“Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.”

Quoted in Hilbert's Die Grundlagen der Mathematik (1927)

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a s…" by David Hilbert?
David Hilbert photo
David Hilbert 30
German prominent mathematician 1862–1943

Related quotes

“The assumptions and definitions of mathematics and science come from our intuition”

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

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)
Context: The assumptions and definitions of mathematics and science come from our intuition, which is based ultimately on experience. They then get shaped by further experience in using them and are occasionally revised. They are not fixed for all eternity.

“Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition.”

Joshua Girling Fitch (1824–1903) British educationalist

The Fourth Dimension simply Explained. (New York, 1910), p. 58. Reported in Moritz (1914); Also cited in: Howard Eves (2012), Foundations and Fundamental Concepts of Mathematics, p. 167

Lucio Russo photo

“Euclid … manages to obtain a rigorous proof without ever dealing with infinity, by reducing the problem [of the infinitude of primes] to the study of finite numbers. This is exactly what contemporary mathematical analysis does.”

Lucio Russo (1944) Italian historian and scientist

2.4, "Discrete Mathematics and the Notion of Infinity", p. 45
The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn (2004)

David Hilbert photo

“The art of doing mathematics consists in finding that special case which contains all the germs of generality.”

David Hilbert (1862–1943) German prominent mathematician

Quoted in Constance Reid, "Hilbert" (1970)

Johannes Kepler photo

“Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression… [Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations.”

Jay Lemke (1946) American academic

Jay Lemke (2003), "Teaching all the languages of science: Words , symbols, images and actions," p. 3; as cited in: Scott, Phil, Hilary Asoko, and John Leach. "Student conceptions and conceptual learning in science." Handbook of research on science education (2007): 31-56.

Paul Bernays photo
George Gilder photo

“Gödel demonstrated that every logical scheme, including mathematics, is dependent upon axioms that it cannot prove and that cannot be reduced to the scheme itself.”

George Gilder (1939) technology writer

Knowledge and Power : The Information Theory of Capitalism and How it is Revolutionizing our World (2013), Ch. 10: Romer's Recipes and Their Limits <!-- Regnery Publishing -->
Context: Academic scientists of any sort expect to be struck by lightning if they celebrate real creation de novo in the world. One does not expect modern scientists to address creation by God. They have a right to their professional figments such as infinite multiple parallel universes. But it is a strange testimony to our academic life that they also feel it necessary of entrepreneurship to chemistry and cuisine, Romer finally succumbs to the materialist supersition: the idea that human beings and their ideas are ultimately material. Out of the scientistic fog there emerged in the middle of the last century the countervailling ideas if information theory and computer science. The progenitor of information theory, and perhaps the pivotal figure in the recent history of human thought, was Kurt Gödel, the eccentric Austrian genius and intimate of Einstein who drove determinism from its strongest and most indispensable redoubt; the coherence, consistency, and self-sufficiency of mathematics.
Gödel demonstrated that every logical scheme, including mathematics, is dependent upon axioms that it cannot prove and that cannot be reduced to the scheme itself. In an elegant mathematical proof, introduced to the world by the great mathematician and computer scientist John von Neumann in September 1930, Gödel demonstrated that mathematics was intrinsically incomplete. Gödel was reportedly concerned that he might have inadvertently proved the existence of God, a faux pas in his Viennese and Princeton circle. It was one of the famously paranoid Gödel's more reasonable fears. As the economist Steven Landsberg, an academic atheist, put it, "Mathematics is the only faith-based science that can prove it."

George Pólya photo

“Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.”

George Pólya (1887–1985) Hungarian mathematician

[Jon Fripp, Michael Fripp, Deborah Fripp, Speaking of Science: Notable Quotes on Science, Engineering, and the Environment, https://books.google.com/books?id=44ihCUS1XQMC&pg=PA45, 2000, Newnes, 978-1-878707-51-2, 45]

Related topics