Making science relative to a cognitive subject by vincent douzal

Dear Luca,

Here are some answers before I read you prose.

1. Indeed, I did not mention quantum mechanics because this elaborated theory is `at the leaves of the tree', while I really start from the base of the trunk (and even try to show some roots).

However, with more room available, an interesting remark was possible. I noted that mathematically, any structure is characterised by its invariants. An accomplished theory is axiomatised; and the structure of the axioms carries its invariants in all the further developments. Gilles Cohen-Tannoudji gives an illuminating interpretation of universal constants of physics in terms of such invariants, for which he uses concept of horizon, inspired by Ferdinand Gonseth (see his contribution, alongside those of Bohr, Einstein, de Broglie, Heisenberg, Reichenbach notably, in Dialectica 2:3/4 (1948) pp. 305--424).

Now, the very first hypothesis of physics expels the subject (cognitive subject). That applies to all physics, with consequent limitations I have pointed at. It could only happen that in some cases, it would become absolutely necessary to reintegrate the observer. That has happened in quantum mechanics. The most spectacular illustration is the von Neummann-Wigner interpretation of quantum mechanics or any of its variants, whereby to make it short, consciousness causes the collapse. (This is not the place to discuss whether this interpretation is right or not. Suffices to say that many prominent actors of physics took it seriously, and that its core still holds.) My comment is that it is no surprise that when it becomes absolutely necessary to take the subject into account --and this is very closely connected to measurement-- he suddenly lands in the middle of the discourse. (The close connection to measurement is clearly seen in quantum mechanics, and this is why in the essay I have mentioned the relative nature of any measurement, and the formal closeness between measurement, modelling, perception, and formalisation, which is connected to the subject making categories --symmetries--, and going from what is given in the theory as a description at infinite resolution --in real numbers--, to a reduced, categorised view --the transition to macroscopic in a quantum mechanics measurement.)

So there could be a criticism addressed to `quantum mechanics' --say von Neumann: that he should have explicitly stated that a subject was to be added as an additional basic element, to his axioms of quantum mechanics, from the moment he was summoned to effectively make something happen, to collapse the wave function.

Adding the subject is basically what I do. But in the same movement, I find unnecessary to carry with me all the usual established axioms, I proceed straightforwardly by restarting from scratch.

2. With these topics of perception, measurement, I have a natural transition to time, your next question, with this quotation of Heisenberg, in an interview by Paul Buckley, in Glimpsing reality: ideas in physics and the link to biology:

PB--How does quantum mechanics deal with time flow or does it in fact say anything at all about it?

WH--I would have to repeat what C. von Weizsäcker said in his papers: that time is the precondition of quantum mechanics, because we want to go from one experiment to another, that is from one time to another. But this is too complicated to go into in detail. I would simply say that the concept of time is really a precondition of quantum theory.

It makes strikingly clear that Heisenberg would never have considered non-existence of time as a conceivable consequence of any theory of physics, for the very reasons I have raised in the essay, and that his position is just one step away from subjective relativity.

3. As to whether time is discrete or continuous (is perceived as, or represented as), I would refer to the definition of a perceptive continuum by Poincaré (he writes physical continuum), which is mentioned in a previous answer of mine:

It has, for instance, been observed that a weight A of 10 grammes and a weight B of 11 grammes produced identical sensations, that the weight B could no longer be distinguished from a weight C of 12 grammes, but that the weight A was readily distinguished from the weight C. Thus the rough results of the experiments may be expressed by the following relations: A = B, B = C, A < C, which may be regarded as the formula of the physical continuum.

The reported effect is quite common, and works as well for mere categories (not only something that purports to a scalar, as in the given example). It works, e.g., for our concept of species: a man is not a monkey, however, if men beget men, and monkeys beget monkeys, we admit since Darwin that man is without interrupt in the descent of monkeys. So monkey = monkey = [...] = man, man ≠ monkey. Of course that property conflicts with objects we can calculate with. (There is a contradiction.)

To sum up, the world is perceptively a continuum. (Perceived) space is continuous, (perceived) time is continuous. But as explained in the essay, a continuous, homogeneous space is only apprehended by its discontinuities: you could not measure a span of space without objects in it, breaking is homogeneity, breaking its invariance under displacement. This is completely consistent with the fact that you cannot see, or feel any sort of space, except by the object that are obstacles to your sight, or your movement, any sort of sensitivity. As for time, these singularities are called events. (Thom has made interesting discussions on saillances and prégnances, discrete objects out of a continuum, but not up to that stage.)

However, as soon as I want to speak about anything, time for instance, I have to pass through categorisation. If, to make it really clear, I put it in writing, I shall attach symbols to categories, and the discrete nature of my discourse will be clearly reflected in a succession of symbols on a paper.

