Wednesday, August 01, 2001
Answering the critics
One thing that I find appalling is how easily some of the posters on this newsgroup distort the truth. Since they claim that I'm the one doing so, I will stick to the concrete from here on out in this post, so that you the reader can properly evaluate what's going on.
First off, let me return to the issue of rings.
My understanding is that some have claimed that you can't have a proof of anything without specifying the ring.
I have claimed that a particular number can be a factor without any other information except that the number is in a ring.
Rather than endlessly debate the issue, I presented a quadratic to explain my position:
fx^2 + gx + h = (a1 x + b1)(a2 x + b2), with f, g and h integers.
(For instance, x^2 + 4x + 4 = (x+2)(x+2), where f=1, g=h=4, and a1=a2=1, and b1=b2=2.)
And them I repeatedly emphasized that DESPITE knowing that f,g and h are integers, you CANNOT specify a particular ring to cover all possibilities for a1, a2, b1 and b2.
However, my contention is that a1 and a2 are factors of f, DESPITE that, and similarly b1 and b2 are factors of h, despite the fact that you CANNOT specify a ring that covers all cases.
Some posters have apparently believed that you could simply say that a1, a2, b1 and b2 are algebraic numbers; however, I pointed out that a1 and a2 could be quaternions.
The issue has to do with use of factor in a general sense without knowing anything other than that you're in a ring.
Replies by posters to my posts seem to push the extreme position that it is not possible to have a proof without specifying a ring, so they would seem to indicate that a1 and a2 cannot be handled in a mathematical proof without specifying them to a particular ring (like saying a1=a2=1, as I did in my example before).
However, I'd like this position to be backed up by axioms, as it appears to me to be convention.
I think this point is a great place to start in resolving these discussions.
I would challenge those that reply to do so objectively, and without malice.
First off, let me return to the issue of rings.
My understanding is that some have claimed that you can't have a proof of anything without specifying the ring.
I have claimed that a particular number can be a factor without any other information except that the number is in a ring.
Rather than endlessly debate the issue, I presented a quadratic to explain my position:
fx^2 + gx + h = (a1 x + b1)(a2 x + b2), with f, g and h integers.
(For instance, x^2 + 4x + 4 = (x+2)(x+2), where f=1, g=h=4, and a1=a2=1, and b1=b2=2.)
And them I repeatedly emphasized that DESPITE knowing that f,g and h are integers, you CANNOT specify a particular ring to cover all possibilities for a1, a2, b1 and b2.
However, my contention is that a1 and a2 are factors of f, DESPITE that, and similarly b1 and b2 are factors of h, despite the fact that you CANNOT specify a ring that covers all cases.
Some posters have apparently believed that you could simply say that a1, a2, b1 and b2 are algebraic numbers; however, I pointed out that a1 and a2 could be quaternions.
The issue has to do with use of factor in a general sense without knowing anything other than that you're in a ring.
Replies by posters to my posts seem to push the extreme position that it is not possible to have a proof without specifying a ring, so they would seem to indicate that a1 and a2 cannot be handled in a mathematical proof without specifying them to a particular ring (like saying a1=a2=1, as I did in my example before).
However, I'd like this position to be backed up by axioms, as it appears to me to be convention.
I think this point is a great place to start in resolving these discussions.
I would challenge those that reply to do so objectively, and without malice.
# posted by James Harris @ 10:15 
