A philosophical theory of numbers. Consisting of the thesis that the propositions of mathematics are reducible to, or are no more informative than, what is found in the axioms and principles of logic. You can get an idea of what motivates logicism if you think through this (an example reminiscent of, though perhaps not strictly representative of, Gottlob Frege's pioneering thinking):

To what do we refer when we use the symbol '2'? Perhaps we refer to a set; most obviously (i.e., after some thought), this would be the set of all sets with two members.

Now set theory is the modern boundary between mathematics and logic: it suggests, for the logicist, the unity of the two areas. For consider the statement,

'2 plus 2 equals 4.'
Let 2 and 4 be sets, and the statement be rephrased as
'Any set that is a member of 2, in disjoint union with another such set, forms a set equinumerous with any set that is a member of 4.'
The long-winded reconstruction can be said to use just logical axioms, concepts of sets and quantification to achieve a logical formalization of the first statement. (Just as long as set theory can properly be considered part of logic.)

These ideas of were an important part of Bertrand Russell's essentially positivist philosophy in Principia Mathematica, an attempt at the definitive formalization of mathematics -- the placing of mathematics on a sound footing, as it were. He believed that

  • The theorems of mathematics constitute a proper subset of those of logic.
  • The vocabulary of mathematics is constructible, by mere sequences of definitions, out of logical vocabulary.

As is frequently noted in discussion, Russell's own paradox helped to defeat the bold aims of the Principia. The complexity introduced by the need to cope with these paradoxes of set theory forced increasingly ad hoc-seeming modifications. Furthermore, Gödel's Theorem, a 'death knell' for the axiomatic method, denied any prospect of finding logical proofs (within a single system) of all mathematical truths. Because there is no satisfactory proof of arithmetic's consistency, there is no meeting the intuitive demand for the logicist to reduce mathematics to something 'less complex' than itself -- something consistent, axiomatized and complete.

However, it may be salutary to consider how things look if the problems revealed by the Principia in set theory (not just the mere presence of paradoxes but the forced addition of questionable axioms such as AC and the axiom of infinity) are set aside. That is, what if logicism's motivation was correct? Could it be that mathematics is just logic?

It seems to me that even without the paradoxes inherited from set theory, logicism says crazy things about what mathematics is. Accepting its argument -- mathematical truths are analytic, because mathematical concepts are logical concepts -- just doesn't seem valid or sound. There is a flaw (a flaw perhaps exposed in the writings of Wittgenstein, and Quine) in the positivist, which is to say Russellian, denial of the existence of the synthetic a priori. The meanings of mathematical terms don't just reduce to their formal role or to some particular embedding into logic; they have substantive meaning in their own right.

Logicism is the thesis in Philosophy of Mathematics, which maintains that:

1. All mathematical formulae are reducible to logical formulae by use of explicit definitions; and

2. All mathematical truths (axioms) are reducible to logical axioms by use of explicit definitions.

It would be more accurate, however, to state that logicism, as it was advocated by Frege concerned itself with arithmetic more so than with the whole of mathematics, given that Frege himself conceded that geometry was within the domain of intuition and that its truths were synthetic a priori (see his Die Grundlagen der Arithmetik - The Foundations of Arithmetic). This resulted from a conviction he shared with Kant that Euclidean geometry was true.

His point of departure with Kant concerned the way in which mathematical formulae such as '7 + 5 = 12' were true. Kant held that they were synthetic a priori, while Frege claimed that they were analytic, which means, that they are provable with the use of explicit definitions of '7' , '5', and the concepts of equality, and addition.

Frege's more significant and fundamental concerns dealt with the nature of mathematical objects, i.e. numbers. Thoroughly dissatisfied with previous and contemporary attempts to define numbers, Frege formulated his own, which heavily relied on his notions of objects (Gegenstände) and concepts (Begriffe). Objects were, for Frege, self-standing and complete (vollendet) or saturated (gesättigt), while concepts were incomplete (unvollendet) or unsaturated (ungesättigt). Objects are the referents (Bedeutungen) of names, which include entire sentences, whose referents are truth values(the True and The False, which are also objects), while concepts are the referents of predicates. Now objects "fall under" concepts, i.e. they satisfy the propositional function 'x is P' where 'x' is the object and 'P' is the predicate whose referent is the concept in question. For every well-formed concept, there is an extension of that concept (or set), which contains all the objects that satisfy the related propositional function. Concepts are, as well, a special type of function, which take objects as arguments and yield truth values as values.

