93 thoughts on “Mathematics and logic”

  1. I have come to a hostile spirit against math as being as intellectually equal with logic or sciences.
    I see math as just a language of rerality and not a investigative tool. Yet it can be a tool because of otherwise incompetent human investigation.
    Newton stressed math as getting credit for his ideas but I think this is not true. It was a apple hunch/others suggestions that move his imagination.
    Math aside from new math discoveries , is a mere matter of memorization and a wee bit of careful understanding.
    Logic is only logical after all presumptions are discovered and so its more complicated.
    Math deals in minor matters of nature even though more accurately.
    God does logic. Math to God is chump change of boundaries.
    Logic hurts evolution. Math is neutral on it. Nothing to add.
    Except the odds needed to change bugs into buffalos.

  3. It is turtles all the way down.

    Or, in other words, nothing is fundamental.

    Mathematical logic is a branch of mathematics. Maybe I should insert “only” before “a branch.” Philosophic logic is just weird.

  4. What got me thinking about this is the fact that computer arithmetic circuits can be built out of logic gates, and the logical operations carried out by the gates can be expressed in terms of Boolean algebra. So you have an interesting case of mathematics based on logic based on mathematics.

    My own hunch is that either one can be more fundamental, depending on your choice of axioms. If that’s right, an interesting follow-up question is whether one approach would require fewer axioms than the other.

  6. Robert Byers:

    I have come to a hostile spirit against math as being as intellectually equal with logic or sciences.

    You seem to have a hostile spirit toward all forms of non-biblical learning.

    I see math as just a language of rerality and not a investigative tool. Yet it can be a tool because of otherwise incompetent human investigation.

    Whaat?

    Newton stressed math as getting credit for his ideas but I think this is not true. It was a apple hunch/others suggestions that move his imagination.

    Pretty hard to express Gm1m2/r^2 without mathematics.

    Math aside from new math discoveries , is a mere matter of memorization and a wee bit of careful understanding.

    Yes, when learning your multiplication tables. It gets a little more complicated after that.

    Logic is only logical after all presumptions are discovered and so its more complicated.

    What is logic before it becomes logical? Illogical, or alogical?

    Math deals in minor matters of nature even though more accurately.

    Minor matters like when did it all begin? Will it keep expanding forever, or will it collapse on itself? Why are atoms stable?

    God does logic. Math to God is chump change of boundaries.

    But which one is more fundamental?

    Logic hurts evolution. Math is neutral on it. Nothing to add.

    So much for the Wistar conference.

    Except the odds needed to change bugs into buffalos.

    What are the odds of changing ribs into women?

  7. keiths: What got me thinking about this is the fact that computer arithmetic circuits can be built out of logic gates, and the logical operations carried out by the gates can be expressed in terms of Boolean algebra.

    But arithmetic isn’t all of mathematics. It is only a tiny part.

    This is really the argument between logicism (mathematics is a branch or part of logic), and platonism (mathematics is far more than logic).

    For myself, I’m a fictionalist (mathematical objects are useful fictions). I don’t accept the exorbitant ontology of platonism. But the way I do mathematics is similar to the way that platonists do it. In particular, I agree that mathematics is far more than logic.

    As far as I know, Frege was a proponent of logicism, as were Russell & Whitehead. Quine probably was, too. But Gödel was a platonist.

    In my opinion, Quine’s argument in “Truth by convention” fails as an argument against mathematical truth by convention, but works pretty well as an argument against logicism. In simple terms, the use of convention in mathematics (think axiom systems) is far broader than Quine assumes.

  8. keiths: We stop when we get to axioms.

    No, we don’t. We can come up with other (different) axiom systems. There does not appear to be any limit to the axiom systems we can invent.

  9. Neil,

    Parallel systems of axioms would be “turtles off to the side”, not “turtles all the way down.”

  10. Neil,

    I don’t see mathematical platonism as necessarily opposed to logicism, for the following reason: You could grant the mind-independent reality of mathematical objects, yet still maintain that they are describable in terms of logical primitives, just as you can grant the reality of matter and energy, yet maintain that they are describable in terms of physical theories.

