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Mathematics to be simply an extension of logic

  1. Jun 2, 2013 #1
    Hello,

    I read that many people believe mathematics to be simply an extension of logic and therefore some or all of math to be reducible to logic. I thought this was an obvious fact for the longest time. I was wondering if there was any flaw with such an argument or what else there is which can make math not be partially or completely a subset of logic.

    Thanks.
     
  2. jcsd
  3. Jun 2, 2013 #2

    MathematicalPhysicist

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    existence of numbers is not a logical implication of logic, and thus the induction theorem or axiom is not a logical axiom or theorem.
     
  4. Jun 2, 2013 #3

    mathman

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    Mathematics is an application of logic, not an extension.
     
  5. Jun 2, 2013 #4

    Stephen Tashi

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    You could say that all drawing is the application of some substance to a substrate. That may be true, but the ability to apply a substance to a substrate doesn't give you the ability to draw well. Being good at applying logic doesn't guarantee you will be good at doing mathematics. To do mathematics you need to be inventive, or at least be able to appreciate inventiveness. Most of mathematics (perhaps all of it) is merely working out the consequences of definitions. (For example, from the viewpoint of mathematical logic, the real numbers aren't a collection of physical objects that have certain "properties". Instead the "properties" of the real numbers are simply part of the definition of the real numbers.) Coming up with useful definitions is an art.
     
  6. Jun 3, 2013 #5
    @MathematicalPhysicist: That's interesting. But I still don't know if there's more to math than logic from that perspective. I'm only informed that there's more to logic than what can be contained in logic at any one point in time. We already knew that from Godel's theorem or the CH. But the math we can know of, is it all contained inside of logic? Or is there things in math which logic cannot do?

    @mathman: Sounds good, thanks. Now the problem is, some people do believe there is more to math than what is contained in logic. The realists believe numbers are real things that can be sensed just like the eye see the light.

    @StephenTashi: 1. You can draw a number in the air. 2. I don't see how stating that you need to be inventive to do math help understand if math is an extension of logic. Plus, settling all math questions as simply being an innate characteristic seem discriminatory in some way. Moreover, not all inventive persons do math.

    Thank you for the answers.
     
  7. Jun 3, 2013 #6

    Stephen Tashi

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    Do you understand what logic is? In common language, people use the adjective "logical" to mean something that is "reasonable", "plausibe" or "true". However, this is not the technically correct definition of Logic.

    Logic is only the study of reliable methods of reasoning. These methods of reasoning don't specify any particular topic to be reasoned about. The methods of logic do not involve determining the objective truth or falsity of anything in the real or the mathematical world. Logic only involves procedures for deducing the consequences of assumptions. It doesn't tell you what assumptions should be made in physics or mathematics.

    When you study a particular set of assumptions (such as set theory or the axioms for the real numbers) this is doing mathematics, not doing Logic. The creation of particular assumptions isn't a matter of Logic.

    If current methods of logic were powerful enough to tell how to prove or disprove any possible consequence of a set of assumptions then there wouldn't be any famous "unsolved" mathematical problems. Proving mathematical theorems involves applying Logic to a set of assumptions. However the current science of Logic doesn't explain how to apply methods of reasoning to get to a particular goal. It only explains how to evaluate a proof that a person has created. Creating the proof involves creativity and invention.
     
  8. Jun 3, 2013 #7
    Nile3

    The answer to your question depends on what you take a reduction of mathematics to logic to consist in.

    There are many mathematical theories whose axioms are not reducible, recoverable or interpretable as theorems or axioms of logic. As MathematicalPhysicist said, logic doesn't imply the existence of any entities while, for example, the axioms of number theory and set-theory do imply the existence of (infinitely many) entities. (Though I'm not sure that the induction scheme mentioned by MathPhys is a good example, as it is really a conditional statement).

    It's not entirely clear cut, because there's some debate about what precisely counts as a logical truth -- Frege and Russell thought that second order logic - quantification into predicate position -- did entail the existence of set-like entities.

    However, you might think that mathematics doesn't really assert the existence of numbers and sets; rather, mathematicians are concerned with what would be true if a certain collection of axioms were true - the axioms defining number theory, or the axioms defining set-theory. From that point of view, there is a sense in which most modern mathematics, and certainly most standard modern mathematics, can be regarded as a kind of logic -- the art of drawing out ever more subtle and difficult logical consequences of a set of axioms.

