Frank Plumpton Ramsey (1903-1930)

If I was to write a Weltanschaung I should call it not “What I believe” but “What I feel.” This is connected with Wittgenstein’s view that philosophy does not give us beliefs, but merely relieves feelings of intellectual discomfort....

I don’t feel the least humble before the vastness of the heavens. The stars may be large, but they cannot think or love; and these are qualities which impress me far more than size does. (Ramsey 1931: 290-91)

In one of those extraordinary serendipities with which greatness is tinged, and Frank Ramsey - despite the cruelty of a life cut short at its prime - was blessed with an abundance of it, he was born in Cambridge on 22 February 1903, the year the defining works of the twentieth century in ethics and the foundations of mathematics were published: Principia Ethica by G.E. Moore and the Principles of Mathematics by Bertrand Russell, the latter presaging the monumental Principia Mathematica (written jointly with Whitehead). With equal irony, Ramsey died in London, on 19 January 1930, the year Kurt Gδdel announced his famous incompleteness theorem(s) in Kδnigsberg. With the celebrated Paris-Harrington results (Paris and Harrington 1977), the connection between Ramsey’s posthumously published classic, On a Problem of Formal Logic (Ramsey 1928a, 1928b), and Gδdel's pioneering results have been shown to be woven from the same foundational fabric of the Entscheidungsproblem that Hilbert had formulated, to settle, decisively, the grundla- genkrise of the 1920s, precipitated by Brouwer’s intuitionistic and constructive challenges. It may be that Ramsey himself veered towards intuitionism, in mathematical philosophy at the end of his short life (Braithwaite’s ‘Introduction’ to Ramsey 1991; Ramsey 1991: chs 53, 54).

The bare bones of Frank Ramsey’s family background are simple enough to document, although behind this simplistic account there is a richness that deserves to be expanded.

He was the eldest of Agnes Mary - nee Wilson - and Arthur Stanley Ramsey’s four children. His brother, Michael, later to become the Archbishop of Canterbury, was born one year later and his sisters, Bridget and Margaret were, respectively, four and 14 years younger. Frank’s father, Arthur Stanley, was also a mathematician, Fellow, Tutor, Bursar and President of Magdalene College, Cambridge. Frank himself, at the time of his death, was a Fellow of King’s College, Cambridge. He had been a scholar at Winchester, from 1915 and went up to Trinity College, Cambridge, in 1920, graduating as a Wrangler in 1923. He was elected to a Fellowship at King’s College in 1924, to a University Lectureship in Mathematics in 1926 and took over as the Director of Studies in Mathematics at King’s. He married Lettice Barker (d. 12 July 1985), five years his senior in age, in September 1925 and they had two daughters, Jane and Sarah; only the latter survived the deaths of her parents, separated by 55 years, to produce three grandchildren, Stephen, Belinda and Matthew Burch.

Moore’s Principia Ethica, Russell’s (and Whitehead’s 1913) Principia Mathematica, Hardy’s Pure Mathematics (1908), Keynes’ Treatise on Probability (1921) and Wittgenstein’s Tractatus Logico Philosophicus (1921-22; the manuscript version of which was placed at Frank Ramsey’s disposal by C.K. Ogden, with whom he eventually translated it while still a teenager) redefined the intellectual landscape of their respective fields, as Ramsey himself reached his precociously youthful maturity in the approaching postwar period. To these themes, in which he excelled and made what can only be described as outstanding contributions, he added, under the influence of Maynard Keynes (and A.C. Pigou, also a Fellow at King’s), a mastery of economics that few, then or since, matched for analytical brilliance and originality.

The “Eponymous F.P. Ramsey” (Mellor 1983) has given birth to Ramsey economics, Ramsey pricing and the Ramsey taxation formula, the Ramsey sentence, the Ramsey theorem (Wang 1981), Ramsey problem, Ramsey theory and Ramsey number.

Beyond these, there are the contributions to a simplified theory of types (Ramsey 1925, 1926; Chwistek 1949: ch. 6, whose work Ramsey was acquainted with, claims priority on this point) to resolve the perplexities of the logical and semantic paradoxes that plagued Principia Mathematica and the discussions and debates with Wittgenstein on finitism and the role of tautologies and the definition of identity in mathematics. Finally, there is the acknowledged influence of Ramsey on Production of Commodities by Means of Commodities (Sraffa 1960).

Since it is impossible to discuss all of the above in any kind of justifiable detail within the restricted space we have at our disposal, we confine ourselves to five core contributions, with economic relevance, past, present and, we are sure, future.

“Officially”, in the past and the present, Ramsey’s contributions to economics have been confined to his three articles on “Truth and probability” (1926), “A contribution to the theory of taxation” (1927a) and “A mathematical theory of saving” (1928c). With hindsight at our disposal, we can see that “Ramsey’s theorem” (1928b) has become relevant for various frontiers in economic theory. Then, there is the explicit acknowledgement by Sraffa, in the preface to his magnum opus, of Ramsey’s “mathematical help”.

