Can mathematics be constructed from logic?

Question

This may be a rather naive question in this day and age, but has mathematics been reduced to logic? In other words, can all of mathematics be constructed from logic? My understanding is that Russell and Whitehead attempted to show that math could be derived from logic, but their attempt failed. But that was over 100 years ago, so maybe things have changed.

Answer 1

Yes. Practically all of mathematics is formalized as a theory of sets in first-order logic using the axioms of ZFC.

Answer 2

While the answer to your question must depend upon your presuppositions concerning both logic and mathematics, there are some pragmatic considerations which seem independent. Logic demands ontology. The received views incorporate ontology under the guise of "self-identity" which declares the so-called law of identity to be axiomatic. From where does this presupposition actually arise? It is the consequence of bias against infinite regress.

This question applies to the distinction between comprehensionalist set theory and the category-theoretic notion of set because the latter relies upon inclusion, and, inclusion is interpreted as "second-order" by proponents of comprehensionalist set theory. As second-order logic is considered problematic, the comprehensionalist will reject the foundational import of the latter. This is explicit in the comparison of Dedekind's work and Peano's introduction of a symbol for membership. Russell points this out initially; you can find its modern account in Potter's book on set theory and its philosophy.

The comprehensionalist, however, relies on a dubious account of mathematics. The so-called crisis in geometry motivating the arithmetization of mathematics had actually been a crisis only for those who held unwarranted beliefs about geometry and material reality. Arithmetization can then be understood as replacing an unwarranted belief about geometry and material reality with an unwarranted belief about arithmetic and material reality. Both fall under unwarranted belief about mathematics and reality.

Formalism provides a means of distancing this problem from the practice of mathematics. But, it is not innocent . Hilbert promoted formalism to support the use of mathematics for applications. The idea of formalist mathematics as a mere game of symbols has a different origin that conflates Hilbert's formal axiomatics with his metamathematical approach. Following the notion of sensible a priori intuition introduced by Kant, Hilbert simply restricted it to the sensible impressions of marks made on a page.

The problem with the modern account of formalist mathematics is that the dichotomy between syntax and semantics does not address the applicability of mathematics. This is because neither addresses the relationship between language and the users of language. A proper term for this comes from Carnap's response to the extreme syntactic nature of his early work. Namely, pragmatics.

Pragmatics enters in to your question because of how the teaching of geometry had been deprecated in favor of logic at the end of the nineteenth century in many places. Regardless of rhetoric on the part of logicians, most people learn about logical connectives through naive geometric incidence from Venn diagrams. The idea behind membership relating individuals to classes in logic is not geometric incidence.

So, now you have a problem because pragmatics is ignored.

Now consider "self-identity" in relation to 'true' and 'false'. If one is provided with symbols representing truth objects different from the natural language words that carry pragmatic import, how shall the truth value constants be assigned? Max Black wrote a paper on the problem of assigning names to a symmetric configuration.

You will find yourself involved in an infinite regress.