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Thursday, September 24, 2009

Metamathematics

I asked my calculus professor the following question earlier today:

"Is there a mathematical relationship between the inverses of different operations? "

He thought the question didn't really make sense, and I suppose it doesn't. I guess what I'm trying to get at was this: is there a common algorithm for taking the inverses using different operations?

For instance, say we start with the point (2,2).
Its additive inverse, the negative, is (-2,-2).
Its multiplicative inverse, the reciprocal, is (1/2, 1/2)
We could define other inverse operations, but these seem to be the ones that make most sense for the purpose of considering points on a plane (or hyperplane).

The question then is: is there some function or algorithm that you could pass "multiplicative" or "additive" to, and have it return the inverses? I'm looking for the same algorithm to be applied with different parameters, not just a look-up table that specifies different methods (though I suppose that might be the only solution). Specifically: I want to understand how these sorts of operations are possible with neural architecture alone. Is it the same algorithm? Does it use a "lookup table?"

This is all aimed at generalizing Dr. Bickle's result with the "negative echo" phenomenon in visual neural fields.

The following rekindles my faith in and love for technology: its the USPS tracking history for the derailleur hangers I ordered online earlier today.

Bullet Processed through Sort Facility, September 24, 2009, 6:31 pm, DENVER, CO 80266
Bullet Acceptance, September 24, 2009, 4:44 pm, LOUISVILLE, CO 80027

It's already on its way through priority mail. I can imagine it speeding towards me through the clouds now. (I actually ordered two in anticipation of breaking another one)

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