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

      This may be interesting to you. Michael Hansmeyer talking about building shapes that can not be imagined because they are too complex and at a scale of folding that can't be carried out by human beings. He shows that how to produce these shapes can be thought about quite simply, even though the output itself is unimaginable. It is a very simple process likened to morphogenesis and he mentions breeding of types to produce new designs. He also mentions designing processes rather than shapes in the future. So it seems we are not limited by our imagination. Michael Hansmeyer: Building unimaginable shapes, kind regards Georgina

      Dear Georgina,

      Thank you for messages.

      I did read your earlier message and did find it an interesting point of view. Since you didn't ask any questions (and - as a finite being - I am always short on time) I did not answer back then. ;)

      Your comments remind me of fractals: at the same time, easy to define and impossible to image in all their intricacies. It also reminds me of so-called 'intangible objects' of which no explicit example can be given (because their existence relies crucially on a non-constructive axiom). For instance a free ultrafilter, which is crucial for the infinitesimals that I mention in my essay.

      In one way of seeing it, such objects cannot be imagined, but in another way they can be: I can understand the definition of a free ultrafilter and understand the existence proof, and at the same time understand that no explicit example can be given. If there is at least one viewpoint in which an object can be imagined (in this case via the definition), I consider it to be 'imaginable': this definition was developed by people and in that sense imagined, even though we do not necessarily grasp all the consequences. I do not require this kind of transparancy or omniscience for using the word 'imaginable'.

      Thank you for the link to the TED-talk; I will watch it this evening. :)

      Best wishes,

      Sylvia

      Dear Silvia Wenmackers,

      Neither the title nor the abstract of your essay sparked my interest. I was surprised how clever you managed to precisely tackle the topic in your essay and just to ignore pointless skeptical comments.

      However, I wonder if you are in position to substantiate what you wrote on p. 6: "NSA [you referred to Non Standard Analysis, not to Natural Science Alliance, not to National Security Agency, ;-)) ] seems a very appealing framework for theoretical physics: it respects how physicists are already thinking of derivatives, differential equations, series expansions, and the like, and it is fully rigorous."

      Appealing to you might not be enough to me. I am claiming that there are compelling reasons to abandon putatively rigorous but actually unwarranted naive set theory. That's why I consider Abraham Robinso(h)n not even wrong when he wrote: "any mention, or purported mention, of infinite totalities is, literaly, meaningless". Isn't NSA meaningless?

      Eckard Blumschein

        Dear Eckard Blumschein,

        Thank you for your comment and question.

        The main point of 2.4 is the observation that in applying mathematics to something non-mathematical, there are many degrees of freedom. This point applies even if one would reject some parts of mathematics.

        I regard almost all of mathematical statements as expressing suppositional knowledge: assuming these axioms, this follows; assumming those axioms, that follows; etc. (In practice, the first part is often silent and fixed by context only.) In that sense, one does not need to accept any set of axioms, merely check what follows from them. The resulting suppositional statement can be meaningful even if one is not willing to accept any of the axioms. So, I would not say that NSA is meaningless, rather that both standard and non-standard analysis are 'useful fictions' that do produce suppositional knowledge.

        Best wishes,

        Sylvia

        3 months later

        Hello dear Ms Wenmackers

        Congratulations for your essay and your humility Inside this universal sphere in spherisation. We are just indeed simple and we continue a humble road.

        The philosophy shows the road of this universal love in improvement, spherisation for me.Indeed when we understand what we want, it is simple to understand that we are just Young annimals inteeligent in evolution spherisation.I am asking me where the humility is encoded in our quantum sphères :)

        congratulations for your work , it is relevant

        Best Regards

        7 months later

        Sylvia, on NSA--

        properTime = (clockTime, properTime)

        and

        clockTime = (nonstandardPast, standardPresent, nonstandardFuture)

        There is a bi-conditional suggesting an exchange of information between NSPast and NSFuture,

        thus generating a Born infomorphism--- all made possible by the simplest possible application of NSA.

        Can you suggest a next step?

        http://fqxi.org/community/forum/topic/2420

        5 months later

        Sylvia,

        Sorry, way too general. Here is a more specific question--

        There is a "logical black hole" in the standard analysis of the Standard Model of Particle Physics.

        Do you think it might be worthwhile to model this logical black hole as a physical black hole?

        The logical black hole is implicit on page 2 of Abraham Robinson's book "Non-standard Analysis," where he wrote:

        "To this question we may expect the answer that our definition may be simpler in appearance but unfortunately it is also meaningless."

        In the previous paragraphs, Robinson had translated an epsilon-delta definition from standard analysis into a meaningful definition using non-standard numbers.

        Like a physical black hole, his translation took us inside of H. Jerome Keisler's "infinite microscope." But once inside we cannot get back out, because of the above quote.

        Although statements about very small real numbers like epsilon and delta might be meaningfully translated (as Robinson did) into statements about non-standard infinitesimals, the statements in question about non-standard infinitesimals cannot be meaningfully translated into statements in standard analysis about real numbers like epsilon and delta.

        It is a "logical black hole."

        Is it physical?

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