How not to factor a miracle. by Derek K Wise

Hi Derek,

I enjoyed your essay. You wrote:

"However, it is worth pointing out that, for example, calculus and a large part of differential equations -still among the physicist's most important tools-were designed precisely for physical applications."

I agree, but do you think that achieving an infinite is actually possible in the physical world? Would potential infinities be a more logical concept for physicists to employ? Or are you fine with physics only very closely approximating the real world?

"Moreover, as `quantum mathematicians' we would presumably not prove theorems according to the familiar

rules of logic, which are deeply tied to set theory, but according to rules of quantum logic which reflect the fuzzy, indistinct nature of propositions about fundamentally quantum mechanical systems."

This is idea I was trying to get at in my question #4 which asks, "If quantum mechanics is a world where things can be both "yes" and "no" at the same time, should experimental results be analyzed with Zen Koans instead of logical inferences?"

I also like you questions regarding the choices of mathematical foundations, which I think is well... foundational.

Also, your talk of sub-objects and quotients of a set make me think that you might like to weigh in on a question I asked (#3): "Is there an analogy between the following relationships: a "class" vs. a "set" and "true" vs. "provable"?"

Please check out my Digital Physics movie essay if you have a chance.

Thanks,

Jon

Hi Derek, I enjoyed your essay and found it educational. Re. in your conclusion -"It is one thing for the universe to be sensible in some precise way, and quite another for some entity within the universe to make sense of it."Here's a thought - Have we made sense of it or have we fabricated a sense ( or a number of different types of sense) that we then presume the universe to correspond to. Only because it correlates with the data received, though the sense made is not necessarily the causal reality?" That could apply to personal misinterpretation of events or sense as widely accepted as the space-time continuum.

Good Luck, Georgina

Nice essay Derek, and I've given it an appropriate community rating. I do hope you find something in this forum that helps you resolve our place in the scheme of things,

Cheers,

Rowan

Derek,

Like your surname, your essay is pithy yet profound. I too believe that you cannot factor out the human element, mentioning the connections of math, mind, and physics, as well as the quantum with the classical, but eschewing reductionism.

The book "Outlier" documents how mastery of any endeavor takes years of dedication. Having a primitive understanding of mathematics, I can certainly agree with math requiring an added level of effort.

Great job.

Jim

Dear Derek,

I really like some of the points you make in this very nicely written essay.

I think your rebuttal of Wigner's arguments are persuasive. His characterization of mathematics is clearly very poor. You are also surely right to comment that that large parts of mathematics (the differential and integral calculus) were invented precisely to describe physical effects, so it is hardly surprising that they do work well in that field.

I would like to try to add to your comments a further reason why mathematics has proved to be so effective in physics. It relates to your remark "that we can find useful mathematical abstractions of the simplest (you italicize this word) objects in our experience" and Einstein's claim that the comprehensibility of the world is "eternally incomprehensible". I think he discounted the fact that the mind is certainly not independent of the physical processes that take place in our brains and bodies. Indeed, much indicates that it and all our thoughts are strongly and perhaps totally dependent on them.

My comments derive in part from a book about cognitive science with the somewhat daunting title Philosophy in the Flesh with subtitle The Embodied Mind and its Challenge to Western Thought. The authors are George Lakoff and Mark Johnson. Lakoff has also written a book called Where Mathematics Comes From. I read about the first quarter of the first of these books a couple of years ago and found it illuminating. The book's thrust is well caught in these passages from the opening pages:

"Reason is not disembodied ... but arises from the nature of our brains, bodies, and bodily experiences. ... the very structure of reason itself comes from the details of our embodiment. The same neural and cognitive mechanisms that allow us to perceive and move around also create our conceptual systems and modes of reason."

Although I do not recall the authors arguing exactly as I do now, I parse these sentences as follows: we are each a self-aware physical system that forms key concepts based precisely on the most important physical facts that enable us to function and maintain our more or less autonomous existence. We have a direct 'feel' for the physical world precisely because we are part of it. Darwinian evolution means that some of the most important facts about how we function in the world as physical systems (and what aspects of the world are crucial to our survival) are hard wired into our neural makeup. Some of these deep truths are revealed to us in consciousness, above all the properties of the integers, the notion of continuity and basic geometrical notions such as point, line, angle and distance. As physical systems, we are finely attuned to vitally important simple verities that are simultaneously physical and mathematical in nature. We could not have survived as a species if that were not so.

