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Monday, April 02, 2018

Where identity is key

One of those things did years ago, had me wondering why I bothered. But again shows how tautological spaces which are what I call complex identities can do MUCH when it comes to algebra.

Copying from my post September, 28, 2011:

...result that in general x2 - y2 = (x+y)(x-y), which can be constructed from within the tautological space as an intrinsic property:

x+y+vz = x+y+vz, x= -y - vz + (x+y+vz), squaring both sides, gives:

x2 = y2 + 2yvz + v2 z2 - 2(y+vz)(x+y+vz) + (x+y+vz)2

x2 - y2 = 2yvz + v2 z2 - 2(y+vz)(x+y+vz) + (x+y+vz)2, and setting v = 0, I have:

x2 - y2 = - 2y(x+y) + (x+y)2, which is: x2 - y2 = (x+y)(-2y + x +y),

so: x2 - y2 = (x+y)(x-y).

Source: Under the hood

Was like the coolest thing to me when realized was possible. Have pondered and wouldn't be surprised if you can build every part of algebra using tautological spaces, and in so doing relate everything back to identity.

So, given: x+y+vz = x+y+vz, you have x+y, and x-y, and a standard result. Follows logically from the tautological space itself. Oh yeah, also need some basic operations of addition, multiplication and need the distributive property. And the integers, should not forget. But realized also just need -1, 0, and 1, actually as of course from there can get the rest.

Reducing mathematics to identity? Resolves questions of truth.

That basis in truth then conceivably is the basis for mathematics itself: identity.

If so, then you just need identity and mathematical operations, and can build ALL of mathematics.

I like that idea. May as well make a post that goes ahead and states something else have been pondering for years now.


James Harris

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