
The physics of infinity - okket
https://www.nature.com/articles/s41567-018-0238-1.epdf?shared_access_token=nhIyZJldj4QzWZs7LvZIttRgN0jAjWel9jnR3ZoTv0PMOaEWTfe0Iq_Ol3Eo9bd6Lh9xyPK-ya44kxWDxYi4IQo2Zqj-Ymd6yZVANNbW9FXmT1HwoVMnEtM00qpXT48gLDqpQXX3mvS3gRH22aRhLs-Cf_4dd6NkVcLZZP3rPbg%3D
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mannykannot
I don't get the dual argument about zero, for which the impossibility of a
completely empty region of spacetime is given as an example. That may be so,
but a region empty of countable things like atoms seems physically
conceivable. I am writing from a room containing zero elephants.

~~~
okket
Rules of the macroscopic world do not apply to the subatomic (or even
molecular) world. There are no two identical elephants in the world. On the
other hand, you will have a hard time to tell two electrons or even two
molecules of the same configuration apart from another.

A total empty region, void of any atoms etc., still contains 'quantum noise'
and thus virtual particles. The world, the universe as we know it so far has
become inconceivable without this observation, known as the uncertainty
principle.

I find the argument quite convincing.

~~~
naasking
> The world, the universe as we know it so far has become inconceivable
> without this observation, known as the uncertainty principle.

Not really, deterministic interpretations of QM don't require such mental
contortions.

~~~
BoiledCabbage
I'm far from an expert but any deterministic interpretation of QM (including
the popular De Broglie–Bohm) is required to still maintain the uncertainty
principle - otherwise it would be falsified by showing it conflicts with the
results of hundreds of known experiments.

~~~
naasking
Yes, but it violates the original claim that rules of the macroscopic world
don't apply in this realm. de Broglie-Bohm is a classical theory with a extra
term to account for quantum behaviour.

The field theory covering "quantum noise", the existence of virtual particles,
and the vacuum is also completely different.

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andyjohnson0
From the article:

> There is a duality between zero and infinity, expressed in the elementary
> identity 1/0 = ∞.

I understand what they're saying here - but I was always under the impression
that, rather than yielding infinity, division by zero was undefined. Wikipedia
[1] says that this is true "In ordinary arithmetic", but goes on to say that
it is "sometimes useful" to think of a "formal calculation" involving division
by zero as evaluating to infinity. A formal calculation seems to be one that
ignores whether the result is well defined.

Would anyone care to comment on whether the mathematical perspective above
carries over into the physics domain?

[1]
[https://en.wikipedia.org/wiki/Division_by_zero](https://en.wikipedia.org/wiki/Division_by_zero)

~~~
throwawaymath
Division by 0 is not possible in _fields_ [1], which are the algebraic
structure most commonly associated with “ordinary arithmetic.” For example,
the real numbers comprise an ordered and complete field. You cannot divide by
0 in a field essentially by definition, so it’s undefined.

However, you can define meaningful algebras in which division by 0 is not only
trivially allowed, but also equal to infinity. This is possible by extending
the complexes, for example. More generally, see [2].

Note I’m speaking strictly of the math here. I can’t comment on this usage in
physics specifically. But there’s no issue with defining 1/0 = ∞ if the rest
of the theory remains fundamentally consistent.

______

1\.
[https://en.m.wikipedia.org/wiki/Field_(mathematics)](https://en.m.wikipedia.org/wiki/Field_\(mathematics\))

2\.
[https://en.m.wikipedia.org/wiki/Wheel_theory](https://en.m.wikipedia.org/wiki/Wheel_theory)

