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Who wants to be a logician? (anybody?)

  1. Aug 25, 2011 #1
    I'm interested in finding out who on this board has been afflicted with an unfortunate desire to become a logician, and what your interests are.

    I'll start:

    I'm interested in proof theory (presently I'm trying to learn about the Curry Howard Isomorphism), applications of model theory to number theory/algebraic geometry, reverse mathematics and some ideas from category theory/categorical logic.

    I'm also very open to becoming an algebraic geometer, if I end up being a mathematician rather than a logician.

    So, does anyone else like logic?
     
  2. jcsd
  3. Aug 25, 2011 #2

    micromass

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    Yes, I like logic very much. Especially things like set theory. Unfortunately, I did not have the chance to go into those fields, but I still have a lot of affinity for the topic.

    I'm quite interested in things like:
    - Model theory
    - Universal algebra
    - Boolean valued models

    I don't claim to know much about these things, but it interests me. I wish I had more time to study it.
     
  4. Aug 25, 2011 #3
    What I like about logic is the non-intuitive things you can derive that appear inconsistent but are purely clear in the mathematics i.e Banach-Tarski Paradox.
     
  5. Aug 25, 2011 #4
    I guess I'm a kind of logician, though most of the stuff I've done has been on lattices and posets, the justification for looking at them (at least as far as my funding was concerned) was their uses in the model theory of alternative logics (though in truth I don't actually know very much about this). I use classical model theory quite a lot (saturation, compactness, ultraproducts etc.), and occasionally it turns out something is dependent on the existence of large cardinals so I have to know a little set theory, but I doubt I'll be making any contributions to either of these fields. I've also done a small amount of work in classical algebraic logic (relation algebras, cylindric algebras etc.). I have to know a little category theory (some topology too) as it crops up all the time in duality results (such as Stone's theorem and its generalizations).
     
  6. Aug 25, 2011 #5
    I plan to become Algebraic geometer or category theorist/algebraic topology.

    Could never study logic on it's own through. I have doubts that everything I'm doing is abstract nonsense and should be studying physics, if I do logic that would probably be amplified.
     
  7. Aug 25, 2011 #6
    It wouldn't be amplified if you learned about computability theory and computer science more generally. Logic is actually quite applicable, significantly more so than category theory (at least as far as I know, though category theory is certainly applicable and crosses over with computer science).

    Automatic theorem proving, computability theory, computational complexity, formal verification methods, applications to problem solving in mathematics, logic programming, multi-agent protocols etc.
     
  8. Aug 25, 2011 #7
    That's actually an area I have some interest in learning about (model theory of alternative logics), I've been meaning to start reading Model-Theoretic Logics but haven't gotten around to it (between my courses, the math subject GRE looming overhead, trying to study algebraic geometry and the Curry Howard Isomorphism stuff... j'ai pas le temps :smile:)


    I'm familiar with compactness and ultraproducts (we did an ultrapower construction of the non-standard reals beck when I took mathematical logic 2), but we didn't touch on saturation (nor did we, to my recollection, actually prove the transfer principle since we were only dealing with the topic in passing).

    Truthfully my model theory is somewhat rusty, though I'm quite interested in the subject and its applications. My proof theory background is significantly more robust. I plan to work through Chang and Keisler if time permits... though Wilfrid Hodges book also looks nice.

    ETA: If you don't mind sharing; where (are you attending)/(did you attend) for your PhD?
     
    Last edited: Aug 25, 2011
  9. Aug 25, 2011 #8
    Well, as I said, I don't actually know much about this, but my understanding is that you can use lattices and posets extended by additional operations to provide algebraic semantics for various non-classical logics. For example, I'm interested in a construction called the 'canonical extension', which you can use to embed a poset or lattice expansion into a complete lattice. It turns out that the closure of the class of algebraic models for a logic under taking canonical extensions gives you complete relational semantics for that logic, which is good (or so I'm told).

