HALF A PHILOSOPHER, HALF A MATHEMATICIAN

The great mathematician, logician, philosopher and linguist Friedrich Ludwig Gottlob Frege (1848-1925) once said that “Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician.” This may not be true in each and every case, but this is certainly true of Frege, whose brilliant achievements not just in mathematics and logic, but, specifically, in the philosophy of logic, philosophy of mathematics, and philosophy of language, are of outstanding importance.

My first acquaintance with Frege was at an early age through reading Bertrand Russell’s History of Western Philosophy, where, in the chapter The Philosophy of Logical Analysis, Russell (who happened to be one of very few people capable of appreciating Frege before anyone else did) writes the following: “In philosophy ever since the time of Pythagoras there has been an opposition between the men whose thought was mainly inspired by mathematics and those who were more influenced by the empirical sciences. Plato, Thomas Aquinas, Spinoza, and Kant, belong to what may be called the mathematical party; Democritus, Aristotle, and the modern empiricists, from Locke onwards, belong to the opposite party. In our day, a school of philosophy has arisen which sets to work to eliminate Pythagoreanism from the principles of mathematics, and to combine empiricism with an interest in the deductive parts of human knowledge. The aims of this school are less spectacular than those of most philosophers in the past, but some of its achievements are as solid as those of the men of science.

The origin of this philosophy is in the achievements of mathematicians who set to work to purge their subject of fallacies and slipshod reasoning. Russell now begins his list of such mathematicians with Leibniz and Cantor. The next man of importance was Frege, who published his first work (Begriffsschrift) in 1879, and his definition of “number” in 1884 (in The Foundations of Arithmetic); but in spite of the epoch-making nature of his discoveries, he remained wholly without recognition until I drew attention to him in 1903. It is remarkable that before Frege every definition of number that had been suggested contained very elementary logical blunders. It was customary to identify “number” with “plurality.” But an instance of “number” is a particular number, say 3, and an instance of 3 is a particular triad. The triad is a plurality, but the class of all triads which Frege identified with the number 3 is a plurality of pluralities and number in general of which 3 is an instance is a plurality of pluralities of pluralities. The elementary grammatical mistake of confounding this with the simple plurality of a given triad made the whole philosophy of number before Frege a tissue of nonsense in the strictest sense of the term “nonsense.”

From Frege’s work it followed that arithmetic and pure mathematics generally, is nothing but a prolongation of deductive logic. This disproved Kant’s theory that arithmetic propositions are “synthetic” and involve a reference to time. The development of pure mathematics from logic was set forth in detail in the Principia Mathematica, by Whitehead and myself.

Developing Russell’s explanation of Frege’s historical accomplishment, I may paraphrase it by saying that, prior to Frege, number was seen even by the greatest mathematical minds of all time as a simple plurality, or a plurality of the first order, whereas three men in a boat, or three oranges, or three sisters are all such simple pluralities, while 3, as a number common to all, of them becomes in reality a second-order plurality, and number as-such, containing 3; 4; 5; 6; etc., becomes a third-order plurality. For someone untutored in complex mathematical labyrinths, this distinction may not be such a big deal at all. But once we enter into the realm of formal logic, Predicate Calculus, and Quantification--- without which modern mathematics is practically helpless, the Frege Revolution becomes comprehensible, and it is fairly found comparable with the Cartesian Revolution of the seventeenth century.

Talking about the Frege revolution, I am fortunate enough to know what I am talking about, as Frege, and his revolutionary mathematical logic, was one of my subjects at Moscow University.  This is obviously not the proper place to perplex the reader any further on either the technicalities or even generalities of Frege’s mathematical logic, but one thing must be repeated from the things said before, that here in this sphere, associated with the broader sphere of logical analysis, mathematics and philosophy merge into one, confirming, at least for these particular intents and purposes, the wisdom of Frege’s observation, which has been condensed into the title of this entry.