    When you say that “mathematics is far more than logic”, do you mean that you think there are parts of mathematics that cannot be expressed solely in terms of logical axioms and operations?

  11. keiths:
    Neil,

    I don’t see mathematical platonism as necessarily opposed to logicism, for the following reason:You could grant the mind-independent reality of mathematical objects,

    As I understand the way math philosophers use the term, once you accept that mathematical objects exist, you are a platonist. For philosophers usage of the term, I think logicism would have to involve denying their existence.

    I’m not really sure about the purpose of this thread, There is at least 100 years of math philosophy with arguments and counter-arguments about the relationship of math and logic and set theory. How does that work come into play in this thread?

    On another of your posts above: I believe that RB cannot be refuted by ordinary Western logic; one must understand his comments as similar to Zen Koans. Of course, he will deny that, thus proving my point.

  12. keiths:
    Neil,

    Parallel systems of axioms would be “turtles off to the side”, not “turtles all the way down.”

    How is that relevant to the question of whether there is an underlying fundamental?

    We started with arithmetic. The we generalized that to allow rationals. Then we generalized to allow reals. Then we generalized to all complex numbers. Then we relaxed the axiom set to allow non-commutative algebras, then non-associative algebras. Those kinds of changes don’t seem to all be “turtles off to the side”.

  13. keiths: When you say that “mathematics is far more than logic”, do you mean that you think there are parts of mathematics that cannot be expressed solely in terms of logical axioms and operations?

    Geometry, particularly the kind of geometric thinking that precedes axiomatization.

  14. Neil Rickert:
    It is turtles all the way down.

    Or, in other words, nothing is fundamental.

    Mathematical logic is a branch of mathematics.Maybe I should insert “only” before “a branch.”Philosophic logic is just weird.

    What is philosophic logic? The Square of Opposition? Truth tables? Venn diagrams?

  15. Neil Rickert: Geometry, particularly the kind of geometric thinking that precedes axiomatization.

    Hm. Could you give an example of this sort of pre-axiomatic geometric thinking?

    My view is that the axiom systems are still there, but just not stated explicitly. Otherwise, how do you know which steps are allowed? How would two people with differing answers decide which, if either, is correct?

    I don’t see how one can do mathematics without knowing (at least subconsciously) what the rules of the game are.

  16. walto: What is philosophic logic?

    I’m not quite sure. That’s part of what make it weird.

    Some philosophy books are chock full of unpersuasive logic arguments. I’m inclined to say that it is the attempt to apply logic directly to things in the world (or in theology), instead of restricting the use of logic to a formal model.

  17. What exactly do you mean by “fundamental”? It is fairly trivial (in the conceptual sense, not in the technical details) to derive, say, arithmetics from an appropriately axiomatized logic system. The controversies in the philosophy of mathematics, starting from Frege’s monumental work, turn on what constitutes a [i]fundamental[/i] axiomatization. Frege contended that his axioms were analytical a priori, known to us independently of experience. That contention has been attacked from various angles. His original axiomatization also led to inconsistencies, which could be fixed, but at the cost of substituting a more powerful and less intuitive axiom.

  18. Neil Rickert: I’m not quite sure.That’s part of what make it weird.

    Some philosophy books are chock full of unpersuasive logic arguments.I’m inclined to say that it is the attempt to apply logic directly to things in the world (or in theology), instead of restricting the use of logic to a formal model.

    I really have no idea what you mean (although I’m familiar with your frequent and weird hostility to philosophy). “Philosophical logic” is stuff like like Aristotle on syllogisms, Euler and Venn, Russell and Whitehead, Godel, modal logic. It’s taught in Critical Thinking and Foundations of Math classes. (Mis)-use of that stuff in theology or elsewhere isn’t any more logic than my house is an example of hammer and saw making.

    Did a philosophy professor slap you in front of the class once or something?

  19. BruceS: For philosophers usage of the term, I think logicism would have to involve denying their existence.

    I should not have gone by memory on that. I quick check of SEP and I find that logicism is strictly speaking ontologically neutral.

