Fuck Flash Cards

 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.

Philosophy of Logic

 

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 what a right triangle is and how to calculate area. Euclid wrote an entire book showing how to derive the Pythagorean theorem from postulates. Euclid had to choose the postulates that would prove his theorems.

Thousands of years later, propositional calculus was created to describe the process of proof. They had operators for or, and, and if then.  Their conceit was that they dodged causality. Simply because I can create a truth table for these operators does not give me inference. The sky is blue, there is sand in the earth, connect them as you wish and so what? There will always be a point in an argument where you challenge the other party.   What else could it be? What’s a better argument?

Frege compares the morning star to the evening star. Let us take when the moon and sun are both in the sky. You can see the reflection of the sun on the moon. Everything is kind of round, why was this so difficult? Notice that the reflection of the sun on the moon does not correspond to the position of the sun in the sky. Why is everything so complicated? How do we figure out anything?

I asked my tutor the Chicago question about language: is supporting a large block above you equivalent to telling you to move? The tutor failed me. Who won that argument?

Thankfully, Gödel using Cantor’s technique, argued that even if you could construct such a propositional system there would be undecidable results. Of course, you already knew that.

 

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[i] No?

Countable

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 difficult. So if you don’t get this, don’t feel bad, it just means you are old.
She then told us that it is impossible to prove that there is not an order of infinity between the countable and uncountable. It turns out that almost anytime you can’t prove anything in mathematics it is equivalent to this continuum hypothesis. So after all this work, we are left with yet another metaphor for life. Which seems like a lot of work to get there and it isn’t like there is a shortage of metaphors for life.
Back in second grade a kid came up to me and said:
-If you take all the real numbers of one decimal place, .0 through .9, there aren’t any others. You can’t insert any. Then if you go up by number of decimal places, .00 through .99 and so on, you will get to infinity which makes the real numbers countable.
-Those aren’t numbers, they are parts of numbers.
I told him.
-Then I will pad them with zeroes.
-Then I will insert a new one.
-Not if you follow my rules. It is just a matter of definition.
-Is not.
-Is so.
If you want to make a mathematician uncomfortable, tell them about this assertion made back in second grade. Part of the reason that they will be uncomfortable is that the foundations are shaky. I believe that this assertion is not equivalent to the continuum hypothesis, because it obviates the hypothesis. I suppose it would be boring if the real numbers were countable.
If one day this assertion is proven, proven would mean that most mathematicians agreed with it, which might happen if countable real numbers solved some other problem. If that happened it doesn’t mean that something else isn’t uncountable. But another reason that they would be uncomfortable is that it suggests that most of the high level mathematics done in the last century was a huge waste of time. My girlfriend Christine could have told you that in the first place.

Counting the Real Numbers

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 101 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 102, there are 100 of them, the numbers 0.00, 0.01, through 0.99. For each decimal place, there is a matrix 10n that completely lists all the real numbers to that many decimal places. As n approaches infinity, the real numbers are completely listed.