It is moreover striking how much of the advanced mathematics essential for theoretical physics grew out of relatively simple and even obvious generalization of notions we acquire as children. I will mention only the transformation of Euclidean geometry, the understanding of which at both neural and conscious level was essential for our survival as hunter gatherers, by the concept of curvature by first Gauss and then Riemann. Curvature is something we literally see and the attempt to quantify it is very natural.

Thus, it seems to me that Wigner could hardly be further from the truth in defining mathematics as "the science of skillful operations with concepts invented just for this purpose". The brain's very development means that it operates on the basis of deep but simple truths and has a highly honed sense for what is important. Survival has built skill into our mental capacity. It is not used to play games but to sift the essential from the inessential. Since profound truths are nearly always simple, you are right to suggest it is "not surprising that we can find useful mathematical abstractions of the simplest objects in our experience."

I even think Einstein got it wrong. It is not eternally incomprehensible that the world is comprehensible. The world is manifestly aware of itself, for we are part of the world and are self-aware. The workings of the brain must reflect the workings of the larger world. Otherwise it would not survive and function. We would not be here if sensory experience were not organized in our minds. That is what thinking is. The eternally hard problem is the existence of consciousness in a world that does not seem to need it.

Perhaps the puzzlement felt by Einstein and Wigner arose because they died before cognitive science got into its stride.

Julian Barbour.

I feel there's a conjugation of all the applied theories of succession of this pertained Mystery.

- With regards,

Miss. Sujatha Jagannathan

Dear Sir, your essay started out with a lot of promise, but alas ended with not much of a resolution. There were some good ideas at the start but they were not carried through to a conclusion that I found to be acceptable. In my opinion there is a simple answer to this problem. Mathematics is effective in physics because we humans want it to be so. That is we invent the world to make it comprehensible to us. If magic were fashionable we would still have a world of magic. In some respects the world of physics is magic and that is why mathematics is so effective. We invent a magic of mathematics to make the world behave as we desire it to be in our imagination, and if the math doesn't work out as it should we make the world fit the mathematics. So it is inevitable that mathematics is unreasonably effective in physics, because we humans require it to be so. I wish academics would see the world as it is rather than trying to make the world into what is the politically correct current cultural opinion of the right fashionable thing to belief.

Dear Derek,

Your essay reads very well and clear, so it invites many comments. In addition, I feel very close to many of your ideas.

1. You seem to depart from Wigner's definition of mathematics, and to avoid giving one, till you finally wonder how we can factor out the human element. What do you think of Grothendieck's testimony, in Récoltes et semailles? He describes mathematics as something which is to be explored, much like an unknown continent. Someone who does maths is in a relationship similar to his relationship to the world in general, though of course what we call mathematical objects are a special case of objects within the whole world. Wigner focuses on the final result, which Grothendieck criticises as missing the living ingredient, the active, the discovery, the history part. It is good that the mathematical discourse is eventually abstracted --and certainly Grothendieck does not refuse abstraction--, but if abstraction is presented at once, naked, it cannot be understood. Hence the stress is on abstraction, as something built by a subject. Poincaré has insisted on exactly the same points about teaching mathematics (La logique et l'intuition dans la science mathématique et l'enseignement, in L'Enseignement mathématique, 1899):

``By becoming rigorous, the mathematical science takes an artificial character that will strike everybody; it forgets its historical origins; one sees how to solve questions, not any more how, and why questions are posed at all.''

Do I follow you well, would you see that fit with the spirit of your essay?

2. You define reductionism as `ignoring'. I would point that discarding, or ignoring many aspects of a situation is a good definition of abstraction; in your own words: `stripping away all inessential details'. This aligns the two terms that you keep separate.

It is a very common observation that problems get simpler when they are seen in a more general frame: when they are (pertinently) abstracted.