    So your points about the possible compatibility of the logicism and platonism are valid, I think.

  20. walto: Euler and Venn, Russell and Whitehead, Godel, modal logic.

    I think Godel would be considered a mathematician these days.

    The basic issue is this: suppose you are studying in this field. What are you going to call yourself in order to get a paying job? No hard feelings, I hope.

    So we’ll take Godel, and of course there was never any issue about what to call Euler. You can keep Russell and Whitehead.

  22. socle: Hm. Could you give an example of this sort of pre-axiomatic geometric thinking?

    A lot of mathematics is idealization. This includes idealization of methods from science.

    I take categorization (dividing the world into two parts) as a basic empirical method. That’s about what is used in the Dedekind construction of the reals, and in the Bolzano-Weierstrass theorem (compactness of closed bounded sets of reals). Try “The hunting of big game” for a light hearted illustration.

  23. Do we have examples of one without the other (after carefully examining unstated premises such as Uniformitarianism)?

  25. Richard,

    Do we have examples of one without the other…

    You mean, of math without logic or logic without math?

    … (after carefully examining unstated premises such as Uniformitarianism)?

    Could you clarify what you mean by “Uniformitarianism”? I’m only familiar with its usage in a geologic or scientific context.

  27. socle:

    Hm. Could you give an example of this sort of pre-axiomatic geometric thinking?

    My view is that the axiom systems are still there, but just not stated explicitly.

    Neil:

    I take categorization (dividing the world into two parts) as a basic empirical method.

    But categorization is itself implicitly axiomatic. For example, if X is part of the world W, and we divide W into two parts A and B, then X is either part of A or else it is part of B, provided that X itself is not divided when W is divided.

  28. keiths:

    In fairness to walto, Gödel also did quite a bit of philosophical work,

    Of course you are right, Keith, but the point is, if that Institute for Advanced Study would not have had him, or if he decided that he wanted to eat rather than starve to death and so had to pay for food, then he may have wanted a paying job. Hence, he would have been careful to call himself something that people might hire. Although “quant” might be even better.

    Yes, it was a joke. Perhaps at Walt’s expense, I know, but meant in fun.

  29. Neil Rickert: A lot of mathematics is idealization.This includes idealization of methods from science.

    I take categorization (dividing the world into two parts) as a basic empirical method.That’s about what is used in the Dedekind construction of the reals, and in the Bolzano-Weierstrass theorem (compactness of closed bounded sets of reals).Try “The hunting of big game” for a light hearted illustration.

    Maybe I misunderstood your statement—clearly the construction of the real numbers via Dedekind cuts can be expressed solely in terms of an axiomatic system, no?

    If you are rather saying that mathematicians in practice idealize arguments based on physical models (e.g., cutting a piece of paper in half over and over is like bisecting the plane repeatedly), then I would agree.

  30. SophistiCat:

    What exactly do you mean by “fundamental”?

    I deliberately left it unspecified, because I am interested in whether there is any defensible sense in which math or logic can be considered more fundamental than the other.

    It is fairly trivial (in the conceptual sense, not in the technical details) to derive, say, arithmetics from an appropriately axiomatized logic system.

    It is also fairly trivial conceptually to derive logic from mathematics, though, so “Y can be derived from X” is, by itself, insufficient to demonstrate that “X is more fundamental than Y”.

    That’s why I asked my follow-up question:

    My own hunch is that either one can be more fundamental, depending on your choice of axioms. If that’s right, an interesting follow-up question is whether one approach would require fewer [or simpler] axioms than the other.

  31. walto:

    The ‘weirdest’ one in that list is Godel, the guy you say was doing math.

    I believe I respond to that point in detail in my reply to Keith.

    Of course, I wholeheartedly agree Godel was one weird guy. Rather intelligent, but still, weird. (There’s that story about him finding a logical inconsistency in the US constitution just before his citizen hearing, for example.)

  32. Bruce,

    Of course you are right, Keith, but the point is, if that Institute for Advanced Study would not have had him, or if he decided that he wanted to eat rather than starve to death and so had to pay for food…

    The sad thing is that he really did eventually decide to starve to death.

