“If scientific reasoning were limited to the logical processes of arithmetic, we should not get far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.”

— Vannevar Bush, book As We May Think

As We May Think (1945)
Context: If scientific reasoning were limited to the logical processes of arithmetic, we should not get far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability. The abacus, with its beads strung on parallel wires, led the Arabs to positional numeration and the concept of zero many centuries before the rest of the world; and it was a useful tool — so useful that it still exists.

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Vannevar Bush57

American electrical engineer and science administrator 1890–1974

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“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.”

Gottlob Frege book The Foundations of Arithmetic

Gottlob Frege (1950 [1884]). The Foundations of Arithmetic. p. 99.

“Although the whole of this life were said to be nothing but a dream and the physical world nothing but a phantasm, I should call this dream or phantasm real enough, if, using reason well, we were never deceived by it.”

Gottfried Leibniz (1646–1716) German mathematician and philosopher

As quoted in The World of Mathematics (1956) by J. R. Newman, p. 1832

“Science and mathematics… have added little to our understanding of such things as Truth, Beauty, and Justice. There may be definite limits to the applicability of the scientific method.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“The idealistic tinge in my conception of the physical world arose out of mathematical researches on the relativity theory. In so far as I had any earlier philosophical views, they were of an entirely different complexion.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

The Nature of the Physical World (1928) Context: The idealistic tinge in my conception of the physical world arose out of mathematical researches on the relativity theory. In so far as I had any earlier philosophical views, they were of an entirely different complexion. From the beginning I have been doubtful whether it was desirable for a scientist to venture so far into extra-scientific territory. The primary justification for such an expedition is that it may afford a better view of his own scientific domain. Preface http://www-groups.dcs.st-and.ac.uk/~history/Extras/Eddington_Gifford.html

“I thought we should require physical determinations, and not abstract integrations. A pernicious taste begins to infiltrate, from which real science will suffer far more than it will progress, and it would be often better for the true physics if there were no mathematics in the world.”

Daniel Bernoulli

Ich vermeinte, man verlange physische Determinationen und nicht abstracte integrationes. Es fängt sich ein verderblicher goût an einzuschleichen, durch welchen die wahren Wissenschaften viel mehr leiden, als sie avancirt werden, und wäre es oft besser für die realem physicam, wenn keine Mathematik auf der Welt wäre.
Letter to Leonhard Euler, 26 January 1750, published in [Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle, P. H. Fuss, Saint Petersburg, 1843, 650]

“It might seem limited to impose our human perception to try to deduce the grandest cosmic code. But we are the product of the universe and I think it can be argued that the entire cosmic code is imprinted in us. Just as our genes carry the memory of our biological ancestors, our logic carries the memory of our cosmological ancestry. We are not just imposing human-centric notions on a cosmos independent of us. We are progeny of the cosmos and our ability to understand it is an inheritance.”

Janna Levin (1967) American theoretical cosmologist

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“Our mathematical concepts, structures, ideas have been invented as tools to organise the phenomena of the physical, social and mental world. Phenomenology of a mathematical concept, structure, or idea means describing it in its relation to the phenomena for which it was created, and to which it has extended in the learning process of mankind, and, as far as this description is concerned with the learning process of the young generation, it is didactical phenomenology, a way to show the teacher the places where the learner might step into the learning process of mankind.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (1961), p. ix

“When people start rejecting the fundamental understanding of the world that scientific process brings you, well, that's bad in my opinion.”

Bill Nye (1955) American science educator, comedian, television host, actor, writer, scientist and former mechanical engineer

[NewsBank, D-01, Bill Nye, the Science Guy, brings humor to normally serious field, The Daily Gazette, Schenectady, New York, March 9, 2005, Bill Buell]

“People can be fooled into thinking of mathematics as logical, formal, symbolic reasoning. But this is far from reality...computers are far better at formal computation and formal reasoning, but humans are far better mathematicians.”

William Thurston (1946–2012) mathematician

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“I must tell you that mathematicians are not scientists, they are artists…. Apart from the most elementary mathematics, like arithmetic or high school algebra, the symbols, formulas and words of mathematics have no meaning at all. The entire structure of pure mathematics is a monstrous swindle, simply a game, a reckless prank. You may well ask: "Are there no renegades to reveal the truth?" Yes, of course. But the facts are so incredible that no one takes them seriously. So the secret is in no danger.”

Theodore Kaczynski (1942) American domestic terrorist, mathematician and anarchist

The Net: The Unabomber, LSD and the Internet https://www.youtube.com/watch?v=xLqrVCi3l6E Interviews

“If scientific reasoning were limited to the logical processes of arithmetic, we should not get far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.”

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Vannevar Bush57

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