It's a lot of prose with hand-wavy analogies. My takeaway is that the author seems to have become enamored with intuitionistic logic but seems to lack much concrete experience working with it.
For anyone intrigued by constructive mathematics, there's a nice talk and paper by Andrej Bauer (nice coincidence) called "Five Stages of Accepting Constructive Mathematics." It's a nice mix of prose and rigor of varying levels:
The article argues for intuitionistic mathematics, not constructive mathematics. This is important. Just as classical mathematics is constructive math plus some arbitrary unprovable assumption (law of excluding the middle), intuitionistic mathematics is constructive math with another unprovable assumption (the existence of the choice sequence).
The article also made a claim that physics assumes classical mathematics. Which is wrong. The equations of physical models and their solution stays exactly the same in constructive mathematics as in classical one.
I'm not sure the original paper has enough substance to precisely pin down the author's preferred axioms. In practice, the adjectives "intuitionistic" and "constructive" are used with enough author-specific meanings that you just need to read their definition each time.
FWIW, the term "intuitionistic mathematics" sounds a bit odd to my ear. Usually you hear about "intuitionistic logic" or "constructive logic" which are both part of the larger program of "constructive mathematics."
Anyway, the "Five Stages" paper I link does a good job of introducing the broader ideas of constructive mathematics and, perhaps, gives a taste of what Gisin is excited about.
> The equations of physical models and their solution stays exactly the same in constructive mathematics as in classical one.
I'm not a cosmologist, but AFAIU the Hilbert formalism of QM relies on Hilbert spaces always having a basis, which is famously equivalent to the Axiom of Choice. I'm not sure you can formulate the concept of self-adjoint operators without that, at least for arbitrary Hilbert spaces.
My suspicion is that AoC simply broadens the class of phase spaces to include pathological ones that "don't actually matter," so your point probably stands in all practical applications. However, it would take (hard) work to carefully extricate AoC from the current formalisms.
EDIT to clarify the parent comment. Essentially the paper claims that using classical mathematical analysis (which is the constructive analysis plus the Axiom of Choice) when interpreting the equations of physical models is poor choice. It is better to use intuition mathematical analysis (which is the constructive analysis plus axioms of choice sequence) as it provides better match with our intuition.
But for me the extra axioms are still arbitrary. As all our measurements have finite precision, all those extra axioms does not matter in real calculations. If some extra axioms helps to discover results that are also applicable to the world of finite precision, then go for it. But not claim that the world is such.
Thanks. I gave up about 20% in, thinking that whatever it is, it's either BS or over my head. Which is where much cosmology and theoretical physics leaves me. And yes, that probably says more about me than them ;)
But whatever, I rather think that subjective time involves movement at the speed of light through some n-dimensional manifold. But maybe that just reflects the SF that I've read. Such as Stephenson and Egan.
> We envision two possible mechanisms that could explain the
actualization of the variables:
1. The actualizations happens spontaneously as time
passes. This view is compatible with reductionism and it
does not necessarily require any effects of top-down causation. Note that this mechanism resembles, in the context of
quantum mechanics, objective collapse models such as the
“continuous spontaneous localization” (CSL) [23, 25].
2. The actualization happens when a higher level requires
it. This means that when a higher level of description (e.g., the
macroscopic measurement apparatus) requires some physical
quantity pertaining to the lower-level description to acquire a
determined value, then the lower level must get determined. In
quantum mechanics a similar explanation is provided by the
Copenhagen interpretation and, more explicitly, by the model
in Ref. [37].
Yeah I did have this idea at one point that if we’re in a simulation, than there is a cost for simulating things at finer and finer detail, which could manifest itself in higher energy requirements that we see in particle accelerators, but that we aren’t actually learning anything new at those levels, maybe we’ll find some new fundamental particles that make up the electron for example, And we’ll find new particles that make up those, none of which even existed until we started poking at them, since they aren’t necessary for the simulation to run.
