Pure Mathematics is a system of (nontrivial) tautologies. Roughly speaking, a tautology[2] is an analytic statement.

For example, consider this mathematics statement:

2 + 2 = 4

You do not have to do any kind of real world experiments in order to verify the statement. As long as you know the meanings of the symbols “2”, “+”, “=”, and “4”, you know that the statement is correct, and always. Of course, it says nothing about the physical world.

In pure mathematics, since you cannot and do not have to say anything about the real physical world, you can do[3] anything you like. Just like what you do when designing the rules of chess.[4] You can do anythings as long as they are

consistent and

interesting.[5]

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Is a tautology just a nonsense?

Maybe, maybe not. It depends on context:

When you present an analytic statement as an analytic statement, it is not a nonsense.

When you present an analytic statement as a synthetic statement, it is a nonsense.

[2] 重言句, 恆真式
[3] define
[4] or when programming a software
[5] i.e. useful

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Imagine the following scenes.

1.1.3 Scene One

A primary school student wrote 2 + 2 = 5 in his homework. His mathematics teacher told him that 2 + 2 = 5 was incorrect, “Two plus two should equal Four.” In such a context, the statement is, although analytic, not a nonsense.

1.1.4 Scene Two

After 30 years of research, a physicist declared his research result, “Two plus Two equals Four!!!” In such a context, the statement is, although true, a nonsense.

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2008.05.12 Monday