“Gödel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.”

— Freeman Dyson

The Scientist As Rebel (2006)

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Freeman Dyson 90

theoretical physicist and mathematician 1923

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“If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.”

John D. Barrow (1952–2020) British scientist

The Artful Universe (1995)
Context: If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.<!-- Ch. 5, p. 211

“Fifty years ago Kurt Gödel… proved that the world of pure mathematics is inexhaustible. … I hope that the notion of a final statement of the laws of physics will prove as illusory as the notion of a formal decision process for all mathematics.”

Freeman Dyson book Infinite in All Directions

Source: Infinite in All Directions (1988), Ch. 3 : Manchester and Athens
Context: Fifty years ago Kurt Gödel... proved that the world of pure mathematics is inexhaustible. … I hope that the notion of a final statement of the laws of physics will prove as illusory as the notion of a formal decision process for all mathematics. If it should turn out that the whole of physical reality can be described by a finite set of equations, I would be disappointed, I would feel that the Creator had been uncharacteristically lacking in imagination.

“The object of mathematics is to discover "true" theorems. We shall use the term "valid" to describe statements formed according to certain rules and then shall discuss how this notion compares with the intuitive idea of "true."”

Paul Cohen (1934–2007) American mathematician

Set theory and the continuum hypothesis, p. 8. https://books.google.com/books?id=Z4NCAwAAQBAJ&pg=PA8
Set Theory and the Continuum Hypothesis (1966)

“Only by a study of the development of mathematics can its contemporary significance be understood.”

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Context: The professional mathematician can scarcely avoid specialization and needs to transcend his private interests and take a wide synoptic view of the whole landscape of contemporary mathematics. His scientific colleagues are continually seeking enlightenment on the relevance of mathematical abstractions. The undergraduate needs a guidebook to the topography of the immense and expanding world of mathematics. There seems to be only one way to satisfy these varied interests... a concise historical account of the main currents... Only by a study of the development of mathematics can its contemporary significance be understood.

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Hans Freudenthal (1905–1990) Dutch mathematician

Norbert Wiener (2008)

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

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“We are told … statements that are not mathematical or logical formulae may look as if they were necessarily or certainly true, but they only look like that. They cannot really be either necessary or certain.”

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Albert Einstein (1879–1955) German-born physicist and founder of the theory of relativity

1930s, Obituary for Emmy Noether (1935)
Context: Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature.

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Freeman Dyson 90

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