Regarding the cost of simulation, you might find what Dr. Michio Kaku said during his AMA.
>I mentioned that a digital computer cannot simulate even a simple reality, since there are too many molecules to keep track of, far greater than the capabilities of any digital computer. We need a quantum computer to simulate quantum reality, and hence, once again, the weather is the smallest object that can simulate the weather. Therefore, I don’t think we live in a simulation, unless the simulation is the universe itself. -Dr.Michio Kaku
I guess it boils down to whether or not we're willing to accept infinite regress or not. Of course, if not, one must necessarily accept that reality 'comes from' nothing.
Although I'm a VR developer, I can't believe it's "all turtles all the way down" :@)
Actually I've been working on a theory to define randomness from complete uncertainty, without the "fair coin" metaphor based on constructivism, which is deterministic.
But wouldn't a simulation be simulating the laws of an "actual" world? Why would the actual world have such a cost? Oppositely, if the actual world had such a cost, that's equal evidence that we are in said actual world.
Sort of a "turtles all of the way down", but only if you actually go looking for the turtles. This entire article makes my brain hurt in ways I'm really not used to. (I'm used to brain-hurt, you can't be intellectually curious without experiencing that occasional side-effect. It's just this particular flavor of it is... disconcerting)
The question I always end up with - how many simulations deep are we? Universe A builds a simulator to see how it works, it then builds a simulator and so on. We can never communicate with the original universe, so we build our own simulation.
As always if simulating universes is possible then the probability we're in level 0 tends to 0.
I often think that the flow of time is really just an artifact. At any given point in time, a human has a memory whose state is dependent upon the points in time before it. So at any given point in time, it appears as though we've travelled through time up until that point. However, as long as causality is preserved, I don't really see any necessity at all for movement through time. I mean, the phrase is self-referential - "movement through time". Movement is defined by change in position over time. I think from a practical perspective, the concept of time "flowing" or "moving" is really just saying that causality is preserved. There is an order. That's all. And from an experiential perspective, as long as you have causality and memory, it's going to appear as if time "flows". I'm not sure there is any meaning to be derived beyond that.
> I often think that the flow of time is really just an artifact
All human concepts are artifacts. "Flow" is a concept that presupposes the concept of "time", i.e. it is self-referential as you say. "Movement" presupposes "time". Citation marks are used here to highlight the conceptual nature of these terms - to say that time is something other than our concepts is true since the phenomenon is different than the concept, but once you conceptualize those differences you are again creating artifacts.
This process flows a common pattern, but usually just means one has found a new way of looking at a phenomenon which is different than conventional concepts, usually involving the removal of some dimension from view and viewing it as a two-dimensional or three-dimensional object, or similar (even 11-dimensional). It can be a truer view, if it explains something causally that previously was just guessed or misunderstood, but usually it just an expression of some hoping-for-a-goldmine theoretical framework that is going nowhere.
Some phenomena are especially prone to these kinds of gymnastics - economics comes to mind, attempts at creating historical models another. They take extreme complex and volatile phenomena and extract (reduce it to) a few dimensions, which may explain things with in a very limited frame of reference, not in its totality. It is certainly true of that which we denote as "time", a phenomenon that we have no problem communication about (we know what "time" is) but which at the core as a phenomenon is a mystery to us.
The insight that terms differ from, and can never fully explain phenomena in and of themselves, is an insight that comes with philosophy. At least since Kant. Science tends to downplay this, to the detriment of our understanding of the world.
I don't think it's necessary to introduce human perception to illustrate the directionality of time.
Mathematically, there are innumerable physical laws where time is a variable, where the current state of something is dependent upon the previous state - and specifically _independent_ of the future state.
...or maybe that too is just an illusion. Perhaps we misinterpret those equations (eg simple Dynamics in Physics I).
...but regardless, discussing human perception is an unnecessary complication in the discussion.
True. Why now? Why not any other "time"? These are questions that are very similar to "Why 'I'?", "Why not anyone else?". Maybe time and consciousness are deeply related, or even based on the same principle?
