
Does time really flow? New clues from intuitionist math - nikolasavic
https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407/
======
xelxebar
Here's what looks like the original paper; it's a mere 3 pages long:

[https://arxiv.org/abs/2002.01653](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...](http://math.andrej.com/2016/10/10/five-stages-of-accepting-
constructive-mathematics/)

The metamath[0] proof verifier also has a database of theorems on
intuitionistic logic:

[http://us.metamath.org/ileuni/mmil.html](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/](http://us.metamath.org/)

~~~
fpoling
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.

~~~
xelxebar
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.

~~~
fpoling
I got impression that intuitionistic mathematical analysis refers to original
Boyer papers. Which is in modern terminology is constructive mathematical
analysis plus an axiom of the choice sequence. Intuitionistic logic on the
other hand is a classical logic without the law of excluded middle. This is
what the constructive mathematics uses and which was also used by Boyer.

And yes, as in practice all physically measurable spaces are of finite
dimensions, so using a formalism that works assuming AoC for arbitrary spaces
is OK. The only danger is that one reads too much from it and assumes that the
real world is like that.

------
ffhhj
> 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].

[https://arxiv.org/abs/1909.03697](https://arxiv.org/abs/1909.03697)

Seems like a framework for the Simulation Theory.

~~~
empath75
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.

~~~
Abishek_Muthian
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

~~~
andrewon
But a digital computer at present in this world may not be a good reference to
the capability of the computer that supposedly simulate this world. We know
nothing about the "real" world that computer resided, and nothing about that
computer.

~~~
stromgo
We do know that the real world is richer than the simulated world, since it
holds a computer that runs the simulated world. Therefore if you exist, then
it's more likely that you're the result of evolution in the real world than
the result of evolution in the simulated world.

Imagine the warehouse-size computer that is needed to simulate a bacterium
here on Earth. Computers are dusty, and dust contains bacteria, so if you're a
bacterium, then it's more likely that you're one of the billions of bacteria
in the dust _on_ the computer, than the bacterium being simulated _by_ the
computer. The same reasoning should hold for other worlds.

~~~
fpoling
The assumption that real world is richer than simulated world is just that, an
assumption. For one it assumes that both are finite.

~~~
pegasus
It's logically necessary, not just an assumption. The simulated world with all
its richness is by definition a strict subset of the simulating world. So the
latter must be richer than the former.

~~~
vidarh
Only if you talk about the simulated features of the simulated world, rather
than compare the "simulated world as seen by its inhabitants" with the
simulating world.

We don't have dragons on earth, but I can simulate dragons.

In the sense that this simulation exist in our world, you are right that the
simulating world will then always be "richer" because it contains the
simulation.

But if I could enter the simulated world, I could ride dragons. I can't ride
dragons in "our" world, so in that sense it is clear that we can simulate
things that do not have a concrete existence in our own world, and I to me at
least that would make the simulated world "richer" in that respect by making
things possible in the simulation that requires you to be in the simulation
for it to be possible.

Similarly, we can clearly simulate something with more detail - e.g. we could
simulate a world where our elementary particles can be subdivided endlessly,
if we choose to. In the simulating world this would "just" be a simulation,
but in the simulated world it would be that worlds reality.

There is even no reason why, with sufficient resources and time dilation, it
would not be possible for the simulating world to simulate a world equivalent
to the simulating world, so it could well be turtles all the way down.

------
mikekchar
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.

~~~
twomoretime
That doesn't explain why humans have the concept and sensation of "present".

~~~
jonsen
This is due to the (extremely) limited capacity of consciousness. It is only
able to handle a window of a few seconds.

~~~
mgamache
I think the 'frame rate' of the human mind is around 13 milliseconds. We fill
in the gaps so our perception is of a continuous reality.

~~~
mgamache
I can't find the source, but here's a reference:

[https://www.dailymail.co.uk/sciencetech/article-2542583/Scie...](https://www.dailymail.co.uk/sciencetech/article-2542583/Scientists-
record-fastest-time-human-image-takes-just-13-milliseconds.html)

------
jackhalford
> 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.

~~~
dTal
>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.

~~~
ineedasername
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_?

~~~
roywiggins
You can write down an algorithm to generate the digits of .33..., so that set
of digits exists as a "potential infinity". Same with Pi and the square root
of two.

It is numbers that _haven 't_ been constructed that intuitionist mathematics
doesn't generally think have been proven to exist.