  33. keiths:

    I deliberately left it unspecified, because I am interested in whether there is any defensible sense in which math or logic can be considered more fundamental than the other.

    Keith:
    The reasons I asked originally about the prior art, so to speak, was that I thought it was basically a settled matter that all of conventional math could be reduced to set theory with the right axioms plus predicate logic. But math cannot be reduced to predicate logic. Although you do need predicate logic to do math.

    I realize there are likely lots of if, ands, and buts to that broad statement, but I suspect that are too technical except for the experts.

    Are there any mathematical philosophers in the house to set me straight on that?

  34. keiths:
    Bruce,

    The sad thing is that he really did eventually decide to starve to death.

    Yes, I knew that, of course. My whole post was meant as food for thought, so to speak.

  35. Neil,

    We started with arithmetic. The we generalized that to allow rationals. Then we generalized to allow reals. Then we generalized to all complex numbers. Then we relaxed the axiom set to allow non-commutative algebras, then non-associative algebras. Those kinds of changes don’t seem to all be “turtles off to the side”.

    They don’t seem to be “turtles below”, unless you claim that “X encompasses Y” means that “X is more fundamental than Y”. That doesn’t seem right to me, because (for example) it would be odd to say that the set {0,1,2,3,4,5} is more fundamental than the set {0,1}.

  36. keiths: For example, if X is part of the world W, and we divide W into two parts A and B, then X is either part of A or else it is part of B, provided that X itself is not divided when W is divided.

    I’m not seeing the relevance.

  37. Bruce,

    I’m not really sure about the purpose of this thread,

    The purpose is to discuss a question I think is interesting.

    There is at least 100 years of math philosophy with arguments and counter-arguments about the relationship of math and logic and set theory. How does that work come into play in this thread?

    It comes into play when someone introduces it!

    The reasons I asked originally about the prior art, so to speak, was that I thought it was basically a settled matter that all of conventional math could be reduced to set theory with the right axioms plus predicate logic.

    Yes, I think that’s correct.

    But math cannot be reduced to predicate logic.

    If so, I didn’t know that! (Which is one of the reasons for having these discussions.)

  38. socle: Maybe I misunderstood your statement—clearly the construction of the real numbers via Dedekind cuts can be expressed solely in terms of an axiomatic system, no?

    Which came first – the idea for the construction or the axiomatization?

    What are we discussing here? Are we trying to find the source of new mathematical ideas? Or are we trying to construct a Whig history, such that we can describe everything as if we started with axioms?

  40. Neil,

    Or are we trying to construct a Whig history, such that we can describe everything as if we started with axioms?

    We aren’t doing history at all, unless you think that “X is more fundamental than Y” means “X preceded Y historically”.

  41. BruceS: The reasons I asked originally about the prior art, so to speak, was that I thought it was basically a settled matter that all of conventional math could be reduced to set theory with the right axioms plus predicate logic.

    I tend toward skepticism about that idea of reduction. I see set theory as a kind of universal modeling clay. But the set theory itself does not tell us what to model or how to do the modeling.

  42. Here’s what the SEP says about the current status of the logicist project:

    And there matters stood for more than fifty years. In 1983, Crispin Wright’s book on Frege’s theory of the natural numbers appeared (Wright 1983). In it, Wright breathes new life into the logicist project. He observes that Frege’s derivation of second-order Peano Arithmetic can be broken down in two stages. In a first stage, Frege uses the inconsistent Basic Law V to derive what has come to be known as Hume’s Principle:

    The number of the Fs = the number of the Gs ≡ F≈G,
    where F≈G means that the Fs and the Gs stand in one-to-one correspondence with each other. (This relation of one-to-one correspondence can be expressed in second-order logic.) Then, in a second stage, the principles of second-order Peano Arithmetic are derived from Hume’s Principle and the accepted principles of second-order logic. In particular, Basic Law V is not needed in the second part of the derivation. Moreover, Wright conjectured that in contrast to Frege’s Basic Law V, Hume’s Principle is consistent. George Boolos and others observed that Hume’s Principle is indeed consistent (Boolos 1987). Wright went on to claim that Hume’s Principle can be regarded as a truth of logic. If that is so, then at least second-order Peano arithmetic is reducible to logic alone. Thus a new form of logicism was born; today this view is known as neo-logicism (Hale & Wright 2001).

