VI Arnold On Models

Thanks to reader Victor Domin for pointing us to the paper “On teaching mathematics” by V.I. Arnold. Here’s an extended quote on models (do read the whole thing).

At this point a special technique has been developed in mathematics. This technique, when applied to the real world, is sometimes useful, but can sometimes also lead to self-deception. This technique is called modelling. When constructing a model, the following idealisation is made: certain facts which are only known with a certain degree of probability or with a certain degree of accuracy, are considered to be “absolutely” correct and are accepted as “axioms”. The sense of this “absoluteness” lies precisely in the fact that we allow ourselves to use these “facts” according to the rules of formal logic, in the process declaring as “theorems” all that we can derive from them.

This follows along our recent weight of evidence post, and the crucial distinction (discussed in loving detail here) between local and universal or necessary truths. Local truths are when the evidence used to prove the truth of a proposition is itself not true or known to be true. Universal or necessary truths proof propositions based on evidence that is itself a necessary truth. Mathematical, logical, metaphysical, and revelational truths fall into this category.

Probabilities or propositions deduced from ad hoc models, or models which are tentative in the sense Arnold states, are true. But they are local truths. That doesn’t mean they aren’t useful, but usefulness is not the same as truthfulness. Mistaking these things accounts for nitwits running around screeching “Denier!” Scidolatry, of course, is to blame.

It is obvious that in any real-life activity it is impossible to wholly rely on such deductions. The reason is at least that the parameters of the studied phenomena are never known absolutely exactly and a small change in parameters (for example, the initial conditions of a process) can totally change the result. Say, for this reason a reliable long-term weather forecast is impossible and will remain impossible, no matter how much we develop computers and devices which record initial conditions.

Denier!

This sensitivity to initial conditions has the official name of chaos theory. Giving it a name does not take away from Arnold’s correct conclusion, however. There are some tricks to not improve guesses from these models not in an absolute sense, but to better nail the uncertainty. These are to be admired. But they can’t do what can’t be done.

In exactly the same way a small change in axioms (of which we cannot be completely sure) is capable, generally speaking, of leading to completely different conclusions than those that are obtained from theorems which have been deduced from the accepted axioms. The longer and fancier is the chain of deductions (“proofs”), the less reliable is the final result.

This exactly fits into the weight of probability discussion. There is only probability, or logic, but it makes sense to discuss how influential evidence is.

It’s not that the length of the chain of deductions matters per se, but the more crap you stick on the right hand side of the logic/probability equation, the greater the chance (based on experience) one of those premises won’t itself be true. One weak link is all takes to turn a necessary truth into local truth, i.e. a falsity.

Complex models are rarely useful (unless for those writing their dissertations).

The mathematical technique of modelling consists of ignoring this trouble and speaking about your deductive model in such a way as if it coincided with reality. The fact that this path, which is obviously incorrect from the point of view of natural science, often leads to useful results in physics is called “the inconceivable effectiveness of mathematics in natural sciences” (or “the Wigner principle”).

That has to be because the universe is designed.

“Dude, that makes you a creationist.”

Yep. It do. But consider that most people misunderstand this, thinking creation applies only to the physical word, which supposedly “evolution” and “laws of physics” own. Creation does apply to that, for the “laws” of physics (which is a metaphor) must themselves have been created. They cannot have come from nowhere.

Forget that and think of this: creation also applies to mathematics itself. There must be a reason math is the way it is. Math could not have created itself. Math has a cause, all those theorems have a cause.

So it’s no wonder physics often aligns with mathematics the closer physics gets to its fundaments. This is because both must have the same creator.

There is no escaping this conclusion.

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23 Thoughts

  1. “the more crap you stick on the right hand side of the logic/probability equation, the greater the chance (based on experience) one of those premises won’t itself be true.” – what William of Ockham was trying to say.

  2. “Universal or necessary truths proof propositions based on evidence that is itself a necessary truth.”

    Isn’t this refuted by the analytic philosopher Saul Kripke’s famous a posteriori necessity–i.e., necessary truths discoverable only a posterior? E.g., “H2o is water” is an a posteriori necessary truth.

  3. James,

    No. To see that, write Y = “H20 is water” and then find the X = “…” which allows that deduction.

  4. I don’t think that you bods in your ivory towers are ever going to be able to make a coherent, congruent case for anything but more fantastic speculations unless you get to make the distinctions between hypothesis (whether you call it “modelling”, “local truths”, “theory”, or anything else) and valid premises. To plug an “hypothetical factor” into an equation to “prove” or even justify an hypothesis is an exercise in duplicity that can only be described as a logical “no no” usually called “circular reasoning”.

  5. Oldavid,
    In Briggs’ baby book about statistics, see ancient posts he explains how it is premises which people argue about. Auditors argue about the maths.
    It seems that there is circularity in any argument where the starting point isn’t agreed upon.
    What you seem to complain about is that everybody’s argument doesn’t start with, “given that God exists.”
    Yet nobody can know that premise for sure.

  6. Ah! y’ dear ole thing.
    A proper premise is inarguable from any reasonable (the science of logic) stand.

    Sure, you might argue the sophistic point that “you can’t prove that God exists” (as in some empirical “test”) but the alternative that Nothing turns itself into Everything is self contradictory and thus absurd.

    No, dear, my premise is not “given that God exists.” It is that things exist and they require a cause that is not themselves. The silly “turtles all the way down” might be conveniently assumed by narcissistic ideologues but it doesn’t make any sense. There must be an “uncaused first cause”.

  7. I am commenting to tick the box and receive e-mail notifications, in case someone says something interesting about the Fifth Way.

  8. Oldavid,
    I told you it’s premises people argue about.
    However, when you respond, you like to have your own set of premises for your opponent to kick things off.
    Today, I am singing Rivers Of Babylon. Up REALLY LOUD.
    There’s always the Flood, Take That you old avid you.

