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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.

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  • $\begingroup$ Did you read the answer? You can't pick $a=c$ because it would make a denominator 0, and if $a<c$ then it contains no points. There's no lack of rigor here. $\endgroup$ Commented Jan 14, 2016 at 5:07
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    $\begingroup$ Want all the assumptions? Have a go at Principia Mathematica instead. $\endgroup$ Commented Jan 14, 2016 at 5:10
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    $\begingroup$ As to the complex numbers comment, we don't use complex numbers for the same reason we don't use octonions or polynomials...Just because something exists doesn't mean it's okay to plug it into the equation. We're considering real numbers here. $\endgroup$ Commented Jan 14, 2016 at 5:10
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    $\begingroup$ I think the Principia Mathematica comment was sarcastic but I will try to get a hold of it anyway. As for the other comments, as far as I understand it, you can only create that formula by assuming $a$ doesn't equal $c$. The equality isn't precluded by the formula, the formula assumes an inequality. And it would be good if he explained why it's not okay - obviously there are some rules operating that are not explained. This is a book that stated an assumption that $1$ did not equal $0$ - the drop off in transparency seemed a bit sharp. $\endgroup$ Commented Jan 14, 2016 at 6:12

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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.

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    $\begingroup$ "Each statement would have to be proven from axioms ..." Not so. Statements could still be built on known theorems and definitions. $\endgroup$ Commented Jan 14, 2016 at 11:11
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    $\begingroup$ Yes, but where do the theorems come from? $\endgroup$
    – zz20s
    Commented Jan 14, 2016 at 12:51
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    $\begingroup$ @zz20s Yeah but once you proved a theorem (from axioms) you can use it instead of repeating the argument every time. So I agree with Jorgen Fogh. Still a good answer though :) $\endgroup$
    – Ant
    Commented Jan 14, 2016 at 14:08
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    $\begingroup$ "The reality is that, if we are 100% rigorous in everything we say, any textbook would be thousands of pages long." - Rudin called, he wants a word. Apparently something about word economy and small type or something. $\endgroup$ Commented Jan 14, 2016 at 19:07
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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.

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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.

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  • $\begingroup$ -1: the "we must clearly choose $a>c$" refers to the formula, where $a^2-c^2$ appears as a denominator. So it is clear, indeed. $\endgroup$ Commented Jan 9, 2021 at 20:29

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