The opening of Felix S. Cohen, “Transcendental Nonsense and the Functional Approach” (1935):
Some fifty years ago a great German jurist had a curious dream. He dreamed that he died and was taken to a special heaven reserved for the theoreticians of the law. In this heaven one met, face to face, the many concepts of jurisprudence in their absolute purity, freed from all entangling alliances with human life. Here were the disembodied spirits of good faith and bad faith, property, possession, laches, and rights in rem. Here were all the logical instruments needed to manipulate and transform these legal concepts and thus to create and to solve the most beautiful of legal problems. Here one found a dialectic-hydraulic-interpretation press, which could press an indefinite number of meanings out of any text or statute, an apparatus for constructing fictions, and a hair-splitting machine that could divide a single hair into 999,999 equal parts and, when operated by the most expert jurists, could split each of these parts again into 999,999 equal parts. The boundless opportunities of this heaven of legal concepts were open to all properly qualified jurists, provided only they drank the Lethean draught which induced forgetfulness of terrestrial human affairs. But for the most accomplished jurists the Lethean draught was entirely superfluous. They had nothing to forget.
Von Jhering’s dream has been retold, in recent years, in the chapels of sociological, functional, institutional, scientific, experimental, realistic, and neo-realistic jurisprudence. The question is raised, “How much of contemporary legal thought moves in the pure ether of Von Jhering’s heaven of legal concepts?” One turns to our leading legal textbooks and to the opinions of our courts for answer. May the Shade of Von Jhering be our guide.
A striking passage from Frege’s unpublished essay “Boole’s logical calculus and the concept-script,” which he submitted in 1881 first to Zeitschrift für Mathematik und Physik, then the Mathematischen Annalen, and finally the Zeitschrift für Philosophie und philophische Kritik, but was rejected each time:
I now return once more to the examples mentioned earlier, so as to point out the sort of concept formation that is to be seen in those accounts. The fourth example gives us the concept of a multiple of 4 […]. The eighth example gives us the concept of the congruence of two numbers with respect to a modulus, the 13th that of the continuity of a function at a point etc. All these concepts have been developed in science and have proved their fruitfulness. For this reason what we may discover in them has a far higher claim on our attention than anything that our everyday trains of thought might offer. For fruitfulness is the acid test of concepts, and scientific workshops the true field of study for logic.
This is pp. 32-33 in the translation by Peter Long and Roger White, with the assistance of Raymond Hargreaves, in Posthumous Writings, edited by Hans Hermes et al (Blackwell, 1979).
This remains one of the lesser appreciated themes in Frege’s work. The standard fare in a philosophy major mentions sense and reference, concept and anti-psychologism, but I’d bet even very few working philosophers are familiar with these suggestive passages on fruitfulness. Of course, every time I read the word I can only think of Keats, on autumn: “Season of mists and mellow fruitfulness…”
Some helpful discussions appear in:
- Jamie Tappenden, “Fruitfulness as a Theme in the Philosophy of Mathematics,” The Journal of Philosophy (2012)
- Jamie Tappenden, “The Mathematical and Logical Background to Analytic Philosophy,” The Oxford Handbook of the History of Analytic Philosophy
- Jamie Tappenden, “Extending knowledge and ‘fruitful concepts’: Fregean themes in the philosophy of mathematics,” in Gottlob Frege: Critical Assessments of Leading Philosophers, Vol. III, Frege’s philosophy of mathematics, edited by Michael Beany and Erich Reck
A charming discussion of what should be called the fundamental theorem of computation theory, in Epstein and Carnielli, Computability: Computable Functions, Logic, and the Foundations of Mathematics (2008):
We have studied one formalization of the notion of computability. In succeeding chapters we will study two more: recursive functions and functions representable in a formal system.
The Most Amazing Fact
All the attempts at formalizing the intuitive notion of computable function yield exactly the same class of functions.
So if a function is Turing machine computable, it can also be computed in any of the other systems described in Chapter 8.E. This is a mathematical fact which requires a proof. […] Odifreddi, 1989 establishes all the equivalences. […]
The Most Amazing Fact is stated about an extensional class of functions, but it can be stated constructively: Any computation procedure for any of the attempts at formalizing the intuitive notion of computable function can be translated into any other formalization in such a way that the two formalizations have the same outputs for the same inputs.
In 1936, even before these equivalences were established, Church said,
We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambda-definable function of positive integers). This definition is thought to be justified by the considerations which follow, so far as positive justification can ever be obtained for the selection of a formal definition to correspond to an intuitive notion.
So we have
Church’s Thesis: A function is computable iff it is lambda-definable.
This is a nonmathematical thesis: it equates an intuitive notion (computability) with a precise, formal one (lambda-definability). By our amazing fact this thesis is equivalent to
A function is computable iff it is Turing machine computable.
Turing devised his machines in a conscious attempt to capture in simplest terms what computability is. That his model turned out to give the same class of functions as Church’s (as established by Turing in the paper cited above) was strong evidence that it was the “right” class. Later we will consider some criticisms of Church’s Thesis in that the notion of computability should coincide with either a larger or a small class than the Turing machine computable ones.