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I write about random topics that are interesting. So far I have some notes about MIX and some recommended reading links.
Walter Gautschi's Leonhard Euler: His Life,the Man,and His Works remarks how Euler was apt to share in the joy of discovery and give credit to others, writing (pg 30):
Feelings of rancor, due to either priority issues or unfair criticism, were totally foreign to Euler. When Maclaurin, for example, discovered the well-known summation formula which Euler obtained six years earlier, Euler did not object, let alone complain, when for some time the formula was generally referred to as the "Maclaurin summation formula." It may even have pleased him that others hit upon the same fortunate idea. In due time, of course, the formula became justly known as the Euler-Maclaurin summation formula. Another example is Maupertuis's claim for the principle of least action (cf. section 2.3), which Euler had already enunciated before, much more clearly and exhaustively; yet Euler remained supportive of Maupertuis. Euler's forgiving way of reacting to Robins's criticism of the Mechanica has already been mentioned in section 3.3.1.
Sharing ideas with others and letting others take part in the process of discovery is another noble trait of Euler. A case in point is the way he put on hold his already extensive work on hydrodynamics, so that his friend Daniel Bernoulli, who was working on the same topic, could complete and publish his own Hydrodynamics first! It became a classic.
As for Euler's qualities as expositor and thinker, he had what Vladmir Arnold called "the Russian approach" always in mind. Well, Arnold writes in his article "Polymathematics" (fn 2, pg 5):
The Russian way to formulate problems is to mention the first nontrivial case (in a way that no one would be able to simplify it). The French way is to formulate it in the most general form making impossible any further generalisation.
Gautschi continues (pg 31, emphasis mine)
Euler's writings have the marks of a superb expositor. He always strove for utmost clarity and simplicity, and he often revisited earlier work when he felt they were lacking in these qualities. Characteristically, he will proceed from very simple examples to ever more complicated ones before eventually revealing the underlying theory in its full splendor. Yet, in his quest for discovery, he could be fearless, even reckless, but owing to his secure instinct, he rarely went astray when his argumentation became hasty. He had an eye for what is essential and unifying. In mechanics, Gleb Konstantinovich Mikhailov [20, p. 67] writes, "Euler possessed a rare gift of systematizing and generalizing scientific ideas, which allowed him to present large parts of mechanics in a relatively definitive form." Euler was open and receptive to new ideas. In the words of André Weil [30, pp. 132-133],
...what at first is striking about Euler is his extraordinary quickness in catching hold of any suggestion, wherever it came from [...]. There is not one of these suggestions which in Euler's hands has not become the point of departure of an impressive series of researches[...]. Another thing, not less striking, is that Euler never abandons a research topic, once it has excited his curiosity; on the contrary, he returns to it, relentlessly, in order to deepen and broaden it on each revisit. Even if all problems related to such a topic seem to be resolved, he never ceases until the end of his life to find proofs that are "more natural," "simpler," "more direct."
I find these qualities very admirable.
Jeffrey Lagrias writes in "Euler's constant: Euler's work and modern developments" pg 28
To conclude our discussion of Euler's work, it seems useful to contemplate the approach to research used by Euler in his long successful career. C. Truesdell [303, Essay 10], [304, pp. 91–92], makes the following observations about the methods used by Euler, his teacher Johann Bernoulli, and Johann’s teacher and brother Jacob Bernoulli.
Always attack a special problem. If possible solve the special problem in a way that leads to a general method.
Read and digest every earlier attempts at a theory of the phenomenon in question.
Let a key problem solved be a father to a key problem posed. The new problem finds its place on the structure provided by the solution of the old; its solution in turn will provide further structure.
If two special problems solved seem cognate to each other, try to unite them in a general scheme. To do so, set aside the differences, and try to build a structure on the common features.
Never rest content with an imperfect or incomplete argument. If you cannot complete and perfect it yourself, lay bare its flaws for others to see.
Never abandon a problem you have solved. There are always better ways. Keep searching for them, for they lead to a fuller understanding. While broadening, deepen and simplify.
My interest lies in puzzles and playing with examples. Arnold notes ("Polymathematics", pg 2)
Sylvester (1876) already described as an astonishing intellectual phenomenon, the fact that general statements are simpler than their particular cases. The antibourbakist conclusion that he drew from this observation is even more striking. According to Sylvester, a mathematical idea should not be petrified in a formalised axiomatic setting, but should be considered instead as flowing as a river. One should always be ready to change the axioms, preserving the informal idea.
Even a wrong example is interesting and has merit. Again, Arnold continues ("Polymathematics", pg 6):
Mistakes are an important and instructive part of mathematics, perhaps as important a part as the proofs. Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of characters.
At a conference, Arnold remarked ("From Hilbert's Superposition Problem to Dynamical Systems" in The Arnoldfest (Toronto, 1997), Fields Inst. Commun. 24; Providence, American Mathematical Society, 1999, pg 1):
When you are collecting mushrooms, you only see the mushroom itself. But if you are a mycologist, you know that the real mushroom is in the earth. There's an enormous thing down there, and you just see the fruit, the body that you eat. In mathematics, the upper part of the mushroom corresponds to theorems that you see, but you don't see the things that are below, that is: problems, conjectures, mistakes, ideas, etc.
No one supports the blink tag anymore. So, to annoy people, you need the following CSS3 code:
@keyframes blink {
0% { opacity:1; } 75% { opacity:1; } 76% { opacity:0; } 100% { opacity:0; }}
@-webkit-keyframes blink {
0% { opacity:1; } 75% { opacity:1; } 76% { opacity:0; } 100% { opacity:0; }}
@-moz-keyframes blink {
0% { opacity:1; } 75% { opacity:1; } 76% { opacity:0; } 100% { opacity:0; }}
@-ms-keyframes blink {
0% { opacity:1; } 75% { opacity:1; } 76% { opacity:0; } 100% { opacity:0; }}
@-o-keyframes blink {
0% { opacity:1; } 75% { opacity:1; } 76% { opacity:0; } 100% { opacity:0; }}
blink, .blink {
text-decoration: inherit;
animation: blink 0.75s ease-in infinite alternate;
-webkit-animation: blink 0.75s ease-in infinite alternate;
-moz-animation: blink 0.75s ease-in infinite alternate;
-ms-animation: blink 0.75s ease-in infinite alternate;
-o-animation: blink 0.75s ease-in infinite alternate;
}This isn't original, I took from somewhere...but I forgot where...