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Is it worth studying mathematical logic?

  1. Oct 13, 2013 #1
    I want to address this post primarily to people who have already studied mathematical logic, or are currently studying it.

    Since a while I've immersed myself in studying some math, and I must say that I started to enjoy pure mathematics. However, sometimes I don't really feel comfortable with some of the reasoning; for instance, when using proof by contradiction.

    As you may have guessed, my question is the following: For you guys who have studied, or are studying mathematical logic, do these types of questions get answered in there?

    Also, does mathematical logic show that the very foundations of mathematics are sound, and that all of the typical mathematical reasoning is air-tight?
     
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  3. Oct 13, 2013 #2

    UltrafastPED

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    I took an advanced course in symbolic logic when I was a freshman in college. I really liked it - after learning the fundamentals (modus ponens, etc.) we went on to a systematic program which lead to a proof (by the instructor, in easy stages) of Goedel's theorem. Then we studied a system with a finite set of elements - I think it was due to Fitch. In this system you could actually either prove or disprove every valid statement, and we proved this.

    I'm not sure that all of my classmates appreciated the course as much as I did! :-)

    This grounding in logic certainly made me better at computer programming - including analysis and design. The downside is that my wife says that I am too logical ...
     
  4. Oct 13, 2013 #3

    Mark44

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    The higher you go in math, the more you'll see proofs presented, and the more you will be expected to present proofs of various statements. A little mathematical logic will go a long ways in these endeavors. For example, in a basic implication P ##\Rightarrow## Q, by definition, the only values for which this implication is false occurs when P is true and Q is false. When you assume a certain implication is false, you are assuming that P (the hypothesis) is true, but that Q (the conclusion) is false. If you start with this set of assumptions and reach a contradiction, that means that your initial assumption that the implication was false, was itself incorrect. Therefore, the implication must be true. In summary, this is what a proof by contradiction is.

    Another type of proof that you will often see is the contrapositive. For a statement P ##\Rightarrow## Q, the contrapositive is ~Q ##\Rightarrow## ~P, where the ~ denotes "not". A statement and its contrapositive are always equivalent, which means that they have the same truth values. In contrast, the converse of a statement P ##\Rightarrow## Q is the statement Q ##\Rightarrow## P. These statements are not generally equivalent.

    I am assuming that you have some familiarity with mathematical statements, so I haven't provided any expository examples.
     
    Last edited: Nov 21, 2013
  5. Oct 14, 2013 #4
    Thank you both Mark44 and UltrafastPED. Your comments were enlightening.

    I guess I will study some mathematical logic after I have studied quite a vast amount of the other pure mathematics areas; this should convince me, in the end, that everything is really consistent.
     
  6. Oct 14, 2013 #5

    verty

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    Mathematics and logic are two different things. Mathematical logic is mathematical because sets are used to define the objects of logic. So logic questions become math questions, if that makes sense. But you would still be learning about logic, so your questions about logic would be answered.

    Mathematics, through set theory, is translatable into logic, so think of first order logic as the language or system used to write math (including set theory). You want to be able to read math like a language, this is my point of view, and knowing logic helps with this. I won't say you need to study logic, it is something you can pick up on your own, but studying it would give you a much deeper understanding, and you could proceed to learn about topics like Kripke semantics that more casual learners (like myself) would struggle with.

    While I recommend reading about mathematical logic to anyone interested, I would recommend studying it only to those who want the most complete and functional understanding. I think it is probably a common feeling that student have, during the second half of such courses, that they think to themselves, I really don't care about this. What I cared to know I learned during the first half.

    PS. I took "study" to mean enrollment in a proof-based course with exams/testing.
     
  7. Oct 14, 2013 #6
    Thank you for your help and for sharing.

    I think I will only read a little about logic till I'm satisfied with it. From what you said, I guess that I fall in the category of people who would get bored on the second half.
     
  8. Oct 14, 2013 #7
    It's ok to not like proof by contradiction. I notice a lot of students jump to it immediately where as I always try for a direct proof first.

    Now, proof by contradiction is completely valid. But the reason I (we?) don't like it is that it often seems to reveal less about the thing we are trying to prove. It doesn't really show you why something is true, but rather, it tells you why it isn't false. That can be a bit unsatisfying, but sometimes it is the only way to go.

    Remember that the reason we do proofs is not just to show things are true, but to try and understand them better.

    (On the other hand, the logic of proof by contradiction is kind of beautiful in it's own way).

    I'm not sure what your questions are, or what you're calling mathematical logic exactly, but you'll touch on a lot of this foundational stuff in analysis, abstract algebra, and beyond. These things will be discussed both formally and informally. My professors love to talk about this stuff.

    Most people would say "no," and refer you to Goedel's incompleteness theorem. But it's really about how far back you want to draw the line in the sand. (Maybe not the best analogy). For an engineer, the math just works, so it's justified. The mathematician needs some more justification. He or she knows about incompleteness but also says "Yes, but it still works." A real mathematical logician might not be happy with that answer, and so on. Mathematicians who work at this very foundational level are often accused of not doing real mathematics and doing something more like philosophy.

    -Dave K
     
  9. Oct 15, 2013 #8
    Oh, alright then! :smile: I thought those topics (analysis, algebra, etc.) would simply expect one to implicitly assume the foundational stuff. That's the main reason why was so bothered in the first place.
     
  10. Oct 15, 2013 #9
    Professors will expect some of that mysterious quality known as "mathematical maturity" by the time you get to these classes, but they also know that you haven't been formally introduced to any of these topics. So they won't assume you know it. They will basically give you what you need to get through the class, and, at least in my experience, they will give you a little bit more, just because they are math nerds at heart who love talking about it.

    -Dave K
     
  11. Nov 21, 2013 #10
    Mathematical Logic (ML) is a theory of "how to write logic", ie, it is a specific formal language theory and it mimetizes what you allready know on Logic. The human reason is behind all, behind the very modus ponens and behide every mathematical proof. Your problem is not with ML but with some philosophy. It is a philosofical question too if ML models good enough our human reason.
     
  12. Nov 24, 2013 #11
    Yes I somewhat agree that my problem is not really with the math, but it is the philosophy part that bothers me rather.

    I once asked one of my friends(who's an aspiring mathematician) about proof by contradiction and its validity, but he could not come up with a good enough argument. In the end, he just told me that maybe I should reconsider my plans about pursuing pure mathematics. Well, of course I got a little offended, but still, he was quite right.

    But, oh well... I'll continue in that direction anyway and see where it leads to.
     
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