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

+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.