Why Mathematical Reasoning Cannot Be a Simple Matter of Definitions and Formal Rules

by Gene Callahan

The point I wish to make here has been made before, notably, by Lewis Carroll in his essay “What the Tortoise Said to Achilles“, as well as by Wittgenstein, in his work on what it means to “follow a rule,” and by Gödel in his famous paper on undecideability. But, as I recently encountered a very, very bright young philosopher who seemed unaware of the import of such arguments, it is, perhaps, a point worth making once again.

The contention at hand is that, contrary to those who hold that mathematical knowledge offers us an example of objective truths that are of a non-physical nature, mathematical truth is “simply” a matter of positing some arbitrary set of definitions and rules for drawing conclusions from them. The budding philosopher in question contended that the truth of the proposition “2 + 2 = 4” says nothing about the existence of non-material yet objective truths, since its truth is a mere consequence of the accepted definitions of the symbols ‘2’, ‘+’, ‘=’, and ‘4’. No metaphysical truth is revealed in the proposition, anymore than would be revealed by a rule that declared, if you hear “waaaa,” then you ought to respond “xeeee.”

As I see it, and as Carroll, Wittgenstein, and Gödel so ably indicated, the problem with this contention is that nothing at all follows from accepting the definition of each of those terms concerning the truth of “2 + 2 = 4,” in the absence of embracing a notion of ‘truth’ that lies beyond any formal system itself. Why?

Let’s say you are conversing with a chap who doubts that this is all a matter of definition. You take out your copy of Principia Mathematica and explain to him how, in that system, ‘2’, ‘+’, ‘=’, and ‘4’ are defined. He declares that he understands all of those definitions, and is willing to accept them during your discussion. “Therefore,” you declare, “you must acknowledge that 2 + 2 = 4!”

“No,” he says, “I believe 2 + 2 = 5.”

“How can you say that?” you respond. “You accepted all of my definitions — don’t you see my result follows from them?”

“I’m sorry,” he replies, “but I don’t recall you offering me a definition of ‘follows from,’ and, since you are contending that all of this is simply a matter of accepting certain, purely conventional, definitions, I can’t understand why you think what ‘follows from’ these definitions must be the same for me as it is for you.”

As Carroll demonstrated, any attempt to supply more and more formal definitions and rules to patch up this difficulty simply leads to an infinite regress. Without some non-formal sense of what is ‘implied’ by any formal operation or system, i.e., some inkling that there are non-physical facts of the matter that are objectively true, there is simply no way to determine whether or not someone has drawn the ‘correct’ conclusions from a formal system.

17 thoughts on “Why Mathematical Reasoning Cannot Be a Simple Matter of Definitions and Formal Rules

  1. *blushes* Thank you, Gene.

    I think that perhaps the point I was trying to make in our earlier discussion was not advanced properly… I do not deny that the rules of logic and mathematics tell us things that are true in an objective sense, insofar as there are no problems with our premises and our arguments are valid. I similarly do not deny that we can say that we “know” that the conclusions of a valid argument are true if the premises are true.

    What I said was this:

    Gene, 2+2=4 is just an application of definitions of those symbols. It is simply a feature of what a “2” is that when a “2” is “added” to another “2,” they “equal” a “4.” If someone wanted to deny that, then they simply would be demonstrating a lack of understanding of the meaning of at least one of the relevant terms.

    I’m not sure that the same thing is clearly true of something like “Capricious killing is wrong” or “You ought not to kill capriciously.” It’s at least not so obviously true that you can just assert it as if it’s self-evident.

    I can see why you might have took my first paragraph to suggest that these are mere definitional truths, but that’s not what I meant. I just meant that “4” being equal to “2+2” simply cannot be coherently denied — to do so would be to betray a lack of understanding of what the constitutive terms mean (or, as you point out here, a lack of understanding of how logic works). I acknowledge that I may not be articulating this well…hopefully you now understand what I’m trying to say?

    In any case, the more substantive point I was trying to make was that moral statements, such as “capricious killing is wrong,” seem very different from mathematical statements: they can coherently be denied without rejecting the meaning of capricious killing, wrongness, or logic. The reason I took this to be important was that you had just said:

    Yes, Dan, #1 is certainly wrong, just as it is true that, without mathematicians, 2 + 2 would still equal 4.

    (Where #1 was my claim that “Value is a mental phenomenon which proceeds from evaluation (conscious or unconscious); without evaluating minds it would be incoherent to speak of value.”)

    Does that help? Am I still missing the point? Thanks for taking the time to hash this out! Worst case scenario, we can have an epic showdown in Michigan in a few weeks 😛

  2. Danny, I was carefully hiding your identity so that, later in life, you would not have to own up to once having had such views! And now you’ve gone and blown the cover I was providing you.

