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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?

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Logicism: – Doug Spoonwood Dec 7 '12 at 4:16
Perfect Thanks! – vanattab Dec 7 '12 at 4:18

5 Answers 5

up vote 7 down vote accepted

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.

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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.

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Also, you can think of both mathematics and logic as subsets of the Philosophy of Proof.

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[(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.
Philosophy of Math is a branch of Philosophy, which is also of interest to (some) mathematicians.

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Nice answer, but every theorem can be rewritten using propositional calculus, for example. We can, of course, think of rules of inference as mathematical operations. – glebovg Dec 7 '12 at 4:28
Thanks, @glebovg. Of course, I root for philosophy as overarching ALL. Every discipline is (or was at one time) the birthchild of Philosophy! – amWhy Dec 7 '12 at 4:34

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.

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