Intuitionist Maths might help reconcile quantum physics and relativity

IdleRich

IdleRich
I don't know much (actually anything) about Intuitionist Maths but apparently, going back to the start of the 20th century when you had people trying to axiomatise mathematics through set theory, the Hilbert Programme etc some Dutch guy came up with this particular idea of a type of maths that was entirely constructed, making explicit the argument that it is NOT platonic ie it's something we invent rather than discover. Maths is basically a system we invent and then things within it can only exist if we can invent them from what went before.
So apparently he came up with a new way of doing things, he rejected the idea of infinite sets in favour of potentially infinite lists (I forget what he called them) - distinction seems slight but I think he's saying that instead of there being an infinitely large (countably infinite) number of natural numbers which form an infinite set, there is no such set, just a list that could go forever if we wanted it to. Without going into it too much that means that you don't get the Russell Paradox which was such a problem for Frege etc and you get a whole weird new kind of maths. In this new maths two big distinctions are the law of the excluded middle doesn't apply, and also the treatment of Irrational but Real numbers is different. The latter follows from the infinite thing, an irrational number being one with a non-recurring and infinitely long list of numbers following on behind a decimal point. Basically in Intuitionist Maths it seems that (apart from certain special irrational numbers that we can generate according to a given rule) most irrational numbers are not really described, or are not fully described. You can't necessarily say which is larger than the other at any given expansion and so you get this kind of indeterminism of which numbers are greater than or less than others, you lose the rule which says for any number, any other number is either less than it or the same as it or greater than it.
Anyway, this idea was invented early 20th century and although it never kinda became the main maths, it has been studied almost in parallel for a long time, it actually turns out to be more rigorous (ironically given its name) and require more conditions and so on for given theorems as you need to deal with approximations and so on, but turns out a lot of theorems in normal maths have a pretty decent equivalent in IM.
So, interesting stuff (I think), but, long story short, I read an article saying that some physicist read about this recently and was like hang on a minute, if I use this kind of maths with relativity it introduces a non-deterministic element that allows me to marry it with quantum physics. Now I don't know much about physics but I think that reconciling those two theories is THE problem in the field.
Another thing I like (mentioned in the article below) is that he argues it's an explanation of how we experience time. In physics you have the past and the future and so on, but the present is an instantaneous and infinitesimally small moment, yet we experience all our lives in a continuous present. I think he's arguing that irrational real numbers having this kind of fuzziness when described this way make the instant not infinitesimally small, cos that's not properly defined. Something seems intuitively right there somehow (although I've not explained it properly).
So... here's the article on the physics

https://www.theatlantic.com/science/archive/2020/04/passage-of-time-relativity-physics/609841/

Some background reading on Intuitionist Maths (what I read yesterday in other words)

https://en.wikipedia.org/wiki/Intuitionism

https://plato.stanford.edu/entries/intuitionism/ (this one is a bit complicated so I kinda skimmed it)
 

IdleRich

IdleRich
Could have gone in thought I guess.
With Frege and Russell's Paradox. Kinda sad, Frege was writing his life's work, a three volume treatise on the axiomatisation of set theory, and was on to volume 2, almost finished it in fact, when Russell spotted his paradox and realised it fucked it all up, sent a letter to Frege who basically had to admit he was right, published volume 2 anyhow but with an appendix saying "Oh, but Russell has pointed this out which is correct and means everything up to now is bollocks" - he then fell into a depression and never did anything again. The thing to me though is that (if I remember correctly) when Zermelo saw Russell's Paradox written out he was like "Yeah, that's obvious, I noticed that years ago" - so I'm thinking, why the fuck didn't he tell Frege?
 

Mr. Tea

Shub-Niggurath, Please
Fascinating stuff. I'd never heard of this whole other approach to maths - then again, the only foundations-of-mathematics type course I ever took was an option in symbolic logic, which I didn't get on with at all and almost failed. (In my defence, the lecturer was a Czech guy with a very strong accent and incredibly small, crabbed handwriting, so a large part of my brain's runtime was taken up by trying to work out what he was saying or writing at any given time.)

The idea that it can be ambiguous which of two non-identical numbers is the larger is a real headfuck, isnt' it?

when Zermelo saw Russell's Paradox written out he was like "Yeah, that's obvious, I noticed that years ago" - so I'm thinking, why the fuck didn't he tell Frege?
 

IdleRich

IdleRich
Yeah it's a real headfuck... sort of makes sense I guess in that you don't totally know the value of either at any given finite expansion. Though at a given point obviously they may diverge and tell you which is largest I think, if I'm understanding it properly.
 

Mr. Tea

Shub-Niggurath, Please
Different approach which tackles the same problem, shows great promise. The simplest rules can be iterates as finitum to achieve structure and complexity. Nothing new there, but Wolfram is busy throwing computing resources at it a novel way.

https://writings.stephenwolfram.com...damental-theory-of-physics-and-its-beautiful/
Wolfram is a smart guy and his ideas on cellular automata are interesting, but from what I've read about him, his whole "I've invented a brand new approach to physics* that explains the whole universe all by myself!" schtick that he's been plugging for a couple of decades hasn't won him a lot of friends in the physics community, and he's widely regarded as a self-aggrandising crank. Which is not to say he's wrong as such, of course. Let's just say that if and when his approach can be used to calculate ab initio the mass of an electron or the strong coupling constant - which the string theorists have conspicuously failed to do, in the better part of forty years - then I'll be suitably impressed.