    I believe God exists but I also know it is not provable.
    If you’r sure, that’s what’s making you angry. God exists *if* he didn’t make it patently obvious and probable, with good reasons that need not to be explained.

  9. provable, not probable.
    (Monkey still in the middle of comment box.)
    Proved only from a premise that’s agreed upon.
    Enough!

  10. Dav,
    After fifteen lines of the pdf article, it’s just water, water everywhere.
    I preferred the cat knocking on doors. I wonder what he thinks about water.
    Luckily, he can never tell us.

  11. Good on you, Thiago, but the Ivory Tower is issuing goads and insults in today’s article so I might have to duck over there and give this one a bit of a miss.

  12. Cop-out, Joy.

    I used the wrong word above. Valid premises are indisputable according to the science of logic by their very nature. The only premises that are open to argument are those based in speculation or “hypothesis”.

    A truly valid, sure and certain, premise is a proposition to which the only alternative is its contrary and which contrary is self contradictory and thus absurd; the most important of which (in today’s fantasies) is the self evident “a thing that does not exist cannot cause itself to exist” or, more succinctly, “Nothing cannot turn itself into anything”.

  13. @ Oldavid,

    “It is that things exist and they require a cause that is not themselves […] There must be an “uncaused first cause”.”

    If everything must have a cause, then God must have a cause. If God doesn’t have a cause, then everything must not have a cause. Your argument is self-contradictory and therefore invalid.

    “Nothing cannot turn itself into anything.”

    According to what law of logic can you attribute qualities to a state (Nothing) which has no qualities, by definition?

  14. You silly fellow! Is there nothing so irrational that you can’t claim it to defend your ideology?

    [quote] If everything must have a cause, then God must have a cause. If God doesn’t have a cause, then everything must not have a cause. Your argument is self-contradictory and therefore invalid.If everything must have a cause, then God must have a cause. If God doesn’t have a cause, then everything must not have a cause. Your argument is self-contradictory and therefore invalid.[/quote]

    Which part of “uncaused first cause” do you not understand?

    [quote]According to what law of logic can you attribute qualities to a state (Nothing) which has no qualities, by definition?[/quote]

    The point is that Nothing can’t do anything. It’s only you and your lot that claim that Nothing turns itself into Everything.

    We ordinary “unenlightened by impossible b.s.” people can easily understand that the perfection of everything can cause, or produce, anything less than itself.

  15. @ Oldavid,

    “Which part of “uncaused first cause” do you not understand?”

    The part which proves that this uncaused cause can’t be the cosmos, the part which proves that the cosmos requires a cause, and the part which proves an uncaused cause is possible in the first place. That should do for starters.

    “The point is that Nothing can’t do anything. It’s only you and your lot that claim that Nothing turns itself into Everything.”

    If Nothing has no qualities, it isn’t possible to make any statements about what can or cannot happen given such a state. I do not claim that Nothing turns itself into Everything, I suggest that the Cosmos (Everything) may be a self-contained structure.

    Having said that, I actually don’t think it’s possible to reach definitive conclusions about things which are so far removed from human experience. I would have thought it more likely that the “real” answer to why the Cosmos exists would be something which humans can’t understand at all.

  16. Copout, Fishy.
    The “Cosmos” is changeable and changing… it cannot be eternal because it’s never what it was or will be.

  17. My comments must be swiftly executed and brief because it seems that there is a malicious censor out there.

  18. (Olddavid, if you’re still having trouble, email the comment you’re trying to post and tell me what you’re seeing. Thanks.)

  19. @ Oldavid,

    “The “Cosmos” is changeable and changing… it cannot be eternal because it’s never what it was or will be.”

    I didn’t say that the cosmos is eternal, and I’m not clear how this changes anything. If by ‘eternal’, you mean ‘outside of time’, then I would have thought that from ‘outside of time’, the cosmos would appear to be a static object which ‘just exists’. AFAIK, it would also appear to be a static object from the point of view of an observer travelling at the speed of light.

    Also, God must change, otherwise how can he do anything? If he can change but still be eternal, then why can’t the cosmos do the same?

  20. Your “outside of time” seems to suggest that “time” is some kind of “thing” or “stuff” like a soothsayer’s crystal ball in which “things” just happen… randomly. My consistent claim is that time is only a succession of events which we “measure” by comparing some reasonably regular succession of events to other successions of events. It’s not “stuff”. Any succession must have a “before and after” that has to start somewhere.

    Yair, I know about “Mobius rings” and “Klein bottles” that try to befuddle reasonable thought with vague insinuations that beginnings and contents are irrelevant.

    Your observation that: “Also, God must change, otherwise how can he do anything? ” is a good one. It’s something that I almost brought up in another conversation but pulled back thinking that it was an unnecessary complication in the context.

    We need to make a distinction between “eternal” as in no beginning and no end and “eternal” as a beginning without end… an endless continuation of events as in Heaven and Hell.

    The first kind of eternity admits no kind of change or “movement” as everything is eternally the present, never changing. It has been described by great thinkers as “God lives in an eternal NOW”. Everything always exists in the Author of Everything but Himself. So how can He “interfere” in things of a temporal nature without “doing something” Himself? The answer is that just as He created (from eternity) all the material stuff that we recognise as the Universe He created all the intelligent powers that control it. (Usually called Angels “down here”; some of which refused their commission).

    Anyhow, this is getting too long. I’m old and tired. Feel free to quibble on any clumsily presented points.

  21. @ Oldavid,

    “Anyhow, this is getting too long. I’m old and tired. Feel free to quibble on any clumsily presented points.”

    Fair enough, I’m pretty old and tired myself. Have a good day.

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