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The Unknown and the Unknowable
A Talk With Joseph Traub

Clifford Pickover on The Unknown and the Unknowable by Joseph Traub

From: Clifford A. Pickover

Like Joseph Traub, I too am interested in the unknown and the unknowable. Perhaps that's why many of the questions I pose in my books are unanswered. Some may be unanswerable. As Stanford psychologist Roger Shepard recently noted at a Santa Fe Institute workshop on the limits of scientific knowledge, even if our computers and mathematical tools continue to improve, we may not understand the world any better. He says, "We may be headed toward a situation where knowledge is too complicated to understand." Princeton astrophysicist Piet Hut has pointed out that the structure of the physical universe may represent the ultimate limit on human knowledge. John Horgan (Scientific American) believes that particle physicists may never be able to test theories that unify gravity and the other forces of nature because the predicted effects become apparent beyond the range of any conceivable experiment.

Our brain is our biggest limitation. We can hardly imagine a chimpanzee understanding the significance of prime numbers, yet the chimpanzee's genetic makeup differs from ours by only a few percentage points. These minuscule genetic differences in turn produce differences in our brains. Additional alterations of our brains would admit a variety of profound concepts to which we are now totally closed. What mathematics is lurking out there which we can never understand? How do our brains affect our ability to contemplate God? What new aspects of reality could we absorb with extra cerebrum tissue? And what exotic formulas could swim within the additional folds? Philosophers of the past have admitted that the human mind is unable to find answers to some of the most important questions, but these same philosophers rarely thought that our lack of knowledge was due to an organic deficiency shielding our psyches from higher knowledge.

If the Yucca moth, with only a few ganglia for its brain, can recognize the geometry of the yucca flower from birth, how much of our mathematical capacity is hardwired into our convolutions of cortex? Obviously specific higher mathematics is not inborn, because acquired knowledge is not inherited, but our mathematical capacity *is* a function of our brain. There is an organic limit to our mathematical depth.

How much mathematics can we know? The body of mathematics has generally increased from ancient times, although this has not always been true. Mathematicians in Europe during the 1500's knew less than Grecian mathematicians at the time of Archimedes. However, since the 1500's humans have made tremendous excursions along the vast tapestry of mathematics. Today there are probably around 300,000 mathematical theorems proved each year.

On a similar line of thought, a dog cannot understand Fourier transforms or gravitational wave theory. Human forebrains are a few ounces bigger than a dog's, and we can ask many more questions than a dog. Linguist Noam Chomsky once noted that a rat can learn to turn left at every second fork in a maze, but not at every fork corresponding to a prime number. The human mind, limited by the same kinds of biological constraints as the rat, may reach the edge of its ability to comprehend. We are flesh and blood, not Gods. Are there facets of the universe we can never know? Are there questions we can't ask? Our brains, which evolved to help us find food on the African plains, are not constructed to penetrate all the enigmas in the infinite mathematical cloak of our universe.

Note, however, we do have a chance of understanding a great deal about the universe. The fact that reality can be described or approximated by simple mathematical expressions suggests to me that nature has mathematics at its core. Formulas like E = mc**2 , F = m*a , 1 + e**(i*pi) = 0 , and lambda = h/mv all boggle the mind with their compactness and profundity.

The shape assumed by a delicate spider web suspended from fixed points, or the cross-section of sails bellying in the wind, is a catenary -- a simple curve defined by a simple formula. Seashells, animal's horns, and the cochlea of the ear are logarithmic spirals which can be generated using a mathematical constant known as the golden ratio. Mountains and the branching patterns of blood vessels and plants are fractals, a class of shapes which exhibit similar structures at different magnifications. Einstein's E = mc**2 defines the fundamental relationship between energy and matter. And a few simple constants -- the gravitational constant, Planck's constant, and the speed of light -- control the destiny of the universe. I do not know if God is a mathematician, but mathematics is the loom upon which God weaves the fabric of the universe.


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