Sunday, December 24, 2006
JSH: Re-visiting planet Contrary
After years of acting as if expressions like 1+sqrt(2) give you one number so that 1 - sqrt(2) is a different number, many of you have no clue about the mistakes in mathematical reasoning you are making so I will re-visit the concept of a strange planet I call Contrary where people do things differently than on planet Earth.
Like on planet Contrary philosophers recently decided it was immoral to resolve the square root when you use the quadratic formula, so when they use it on
x^2 - 5x + 4 = 0
they get
x = (5 +/- sqrt(9))/2
and they say they have (5 + sqrt(9))/2 and (5 - sqrt(9))/2 and that each is a number, which you may think, hey, that's ok! Because to you they ARE single numbers where one is 4 and the other is 1, but the people on planet Contrary have one upped you, as thinking those are single numbers they have developed Contrary Galois Theory which is just like that used here on Earth except they have "numbers" in their Contrary Galois Theory where we'd just resolve the freaking square root.
And with their Contrary Galois Theory they can "prove" all kinds of things!
Like they can "prove" to you that no integer unit can be the root of a monic primitive with integer coefficients of degree 2 or higher, and they can even "factor" roots of monic polynomials with integer coefficients with remarkable adeptness giving convoluted expressions that are roots of polynomials we can just work out to integer
roots—but on Contrary they look get things like
(3 + sqrt(25))^{1/3}
and ooh and awe over the wonder of such complexity!!!
Recently things are restless though on planet Contrary as one person has dared to stand up to the philosopher overlords and proclaimed that they are silly and that
5 + sqrt(9)
is TWO NUMBERS and not just one, and that no, it's NOT true that it can't be the root of some other monic primitive with integer coefficients.
In fact he can PROVE that ONE of its solutions can be the root of <gasp>
x^2 + 9x + 8 = 0
and that it is just silly anyway to keep saying that sqrt(9) can't be evaluated, as in fact it can be evaluated to give two integers and people should toss out the ruling philosopher overlords for just being nasty turkeys.
Well the furor that has erupted on planet Contrary is something to behold!
Angry mathematicians and philosophers rant and rave about the crackpot who dares to say square roots should be evaluated, and foam at the mouth over the brazen heresy that the venerable square root function would be such a lowly beast as to give TWO SOLUTIONS??!!!
No way they argue, and the debate goes back and forth on planet Contrary where some philosophers DEFINED the square root function as too holy to give ANY solution at all!!!
[A reply to someone who wrote that sqrt(2) = 1.414213562…, 1 + sqrt(2) = 2.414213562…, and 1 − sqrt(2) = 0.414213562&hellip, and then asked James why it was so hard for him to understanding that.]
It's not true.
Consider
(3 + sqrt(25))^{1/3}
and if you could not do that simple thing of evaluating sqrt(25) and later evaluating one solution to find a cubic, you could go on and on about this special number that is an algebraic integer root of some other number as if you were doing something major, when you're not.
But when you can resolve the square root—you do it.
So people don't go around talking about 3 + sqrt(25), as instead they use A NUMBER.
But if it's sqrt(2) or some other expression where you haven't figured out a way to get beyond that square root, you drag it around in expression like 1 + sqrt(2).
Now you wouldn't drag around sqrt(4) in 1 + sqrt(4) because you can evaluate out of the square root and pick a solution.
But you DO drag around something like sqrt(5) because you CANNOT.
Get it, yet?
And in number theory approximations are no solution at all.
People drag around expressions asking to be evaluated, when they don't quite have the mathematical machinery in place to evaluate them.
So they'd resolve 1 + sqrt(25) but drag around 1 + sqrt(5).
Understand yet? Has anything seeped through the wall of stupidity which may be protecting you from basic knowledge about mathematics?