> A real number with infinite digits can’t be physically relevant.
but
> Popescu objects to the idea that digits of real numbers count as information.
I don't know where to stand, what about information encoded in geometry, like pi. If I get a spherical system, in a small enough space - no, in any space - then there's a "cutoff" to the actual number of digits to pi. Because a chunk of spacetime can't contain infinite information? sounds good.
> Quantum math bundles energy and other quantities into packets, which are more like whole numbers rather than a continuum. And infinite numbers get truncated inside black holes.
Layman here, but AFAIK concepts like black holes aren't consistent with quantum mechanics so I'm not sure it's wise to use concepts from both theories at the same time. (i.e QM predicts that wave functions evolve deterministically but GR predicts information loss in black holes, these two views conflict).
It's way beyond my grasp but some theories seems to quantize space, I wonder how those agree with the notion of "thickness".
I'm disappointed that the article doesn't point into any mathematical theory that models the "thickness" that comes from removing the empty middle theorem.
>then there's a "cutoff" to the actual number of digits to pi. Because a chunk of spacetime can't contain infinite information? sounds good.
It's more subtle than that. The "infinite digits" of pi isn't information, no more so than the endless decimal 1/3 = 0.333... is "infinite information". You can't use it to "store" anything. This is a distinct notion from the practical reality that real spacetime is quantized. An alternate universe with un-quantized spacetime might, or might not, allow you to store infinite information in a chunk - but every digit of pi would be relevant there.
Would it be correct to say that, under intuitionist thinking, actually constructing 1/3 = 0.333... (on paper, in a computer, whatever) would take infinite information (not to mention energy and space)?
Though if I understand correctly, intuitionist math would also hold that true infinite 0.3333.... also cannot be constructed?
Pi is computable. What is computable, can be represented in finite information (the algorithm used to compute the number). Almost all real numbers are not computable.
Maybe this will help people, maybe just make things worse, but for what it's worth, here's my $0.02: I read the first half (with an early tangent to wikipedia for "intuitionist math") feeling profoundly uncomfortable with the entire premise.
Then at the halfway mark, I realized that intuitionist math feels a lot like David Hume's approach to metaphysics and epistemology, which always felt right to me.
Intuitionist math still makes me feel uncomfortable, but now at least it also seems consistent with a framework of thought that doesn't. I'm not sure I've ever been quite so profoundly intellectually ambivalent.
The first smell for me was the name "intuitionist". The concepts involved make a lot of sense to me, though. This kind of number system follows the rule of "TANSTAAFL" (Robert Heinlein's "there ain't no such thing as a free lunch"), namely that you cannot have zero-cost infinities as real numbers require.
As a side note, if this notion pans out... welcome back Free Will.
Intuitionism should be familiar: it's effectively the logic underlying most programming. It's basically what allowed theorem provers like Coq to extract a runnable OCaml program from a logic proof.
I watched Conway's 6th lecture about Free Will today, and was thinking about the consequences of his theorem. As I understand, he proved that Free Will is mutually exclusive with determinism: "Everything happens for a reason."
Then I started thinking about time and causality. Does time really exist? Time dilation really exists, after all. What if there were an elementary particle that has no Free Will? Would that make it eternal?
I feel that eternity and infinity are deeply connected, but I'm not mathematically smart enough to prove it. If you'd like to discuss further though, please send me an email!
> If numbers are finite and limited in their precision, then nature itself is inherently imprecise, and thus unpredictable.
Is this not what Planck's constant implies? We can only know position and/or motion to a certain degree, and not exactly? Does not quantum mechanics already include this idea?
If spacetime is quantized, then the speed of light would be 1 planck length / 1 planck time. Assuming spacetime is actually quantized to that metric, we can then ask: How does something move at 2/3c? Or two discrete planck lenghts in 3 discrete planck times?
