Sylvia,

A very stimulating essay. I differ on the selection process. For 95% of human existence humans, like their hominid predecessors lived in small Hunter-Gatherer groups. Here Darwinian evolution was operative in a way that it is not now operative. Most individuals did not live long enough to reproduce. This inculcated the dispositions needed for survival, willingness to sacrifice for the sake of the group and some form of altruism within the group. Mathematics could develop only when a more sedentary life style developed. In the very different universe you consider I doubt if hominids could have survived.

Edward MacKinnon

9 days later

In accordance with your work, I would like to get answers from you :

If MAN is born only because mathematical bound how the physics came in this picture?

If everything are here in numbers with no texture and color which inanimate and animate would live?

Regards,

Miss. Sujatha Jagannathan

Dear Sylvia, Good job on this essay. It is a better essay that the ones I have read so far. However, as I read, I became less and less convinced of you thesis and at the end felt unsatisfied. You are on the right track regarding the selection effect. In my opinion this issue is not complicated. We invent the universe as we imagine it to be. That process involves mathematical imagination. A lot of that is not effective as being persuasive in terms of explanation. Yet some of the imagination works well as in classical physics. But that is based upon infinitesimal calculus. As you show that is not correct mathematics. So we invent some new mathematics that fixes the problem. I think the real question ought to be how is it possible for wrong mathematics to be so effective in physical science. That is because as we learn we discover that the math used before is wrong in many ways. So we are constantly reinventing the math to fit the universe. I think this is because humans want a mathematical universe, not because the universe is mathematical.

12 days later

Sylvia,

Time grows short, so I am revisiting essays I've read to assure I've rated them. I find that I rated yours on 3/13, rating it as one I could immediately relate to. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345.

Jim

    7 days later

    Hello. In your essay you wondered how "to imagine a world that would defy our mathematical prowess", in fact you mean : that would be anything else than highly mathematical in the way it was found.

    In my essay I gave some precise expressions of how remarkably mathematical the universe is, what does this precisely mean. Moreover I do hold that, aside the remarkably mathematical aspects, there are also non-mathematical aspects, and I gave a precise sense to this claim.

    Other descriptions of the remarkably mathematical character of the universe, showing that it really means much more than tautological consequences of the human ability to do mathematics (which you unimaginatively assume to be all what it might mean), are given in other essays: those of Peter Woit, Alexey and Lev Burov and Martin Seltmann.

    Sylvia,

    Here are a few points for you to consider.

    There are patterns in nature, but only reproducible patterns can be modeled with mathematics.

    When there are reproducible patterns that we cannot model with our mathematics, we invent new mathematics to do the job. That is where most math comes from and why it works so well.

    You say that probability and statistics informs statistical mechanics. I submit that the opposite is closer to the truth. On this point see

    "Probability Theory, the logic of science" by E.T. Jaynes

    -- one of the great books of the twentieth century!

    .....David

      Dear Sylvia,

      I enjoyed reading your essay. Indeed, we understand the world, at least as much as we do, because we are its children, and this may explain why our math is effective in our physics. The four elements, in particular selection, support very well your thesis, and the example of non-standard analysis is well chosen. The style is eloquent, pleasant and funny just as much as it should be (I loved the disclaimer). I am very glad I didn't miss your essay!

      Best wishes,

      Cristi Stoica

        Oups: "it is not nature, it is scientists that are simple".

        Thanks Sylvia, I agree with most you are writing, a truly Darwinian essay.

        Michel

          Dear David Hestenes,

          I am grateful for your comment.

          I am very fond of the work of E.T. Jaynes! For instance, in my course on the philosophy of probability theory, I teach his analysis of the chord paradox. But I take your point: my statement about statistical mechanics was sloppy at best.

          Best wishes,

          Sylvia

          Dear Christi,

          Thank you for your positive comment!

          I had already read your text, but waited for the deadline to submit all my comments in one batch: it is now in your forum.

          Best wishes,

          Sylvia

          Dear Michel,

          Thanks for your kind comment. I also wrote a reply to yours.

          Best wishes,

          Sylvia

          Dear Jim,

          Thank you for both of your comments. I finally got round to reading and commenting on your essay as well.

