Date: Nov 27, 2012 5:51 PM
Author: Joe Niederberger
Subject: Re: Some important demonstrations on negative numbers

>Again, there is another layer below all that. We call that layer "mathematics".

Thank you Mr. Hansen. By the way, there be turtles under that layer, and all the way down.

Your war against common sense though, has reminded me of something I wanted to post in the last common sense related thread about "continuity". The English translation of the classic Dedekind essay that brought us one of the earliest rigorous expositions on the concept of continuity is freely available, here is one:

If anyone takes time to read (it not difficult) you may find this gem:

...
"the majority may find it substance very *commonplace*" (emphasis mine)
...
"If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this severing of the straight line into two portions. "

(Sometime being common-sensical simply means finding just the right formulation, and stating it simply. Not always so easy.)

He goes on:
"... I think I shall not err in assuming everyone will at once grant the truth of this statement."

If that's not a statement affirming the common sensibility of what he is proposing I don't know what is.
Why would he bother, if he viewed "math" anything like Mr. Hansen does? Of course, he wouldn't.

"To this I may add that glad if everyone finds the above principle so obvious and so in harmony with his own ideas of a line, for I am utterly unable to adduce any proof of its correctness, nor has any one the power."

Of course, the whole trend towards math "foundations" of the second half of the 19th and early part of the 20th centuries was initially driven by a desire to find common sense in what was confusing, obscure or contentious in the very rapid, and sometimes purely "formal", development of math from the 15th century to that point. The surprises that lay in store for the foundation seekers would take some time yet, and in my opinion they have even been adequately "processed" to date, leading some to at least pay homage to absurdist views (like the famous Russell quote: "Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." -- he had a sense of humor.)

Cheers,
Joe N