In one instance it could be:
t=0,x=0, t=1,x=0, t=2,x=1, t=3,x=2
It could also do:
t=0,x=0, t=1,x=1, t=2,x=1, t=3,x=2.
It implies a hidden variable, or at the very least a hidden phase of some sort. All sorts of oddness abounds when you consider all velocities are then quantized fractional values of c.
Readit News
https://arxiv.org/abs/2002.01653
It's a lot of prose with hand-wavy analogies. My takeaway is that the author seems to have become enamored with intuitionistic logic but seems to lack much concrete experience working with it.
For anyone intrigued by constructive mathematics, there's a nice talk and paper by Andrej Bauer (nice coincidence) called "Five Stages of Accepting Constructive Mathematics." It's a nice mix of prose and rigor of varying levels:
http://math.andrej.com/2016/10/10/five-stages-of-accepting-c...
The metamath[0] proof verifier also has a database of theorems on intuitionistic logic:
http://us.metamath.org/ileuni/mmil.html
It can be neat to compare proofs and theorems there with their counterparts in the classical logic database.
[0]:http://us.metamath.org/
The article also made a claim that physics assumes classical mathematics. Which is wrong. The equations of physical models and their solution stays exactly the same in constructive mathematics as in classical one.
FWIW, the term "intuitionistic mathematics" sounds a bit odd to my ear. Usually you hear about "intuitionistic logic" or "constructive logic" which are both part of the larger program of "constructive mathematics."
Anyway, the "Five Stages" paper I link does a good job of introducing the broader ideas of constructive mathematics and, perhaps, gives a taste of what Gisin is excited about.
> The equations of physical models and their solution stays exactly the same in constructive mathematics as in classical one.
I'm not a cosmologist, but AFAIU the Hilbert formalism of QM relies on Hilbert spaces always having a basis, which is famously equivalent to the Axiom of Choice. I'm not sure you can formulate the concept of self-adjoint operators without that, at least for arbitrary Hilbert spaces.
My suspicion is that AoC simply broadens the class of phase spaces to include pathological ones that "don't actually matter," so your point probably stands in all practical applications. However, it would take (hard) work to carefully extricate AoC from the current formalisms.
But for me the extra axioms are still arbitrary. As all our measurements have finite precision, all those extra axioms does not matter in real calculations. If some extra axioms helps to discover results that are also applicable to the world of finite precision, then go for it. But not claim that the world is such.
But whatever, I rather think that subjective time involves movement at the speed of light through some n-dimensional manifold. But maybe that just reflects the SF that I've read. Such as Stephenson and Egan.
https://arxiv.org/abs/1909.03697
Seems like a framework for the Simulation Theory.
>I mentioned that a digital computer cannot simulate even a simple reality, since there are too many molecules to keep track of, far greater than the capabilities of any digital computer. We need a quantum computer to simulate quantum reality, and hence, once again, the weather is the smallest object that can simulate the weather. Therefore, I don’t think we live in a simulation, unless the simulation is the universe itself. -Dr.Michio Kaku
Although I'm a VR developer, I can't believe it's "all turtles all the way down" :@)
Actually I've been working on a theory to define randomness from complete uncertainty, without the "fair coin" metaphor based on constructivism, which is deterministic.
As always if simulating universes is possible then the probability we're in level 0 tends to 0.
https://www.smbc-comics.com/comic/2010-11-09
All human concepts are artifacts. "Flow" is a concept that presupposes the concept of "time", i.e. it is self-referential as you say. "Movement" presupposes "time". Citation marks are used here to highlight the conceptual nature of these terms - to say that time is something other than our concepts is true since the phenomenon is different than the concept, but once you conceptualize those differences you are again creating artifacts.
This process flows a common pattern, but usually just means one has found a new way of looking at a phenomenon which is different than conventional concepts, usually involving the removal of some dimension from view and viewing it as a two-dimensional or three-dimensional object, or similar (even 11-dimensional). It can be a truer view, if it explains something causally that previously was just guessed or misunderstood, but usually it just an expression of some hoping-for-a-goldmine theoretical framework that is going nowhere.
