> Tad wrote:
>
> >Sorry to keep you waiting, I was trying to understand Wittengenstein
> >and
> >the Zen philosophy. We have agreed that Objective Reality has this
> >property of incredible consistency: pi is always 3.14... , OR "always
> >gives
> >off photons in the same way, and always has the same magnetic field,
> >and always exhibits the same time dependent behaviour, no
> >matter what the observer is."
>
> Well, YOU may have agreed on that, but I certainly haven't. Pi is a
> human concept. It's a distinction-meme, a very useful one. But it
> doesn't exist anywhere outside human culture. Likewise, statements of
> consistency are components (memes) of our current scientific thought,
> NOT Truthful descriptions of Objective Reality. They are all
> approximations.
This plunges in "where angels fear to tread".
The existence, or lack thereof, of mathematical objects, is extremely
opinionating.
Brodie, I'd like you to consider this example [don't worry about details,
just principles. The cubic and quartic formulae are monstrosities,
that's why we don't teach them in College algebra.]
====
Which card is this: The Hanged Man?
Quick review of Algebra:
These four have formulae that don't need trigonometry or worse, just
+|-|*|/|roots:
Linear equation: 0 = a_0 + a_1*X
Quadratic equation: 0 = a_0 + a_1*X + a_2*X^2
Cubic equation: 0 = a_0 + a_1*X + a_2*X^2 + a_3*X^3
Quartic equation: 0 = a_0 + a_1*X + a_2*X^2 + a_3*X^3 + a_4*X^4
This [and all higher-degree polynomials]: DO NOT have such nice formulae.
Quintic equation: 0 = a_0 + a_1*X + a_2*X^2 + a_3*X^3 + a_4*X^4 + a_5*X^5
====
People spent over 200 years between the discovery of the cubic and
quartic formulae in the early 1600's, and the nonexistence of the 5th
degree formula [in those terms] just after 1880.
The cubic and quartic formulae came fairly quickly after the use of
complex numbers was introduced. This required some unusual [at the time]
definitions; that's why we have the 'real' and 'imaginary' parts to a
complex number. "Imaginary numbers" were, to the 16th century mind first
introduced to them, truly imaginary.
The nonexistence of the quintic formula was immediate, once Evariste
Galois came up with the appropriate definitions linking the roots of a
polynomial equation to the methods of constructing them. [I'm skimming a
month of 800 Algebra; this MUST be sketchy.]
Now, do you have an opinion on this:
Did Evariste Galois, by his definitions, wipe out something [the nice
quintic formula] that could have existed before he got around to it?
Or did he merely create appropriate tools for documenting what was
physically real: the nonexistence of the nice quintic formula?
The claim (I am responding to) superficially directly answers the second
question as No, and the first question as Yes. Are these answers plausible?
[CLIP]
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/ Towards the conversion of data into information....
/
/ Kenneth Boyd
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