1. Dear Roberto,
I read our essay with the ease that a clearly flowing exposition adds to finding ideas congruent with my own.
The aim of my essay was to outline a formal system that is consistent with much of what you argue for, but looking back at it, I see it is probably very hard to follow the thread of too many ideas packed into a few pages.
Here are some of these strong congruences between your terms and my essay (搂聽6):
``The one real world'' -> the universe is the set of all perceptible events at a `given moment'. (All that is accessible, by any means.)
``Particularity'', ``individual phenomena'', ``time-bound particulars'' -> events are fixed entities of space-time.
``Time holds sway'' -> time is defined as memory in my framework, that is, time is observed (even measured) only through memory. Memory is sedimented events, sedimented time. Sedimented events are what we call space, or structure. This definition works perfectly well to describe any basic cognitive operation, which in elementary form consists in connecting two memories, and up to describing any structure --that we may term mathematical or not.
In this formal framework, the `evisceration of time' you speak about is clearly and naturally displayed.
It does so by including cognition, which could be a natural way to expand your framework:
If you define symmetry as immunity to a possible change, you have the seed of a very apt concept to describe perception, as well as any law in the most general sense: A perception is a wager, a bet; perception works because when you encounter a situation, beyond being completely unique --particular--, it has changed aspects, but also aspects which are conserved, from previously known situations, and so you decide how to judge, to behave, to act. Hence a perception is a symmetry. It is inherently a categorisation. It is also necessarily a simplification, a reduction of the world. It is also an analogy, and all these terms can be seen as occurrences of symmetries. By building classes, or categories, perception `averages' on the whole class, hence a perception is a smoothing of the world. This is the only way to be able to predict, in a set of a priori particular, singular events (because prediction supposes some sort of recurrence, of reproducibility, both again are symmetries). So with symmetry we have a general expression for any perception, and any sort of law, included those we derive in physics. Included, also, any mathematical structure.
This formalisation clearly shows how me move from particulars to laws, or structure, or predictions. So moving out of particulars is in the very nature of a prediction, a concept that is evidently so pivotal for a living individual --were it strongly cognitive like humans are, or not: to live is to display a know-how, an ability to de facto predict correctly.
2. As a consequence, we also see that a perception operates under a particular interpretation, that is, a structure that you endow this singular situation with (singular situation that I represent by an a priori completely amorphous set of events).
This is exactly how I understand your beautiful expression of second-order facts, in a precise sense.
Perceptions are second-order facts; they are interpretations, they are formed in the frame of reference of the subject, the living being whose survival is engaged in perception-action loops. And the basic criterion for `success' is that the bet that each such perception, each such interpretation is, does not lead to death, so that the game can go on.
Interpretations are always made in the reference frame of a subject. Moreover, a cognitive subject is only able to make interpretations, never to express anything absolute about the world. All his statements are relative to himself, and his own experience. All we can express are interpretations about the world, not first-class knowledge.
A statement like ``the world is mathematical'' is absolute, it affirms something ontological, first-class, and as such, it does not comply to Popper's criterion of scientificity. Whereas taking it as a working hypothesis, to see the consequences, is acceptable. It is striking that so many physicists fall into this trap.
The same can be said about most presentations of the idea of a multiverse, and, of course, of extreme Platonist positions according to which the mathematical realm is completely disconnected, pure and independent of our world. Or are non-scientific because they are absolute, they purport to an act of faith. And, made apparent when we include the cognitive subject in our formalisation, they deny that knowledge is relative to a cognitive subject; that, in a strong sense, a subject is a reference frame for knowledge.
In passing: when you say that according to one thread of ideas, mathematics is discovery, perhaps `recollected', is it an allusion to Plato's conception of knowledge?
It has in common with the idea of multiverse that it naturally tends to completely unfold all that is conceived of, even remotely conceived of, into a single, static space (devoid of time).
