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@jeffcliff @ceo_of_monoeye_dating @plotinusgroyper do NOT tell Jeff about the octonions. I repeat, do NOT put a needle in Jeff's hand and tell him about the octonions
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@ceo_of_monoeye_dating @plotinusgroyper give me one fixed needle and i will move the world with it
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@jeffcliff @plotinusgroyper Mathematical equivalent of making your foundation a needle and trying to build a house on top of it
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@jeffcliff @plotinusgroyper Jeff are you sure they put those jabs in your arm?
It sounds like they put a needle in your damn brain what the fuck.
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@ceo_of_monoeye_dating @plotinusgroyper
> If we do it your way and Complex Numbers become just fucking dust, then *all* of our math becomes garbage and we're totally fucked. The thing is that it is *entirely possible* that we're just completely fucking wrong about our axioms - they're *axioms* after all. We'd like very much for as much math to stay intact if we have to give up on an axiom, and we build it the way we do because we're more willing to accept "Real Numbers are garbage" than "no there aren't actually Natural Numbers."
of course, natural numbers are just complex numbers of a certain kind: complex numbers with angle = 0 / imaginary component = 0 / 2nd column = 0 /etc and which the "real" part / magnitude >= 0
ie if you grant complex numbers you can easily define reals / natural numbers. ie complex numbers are more fundamental.
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...though i'm starting to warm to the idea of using quaternions as the fundamental building block with time
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@jeffcliff @plotinusgroyper >If ease of construction is more important than truth connectiveness to reality. It's more of a preference thing.
Well, the way we think of it is like this:
Let's say, hypothetically, that ZFC is completely fucked somehow and we find out that we can't *actually* construct the Reals. Then what would happen is all of Real Analysis and Complex Analysis becomes fucking dust overnight, but we'd still have quite a lot of useful math lying around.
If we do it your way and Complex Numbers become just fucking dust, then *all* of our math becomes garbage and we're totally fucked.
The thing is that it is *entirely possible* that we're just completely fucking wrong about our axioms - they're *axioms* after all. We'd like very much for as much math to stay intact if we have to give up on an axiom, and we build it the way we do because we're more willing to accept "Real Numbers are garbage" than "no there aren't actually Natural Numbers."
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@ceo_of_monoeye_dating @plotinusgroyper
> The standard way we construct the Complex Numbers *normally* goes kinda like this:
well yeah you can construct complex numbers but i'm assuming them as the fundamental counting blocks that all other mathematics are constructed with
> We don't "start" with the Complex Numbers, we actually do build them from other parts.
I mean, you could do it that way, too. If ease of construction is more important than truth connectiveness to reality. It's more of a preference thing.
ie you're starting with set theory and getting to complex numbers
i'm saying: start with complex numbers and work your way to set theory
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> To even claim the existence of negative numbers is to use a subtraction operator somewhere before you add it.
this is silly: if you have complex numbers, and multiplication, you can get subtraction out of it, and complex numbers are in a sense more fundamental than natural numbers so you should always start with complex numbers
if you start with complex numbers, you can get to negative numbers by i*i
i*i 2*i*i 3*i*i 4*i*i . . .
and rational negative numbers by
\sum _k=1 ^\infty \sum _j ^ \infty =0 i*i j / k etc
1 - 1 = 1+ (-1) = 1 + i*i*1
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@jeffcliff @plotinusgroyper Oh, this is a strong misunderstanding of how these things are built but that's normal for someone who hasn't studied it extensively.
The standard way we construct the Complex Numbers *normally* goes kinda like this:
1) Start with the Natural Numbers (built somehow using set theory).
2) Build the Integers by giving the Natural Numbers a group structure (additive inverses, a zero, and addition)
3) Build the Rational Numbers by giving the Integers a field structure (multiplication, multiplicative inverses, and a one.)
4) Build the Real Numbers by Dedekind Cuts or similar
5) Build the Complex Numbers as the Algebraic Closure of the Reals (Add in all of the roots of polynomials which have real coefficients).
We don't "start" with the Complex Numbers, we actually do build them from other parts. I don't actually see how you'd build the Complex Numbers *other* than as the Algebraic Closure of the Reals.
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I think the order of which we primitively arrive to arithmetic operations is subtraction then addition then division as repeated subtraction or addition, and then multiplication as reciprocal division.
You cannot substitute subtraction with addition.
To even claim the existence of negative numbers is to use a subtraction operator somewhere before you add it.
Division can have the resultant effect of multiplication as reciprocal dividend, just as a single cell divides itself multiplies itself by two cell units.
In other words, just another reason I don't like Set Theory crap
mist
Jeff "never puts away anything, especially oven mitts" Cliff, Bringer of Nightmares 🦝🐙🇱🇧🧯
CEO of Monoeye Dating
Plotin Groyp
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