"317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way."
- Godfrey Hardy (1877-1947)
This observation by Godfrey Hardy got me thinking. Once I was in a discussion with a certain guy, and in the course of an argument, I put forth that two plus two is four, irrespective of what kind of mathematics we are using.
Natural numbers do have a kind of physical reality. Even a primitive level of intellect (e.g., as in animals) is enough to comprehend the fundamentals of counting. The ability to differentiate between distinct objects is the origin of counting and of numbers.
However the genius I was discussing with questioned this statement vigorously. He said that two plus two is four only because our system assumes it. Two and two can make five if we had a different system of mathematics! ... My point was that whatever 'names' we give to the numbers are not important here. Calling 4 as 5 does not make it 5. Once we have agreed upon the language, this fact (i.e., 2 + 2 = 4) will be consistent throughout.
The point of this whole discussion, I do admit, was quite vague. And also difficult to convey. What my friend meant was that everything around us has an interpretation given by us, which may or may not be true (and can not be reliable as a basis for argument). I agreed, but I also said there are some things which may have an existence quite independent of our perception. We have just given them some convenient names.
I had to sidetrack our main topic (which was incidentally a different matter altogether!) for sorting this point out(requiring a fair bit of time and energy). But my friend steadfastly refused to be budged by any of the arguments, however logical they may be! And finally after a seemingly long and utterly frustrating effort, I had to admit defeat. There were no premises on which to base my arguments, once that guy nullified the fact that two and two make four!
(I removed the rest of the post from here, because it was a completely unrelated matter and would be dealt elsewhere. More importantly, it was indeed interfering with the coherence of this piece.)
