Does logic "come before" mathematics?

Question

I always thought of mathematics as being founded on logic. After all, even the most basic mathematical definition is based on logic. When we enunciate ZFC axioms, we're relying on the concepts of "axiom" and "axiomatic system", which are concepts of logic.

However, I've seen sometimes logical systems founded upon mathematics (see cathegorical logic). I always see mathematical tools used to prove statements about logical systems which, according to my previous (and clearly wrong) understanding are what mathematics is founded upon (the most trivial example being using mathematical induction over a set of propositions).

I understand that my question is quite vague, but I'm really no expert in logic, so as a beginner I'm very lost right now. My request is, can someone enlighten me about this issue I'm having, by clarifying the relationship between logic and mathematics?

EDIT. Upon reading the answers and comments, I would like to ask the same question in the title, replacing "logic" with "metalogic".

Answer 1

Historically, logic and mathematics were separate disciplines and had little to do with one another. Apart from Euclid, few mathematicians were concerned with axioms. Some scholars were good at both logic and mathematics, and a few of those saw a connection between them. Leibniz, for example, had quite a modern approach to logic and supposed that it might be possible to devise a calculus for language that did the same job for logic that mathematics did for arithmetic and geometry.

Arguably, this idea came to fruition in the 19th century with Frege and Peirce and the development of quantifier logic. At the turn of the 20th century, non-Euclidean geometry and Russell's paradox created a crisis in the epistemology of mathematics. Frege and Russell proposed to create a firm foundation for mathematics by showing how it could all be reduced to logic. This project was called logicism. On this account, logic is fundamental and mathematics is founded upon logic axiomatically.

Not everyone agreed. Brouwer, the inventor of intuitionism, considered that mathematics is fundamental and logic is of little interest to mathematicians. Other intuitionists, Heyting and Kleene, developed a distinctive intuitionistic logic that differed from the classical logic of Frege and Russell.

Gödel’s incompleteness results are widely understood to have blown a big hole in the logicist program. There are some neo-logicists who defend a weaker version of the idea that mathematics is founded entirely upon logic, but it is no longer a popular view. Also, some of the axioms that Russell proposed to rely upon, such as the axiom of reducibility and the axiom of infinity are difficult to justify.

Today, intuitionism also is a minority position. Some mathematicians are platonists, others formalists, and there are other options as well. The ability to express a mathematical theory as an axiom system is nice, but hardly essential for it to qualify as mathematics. Gödel’s theorems demonstrate that there are fundamental limitations to what can be achieved using axiom systems.

The upshot is that logic and mathematics are separate but interrelated. Mathematical methods can be used to study systems of logic and say interesting things about their properties. And logic can be used to study the foundations of mathematics. Which you consider to be fundamental is rather a matter of perspective.

Answer 2

A few points:

• The notion of an 'axiom' is something that certainly has a definition in what we now call 'logic', but as a concept, the former long predates the latter. Wikipedia:

The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning.

Thus, we don't need 'logic' to have a notion of an 'axiom'.

• Zermelo set theory was created in 1908. (And note that ZF(C) alone doesn't give us a foundation for mathematics; it's typically paired with First-Order Logic, FOL, for that.) What was the foundation of maths prior to that? One might argue that it was Euclid's "Elements", but that was primarily about geometry, and certainly wasn't present for Babylonian mathematics. i would suggest that, for most of history, mathematics has been 'founded' on abstractions of empirical data about physical relationships, from simple counting through to calculus (which also well predates the establishment of FOL).

• Some relevant quotes by John Baez:

I hope the math community has reached the point of realizing that we really need not one foundation of mathematics, but many, together with clearly described relations between them. Indeed at this point the word ‘foundation’ is perhaps less helpful than something else… like maybe ‘entrance’.

-- https://golem.ph.utexas.edu/category/2012/12/rethinking_set_theory.html#c042716

[M]y opinion is mainly that people should try all sorts of foundations and see what happens.

-- https://golem.ph.utexas.edu/category/2009/11/feferman_set_theory.html#c029804

and Andrej Bauer:

Any attempt to bring mathematics within the scope of a single foundation necessarily limits mathematics in unacceptable ways. A mathematician who sticks to just one mathematical world (probably because of his education) is a bit like a geometer who only knows Euclidean geometry. This holds equally well for classical mathematicians, who are not willing to give up their precious law of excluded middle, and for Bishop-style mathematicians, who pursue the noble cause of not opposing anyone.

-- https://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/

• Finally, as another example of a mathematical system providing 'logic', there's type theory, as described in a post by Mike Shulman:

[T]ype theory is not built on top of first-order logic; it does not require the imposing superstructure of connectives, quantifiers, and inference to be built up before we start to state axioms. Of course, type theory has first-order logic, which is a necessity for doing mathematics. But first-order logic in type theory is just a special case of the type-forming rules. A proposition is merely a certain type; to prove it is to exhibit an element of that type. When applied to types that are propositions, the type-forming operations of cartesian product, disjoint union, and function types reduce to the logical connectives of conjunction, disjunction, and implication; the quantifiers arise similarly from dependent sums and products. Thus, type theory is not an alternative to set theory built on the same “sub-foundations”; instead it has re-excavated those sub-foundations and incorporated them into the foundational theory itself. So not only is it more faithful to mathematical practice than either kind of set theory, it is literally simpler as well.