Abstraction is not easy. The simplest example I can think of is the transition from numbers-of (something) to numbers (`pure'). Adherence to numbers-of is still clearly perceptible in the late 19tr century writings of Lazare Carnot, who violently dismissed negative numbers, because ``they where obtained by removing something from zero, an impossible operation'', and besides, ``-3 would be less than 2, and though (-3)^2 would be greater than 2^2, that is the square of the largest would be smaller than the square of the smallest, which shocks any clear idea of quantity''. (This opinion, in a given viewpoint, is perfectly fair: how can you take three sheep out of an empty field?). And here we see what is wrong in his view: he cannot forget that numbers used to represent quantities, so he enforces on them to carry on satisfying quantities constraints. At the same time, he sees clearly what the rules to operate on negative numbers are, to be coherent. Thus seen from today's commonplace thinking, most of us would wonder how he could both understand and not understand the issue of negative numbers. The answer is that abstraction is not easy. There are countless examples of such obstinate resistance to abstraction. There are rules you must abandon in some cases, to be able to extend the range of things you can represent. Also, we see that the full extension of the rules to manipulate numbers is not visible from the onset: it must be explored, with trial and error.

The underlying idea is very close to what Wigner captures in his definition: yes, the rules of composing numbers are like invented just for he purpose of being consistent, that's the point we reach when we abstract them more. They forget that we came to numbers by an

A formalised piece of mathematics does not exhaust the concepts. Neither in the mind of the mathematician, nor (even less), in a computing machine, for cases where an implementation is possible or makes sense. The implementation is bound to inputs and outputs in such a way that it cannot be applied to any sort of object without an adaptation.

3. So, here are my questions:

Do you imply that there is any other way to build a science, than reductionism, as you have defined it? (And its generalisation, I have pointed to.) I would like to read your vision about it.

4. To elaborate,

what I have called the basic hypothesis of physics is actually a reduction: that the observer can be considered completely separated from the world (and then, even more, totally elided). (You may refer to my essay, should you need to.) It is not obligatory to follow the usual interpretation that physics makes of this necessary reduction, though. I have outlined that you cannot do without a reference frame to describe any situation, and the ultimate reference frame is the individual. (And, being the observer who enunciates what we call science, there is nothing problematic in taking the existence of the living subject as granted, as an axiom.)

In addition, any law is necessarily expressed in terms of classes (if approximate). It is easily seen if we accept a space time-model of the universe (any{ sort of space-time): by construction, events are fixed entities in such a representation. Hence when we describe an experiment, and say is is reproducible, we sate that some set in space-time shows, from a given point of view (forgetting, abstracting all the `unnecessary'), identical to many other sets. Hence we describe classes in the huge sets of events. This means that any law intrinsically forgets many things that it deems unimportant, unnecessary... and thus is a reduction, in a sense that is perfectly consistent with your use.

If a situation is, in our theory, liable to the same law as another one, then it is in the name of a reduction. There is no sort of law that cannot be a reduction. We can even go even one step further: any perception proceeds by the building of categories, and from this abstract vantage point, perception and science appear in continuity (incidentally, this has been widely acknowledged in psychology --See e.g. Eleanor Rosch, or Lakoff, to whom Julian Barbour already referred to.) Or, said in other, less general words (those of mathematical analysis), perception is a smoothing of the world. For instance, when you see the cliff from a distance, you are warned in advance you are informed by a signal which is not as sharp as the cliff itself, it is exactly a smoothing of the cliff's curve.

Would you care to answer these questions, or react to these comments?

Derek,

Time grows short, so I am revisiting essays I've read to assure I've rated them. I find that I rated yours on 3/21, 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

Dear Derek,

I liked your short essay very much and I keep

"The real miracle is the level of complexity of the world relative to our own intelligence".

As I am a mathematical physicist, I am interested by your opinion that "quantum groupoid" is a right language, I follow you with groupoids and I am not too far with the G-set structure of dessins d'enfants, I am less convinced by the quantum groups whose role in quantum information (and more generally physics) is questionable although topological quantum error correction is a possible instance.

I tend not to put too much abstraction and I favour constructive mathematics.

Thanks for a pleasant reading.

Best regards,

Michel

Dear Derek,

I liked your distinctions in the way of ignoring, which I think is not sufficiently appreciated. The terminology "co-reductionism" may help towards mending this, but I don't know how widely it is used. My suspicion is that most people would lump both together as examples of a broader conception of reductionism, but this would obscure that important differences between them, as you rightly point out.

If I may make one critical comment, given the title you chose for your essay, I felt that there was more that could have been said in the last section of the essay, particularly 1) why it is so amazing that the universe is sensible to us 2) why you see no sensible way to factor it out?

Overall your essay was very clearly written, and your argument that how we conceive the interplay between mathematics and physics affects how we study their interface resonates with me.

Best wishes,

Armin