In A Philosophical Essay on Probabilities, published in 1814, Pierre-Simon Laplace introduced a notorious hypothetical creature: a “vast intelligence” that knew the complete physical state of the present universe. For such an entity, dubbed “Laplace’s demon” by subsequent commentators, there would be no mystery about what had happened in the past or what would happen at any time in the future. According to the clockwork universe described by Isaac Newton, the past and future are exactly determined by the present. ...I believe that this view is mistaken.

A century later, quantum mechanics changed everything.

I don't just mean that some classical theories use probability, like statistical mechanics. Or that quantum mechanics sometimes predicts a sure result.

I mean that determinism is not a genuine difference between classical and quantum mechanics.

A couple of recent papers by Flavio Del Santo and Nicolas Gisin make this point.

One says:

Classical physics is generally regarded as deterministic, as opposed to quantum mechanics that is considered the first theory to have introduced genuine indeterminism into physics. We challenge this view by arguing that the alleged determinism of classical physics relies on the tacit, metaphysical assumption that there exists an actual value of every physical quantity, with its infinite predetermined digits (which we name "principle of infinite precision").Also:

Classical physics is generally regarded as deterministic, as opposed to quantum mechanics that is considered the first theory to have introduced genuine indeterminism into physics. We challenge this view by arguing that the alleged determinism of classical physics relies on the tacit, metaphysical assumption that there exists an actual value of every physical quantity, with its infinite predetermined digits (which we name "principle of infinite precision"). Building on recent information-theoretic arguments showing that the principle of infinite precision (which translates into the attribution of a physical meaning to mathematical real numbers) leads to unphysical consequences, we consider possible alternative indeterministic interpretations of classical physics. We also link those to well-known interpretations of quantum mechanics. In particular, we propose a model of classical indeterminism based on "finite information quantities" (FIQs). Moreover, we discuss the perspectives that an indeterministic physics could open (such as strong emergence), as well as some potential problematic issues. Finally, we make evident that any indeterministic interpretation of physics would have to deal with the problem of explaining how the indeterminate values become determinate, a problem known in the context of quantum mechanics as (part of) the ``quantum measurement problem''. We discuss some similarities between the classical and the quantum measurement problems, and propose ideas for possible solutions (e.g., ``collapse models'' and ``top-down causation'').Another:

Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with quantum theory and conclude that the common real numbers are, de facto, the hidden variables of classical physics. Consequently, real numbers should not be considered as "physically real" and classical mechanics, like quantum physics, is indeterministic.The point here is that any deterministic theory involving real numbers becomes indeterministic if you use finitary measurements and representations of those reals. In practice, all those theories are indeterministic.

Also, any indeterministic theory can be made deterministic by including the future observables in the present state. Quantum mechanical states are usually unknowable, and people accept that, so one could add the future (perhaps unknowable) being in the present state.

Thus whether a physical theory is deterministic is just an artifact of how the theory is presented. It has no more meaning than that.

One can argue that Classical Mechanics

ReplyDeleteas an idealis deterministic, because in mathematical models that's OK. The quote "the alleged determinism of classical physics relies on the tacit, metaphysical assumption that there exists an actual value of every physical quantity" is unfortunate, in that it conflates the model and things modeled. In a Hamiltonian model ---as mathematics, with no metaphysics in sight--- there does exist an actual value of every function of the position and momentum variables, however I very much doubt that any such model is exactly the same as any part of the real-world, which in any case we cannot measure with infinite accuracy and precision.For all that, FWIW, I agree with you that in practice theories are quite different from theories in the ideal. With care, particularly with an acknowledgement that we can't measure anything precisely, so we have to introduce probability of

somekind, an abstract classical measurement theory looks very close to the same as an abstract quantum measurement theory. I'm pretty sure I've posted the link to your blog before, but you might try arXiv:1901.00526, "Unary Classical Mechanics", which takes a Liouville state approach.Thanks for the link.

ReplyDeleteMore and more I am taking the view that there are no real numbers in the real world. Saying that we cannot measure to arbitrary precision is true, but it suggests reality is a thing with arbitrary precision real numbers, just like that Hamiltonian model.

I say that's wrong. A real number could encode all the world's information, plus oracles for unsolvable problems. Why would we find such a thing in nature? Once you abandon that, the determinism is meaningless.

I agree, insofar as models, much like maps, are not

Deleteperfectlylike the world, aka the territory, and yet, a mathematical model is ipso facto a systematic, formal development from some starting point. That's not determinism in the evolution over time sense, but consequences are determined by the rules and premises as they are given.I feel more-or-less OK using anything I like in a mathematical model, including "real" numbers, but I can't see that whatever structures I use will correspond more than loosely, albeit usefuly, as of map to the territory, to the real world. I'd hate not to be allowed to use real numbers, integration, and differentiation and other formal rules in models just because they're not real. It's enough that they're useful, provided we keep some humility about us.

The Banach-Tarski paradox should be evidence enough that real numbers are a flimsy model. They assume a completed infinity and that leads not just to paradox but outright contradiction. The problem I have is that there's a false dichotomy in mathematics between continuity and discreteness. Continuity is defined as degenerate (infinite) discreteness, so mathematics does not really have a distinction between parts and wholes. Zeno's paradox still stands. Joseph Mazur pointed this out in his book The Motion Paradox. Certainly, mathematics is not physics. Try telling the qubit people that.

DeleteDear Roger,

ReplyDeleteThe thing is: even real numbers are, in the end, only numbers.