    Most philosophers of mathematics today doubt that Hume’s Principle is a principle of logic. Indeed, even Wright has in recent years sought to qualify this claim: he now argues that Hume’s Principle is analytic of our concept of number, and therefore at least a law of reason.

  43. Neil,

    I tend toward skepticism about that idea of reduction. I see set theory as a kind of universal modeling clay. But the set theory itself does not tell us what to model or how to do the modeling.

    Why would set theory need to “tell us what to model” in order for mathematics to be reducible to set theory and predicate logic?

  44. Neil Rickert: I tend toward skepticism about that idea of reduction.

    OK, I think there is a technical sense of “reduction” in which my statement is held to be true but the technicalities would be beyond my pay grade.

    I would certainly agree that it makes no difference to what most mathematicians do. Same as Godel’s Incompleteness work, I guess.

  45. Neil Rickert: Which came first – the idea for the construction or the axiomatization?

    The idea for the construction.

    What are we discussing here? Are we trying to find the source of new mathematical ideas? Or are we trying to construct a Whig history, such that we can describe everything as if we started with axioms?

    That’s essentially the question I was alluding to in my previous post. This is what I originally responded to:

    keiths: When you say that “mathematics is far more than logic”, do you mean that you think there are parts of mathematics that cannot be expressed solely in terms of logical axioms and operations?

    Neil Rickert: Geometry, particularly the kind of geometric thinking that precedes axiomatization.

    which I (incorrectly, it appears) interpreted to mean that you believed some kinds of geometry cannot, in principle, be axiomatized.

  46. keiths: Why would set theory need to “tell us what to model” in order for mathematics to be reducible to set theory and predicate logic?

    It depends on what you mean by “reducible.”

    If you know everything that is to be known about set theory, it does not follow that you know all of mathematics.

  47. Something I can’t imagine ever opening: Crispin Wright’s book on Frege’s theory of the natural numbers

    Not so much because of the subject matter (though that’s pretty far out of my wheelhouse), but I find Wright really hard to follow even when he’s talking about stuff I think I should understand. IIRC, somebody on some web phil. forum I belong to had a class with him and said he wasn’t so bad in person. But his writing is really dense, hard-to-follow and confusing, I think. And I never feel like I’ve learned something useful when I slog through it.

    Maybe one important way in which philosophy differs from science involves this matter of style. My sense is it doesn’t have the same sort of importance where there are, you know….real empirical results. In philosophy it’s nice to be entertaining (though maybe not as entertaining as Fodor), as well as convincing, clear, etc. (And maybe you get fans/groupies of maybe a slightly different type?)

  48. Neil Rickert: It depends on what you mean by “reducible.”

    SEP article on set theory gives the following informal description of reducibility:

    [A]ssuming the existence of infinite sets, one can define numbers, including integers, real numbers and complex numbers, as well as functions, functionals, geometric and topological concepts, and all objects studied in mathematics. In this sense, set theory serves as Foundations of Mathematics. The significance of this is that all questions of provability (or unprovability) of mathematical statements can be in principle reduced to formal questions of formal derivability from the generally accepted axioms of Set Theory.

  49. This discussion of set theory as foundational to mathematics arouses traumatic memories of having been subjected to “the New Math” as an elementary school kid in the late 1950s and early 1960s.

    *twitches*

  50. BruceS: The significance of this is that all questions of provability (or unprovability) of mathematical statements can be in principle reduced to formal questions of formal derivability from the generally accepted axioms of Set Theory.

    Fair enough. However, I do not take questions of provability as the entirety of mathematics.

    Mathematics is far older than “foundations of mathematics.” In set theory, some of the main examples of sets (particularly of infinite sets) are sets of numbers or sets of other mathematical objects. So the idea of reducing mathematics to set theory, means that set theory has to get under way without its main examples.

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