics exists independently of humans, the way humans construct theories shapes what is being perceived.





Posted by jasoncstone  on  01/18  at  05:59 PM

It’s also interesting to consider the idea that the only things we can be aware of are those things that our consciousness organizes in a certain way. Maybe all we can be aware of is a mental construct that is being actively made compatible with our human facility of awareness.

For instance, what if all our mind could ever perceive were things that were displayed on a mosaic tiling of a certain kind. Mathematics would then be the study of all possible patterns that could appear within that mosaic system and science would be the study of how frequently different patterns appear and, possibly, how they relate to one another when they do appear.

Maybe we really are converging on a perfected knowledge of all the patterns we can and do perceive and that has great utility for us as individuals and a society. The Universe does not have to be a Platonic reality just because all that we can perceive is conditioned by a mathematical-like framework that limits what our minds are capable of perceiving.

I find this perspective somewhat compatible with all three views you shared above.





Posted by rms  on  01/31  at  04:55 AM

Humans can learn to predict where a moving stone will go, and since
this was useful during prehistory for hunting, it is plausible that
this faculty was selected for.  But I doubt that it depends on
mathematical appreciation of something as specific as a parabola—
especially since doing a really good job of this prediction involves
taking account of effects that make the trajectory vary from a
parabola, such as air resistance, wind, and rotation.  What equation
defines the motion of a thrown baseball?  I don’t think ballplayers
need to know it.  What they do depend on is reproducibility of the
motion.

Tarsiers hunt by jumping from one tree branch to another very
accurately.  Perhaps they use the same mental faculty.






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