I used to worry that our mathematical and physical descriptions of the universe grow forever, but our brains and language skills remain entrenched. Some of our computer chips and software are becoming mind-numbingly complex. New kinds of mathematics and physics are being discovered or created all the time, but we need fresh ways to think and to understand.

I used to worry that we will understand less and less about more and more. For example, in the last few years, mathematical proofs have been offered for famous problems in the history of mathematics, but the arguments have been far too long and complicated for experts to be certain they are correct. Mathematician Thomas Hales had to wait *five years* before expert reviewers of his geometry paper—submitted to the journal *Annals of Mathematics*—finally decided that they could find no errors and that the journal should publish Hale's proof, but only with the disclaimer saying they were not certain it was right! Moreover, mathematicians like Keith Devlin have admitted in *The New York Times* that "the story of mathematics has reached a stage of such abstraction that many of its frontier problems cannot even be understood by the experts."

Israeli mathematician Doron Zeilberger recently observed that contemporary mathematics meetings were venues where few people could understand one another, nor even made an attempt, nor even cared. The "burned out" mathematicians just amble from talk to talk where "they didn't understand a word." Zeilberger wrote in 2009, "I just came back from attending the 1052nd AMS [American Mathematical Society] sectional meeting at Penn State, last weekend, and realized that the Kingdom of Mathematics is dead. Instead, we have a disjoint union of narrow specialties… Not only do [the mathematicians] know nothing besides their narrow expertise, they don't care!" Incidentally, Zeilberger considers himself to be an *ultrafinitist*, an adherent of the mathematical philosophy that denies the existence of the infinite set of natural numbers (the ordinary whole numbers used for counting). More startlingly, he suggests that even very large numbers do not exist—say numbers greater than 10 raised to the power of 10 raised to the power of 10 raised to the power of 10. In Zeilberger's universe, when we start counting, 1, 2, 3, 4, etc., we can seemingly count forever; however, eventually we will reach the largest number, and when we add 1 to it, we return to zero!

At the risk of a further digression into the edges of understanding, consider the work of Japanese mathematician Shinichi Mochizuki. Some of his key proofs are based on "inter-universal Teichmüller theory." As he develops future proofs based on this mathematical machinery, which was developed over decades in many hundreds of pages, how many humans could possibly understand this work? What does it even mean to "understand" in contexts such as these? As mathematics and subatomic physics progresses in the 21st century, the meaning of "understanding" obviously must morph, like a caterpillar into a butterfly. The limited, wet human brain is the caterpillar. The butterflies are our brains aided by computer prosthetics.

*Should* we be so worried that we will not *really* be able to understand subatomic physics, quantum theory, cosmology, or the deep recesses of mathematics and philosophy? Perhaps we can let our worries slightly recede and just accept our models of the universe when they are *useful*. Today, we employ computers to help us reason beyond the limitations of our own intuition. In fact, experiments with computers are leading mathematicians to discoveries and insights never dreamed of before the ubiquity of computers. Computers and computer graphics allow mathematicians to discover results long before they can prove them formally, and open entirely new fields of mathematics. American educator David Berlinski once wrote, "The computer has… changed the very nature of mathematical experience, suggesting for the first time that mathematics, like physics, may yet become an empirical discipline, a place where things are discovered because they are seen." W. Mark Richardson fully understands that scientists and mathematicians must learn to live with mystery. He wrote, "As the island of knowledge grows, the surface that makes contact with mystery expands. When major theories are overturned, what we thought was certain knowledge gives way, and knowledge touches upon mystery differently. This newly uncovered mystery may be humbling and unsettling, but it is the cost of truth. Creative scientists, philosophers, and poets thrive at this shoreline."