The crux is that this unfolding is of little use. What do you gain once you say that there is a set of all the possible chess games, and evaluate the absolutely huge combinatorics implied? There are more illuminating ideas in Borges's Library of Babel, which, perhaps, contains all the possible books of a given size, included the one containing the true theory of the library, and many others differing by a single character, and all its possible refutations: all this profusion is of no use, past the profitable head-spinning phase that the perspective provokes. And that realisation is already present I believe in the old Hebrew tradition, when it says that everything that can be expressed is within the possible combinations of the 23 letters of the alphabet.
To return to the multiverse fiction, if I use my definitions, `unobservable universes' is just an oxymoron. Good for literature or poetry, bad for a built, scientific knowledge.
Reading Smolin's essay, I wrote down a question that I eventually did not ask in my comments. It ran: As much as you dislike a Platonic position, that is to say, a static vision of the (mathematical) world, I suppose you will not either like a multiverse theory, which in the same way unfolds a sort of full combinatorics of all the possible worlds? I suppose my interpretation was right, considering the closeness of your positions.
3. I feel close to you again when you raise the idea that our concept of time is closely related to the issue of the effectiveness of mathematics. I can certainly listen to physicists who claim that time is nonexistent, if that is a hypothesis they enjoy (Einstein is famous for having insisted that time is an illusion). But firstly they certainly cannot state that in an absolute way --which they dare do. Secondly, how fruitful is that hypothesis? It is certainly sterile, in that it does not speak at all about all our experience, without any exception: events cannot be removed. Things done cannot be undone.
It is worth quoting (again, since I have inserted it in the comment on another essay) an interview of Heisenberg in
Glimpsing reality: ideas in physics and the link to biology Paul Buckley and F. David Peat, editors University of Toronto Press Revised edition (March 1996)
``PB How does quantum mechanics deal with time flow or does it in fact say anything at all about it?
I would have to repeat what C. von Weizs盲cker 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.''
What I say is that all what we live is under the seal of irreversibility, which means exactly that past events cannot even be accessed; however, all the matter of cognition is to use the past to bet about the future. This is possible with memory, by retaining a record of events, that can be reversibly accessed (that is, you can read again and again the record of any event, at will). Which means that you can try arbitrary connections between events. By turning time (always occurring events) into memory (or space, or structure, or writing: all forms of sedimented time), reversible access to images of events is possible. The whole matter of cognition, of reasoning, is reversibility. Hence the deep involvement with time.
In this view, the ``basic asymmetry in the relation of mathematics to space and to time'' that you raise appears as a symmetry breaking inherent in sedimenting time into space, by leaving traces of events in the world.
4. I have slight reservations, on what I feel are good points that you make, but without completely going to the end of the argument.
You say that mathematical propositions are `outside of time' and `removed from nature' (p.聽4) quite convincingly, but you expose yourself to the criticism that this eventually means nothing different from a timeless, independent realm of extreme Platonism. So a Platonist would say that you just seem to mean something different. The same applies to your denying mathematics being `objects of a special type' (p.聽5), which mathematical objects definitively are, in your own words --though again I agree with what you mean.
I have not been able to find a convenient answer to this attack, placing myself within the confines of your essay. While I agree with what you mean, I cannot use your text as a demonstration that I could reuse to convince someone else.
On the particular point of the relation to time, I believe that the formalisation I have just restated (hopefully, in an understandable way) is very fruitful to position cognitive elaborations and the world in a common theoretical framework.
5. Three small points.
I have a question聽: when you write ``The enigma solved: the effectiveness of mathematics in physics is reasonable because it is relative'', what is it relative to exactly?
While I understand the connection you do between recursive as you explain it and taking ``itself for a subject'', the latter is more appropriately called reflexive, and the concept is much larger.
I would avoid invoking causality as you do, a concept which can only blur your text, since as I understand it it is completely unnecessary to your explanations. I have tried to explain why it is tricky to invoke causality in my comments to Lee Smolin's essay.