[x42bn6]

Originally Posted by B~E

So this is all just a thought experiment, like a social contract or an "original position" in social sciences. But what's the point of it?

2 is defined as the successor of 1 (proof can vary from a couple of lines to about 67,000 lines). You could define "+" as a totally different thing. One can define the "+" in "a+b" to be "a+b+ab" where these "+"s are the addition symbols we recognise by default. It can be proved that this still is a mathematical group and hence mathematical structures can be built from this. I'm not sure it's a field or a ring so it might not have full use (division, for example, isn't a ring because division by zero is undefined, but addition operations are rings, fields and groups).

We could've defined "1+1" to equate to "1", because "a*b=^defa+b-1" where "*" is our new addition operator, and had we done so when the first mathematical and logical systems were created, it would've made sense, if tedious. Use is a different matter. If I have 2 tins and add 2 more tins to my collection, I would have 4, but "2+2=3" with my new definition. Not really useful, which only really implies addition in this sense isn't quite the same as concatenating loads of 1s together.

Of course, one can then say that we are making our own logic system. Which is true - we can make pretty much any logic or mathematical system we want, but you will be limited by the properties such as reflexivity (what would be the implications of a system where a=a was false?), rings, groups, fields, associativity, and so on. The question is whether someone up there gave us this system, or the idea.

To me, (mathematical) logic arose because of intuition. Caveman A thought that if he made another spear, he would have more spears so if one broke, he had another spare. Enter Caveman B, who has hundreds of spears, stolen. He steals one of Caveman A's spears. Caveman A sees this and thinks, "Ah! One of my spears is gone!" Addition and subtraction were born.

Hence I think that logical systems are independent of the Universe. Did we create logic systems? Of course. We created the notion of truth and used it to make decisions. Proof by contradiction, reflexivity, etc. are direct results of this. Unless the person up there gave us the idea of a logic system, I am guessing we created it. But did we create logic? I'm not sure that's possible. What we think of as true or false is a logic system. So I'm taking the stance that logic is a human creation, unless God gave that knowledge to us. And, of course, that is going to be difficult to prove or disprove.

Then there is the problem that the logic system we use at the moment may not be able to solve all the problems we come across (Godel's Theorem?). Then we could shift our logic system and solve this problem. Of course, we might have to redefine 1+1 to be 3, but then again, these systems are only as useful as we think they are.*