Frege’s Logicism: A Short Portrait

In 1893, Gottlob Frege published the first volume of Gottlob Frege: Basic Laws of Arithmetic.Using the logical notation, logical axioms, and axiomatic predicate logic (first- and second-level quantification, which he himself had invented!), from his earlier works (the Begriffsschrift and The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number), he meticulously laid out his ontology, rules of inference for his theory, and an analysis of the natural numbers. His logicist project seemed successful until 1902, when, just as the second volume of The Basic Laws of Arithmetic was being printed, Bertrand Russell presented him with what has come to be known as “Russell’s Paradox”, a contradiction that arose from one of Frege’s assumptions – that every concept that has an extension  – is false. It soon became apparent (even to Frege) that Russell’s Paradox had shown that Frege’s project was inconsistent, and Frege’s logicist programme came to an end.

Historical and Epistemological Motivations for Logicism

In order to understand why Frege wanted to create a reduction of mathematics to logic, it is first necessary to understand the most influential theories regarding the foundations of mathematics that were prominent during his time: those of John Stuart Mill and Immanuel Kant. In 1843, Mill had proposed that all mathematical truths arose from and were justified in sensory experience. In 1783, Kant had claimed that mathematical judgements were both synthetic (they provided us with some sort of knowledge about the physical world) and a priori (they could be known apart from experience). Frege, while acknowledging that Mill and Kant (along with others) had given interesting accounts of the foundations of mathematical truths, believed that these accounts had fallen short: justification of mathematical truths, Frege argued, had nothing to do with sensory experience (as Mill’s account required) or intuitions (as Kant had proposed).

Frege’s rejection of those theories led him to seek an answer to the epistemological problem regarding mathematics: the problem of how we can know about mathematics (specifically, the natural numbers and the relationships between the natural numbers, and how it is possible to provide justification for mathematical truths). Because he believed that mathematical truths could not be justified empirically, his answer to this epistemological problem was deceptively simple: there was nothing mysterious about mathematical truths – no epistemological problem existed, because mathematics could be derived solely from logic, which was, he believed, the structure of our thoughts. It is, of course, unclear why Frege thought it unacceptable to assume mathematics while, at the same time, providing no justification for his assumption of logic.

The Reduction of Mathematics to Logic

Frege’s ontology contained only objects and concepts, objects being the semantic values of names and concepts being the semantic values of predicates. He assumed that every concept had an extension, and included both first-level and second-level concepts and predicates. First-level predicates can be formed by taking subject-predicate sentences of the form Fa (such as “Susan is tired”), in which  is a name (whose semantic value is an object) and is a predicate (whose semantic value is a concept), and getting rid of a. One you have removed a, you are left with a first-level predicate, F (or “is tired”), whose semantic value is a first-level concept whose extension contains all objects (like “Susan”) that fall under this concept. Second-level predicates, on the other hand, are formed by taking sentences of the form “There is some x such that x is F“, and removing the first-level predicate (“is F”). After removing the first-level predicate, you’ve formed a second-level predicate whose extension contains a first-level concept. Whew.

To derive zero and the number succeeding it, Frege took the expression “the number of Fs”, a name which has the semantic value of an object, and defined it as “the extension of equinumerous with F“. Any concept could be equinumerous with F so long as there existed a one-to-one correspondence between the things in their extension. For example, the “number of tires on a car” and “the number of Aces in a deck” are equinumerous because you can create a one-to-one correspondence between the things in their extension.

To get “0”, Frege simply defined it as the number of not self-identical, meaning it had as its extension all concepts that were not equinumerous with anything (i.e., concepts like “unicorn”, which, because none existed, could not be equinumerous with anything). He then defined “1” as the number of identical to 0. The extension of “1” was “0”, and “1” is equinumerous with “0” because a one-to-one correspondence relation existed between the things in the extension of “1” (which is “1”), and the things in the extension of “0” (which is “not-self-identical”). For the rest of the numbers, he defined each one as the number of _____, in which _____ contained all of the numbers that had come before it.

In this way, he had derived numbers from logic by defining numbers as sets of concepts. However, if this was to serve as a good definition of numbers, Frege had to provide a relation between the numbers that would make the numbers succeed each other (“1” had to come after “0”, “2” had to come after “1”, “3” had to come before “4”, and so on). This relation had to be derived by pure logic.

