Fun quotes

The only excuse in the XXIth century for indulging in «foundations», is a «grain de folie», i.e., a slight madness.

— Jean-Yves Girard, The Blind Spot, 2011

Infinity

Infinity is merely a way of speaking, the true meaning being a limit.

— Carl Frederich Gauss, 1831

Not only in probability theory, but in all mathematics, it is the careless use of infinite sets, and of infinite and infinitesimal quantities, that generates most paradoxes.

— E.T. Jaynes, Probability Theory: The Logic of Science, 2004

The totality of mathematical theorems is, among other things, also a set which is denumerable but never finished.

— L.E.J. Brouwer (translated from Dutch)

The notion of the actual infinity of all numbers is a product of human imagination; the story is simply made up. The tale of ω\omega even has the structure of the traditional fairy tale: “Once upon a time there was a number called 0. It had a successor, which in turn had a successor, and all the successors had successors happily ever after.”

— Edward Nelson, Warning Signs of a Possible Collapse of Contemporary Mathematics, 2006

Constructive mathematics

In its fight mathematics appears as a gigantic “paper economy”. Real value, comparable to food products in the national economy, has only the direct, downright singular; everything general and all the existential statements participate only indirectly. And yet we mathematicians seldom think of cashing in this “paper money”! Not the existence theorem is the valuable thing, but the construction carried out in the proof. Mathematics is, as Brouwer sometimes says, more action than theory.

— Hermann Weyl, 1921

Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether.

— David Hilbert, 1928

The excluded middle, once translated as ¬¬(A¬A)\neg \neg \left( A \vee \neg A), becomes intuitionistically provable and it is not more «risky» to add it. However, one can answer back that it is not because one is warranted impunity (consistency) that one is entitled to commit a crime (enunciate an unjustified principle).

— Jean-Yves Girard, The Blind Spot, 2011

If a function is defined as a binary relation satisfying the usual existence and unicity conditions, whereby classical reasoning is allowed in the existence proof… then a function cannot be the same kind of thing as a computer program.

Similarly, if a set is understood in Zermelo’s way as a member of the cumulative hierarchy, then a set cannot be the same kind of thing as a data type.

— Per Martin Löf, Constructive Mathematics and Computer Programming, 1984

Scientists imagine that in different universes one might encounter different fundamental constants, such as the strength of gravity or the Planck constant. But easy as it may be to imagine a universe where gravity differs, it is difficult to conceive of a universe where fundamental rules of logic fail to apply. Natural deduction, and hence lambda calculus, should not only be known by aliens throughout our universe, but also throughout others. So we may conclude it would be a mistake to characterise lambda calculus as a universal language, because calling it universal would be too limiting.

— Philip Wadler, Propositions as Types, 2014

When a notion of knowledge and truth as harmony with nature begins to be part of one’s view of the world, then one begins to realize that, contrary to a common belief, it is the classical approach which is less free, since it forces reality into unnatural principles. The sharp division of the world between good-truth and evil-falsity appears as the mathematical continuation of a childish wish of omnipotence. To justify it, one is bound to adhere to some kind of faith, like the existence of platonic ideas, or even worse to divide the self, that is split body from soul, form from content, and retain only the shadow of truth which is materially perceivable in formulae.

. . .

Exploring the world of mathematics with a weak foundation is like travelling on foot or riding a bicycle, rather than by car or plane. The pleasure is not of possessing as many places as possible by passing through them, but of getting familiar with the landscape and harmonious with nature. In this way, once can observe many facts which otherwise, due to speed or distance, would remain unnoticed. This means acquiring a kind of knowledge which is impossible otherwise. Moreover, a considerable side effect is that of avoiding crowds (up to now, at least).

— Giovanni Sambin, Some points in formal topology, 2003

Computation

So parentheses are the basis of computation! [Referring to the universal X combinator from combinatory logic]

— Bob Constable, 2015 (at OPLSS)

Gödel was a pretty good programmer. [Referring to Gödel’s incompleteness theorems]

— Bob Constable, 2015 (at OPLSS)

Probability

How dare we speak of the laws of chance? Is not chance the antithesis of all law?

— Joseph Bertrand, Calcul des probabilités, 1889 (translated from French)

Foundations

God made the integers, all else is the work of man.

— Leopold Kronecker, 1893 (translated from German)

The famous saying by Kronecker that God created the numbers, all else is the work of Man, presumably was not meant to be taken seriously. Nowhere in the book of Genesis do we find the passage: And God said, let there be numbers, and there were numbers; odd and even created he them, and he said unto them, be fruitful and multiply; and he commanded them to keep the laws of induction.

— Edward Nelson, Predicative Arithmetic, 1986

Mathematics belongs to man, not to God. We are not interested in the properties of the positive integers that have no descriptive meaning for the finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.

— Errett Bishop and Douglas Bridges, Constructive Analysis, 1985

The Axiom of Choice is obviously true, the Well–ordering theorem is obviously false; and who can tell about Zorn’s Lemma?

— Jerry Bona

In Brouwer’s case there seems to have been a nagging suspicion that unless he personally intervened to prevent it, the continuum would turn out to be discrete.

— Errett Bishop and Douglas Bridges, Constructive Analysis, 1985

Mathematical pathology

Logic sometimes breeds monsters. For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemblance as possible to honest functions that are of some use. No more continuity, or else continuity but no derivatives, etc. … Formerly, when a new function was invented, it was in view of some practical end. Today they are invented on purpose to show our ancestors’ reasonings at fault, and we shall never get anything more out of them.

— Henri Poincaré, Mathematical definitions and education, 1906