*I mean he wrote a book titled A New Kind of Science!
 

IdleRich

IdleRich
Couldn't sleep last night and was thinking about this trying to drop off... it still seems that there is some kind of problematic paradox there in that the guy invented Intuitionist Maths deliberately as an abstract rule-based system, explicitly as part of an argument (or belief system) that numbers are entirely something we invent and use to sort of model the world, but very much are not there in the world... and yet they are now using it to explain time by time saying that time IS a continuum of real numbers and those real numbers ARE as described by IM and so time behaves as those kinda sticky undefined numbers in that system.
I mean maybe that'a a kind of general paradox in modelling - but the contradiction. seems to run deeper here somehow.
 

Mr. Tea

Shub-Niggurath, Please
Sounds like the kind of thing you could easily drive yourself mad by thinking about, which I guess is probably what happened to Georg Cantor.
 

IdleRich

IdleRich
I remember being told by a lecturer that a lot of logicians go mad... trying to use a language to talk about the limits of language (and so on) is intrinsically head-fucking I guess.
But do you understand the particular point I'm making?
I mean if you say "The thing" or whatever can't move like that cos the equations have no solutions that would allow it. Then is that cos the equations model how it's moved before (or an approximation of that) and so they can be expected to be right again, or is it cos the maths is fundamental to the movement and so the model IS the movement? In a sense it doesn't matter as long as it works... but in the example above they are taking a type of maths specifically designed to fit into the first world and making an argument that depends on the second case being correct. I think.
 

Mr. Tea

Shub-Niggurath, Please
Yeah, I guess this must be one of the core questions in metaphysics: namely, does mathematics merely describe a natural universe that is "blind" to maths and just does its own thing - albeit describe it with arbitrarily good accuracy, depending on how good your measurements are and how powerful your computer is - or do the laws of physics proceed directly from those of mathematics in the same way chemistry proceeds from physics? In the end I suppose it almost comes down to a question of taste, or aesthetics, as to which you prefer.
 

Mr. Tea

Shub-Niggurath, Please
Although a thorny problem for the former outlook is surely: just because "kinetic energy equals half of mass times velocity squared" seems a good descriptor of the universe today, for what reason doesn't kinetic energy equal three quarters of mass times velocity squared tomorrow? Or velocity cubed, for that matter?
 

IdleRich

IdleRich
Yeah of course that kind of inductive problem is another whole kettle of fish. Although I guess Wolfram is saying that he's found a model of the universe that precludes that (unless the model itself somehow fails inductively).
But anyway, what you're saying here...

Yeah, I guess this must be one of the core questions in metaphysics: namely, does mathematics merely describe a natural universe that is "blind" to maths and just does its own thing - albeit describe it with arbitrarily good accuracy, depending on how good your measurements are and how powerful your computer is - or do the laws of physics proceed directly from those of mathematics in the same way chemistry proceeds from physics? In the end I suppose it almost comes down to a question of taste, or aesthetics, as to which you prefer.
Is exactly the question. But I'm saying that there is an issue with that Intuitionist Maths explanation (linked to above) for time and relativity in that it - the explanation itself - seems to depend on the view "physics proceeds directly from mathematics" and yet is based on a type of maths which is explicitly designed as maths that "merely describes a natural universe" and denies that very view.
Am I a) making sense? and if so b) correct in saying that seems problematic?
 

IdleRich

IdleRich
Certainly kinda odd... if not fully paradoxical it surely raises some questions. Maybe it could be an argument for Intuitionist Maths being actually literally how numbers are and the numbers being literally the world. So it was designed with the opposite idea in mind but by fluke landed on an absolutely true description of how platonic maths really is. Sounds ridiculous.
 

Mr. Tea

Shub-Niggurath, Please
My instinct is that there is a fundamental mathematical basis for physical law, but that it is "unknowable" in its pure and original form, in the way the "true" name of God is unsayable or unthinkable in some traditions. What we can do is create - in a way that will be entirely culturally contingent and limited by our language and powers of perception - sets of mathematical axioms that reflect the "original" mathematics. Two or more sets of axioms may appear to be very different but will (edit: I mean may), in the end, be equivalent, since they can be used to derive the same laws of algebra, geometry and all the rest. On a very trivial level, we count in base 10, computers in base 2, the Mayans used base 20 and the Babylonians base 60. You can use any natural number as the base for counting, and there's an infinite number of them, so in principle there's an infinite number of different ways you can count things.

This is already familiar from physics, because for example you can construct quantum mechanics either from Heisenberg's matrix mechanics or from Schroedinger's wave mechanics. They look very different and there are different sorts of problem in which one approach might be much easier and more natural than the other, but it's useless to ask which is "right" and which is "wrong", because they give the same answer.

Now it would seem that this intuitionist maths is, on a fundamental level, different from the standard basis of maths that assumes an infinitely divisible continuum of real numbers. If it can be used to resolve paradoxes that appear in standard maths and be used as the basis for constructing laws of physics ab initio without requiring the assumptions or empirical observations (e.g. that there are three space dimensions) that are necessary to physics as we know it, then there would be a good case for saying that intuitionist maths is a better match to the 'unknowable' ur-mathematics.
 
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IdleRich

IdleRich
Yes that's what I take from it. Which would be ironic though as it is part of an argument that there is no such ur-maths.
 
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