Mathematics versus reality

In the comments to my posting on the incoherence of Narnia, it has become apparent that some of the respondents are deeply confused about the relationship between mathematical and empirical truth. Symptoms of this confusion have included a superficially plausible but mistaken application of the Law of the Excluded Middle and an an attempt to invoke Gödel’s Incompleteness Theorem to suggest constraints on our ability to obtain empirical truth.

It’s time to bust some myths…

I think what these confusions mainly demonstrate is that even most of the bright college-educated geeks who comment on my blog have managed to avoid learning anything about philosophy or foundational mathematics. This is mostly not their fault; foundational mathematics is pretty recondite stuff, and the way philosophy is taught at most American universities (e.g. as a branch of literature, or worse as a kind of political indoctrination) is such that a lot of people walk out of philosophy courses less able to reason effectively than when they walked in.

I was, at one time, a mathematician with a strong interest in foundational mathematics and philosophy. Before that I had the good fortune to learn General Semantics from my grandfather at the age of twelve, an experience that gave me extremely effective filters against the kinds of map-vs.-territory confusion endemic in traditional philosophy. The effect of this background (and a lot of thinking) is that I happen to be an expert on the specific philosophical issue of the relationship of mathematical truth to empirical truth, and on some closely related areas in confirmation theory and epistemology.

All this laying-out of credentials is to motivate you to go read The Utility Of Mathematics, an essay on this topic I wrote back in the 1990s. Go read it now…

…and now that you have, I hope it will be clearer why the errors I called out above are in fact errors. In case it isn’t, a bit more education follows.

The “Law” of the Excluded Middle (“For any propositions x, either x is true or not-x is true”) is a specific property of some two-valued logics. Before you can use this “Law” to reason about the empirical world, you have to establish that the essential features of the empirical system you are describing are in fact captured sufficiently well for predictive purposes by a two-valued (or “Aristotelian”) logic.

Usually this will not be the case. We are commonly fooled into thinking that the empirical world obeys Aristotelian logic because the two-value assumption is wired into almost all natural languages. But in most cases Aristotelian logic is as wretched a guide as Aristotelian physics. (Most of General Semantics is techniques for getting rid of Aristotelian mental reflexes conditioned in by our birth
languages.)

My essay should also make clear why trying to draw conclusions straight from Gödel’s Incompleteness Theorem about the limits of human empirical knowledge is doomed. The GIT is about marks on paper, formulae in a formal system. To make it suggest anything about empirical reality, you have to ring in a lot of complicated and fragile assumptions about how exactly pieces of reality behave as models of certain specific and complex formal systems.

In the words of that famous recipe book: first, catch your rabbit. To apply GIT to human empirical reasoning, you’d have to start by showing that Peano arithmetic and classical two-valued predicate logic are a sufficient formalism to capture the entirety of human empirical reasoning.

Three words: Not. Gonna. Happen.

Be careful out there. The relationship between logic and/or formal mathematics (on the one hand) and observable empirical reality (on the other) is a helluva lot thinner and more contingent than most people think. Please consult a qualified expert before putting it under load.

24 thoughts on “Mathematics versus reality

  1. Wow! A lot of things make sense to me after reading your “The Utility of Mathematics”. I would be greatful for pointers to some goods books (or articles) that elaborate further on this subject matter. Thanks a million!

  2. “The Utility of Mathematics” was interesting. Thanks.

    Though the title strikes me as a bit of a(n) (intentional?) misnomer. Most of the essay talks, not about how mathematics is useful, but about the limitations on its usefulness.

    I never assumed, unlike many people, that mathematics can truly describe the universe. But I did think of it as the “language of science”–the basic tool for how most sciences handled the concepts in which they specialize. Your essay reminded me that mathematics is not one “language,” so to speak, but includes whole families of languages many of which are mutually unintelligible to each other.

    Your essay also gave me one important insight. No matter how imperfect mathematics is, attempting to describe phenomena in mathematical terms can reveal flaws in the scientific model underlying the description. To say it in terms closer to General Semantics, being forced to bridge the gap between map and territory can lead to new and more useful models for understanding empirical phenomena.

  3. My “contradiction sensor” is going off.

    ESR sez:
    “Because I’m a rationalist, I’m very skeptical that category 4 is more than an empty set;

    4. Phenomena we will never comprehend because they are intrinsically inscrutable to the human mind.”

    which my natural language neuro processor translates to: “I believe there is nothing man can not understand.”

    And from the (interesting) utility document:

    “There are many phenomenal systems for which no such exact predictive formalism has been found, nor for which one seems likely”

    If you can’t predict it, how can you believe it is scrutable? Unless you believe the mind is so powerful it will predict things when no system man can make is able to, in which case I would say the human mind is therefore inscrutable.

