Is logic part of Philosophy or Mathematics?

Question

Is logic part of Philosophy or Mathematics?

I asked this question "Does programming use logic more or mathematics more" and users on some site insisted that logic was part of mathematics, I checked Wikipedia and it listed logic(symbolic, first-order, formal, etc) as part of Philosophy, not mathematics. So it has to mean logic is part of philosophy, right?

When I look at source code of programs, most of it seems to contain only symbolic logic and first order logic, math is used very sparsely in most programming, there are areas of programming like graphics, cryptography, AI etc where math is used a lot but rest of the programming is only symbolic logic.

Answer 1

"Logic" is a part of mathematics as well as it is a part of philosophy!

Now, the explanation. I suppose you are well aware of the fact that a term "logic", as more or less any term, can be interpreted differently by any person. But let's put aside the "non-canonical" interpretations...

The term "logic" has an uncertain origin. The Greeks say it was Parmenides (5th century bce) who coined the term. What's certain is that the first system of logic (namely the syllogistic logic, the logic of classes and categories) comes from Aristotle, who was certainly a philosopher, but he was also involved in natural sciences to which he applied his system (i.e. his zoological systematization).

Then were the stoics, who invented the sentential calculus, which, much later, was discovered by mathematicians to be well suited for their metamathematics.

The so called "traditional logic" was certainly considered a part of philosophy.

I'd say that our "philosophy or mathematics" dillema started when George Boole introduced his algebra. It meant that from that moment on logic can not only evaluate mathematical problems, but also can be interpreted in it.

Of course I have to skip many significant breakthroughs and theorems that strenghtened the position of logic in mathematics, I'll just mention here: Hilbert, who wanted to build the whole mathematics solely on logical axioms; Goedel, who proved Hilbert wrong; Tarski, who developed many metalogical guidelines that are well-respected to this day.

The real question here is not: "Is mathematics deducible from logic?", but: "What are the differences between (pure) logic and mathematics?". I'd say that the definitions of terms: "logic", "mathematics", "philosophy" are always the results of some conventions and so cannot be understood unambiguously without more context.

Answer 2

It is a mistake to assume that various academic discplines constitute a partition. Something being part of mathematics does not rule out it also being philosophy, computer science, etc.

I do research in logic. Most of my peers are employed at either CS or maths departments (roughly even split), but there is a meaningful number of people at philosophy departments, too. Of those at CS departments, probably quite a few identify more as mathematicians than as computer scientists (I do). I'd reckon almost everyone identifies as a mathematicians at least to some extent. I don't know how much identification with philosophy there is, but I'd guess much less.

Of the stuff I would recognize as logic, I'd also recognize everything as mathematics. Some of it is also clearly philosophy, others is also clearly CS. There is certainly also overlap with other disciplines.

Summary: Logic is definitely part of mathematics, but that does not preclude some or all of it also being part of philosophy.

Answer 3

Yes and no, depending on what you mean by "being part of".

Logic is part of philosophy, but can be applied to various disciplines. When you find logic in philosophy of mathematics, that's just an application of logic, and does not mean that logic is subordinated to that particular discipline. In fact, one may even argue that mathematics itself is an application of logic.

So, is logic part of philosophy of mathematics in the sense that there a significant amount of logic in philosophy of mathematics? Yes. Is logic as a whole a sub-discipline of philosophy of mathematics? Nope.

Answer 4

I like the definition that Wikipedia gives

"Philosophy (from Greek: φιλοσοφία, philosophia, 'love of wisdom') is the study of general and fundamental questions, such as those about reason, existence, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. The term was probably coined by Pythagoras (c. 570 – c. 495 BCE). Philosophical methods include questioning, critical discussion, rational argument, and systematic presentation."

"Historically, philosophy encompassed all bodies of knowledge and a practitioner was known as a philosopher. From the time of Ancient Greek philosopher Aristotle to the 19th century, "natural philosophy" encompassed astronomy, medicine, and physics. For example, Newton's 1687 Mathematical Principles of Natural Philosophy later became classified as a book of physics."

I don't know who said that philosophy was the mother of all sciences but he has my support.

I think the difference between a philosopher and a scientist is that each scientist is specialized in a specific area but even not long ago they were called natural philosophers.