Lecture: Numbers and Logic

This blog is on the lecture concerning numbers and their relationship with logic.

The use of numbers originated from a very simple tribal system. Instead of numbers apes or stone-age tribesman would refer to objects as ‘one thing’ or ‘more than one thing’ as well as ‘many things’. Small numbers could be considered to be completely different from large numbers. Small numbers are generally considered countable up to the number 7. Large numbers can be considered to be any number above 7. The difference lies in the size of the number, for example, if you went to a friends house and 3 of your friends were there, you would instantly be able to count how many people were there. However if you went to a football match, it would be near impossible to be able to count how many people would be at the game.

Ancient civilisations had hieroglyphs for numbers and the number 0 did not exist, even the number 1 wasn’t regarded as a number, therefore to begin counting you would have to start with the number 2. It was the concept of zero which meant it didn’t exist as it refers to the paradox that nothing can exist as something, and 0 equals nothing. This contradicted with Aristotle’s law of Contradiction which was the foundation of all logic.

Leibnitz solved this problem by saying that an object can contain its own negation and the number zero can mean that there are no numbers (of whatever it is being counted) in the room opposed to literally nothing.

Numbers were seen as magic, floating, prefect platonic forms, this was especially considered with the numbers 3, 7, 12 and 13.

Bertrand Russell believed that numbers were synthetic a priori propositions and can in principle be defined logically from a limited set of axioms.

Peano’s Axioms;

1)     

      constant zero is a natural number and you can count using zero. It doesn’t mean nothing in the metaphysical sense. 

For Russell this fact about zero was assumed and not proven which he believed was a significant weakness.

2)      X=X, every number is its own equivalent

3)      Every natural number has a successor number, this was known as the infinity paradox

4)      No Natural number exists who’s successor is zero (negative numbers)

5)      If the successor of N is equal to the successor of M, then N is equal to M

Zero, Number and Successor were never explained by Peano and Bertrand Russell wanted to complete this theory further.