Shaky Danny

  Next time you see some little darling tormented with multiplication tables, have them make up a grid and fill it in themselves. It is easier to remember with context and seeing the relations between numbers. Be sure to point out the square’s diagonal. They can also build grids for the other operations, just for fun.

  One of the questions of mathematics is are we uncovering the ideal structure of reality or creating language. Yes. [i] I think of mathematics as a collection of tools or techniques. Socrates argued that all knowledge is innate. He took a young slave boy and interrogated him as to a proof of the Pythagorean theorem. Since the boy kept agreeing with Socrates, he must have known of this proof already. Socrates described a right triangle, then showed a square constructed from 4 of these identical triangles, then set the area of the square to the areas of the 4 triangles and the square contained in them. Then he solved for the Pythagorean theorem. Socrates was in the impossible position of arguing for ethics and logic in a polytheistic world, surrounded by the arbitrary gods. By proving that a slave had the same innate knowledge as the rest of us he was calling into question slavery. Euclid hated this proof of the Pythagorean theorem. The proof requires that you already know w...

In second grade our teacher Ms. Bowers introduced us to Cantor’s diagonal proofs.  You are already familiar with this, of course.  The rational numbers, the fractions, are listed with 1/1, ½, 1/3… on the top row, 1/1, 2/1, 3/1… in the first column and the diagonals always equal to 1: 1/1, 2/2, 3/3… and all the fractions in between. Then Cantor counts them by going up and down diagonally, zig-zagging between them.  All the second graders accepted that.  Then she showed us that the real numbers, say all the real numbers between .0000… and .9999…., were uncountable because no matter which way we listed them, she could generate a new one by going down the diagonal going on to infinity and generating a new one.  Cantor liked diagonals. I may have lost some of you.  I think the reason we got this as second graders is because we knew that if Ms. Bowers was explaining this to us, it couldn’t be that complicated.  But adults believe that this stuff should be...

The real numbers as described by the decimals can be completely counted for each order of decimal place.    Take all the real numbers from zero through one.    For one decimal place, they are the numbers 0.0 through 0.9, there are ten of them or 10 1  real numbers and this list is exhaustive.    That is, there is no real number that can be generated from this list that would not be redundant.   For two decimal places 10 2 , there are 100 of them, the numbers 0.00, 0.01, through 0.99.    For each decimal place, there is a matrix 10 n  that completely lists all the real numbers to that many decimal places.    As n approaches infinity, the real numbers are completely listed.