Logic as subset of mathematics and mathematics as subset of logic

Question

Is logic a subset of mathematics or is mathematics a subset of logic? I have heard the former view, but is there any argument for the latter?

Answer 1

Mathematical logic is a branch of mathematics. But mathematical logic is by no means all of logic.

There have been recurrent attempts, from Frege through Whitehead/Russell and others, to develop mathematics within what they thought of as logic. The attempts failed, we have moved on.

Answer 2

[(logic) $\cap$ (math) $\neq \varnothing$] $\;\land\;$ [(logic)$\setminus$(math) $\neq \varnothing$] $\;\land\;$ [(math)$\setminus$ (logic) $\neq \varnothing$].

That is, the intersection of logic and math is clearly not empty, but I think it is also the case that neither one completely encompasses (contains) the other.

Also note:
Mathematical Logic is a branch of mathematics, and is also of interest to (some) philosophers.
Likewise,
Philosophy of Math is a branch of Philosophy, which is also of interest to (some) mathematicians.

Answer 3

It is actually the other way around. Philosophy is the root of all sciences, including mathematics. You can think of mathematics as an application of logic.

Answer 4

I am not a mathematician, but here's my 2 cents.

Any mathematical entity must be in accordance with logic. There's not even such a concept in mathematic as "illogical" theoremes, proofs, equations or whatever. Any attempt to think of what such a, say, theorem, might look like, fails. Thus, it is natural to accept that all of mathematics is built within the body of logic.

However, there's no way of defining logic without mathematics. I.e. the only proper way to designate logic, is through mathematics.

So, it happens to be that mathematics is simply a kind of logic, and that logic can be used to describe itself to some extent. This can especially well be observed in gourps and sets theory, where borders between logic and mathematics become less pronounced.

At this point one would naturally come to the conclusion that despite that the original question seems sound, one has to define what mathematics and logic are.

What could serve better for that, than a little historical excourse?

Math appeared out of natural needs of humans. Originally it was made to improve the quality of life and organization. It was our minds predestination to find patterns in the surrounding world, that played the key role, and we began to develop it. So it was the product of interaction of our mind with the world. The product which we turned into a tool.

Eventually, due to the nature of perception, reframing occured and instead of finding patterns in the surrounding world, we began noticing patterns in patterns, then patterns in patterns of the patterns, and so on.. mathematics was developing and it essentially was a product of interaction of our mind with its previous products of interaction, with itself.

Logic was never different from that. However the term "logic" is mostly used to describe human actions and perceptions, where quantity is of a lesser value. It is also thought of as more accessible to an average person than mathematics. The ability of "logical" thinking is an essential ability of a healthy mind. In other words, logic is generally thought of as being "grounded" to the world of material things and everyday perception.

All that being said, there are different ways of using the word "logic". For example, there's Boolean logic which has no big meaning away from mathematics. And actually it is an algebra. So there's more a play of words than an actual well-defined difference between mathematics and logic.

Answer 5

+1 to @glevobg although the answer is not properly developed. This should be it:

• Philosophy: "the mother of all sciences"

  • Seeks for all types of truth, physical and metaphysical (e.g. liquids occupy space, or 1/0=infinite).
  • Therefore, it is said that philosophy targets final truth, even if such idea is a utopy.

• Science: knowledge obtained using the scientific method

  • Main target is empirical verifiability (implies using the five senses)
  • Science targets empirical truth, for example, the idea that the earth is flat is empirically true for many. It can't be said that they don't use the scientific method, they might use it in many cases, it is just they don't trust on past science, which is an acceptable criteria of skepticism.

• Logic: The study of reasoning

  • Cannot be empirically verified
  • It is part of philosophy, due to the previous reason, and due to it is historically accepted as so.
  • Therefore it is not a science.
  • Is formal in part (see the definition of formal in the next point)
    • E.g. formal logic
  • In the other part is considered to depend on metaphysical considerations
    • language
    • See Kant's Critique Of Pure Reason for more: intuition (mental representation, in case of of logic concepts), imagination (conception of logical entities by means of "images", meaning sense impressions), etc.

• Mathematics: A formal system based in logical rules.

  • Formal system: essentially a system, a set of interrelated parts,
    • Sustained on axioms and concepts
    • Able to produce theorems based on them, by mean of a specific domain calculus
  • Described by a formal language (which is also another formal system)
  • Cannot be empirically verified (how to verify the result of ${\pi}^{182}+1$)?
  • Therefore it is not a science
  • Therefore it is part of philosophy

So, in a case including them all, it can be said that sciences are dependent on mathematics, which is part of logic, which is part of philosophy.