The Law of Unreason

Not much time to post today, so I thought I’d just put up a couple of nice little quotes about the Central Limit Theorem. In case you don’t know it, this theorem explains why so many phenomena result in measurable things whose frequencies of occurrence can be described by the Normal (Gaussian) distribution, with its characteristic Bell-shaped curve. I’ve already mentioned the role that various astronomers played in the development of this bit of mathematics, so I won’t repeat the story in this post.

In fact I was asked to prove the theorem during my PhD viva, and struggled to remember how to do it, but it’s such an important thing that it was quite reasonable for my examiners  to ask the question and quite reasonable for them to have expected me to answer it! If you want to know how to do it, then I’ll give you a hint: it involves a Fourier transform!

Any of you who took a peep at Joan Magueijo’s lecture that I posted about yesterday will know that the title of his talk was Anarchy and Physical Laws. The main issue he addressed was whether the existence of laws of physics requires that the Universe must have been designed or whether mathematical regularities could somehow emerge from a state of lawlessness. Why the Universe is lawful is of course one of the greatest mysteries of all, and one that, for some at least, transcends science and crosses over into the realm of theology.

In my little address at the end of Joao’s talk I drew an analogy with the Central Limit Theorem which is an example of an emergent mathematical law that describes situations which are apparently extremely chaotic. I just wanted to make the point that there are well-known examples of such things, even if the audience were sceptical about applying such notions to the entire Universe.

The quotation I picked was this one from Sir Francis Galton:

I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the “Law of Frequency of Error”. The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along

However, it is worth remembering also that not everything has a normal distribution: the central limit theorem requires linear, additive behaviour of the variables involved. I posted about an example where this is not the case here. Theorists love to make the Gaussian assumption when dealing with phenomena that they want to model with stochastic processes because these make many calculations tractable that otherwise would be too difficult. In cosmology, for example, we usually assume that the primordial density perturbations that seeded the formation of large-scale structure obeyed Gaussian statistics. Observers and experimentalists frequently assume Gaussian measurement errors in order to apply off-the-shelf statistical methods to their results. Often nature is kind to us but every now and again we find anomalies that are inconsistent with the normal distribution. Those exceptions usually lead to clues that something interesting is going on that violates the terms of the Central Limit Theorem. There are inklings that this may be the case in cosmology.

So to balance Galton’s remarks, I add this quote by Gabriel Lippmann which I’ve taken the liberty of translating from the original French.

Everyone believes in the [normal] law of errors: the mathematicians, because they think it is an experimental fact; and the experimenters, because they suppose it is a theorem of mathematics

There are more things in heaven and earth than are described by the Gaussian distribution!

23 Responses to “The Law of Unreason”

  1. Anton Garrett Says:

    Galton and Lippmann are reconciled upon realising that probability is epistemological, not ontological.

    Does “the existence of laws of physics require that the Universe ha[s] been designed”? With the word “require” included in that sentence, I don’t know. But I challenge anybody to find a better explanation for why the laws of physics are *beautiful*. Don’t tell me that there is a survival edge in seeing beauty in Maxwell’s equations of electromagnetism (which were known only in the last 140 years out of many millions, and then only to a tiny minority of persons).

    Anton

  2. telescoper Says:

    Anton,

    I agree that Maxwell’s equations, the Dirac equation, the Einstein equations and so on are beautiful but I don’t really know what is meant by that. Perhaps its primarily because they’re compact that we are impressed: they capture so much using so little ink.

    Peter

  3. Anton Garrett Says:

    Peter,

    To adapt a quote from David Stove, you have to be very highly educated indeed not to know what beauty is…

    Anton

  4. telescoper Says:

    I like that quote, but I suspect it wasn’t intended to be a compliment!

  5. Anton Garrett Says:

    The exact quote is: “You have to be very learned indeed to find things as hard to understand as Kuhn does.” (From David Stove’s essay “Karl Popper and the Jazz Age.” I should add that trad jazz, at least, was a point of difference between David and myself.)

    Anton

  6. Anton Garrett Says:

    You might win a Nobel Prize!

  7. I don’t know. But I challenge anybody to find a better explanation for why the laws of physics are *beautiful*.

    I don’t know about “beautiful”, but I can tell you exactly why these highly practical laws are the way that they are. Does that count, and would you believe it even if made perfect cosmological sense?