Given that rather cursory explanation of Frege's slightly more complex theory of objects and concepts, his definition of Number (Anzahl) is:

"the number which belongs to the concept F is the extension of the concept 'equal to the concept F'". (The Foundations of Arithmetic, section 68, pp. 79 - 80)

What the above defintion amounts to is that the number of F things is the set of all things that have as many members as F has members, i.e. if F has exactly two members, then the number of F things is (in the strong sense - is identical with) the set of all 2 membered things.

To say that n is a number is to say that:

"there exists a concept such that n is the number which belongs to it." (The Foundations of Arithmetic section 72, page 85)

This might seem odd, but for Frege concepts are distinct from ideas and are objective.

Specific applications of these definitions can be found in Frege's definitions of the numbers 0 and 1:

"0 is the number which belongs to the concept 'not identical with itself'." (The Foundations of Arithmetic section 74, page 87)

"1 is the number which belongs to the concept 'identical with 0'." (The Foundations of Arithmetic, section 77, page 90)

Frege's use of the concept of identity as a means of defining 0 is due to his belief that arithmetic and logic were analytic, and since identity is, for Frege a logical concept (as explained in his Begriffsschrift - Concept Notation), it too is analytic (any other concept could serve as a means of defining 0, e.g. 'unicorn' or 'square triangle', but they would not have been analytic.

The difficulties for Frege arose in 1903 immediately before the publication of the second volume of his Die Grundgesetze der Arithmetik - The Basic Laws of Arithmetic, when Russell wrote him a letter revealing that his Basic Law V allowed for a paradox, i.e. that one could show that the set of all sets that do not contain themselves contains itself if and only if it does not.

This has led many set theorist to deny the existence of a universal set and the principle of Basic Law V, which states that extensions of two concepts are indentical if and only if the values yielded by each concept for the same argument are the same, and, its underlying assumptions, that for every well-formed concept there is an extension corresponding to it, the domain of concepts is larger than extensions (seen from Basic Law V), and also that the domain of concepts and the domain of extensions are equal.

This did now, however, undermine Russell and Whitehead's work in the Principia Mathematica (3 volumes, 1910 - 1913), given that its first volume was published seven years after Russell's discovery of the paradox. It did, of course, lead to Russell's creation of the theory of types and, later, the ramified theory of types, which both greatly weakened his logicist programme.

A greater blow to logicism were Gödel's incompleteness theorems, which proved that any system strong enough to contain first-order arithmetic was incomplete if it was consistent and inconsistent if it was complete. Consistency proofs for first-order arithmetic abound, which leaves the logicist with the unfortunate situation of its being incomplete.

Some important works in logicism are:

Books:

Frege, Gottlob. Begriffsschrift, eine der arithmetischen nachgebildete Fremdsprache des reinen Denkens (Concept Notation, a formula language, modeled upon that of arithmetic, for pure thought) (1879) reprinted in From Frege to Gödel: A Sourcebook in Mathematical Logic 1879 - 1931 ed. Jean van Heijenoort. Cambridge, Mass.: Harvard University Press, 1967. 1 - 82.

Frege, Gottlob. The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number (Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl (1884)). Trans. J.L. Austin. Evanston: Northwestern University Press, 1950.

Frege, Gottlob. The Basic Laws of Arithmetic (Die Grundgesetze der Arithmetik (zwei Bände 1893/1903)). Trans. Montgomery Furth. Berkeley: University of California Press, 1964. (Note: this is only a partial translation of volume I)

Russell, Bertrand. The Principles of Mathematics. Cambridge: Cambridge University Press, 1903.

Russell, Bertrand. Introduction to Mathematical Philosophy. London: George Allen & Unwin, 1919.

Whitehead, Alfred North., and Bertrand Russell. Principia Mathematica. 3 vols. Cambridge: Cambridge University Press, 1910 - 1913.

Whitehead, Alfred North., and Bertrand Russell. Principia Mathematica to *56. Cambridge: Cambridge University Press, 1962.

Wright, Crispin. Frege's Conception of Numbers as Objects. Aberdeen: Aberdeen University Press, 1983.

Articles:

Benacerraf, Paul. "Frege: The Last Logicist." Midwest Studies in Philosophy. 6 (1981): 17 - 35.

Boolos, George. "The consistency of Frege's Foundations of Arithmetic." in On Being and Saying: Essays for Richard Cartwright. Cambridge, Mass.: MIT Press, 1987. 3 - 20.

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