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I do not understand the discipline in mathematics that is called "mathematical logic". For me, all of mathematics is based on logic and that is what makes it the exact science. Every theorem or lemma or result should be based on correct reasoning, and there should be some true logic behind it. Or am I mistaken?

Are there some results that cannot be treated with logic? What is the difference between the discipline of "mathematical logic" and the logic used in mathematics? Are there known results in mathematics that are not based on logic?

For example:

Find $x$ in $(1)$: $$(1)\quad x+1=0.$$ The logic is to find $x=-1$ by subtracting $-1$ and no one else can say otherwise.

For me, this is mathematics. Even though a lot of theorems are hard to understand (at least for me), there must be some kind of logic behind them. Am I right?

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One way to find out is to actually take a course "Logic" and see what it is about. Perhaps it would answer your question as to what makes an actual Logic course (and the thinking requirement) different from mathematics. –  imranfat Apr 25 at 15:26
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For mathematical logic see en.wikipedia.org/wiki/Mathematical_logic. This is just a specific area of mathematics and should not be confused with logic in the sense you have in mind. –  Dietrich Burde Apr 25 at 15:27
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Mathematical logic is not the logical study of mathematics. It is instead the mathematical study of logic. –  Lee Mosher Apr 25 at 15:27
    
There are multiple varieties of logic. For example, smooth infinitesimal analysis is most naturally developed in non-Aristotelian logic. –  Ben Crowell Apr 27 at 3:41

4 Answers 4

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Generally speaking, mathematical logic investigates the nature and limitations of logical systems. For example, consider the question: "can every true statement be proved?" This is a problem about logic. When we study geometry, algebra and other branches of mathematics we are interested in using logical thinking, but the purpose is to understand something about shapes or symmetries.

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The answer to this question is "no". Mathematicians use logic as a language to express mathematical proofs. But to say that these proofs are "based" on logic is analogous to saying that "War and Peace" is based on Russian. The content of the proof is based on "extra-logical" axioms that describe the mathematical structure the theorem is about. These extra-logical axioms are "hypothetical" -- they only hold for some kinds of structures. They do not fit in the rubric of logic, because they are not universal.

So, what is mathematical logic? Basically, it is the practice of studying logic with the tools of mathematics. Because mathematics have very rich methods, the subject is very rich, and includes "recursion" about the topics. So, for example, we use set theory to make theories of how a logic should work. And then we analyze what the new logic can prove. And, for example, we might show that the new logic can encode set theory. Or that it can't. It depends on what kinds of extra-logical axioms we are using to describe the logic.

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+1 for the literature vs. language analogy. –  Marnix Klooster Apr 26 at 15:45

Are there known results in mathematics that are not based on logic?

There are many results in mathematics which are not based exclusively on (first order) logic. A large portion of mathematics is based on set theory, especially ZFC. It would probably be possible to achieve the same results with second order logic (or higher oder logic/type theory) with Henkin semantics, and impredicative comprehension axioms emulating the ZFC axioms. However, this is not the current practice how second order logic is used.

Instead of using impredicative comprehension axioms emulating the ZFC axioms, one normally uses much weaker comprehension axioms when using second order logic. The reason is that the people using second order logic instead of set theory do it in the context of reverse mathematics. Here the goal is not just to prove a result based on certain axioms, but also to show that these axioms are really required for proving the result, given a certain base theory like primitive recursive arithmetic.


Given that the asker doesn't understand "mathematical logic", it might seem strange to give an answer based on the differences between first order logic, ZFC set theory and second order logic with Henkin semantics. But maybe this answer helps the asker to realize what he doesn't understand about mathematical logic that prevents him from seeing that the statement "all mathematics are based on logic" might be wrong, if interpreted as "all mathematics are based exclusively on logic".

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Thomas please note that the common name for the person asking a question is OP (Original Poster) –  magma May 1 at 9:31

Are there known results in mathematics that are not based on logic?

Well, there's always Ramanujan's approximation of $\pi$ as $\sqrt[4]{\dfrac{2143}{22}}$ , based on a dream which he had one night about a certain Hindu goddess that his family worshipped. :-)

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Honestly, you can literally answer this question with almost anything Ramanujan came up with. He had some weird dreams. :| –  Shahar Apr 26 at 0:31
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Or with Gauss' Easter Algorithm. It gives me a headache whenever I think about it. –  Dunno Apr 26 at 10:03

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