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To clarify what I am asking, it is best to make an analogy with another mathematical discipline: geometry.

In old days there was no clear separation between mathematics and real world science e.g. physics. On the other hand, euclidian geometry was usually considered as absolutely true in our physical world - Kant has believed this ...

Then came Lobachevski / Bolyai (and Gauss unpublished) ... it became more and more clear that there "exist" several geometries (incompatible with one another) - existence meaning roughly consistency ... so one could consider 1. a pure math approach and 2. the geometry of the physical world. In 1. you had first two then an infinity of geometries (Riemann manifolds) and then math. made itself free of the limitation to at most 3 dimensions, finally there could be infinte-dimensional space (Hilbert spaces are the ones most similar to classical euclidian geometry) but few thought 2. could have a question (on that side Euclide was still absolutely right for most scientists)

Then came Einstein with his general relativity: the physical world was no more exactly euclidian - near heavy objects (huge dense stars etc) even wildly non-euclidian. And Hilbert spaces came into physics via quantum mechanics ... but not replacing 3D, so now there is also more than one geometry used in physics.

So geometry is now two things: 1) the math. geometry subjects 2) several applications to the real world, using math. spaces as models AND that is the important point for my question SCIENCE about those geometries that physic theory & experience / observation describe as existing in real world.

But what are the analogues concerning probability? On one side we have a well-developed math. theory - essentially part of measure theory, the measures concerning are only restricted slightly: positive and with total mass 1 - every thing else that general measure theory doesn't treat being only there because of practical interest or due to special aspects that would not be nice where the restriction is not be made (things about stat. independance, law of large numbers, normal / the central limit theorem and the like down to Markov chains / factor analysis etc.) ... and what is on the other side? Mainly literature about avoiding fallacies and philosophy about what probabilities really MEAN alias how they have to be evaluated in real world (= the interpretations). But it seems to me: apparently NO SCIENCE at all! (This is BTW the reason why probability is often considered as math' only.) My idea is that this is largely due to the fact that all results of the theory are calculations of probabilities of complicated combinations so that no assertion is falsifiable by experiment / observation (as usual in real world science) in a strict sense - in principle even when there is a probability 0 for a not impossible event / 1 for an uncertain event (this being anyway rather abstract due to limited precision of all measurements and the impossibility to really produce infinite trial series ...)

What I am mainly wondering about is: do some people consider that there IS in fact real science there or will be in some (far?) future? In the first case (it exists already) what is it?