~~~
paganel
> exists as a "potential infinity".

Not a mathematician nor a physical theorist by any means but some might regard
the putting together of "exists" and "potential infinity" (even if using ") as
an oxymoron. It's an endless discussion, of course, I personally think it all
boils down to Zeno's paradox remaining, well, an unsolved paradox for the
foreseeable future.

~~~
shrimpx
Imagine the lazy Fibonacci series. As long as you keep taking a number, the
next one in the series is generated. It’s not incorrect to say it’s
“potentially infinite”. And it exists, as the live algorithm that keeps
cranking as long as you put in energy.

~~~
paganel
There must be a philosophical term for it but imho "potentially existing" (or
having the potentiality of being constructed) is not the same thing as
actually "existing" (in the reality that surrounds us).

Leaving aside the fact that we're not even sure numbers "exist", for better or
worse, their "existence" is just us abstracting away some quantitates for
different stuff (we've passed from counting cows or sheep on clay tablets 5000
years ago to believing that there could actually be an infinite number for us
to count to).

And yes, I do believe there's a huge impedance mismatch between the world as
we experience it around us and the different theoretical constructs that we
now call physics or maths. I'm a Hume-ian, a guy who didn't take mathematical
induction for granted (presumably not the Fibonacci series either).

------
ineedasername
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_.

~~~
slowmovintarget
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.

~~~
naasking
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.

------
aunlu
“Information is destroyed as you go forward in time; it’s not destroyed as you
move through space,” Oppenheim said.

Strangely, this reminds me Stephen King's "The Langoliers".

~~~
calebm
Destruction is part of the process of creation.

------
bill_from_tampa
> 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?

~~~
Koshkin
How does discreetness imply imprecision and randomness? If anything, it should
indicate some degree of certainty.

~~~
akira2501
I'm an amateur, but I like to think about this:

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.

~~~
Ono-Sendai
If you can have real numbers at each spacetime point (as opposed to boolean
values) then you can easily get a speed less than c. This is similar to
simulating the wave equation on a grid on a computer.

------
DeathArrow
Pretty interesting:

>"The dependence of intuitionism on time is essential: statements can become
provable in the course of time and therefore might become intuitionistically
valid while not having been so before."
[https://plato.stanford.edu/entries/intuitionism/](https://plato.stanford.edu/entries/intuitionism/)

------
jeromebaek
Now tie this with computability theory. Almost all real numbers are
uncomputable. It's obvious that they don't "exist" in any sense if by "exist"
we mean "computable". Now consider that entropy, is information, is the
measure of randomness, or uncomputability, is time. What is uncomputable,
coming into existence, is the passage of time.

------
posterboy
> Hilbert and his supporters clearly won that second debate. Hence, time was
> expulsed from mathematics and mathematical objects came to be seen as
> existing in some idealized Platonistic world.

I guess that was so because the view was common, before. It should be
difficult to find and prove that the line had better read "... continued to be
seen as ..."

------
zone411
We know that our experience of physical reality is flawed (thanks to quantum
mechanics and relativity). We also know that it's not the only possible way to
experience it (thanks to drugs). Does it matter then if physics is formulated
in a way that's a closer match to our experience?

~~~
rmdashrfstar
Not the only possible way to experience it, or is it messing with a
functioning motor?

------
totemandtoken
I do like the idea of viewing time in the context of physical information.
That makes a lot of intuitive* sense to me as a layperson

*Intuitive in the colloquial sense of the term, not in the way it's being used in this article

------
lonelappde
Arxiv is great, but why it the media writing up non peer reviewed articles?

------
russellbeattie
My key takeaway from the article: Ooooh. That looks like a great chair!

------
Koshkin
TL;DR A reformulation of physics based on intuitionist mathematics seems to
aid human intuition, makes no impact on calculations.

~~~
ineedasername
It might make no impact on calculations (I'm not really qualified to say that)
but it seems like it would make a big impact on epistemology and determinism.
I think?

~~~
madhadron
Not really. The math is chosen to fit the physics, remember, not the physics
chosen to fit the math.

~~~
ineedasername
I think you're right about that for determinism, but the acceptance of
intuitionist math _requires_ a different epistemology. It is literally a
different set of assumptions on what is knowable and what is true.

