Archive for the 'Mathematics' Category

July 6, 2020

“Why do I call [Newton] a magician? Because he looked on the whole universe and all that is in it as a riddle, as a secret which could be read by applying pure thought to certain evidence, certain mystic clues which god had laid about the world to allow a sort of philosopher’s treasure hunt to the esoteric brotherhood. He believed that these clues were to be found partly in the evidence of the heavens and in the constitution of elements (and that is what gives the false suggestion of his being an experimental natural philosopher), but also partly in certain papers and traditions handed down by the brethren in an unbroken chain back to the original cryptic revelation in Babylonia. He regarded the universe as a cryptogram set by the almighty – just as he himself wrapt the discovery of the calculus in a cryptogram when he communicated with Leibniz. By pure thought, by concentration of mind, the riddle, he believed, would be revealed to the initiate.”

“All would be revealed to him if only he could persevere to the end, uninterrupted, by himself, no one coming into the room, reading, copying, testing — all by himself, no interruption for god’s sake, no disclosure, no discordant breakings in or criticism, with fear and shrinking as he assailed these half-ordained, half-forbidden things, creeping back into the bosom of the godhead as into his mother’s womb. ‘Voyaging through strange seas of thought alone’, not as Charles Lamb ‘a fellow who believed nothing unless it was as clear as the three sides of a triangle’.”

“There is an unusual number of manuscripts of the early English alchemists in the libraries of Cambridge. It may be that there was some continuous esoteric tradition within the University which sprang into activity again in the twenty years from 1650 to 1670. At any rate, Newton was clearly an unbridled addict. It is this with which he was occupied ‘about 6 weeks at spring and 6 at the fall when the fire in the elaboratory scarcely went out’ at the very years when he was composing the Principia – and about this he told Humphrey Newton not a word. Moreover, he was almost entirely concerned, not in serious experiment, but in trying to read the riddle of tradition, to find meaning in cryptic verses, to imitate the alleged but largely imaginary experiments of the initiates of past centuries. Newton has left behind him a vast mass of records of these studies. I believe that the greater part are translations and copies made by him of existing books and manuscripts. But there are also extensive records of experiments. I have glanced through a great quantity of this at least 100,000 words, I should say. It is utterly impossible to deny that it is wholly magical and wholly devoid of scientific value; and also impossible not to admit that Newton devoted years of work to it.”

~ “Newton, the Man”. John Maynard Keynes.

July 6, 2020

“[…] ‘He was so happy in his conjectures’, said De Morgan, ‘as to seem to know more than he could possibly have any means of proving’. The proofs, for what they are worth, were, as I have said, dressed up afterwards — they were not the instrument of discovery.”

“There is the story of how he informed Halley of one of his most fundamental discoveries of planetary motion. ‘Yes,’ replied Halley, ‘but how do you know that? Have you proved it?’ Newton was taken aback — ‘Why, I’ve known it for years’, he replied. ‘If you’ll give me a few days, I’ll certainly find you a proof of it’ — as in due course he did.”

“Again, there is some evidence that Newton in preparing the Principia was held up almost to the last moment by lack of proof that you could treat a solid sphere as though all its mass was concentrated at the centre, and only hit on the proof a year before publication. But this was a truth which he had known for certain and had always assumed for many years.”

“Certainly there can be no doubt that the peculiar geometrical form in which the exposition of the Principia is dressed up bears no resemblance at all to the mental processes by which Newton actually arrived at his conclusions.”

“His experiments were always, I suspect, a means, not of discovery, but always of verifying what he knew already.”

~ “Newton, the Man”. John Maynard Keynes.

Worrying

May 31, 2020

“At the moment when I put my foot on the step of the bus, the idea came to me, without anything in my former thoughts seeming to have paved the way for it.”
~ “Science and method”, Poincaré.