          Best wishes,

          Sylvia

          Dear Sylvia,

          Your paper really needs more comments than I was able to deliver in such a short time left to us. I love your disclaimer. But I also enjoy concepts as the multiverse, the maxiverse, the megaverse, the babyverse, the monsterverse., everything chaotic, exotic, sporadic, anomalous probability distributions... With them it seems that we are are closer to the complexity of the world internal or external to us. Thank you for your read of my dialogue and your positive appreciation

          Best wishes,

          Michel

          Hi Sylvia--

          Your essay is superb! It is both creatively crafted and well argued. Moreover, I agree with both your main argument and your supporting elements. Of course, my admiration for your essay may thus be a product of mere "selection bias".

          Speaking of selection bias, I whole-heartedly subscribe to your point-of-view that we are blind to "ubiquitous failures" when assessing the efficacy of mathematics. Before reading your essay today, I had a very polite back-and-forth with Cristi Stoica on this very subject regarding his essay. I made similar comments on Lee Smolin's threads. Amusingly, before settling on the essay that I posted, I had considered doing an essay for this contest that forthrightly addressed the many ways in which mathematics fails to efficaciously describe the physical world. I had tentatively entitled the piece, "On the unreasonable ineffectiveness of mathematics in the natural sciences". Given Section 2.3 of your essay, I'm glad that I moved in another direction.

          The only area in which we seem to disagree is on the subject of the "unthinkable", especially with respect to randomness. For starters, I bristle at such phrases as "totally random" or "pure randomness". That's like saying that a man is "totally dead" or that a women has a "pure pregnancy". Something (an event) is either random or it is not. Statistical distributions, such as the Gaussian, are an entirely different kettle of fish (which, I think, was the point you were trying to make).

          Furthermore, I do not agree that "unthinkable" worlds are so unthinkable. Here's one: A world of "white noise" in which all variables have an amplitude greater than your field-of-view. Sure, we can define "white noise" from the outside. But living it, on the inside, would be another matter. You'd probably be wiped out in the ensuing chaos before you could even voice the thought, "Wow, this world may be based on white ... argh!'. Here's another: Your "Daliesque" world, except one in which the "laws of nature" change randomly and drastically (and not as in, say, a Gaussian way with small-scale random effects) and do so at random times. There would be no meta-regularities. Here's a third: A world in which there were no discernable "events" or "objects"; there's just amorphous "stuff".

          We are fortunate to live in a Universe that consistently displays regularities. This enables us to engage in reliable pattern recognition and, hence, algorithmic compression. Which allows us to do mathematics. Which then allows us to do mathematical physics. What a beautiful selection effect!

          For reasons that escape me, your essay is terribly under-rated. I shall now try to adjust that.

          Very best regards,

          Bill.

          Dear Bill,

          Thank you for your detailed and constructive feedback! It is sort of reassuring that you considered similar ideas for your essay. And even nicer that -in the end- we didn't come to this party wearing an identical outfit. ;)

          I do think that it makes sense to speak of "totally random" as opposed to "partially random". I use the term "totally random" in situations where there are equal probabilities pertaining to two or more possible outcomes and "partially random" when there are non-equal probabilities. On this view, pure randomness hits a strange equilibrium between knowledge and uncertainty: it does not represent total lack of knowledge (because then we wouldn't even know what the possible outcomes are*), yet it does represent maximal uncertainty regarding which of the possible outcomes will be realized at the next instance of the relevant process. Still, at the group level, we do know a lot about random events (both for total and partial randomness). I do think that this sense of total randomness is an idealization, and of course we can never demonstrate something to be totally random (or even falsify it: a fair coin may keep coming up heads for however long we try, it's just exceedingly unlikely).

          [*: unless the randomness is implemented at a higher level, as in a world randomly switching between laws, as you proposed.]