Numbers themselves are only mathematical concepts. All mathematical objects (points, lines, surfaces, numbers, operators, functions, whatever) are just results of the methods of measurement we have invented; they are concepts of methods, not of physically existing objects.

The entirety of mathematics is what we have devised/invented to be able to measure the actually existing sizes of the actually existing physical objects in the real world out there. So, sizes exist. But numbers don't.

If we all get annihilated, numbers would cease to exist, but physical objects will continue to exist---with the same sizes with which they now exist. The moon's diameter will not suddenly change or become zero simply because we cease to exist---and with us also ceases all knowledge, including the bit that says that there is a number (real, rational fraction, or integer) which represents that size (that diameter) in some units.

Thus, the very questions "Can real numbers exist?" or "Can physically existing numbers be such that they have an infinity of digits in some (decimal or other) representation?" are basically wrongly formulated.

Numbers don't exist; sizes/magnitudes do. Numbers are just a means to measure the existing sizes according to some method of measurements (and according to some standard, and within some context).

In this context, the real utility of the real numbers system is that they never can run out of supply, for any arbitrary demand on precision while exercising any method of calculations.

Notice, no one has ever *defined* real numbers *as* an end-product of some *specific* method. Integers are defined as an end-product of counting; fractions via divisions, themselves as inverses of multiplications, themselves as repeated additions; some of the irrationals via certain geometric constructions and their algebraic representations, etc.

But the real number system has not been defined as the end product of any one, specific, method of constructing numbers, period.

The real number system (aka continuum) is a mere place-holder for saying that no matter what method of calculations or measurement you might ever devise, if its logic can also produce the other known numbers (like integers, fractions, known irrationals, etc.), then *any* other product it produces via the same logic *also* is a *number*, and this one must lie somewhere between two otherwise known numbers (i.e. in an interval of vanishing size, via limiting arguments, i.e., effectively, on the so-called real number line).

So, real numbers are just a convenience of maths, a reassurance that we will never run out of the precision in calculations. They originally have nothing to do with the sizes of objects existing out there. If they were to have something to do with the actual sizes, they would be too restricted a set, and hence, wouldn't be useful for arbitrary calculations.

Ditto, for the continuum of solids, fluids, or of EM. I think I wrote a PDF doc about the real number system a while ago. May be I should update it further to include points such as those noted here.

Best,

--Ajit

PS: Sorry for a long reply.

@Ajit,

ReplyDeleteAt last. About time someone put the pretentious metaphysical Platonic nonsense of the 'math/numbers informs the universe' trope to rest. What we call Math clearly is an abstract construct that required quite some time for humans to develop it, it did not fall from the sky or spontaneously appear. Math followed numbers which followed logic which followed simple counting which followed after humanity, it did not proceed it, obviously. It also DOES NOT exist independently of mental abstraction, and never has.

Dear CFT,

DeleteI am delighted!

...The first time I heard the idea that the universe is, fundamentally, made of numbers was in my 11th standard. I found the very idea itself very stupefying back then. Stupefying enough that I became distinctly suspicious about it right then and there. But I didn't know how to think about it. (Fortunately, our coaching class teacher back then, one Mr. Wani, an MSc in Maths, himself was inclined to think that the idea was ridiculous, and even said so!)

...It took me a lot of time (3--4 decades), and also studies of philosophy (esp. epistemology by Ayn Rand, and commentaries by Dr. Binswanger), plus going through history of maths, hitting calculus books once again (even after being in graduate school), etc. But finally, I got it!

The idea that if all people are annihilated, so would all knowledge, all concepts, including mathematical ones, isn't mine; I got it from Dr. Binswanger.

Coming back to maths, personally, the most difficult part for me was getting to realize that there *is* this distinction between sizes (which can be found in the physical reality, as sizes of attributes of objects), and numbers or other mathematical concepts (which can't be).

The second most difficult part was realizing that in counting, the idea of a *set* of $n$ objects is a purely man-made construct. Physically, $n$ objects might exist in a bowl, but their *set* doesn't. It is us who deliberately decide to mentally club *all* of them together. But, just because all these objects physically exist in the same bowl, you are not therefore obliged to take them all into a single set; you can always partition the same collection into two (or more) distinct sets. Tracing the logic back, you get the distinction between the sets suggesting the numbers 2 and 1. So, the idea that even when there is just one object in a bowl, you still have to take a *set* of this singleton (even if only implicitly) before you can define the counting operation---the idea that in counting you are basically comparing only *sets* (mentally constructed things)---the idea that concrete objects don't naturally form groups...

It now seems ridiculously obvious, but this idea *also* was pretty hard for me to get at. But yes, finally, I got it! (And, no, I didn't find it mentioned so explicitly anywhere! Descriptions everywhere made direct references to objects but not to sets.) So, in that sense, I am happy to have got it right.

Pedagogy does stand to benefit a lot if the nature of maths is spelt out clearly, removing from it all traces of Platonism.

Thanks, anyway!

Best,

--Ajit

PS: Once again a very lengthy reply! Sorry!

But I am inclined to agree with Laplace. How to reconcile indeterminism or discontinuity with Noether’s theorems, laws of conservation and and the wavelike nature of underlying microprocesses. There is also a problem here for the notion of intrinsic probability in standard quantum mechanics.

ReplyDeleteWe have those same conservation laws in classical and quantum mechanics, so I don't see what there is to reconcile.

ReplyDeleteIntrinsic probability is a funny notion. We don't know that there is any such thing, or that the probabilities in quantum mechanics are any more intrinsic than the ones in classical mechanics.