Mathematics and logic, historically speaking, have been entirely distinct studies. Mathematics has been connected with science, logic with Greek. But both have developed in modern times: logic has become more mathematical and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one. They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic. This view is resented by logicians who, having spent their time in the study of classical texts, are incapable of following a piece of symbolic reasoning, and by mathematicians who have learnt a technique without troubling to inquire into its meaning or justification. Both types are now fortunately growing rarer. [1]
What is interesting is that Russell held onto quasi-mystical views of logic that were inherited from the classical philosophical tradition he is referring to; logic as the “Laws of Thought”, as something which is a precondition for any rational thought whatever. With this conception, it is impossible to go “beyond logic”: How can you reason about logic when logic is a precondition for reasoning itself? Logic had to liberate itself of its dogmatic past, had to break the shackles that philosophy placed around its neck. This was accomplished by Hilbert and Bernays.
So, how did Hilbert and Bernays go “beyond logic”? The solution is that one can treat logic as a “formal system”. Strictly speaking, a formal system deals only with symbols on a page. It is a set of rules that tells you which sequence of symbols is a string, and which sequence of strings is a sentence, and which sequence of sentences is a proof. Mathematically, a formal system consists of three parts: a language, a list of axioms, and a list of deduction rules. These can be given precise definitions, but you should have a rough idea of what these notions are supposed to be.
Once you have all of this set up you have already solved the problem; you can go “beyond logic” by dealing only with the symbols on the page themselves, and the rules of the symbol manipulation are so basic that they can (in principle) be carried out entirely by hand (or, more importantly for formal verification, on a computer).
Now, why would one really care about such a thing? What is the motivation for regarding logic as a “formal system”? There are a couple:
- We can now reason about logic itself mathematically. We can formulate statements about symbols on the page such as “there is no proof of the sequence of symbols XYZ in the formal system of first-order logic”. These statements can be given mathematical proofs. Hence, we can have “meta-theorems” about logic, e.g. whether logic is capable of proving or disproving certain statements.
- There was a foundational crisis in early 20th century mathematics about the validity of certain axioms. If we take formal logic as the language of math, and logic as a formal system, mathematics itself can be regarded as a formal system. Hence, philosophical debates about infinity, continuity, etc. can be entirely disregarded: we only need to focus on the symbols on the page.
As an example of (2), the axiom of choice in mathematics is often thought to commit one to the philosophical belief that humans can make an infinite number of choices. If we conceive of mathematics as a formal system, and nothing more, the axiom of choice just becomes a sequence of symbols; it has no content, and hence no philosophical significance.
Given all of the above facts, we can ask the same questions Hilbert asked: (i) Does there exist a formal system of mathematics that can prove or disprove any statement? (ii) Can we show that a proposed formalism of mathematics is consistent? If the answer is in the affirmative to both questions, we can tell those annoying philosophers (and sometimes mathematicians with strong philosophical opinions) that all of their philosophical arguments about mathematical axioms are irrelevant. We can conceive of mathematics as a game of symbols, and we manipulate these symbols with finitary tools that no one can object to; furthermore, we can prove that this game does not lead to contradiction, and that every mathematical problem has a solution inside this game, i.e., we can show that this game captures everything you could ever want about mathematics.
It turns out that the answer to Hilbert’s questions is a resounding “no”. This is the content of Gödel’s Incompleteness Theorems.
[1] Russell, Bertrand. Introduction to Mathematical Philosophy, Chapter XVIII