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Math = Truth

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Posted by Michael Levine on July 17, 2003 15:58:04 UTC

I brought up meaning because you said: "Actually, the term 'true' has a very specific meaning in math." An undefined term can acquire meaning, it just can't be reduced (in mathematics) any further. This is why my response was in regards to how 'true' can have meaning in math and still remain an undefined term.

I think I missed your point, sorry about that. In any case, the term "true" has very specific meaning and definition in math. The definition is given by the symbol '='. The meaning of 'true' in math is the same as the meaning of math itself. Truth, and the equal sign, are meta-concepts in math, as they refer to math itself.

Notice how the '=' sign embodies the notion of truth: to say "it is true that 2 + 2 = 4" is exactly equivalent to saying "2 + 2 = 4". "It is true that..." is redundant, as the concept of truth is already expressed by the '=' sign. That is why it makes no sense to question whether "it is true that 2 + 2 = 4", as it is exactly equivalent to asking "given that X is a true assertion, is it true that assertion X is true?".

Right. But, what I said is that the term 'true' is never defined in mathematics. The conditions of truth are defined, and this is done in formal systems with axioms, rules of inference, etc. This is why I said that the term true "is simply an undefined term that is used under certain conditions".

I think you are confusing the mathematical notion of truth, as expressed by the '=' sign, with the philosophical notion of truth, which concerns statements about reality. People tend to get confused by that; I suppose it's hard to understand that mathematics is not concerned with reality at all. People learn at a very young age that "you're not supposed to add apples and oranges", when in fact you're not even supposed to add apples and apples. The only thing you can add is numbers. By the time one is old enough to assimilate that, the notion that numbers refer to "things" is too deeply ingrained in one's consciousness.

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