Epstein Secrets

more complex than they first appear. They are infinite, yet there are different infinities, and they have meaning. The smallpox example above and the Turing numbers we will discover shortly suggest numbers do have meaning independent of culture and language. The next two chapters will show us what happens

away their secrets. The Enigma machine was an elegant compromise between a truly unbreakable code and a simple cipher. Unfortunately for the Germans, Turing was on the side of the Allies. In the 1930s almost all mathematics, accounting, and code-breaking were performed by humans using pencil and paper. I

r the occupants of the bus is larger than infinity. Real Numbers What about the real world we live in? Is the larger infinity we failed to fit into Hilbert’s Hotel present, or was it just a mathematical fiction? Hold up your thumb and index finger for a moment. The gap between them is a distance. Most li

g. Furthermore, solving the problems would have a major impact on enriching mathematics. They were very different from the imitation game, which, as Turing explained, had no particular intellectual interest but could be useful in encouraging the development of better hardware/software. Incidentally, as

rld country. Paris is better, I'm sure. These are interesting suggestions, but I'm not sure they are the right ones. They're very different from the Hilbert questions. These were deeply embedded in rich theoretical understanding, and were at the borders of such understanding. Furthermore, solving the pr

g. Furthermore, solving the problems would have a major impact on enrichi=g mathematics. They were very different from the imi=ation game, which, as Turing explained, had no particular intellectual int=rest but could be useful in encouraging the development of better hardware/software. Incidentally, as

ld country. Paris is better, I'm sure. These are interesting suggestions, bu= I'm not sure they are the right ones. They're very =ifferent from the Hilbert questions. These were deeply embedded in rich theoretical understanding, and were at the borders of such understand=ng. Furthermore, solving the pro

but, using this simple idea, Bertrand Russell changed the course of mathematical history and it is the fundamental paradox used to show computers are Turing limited. The Russell Paradox In the late 19 century, mathematicians began to think about the nature of numbers. What is a number? It is certainly

ain infinity to a child they immediately ask “What's infinity plus one.” A particularly smart kid I met, Dermot, asked, “What's infinity plus three?” Hilbert’s Hotel allows us to answer this problem in a way we can visualize. HOUSE_OVERSIGHT_015872 vhylbyyy | | tt tt VAVAVAVA A MY YI | i tt tt tt tt tT

se could be preprogrammed with every theory we could ever discover within it. (There would be no arbitrary problems.) This argument neatly sidesteps Turing’s theorem by specifying there is no such thing as an arbitrary problem - a random problem picked from the infinite set of problems. At the same time,

disproof of P =NP @ 1.5 The Poincaré Conjecture @ —- Solved 1.6 Riemann Hypothesis °° 1.7 Yang-Mills Theory @ 1.8 Can we understand and solve all 23 Hilbert Problems ™** 1.9 Goldbach Conjecture ™ 1.10 Is mathematics fundamental to or simply a good model of our Universe? "° 1.11 Is mathematics an emergent