Essay Abstract

Essential tensions remain in our understanding of the reasons underlying the striking success achieved in science by applying mathematics. Wigner and many likeminded scientists and philosophers conclude that this success is a miracle, a ``wonderful gift which we neither deserve nor understand.'' This essay seeks to dissipate that aura of mystery and bring the factors underlying the success of applied mathematics into the fold of scientific rationality.

Author Bio

Nic is an assistant professor of philosophy and member of the center for scientific computing at Simon Fraser University, Canada. His work focuses on philosophy of science, philosophy of mathematics, logic, and numerical analysis.

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Dear Nicolas Fillion,

A very interesting a readable essay. Your comment "A model or theory that contained "the whole truth and nothing but the truth" would quite simply be too true to be good." hit the exact theme I started my essay with.

As an aside, from your comment "Let me first illustrate the point with a modern approach to understanding the impact of computational error as it occurs in computer simulations.", I wonder if you have consider cumulative errors. The error that makes many computer simulations so difficult is the compounding errors of using the output of one calculation as the input of the next calculation, ie. errors that result from f(f(f(x))). Modelling the "trajectory of Uranus" or the expect path of a space craft under the influence of several masses (where they cannot be solved as a n-body problem) require an amazing level of accuracy for the initial conditions and tend to diverge pretty rapidly.

My argument concentrated more on the "usefulness" of a model rather then the "accuracy" of the model, but I certainly enjoyed detailed reading of your essay. Hope you get a chance to have a look at mine here.

Best of luck in the contest.

Regards,

Ed Unverricht

    Dear Ed (if I may),

    Thanks for reading my essay and for your comment and question! I will certainly return the favour some time next week as I'll have a bit more time by then. Your worry about cumulative errors is very relevant, and I have have a long story to tell about that! In fact, I've come to the philosophical argument included in my essay mostly by thinking about accumulating numerical error! I tried to keep the essay as nontechnical as possible, but I may have gone a bit too far in that direction.

    I take it that there are two dimensions to accumulating errors that are relevant here. The first is, as suggested, essentially numerical. These days, numerical solvers have become quite good at controlling accumulating numerical error. Part of it is that we use higher-order methods (say, for ODEs and PDEs) since we have more computational powers, and part of it is that we've become better at designing automated adaptive discretization methods that lead to more accurate results. But also, numerical interpolation methods have become really good too, so we can interpolate discrete solutions to find what I call the residual in the paper, and we can do this "live" to gauge the numerical error as we go. If you'd like technical details, please email me (nfillion@sfu.ca) and I'll be happy to send you a textbook chapter I wrote on this!

    The second part of it is that whether accumulating numerical error will have a big impact on the quality of computed solutions has to do with sensitivity under perturbations---and here there's no reason not to consider numerical error as a perturbation, since very stable systems will quickly damp it while sensitive systems will magnify it. In the approach to numerical analysis I favour---the so-called backward-error analysis---the sensitivity is measured by a quantity called the condition number. And here, we can address the question you ask about composition f(f(f(x))). There is a nice theorem by Deuflhard about the submultiplicativity of the condition numbers under composition. So, if all functions are only moderately sensitive, the composed function will also be. Of course, when there's high sensitivity (as e.g. in chaotic systems, or near singularities), no algorithms will save you, however accurate it is. But this, at least, is something we can reliably detect thanks to residual analysis. And, if the stability of a problem is such that is amplifies numerical errors, then it would also amplify any other kinds of perturbations. So, as long as our numerical error is less than what comes from our modelling practice or from the system's environment, the algorithm isn't to blame for predictive error!

    Finally, to address your last point, I think usefulness and accuracy are closely related to each other. And I think there's a principled mathematical reason for that, which goes far beyond pragmatic reassurance. I'll comment about that later in your thread after I read your paper!

    Cheers, and talk to you next week!

    Nic

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    • Post #4

    Dear Nicolas,

    Like you, I see no mystery here. Math is Math and Physics should be too!

    I have argued in past and present essays all Basic Laws of Physics are Mathematical Truisms. It is our physical interpretations of these mathematical truths (based on the 'physical view' we have) that "trick" us into believing non-sense. We do not need to know the "magician's trick", however, to know the magic is not real!

