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degrees Centigrade. The theorists were on their own here; experimentalists could offer little more than good wishes. All during 1944 the computation effort grew. John von Neumann served as a traveling consultant with an eye on the postwar future. Von Neumann—mathematician, logician, game theorist (he was more and more a fixture in the extraordinary Los Alamos poker game), and one of the fathers of modern computing—talked with Feynman while they worked on the IBM machines or walked though the canyons. He left Feynman with two enduring memories. One was the notion that a scientist need not be responsible for the entire world, that social irresponsibility might be a reasonable stance. The other was a faint, early recognition of the mathematical phenomena that would later be called chaos: a persistent, repeatable irregularity in certain equations as they prepared to run them through their primitive computers. As a shock wave, for example, passed though a material, it left oscillations in its wake. Feynman thought at first that the irregular wiggles must be numerical errors. Von Neumann told him that the wiggles were actually features of interest.

Von Neumann also kept these new computer specialists up to date with the other sites he visited. He brought news of an electromechanical Mark I under construction at Harvard, a relay calculator at Bell Laboratories, human neuronal research at the University of Illinois, and at the Aberdeen Proving Ground in Maryland, where problems of ballistic trajectories motivated the calculators, a more radical device with a new kind of acronym: ENIAC, for Electronic Numerical Integrator and Computer, a machine composed of eighteen thousand vacuum tubes. The tubes controlled binary on-off flip-flops; in a bow to the past, the flip-flops were arranged in rings of ten, to simulate the mechanical wheels used in decimal calculating machines. The ENIAC had too many tubes to survive. Von Neumann estimated: “Each time it is turned on, it blows two tubes.” The army stationed soldiers carrying spare tubes in grocery baskets. The operators borrowed mean free path terminology from the ricocheting particles of diffusion theory; the computer’s mean free path was its average time between failures.

Meanwhile, under the influence of this primal dissection of mathematics, Feynman retreated from pragmatic engineering long enough to put together a public lecture on “Some Interesting Properties of Numbers.” It was a stunning exercise in arithmetic, logic, and—though he would never have used the word—philosophy. He invited his distinguished audience (“all the mighty minds,” he wrote his mother a few days later) to discard all knowledge of mathematics and begin from first principles—specifically, from a child’s knowledge of counting in units. He defined addition, a + b, as the operation of counting b units from a starting point, a. He defined multiplication (counting b times). He defined exponentiation (multiplying b times). He derived the simple laws of the kind a + b = b + a and (a + b) + c = a + (b + c), laws that were usually assumed unconsciously, though quantum mechanics itself had shown how crucially some mathematical operations did depend on their ordering. Still taking nothing for granted, Feynman showed how pure logic made it necessary to conceive of inverse operations: subtraction, division, and the taking of logarithms. He could always ask a new question that perforce required a new arithmetical invention. Thus he broadened the class of objects represented by his letters a, b, and c and the class of rules by which he was manipulating them. By his original definition, negative numbers meant nothing. Fractions, fractional exponents, imaginary roots of negative numbers—these had no immediate connection to counting, but Feynman continued pulling them from his silvery logical engine. He turned to irrational numbers and complex numbers and complex powers of complex numbers—these came inexorably as soon as one from facing up to the question: What number, i, when multiplied by itself, equals negative one? He reminded