I write this in response to ongoing discussions about proving calculation techniques.
What is a number?
Most people have no clue how to answer. They might pick up a few coins and try to demonstrate how they use them, but they can’t explain what they are. It’s a fascinating fact that this question was not really tackled until the 19th century by men like Cantor, Cauchy, Dedekind… The mathematical development of number systems that include fractions and irrational is very technical and deep but the Natural numbers and then Integers are simpler so I will talk about those examples.
The modern Integers are a set of abstract objects whose nature depends entirely on their formal definitions. The Integers are given by the Construction of the Integers which are the rules that make them exist and govern their behavior.
Consider a close parallel, a game like chess, checkers, or card games. Then what is a chess piece? What actually is a Knight. The little figurine is superfluous. It’s just a notation convenience. It could be a piece of paper with a K on it. It need not have any physical form at all. When one plays online, there are no real pieces. Strong chess players can play without sight of the board. You don’t need a chess set to play chess - all you need is the rules and a brain to explore them. A chess piece’s nature, it’s very existence comes out of the rules of the game. A knight is entirely defined by its relationship to other chess pieces and the board. A knight is nothing more than a web of relationships with other chess pieces. *It is self contained. It makes no reference to the physical world".
Likewise with numbers, an integer’s properties, its existence is defined by its relationships to the other numbers. A web of relationships with nothing at the vertex. But numbers have value. There is something there surely - 7 has a value. But 7’s claim to a value is founded on it’s being a multiple of 1’s and one’s have a value of 1 because they are defined that way. The value is created by definition!. Ex Nihilo!
Might seem outrageous at first but money works the same way and we are quite comfortable with it. $1 has value by definition. It is not defined in terms of any other currency or anything else of value.
So what about their application to the physcial world? Chess has no direct correlation. Mathematicians construct endless numbers of these systems of abstract objects and play with them because they are fun, like chess. That’s pure mathematics. But we use numbers to think about the physical world. We use numbers as a model of things in the physical world. They allow us to make useful predictions. Instead of putting all the sheep in one pen to count them, one can figure, there were 21 sheep in that pen and 15 in the other - that’ll be 36 altogether.
But if one pen holds chickens and the other foxes, then numbers are not a good model for predicting what happens when you put them together. Sometimes things are close enough to be useful, with care. Quantities of money are not true numbers. The 1 in $1 is only divisible by 100! If I divide $1 by 300 I get pieces that are $0.00333. There is no such unit of money. $0.00333 is nothing. This is a real problem in financial transactions and a number of scams have been pulled off using this trick. If I am a bank holding your money, I can ‘process’ it by splitting it up into tiny fractions of a cent, round them all to zero and declare you have a zero balance.
When you learn to play chess you have to put aside what you know about horses and bishops. You think only about the rules and what can be done with them.
With numbers too, it can be treacherous if you use practical real world examples as your main understanding. The reason 1+1 = 2 should not be validated by a reference to some practical example, it should be explained by the rules of the game.
Edited- stupid error in one line.