Some phenomena are especially prone to these kinds of gymnastics - economics comes to mind, attempts at creating historical models another. They take extreme complex and volatile phenomena and extract (reduce it to) a few dimensions, which may explain things with in a very limited frame of reference, not in its totality. It is certainly true of that which we denote as "time", a phenomenon that we have no problem communication about (we know what "time" is) but which at the core as a phenomenon is a mystery to us.
The insight that terms differ from, and can never fully explain phenomena in and of themselves, is an insight that comes with philosophy. At least since Kant. Science tends to downplay this, to the detriment of our understanding of the world.
Mathematically, there are innumerable physical laws where time is a variable, where the current state of something is dependent upon the previous state - and specifically _independent_ of the future state.
...or maybe that too is just an illusion. Perhaps we misinterpret those equations (eg simple Dynamics in Physics I).
...but regardless, discussing human perception is an unnecessary complication in the discussion.
but
> Popescu objects to the idea that digits of real numbers count as information.
I don't know where to stand, what about information encoded in geometry, like pi. If I get a spherical system, in a small enough space - no, in any space - then there's a "cutoff" to the actual number of digits to pi. Because a chunk of spacetime can't contain infinite information? sounds good.
> Quantum math bundles energy and other quantities into packets, which are more like whole numbers rather than a continuum. And infinite numbers get truncated inside black holes.
Layman here, but AFAIK concepts like black holes aren't consistent with quantum mechanics so I'm not sure it's wise to use concepts from both theories at the same time. (i.e QM predicts that wave functions evolve deterministically but GR predicts information loss in black holes, these two views conflict).
It's way beyond my grasp but some theories seems to quantize space, I wonder how those agree with the notion of "thickness".
I'm disappointed that the article doesn't point into any mathematical theory that models the "thickness" that comes from removing the empty middle theorem.
It's more subtle than that. The "infinite digits" of pi isn't information, no more so than the endless decimal 1/3 = 0.333... is "infinite information". You can't use it to "store" anything. This is a distinct notion from the practical reality that real spacetime is quantized. An alternate universe with un-quantized spacetime might, or might not, allow you to store infinite information in a chunk - but every digit of pi would be relevant there.
Though if I understand correctly, intuitionist math would also hold that true infinite 0.3333.... also cannot be constructed?
Then at the halfway mark, I realized that intuitionist math feels a lot like David Hume's approach to metaphysics and epistemology, which always felt right to me.
Intuitionist math still makes me feel uncomfortable, but now at least it also seems consistent with a framework of thought that doesn't. I'm not sure I've ever been quite so profoundly intellectually ambivalent.
As a side note, if this notion pans out... welcome back Free Will.
Then I started thinking about time and causality. Does time really exist? Time dilation really exists, after all. What if there were an elementary particle that has no Free Will? Would that make it eternal?
I feel that eternity and infinity are deeply connected, but I'm not mathematically smart enough to prove it. If you'd like to discuss further though, please send me an email!
https://www.youtube.com/watch?v=IgvkhgE1Cps&list=PLhsb6tmzSp...
Strangely, this reminds me Stephen King's "The Langoliers".
Is this not what Planck's constant implies? We can only know position and/or motion to a certain degree, and not exactly? Does not quantum mechanics already include this idea?
If spacetime is quantized, then the speed of light would be 1 planck length / 1 planck time. Assuming spacetime is actually quantized to that metric, we can then ask: How does something move at 2/3c? Or two discrete planck lenghts in 3 discrete planck times?
In one instance it could be:
t=0,x=0, t=1,x=0, t=2,x=1, t=3,x=2
It could also do:
t=0,x=0, t=1,x=1, t=2,x=1, t=3,x=2.
It implies a hidden variable, or at the very least a hidden phase of some sort. All sorts of oddness abounds when you consider all velocities are then quantized fractional values of c.
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