    I doubt it.

  8. telescoper Says:

    Try me.

  9. Okay, the universe is configured to maximize energy via it’s observed extreme low entropy structuring, which enables more energy to be used to do work before it goes inert from heat death.

    It’s an energy conservation law that works like an optimization problem, but the fate of the universe is far from what is assumed.

  10. Like I said, perfect sense… and disbelief… 😉

  11. Anton Garrett Says:

    Peter,

    To be a bit more constructive, I am suggesting that beauty is as objective a concept as anything that physicists play with, but it lies in a different space.

    Anton

  12. telescoper Says:

    Anton,

    I don’t agree that beauty is objective, or at least I think not all forms of beauty are objective. Brevity certainly is though, as long as you used the right coordinates.

    Peter

  13. telescoper Says:

    Ricky,

    I don’t think you’re due for a trip to Stockholm just yet.

    In the meantime perhaps you could explain why, if the Universe maximises its energy, why it doesn’t have more energy than we find?

    Peter

  14. Anton Garrett Says:

    Peter: You’re up early! I agree that not all forms of beauty are objective, for cultural counter-examples exist. Perhaps I should have said that beauty exists as a thing-in-itself. Beauty in a landscape has nothing to do with economy/brevity, though – Anton

  15. When I say that it maximizes energy, I mean that it is more-efficiently configured to waste less energy than a more rapidly expanding universe would. This is quite obviously an energy conservation law.

    Energy is also more uniformly disseminated in this manner… again… an obvious energy conservation law if you don’t automatically assume that the unproven speculations of the cutting edge are anything more than that without the complete theory that they don’t have to justify these theoretical conjectures.

  16. Anton Garrett Says:

    Ricky: You clearly mean a variational principle. In what way does this differ from variational principles from which general relativity can be derived? Anton

  17. More of a maximum action principle, or a maximum possible entropy principle, I should think.

    • telescoper Says:

      Is there somewhere interested people can learn about the rigorous mathematical basis of your theory and compare its quantitative predictions with the extensive observational constraints now available in cosmology?

  18. I have trouble understanding how this observation is a theory, but you can click on my name and find some helpful links at the bottom of that page. That stuff is more involved than the observation that we have been talking about, and I have been told by experimentalists and astronomers that I need to write a paper. Theorists typically don’t like to give much credence to my point, because they are so sure that they are already on the right track, but they’ve also never been able to give me a simple refutation either, and will often admit that my claims are correct, at least, in context with the cosmological model that I believe will ultimately win the day and everything else.

    I would need help writing and submitting anything, but nobody that I’ve approached wants to stick their necks out to co-author with me.

    Things might change though if the LHC comes up empty handed and particle theory fails at the Higgs scale.

    In the mean time…. I have fun using to resolve all these “hard” problems of science.

    • telescoper Says:

      Ricky

      I did click your name but didn’t find any actual predictions on your site, or indeed any calculations that could be used to make a comparison with data.

      Anyway, if you know all the answers why would you need a co-author? Write it up, submit it to a journal and see what the referees say.

      Peter

  19. Peter, I said that there were some helpful links at the bottom of that page and I can’t handle the format or I would do it myself. I’ve tried… I’ve tried… I’ve tried.

    The following link is very helpful in understanding this as it applies to Einstein’s cosmological model, and there are several more relevant followup statements that are linked at the bottom that I made (with increasing boldness and earnest as the lack of response became deafening), to the theorists in the moderated research group that I used to frequent.

    http://www.lns.cornell.edu/spr/2006-02/msg0073320.html

    The difference with this model is that tension between the vacuum and ordinary matter increases as the universe expands, so the second law ant the arrow of time are inevitably preserved when growing tension compromises the forces that bind the universe and we have another big bang.

    Energy is perpetually conserved in this manner, but heat death isn’t a feature of it, although maximum energy is the ultimate unattainable “goal” of this evolutionary process.

    This is how it applies to the negative mass absurdity:
    http://www.lns.cornell.edu/spr/2005-06/msg0069755.html

    A few equations and a long explanation:
    http://dorigo.wordpress.com/2007/10/18/

  20. I have a comment with links in moderation.

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