~~~
fusiongyro
I disagree, only because infinity is not an element of the real or the complex
numbers. You can make extended number systems (the hyperreals, for instance)
but it isn't commonly done. Infinity as a notation in the limit is not the
same thing as infinity as an element of a set.

~~~
throwawaymath
Can you clarify what you're disagreeing with? I don't think I stated anything
controversial.

No, ∞ is not an element of the real or complex _fields_ , that's correct. But
it _is_ an element of the extended real and complex number systems, according
to the following rules:

    
    
        for all x ∈ ℝ:
            1) - ∞ < x < + ∞,
            2) x + ∞ = + ∞, x - ∞ = - ∞,
            3) (x / + ∞) = (x / - ∞) = 0,
            4) x > 0 ==> x(+ ∞) = + ∞, x(- ∞) = - ∞,
            5) x < 0 ==> x(+ ∞) = - ∞, x(- ∞) = + ∞
    

Infinity is not just an artifact of limit notation, because the algebraic
rules defined here formalize the analytic theory underlying calculus. This set
is distinct from the hyperreals for a number of reasons (most importantly,
it's not a field). But just because it's not a field doesn't mean it's not a
coherent algebra. Which really circles back to the original point: you can
usefully define an algebra such that division by 0 is not undefined.

In particular, note that by augmenting ℝ with +∞, -∞, we can derive (x / 0) =
∞ from the third rule. This behavior is only very slightly pathological
because 1) it's (mostly) contained to the infinities, and 2) it still mostly
preserves the order, completeness, addition and multiplication properties of
the real numbers themselves. It's just not a field because we cannot do some
propositional things that follow from Peano arithmetic (e.g. cancellation) and
we can't e.g. take square roots of infinity.

To make this point even more technical, we can choose to define numbers as
cardinalities of sets under Zermelo-Fraenkel set theory instead of under Peano
arithmetic. This allows us to neatly define arithmetic involving infinities
via operations on infinite sets.

~~~
fusiongyro
Upon rereading what you are saying, I realize I am not disagreeing with you. I
am, however, easily triggered by this as an armchair student of math, because
I think infinity forms a kind of a trap for amateurs like me. "Actual"
infinity is not a member of the reals and it is not necessary to augment the
reals to have a functioning real analysis. But you are right that it can be
done, it can be added to the reals to form interesting algebraic structures
and there are even books on nonstandard analysis that start from here
(Elementary Calculus, which I have read part of but not finished).

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pphysch
The more sensible question, IMO, is "Is the universe fundamentally
quantized?". Can you concretely say where the boundary between an apple and
the surrounding air is? Is a particular electron on the boundary part of the
apple or the air?

If the universe is not truly quantized, then there is nothing to count. By
extension, mathematics is not related to physical reality, and it makes no
sense to apply mathematical notions of infinity to the universe in such
speculative ways.

~~~
CompelTechnic
Even if the universe is not discretized, there is no reason to completely
discount the value of mathematics. Continuous probability distributions of
particle locations still have meaning, individual entities can still be
disambiguated as local maxima of density gradients, counting can still be
built up from raw axioms and set theory.

Even for people don't accept the "monism" (to use a metaphor to the monism/
dualism debate of the human mind and body) of physics and mathematics, these
same people cannot avoid accepting that correctly applied theoretical
manipulations of mathematical symbols jive perfectly well with the predicted,
physical outcome they represent. Does claiming that the numbers don't mean
anything mean anything?

~~~
pphysch
I don't know where you read that I "completely discount the value of
mathematics".

>"...cannot avoid accepting that correctly applied theoretical manipulations
of mathematical symbols jive perfectly well with the predicted, physical
outcome they represent."

Close, except for "perfectly". Mathematics and other scientific models are
approximations of reality and nothing more. They are fantastic tools but don't
be misled into thinking they give a "perfect" picture.

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techno_modus
> David Hilbert famously argued that infinity cannot exist in physical
> reality.

This statement reduces the problem to the definitions of "existence" and
"physical reality" which of course may have quite different interpretations.
For example, the existence of particles (with finite mass and finite
coordinates which can be "touched") and the existence of waves are rather
different notions.

~~~
jamesrcole
It seems to me that something can only have existence for us if it can have a
transitive causal influence upon us, and surely both particles and waves exist
in that same sense.

~~~
whatshisface
If by existence you also mean, "something that continues to exist upon closer
inspection," then neither particles nor waves exist, you can always design an
experiment to refract something around a corner and meausre individual quanta
of it when it gets to the detector.

~~~
simonh
The comment gave a meaning of existence: "...if it can have a transitive
causal influence upon us", so there is no need to imagine up straw ones.