    Anyway, I have a different view of the subject. Mathematics is useful for predicting quantum phenonmenon because we have reached the edge of our capacity to understand things. I saw a great PBS show in which this guy was giving a lecture in relatvity, to a dog. During the rest of the show they dragged out a bunch of physicists who claimed quantum mechanics was just not intuitive, which is to say that humans don’t have the capacity to understand it, and have to rely on external tools in order to work around its edges, much the same way that my dog has learned to “up-sit” to get treats from me.

    Time for the omega step. Read more Clarke.

  4. Predicting the exact outcome of a system in every particular, and understanding how the system works in general, are different

    Eric, what is General Semantics? Googling has revealed nothing that I can understand at all.

  5. > Predicting the exact outcome of a system in every particular, and understanding how the system works in general, are different

    General isn’t good enough, and predict has no connotation of exact to it, actually, more like possibility.

    In fact, predicting doesn’t imply understanding. In my view, it is a lesser, though necessary, achievement on the path to understanding.

  6. Examples of systems which are in principle comprehensible but not exactly predictable are dead easy to find. The classical N-body problem is a good one.

    The Wikipedia page on GS, here, looks like a start.

  7. The Law of the Excluded Middle demonstrated to be philosophically irrational and not describing reality by Indian philosopher Nagarjuna. Nagarjuna shows reality best described by “not true, not false, not both of them and not none of them”.

    Of course it is highly unlikely anyone could build any mathematics based on it, or actually it could be used in any scientific process – or at least as long as our current view on what science is and what is it not remains the same as it is now.

  8. I’ve first had contact with some principles of GS from a professor at my university. I didn’t expected to have classes about this on a Computer Science course, but fortunately I had. II really liked how it changed the way I see all the models we use everyday.

    What scares me is that, without this professor, I could live my whole life without knowing it, and not being able to see lots of map-territory confusions out there.

  9. The difficulty, I believe, comes from trying to assume the vocabulary of the audience. That is, if I as a Christian want to discuss The Great Unknowns of the Universe with a skeptic, I’ll make more progress if I can speak in the skeptic’s language.

    Not that that’s an excuse, just an observation.

    Which, oddly enough, brings us back to C.S. Lewis’s attempts to convince the skeptics with “Mere Christianinty,” the children with “Narnia” and the faithful with “the Screwtape Letters.”

  10. Hey Eric, I lost track of the thread in your first Narnia post, but I did want to clarify something for the record.

    You said:
    > You can repair your argument by establishing
    > that “fully God” and “fully man” are Aristotelian predicates,
    > and this should not actually be difficult. But understand that
    > it does need repair.

    Agreed, absolutely. That’s what I meant by “If ‘god’ and ‘man’ are mutually exclusive” in my original comment. I wasn’t rigorous enough in my phrasing to make it clear that I was using “mutually exclusive” in such strong terms.

    Whether or not “fully god” and “fully man” are actually Aristotelian predicates is of course up for debate. But if they are, then I don’t think I was misusing the excluded middle. :)

    (Actually, I was a philosophy major. But after five years as a programmer, these days I’m all about heuristics and fuzzy logic, so I totally agree with your point in this post.)

  11. Wow. The Wikipedia page on General Semantics seems to have been significantly rewritten between last night and right now. A coincidence or the work of somebody who read this thread? Yay Wikipedia!

  12. ” Examples of systems which are in principle comprehensible but not exactly predictable are dead easy to find. The classical N-body problem is a good one.”

    What’s good about it? You’ve already rejected the base system as being perfect (I concur), but all the N-Body problem says is that there isn’t a formula you can plug values into to get your perfect answer (from the imperfect model).

    I would say NASA is having great success in calculating Comet/Asteroid trajectories etc. And even newton if he had enough years could have made reasonable calculations.

    I maintain the ability to predict is necessary, though not sufficient, to understanding.

  13. Neal, the n-body problem is so computationally intractable that they have to use numerical approximations that diverge over a fairly short timeframe. So even though the problem is perfectly scrutable (in the since that humans can comprehend the equations) you can’t get exact predictions out of them.

  14. Maybe something a little simpler will help. When something doesn’t behave as expected, one is surprised. Why? Because there is some part of the system or its interactions/processes one didn’t understand, that affected the outcome.

    With the n-body problem, you can get arbitrarily precise (though not exact) answers by throwing computations at it. If, as some believe, gravity is discrete exchanges of particles, then eventually you could get even more precise answers, though still not exact.