    Sometimes, though, mathematicians can get into a fight as to whether or not an axiom is 'correct'. There used to be disagreement about whether the axiom of choice was an acceptable axiom. Insofar as mathematicians get involved in this discussion, we cannot understand what they are doing as simply doing logic, as they are not just drawing out the consequences of a set of axioms. Some set-theorists take sides on whether the continuum hypothesis is 'true.' But I would say it is doubtful whether such arguments play much part in modern mainstream mathematics. For a start, there are difficulties in using the notion of truth in the mathematical context without finding oneself committed to Platonism.

    It's not clear the degree to which Godel's theorem is really relevant. A common formulation is that Godel's incompleteness theorem shows that any sufficiently strong consistent formal system is incapable of proving all the mathematical truths. But that statement involves the notion of mathematical truth, which is contentious. More formally, Godel's theorem just shows that, for any sufficiently strong consistent formal system, there is a mathematical statement P, such that neither P nor ¬P is provable. Whether this touches upon the reducibility of maths to logic depends upon your attitude towards P. For instance, many mathematicians are unmoved by the unprovability of CH from standard axioms of set-theory.
     
  9. Jun 4, 2013 #8
    @Yossell: Wow, that was amazing. How do we end up knowing anything at all? Is the only way to know if something is true or not simply to bridge the gap with abstractions? You know, I'm starting to think there may be more to this "mathematical realism" than meet the eye at the first sight. Do you have any good resources to keep digging deeper?


    @ Tashi: So I assume you weren't logic in the creation of the assumption that "The creation of particular assumptions isn't a matter of Logic". In that case, you did math because you were inventive. But not logic. But you used logic to deduce your inventive math applied not to logic. Therefore no creation of particular assumptions is made without logic. Therefore I was logical in my creation of the assumption you weren't logic.

    That was just the obvious stuff that came out of your post...
     
  10. Jun 5, 2013 #9

    Mark44

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    Someone isn't "logic" - they can be logical or illogical, depending on how they are reasoning.

    Most of what you said above makes no sense to me, and it seems that you do not understand what Stephen Tashi was saying. A mathematician uses creativity to come up with one or more assumptions, and then uses logic to arrive at some conclusion about those assumptions.

    Logic is not required to produce the assumptions. A good example would be the imaginary numbers, where the initial assumption is that there is a solution to the equation x2 = -1.
     
  11. Jun 5, 2013 #10

    pwsnafu

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    Late to the thread but

    I get the impression that you simply don't know what logic is and how it is used in mathematics. What courses in Logic have you taken?
     
  12. Jun 5, 2013 #11

    lavinia

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    I know nothing of formal systems but i can say that the thought process in mathematics is not only deductive which it what it would be if it were purely an extension of logic. Mathematical truth like much other truth is realized through insight. The logic comes later - like an after thought.

    Also mathematical objects have subtle and complex structures much like Nature itself. These structure come from many sources again much like things in Nature. I do no see how these structures are extensions of logic.
     
  13. Jun 6, 2013 #12

    MathematicalPhysicist

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    You should also define what is "useful" to you.

    But it does seem as an art and it doesn't matter if you do pure maths or applied maths (and for me the distinction is blurred anyways).
     
  14. Jun 6, 2013 #13

    MathematicalPhysicist

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    I am not sure I understand your objection to my example, even if the theorem/axiom is structured by a conditional it's still not a logical axiom, a logical axiom is any tautology in first order classical logic. I mean this axiom doesn't apply for the real numbers for example.
     
  15. Jun 6, 2013 #14
    @Mathematical Physicist
    You originally wrote
    My point was this: since the induction theorem does not imply the existence of numbers, the conclusion of the `thus' does not follow from the fact that the existence of numbers.

    (Aside: Not all logicians agree that the limit of logic is first order logic -- although first order logic can take us very far. Modal logic, temporal logic, infinitary logics are just some extensions of first order logics that logicians regard as part of their domain. In particular, since useful mathematical concepts such as 'there are finitely many', `is a well ordering of' are not expressible in first order logic, the full induction axiom of Peano arirthmetic (as opposed to the weaker first order schema, there are many who argue that logics allowing second order quantification should be part of their domain.)

