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Saturday, October 20, 2007

Mathematical consistency

My prior thinking on logic and then the connection between logic and mathematics through equality finally lead me to realizing that consistency in mathematics is about identity.

So any valid mathematical statement must when numbers are used reduce to an identity, and mathematics is distinguished from the rest of logic only in that its valid statements reduce to numerical tautologies which are identities, like 1=1. That is, mathematics is just a subset of logic.

And mathematical truth can be defined as any mathematical statement where introducing any numbers valid under the conditions given will lead to an identity.

For example, with x=3, y=4 and z=5, it is true that x2 + y2 = z2 as introducing the numbers gives

9 + 16 = 25

which is

25 = 25.

Interestingly, by definition then any valid mathematical statement will give a tautology when numbers are introduced, so any statement that does not, is not a valid mathematical statement.

So by definition, relying merely on the equals being equal, every valid mathematical statement will lead to a tautology, and correct mathematics is consistent.

Extending to logic is straightforward but slightly outside of the scope of this post, though, easily enough I can note that in logic any valid logical statement will lead to a tautology.

And then truth is about tautology, and truth itself can be defined easily enough.

Truth is any unchanging object, concept or thing.

So truth is only about lack of change. Truths are absolutes.


James Harris
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