“Truth and probability”, written as a critical review of A Treatise on Probability (Keynes 1921; incidentally, many insist that Keynes accepted Ramsey’s criticisms and gave up on “partial orders”, which simply cannot be true, given the many ways in which Keynes insisted on the unbridgeable formal gap between “risk” and “uncertainty”, particularly in the General Theory), is justly celebrated as the original contribution to the axiomatization of subjective probability - predating the equally important work of Bruno De Finetti by many years and anticipating the more orthodox von Neumann- Morgenstern work by almost two decades - and providing foundations for expected utility maximization underpinning for rational behaviour.

However, it is rarely - if ever - remembered that Ramsey added an important caveat (De Finetti, was also explicit in his rejection of σ-algebras and the embracing of finite-additivity when axiomatizing subjective probability) to this classic contribution (Ramsey 1926: 85): “[N]othing has been said [in my paper] about degrees of belief when the number of alternatives is infinite. About this I have nothing useful to say, except that I doubt if the mind is capable of contemplating more than a finite number of alternatives.”

Similarly, Ramsey’s injunctions (perhaps “Pigou-inspired”?) against discounting the future, as “a practice which is ethically indefensible and arises merely from the weakness of the imagination” (Ramsey 1928: 261), are no longer even added as a footnote qualification when deriving the Keynes-Ramsey Rule within the framework of the “Ramsey-Cass-Koopmans” model of optimal aggregate economic growth in neoclassical economics. Moreover, the published version of this modern classic omitted two additional sections (document # 006-07-01 in the “Ramsey Collection” held in the Hillman Library of the University of Pittsburgh), where it can be seen that Ramsey thought hard and deeply about the problem of discounting the future. It is our firm belief that Keynes’s Economic Possibilities for our Grandchildren, given not only in 1928, the year Ramsey’s “Saving” paper was published in the Economic Journal, but also at Winchester College, Ramsey’s old school, was consistent with the problem of discounting the infinite future, solved with the notion of “bliss” by Ramsey.

Ramsey (1927a) has spawned two strands of optimal pricing research in economics: the classic optimal taxation literature, linking up with the work that originates with Mirrlees and Diamond; the other, more along the lines of public utility and public goods, pricing has resulted in the Ramsey-Boiteux inverse elasticity formula (which, though famously also derived in Robinson 1933, has no reference to Ramsey!).

However, it remains a puzzle to us that the link with Lindahl pricing has never been developed.

The Ramsey-Sraffa interaction is clearly and, so far as we can tell, exhaustively, presented in Kurz and Salvadori (2001). The only comment we would like to add here is the following. The first of the three comments by Ramsey, summarized by Sraffa on 26 June, 1928 (Kurz and Salvadori 2001: 262), refers essentially to the Ruffini-Abel result on the unsolvability of the quintic, which Ramsey would, as a student of mathematics (particularly of Hardy), have known very well. The point here is the affinity of Abel’s approach to his unsolvability proof with Godel’s later unsolvability proof strategies and the present understanding of the relevance of such things to variations of ‘Ramsey’s theorem’ (cf. Paris and Harrington 1977).

We come now to what we think will be Ramsey’s lasting contribution to economic theory, in the future: the growing influence of Ramsey (1928a, 1928b) on graph theory, combinatorics, recursion theory and computational complexity theory. We are convinced that Ramsey theory - that is, “Ramsey’s Theorem” - will come to play an increasingly important computational role in these fields. To substantiate this conjecture we have to refer to what are called the two principles of Ramsey theory. However, to state them we have, first, to state, in some minimally formal way, the Ramsey theorem:

Ramsey’s theorem (Ramsey, 1928b [1978]: 233, theorem A):

For A C N, where N: the set of natural numbers, [A]ⁿ the class of all n-element subsets of A; Partition [N]ⁿ into finitely many sub-classes, C_i, i = 1, 2,..., p and call the partition P.

H(P): the class of those infinite sets A C N, s.t., [A]ⁿ C C_i for some i.

Then, H(P) is nonempty for every such partition.

Ramsey theory’s first principle:

Structure (of P, for example) is preserved under finite partitions.

Ramsey theory’s second principle:

There is always an appropriate notion of size, s.t., any sufficiently large structure always contains the desired well-organized sub-structure.

Now, suppose we endow P with a recursive structure, then we ask what we can say about H(P), from many points of view: computational complexity theory, for example. Recent research shows that this kind of question can be answered using proof strategies developed in Kolomogorov complexity theory.

Ramsey’s remarkable theorem, in being a source for the derivation of the celebrated Paris-Harrington theorem, also links up with Goodstein’s algorithm and the more recent development in rational-valued nonlinear dynamics (Paris and Tavakol 1993). This latter class of dynamical systems goes beyond the sensitive dependence on initial conditions nonlinear dynamics and a new class, labelled “super sensitive dynamical systems”, has been defined. The attractors of such dynamical systems are invariably trivial.

In conclusion, the Ramseyan precept for intellectual adventures that we have culled out of our research on the remarkable work of this prodigal genius can be summarized by his own paraphrasing of Wittgenstein’s well known aphorism: “What we can’t say, we can’t say and we can’t whistle it either.”

K. Vela Velupillai and Ragupathy Venkatachalam

Frank Plumpton Ramsey (1903-1930)

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