  15. “… I had the good fortune to learn General Semantics from my grandfather at the age of twelve…”

    Ok, Eric, admit it! You’re an instantiated Heinlein character! :-)

  16. Geno Z. writes: “Ok, Eric, admit it! You’re an instantiated Heinlein character! :-)”

    Well, I certainly hope so. I’ve spent thirty years trying to become exactly that.

  17. Sorry for the delay in response. Busy with work.

    Eric, your argument so far seems to be a straw man, which surprises me because you really ought to know better. I’m inclined to think that you’re misinterpreting my argument, rather than deliberately misdirecting the discussion (as I would assume of a lesser person). Alternately, I may not understand your objection.

    So let’s take a look at Godel’s second incompleteness theorem (from Wikipedia):

    “For any formal theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.”

    I rearrange a little: “No consistent formal theory T [...] includes a statement of its own consistency.” Seems sound.

    I rearrange a little more: “No statement of its own consistency is included in any consistent formal theory T [...].”

    Since it is not in T, it must be in not-T. This also seems obvious.

    I propose that any complete explanation of T must include a statement of T’s consistency. This seems obvious.

    Therefore, a complete explanation of T must make reference to something that is not T.

    And thus I conclude (expanding the notion of T somewhat, because it seems appropriate):

    “No rational phenomenon may be completely explained without reference to other phenomena.”

    Is that not a correct interpretation?

    Since you know better than I do whether Godel’s theorem applies in this way, I’ll accept your determination no matter which way it goes. It doesn’t matter.

    I’ll also accept that most people are bad at correlating mathematics with empirical reality; I know I get a little fuzzy once I go outside of algebra, anyway. So if you think I’ve screwed up there, I’ll take your word for it. It doesn’t matter.

    What matters is whether you are proposing that my extrapolated statement is UNTRUE.

    I say you can’t completely describe the natural without reference to the supernatural, any more than you can completely describe dogs without making reference to something that is neither a dog nor a part of a dog. This seems so completely obvious to me, I can’t help but view any statement to the contrary as abject stupidity. It seems an unassailable point, because any attempt to falsify it would almost certainly involve an attempt to legitimise circular logic.

    But if you have a response to that, I’m willing to listen. I could be wrong.

    Now, I have personally chosen to label that something which is neither natural nor a part of nature “God”. Other people can and do label it whatever they like.

    The real meat of my argument is, no matter what I or anyone else may choose to label this something, there *is* a something – and it is the same something for all of us. We’re just using different words. The danger lies in believing that by using (or avoiding) some specific word, you are changing the nature of reality. This is simply not rational.

    There *is* an empirical fact. If there is a God, then there is a God, and no amount of disbelief will make Him go away. Likewise, if there is no God, there is no God… and no amount of belief will make Him appear.

    I don’t believe that fact can be proven, because it would probably get circular… I don’t think you can reference something outside of nature from inside nature without making it part of nature. Then you get the whole brain-baker of whether it was always part of nature and we just discovered it, or whether we have somehow expanded nature to incorporate something supernatural. But just like there are fundamental mathematical truths that cannot be proven, there are fundamental *empirical* truths that cannot be proven. They may only be observed and accepted.

    You just have to accept what they ARE, not what some ancient book says they *ought* to be. Which is where religion so frequently jumps the shark.

  18. >I say you can’t completely describe the natural without reference to the supernatural, any more than you can completely describe dogs without making reference to something that is neither a dog nor a part of a dog. This seems so completely obvious to me, I can’t help but view any statement to the contrary as abject stupidity. It seems an unassailable point, because any attempt to falsify it would almost certainly involve an attempt to legitimise circular logic.

    Sorry, the fact that you find it obvious doesn’t keep it from being dead wrong.

    What you’ve developed here is just a rehash of the old “first cause” argument, and it’s just as fallacious. To see why, let’s carry your logic forward a step. Your premise is that for any system A, a complete description of phenomena in A requires terms from a system, let’s call it desc(A), which is a proper superset of A. At the first step, A = “natural” phenomena, and you posit desc(A) – A = “supernatural” phenomena.

    OK, but now what happens when I apply your premise to desc(A)? Looks like I need another proper
    superset desc(desc(A)). Ooops. Now I apply your premise again…and again…and again….

    Either there’s an infinite regress here, or your premise is false and there is some largest system that doesn’t require outside terms. If there’s an infinite regress, where’s the division between “natural” and “supernatural”? At step 2? Step 2,317? Step googol? And if your premise is broken, why can’t “natural” be the largest system?

