But I’m considering posting something on point set topology. The interesting questions are: What does it mean? How does it relate to epsilon-delta? Why does it work? I.e., what is the underlying principle? I would expect my observations to overlap the new material in Chapter 6.

LikeLike

]]>The challenge of Chapter 6 is to offer a reality-oriented alternative to standard axiomatic set theory and one first step is to distinguish mathematical sets from concepts.

Regarding the concept ‘man’, I don’t consider the existence of borderline cases to be the fundamental issue, nor do I have a problem with sets of specifically demarcated concretes. But a concept, in general, specifies a kind of thing, not a set of things. One can, indeed, specify a set of existing men or even of men who have lived in the past, but to specify a set of men does not end with forming the concept of man.

The units or referents of a concept, such as ‘man’ or ‘balloon’ or ‘galaxy’, include all men or balloons galaxies that have ever existed, exist today, or will ever, in the fullness of time, ever come into existence. The concepts apply to these referents, not by virtue of specifically identifying them, as such, but by virtue of the kind of concretes that these referents are. One can refer to future potential existents, but one cannot specifically identify them.

A mathematical set, by contrast, is a specific demarcation of possible values within a specific constellation of dimensions, pertaining, generally and in some fashion, to measurement. A set is, in this sense, “well-defined” and is distinguished specifically by its membership: Two different characterizations specify the same set if they specify the same elements.

The virtue of a concept is that it is open-ended, that it applies to the not-yet known or encountered and, also, that it embraces characteristics, currently unknown, that may one day be discovered. The virtue of a set is that it is well-defined, as specific values among a specific range of possibilities, possibilities that, by some conceptual means, are identified.

In the course of Chapter 6, I discuss how the concept of set arises in mathematics, what it refers to, why it is needed, how it relates to measurement, and what it accomplishes.

LikeLike

]]>LikeLike

]]>I am trying to figure out why not. Are you saying we cannot determine the referents in some cases? e.g. when a fetus becomes a man? Or when prior species evolved to be called human(i.e. man)? Or is it something else entirely.

On the surface it seems that you want to discount infinite sets for concretes. But if I say the “set of all men,” that may only be finite–most likely finite. It seems we can delimit them based on DNA.

I am not challenging the statement, just trying to follow where you are going with this.

LikeLike

]]>LikeLike

]]>LikeLike

]]>To be accurate, it wasn’t the geometrical interpretation of complex numbers that helped me: it was seeing them as derived mathematical entities with defined operations such that i^2 = (-1,0), a complex number and i=(0,1) another complex number. That left Albert’s strange looking operations as something yet to be defended, but Hobson’s geometric constructions left me with the grasp that they were at least sane.

I read Rand’s comments in ITOE.

I found nothing about complex numbers on page 381 of your book.

The book by Nahin looks fascinating and is in my amazon shopping cart for my next order in July.

LikeLike

]]>• I agree with Ayn Rand’s characterization of them as concepts of method (pp 304-306 of Introduction to Objectivist Epistemology (paperback).

• Historically they arose as a means of finding (real!) roots of third degree polynomial equations. But mathematicians were only happy with them when the geometric interpretation that you mention was discovered. There is an excellent history of imaginary numbers (An Imaginary Tale) by retired Professor of Electrical Engineering Paul J. Nahin.

• I do have some comments on page 381 that I forgot to index.

LikeLike

]]>However, I recently found a savior on the matter in a little book titled “College Algebra” by A. Adrian Albert. He simply defined complex numbers as points in a plane. He then defines operations on these ordered pairs of reals. i turns out to be (0,1) and i^2 is -1 when (0,1)X(0,1) is performed. Mystery basically gone.

Nevertheless, I see a large gap between introducing complex number the way Chrystal does and the way Albert does. So, I’m sure any comments on this issue in your book would have been valuable.

BTW,

I find the quality of this paperback binding to be quite good.

Have you considered setting up a twitter account? You can engage your readers that way, and they may even spread your tweets around to their “followers.” This can help spread your ideas (and Rands’s) and publicise your book.

LikeLike

]]>The text establishes that, for the vectors v1 and v2, and for any two numbers A and B, Av1 + Bv2 = (A + B, -A + 2B, B).

So suppose (a, b, c) = (A + B, -A + 2B, B). Then, by substitution and then simplifying, one calculates a + b – 3c = (A + B) + (-A + 2B) -3(B) = A + B – A + 2B – 3B = 0. This, a + b – 3c = 0, is the asserted relationship.

The point of establishing this formula relating the coordinates is that there are many vectors (a, b, c) for which the equation a + b – 3c = 0 is false. Such vectors cannot be expressed as linear combinations of v1 and v2. So v1 and v2 do not span the vector space of 3-tuples.

LikeLike

]]>