@kennethamy,

kennethamy;164848 wrote:I hope not!................Anyway, for many, love and passion substitute for knowledge and understanding. In fact, many cannot tell the difference.

I had always thought that love and passion fuel our desire to understand and gain knowledge, that and necessity and anxiety and fear and curiosity.

I don't know what you mean by 'substitute' in this context. Substituting love and passion for knowledge and understanding; I can only assume you mean the sophomore who begins to think of his area of learning as mystical and all-encompassing such as the new student who thinks that all life is physics or chemistry or theater etc.

On topic; I think that there is much more to the transcendental number than there is to rational numbers. The transcendental numbers, to my knowledge, first came about from measurements. So we had assumed that the rational numbers would suffice for every measurement, but we found a 'natural' constant, pi. We took our artificial idea of a perfect circle (which is a syntactic approximation of a class of real objects, a way to pass from objects to computations about the objects, I suppose it is an abstraction of a very specific type) and we imposed our system of measurement on it. It did not have a representation; no matter what rational manipulation we carried out we either overshot it or undershot it. Hence the eventual infinite series representation, an algorithm that is guaranteed to bring you closer and closer to the value of the number (and not only closer, but they tell you exactly how many decimal places it is accurate to if you know the error bound formulas).

The main thing that transcendental numbers advised us of was that our concept of number was incomplete with respect to what we wished to use if for. We needed additional mechanics to describe these numbers; and this was achieved in the calculus. Eventually it was given a deeper level of logical structure by Bolzano, Weistrauss, Dedekind, Cantor (indirectly), among others.

In non-standard analysis which sought to realize Leibnitz's intuition for the calculus, we think of every number as an infinite sequence. The mechanics for realizing this intuition were very powerful and came from modern logic as it was in the 1940's. Abraham Robinson was the man responsible for this vindication of Leibniz. The intuition, very roughly, is that a number is a class of infinite sequences of numbers which only disagree at a finite number of places, so if we have, for instance; [1,2,3,4,5,6,7,7,7,8,8,8,8,8,8,8.....8,....8...] and [4,7,6,9,8,0,1,23, 57, 8,8,8,8,8,8,8......8.....8.....] these are considered part of the same 'equivalence class' with the name 8* representing the number 8. However, there are other numbers considered such as [3.1, 3.14, 3.1415.....] which are transcendental and then there are the infinite and infinitesimal numbers such as [1,2,3,4,5,6,7,8,9,10.....] and [1, 1/2, 1/4, 1/16....] which respectively exceed or decrease past any finite equivalence class.

The construction is based on the set theoretic notion of an ultrafilter. If you PM me I can give you some more information on that.

This is very loose and not at all a totally sound representation of Non-standard analysis. One last note; non-standard analysis is logically equivalent to real analysis or 'rigorous calculus' so any first order logical sentence that holds under a non-standard model holds in real analysis. However; statements about things like the least upper bound etc. that have second order definitions do not transfer into non-standard analysis so other tools have to be used there (tools which do not transfer back into real analysis).