Hence time is discrete as long as it is put in most elementary, formal representation: writing. But here is an example where you can see time, at once both continuous and discrete: when you see time [recorded] in a cliff of sedimentary rocks, you have a basic continuum, within which you can delimit discrete layers (that your geologic eye will interpret as sedimentary events).

Dear Joe,

I feel absolutely nothing disrespectful in your comment.

As must have understood, I chose to describe physics as a disciplined way for men to account for some aspects of their world, and abstract physics looks more and more like mathematics. In this sense, actually, physics cannot be said to rule the universe.

I do agree that (which means: in the framework I have outlined), every event, snowflake or other is primarily unique, and our activity as living beings, especially cognitive beings, compiles an organised view of all these events.

However, I have difficulties following the linking of your ensuing statements, probably because you packed too much in just a few lines. Feel free to elaborate.

Regards,

Dear Joe,

Apologies. As you can see from my answer, it is obviously a mistake if your comment has been reported as inappropriate.

I am going to write to forums@fqxi.org, to have that corrected.

Regards.

Michel,

Thank you for your warm comments (it seems that your rating is lost in a mass of low tendency, however...).

I have read early your essay, too, and have been looking at a few of your articles, and have jotted notes about it; I only have not secured time to write a proper comment yet.

If you really like the thread, you may take a look at Derek Wises's http://fqxi.org/community/forum/topic/2494, which touches similar topics. I have just posted a comment to it, where you should be in familiar terrain.

Dear `En',

So you are curious about my name. The origin is quite obscure, I never found any etymological clue, but the unique source seems to be a small village in Tarn, France. There is a variant with an ending `s'. That's in the vicinity of Albi, and during the cruzade against the Albigeois in the early 13th century, many families moved to the area of Strasbourg for instance, and into the territory of (now) Germany, and perhaps further, areas which were more welcoming for protestants. This seems to be the reason why the name is found also there.

I am learning from you that the name is also found up further in Eastern Europe and up to Turkey.

`En passant', I would not dare asking you the origin neither of your first name, nor your surname...

Regards.

Dears Anshu, Tejinder,

Don't blame your limitations. The essay was required to be clear, the author should have done better, and you are generous to say it is well-written.

1. If you want to reason about the effectiveness of mathematics, to evaluate, or even to measure it, you must endow yourself with a frame of reference where you can measure it; you need a standard to which you can compare it. This means you need a larger space, containing mathematics and other things, within which you have at least a possibility to distinguish differences, and say what is more apt to what. This is the etymological sense of to explicate: ex-plicare, to unfold, to show there are different components, and those unfit explain by contrast those fit.

This cannot be done, for instance, in an ultra-Platonist stance, where mathematics is a completely pure, separated realm: With mathematics already split apart, how can you show them distinctively more effective than... what? It inhibits any sort of explanation about the effectiveness of mathematics.

2. To be able to account for the effectiveness of mathematics you must account for processes of perception and cognition, and within them, of mathematics as a specialised activity (or as the product of that specialised cognitive activity).

3. I propose a framework in which all kinds of perceptory or cognitive structures, including mathematical, come into being homogeneously. This should not be a big surprise, if you accept that you capture an important part of mathematics by defining them as the science of structures, or the science of patterns. To perceive is already to find patterns.

So asking about the effectiveness of mathematics is a bit like asking about the effectiveness of perception --any sort of our anticipations, any of our behaviours. Certainly there is some effectiveness, because that is what we live by.

Each perception is a bet, an interpretation, and those that are felt unsuccessful are strongly counter-selected, while we keep using those `correct, up to now'. Similarly we work hard to select only the very small subset of mathematics which works successfully in a given situation. In such a Darwinian context, there is no general reference frame to rate effectiveness, you can only say what your best result is, in the locally explored context.

4. Hence the amazement about the effectiveness of mathematics disappears.

5. (In fact, the mathematics we choose to write in our books, and use, are already a selected subset among a larger world of a priori possible mathematics: we want consistency, i.e., we prefer to describe structures which lend themselves to construction of other structures.

This is certainly a very small subset of the possibles, much like continuous and differentiable functions have been shown to be an infinitesimal subset of functions everywhere continuous, but nowhere differentiable, though we practically speak nearly only of the latter. Do we wonder why most of the laws we write are continuous, differentiable? Or why we require the algorithms we write to be effective, that is, to finish in reasonable time on our current computers? This is exactly the same bias. Already noted by Descartes: ``Good sense is the best shared-out thing in the world; for everyone thinks he has such a good supply of it that he doesn't want more, even if he is extremely hard to please about other things.'')

You said summarise?