          Thank you for your vote: it made my day. :)

          Best wishes,

          Sylvia

          2 months later

          Dear Sylvia,

          I read your essay with great interest when it was posted, but I didn't comment on it while the contest was underway: I had read the disclaimer at the end of your essay, "No parallel universes were postulated during the writing of this essay", and since my own essay postulates an infinite ensemble of parallel universes and multiverses, I preferred to keep a low profile! ;)

          I think you did a great job answering Wigner's question about the usefulness of mathematics in physics:

          "[Mathematics] is a form of human reasoning - the most sophisticated of its kind. When this reasoning is combined with empirical facts, we should not be perplexed that - on occasions - this allows us to effectively describe and even predict features of the natural world. The fact that our reasoning can be applied successfully to this aim is precisely why the traits that enable us to achieve this were selected in biological evolution."

          You are quite right when you say that we need to keep in mind that ""[A]ll our science, measured against reality, is primitive and childlike" and that "it is not nature, it is scientists that are simple". I agree with you when you say that

          "[W]e are creatures that evolved within this Universe, and [...] our pattern finding abilities are selected by this very environment. [...] I think that we throw dust in our own eyes if we do not take into account to which high degree we - as a biological species, including our cognitive abilities that allow us to develop mathematics - have been selected by this reality."

          It is obvious that the mathematics that has been discovered and is being studied by human mathematicians is a product of our cognitive abilities, and is shaped and limited by our biology. But, in my view, it is only a subset of "capital-M" Mathematics. I think this is where our views diverge the most : if I read you correctly, mathematics, in your definition of the term, has to be something that is understandable (in principle) by humans. For instance, you write:

          "It is then often taken to be self-evident that these patterns [that we observe in the world] must be mathematical, but to me this is a substantial additional assumption. On my view of mathematics, the further step amounts to claiming that nature itself is - at least in principle - understandable by humans."

          Of course, limiting the definition of mathematics to what can be understood by humans is a valid approach (that was taken by many participants in this essay contest). I, on the other hand, define Mathematics in a wider sense (in fact, in the widest sense possible) encompassing all abstract structures (finite, infinite and transfinite), including those that are too big, too complex or too irregular to be grasped and studied by human-level minds. Similarly, my definition of Physics encompasses all possible physical realities (human-imaginable or not), and it is within this context that I argue for the possibility that "All-of-Physics" is "generated" by "All-of-Math".

          In a way, the conclusion you reach at the end of your essay calls for transcending your human-limited definition of mathematics to take the larger view:

          "From my view of mathematics as constrained imagination, however, the idea of a mathematical multiverse is still restricted by what is thinkable by us, humans. [...] My diagnosis of the situation is that the speculative questions asks us to boldly go even beyond Tegmark's multiverse and thus to exceed the limits of our cognitive kung fu: even with mathematics, we cannot think the unthinkable."

          The Maxiverse hypothesis that I present in my essay is my attempt to "exceed the limits of our cognitive kung fu". If you have the time, I would be happy to know what you think of it!

          All the best,

          Marc

            Dear Marc,

            Thank you for your detailed and kind reply.

            The distinction you make here, between mathematics and Mathematics, is really helpful for these kinds of discussions. (It would even have been a great starting point for an essay!) When we apply some of the idealizations that go on in mathematics to the field itself, we obtain the concept of Mathematics. This move has been made at least since Plato, and seems to come so natural to us, that it often goes unnoticed. So, it is very helpful to indicate when this is going on. Indeed, I tried to stick to mathematics in the real world, because it is not clear to me that 'Mathematics' refers to anything other than the human concept thereof. But I certainly don't mind to speculate about what it implies if Mathematics would have an indepent existence. So, I will also post a reaction to your essay on your forum.

            Best wishes,

            Sylvia

            Dear Christine,

            Thank you. I remember reading and commenting on your essay. As I wrote then, I quite liked it, so I am not surprised that it got you a prize, too. Congratulations. :)

            Best wishes,

            Sylvia

            Dear Silvia Wenmackers,

            congratulations on your prize.

            I'm sorry that I forgot to say what a good essay. I was preoccupied with trying to make the point Re thinking the unthinkable- we can think about the unthinkable without actually being able to think it. That is probably more interesting to me that to you, as it received no response. Though I am pleased to see that you did respond to some comments by other people.

            Well done, enjoy your prize, kind regards Georgina