Recent content by mathwonk

Here is the situation as I understand it. Stillwell states Euclid's parallel postulate, then gives an argument meant to prove its converse, and then states that "thus" the parallel postulate implies its own converse. One would expect this to mean that his argument has used the parallel...

yes, my drawing labeled the angles oppositely to Stillwell. I have (hopefully) corrected the notation in response to your observation. As to rigor, such arguments require a rigorous discussion of sides of a line, and what happens when a line meets and hence crosses to the other side of a line...

But let's try to prove existence of parallels actually using Euclid's parallel postulate. let p be the point where n meets m in his diagram 1.2, and let q be the point where n meets l, and assume alpha +beta = π. If l meets m on the right side, say at x, then choose a point, say y, further out...

Having found and read Stillwell's account in full, I agree with you that he seems mistaken in his claim. I find that very puzzling, as he has a PhD in logic from MIT under Alonzo Church, and I wonder if I have understood him correctly.

I can't judge a book's logical flow based on selected excerpts. But the ones you have reproduced do puzzle me. If Stillwell is following Euclid, he must know that the existence of parallel lines (Prop. I.27) is proved in Euclid without using the parallel postulate, but based instead on the...

Well again, I am handicapped by not having the book. He visibly states that the result follows from the ASA principle, so that is what he is using in his argument. You might look and see how he proved that principle, e.g. whether he used the parallel axiom there.

It does not appear to be used here. I can only suppose that he does not mean this latter fact actually follows from the parallel postulate, only that it follows from something. In fact it follows from Euclid's exterior angle theorem, and is stated and proved in Prop. 17, Book I, of the...

Fascinating subject. In the 8 years since the previously linked article was published, interest and controversy about this topic, whether glyphosate use is or is not cause for concern by humans, persists. As recently as today in the NYTimes, one can read that a key study supporting its use in...

I try to emphasize to my classes that, to understand any statement, it is crucial to make precise what the terms in it mean, to give not only definitions but also hypotheses, i.e. to say what the symbols represent and what conditions they satisfy. In the statement of linearity of the indefinite...

Exactly for the reason you point out, one should not, in my opinion, ever assume there is a universal definition for any mathematical term. Always consult the specific book you are reading for the definition that will be used ins that book. Even then, some authors are careless and give...

Here is a link to a beautiful and eloquent talk given recently by a friend of mine, Michael Comenetz, containing the gentlest and clearest possible introduction to Galois theory, amazingly aimed at "anyone", i.e. essentially no background at all assumed by the reader, (well, maybe high school...

In terms of quality of exposition versus price, there can perhaps be nothing to match the great, but succinct and theoretical, treatment from 1942 by Emil Artin, (Mike's father), (ed. and supplem. by Arthur Milgram) , available for $7.39 at amazon! Artin throws in a complete elementary...

The good news is: once one appreciates the need for a Riemann sum type definition of a complex path integral, it is possible then to use antiderivatives to evaluate them. I.e. once a path is parametrized, one pulls the integral back to the real parameter interval, and uses the usual real...

real Calculus comes before complex calculus. And when learning real calculus, the topics that will be especially useful in complex calculus are power series and path integration. It helps also to focus on the actual definition of the Riemann integral as approximated by sums. We have a bad...

As to the breadth of Galois' research, the third part of his testamentary letter, on integrals, displays keen insight into basic questions on abelian integrals, later clarified by Riemann, in particular the division of differentials into first, second, and third kind, according to the nature of...