Frank P. Ramsey

"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."

"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."

"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!"

"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."

"[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."

"[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."

"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."

"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."

"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?"

"[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."

"But what we can't say we can't say, and we can't whistle it either."

"[M]y teacher Frank Ramsey... showed that if a scientific system was so completely precise that you could replace every word in it, such as "electrons," by the totality of all observations on the electron, then you could never discover anything new. ...Ramsey's theorem is really equivalent to all the Tarski-Turing theorems in essence because it says that if you push the symbolism even in a word like "mass" so that you say, as operationalists do... mass is everything you do when you weigh the mass, you are never going to discover that mass and energy are interchangeable. You have closed the system to new discoveries."

"The death... is a heavy loss—though his primary interests were in Philosophy and Mathematical Logic—to the pure theory of economics. ...If he had followed the easier path of mere inclination, I am not sure that he would not have exchanged the tormenting exercises of the foundations of thought, where the mind tries to catch its own tail, for the delightful paths of our own most agreeable branch of the moral sciences, in which theory and fact, intuitive imagination and practical judgement, are blended in a manner comfortable to the human intellect."

"When he did descend from his accustomed stony heights, he still lived without effort in a rarer atmosphere than most economists care to breathe, and handled the technical apparatus of our science with the easy grace of someone accustomed to something far more difficult. But he has left behind him in print only two witnesses to his power - his papers published in The Economic Journal on 'A contribution to the Theory of Taxation' in March, 1927, and on 'A Mathematical Theory of Saving' in December, 1928. The latter of these is, I think, one of the most remarkable contributions to mathematical economics ever made, both in respect of the intrinsic importance and difficulty of its subject, the power and elegance of the technical methods employed, and the clear purity of the illumination with which the writer's mind is felt by the reader to play about its subject. The article is terribly difficult reading for an economist, but it is not difficult to appreciate how scientific and aesthetic qualities are combined in it together."

"The loss of Ramsey is... to his friends, for whom his personal qualities joined most harmoniously with his intellectual powers, one which it will take them long to forget. His bulky Johnsonian frame, his spontaneous gurgling laugh, the simplicity of his feelings and reactions... his honesty of mind and heart, his modesty, and the amazing, easy efficiency of the intellectual machine which ground away behind his wide temples and broad smiling face, have been taken from us at the height of their excellence and before their harvest of work and life could be gathered in."

"With Ramsey’s young death, the world of learning was robbed of one of its most glittering stars. It is now time that he receive his due. What is needed is a thorough biography that would describe and place in intellectual history his important contributions to economics, mathematics, and philosophy, while keeping an eye out for what Virginia Woolf called the “fertile facts” that would reveal to us not only the impressive mind, but also the somewhat elusive personality of this extraordinary man."

"He was an extraordinarily clear thinker: no-one could avoid more easily... the sort of confusions of thought to which even the best philosophers are liable, and he was capable of apprehending clearly... the subtlest distinctions. He had... an exceptional power of drawing conclusions from a complicated set of facts. ...his subtlety and ingenuity did not lead him, as it seems to have led some philosophers, to deny obvious facts. ...he could see which problems were the most fundamental, and it was these... which he was most anxious to solve. ...I almost always felt, with regard to any subject which we discussed, that he understood it much better than I did."

"Ramsey... had a most uncommon power of explaining clearly to others what he thought and why... But sometimes... he fails to explain things as clearly... simply because... he does not realize that what to him seems perfectly clear and straightforward may to others, less gifted, offer many puzzles. ...But even where you cannot understand him completely you can often understand him enough to find him extraordinarily interesting. ...even if he was wrong, [he] had very good reasons for the opinions at which he had arrived."