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?