[A reply to someone who said that, despite all the maths books he read, he never saw “proof by invocation of imaginary alien civilization” used as a proof method before]
I think it's an excellent way to try and get people to look at what is being done in a different way, as consider some aliens who go even further than declaring sqrt(4) as only having one solution to saying it has NO solutions, and they can build Contrary Galois Theory with exactly the same machinery human beings have built up their Galois Theory with, so that it'd look exactly the same except you'd look at it and go, hey, wait, you can just evaluate things like sqrt(4) in there.
Taking the silly idea that you can declare away one solution to the square root to the limit of declaring away ALL solutions shows how dumb it is, I hope.
I've proven and proven and proven the mathematical truth.
Yet even publication in a mathematical journal meant nothing to people like you who can explain away just about anything.
But I'll make sure that on some level you understand how stupid your position is, and realize it also ultimately rests on refusing to acknowledge that sqrt(4) is 2 or -2.
Real aliens contemplating the inability of human beings to get over this hurdle and start developing number theory again would probably just conclude that our species has just kind of gone as far as it can go, and might just keep going themselves rather than deal with such an obstinate and clueless species. Oh that and how we keep destroying our own world, of course.
[A reply to someone who said that if d is an integer which is not a square, then there is an automorphism of Q(sqrt(d)) which sends sqrt(d) to −sqrt(d) and that if d is a square, then that's not the case.]
Yes it does—on planet Contrary.
Dude, to get Galois Theory with integers all you have to do is NOT take the square root with squares!!!
That's it!
You get the exact same damn theory with integers if you with sqrt(4) you just leave it as sqrt(4) versus resolving it.
Don't you get the story?
On planet Contrary the weird aliens DEFINE the square root to have NO SOLUTIONS AT ALL, as it's immoral or something according to their philosophers who are terrible overlords.
By DEFINING the square root of squares to not have a solution they managed to create Contrary Galois Theory, and it looks just like Galois Theory on this planet, except you have things like sqrt(25) in there while humans would have sqrt(5).
It's a great analogy showing how stupid it is to define away a solution to the square root as someone can just go further and define away ALL solutions and get—Galois Theory with integers.
Surely you're not too dumb to realize that if you just don't resolve square roots you can get Galois Theory with integers, are you?
Like on planet Contrary philosophers recently decided it was immoral to resolve the square root when you use the quadratic formula, so when they use it on
x^2 - 5x + 4 = 0
they get
x = (5 +/- sqrt(9))/2
and they say they have (5 + sqrt(9))/2 and (5 - sqrt(9))/2 and that each is a number, which you may think, hey, that's ok! Because to you they ARE single numbers where one is 4 and the other is 1, but the people on planet Contrary have one upped you, as thinking those are single numbers they have developed Contrary Galois Theory which is just like that used here on Earth except they have "numbers" in their Contrary Galois Theory where we'd just resolve the freaking square root.
And with their Contrary Galois Theory they can "prove" all kinds of things!
Like they can "prove" to you that no integer unit can be the root of a monic primitive with integer coefficients of degree 2 or higher, and they can even "factor" roots of monic polynomials with integer coefficients with remarkable adeptness giving convoluted expressions that are roots of polynomials we can just work out to integer
roots—but on Contrary they look get things like
(3 + sqrt(25))^{1/3}
and ooh and awe over the wonder of such complexity!!!
Recently things are restless though on planet Contrary as one person has dared to stand up to the philosopher overlords and proclaimed that they are silly and that
5 + sqrt(9)
is TWO NUMBERS and not just one, and that no, it's NOT true that it can't be the root of some other monic primitive with integer coefficients.
In fact he can PROVE that ONE of its solutions can be the root of <gasp>
x^2 + 9x + 8 = 0
and that it is just silly anyway to keep saying that sqrt(9) can't be evaluated, as in fact it can be evaluated to give two integers and people should toss out the ruling philosopher overlords for just being nasty turkeys.
Well the furor that has erupted on planet Contrary is something to behold!
