Tuesday, March 31, 2009

...and then there was truth

From a Tumblr I follow....


“A mathematical truth is timeless; it does not come into being when we discover it. Yet its discovery is a very real event…”

With this Schrödinger notes a Platonic problem: mathematical truths exist apart from us. That is, for example, before humans existed it was still true that “the square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides,” as the Pythagorean theorem states.

This would remain “true” even if the Earth were smashed into rocky mist by an asteroid or humanity annihilated by its own weaponry. It would be true were life never formed: triangular shapes would conform to it. Its truth as a descriptive theorem is not dependent on our minds, we would say.

Yet in the famous words of Richard Rorty:

“Truth cannot be out there—cannot exist independently of the human mind—because sentences cannot so exist, or be out there. The world is out there, but descriptions of the world are not. Only descriptions of the world can be true or false.”

Truth cannot exist without sentences, as truth is a word. It has certain unusual qualities (transitive qualities, symmetry, etc.), but that we call those elements of its syntax ‘mathematical’ or ‘logical’ doesn’t mean they’re not of human (and linguistic) origin. So it would seem that mathematical knowledge is merely a sort of description, right? It is a highly reliable and repeatable description that abstracts forms of the natural world to make them more universal, better for operations, but it remains descriptive. “Two” describes things; “parallel” describes things; “true” describes things.

But Will mentioned circles -perfect circles- and their relationship to the universe. Such circles do not exist: they cannot be said to be descriptive, then; yet laws involving circles are everywhere in effect in our universe. The explanation of such laws by mathematicians has the quality of discovery: we found them! Yet it seems rather that we’ve created them! Yet they exist without us, at least inasmuch as the universe operates according to the principles they establish!

Is this a contradiction? Can you resolve it (in 140 characters)? Are mathematical laws human descriptions or qualities of the universe?


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