[stiff]

Read the following EWD to get sneak-peak at Dijkstra's view of mathematics:

http://www.cs.utexas.edu/users/EWD/transcriptions/EWD12xx/EW...

Dijkstra's dream was reducing mathematics to formal logic, Euler's approach was almost on the opposite end of the spectrum, he used a lot of intuitive arguments sometimes dabbling in areas that didn't get a solid logical foundation for centuries to come.

[ColinWright]

It's certainly true that a lot of Euler's arguments were incomplete and non-rigorous by today's standards, but in many cases they were way ahead of his time. As you say, he was getting results in areas where complete and formal foundations didn't get laid for centuries.

And yet he seemed pretty much always to be right. He didn't fall into any of the traps that await the unwary, and which are the usual reasons for formality being required. It seems clear that he really did understand what was happening at a deeper level, even if the arguments he gave were, in some cases, described now as "superbly reckless."

I'm not convinced Dijkstra would've considered Euler to have been a poor mathematician. Now I wish I'd asked when I had the chance.

[stiff]

There was a lot of irony in my comments that I think not everyone understood. What I am really saying is that everything Dijkstra wrote on the topic of doing good mathematics contradicts the possibility of someone like Euler existing, yet he did exist and did work of unmatched quality. Whatever value might Dijkstra's work ultimately have I have no idea, but I find his insistence of his way being the only way appalling and his writings get really arrogant at times where he picks some very formal nit and makes it appear like a great intellectual achievement.

[btilly]

In Euler's defense, he did his work before Fourier accidentally proved that everything mathematicians believed at that point rested on quicksand.

[gruseom]

Can you expand on this? What did Fourier prove and why was it accidental and what was the quicksand?

[btilly]

What Fourier proved is that every function could be written as a sum of sines and cosines. However his proof worked for "functions" like step functions, which were not at the time accepted as functions, and which could not be sensibly analyzed with the infinitesmal techniques of the day.

It is hard to overstate the shock that came from a sum of nicely behaved functions turning into the pathological step function. If that was possible, what else could go wrong? And if infinitesmals could not be trusted, how could Calculus be put on a rigorous footing?

The question of how to put mathematics on a secure footing were central to 19th century mathematics, and the issues lead directly to set theory, and the unavoidable dead end discovered by Goedel. (Ironically the work in logic that came out of that eventually lead to nonstandard analysis, which in turn justified the infinitesmal approach and most of the infinitesmal arguments. But by then mathematics didn't much care.)

[gruseom]

Fascinating! I know about that story from the point at which the search for rigorous foundations had already begun (i.e. late 19th century), but no awareness of the connection to Fourier. Thanks.

[btilly]

Have you read The Mathematical Experience? Some of its essays address Fourier specifically and what it meant for math. The whole book is fascinating, even if you think you know the material.

[gruseom]

Nope, but I've just added it to my Amazon cart.