“[…] an initial period of concentration — conscious, directed attention — needs to be followed by some amount of unconscious processing. Mathematicians will often speak of the first phase of this process as ‘worrying’ about a problem or idea. It’s a good word, because it evokes anxiety and upset while also conjuring an image of productivity: a dog worrying a bone, chewing at it to get to the marrow — the rich, meaty part of the problem that will lead to its solution. In this view of creative momentum, the key to solving a problem is to take a break from worrying, to move the problem to the back burner, to let the unwatched pot boil.”
~ Dan Rockmore

October 27, 2018

“It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”
~ “An Introduction to Mathematics”, A.N. Whitehead.

December 28, 2016

“συμβόλον symbolon, an object cut in half constituting a sign of recognition when the carriers were able to reassemble the two halves. In ancient Greece, the symbolon, was a shard of pottery which was inscribed and then broken into two pieces which were given to the ambassadors from two allied city states as a record of the alliance.”

December 28, 2016

“Before I knew you, acting was the only reality of my life. It was only in the theatre that I lived. I thought that it was all true. I was Rosalind one night, and Portia the other. The joy of Beatrice was my joy, and the sorrows of Cordelia were mine also. I believed in everything. The common people who acted with me seemed to me to be godlike. The painted scenes were my world. I knew nothing but shadows, and I thought them real.”
~ Wilde

December 28, 2016

“Algorithmic thinking as a novel and productive point of view for understanding and transforming the sciences, started probably with Turing. Natural phenomena in many scientific fields (including mathematics, statistics, physics, astronomy, biology and economics) are intrinsically computational in nature: examples include chemical processes in living cells, the self-organizing behavior of complex systems consisting of many interacting particles, mechanisms governing human evolution, and the collective behavior of competing agents in an economy. These natural science areas should be explored through a computational lens.”

December 26, 2016

“Il modo di pensare scientifico possiede un’ulteriore caratteristica. I concetti che usa per costruire i suoi sistemi coerenti non esprimono delle emozioni. Per lo scienziato esiste solo l’essere, non il desiderio, il valore, il bene, il male, l’aspirazione. Finché restiamo nel dominio della scienza vera e propria, non possiamo mai incontrare una frase del tipo “Tu non mentirai”. Vi e’ come un freno puritano nello scienziato che cerca la verità: egli si tiene lontano da tutto ciò che è di carattere volontaristico o emotivo. Detto per inciso, questo tratto è il risultato di un lento processo, proprio del pensiero moderno occidentale.”
~ Einstein

December 26, 2016
In Leonardo’s view, “art, science, everything, including literature and philosophy, proceeded exclusively from nature. ‘No one should imitate the manner of another, for he would then deserve to be called a grandson of nature, not her son. Given the abundance of natural forms, it is important to go straight to nature rather than to the masters who have learned from her’.”
“In his view, all branches of knowledge were connected and interdependent, and man was always at the center. “Man is the model of the world”, as he put it. […] He discussed botany with the vocabulary of an embryologist or gynecologist, while he tackled anatomy in the spirit of a geographer. […] Everything is in everything, he seems to be saying. […] He was passionately pursuing his interests in twenty or more subjects.”
The ‘fantasie dei vinci’ in the Sala delle Asse are “symbols of both the infinity and the unity of the world, they proclaim that there must be a rule governing everything.” … Everything was connected, like the long echoes in Baudelaire’s poem, “in a deep and dark unity”.
“From 1490 on, Leonardo sought to assimilate and rigorously classify (according to what he called “my mathematical principles”) the whole of human knowledge, by structuring it, correcting it if necessary, and enlarging it if possible.”

~ “Leonardo. The artist and the man”, Serge Bramly.

December 26, 2016

“Many writers have reproached Leonardo for producing a cold, ungenerous art, overreflective and abstract, like an algebraic equation. It lacks “the tears and music of love”, according to André Suarès.”
~ “Leonardo. The artist and the man”, Serge Bramly.