The first attempt aimed at rewriting the established body of mathematics in logical symbolism was made by the German logician and philosopher Gottlob Frege (1848-1925). Frege received his doctorate in mathematics from Göttingen in 1873. The following year he began his teaching career at the University of Jena, where he remained for 45 years. Frege’s mathematical work was almost entirely devoted to mathematical logic and its foundations. His treatise, entitled “Conceptual Notation, a Symbolic Language of Pure Thought Modeled on the Language of Arithmetic”, published in 1879, was a milestone in the history of modern logic. However, more than 20 years passed before Bertrand Russell recognized the magnitude of the result.
As for Dedekind, also for Frege arithmetic must be thought of as closely linked to logic; but while the first
“identified logic and arithmetic meaning that logic was arithmetic… Frege reverses this position: the identification between logic and arithmetic occurs because arithmetic is logical. In this sense, Frege takes a further step, arriving at the logicization of mathematics”. – C. Mangione
The core of the Fregean approach consists in the enunciation of a “logicist program” to be placed at the foundation of mathematics: the mathematical concepts for Frege must be defined in purely logical terms and the “truths” (the theorems) must be derived, through explanatory logical rules, starting from principles of an exclusively logical nature.
The first step towards a logical foundation of mathematics lies in preparing a series of appropriate tools that allow to formalize the logical processes of deduction. This was ensured by Frege with his aforementioned treatise of 1879, which, at the time, did not obtain the recognition that the author expected.
Five years later in 1884, with The Foundations of Arithmetic, Frege, temporarily abandons the complicated symbolism present in his previous work and presents his research program, in which the central moment is represented by the definition in purely logical terms of the concept of number.
Frege is intransigent with the infiltration of psychology into logic and with the question of the foundations of arithmetic. He shows himself closer to the philosophical ideas of Kant, of which he also accepts the terminology that includes analytical, synthetic, a priori, a posteriori judgments.
His opposition to the formalist point of view in mathematics is also decisive. The latter, consistent with the aphorism of Hilbert “in the beginning was the sign”, sees in sign 1 the meaning of the number one, in figure 1+1 the meaning of two, etc. Arithmetic becomes the study of combinations of figures. Such a formalist position will acquire theoretical depth with Hilbert.
Frege opposes the formalists: the numerical sign is nothing more than a way of denoting the conceptual content of the number. He gives his definition of number:
The natural number that belongs to the concept F is nothing more than the extension of the concept “equally numerous to F”, where the important fact is to bring the concept of number back to that of extension, a concept of a decidedly logical nature.
Frege, therefore, defines number as what for Cantor was the cardinality of a set. At the base of the two definitions, we can see the same concept: equipotency of sets for Cantor, equinumerosity of classes for Frege, both based on the notion of two-way correspondence.
However, the theoretical path is different in the two cases. Cantor defines the cardinal number of a set through a process of abstraction – which he leaves intuitively and logically indeterminate, and then considers all sets equipotent to the given set. Frege, on the other hand, first defines the concept of equinumerosity by classes and then, through a logically rigorous definition by abstraction, defines the cardinal number, which is identified with the class of all classes equinumerous to it.
Frege’s naïve set theory
Aside from differences in motivation, Frege developed an approach equivalent to Cantor’s, an approach known as naïve set theory. This theory can be condensed into only two principles that identify sets with the properties that define them. These principles are:
- The principle of extensionality: a set is completely determined by its elements. Thus, two sets with the same elements are equal.
- The principle of comprehension: each property determines a set. The elements belonging to the set are those that satisfy the property; each set is then determined by a property that is to be an object belonging to the set.
The work of Cantor and Frege showed that arithmetic was reducible to set theory, that is, to pure logic. Unfortunately, this simple foundation proved inconsistent, hence the adjective “naïve”. In fact, in 1902, Bertrand Russell showed that the principle of comprehension was contradictory: it was the famous Russel’s paradox. As Piergiorgio Odifreddi explains in his “The mathematics of the twentieth century”:
In essence, the sets of objects are divided into two classes, depending on whether or not they are one of the objects contained in the set itself: in other words, depending on whether or not they belong to themselves. For example, the set of sets with more than one element belongs to itself, because it certainly has more than one element. On the other hand, the set of sets with a single element does not belong to itself, because it certainly also has more than one element.
And it is at this point that Russell’s paradox is inserted with his question: does the set of all sets that do not belong to themselves, belong or not to itself? Again from the book cited:
If yes (it belongs to itself), then it is one of the sets that do not belong to themselves, and therefore cannot belong to their collection, that is, to itself. If not, then it is one of the sets that do not belong to themselves, and therefore belongs to their collection, that is, to itself.
In the last decade of the century, before Russell’s objection, Frege continued to pursue his logicist program by publishing in 1893 the first volume of the Principles of Arithmetic which was followed by the second a few years later.
Bitterness of last years
As mentioned, due to the complexity of the symbolism used and the novelty of the approach, Frege’s works passed almost entirely without recognition until Russell dedicated an appendix to them in his Principles of Mathematics (1903). When the two volumes of the Principles of Arithmetic were in print, Russell found the paradox and communicated it to Frege in a letter.
Frege recognized the contradiction contained in his system, and one has the feeling that he has not since been able to regain his faith in the possibility of a purely logical-formal foundation of arithmetic. In fact, in recent years he abandoned his research and died embittered, convinced that his life’s work had been for the most part a failure. Frege’s death did not arouse much attention from the academic world: an unjust fate for a man who nevertheless was the architect of a revolution in logic.