    Strictly speaking, the induction axiom applies only to a certain kind of mathematical object: it is the *natural numbers* that obey the induction axiom. Accordingly, a regimentation of the induction axiom would contain a predicate `Nx' which would act to restrict the domain of the quantifiers. Now, how to define `Nx'? Well, why not say that to be a number just is to be part of a structure which satisfies the peano axioms. To be a number just is to be one of a collection X, on which there is a relation R which satisfy the following principles: X has an R-least member; each member of X has successor; etc. etc. As we have quantification into predicate position, X and R, this turns out to be a sentence of second order logic. From this point of view, the induction formula is simply part of the definition of what it is to be a number.

    With this in place, we can now regard much work on the Natural numbers as simply working which conditionals of the form -- if there is a set X and relation R satisfying Peano axioms then so and so' are true.

    What this idea won't get you is that mathematical objects exist -- mathematics becomes the working out of logical conditionals of the above form; but logic doesn't entail any of the pure existence theorems of mathematics -- it doesn't entail that there *are* an infinity of primes; only that, if there is an X and an R satisfying Peano axioms, then there are an infinity of primes in X. Since your original objection mentioned the existence of numbers, you appeared to be endorsing an objection of this form.

    However, many are not happy supposing the existence of numbers.
     
  16. Jun 6, 2013 #15
    Well, this is an interesting discussion. But it seems that no matter the path of thinking I take from the ideas you give me, I end up thinking "What makes the math in the physical world, which we describe with math?" This is redundant. Is there a way out?
     
  17. Jun 12, 2013 #16

    Mark44

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    How about this? We (humans) create the mathematics that models or approximates what we observe in the physical world.
     
  18. Jun 15, 2013 #17
    That's simply physics. Which we describe with math.

    Mathematical truth like much other truth is realized through insight. The logic comes later - like an after thought.
     
  19. Jun 17, 2013 #18

    Bacle2

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    I think Stephen Tashi clinched it by mentioning that Logic( Let's assume for now that we're working with sentence logic) does not have any content of its own; it is a tool,and it is as good as the premises one works with
    (This is why I think the magazine called "Reason" is so absurd; reason does not vouch neither for nor against capitalism nor socialism nor any other belief system. Logic is a collection of technicques/methods that are designed to be "truth-preserving" ,i.e., so that if you start with premises that have been determined to be true and apply correctly the rules of sentence logic ,you will end up with a true sentence . The best display of this is in the truth tables for the conditional in which T-->F is F .) The essential concept is that of a valid argument,an argument in which if the premises are true,the conclusion cannot be false.

    And there are ways of knowing that Mathematics is not a subset of sentence logic, since sentence logic is decidable --just use a truth table, while Mathematics is not. Moreover, the truth-value in sentence logic is arbitrary in the sense that it does not depend on the structure of the sentence; sentences here are atoms, whose internal structure has no affect on the sentence's truth value. I don't think this is reflective of , nor in agreement with Mathematics.

    The issue becomes much more complicated when you start working with Predicate Logic.
     
  20. Jun 17, 2013 #19

    Mark44

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    I don't see how this makes sense - that the logic comes later. Do you have any particular example in mind?
     
  21. Dec 11, 2013 #20

    jpz

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    Logicism has long failed, since the formulation of Godel Incompleteness theorem, although we still have a relentless attempt to place logic at the foundation of mathematics. And whether it is or not, depends on what axioms you consider to be part of logic. Logic as traditionally been considered, does not assume equality. However, with a (somewhat ad hoc) model of logic, one that utilizes equality, one can generate the Peano (true math) axioms. Even then, there are still axioms in math, that cannot be defined with traditional logic.

    In any event, Godel marvelous theorem shows, essentially, that in mathematics (and perhaps in any large formal system), the proof for at least one given truth, is never based on the rules that exist to define that problem. This directly implies that there exist axioms, which have gone undiscovered and thus, that humans have some creative /intuition-like ability which allows us to 'realize' these axioms. It will be long time before the common man appreciates the importance of Godel.

    Mod note: Deleted off-topic paragraph and link to web site.
     
    Last edited by a moderator: Dec 11, 2013
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