Re: Proposal: 6NF

Christopher Browne wrote:
> Quoth "Keith H Duggar" <duggar_at_alum.mit.edu>:
> > vc wrote:
> > > Marshall wrote:
> > > > I do not recall learning anything in secondary school
> > > > which would suggest 2 and 2.0 were numerically different
> > > > in any way. Nor can I think of any *arithmetic* way to
> > > > distinguish between 2 and 2.0.
> > >
> > > You have to construct all the real numbers and prove
> > > that 2 is an element of the set.
> >
> > Any mathematical number construct that fails to equate
> > 2.0 and 2, fails to model our most basic common sense or
> > "elemntary school" concept of the number 2.
>
> In abstract algebra, you get groups and other structures where
> 2 may be a meaningful value, but 2.0 isn't

I hold that 2 and 2.0 denote the /same value/. Thus whenever the value denoted by 2 is meaningful so is the value denoted by 2.0. Again, since they denote the same value.

> because there isn't any inherent notion of fractional values.
> Indeed, in the realm of discrete mathematics, it's
> [meaningless] (even undesirable!) to have any values lying
> between 1 and 2 and 2 and 3.
>
> Proof by induction, for instance, depends on the notion that
> there are no intermediate values.

Sure, I understand that we define and use structures wherein only whole numbers such as 1.0, 2.0, 3.0 etc have meaning. But, what bearing does this have on the claim that 1.0, 2.0, ... /denote the same values/ as 1, 2, ... ?

> I don't think that "elemntary school" concepts are of any
> particular relevance when looking at mathematical structures;

The phrase "elementary school" was a dig at VC's condescending language. That said, I would say that "common sense" by which I meant the basic reasoning and logical faculties that all of us /should/ possess, is not only relevant to mathematics it is the entire point of mathematics. In other words, the purpose of mathematics, the aim to which we employ it as a tool, is to abstract and perfect our basic reason, to clarify and codify our logic, to extend our faculty for thought beyond our concrete limitations. Mathematics that defies the most basic sense upon which it is founded has lost it's way.

> they are what they are, irrespective of whether a layman can
> relate them to anything that seems familiar to the layman.

Certainly I agree there are mathematical structures beyond the comprehension of some or many us. However, the concept of the number 2 or 2.0 is not such a structure.

Keith -- Fraud 6

PS. I really enjoyed reading your post. It was a refreshing change from C'mode and VC. And, it would be great if you replied again. I'm particularly interested in what you think regarding the purpose of mathematics as a human tool. Received on Sun Oct 22 2006 - 03:49:11 CEST