Friday, May 19, 2017

Leonhard Euler and pure mathematics

Leonid Euler (1707-1783) was an influential mathematician best known for his contributions modern mathematical notation, calculus and graph theory. Mathematician Carl Boyer says,
The Introductio does not boast an impressive number of editions, yet its influence was pervasive. In originality and in the richness of its scope it ranks among the greatest of textbooks; but it is outstanding also for clarity of exposition. Published two hundred and two years ago, it nevertheless possesses a remarkable modernity of terminology and notation, as well as of viewpoint. Imitation is indeed the sincerest form of flattery. (Introduction to the Analysis of the Infinite, 1950)
François Arago says,
He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air. (Wikiquote)
This post is a collection of quotes from Euler talking about the philosophy of mathematics.

Pure mathematics

"It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations." (Quoted in Induction and Analogy in Mathematics by George Polya)

"...we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful." (Quoted in Induction and Analogy in Mathematics by George Polya)

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate." (Quoted in Calculus Gems by G. Simmons)

Philosophy of science

"Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena." (A conjecture about the nature of air, 1780)

"Since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear... there is absolutely no doubt that every affect in the universe can be explained satisfactorily from final causes, by the aid of the method of maxima and minima, as it can be from the effective causes themselves." (Quoted in The Anthropic Cosmological Principles by Barrow and Tipler)

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