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“It is worth pausing for a moment to consider how far our conclusions are affected by considerations which our simplifying assumptions have forced us to neglect.”

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Frank P. Ramsey

"A Mathematical Theory of Saving", The Economic Journal, Vol. 38, No. 152 (Dec., 1928)

“The object of this paper is to give a satisfactory account of the Foundations of Mathematics in accordance with the general method of Frege, Whitehead and Russell. Following these authorities, I hold that mathematics is part of logic, and so belong to what may be called the logical school as opposed to the formalist and intuitionist schools. I have therefore taken Principia Mathematica as a basis for discussion and ammendment; and believe myself to have discovered how, by using the work of Mr Ludwig Wittgenstein, it can be rendered free from the serious objections which have caused its rejection by the majority of German authorities, who have deserted altogether its line of approach.”

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Frank P. Ramsey

Preface
The Foundations of Mathematics (1925)

“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.”

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Frank P. Ramsey

The Foundations of Mathematics (1925)

“[A]lthough my attempted reconstruction of the view of Whitehead and Russell overcomes, I think, many of the difficulties, it is impossible to regard it as altogether satisfactory.”

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Frank P. Ramsey

As quoted by B. A. Bernstein, Bulletin of the American Mathematical Society 38 (1932), pp. 611-612.
The Foundations of Mathematics (1925)

“[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.”

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Frank P. Ramsey

The Foundations of Mathematics (1925)

“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?”

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Frank P. Ramsey

The Foundations of Mathematics (1925)

“Philosophy must be of some use and we must take it seriously; it must clear our thoughts and so our actions. Or else it is a disposition that we have to check, and an inquiry to see that this is so; i. e. the chief proposition of philosophy is that philosophy is nonsense. And again we must then take seriously that it is nonsense, and not pretend, as Wittgenstein does, that it is important nonsense!”

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Frank P. Ramsey

"Philosophy" (1929) as quoted by Nils-Eric Sahlin, The Philosophy of F. P. Ramsey (1990)

“The first problem I propose to tackle is this: how much of its income should a nation save? To answer this a simple rule is obtained valid under conditions of surprising generality; the rule, which will be further elucidated later, runs as follows.
The rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment of utility falls short of the maximum possible rate of enjoyment.”

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Frank P. Ramsey

"A Mathematical Theory of Saving", The Economic Journal, Vol. 38, No. 152 (Dec., 1928)

“Tautologies and contradictions are not real propositions, but degenerate cases. …Clearly, by negating a contradiction we get a tautology, and by negating a tautology a contradiction. …A genuine proposition asserts something about reality, and it is true if reality is as it is asserted to be. But a tautology is a symbol constructed so as to say nothing whatever about reality, but to express total ignorance by agreeing with every possibility.”

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Frank P. Ramsey

The Foundations of Mathematics (1925)

“[W]e shall be concerned with the general nature of pure mathematics, and how it is distinguished from other sciences. Here there are… two distinct categories of things of which an account must be given—the ideas or concepts of mathematics, and the propositions of mathematics. …the great majority of writers on the subject have concentrated their attention on the explanation of one or the other… and erroneously supposed that a satisfactory explanation of the other would immediately follow.”

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Frank P. Ramsey

Footnote: In the future by 'mathematics' will always be meant 'pure mathematics'.
The Foundations of Mathematics (1925)

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