Lack of rigour in Spivak's Calculus book?

Question

I logged on today with this exact question: Ellipse definition

I found it disconcerting for him to say that it was clear that $a > c$ when $a$ could be equal to $c$ (a straight line) or maybe even less than $c$ (if complex numbers are allowed). So he is assuming that we don't want a straight line, and also that complex numbers aren't allowed. Neither of those assumptions were stated or explained. I don't even know whether complex numbers would work, whether any sum at all could be arrived at. It's also not stated that the formula wouldn't work for a straight line; it's just glossed over by saying it 'clearly' couldn't be a straight line.

I picked up Spivak's book because I had heard it was extremely rigourous, but now I'm wondering a) whether the unstated assumption and lack of addressing conceivable possibilities is common in his book, and b) whether there were any other book recommendations to learn calculus with the requirement of rigour in mind.

I'm a bit hesitant to continue, as I may be unable to tell whether something 'clear' to him is not clear to me due to me not understanding it properly, or due to not being aware of his assumptions. As I'm trying to learn this on my own, that's not a favourable position for me to be in.

Answer 1

I agree with you 100%: Spivak's definition is sloppy. He says:

A close relative of the circle is the ellipse. This is defined as the set of points, the sum of whose distances from two "focus" points is a constant.

By this definition, the line segment between the two foci is an ellipse. When he says later, "we must clearly choose $a > c$", he is contradicting his own definition.

Answer 2

I think we can reasonably state that an ellipse is both $1)$ existent and $2)$ not a straight line.

The reality is that, if we are $100\%$ rigorous in everything we say, any textbook would be thousands of pages long. Each statement would have to be proven from axioms, and nobody would want that. There are certain reasonable things we can and must assume.

I also will say that I understand the frustration. Sometimes textbooks are frustratingly casual where you don't want them to be. But part of writing math well involves knowing which things are okay to exclude.

Answer 3

Looking at the PM will probably be a good experience.

As for "rigor/transparency", as others have mentioned every single line of written math will have implicit assumptions, and these depend on context. When Spivak requests that $1\ne0$ he is discussing the axioms for the real numbers; eventually, the properties of the reals are taken for granted (or do you expect axioms every time he goes from, say, $5x+1=0$ to $x=-1/5$?).

In your concrete case of the ellipse, for each point $(x,y) $ you have a triangle where one side is $2c $ and the sum of the other two is $2a $. This immediately implies that $c <a $. Spivak's reasoning starts from the intuitive idea that you are defining the ellipse in the real plane, so it is sound.

Besides, "we must clearly choose $a>c$" refers to the formula, where $a^2−c^2$ appears as a denominator and so it has to be nonzero. So it is clear, indeed.