    Both those books are good, though I've found myself reaching Chang and Keisler more often than not when I want to look something up. Neither my model theory nor my proof theory is particularly hot but I use model theory more often. I guess that's because I'm approaching things from the algebra side, so model theory comes in handy for constructing things that have properties you want. Also, a significant proportion of my work so far has been trying to find counter examples to closure under ultraroots for various classes of posets and lattices, and so prove they're not elementary. Not particularly glamorous, but surprisingly hard in some cases.
     
  10. Aug 25, 2011 #9

    micromass

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    That's actually very interesting!! Lattices and posets are a very interesting domain of study!

    Judging by your user name, I thought you were into computer science :biggrin:
     
  11. Aug 25, 2011 #10
    Thanks, though it's possibly best to take what I say about the non-classical model theory stuff with a pinch of salt as I've not looked at this area in much depth.

    Technically I think I am some kind of computer scientist as I work/study in a computer science department, but it's not my main interest (though it can be pretty interesting). My choice of user name is a result of having skimmed through a paper linking the canonical extension I was talking about before with DCPO presentations a few days ago. I suppose the word was on my mind at the time; I kind of like the sound of it.
     
  12. Aug 25, 2011 #11

    wukunlin

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    I would love to take some courses on logic but I really have no time. They are so incredibly useful in programming and digital electronics
     
  13. Aug 25, 2011 #12
    Any chance you've played with lambda calculus much?
     
  14. Aug 26, 2011 #13
    I'm afraid not.
     
  15. Aug 26, 2011 #14
    Fair enough. I ask because the only place I've seen DCPOs is in the context of the Scott Topology applied to lambda calculus.
     
  16. Aug 26, 2011 #15
    I'm a huge fan of category theory, and a bit of set theory/model theory. However, I don't think I have time to take many logic courses since my intention (for now) is to concentrate on topology (algebraic, differential, low-d, etc.).
     
  17. Aug 26, 2011 #16
    Category theory is pretty great, are you familiar with categorical logic at all? You can define logics as categories where proof rules are morphisms between formulas. The concept is very intriguing to me; also of interest are cartesian closed categories and toposes because of their application to constructive mathematics.
     
  18. Aug 26, 2011 #17
    I don't see the point of constructive mathematics. Proof by contradiction doesn't seem that bad.
     
  19. Aug 26, 2011 #18

    micromass

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  20. Aug 26, 2011 #19
    Funny you should mention the Scott topology because this crops up in the theory of completions for lattice expansion quite a lot.
     
  21. Aug 26, 2011 #20
    Well, you have to fully consider what constructivism is, it is a fairly deep subject and is worth taking time to consider.

    Proof by contradiction was not explicitly what the constructivist were/are worried about, they were worried about the law of excluded middle. There are plenty of examples of theorems where constructive existence does not follow from classical existence proofs.

    In addition to that, I strongly preferhttp://en.wikipedia.org/wiki/Computable_analysis" [Broken]. I find is somewhat implausible that earlier mathematicians intended there to be numbers that are not only transcendental, but are in fact void of any decimal approximation whatsoever.

    In addition to this, because constructive implication is stronger than classical implication (and constructive proof is stronger than classical proof in general), you can embed classical logic in constructive logic and treat a proof by contradiction to be "evidence" for a construction.

    If you want to learn about constructive mathematics you might lookhttp://plato.stanford.edu/entries/logic-intuitionistic/#RejTerNonDat" It took me a while to get to the point that I wasn't a bit up in arms against constructivism, what really made me like it was the realization that rather than totally throwing out classical logic, it provides more subtle distinctions than are possible in CL alone. It really is a beautiful perspective once you start to "get it".

    Also, (and this will likely irk some people on here, and I'm really not trying to have a debate about it) it appears to me that the mind is computable, so I tend to view non-computable mathematics as computable mathematics obfuscated by the sort of mental hacks and tricks that come with being hairless apes (having not been selected with 'doing math' in "mind").
     
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