Dear Lee,
Here my comments, questions, and suggestions about your essay.
Contents
1 A pleasant essay.
2 Platonisms, and making one's ideas clear.
3 The explanatory power of a formal system.
4 Some room for improvement.
5 A path to improvement.
6 A Question.
7 Two quotations.
1. A pleasant essay.
It is a pleasure and even a relief to find a contribution by an accomplished physicist that integrates time as something substantial.
It pleases me of course because it is closer to my views than many others. I even have the feeling that it is congruent, it combines well with what I have tried to express (in particular, a central rôle of time, as we experiment it, `irreversible').
The essay is well written, it is easy for the reader to follow the author, and... it answers the question, which is worth mentioning since not all essays do.
Most of the statements touch on things that I feel right and I am sympathetic to. All the pieces fit well overall to form a pleasant discourse. The result is then that we ask more from it, we want to put that framework to the test, we want to try using it.
2. Platonisms, and making one's ideas clear.
The form of Platonism that you attack --aptly I believe-- is quite extreme.
Once I witnessed on one occasion Jean-Pierre Changeux trying to take the upper hand against Alain Connes in a conversation, by treating him of Platonist, for the mere reason that he had said that the solution to a mathematical problem is fixed, you can do whatever you want, it won't change, and that's a characteristic of mathematical objects. What Alain Connes was saying is just what everybody has to agree about, as you write: ``if any person can demonstrate it, any one can'', these are ``properties which can be discovered or proved, about which there is no choice''. Mathematical objects show like inescapable invariants.
In this situation, you would have been termed a Platonist. As usual, not much interesting can be said unless we define precisely what we mean by each term we use, and that implies getting closer to a formal system, and giving pragmatic criterions to define the key concepts, so that someone else can virtually reproduce, at least mentally, each experiment and ascertain semantic correspondence with what we signify.
3. The explanatory power of a formal system.
About formal systems, everyone is free to decide what rules he wants. Only, some decisions are really unfruitful.
One way of being unfruitful for a formal system is the inability to explain some classes of facts, by its inherent structure. In the present case, of course, if you make `mathematics' and `the existing universe' two completely separate domains, by construction, you have difficulties articulating them: there is by construction no natural connection between them. That's in summary what happens to Wigner. It is merely a consequence of the axioms.
Only should one be conscious that it is useless asking unanswerable questions, in a given system. So, more specifically, some decisions about a formal system are unfruitful in that they prohibit specific explanatory schemes.
Of course, even while a Platonist states the axiom of complete separation between `mathematics' and`the existing universe', he does enunciate the existence of both, and hence their connection, if only through himself because he names them, he has ways of connecting them, of interacting with each one, so he does recognise they both belong to a larger world. Most of the ensuing confusion stems only from occasional shifts to the general discourse in which you nested your formal system, when you should only be talking from within the formal system itself. If you stick to our axioms, you cannot ask about the effectiveness of mathematics in representing the world. If you want to explain the effectiveness of mathematics, you need a general, abstract domain in which objects that you observe in the world and objects that you meet in mathematics are both produced from a common root, and hence display natural symmetries.
Incidentally, the extreme Platonism you oppose to would fail to pass the test of defining pragmatically the terms it uses. That's another failure.
4. Some room for improvement.
First, you include causality as a constituent part of your principles. Causality is a human concept, in the best case, a theory. To me your sentence would sound better to me ``All that exist is part of a single, connected universe''. Or Simply ``the universe is everything''.
It is worth avoiding causality at this stage, because it is indeed absent of most of (the formal part of) physics. Let us take an example as simple as possible, to avoid being distracted by technicalities: Newton's law F12 = -F21 is completely devoid of causality. It says that the forces between two objects are permanently --and sort of magically, almost fortuitously-- equal in modulus and opposed. According to the equation, things, if they happen to change, change simultaneously; therefore, not causally. Writing an equation means suppressing any idea of causality. In spite of that, most explanations accompanying this and other equations do speak in term of causes, included in physics textbooks. You then have a complete strangeness between the actual theory and the discourse seemingly explaining it. See, e.g. John Clement. Student's preconceptions in introductory mechanics. American Journal of Physics, 56:1 (January), pp. 66-71, 1982. Or, even better (but in French) Laurence Viennot, Raisonner en physique : la part du sens commun. De Boeck Université, 1996.
You may like the following quote, which I I understand well matches you notion of being evoked: ``To say that truth is not out there is simply to say that where there are no sentences there is no truth, that sentences are elements of human languages, and that human languages are human creations.'' (Richard Rorty Contingency, irony and solidarity, 1989.)
Of course, a fastidious reader would require you give a pragmatic definition for `exist'. Mathematical objects, or properties, exist once their question is `evoked', but then they appear as completely invariant. The pattern is: we produce a change, and then we see that invariant phenomena appear, any one at any time would witness it. The pattern is the same when I bump into a stone: I produce a change (a step), and my toes find an invariant, that I call a stone. Any one at any time would find the same. Hence in some respect, the situation is the same, but I don't think we would say that the stone did not exist until it was `evoked' by bumping into it. However, we can certainly say something like that we were unaware of it.
There are similar small difficulties (I don't want to go into those set-theoretical-like paradoxes, whereby I would talk about Borges's Library of Babel which contains all the possible mathematical books, and pretend I evoked all the possible mathematics --that is not correct, since the exact invariants do not show yet): for instance, Ramanujam is famous for having `re-evoked' a lot of mathematics that others knew of already: obviously evocation does not `work' forever. It seems that we must make explicit the relativity of any mathematical object to those who know about it, and maintain it.
5. A path to improvement.
I think there is a way out of this difficulty. It requires, as is suggested by the above parallel, that we include explicitly in our axiomatic framework the processes of perception by which both stones and mathematics can be said to exist: They all appear as a sort of invariant, under particular conditions. This way of making explicit what it means to be `evoked' for mathematical objects is what I have attempted in my essay.
6. A Question.
You take some length to discuss, on one hand, games (typically board games like chess), and on the other hand, formal axiomatic systems.
For me, the two statements ``there are an infinite number of games we might invent'' and ``there is a potential infinity of FAS's (formal axiomatic systems)'' are simply equivalent. When I want to explain to someone not acquainted what a formal system is, I equate it to a board game like chess or go, and I deem it perfectly exact. (There is a sheet of paper in the Turing archive, where he has written down the rules of go strikingly in a formal system, axiomatic manner.)
Do you make any difference? Was that just rhetoric redundancy?
7. Two quotations.
That I feel fit particularly well here.
``And, moreover, we have found that where science has progressed the farthest, the mind has but regained from nature that which the mind has put into nature.
We have found a strange foot-print on the shores of the unknown. We have devised profound theories, one after another, to account for its origin. At last, we have succeeded in reconstructing the creature that made the foot-print. And Lo! it is our own.''
--Sir Arthur Stanley Eddington. Space, Time and Gravitation: An Outline of the General Relativity Theory. Cambridge University Press, 1920.
``A man sets out to draw the world. As the years go by, he peoples a space with images of provinces, kingdoms, mountains, bays, ships, islands, fishes, rooms, instruments, stars, horses, and individuals. A short time before he dies, he discovers that the patient labyrinth of lines traces the lineaments of his own face.''
―Jorge Luis Borges, The Aleph and other stories