Ulysse Keller
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  • "Real science" and "probability theory" are not on the same footing. Probability is a mathematical tool that science may or may not choose to use. There is no specific "science" to probability - it's just maths. It's exactly the same situation with geometry: geometry is just a branch of maths, and some flavors of geometry happen to be useful when trying to describe the world. The world is not particularly "geometric" anymore than it is particularly "probabilistic". It just is, and we use various tools to try and make sense of it. – Frank Jan 29 '23 at 02:14
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    Not even Plato believed that Euclidean geometry is true of our physical world, it was always seen as a (good) mathematical idealization that could be studied theoretically or applied. Plato thought that it comes from an ideal realm and Kant that it comes from our own synthetic intuition. It is the same with arithmetic or theoretical and applied probability, there is even quantum probability, if you want analogs of non-Euclidean geometry – Conifold Jan 29 '23 at 06:25
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    Statistics most dominant orthodox Fisher, as a staunch frequentist, stressed probability is nothing but the ratio of 2 infinite numbers whose limit can be approached asymptotically after performing a “random” test indefinitely times. Thus any probability is essentially a hidden parameter of a physical system and probability is always out there in the real world waiting to be confirmed or falsified. Of course if you're not a frequentist or Popperian propensitist, then it's just like any other (reliable) math tools such as arithmetic which are believed to be metaphysically necessary by many... – Double Knot Jan 29 '23 at 06:50
  • Quantum Mechanics? – Mauro ALLEGRANZA Jan 29 '23 at 08:19
  • It isn't really correct to say that the real world isn't Euclidean. What is true is that the world can be described using non-Euclidean geometry, but it can also be described using Euclidean geometry. – David Gudeman Jan 29 '23 at 09:28
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    @DavidGudeman yes - it depends what you need to do. We use the model that works best for the task at hand, which shows that there is no meaning to statements such as "nature IS Euclidean". – Frank Jan 29 '23 at 19:48
  • @Frank, Yes you may be right. In that case it appears to me as a (relative) change of paradigm. And I still think that the analogy geometry / probability has an essential BUT – Ulysse Keller Jan 29 '23 at 20:34
  • @Conifold after the life data seen in Wikipedia, Euclide was probably not yet born (or still quite young) when Plato died – Ulysse Keller Jan 29 '23 at 20:40
  • BTW à propos Kant: it is an opportunity for me to mention a conjecture of mine about Kant's mind. He may have had quite different reasons to believe / think that there is an intelligible world (behind our experience) which will be (for ever probably) inaccessible to our reason. Different from his arguments derived from his idea that there is a synthetic a priori, but he used these because he thought that his real motive for his concluson would not convince many people. (Continued) – Ulysse Keller Jan 29 '23 at 20:47
  • (cont. of last comm't) It is a large consensus that Kant's synth. a priori has been proven wrong by the advent of non-Euclidian geometry. This does NOT prove that his conclusion about the intellig. world are to be rejected. On the contrary, its likelyhood remains high – Ulysse Keller Jan 29 '23 at 20:51
  • @DavidGudeman, You are certainly right from a theoretical / philosophical point of vue. But most physicists might not agree, I guess. I 'd consider it like this: one can use any "good" map (-ping) to transfer the "image of reality" to a space with another geometry. Such maps exist between Euclidian and many non-Euclidian geometries. But some of them will not include the whole universe (=U). Some mathematics-expert people ask for instance if U has the topology of a 3D torus (the surface of a tyre is a 2D torus), which is compatible with the possibility that it be locally euclidian (continued) – Ulysse Keller Jan 29 '23 at 21:40
  • (cont. of last comm't) On the other hand, if U is a 3D torus or even just a 3D sphere, you will not be able to consider it as an ordinary Euclidian space because "changing the topology" implies to kind of break the thing - a rather uncommon & mentally violent idea ... may-be this must be relativized in the 3D-sphere situation: such becomes topologically an Eucl. space by just removing one single point. But only relativized, because if you want to reach Eucl. geometry (not only topology) you must "inflate" the thing radically in the surrounding of the lost point (radically=quasi infinitely) – Ulysse Keller Jan 29 '23 at 22:01
  • Kant explicitly discusses such "intelligible world" in Critique of Judgment as accessible to intellectus archetypus (intuitive intellect) which we are deprived of, but (presumably) God is not. The trouble is that categories of experience, such as we have in geometry, probability, etc., belong to our intellectus ectypus (discursive intellect) and by their nature cannot apply to the "intelligible world". In other words, geometry of any kind being true of "intelligible world" is not likely or unlikely, it is non-sensical. Like asking about true color of electrons. – Conifold Jan 29 '23 at 22:20
  • To me it remains: if probability is only maths, then all use of it in reality is ultimately senseless, due to impossibility to really falsify even the wildest assignment of probab. to events - like say giving probability 0.9 to the Ukraine war ending before 12.2.23 (I it's now 10.2.23) - imho we cannot reduce this problem to our only approx. knowledge of real world. Approx. theory is math too (general topology) and e.g. math. functions can be approximated uniformly or pointwise (2 among many ways) and with the latter the approx. often don't "look" at all as approaching the limit – Ulysse Keller Feb 10 '23 at 11:11

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As an ex theoretical physicist, I would say that mathematics provides tools we can use to model the world and to make predictions based on those models. Euclidian geometry is a good tool that I have used myself on several occasions- for example, when setting out the foundations of my office and framing the roof. However, Euclidian geometry is not a good model for the wider universe, and there are umpteen other geometries that seem to bear no resemblance whatsoever to the properties of the universe.

Probability theory is no different, in that respect, to mathematics as a whole. It provides us with a box in which we can root in the hope of finding tools we can apply to whatever problem we have at hand.

The main restriction in applying the theorems of probability is that they are subject to the 'garbage in, garbage out' rule, and since for many problems of interest we can only guess what the key input probabilities are, the outputs are only as good as our guesses.

But that is also true of many other applications of mathematics. My PhD was in the development of quantum mechanical models of impurities in transition metals. There is strictly no way in which you can solve the Schrodinger equation for an impure block of zirconium, say, as it contains countless trillions of variables, so we are forced to make drastically simplifying assumptions. The rules of mathematics, if we follow them, prevent us from making mistakes in our calculations, but they do not prevent us from making inappropriate models or starting assumptions.

Marco Ocram
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Look into Karl Friston for the most scientific answer to your question, and always look for a reductive scientific explanation. Philosophers tend to overcomplicate already complicated issues.

Reducing error within probabilistic calculation is an intrinsic part to our survival. Our world has an underlying mechanics demonstrating a physical causality. Our brains are mere physics engines trying to approximate it; epistemically foraging our metaphysics, the true reality outside our perception. I'm a hard determinist myself, believing the probabilities you speak of to be the consequence of hidden variables we've yet to discover. Don't disregard physical causality because a quantum probability exists, remain agnostic. I'll provide a link to another question where I go into detail about causality from a free-energy perspective: What is this idea of causality being articulated?

Will we ever be able to approximate entirely the reality outside our innately human perception? Probably not. Can we get pretty damn close? Hell yes. It's our purpose in life.

kendall.tubbs
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