-- https://golem.ph.utexas.edu/category/2013/01/from_set_theory_to_type_theory.html

Answer 3

Mathematics depends on logic insofar as we’ve agreed on a shared standard of what counts as a rigorous argument or proof. Indeed, historically it’s turned out that mathematics, even more than other sciences, benefits from having a very precisely agreed-on set of basic principles (“logical foundation”), and agreeing that all proof should at base be justified by them. Not everyone necessarily has to agree on exactly the same principles — e.g. some freely assume the axiom of choice, others prefer to avoid it; some prefer set theory, others type theory — but by having them spelled out very precisely in each case, we can translate back and forth between them as necessary, and still work together fruitfully.

But once those principles are spelled out so precisely, we can study them mathematically, with all the tools and ideas of modern maths. This involves both some approaches that are very concrete (logic as formal syntax, strings of symbols from an alphabet, …) and others more abstract (e.g. categorical logic, as you mention). But there’s no circularity, any more than there’s circularity in a textbook on arithmetic having numbered pages. We’re using logic to do mathematics; we’re also studying (an idealised model of) logic inside our mathematics.

Our mathematical study can tell us things about how we expect our logic to behave — just like how theorems of arithmetic tells us how we expect our ordinary counting numbers to behave — and the ways we may choose to use logic (e.g. which foundational axioms/theories/systems to adopt) may well be influenced and clarified by what we learn from studying it mathematically. But using a logic doesn’t “depend on” mathematics, in the sense of requiring mathematics for justification, any more than you need theorems of arithmetic to justify counting eggs.

So overall there’s a strong sense in which mathematics depends on the use of logic — but it doesn’t depend on the mathematical study of logic, so there’s no circularity, just a fruitful feedback loop from the latter to the former.

Answer 4

Because your post does not define “logic” and “mathematics” in a specified sense, I take both terms in a colloquial sense.

1. Then the answer is: Yes, logic comes before mathematics because mathematics presupposes logical thinking.
2. In classical Greek the first explicit thoughts about the concepts of propositions, truth and falsehood as well as about logical implication and syllogisms occur at about the same time period as the first proofs in mathematics about elementary number theory and geometry.

Today both disciplines have the maturity of two different, axiomatized formal theories which many different variants.

Answer 5

The expression 'comes before' could be applied in a historical sense. Or it could imply the first step had to come before the second step.

If logic and mathematics exist separately from the intelligences that try to grasp them, there may be some derivation from First Principles that puts an order to things. In practice, things happen more gradually and more fuzzily for us. For us as individuals, and for humanity as a whole, we gradually refine our understanding of logic and mathematics. It may feel to us as though the rules of integer mathematics, or formal logic were 'always there', but our successes in these fields are largely built on being able to write down our assumptions and our deductions in some formal matter, in such a way that others can understand, and agree (or not). Logic and Mathematics as we understand them probably emerged from the mist together.

The twentieth century saw a shift in philosophy from a pure consideration of ideas to an analysis of the mechanics of language and representation that lets us to share and verify our views. If we met aliens, we would expect them to have a different notation, but their underlying ideas of Logic and Mathematics should be the same.

In the 19th century, the synthesis of urea suggested all life may be chemistry. We have not made whole lifeforms from scratch, but we see no barrier to how close we could come to this. Our early experiments with AI suggest that all intelligence is computing. In time, we may be able to show that many types of intelligence have a common understanding of Logic, and Mathematics. This will not show our understanding is complete, but it is hard to know what better understanding we can hope for.

Answer 6

What does "come before" mean? Are we talking about history, or evolution? Brian Butterworth has argued for a sense of numerosity, which allows answers to questions such as "We have strayed into someone else's territory: fight, hide, or flee?". The active inference folk have argues for Bayesian inference being used by critters to decide whether to go after a possible food source. If these folk are right, some form of maths must have been used since early evolutionary time.

Answer 7

This is too general a question and cannot be answered in one, specific way. Because it depends on how we treat both terms "logic" and "mathematics". They can be treated in their raw, "primitive" form or as fields of systematic and organized study.

The origins of mathematics start when Man started counting. More specifically, math is built on the concepts of number, patterns, magnitude, and form. The origin of more complex mathematics, as an organized field of study, starts with the Babylonians, about 3,000 BC. And we go on from there to a more and more systematic and organized study of math.

On the other hand the origins of logic, in its primitive form, starts when Man started reasoning, which is impossible of course to tell.
But logic as a study and field of systematic analysis starts with Aristotle (384-322 BC).

Maybe a more simple way to say which comes first is to think of a small child. Babies start recognizing shapes and colors and make associations with way before they start counting. And, I believe, even before they form the concepts of number, i.e. distinguish between 2 or 3 objects, etc.

So, in primitive terms --or on a conceptual level, if you like-- I believe that logic "comes before" mathematics.