    There is something perverse and corrupting when we start believing in "magic tricks" like "time travel". Ultimately, our lives get distorted and confused. This idea is encapsulated in the following principle:

    The Anthropocentric Principle : Our Knowledge of the Universe is such as to make Life possible.

    "The 'man-made' Universe"

    Constantinos

      The author focuses on the rational reconstruction of scientific theories rather than on an analysis of particular branches of physics. This abstract approach has a distinct advantage. It is possible to use the sork that philosophers of science have done on modelling, testing theories, perturbation, and epistemology. I think that the generaal position presenrted at thks level is quite reasonable. The disadvantage to this abstract approach is that it does not allow a discussion of differences between the role of mathematics in classical physics, quantum physics, and general relativity. These are significant.

      Ed. MacKinnon

        Dear Constantinos (if I may),

        Thanks for taking the time to read my essay, and I will make sure to return the favour next week as I'll have a bit more time then, and I'll write some comment. I fully agree with you concerning the "magic tricks". I almost included a quote attributed to Einstein, but I couldn't trace it to verify the context in which it was said (if indeed it was said): "There are only two ways to live your life, as if everything is a miracle, or as if nothing is a miracle." (That's from memory, so the phrasing might be off a bit.) Concerning time travel, however, I want to say that it's not entirely to be associated with magic tricks, as there's a meaningful physical notion associated with it, as you probably know, namely the closed timeline curves. But that's for another story! :)

        Best,

        Nic

        Dear Ed,

        Thanks for your comment, and your criticism is very well-taken. In fact, I was myself hesitating about which approach to utilize, as I thought that in such a short paper it wouldn't really be possible to use both. As you point out, it has its strengths and weaknesses, and in any case I didn't see a way to present things without the associated shortcomings. And it is certainly true that much is different between classical physics, quantum physics, and GR, but at the same time some key methods are shared and some of their methodological underpinnings are also shared. That's what I tried to emphasize, and hopefully I haven't failed too miserably! :)

        I look forward to reading your essay! Best,

        Nic

        Nicolas, you write

        "Concerning time travel, however, I want to say that it's not entirely to be associated with magic tricks, as there's a meaningful physical notion associated with it, as you probably know, namely the closed timeline curves."

        What is a "closed timeline curve"?

        The problem always is, we can always give meaning to anything we give thought to. But the question is and always should be, "does it make sense". That is, does it agree with our "sense experience". The proverbial "man in the street" is the "expert" here! Theoretical physicists, no matter how clever they might be, often get it wrong. In fact the more clever, the more wrong!

        I see you are a 'philosopher of science'. Please don't tell me you believe in the "spacetime continuum"!

        Oh! I get it! All "tongue in check"!

        Constantinos

        22 days later

        Dear Nic,

        I thought this was one of the better attempts to deal with this topic. However, at the end I was entirely unsatisfied. In the first place, there is not much evidence to justify the claim of the miracle like effectiveness of mathematics in the first place. You fail to mention Aristotle who I think explains the problem very well in that math deals with quantity in terms of numbers. Nature can be modeled in terms of measurement of quantity. That is all there is to it basically. Physics is making measurements. Once measured in terms of quantity, nature is amenable to math modeling. You don't deal with the counter examples of how physics does a bad job with the math. We have special relativity riddled with paradoxes which are caused by math mistakes yet such is the power of the math over the mind of physics that they can not see that the math is entirely false, while the math errors scream that they are mistaken. They don't see a problem and think it is beautiful and perfect. There is a human problem here. Another problem is unipolar induction where physics can calculate the current but the physics underlying the math result is entirely without justification. In other words they don't have a clue why the math seems to work as it does. You don't address these very real and pressing problems of human failure in science.

        You've mentioned it very rightly, "We" human minds cannot understand this miracle. It is beyond comparison.

        Great!

        - Sincerely,

        Miss. Sujatha Jagannathan

        Dear Nicolas,

        I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

        All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

        Joe Fisher

        3 years later

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        • Post #12

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