Angry mathematicians and philosophers rant and rave about the crackpot who dares to say square roots should be evaluated, and foam at the mouth over the brazen heresy that the venerable square root function would be such a lowly beast as to give TWO SOLUTIONS??!!!
No way they argue, and the debate goes back and forth on planet Contrary where some philosophers DEFINED the square root function as too holy to give ANY solution at all!!!
[A reply to someone who wrote that sqrt(2) = 1.414213562…, 1 + sqrt(2) = 2.414213562…, and 1 − sqrt(2) = 0.414213562&hellip, and then asked James why it was so hard for him to understanding that.]
It's not true.
Consider
(3 + sqrt(25))^{1/3}
and if you could not do that simple thing of evaluating sqrt(25) and later evaluating one solution to find a cubic, you could go on and on about this special number that is an algebraic integer root of some other number as if you were doing something major, when you're not.
But when you can resolve the square root—you do it.
So people don't go around talking about 3 + sqrt(25), as instead they use A NUMBER.
But if it's sqrt(2) or some other expression where you haven't figured out a way to get beyond that square root, you drag it around in expression like 1 + sqrt(2).
Now you wouldn't drag around sqrt(4) in 1 + sqrt(4) because you can evaluate out of the square root and pick a solution.
But you DO drag around something like sqrt(5) because you CANNOT.
Get it, yet?
And in number theory approximations are no solution at all.
People drag around expressions asking to be evaluated, when they don't quite have the mathematical machinery in place to evaluate them.
So they'd resolve 1 + sqrt(25) but drag around 1 + sqrt(5).
Understand yet? Has anything seeped through the wall of stupidity which may be protecting you from basic knowledge about mathematics?
[A reply to someone who said that, despite all the maths books he read, he never saw “proof by invocation of imaginary alien civilization” used as a proof method before]
I think it's an excellent way to try and get people to look at what is being done in a different way, as consider some aliens who go even further than declaring sqrt(4) as only having one solution to saying it has NO solutions, and they can build Contrary Galois Theory with exactly the same machinery human beings have built up their Galois Theory with, so that it'd look exactly the same except you'd look at it and go, hey, wait, you can just evaluate things like sqrt(4) in there.
Taking the silly idea that you can declare away one solution to the square root to the limit of declaring away ALL solutions shows how dumb it is, I hope.
I've proven and proven and proven the mathematical truth.
Yet even publication in a mathematical journal meant nothing to people like you who can explain away just about anything.
But I'll make sure that on some level you understand how stupid your position is, and realize it also ultimately rests on refusing to acknowledge that sqrt(4) is 2 or -2.
Real aliens contemplating the inability of human beings to get over this hurdle and start developing number theory again would probably just conclude that our species has just kind of gone as far as it can go, and might just keep going themselves rather than deal with such an obstinate and clueless species. Oh that and how we keep destroying our own world, of course.
[A reply to someone who said that if d is an integer which is not a square, then there is an automorphism of Q(sqrt(d)) which sends sqrt(d) to −sqrt(d) and that if d is a square, then that's not the case.]
Yes it does—on planet Contrary.
Dude, to get Galois Theory with integers all you have to do is NOT take the square root with squares!!!
That's it!
You get the exact same damn theory with integers if you with sqrt(4) you just leave it as sqrt(4) versus resolving it.
Don't you get the story?
On planet Contrary the weird aliens DEFINE the square root to have NO SOLUTIONS AT ALL, as it's immoral or something according to their philosophers who are terrible overlords.
By DEFINING the square root of squares to not have a solution they managed to create Contrary Galois Theory, and it looks just like Galois Theory on this planet, except you have things like sqrt(25) in there while humans would have sqrt(5).
It's a great analogy showing how stupid it is to define away a solution to the square root as someone can just go further and define away ALL solutions and get—Galois Theory with integers.
Surely you're not too dumb to realize that if you just don't resolve square roots you can get Galois Theory with integers, are you?
# posted by James Harris @ 15:54 