  3. Haha well I still have a whole lot to learn, and this is certainly outside of my area of specialization. Hopefully later in life I’ll be happy to own up to having been very badly mistaken about a whole number of things, including perhaps this issue. After all, I’m still a relative newcomer to this whole “thinking about stuff” thing; I’d be pretty surprised if I already knew what I was talking about!

  4. As Mises said:

    If two and two were four for me and five for someone else, we couldn’t even talk about it.

    “It is impossible to provide conclusive evidence for the propositions that my logic is the logic of all other people and…that the categories of my action are the categories of all human action. However…these propositions work.”

    For we can, in fact, coexist and not just collide with one another.

  5. If Danny describes a formal language with rules of inference where “2 + 2 = 4”, and Gene claims that for Danny’s language “2 + 2 = 5”, then Gene is not following the rules described by Danny.

    It doesn’t matter what Danny or Gene believe, or whether Danny’s formal language can be used to truthfully describe the physical universe, because the object of discussion is not whether the language can truthfully describe what is, but what can be truthfully said about the language. In other words, there is some metalanguage at work, without which we cannot talk about another language, and it is here that concepts of truth, falsity, and entailment reside.

    Formal languages like mathematics and logic are objective. (I need to be careful with my terms here, because the word “objective” to philosophers has more meanings than “freedom” from the mouth of a politician). I suggest that formal languages are objective like the Eiffel Tower is objective, that is, observers of the Eiffel Tower are looking upon the same object, and anything that can truthfully be said by one must be true for the other.

    Although I do not wish to get hung up on ontological questions about whether ideas “really exist”, I will venture so far to point out that so long as two people can think about the same idea, whatever they are thinking about cannot be identical to the thoughts or neural patterns of either. In other words, there must be something objective, like the Eiffel Tower, that we are each thinking about, but which the thoughts of neither are identical to.

  6. This is really important stuff.

    Hayek arrived at the same conclusion, see his “system of rules” paper, etc. Hayek’s route was mostly via neuroscience and the philosophy of law.

    Hayek, of course, was also influenced directly by Wittgenstein and by the “Wittgensteinian” R. S. Peters.

  7. The other side of this is that Frege and Platonic Realism must be wrong:

    “Frege’s Influence on Wittgenstein: Reversing Metaphysics via the Context Principle”, in Early Analytic Philosophy: Frege, Russell, Wittgenstein, W.W. Tait, ed., Chicago: Open Court, 1997, pp. 123-185; reprinted, in abbreviated form, in Gottlob Frege: Critical Assessments of Leading Philosophers, Vol. I, M. Beaney & E. Reck, eds., London: Routledge, 2005, pp. 241-289 (draft)

    You can read it here:


  8. “If Danny describes a formal language with rules of inference where “2 + 2 = 4″, and Gene claims that for Danny’s language “2 + 2 = 5″, then Gene is not following the rules described by Danny.”

    What”s the rule that tells you I am not “following the rules” — in other words, I think you missed the whole point of the post.

  9. “The other side of this is that Frege and Platonic Realism must be wrong”

    I don’t know, Greg — the conclusion I draw is that Plato nailed it.

  10. Greg, I looked at the paper. Man, people who fail to acknowledge the truth of Idealism get themselves all tied up in philosophical knots, don’t they?

  11. Gene,

    There is a difference between being able to define every rule and following every rule. Some rules must always remain unstated, because they are the rules with which we understand the rules. But, just because we cannot state them, it doesn’t mean that someone else can’t follow them.

  12. In regards to the mathematics element, insofar as I can see it math is simply a logical consequence of extension and/or divisibility. Its formulas are true in reality insofar as there is extension to which they could apply. Likewise, non-contradiction is ‘true’ of anything that exists by existing.

    These are not ‘non-physical’ truths, indeed they make no sense outside of physical (ontological) interpretation. What they are are minimal things which are true of any existing reality with extension.

  13. Gene — Read the Rech paper and get back to me.

    Rech’s reading is a brilliant confirmation of my own long-held understanding of the Wittgenstein argument. But decide for yourself.

    Gene wrote:

    “I don’t know, Greg — the conclusion I draw is that Plato nailed it.”

  14. Wittgenstein is pretty clear that the patterned ways of going on together are inside us, and aren’t about entities in an Platonic realm.

    And note also that Wittgenstein agrees with Hayek that behind explicit rules are mere patterns of going on together — patterns we can grope to characterize via rule governed linguistic practices, but which are not fully captured or circumscribed by those linguistic gropings. Wittgenstein talks about these as being natural to us. Hayek and Mises talked about minds with common structures (these do not have to be totally identical structures).

  15. Doesn’t everything point in the other direction?

    Gene writes:

    “Greg, I looked at the paper. Man, people who fail to acknowledge the truth of Idealism get themselves all tied up in philosophical knots, don’t they?”

  16. How do mathematical Platonists handle the independence of the continuum hypothesis? I would have thought that one was a pretty heavy blow to them. OTOH I am not up on my philosophy of math, so maybe that’s a knee slapper for those in the know.

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