    You’ve been bushwhacked by presumptions built into natural language. This is exactly the sort of error that General Semantics trains people to avoid.

    (You’ve also made a serious error in the way you translate the GIT into an assertion about empirical reasoning, but that one is harder to explain. Go reread my essay on this topic, linked from the original article, for detailed discussion.)

  19. > the fact that you find it obvious doesn’t keep it from being dead wrong.

    That’s okay; I was relying more on the fact that it is *right* to keep it from being wrong.

    > What you’ve developed here is just a rehash of the old “first cause” argument, and it’s just as fallacious.

    Um… no. The fallacy in the first cause argument is a false dilemma, namely the incorrect assertion that either there is no first cause at all or there is a *specific* first cause. This ignores the possibility that there may be a first cause other than the one specified.

    This is also known as the fallacy of the excluded middle, which enlightens me as to your repetition of that law. Perhaps it isn’t that I am myself guilty of the fallacy which violates this law, but that you recognise the trappings of a common argument that *is* guilty of that fallacy?

    > To see why, let’s carry your logic forward a step. [...] now what happens when I apply your premise to desc(A)?

    Now, see, this is a sticky point. When you think about it, desc(A) contains both everything that is natural, and everything that is not natural. That’s everything. So given the statement “everything is X”, is that a proper definition of X or not?

    I believe it could be argued that this is in fact a proper definition of X which does not require external reference. That would represent an exception to the rule which should be identified.

    I believe it could also be argued that it is not a proper definition of X at all. That would imply that desc(A) cannot be properly defined, which intuitively strikes me as the more accurate interpretation.

    But either way, applying this rule to desc(A) is inappropriate. If everything is desc(A), then there can’t be any proper superset of desc(A). Unless, of course, we redefine “everything”.

    > why can’t “natural” be the largest system?

    It can. All you have to do is say “everything is natural” and propose that this is a proper definition. I don’t agree with that position, but it’s certainly a valid position to take.

    My major disagreement with this position is that it amounts to a semantic distinction. You won’t be using “natural” and “supernatural” anymore to refer to the two subsets, but the subsets are still there and still need to be differentiated. Nothing really changes.

    It’s every bit as silly as hanging the label “God” on something and expecting it to make a difference in what that something is. Just because I prefer to believe the first cause was God doesn’t make Noah’s flood a historical fact, and no amount of whining will change that. Facts are facts. When you don’t have them, you can hypothesise and develop pet theories. When you do, you just have to take what you get and live with it.

    > You’ve also made a serious error in the way you translate the GIT into an assertion about empirical reasoning, but that one is harder to explain.

    Since you’re much better at formal mathematics than I am, you’re probably right. However, your essay doesn’t shed any light on it, and you’re offering zero information about what error I’ve made. Even if it *is* hard to explain, just making some vague gestures at it would give me some indication of where to start looking.

  20. Wow, as a relative layman, to mathematics and GIT, I find this argument fascinating. It wouldn’t have occured to me to use a statement about formal mathematical systems as a limiting factor in what we can and can’t understand about the universe. Thinking about it, you’d have to accept that the world, or our description of the world, _is_ such a system to say that?

  21. Eric has a definite point that it’s dangerous to try and make one-to-one correlations between mathematics and reality, for precisely the reason you state: you need to propose that reality is itself a mathematical system, which appears absurd. I believe, however, that we can draw parallels between reality and mathematics in useful ways; Eric’s paper actually accepts this fact. The sticking point we’re hitting is whether THIS parallel is useful or erroneous, and it seems that Eric has misunderstood what that parallel is.

    Essentially, when you start going back to the question of origins, you find a hole. You can get there in various ways; you might use thermodynamics, you might use Godel’s incompleteness theorem, or you might just use Newtonian physics. Eric is complaining that the GIT is inappropriate for this, but no matter how you project backward, you come to a point where it’s simply not possible for anything that has happened to happen. The laws contradict reality; while it’s not possible under those laws for any of this to have happened, clearly those things DID happen, so the laws are obviously wrong.

    When logical analysis produces an incorrect result, there are two places the error may reside: it may reside in your precepts or in your process. The scientific view is that we have used an incorrect process, and that we need to correct the laws we have used; the religious view is that we are missing a precept, and that we need to add God to the list. Neither proposition is superior, and both may be correct, but at least one of them *must* be correct.

    Both proposals can be abused, and religion in particular abuses their side. The abuse stems from a failure to recognise that while *something* needs to change, this does not give you a license to change anything and everything just because you would like it better that way – you must change only what needs to be changed, and only so much as it needs to change. Overzealous changes are what make the God hypothesis look so ridiculous most of the time.

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