2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (There are 25 prime numbers below 100.)
2, 3, 5, 13, 89, 233, 1597, 28657, 514229, and 433494437. (Fibonacci Primes, numbers that are BOTH Fibonacci and Prime. There are only 10 of these less than 1 billion. )
Numbers seem simple, and they represent order. We can see nature through numbers, but numbers themselves already create astonishment: 111,111,111 x 111,111,111 = 12,345,678,987,654,321.
The essence of mathematics is never in the particular way it is represented, but in the concept that it brings forth, and the unification of particulars that it embodies. One might hypothesize that any mathematical system will find natural realizations. This is not the same as saying that the mathematics itself is realized. The point of an abstraction is that it is not, as an abstraction, realized. The set { { }, { { } } } has 2 elements, but it is not the number 2. The number 2 is nowhere “in the world”. ~ Louis Kauffman
Wittgenstein, Tractatus Logico-Philosophicus:
5.135 There is no possible way of making an inference from the existence of one situation to the existence of another entirely different situation.
5.136 There is no causal nexus to justify such an inference.
5.1361 We cannot infer the events of the future from those of the present. Belief in the causal nexus is superstition.
"All limits, especially national ones, are contrary to the nature of mathematics… Mathematics knows no races… For mathematics the whole cultural world is a single country" – David Hilbert.
"Face problems with a minimum of blind calculation, a maximum of seeing thought" – Hermann Minkowski
Timothy Morton: "The Ecological Thought: We no longer live within a horizon (did we ever?). We no longer live in a place where the sun comes up and goes down, no matter how much some philosophers insist that we experience things that way. We’ve lost a sense of the significance of events that appear on horizons (did we ever have them?)"
Alasdair MacIntyre: The central epistemological problems of philosophy do not arise primarily from within philosophy at all, but from the recurrence in every area of human thought and practice of rival interpretations, and rival types of interpretation, of events and actions. It is for this reason that every academic discipline is to some degree ineliminably philosophical. The literary critic, the historian and the physicist presuppose, even when they do not explicitly defend, solutions or partial solutions to the problems of representation and justification. Shakespeare and Proust, Macaulay and Charles Beard, Galileo and Bohr cannot be read and responded to adequately without epistemological inquiries and commitments. Moreover, the philosophical problems and solutions in each particular area have a bearing on those in other areas; often enough, indeed, they are the very same problems. Hence the need for a synoptic and systematic discipline concerned with the overall problems of justification and representation. (“Alasdair MacIntyre on the claims of philosophy, ” London Review of Books, Vol. 2 No. 11, pp. 15–16.)
John von Neumann: The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.
Heinz von Foerster: I don’t know where my expertise is; my expertise is no disciplines. I would recommend to drop disciplinarity wherever one can. Disciplines are an outgrowth of academia. In academia you appoint somebody and then in order to give him a name he must be a historian, a physicist, a chemist, a biologist, a biophysicist; he has to have a name. Here is a human being: Joe Smith — he suddenly has a label around the neck: biophysicist. Now he has to live up to that label and push away everything that is not biophysics; otherwise people will doubt that he is a biophysicist. If he’s talking to somebody about astronomy, they will say “I don’t know, you are not talking about your area of competence, you’re talking about astronomy, and there is the department of astronomy, those are the people over there,” and things of that sort. Disciplines are an aftereffect of the institutional situation. [Von Foerster (1995) “Interview Heinz von Foerster” S. Franchi, G. Güzeldere, and E. Minch (eds) in: Constructions of the Mind Volume 4, issue 2. 26 June 1995]
Carl Sagan: We live in a society exquisitely dependent on science and technology, in which hardly anyone knows anything about science and technology.
Adorno/Horkheimer, Dialectic of Enlightenment, 1947:
Myth is already enlightenment; and enlightenment reverts to mythology.
Enlightenment, understood in the widest sense as the advance of thought, has always aimed at liberating human beings from fear and installing them as masters. Yet the wholly enlightened earth radiates under the sign of disaster triumphant.
It is not the victory of science that distinguishes our nineteenth century, but the victory of scientific method over science. Nietzsche, The Will to Power
Georg Cantor: The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though in its highest form it has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.
"Science cannot solve the ultimate mystery of nature. And that is because, in the last analysis, we ourselves are part of the mystery that we are trying to solve." ~ Max Planck
Roger Penrose, (from The Road to Reality): “… no one, not even Benoit Mandelbrot himself [...] had any real preconception of the set’s extraordinary richness. The Mandelbrot set was certainly no invention of any human mind. The set is just objectively there in the mathematics itself. If it has meaning to assign an actual existence to the Mandelbrot set, then that existence is not within our mind, for no one can fully comprehend the set’s endless variety and unlimited complication.”
“The past tells us who we are, without it, we lose our identity.” ” Stephen Hawking
We had this old idea, that there was a universe out there, and here is man, the observer, safely protected from the universe by a six-inch slab of plate glass. Now we learn from the quantum world that even to observe so minuscule an object as an electron we have to shatter the plate glass; we have to reach in there…. John Wheeler, A Question of Physics.
"Science, understood as the discoverer of absolute truth, remains therefore, naturally, a source of disillusion for its lack of absolute truths. If the cold marble idol of a perfect, eternal and universal science that we could only seek to better understand falls and shatters, it is there that suddenly alongside it we find a living creature, a science that our thought freely creates. A living entity: flesh of our flesh, the fruit of our torment, our companion in the struggle . . ." ~ Bruno de Finetti
“I cannot resist the formation of a hypothesis for every subject. But no matter how good I think any hypothesis is, I must not fall in love with it, for it may show itself to be contrary to the facts.” --- Charles Darwin
'The problem is that scientists are supposed to know, but they do not. Science is helpless and covers up this helplessness with a deceptive screen of expert assurance.' Žižek, Slavoj. 'Joe Public v the volcano.' in: New Statesman. April 29, 2010. (English).
It is easier to perceive error than to find truth, for the former lies on the surface and is easily seen, while the latter lies in the depth, where few are willing to search for it. ~ (Johann Wolfgang von Goethe, MuR, §166)