Re: JSH: ?

On Jun 29, 2:30 am, Mark Murray <w.h.o...@xxxxxxxxxxx> wrote:
JSH wrote:
If there are an infinite number of equally valid possibilities then
mathematically there is no single choice that is required as if there
were then you'd have a direct contradiction. I gave explaining
examples so your claim that you don't follow is of interest.

For example, if you multiply x^2 + 3x + 2 = (x+1)(x+2) by 7, there is
no mathematically determined way, for instance:

7(x^2 + 3x + 2) = (7x+7)(x+2)

is one of an infinity of ways to do so as is:

7(x^2 + 3x + 2) = (x+1)(7x+14)

Sure, I understand that, but without resorting to polynomials, there
are an infinity of ways of multiplying two numbers to get (say) 7:

1 x 7
sqrt(7) x sqrt(7)
49 x 1/7
:
etc

and there is no way to prefer one over the other.

I'm still not with you. Is it not the case that if the above were

You're not?

In mathematics if you have an infinity of equally valid possibilities
then one particular one is not mathematically determined.

I understand that. I just don't understand the apparent importance of it.

true, then you just showed that there is a major fault in mathematical
belief. But what is NOT clear is what that fault is.

If mathematicians accepted what I'm saying here as inviolate then they
would not have destroyed SWJPAM and would not still be teaching flawed
mathematics.

Are you saying that your fundamental point is that there are an infinity
of ways of multiplying entities together to get some previously nominated
result?

In other words:

Are you saying that given c = a x b, there are an infinity of (a, b) pairs
and there is no "preferred" pair?

Is that it?

However, it's one thing to agree with a trivial example and another to
accept a basic mathematical principle when you know what's on the
other side of that agreement.
What is on the other side of the agreement?

M

The "core" error.

Please _state_ what this "core" error is.


He probably won't reply to this. Here is the simple
description. The polynomial equation

a^2 - 6a + 35 = 0

has two roots. Both are algebraic integers. Harris
believes that he has an argument which shows that one
of the roots must be divisible by 7, while the other
is coprime to 7. However elementary Galois theory
and elementary algebraic number theory both imply that
neither root is divisible by 7 and neither root is
coprime to 7, in the ring of algebraic integers. Harris'
argument, he thinks, applies to any ring. Therefore,
he thinks, there is something wrong with the ring
of algebraic integers. He has never been able to say
exactly what that something is.

He reasons on the basis of polynomial equations like

a^2 - 12a + 35 = 0

which also has two roots - namely, 5 and 7 - one of
which is divisible by 7 and the other of which is
coprime to 7. Unlike the polynomial I gave above,
this one is reducible over the rationals. Harris thinks
that all factorizations should behave in the same way,
and does not understand that irreducibility makes an
essential difference in the characteristics of the roots.

His argument is a confused mess based on not under-
standing the distributive law. Years of trying to
explain the error to him by a host of people here have
had no effect. He has been shown factorizations which
reduce the problem with his "proof" to ordinary arithmetic.
But he cannot get his head around it. He still thinks
he is right and 150 years of solid elementary mathematics
is wrong.

Marcus.

M

.

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