“The formalists neglected the content altogether and made mathematics meaningless, the logicians neglected the form and made mathematics consist of any true generalizations; only by taking account of both sides and regarding it as composed of tautologous generalizations can we obtain an adequate theory.”

— Frank P. Ramsey

The Foundations of Mathematics (1925)

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Frank P. Ramsey 10

British mathematician, philosopher 1903–1930

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“Like all of mathematics, game theory is a tautology whose conclusions are true because they are contained in the premises.”

Thomas Flanagan (political scientist) (1944) author, academic, and political activist

Source: Game Theory and Canadian Politics (1998), Chapter 10, What Have We Learned?, p. 164.

“[T]he formalist school, of whom the most eminent representative is Hilbert, have concentrated on the propositions of mathematics, such as '2 + 2 = 4'. They have pronounced these to be meaningless formulae to be manipulated according to arbitrary rules, and they hold that mathematical knowledge consists in knowing what formulae can be derived from what others consistently with the rules…. for example…'2' is a meaningless mark occurring in these meaningless formulae. But… '2' occurs not only in '2 + 2 = 4', but also in 'It is 2 miles to the station', which is not a meaningless formulae, but a significant proposition, in which '2' cannot conceivably be a meaningless mark.”

Frank P. Ramsey (1903–1930) British mathematician, philosopher

The Foundations of Mathematics (1925)

“Formal proofs, where there is deliberately no meaning, can convince only formalists, and of the results they themselves seem to deny any meaning. Is that to be the mathematics we are to use to understand the world we live in?”

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

Source: Mathematics on a Distant Planet (1998), p. 645

“The pragmatist regards any theory as a mere mathematical machine for generating numbers which he then compares with experiment. A pragmatist is concerned with results, not reality. The pragmatist refuses on principle to speculate about deep reality, such a concept being meaningless from his point of view. Pragmatism is an intelectually safe but ultimately sterile philosophy.”

Nick Herbert (1936) American physicist

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“A model that unifies all types of selection (chemical, sociological, genetical, and every other kind of selection) may open the way to develop a general ‘Mathematical Theory of Selection’ analogous to communication theory… Selection has been studied mainly in genetics, but of course there is much more to selection than just genetical selection… yet, despite the pervading importance of selection in science and life, there has been no abstraction and generalisation from genetical selection to obtain a general selection theory and general selection mathematics”

George R. Price (1922–1975) American population geneticist

Price, G.R. (1995). "The nature of selection." Journal of Theoretical Biology 175:389-396 (written circa 1971)

“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)

“The most general form of systems theory is a set of logical or mathematical statements about all conceptual systems. A subset of this concerns all concrete systems. A subsubset concerns the very special and very important living systems, i. e., general living systems theory.”

James Grier Miller (1916–2002) biologist

Source: Living systems, 1978, p. 41

“There is, therefore, both positive and negative proof in favour of my mental and suggestive theory, and in opposition to the magnetic, occult, or electric theories of the Mesmerists and electro-biologists. My theory, moreover, has this additional recommendation, that it is level to our comprehension, and adequate to account for all which is demonstrably true, without offering any violence to reason and common sense, or being at variance with generally admitted physiological and psychological principles.”

James Braid (1795–1860) Scottish surgeon, hypnotist, and hypnotherapist

Original Philosophy of Hypnotism The International College of Hypnosis & Hypnotherapy

“The assimilation of tautologies and contradictions with true and false propositions respectively results from the fact that tautologies and contradictions can be taken as truth-functions just like ordinary propositions, and for determining the truth of falsity of the truth-function, tautologies and contradictions among its arguments must be counted as true or false respectively. …Are the propositions of symbolic logic and mathematics tautologies in Mr Wittgenstein's sense?”

Frank P. Ramsey (1903–1930) British mathematician, philosopher

The Foundations of Mathematics (1925)

“In spite of all the foregoing, there is a science of economics, a true, and even exact, science, which reaches laws as universal as those of mathematics and mechanics. The greatest need for the development of economics as a growing body of thought and practice is an adequate appreciation of the meaning, and the limitations, of this body of accurate premises and rigorously established conclusions. It comes about in the same general way as all science, except perhaps in a higher degree, i. e., through abstraction. There are no laws regarding the content of economic behavior, but there are laws universally valid as to its form.”

Frank Knight (1885–1972) American economist

There is an abstract rationale of all conduct which is rational at alt, and a rationale of all social relations arising through the organization of rational activity.
Source: "The limitations of scientific method in economics", 1924, p. 127 (2009 edition)

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