Frege decided to define the number of Fs succeeds the number of Gs as there is an x in F such that the number of Gs is identical to the number of F but not identical to x. To understand this definition, it is necessary to remember the above definitions of numbers (“0” is defined as the number of not-self-identical, etc.). Keeping those definitions of numbers in mind, the definition of “the number of Fs succeeds the number of Gs” can be explained as follows: “the number of 1 succeeds the number of 0” means that 1 succeeds 0 if and only if there is an x in 0 (the x will be 0, because 0 is identical to 0), and the number of “not-self-identical” (which is the extension of 0) is identical to the number of 0 but is not identical to 0; “1” has its extension “0”, therefore “1” succeeds “0”. The same can be done with all the numbers – all you have to do is keep the definitions of the numbers in mind, and put them into the definition of “the number of Fs succeeds the number of Gs.”

In order for these numbers to be natural numbers (the ones used in mathematics), Frege had to derive them from logic. He did so by choosing an arbitrary natural number and giving the following definition of a natural number using only first- and second-order quantification:

n is a natural number = df  F [(F0 ∧ ∀xy ((FxSxy) → Fy)) → Fn]

 Using induction, you can derive ∀xFx from the above definition. Because n is an arbitrary number, under this definition every number falls under F, therefore every one of Frege’s numbers is a natural number.

Frege could now express Peano’s Postulates (axioms for the natural numbers) using the definitions he had derived using only first- and second-order predicate logic. Referring to his definition of numbers as (1), his definition of succession of numbers as (2), and his definition of natural numbers as (3), his model for Peano’s Postulates can be expressed as follows. The First Postulate, “0 is a natural number”, could be shown by (1) and (3). The Second, “If x is a natural number and y succeeds x, then y is a natural number” is shown by (2) and (3). The Third, “0 is not a successor of any natural number”, is shown by (1), (2), and (3). The Fourth, “If x and are natural numbers and is not equal to y, then is not equal to z is succeeds x and succeeds y, by (1), (2), and (3). The Fifth, “If n is a natural number and F0 ∧ ∀x∀y((Fx ∧ Sxy) → Fy)), then F_n” by (2) and (3). These five postulates, when expressed in second-order logic, all come out true. 

Russell’s Paradox

Frege’s project seemed to have been successful: he had derived arithmetic from pure logic, and no longer did mathematics need to be empirically justified. But in 1902, while the second volume of The Basic Laws of Arithmetic were being printed, Bertrand Russell wrote a letter to Frege. In the letter, Russell explained that Frege’s project contained a contradiction (which has come to be known as “Russell’s Paradox”) that stemmed from his assumption that every concept had an extension. This assumption was found in Frege’s fifth basic law of arithmetic, which stated that “for any concepts F and G, the extension of F is equal to the extension of G if and only if precisely the same objects fall under and G” (George and Velleman 2002). Consider, Russell said, the “concept set that is not a member of itself” (George and Velleman 2002): the extension of this concept we will call R, and so “= {x : x is a set and x x}”. If we look at this, and ask if R ∈ R, we arrive at an inconsistency because  R ∈ R only if R ∉ R (i.e., is a member of itself only if R is an extension that is not a member of itself).

This was a devastating blow to Frege’s logicist project (to say the least), and Frege hurriedly added an appendix to the second volume in which he acknowledged Russell’s Paradox and suggested that the contradiction could perhaps be avoiding by revising the fifth basic law of arithmetic. His proposed solution also ended up containing a contradiction, and so Frege abandoned his project.

A Little Summary

Frege’s desire to prove that the foundations of mathematics were not empirical prompted him to create a reduction of mathematics to logic – a project that began in 1879 with the publication of his Begriffsscrift and ended in 1902 with Russell’s Paradox. Using only first- and second-order quantification, axioms of logic, and a simple ontology containing objects and concepts, he attempted to derive truths of mathematics from truths of logic, but was not able to produce a consistent system in which this was possible.

Even though Frege’s reduction failed, the impact it had on subsequent philosophers and mathematicians was enormous: it inspired Russell and Whitehead to embark on their own logicist project (described in their Principia Mathematica, and also failed in its goals), and greatly influenced the work of Rudolf Carnap (a student of Frege’s), Ludwig Wittgenstein, and dozens (perhaps hundreds!) of philosophers and mathematicians.

Bibliography

George, Alexander and Velleman, Daniel. 2002. Philosophies of Mathematics. Malden, Mass.: Blackwell Publishers Inc.

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