ORGANIZE YOUR MIND --- THINK ABOUT NUMBERS ---- Episode 1

                                 WHY THINK ABOUT NUMBERS ?

     If one plays around with numbers, one doesn't have to concern herself with the concrete world and all its contradictions. Toying with number proofs involves perfect abstract generality. Here's an example. It concerns the primes, which---as you probably remember from high school --- are those integers that can't be divided into smaller integers without remainder. One proposition about primes states that there is no largest prime number. { What this means of course is that the number of prime numbers is infinite. But at the time this proposition was first advanced, mathematicians danced around using the term "infinite."} Here is the proof. Assume that there is in fact a largest prime number. Call this number Pn. This means that the sequence of primes (2, 3, 5, 7, 11, . . . Pn) is exhaustive and finite : ( 2, 3, 5, 7, 11, . . . , Pn) is all the primes there are. Now think about the number R, which we're defining as the number you get when you multiply all the primes up to Pn together and then add 1. R is obviously bigger than Pn. But is R prime ? If it is, we have an immediate contradiction, because we already assumed that Pn was the largest possible prime. But if R isn't prime, what can it be divided by? It obviously can't be divided by any of the primes in the sequence (2, 3, 5, . . ., Pn) , because by dividing R by any of these will leave the remainder 1. But this sequence is all the primes there are, and the primes are ultimately the only numbers that a non-prime can be divided by. So if R isn't prime, and if none of the primes (2, 3, 5, . . . , Pn) can divide it, there must be some other prime that divides R. But this contradicts the assumption that (2, 3, 5, 7, . . . , Pn) is exhaustive of all the prime numbers. Either way, we have a clear contradiction.  By removing false assumptions, we conclude that there is no largest prime. This, so far as our minds can currently stretch, is a fact. 

         THE WONDERFUL WORLD OF PRIME NUMBERS

   As we've already seen, the prime numbers are integers like 2, 3, 5, 7, 11, 13, and 17, which are evenly divisible only by themselves and the number 1. We happen to have ten fingers, and our number system is conveniently based on ten digits. But the same primes, with all the same properties, exist in any number system. If we had twenty-six fingers and constructed our number system accordingly, there would still be primes. It's easy to conceive of a culture that doesn't use base 10. We have plenty of examples. Computers use a binary system, and the Babylonians had a base-60 system, vestiges of which are evident in the way we measure time(sixty seconds in a minute, sixty minutes in an hour) . Cumbersome as this sexagesimal system was, it, too, contains the same primes. So does the octary system that Reverend Hugh Jones, a mathematician at the College of William and Mary, championed in the eighteenth century as more natural for women than base-10 because of women's experience in the kitchen working with multiples of 8 (32 ounces in a quart, 16 ounces in a pound ). [ Come on, girls, let's hear it for Rev. Jones !!!! ] 
   G.H. Hardy, a number theorist, believed numbers constituted the true fabric of the universe. In an address to a group of physicists in 1922, he took the provocative position that it is the mathematician who is in "much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real'."  But "a chair or a star is not in the least like what it seems to be ; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it ; but "2" or "317" has nothing to do with sensation, and its properties stand out the more clearly we scrutinize it . . . 317 is a prime,not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way."