I'm a physics student, but I like mathematics. I am currently interested in the foundation of mathematics. I would like to ask a question about the old theory of foundations of the mathematics of Frege and Russell. I know that it leads to paradoxes. Apart from this why it does not have a metalanguage?. In Hilbert's article "On the foundations of logic and arithmetic" he wrote that:

Arithmetic is often considered to be part of logic and the traditional fundamental logical notions are usually presupposed when it is a question of establishing a foundation of arithmetic. If we observe attentively, however, we realize that in the traditional exposition of the laws of logic certain fundamental arithmetic notions are already used, for example, the notion of set and, to some extent, also that of number. Thus we find ourselves turning in a circle, and that is why a partly simultaneous development of the laws of logic and arithmetic is required if paradoxes are to be avoided

The fundamental arithmetic notions are the, so called, finitary arithmetic. In the theory of Frege and Russell the "arity" of functions, relations, etc. and the finiteness of strings in a demonstration are not part of fundamental arithmetic? It seems that they did not introduce the concept of finite arithmetic.

  • $\begingroup$ "Fundamental arithmetical notions" are : zero, successor, sum, product and (for Frege and Rusell) the definition of the "concept" of natural number itself. $\endgroup$ Commented Jul 13, 2019 at 8:01
  • $\begingroup$ See Gottlob Frege and Frege’s Theorem and Foundations for Arithmetic and The Frege-Hilbert Controversy. $\endgroup$ Commented Jul 13, 2019 at 8:02
  • $\begingroup$ You may be interested in reading the posts linked from my profile on "circularity in mathematics" and "building blocks", which should address your inquiries. Frege and Russell didn't really understand foundations of mathematics, hence there's no point trying to pin down what they thought. After all, if their mental conceptions were correct, they wouldn't have designed an inconsistent system! $\endgroup$
    – user21820
    Commented Jun 25, 2022 at 20:32

1 Answer 1


Old theory? The homotopy type theory community claims to be "rediscovering" Frege, and, Martin-Lof found some motivation in Russell.

Russell's paradox relies upon a specific interpretation of the membership relation motivated by logicism -- namely, that it be irreflexive. Under the assumption that it is the sole primitive of a language, an elimination of the universal quantifier yielding a reflexive occurrence of membership will generate an uninstantiable class abstract ( { x | ... } ).

Instead of speaking of "mathematics", one should speak of presuppositions with respect to a paradigm. In category-theoretic foundations, membership can be reflexive. Lawvere maintains that logicism diverged from standard mathematical practice. In addition, category-theoretic set theorists hold that the logicist interpretation is not faithful to Cantor's views. That is something to think about when being told that Cantor gave us set theory.

Russell's foundation for mathematics does not have a metatheory because he actually developed the groundwork for recognizing its possibility. There is a term generation procedure called definite description. Frege used it to speak of "the extension of a concept". This seems to have been a referential use of descriptions. But, Frege's system of logic did not seem to have a classical form. In particular, he preserved the law of identity by taking the empty class as the denotation for fictions. So, names with different intensions would be set equal to one another. Russell attempted to repair this by interpreting definite descriptions attributively. This is the presupposition of the first-order paradigm.

Tarski, impressed by Russell's work, moved away from the Polish school and his mentor Lesniewski. His introduction of a metatheory effectively treats all of the singular terms of a first-order theory as intensions. This is a generalization of the attributive interpretation of definite descriptions.

It is somewhat inappropriate to compare logicist arithmetic with the arithmetic of formalists like Hilbert. As Russell pointed out, how does the formalist explain succession from first principles? Skolem the Great did nothing more than claim it to be "obvious". You might consider looking up the expression "honest toil" to understand Russell's opinion of formalism. To the extent that foundational inquiry is intended to give a clear account of the assumptions used in mathematical proofs, to say that something is simply "obvious" is little more than an insult to one's intelligence. In any case, Frege's account of arithmetic used inclusive disjunction to implement succession for the logicist paradigm.

As for the arithmetical metamathematics of the Hilbert program, if mathematics is restricted to formal axiomatics, the use of numbers external to those axioms is an application of mathematics. The credence one gives to metamathematical claims ought to be seen as an indication of what beliefs one holds concerning mathematics. There are subtle issues involved here.

Galileo, apparently, recognized that one could push reductionism to the spelling of words. He thought it to be nonsensical.

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    $\begingroup$ "The homotopy type theory community claims to be 'rediscovering' Frege" - I'm curious about this, do you have a reference? $\endgroup$ Commented Jul 15, 2019 at 1:34

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