~~~
whatshisface
Zeus and Sherlock Holmes could both be argued to have had transitive causal
influences upon us - so in order to make that statement make sense there must
be some other implied constraints on what is considered to "exist."

~~~
simonh
The fictional character Sherlock Holmes certainly exists.

~~~
jamesrcole
It all depends on what you mean when you say "The fictional character Sherlock
Holmes". I believe that thoughts, memories and knowledge "about Sherlock
Holmes" exist and are physical details, as are Sherlock Holmes books, etc. And
that there's no other reality to "Sherlock Holmes", above and beyond these
things.

But really, this discussion is too large to satisfactorily deal with here.

~~~
lisper
Try this then:

[http://blog.rongarret.info/2015/02/31-flavors-of-
ontology.ht...](http://blog.rongarret.info/2015/02/31-flavors-of-
ontology.html)

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curlcntr
"There is a duality between zero and infinity, expressed in the elementary
identity 1/0 = ∞ . If one side of the duality does not occur in nature, also
the other side ought not to."

It wasn't clear to me that the second sentence was sufficiently proved. At
first glance it seems reasonable, but the authors went a bit too fast past
that claim for me.

~~~
JackFr
I don't think it's proved at all. "Ought" is an appeal to symmetry. At the
same time I don't think that necessarily undermines the argument.

------
j1vms
AFAIK, p-adic quantum mechanics has shown some promise in treating aspects of
the "physics of infinity".

It is the application of p-adic analysis to quantum mechanics.

[0]
[https://en.wikipedia.org/wiki/P-adic_quantum_mechanics](https://en.wikipedia.org/wiki/P-adic_quantum_mechanics)

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monster_group
The article doesn't load for me in Firefox or Chrome. It's stuck at 'Loading
enhanced PDF...'.

~~~
maaark
Doesn't load for me either. Stuck at 'pay us money'.

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nobody_nowhere
Statements like this always bring me back to Goedel's incompleteness
theorem...

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everdev
Infinity in nature sounds like an untestable argument. If space was infinite,
by definition we'd never be able to measure it's boundary to verify the claim.
The closest we could come is "we still haven't discovered a boundary to the
volume of space".

Basically the claims "beyond the farthest point you can measure there is more
space" and "beyond the farthest point you can measure there is nothing" are
equally plausable and untestable.

~~~
psychometry
If you had read the article, you'd learn that the problem is much richer and
more complicated than the way you stated it.

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foobar1962
1/0 is not infinity, it’s undefined.

For the function 1/x as x->0 from the positive side the function tends to
positive infinity, while as x->0 from the negative side the function goes to
negative infinity.

The function cannot be both plus and minus infinity at the same time.

~~~
opmac
While you are technically correct, that's pedantic and doesn't impact the
conclusions of this article one bit, nor does it really add to the discussion.
It's written for the layman, and chose to purposefully ignore technical
details such as lim x->0+ 1/x = +inf and lim x->0- 1/x = -inf.

------
misja111
I find the article a bit pointless. The concept of infinity is made up by men.
Just like the concept of numbers or logic. There doesn't exist something like
a 'four' in reality, this only exists in models that we have created.

Therefore it goes without saying that there is no 'infinity' in physical
reality. On the other hand, nothing is keeping me from creating some model of
reality that has infinity in it.

~~~
simonh
The point in contention isn't that infinities can be imagined, it is whether
they are useful - specifically in the context of cosmology.

Numbers are clearly useful, otherwise it would be difficult to explain the
importance we ascribe to the value of our bank balances. However just because
you can formulate a model of reality including infinities it does not
therefore follow that this model must be useful, or that infinities must be
useful.

~~~
jerf
"The point in contention isn't that infinities can be imagined, it is whether
they are useful - specifically in the context of cosmology."

Going the other way may be more useful; is it possible to create a correct
model of reality that does not at any point use infinities? You could have
alternate correct models that do, but if you can create one without them you'd
have a reasonable basis to call the infinities in the other models unnecessary
and nonexistent.

(I would suggest before anyone rush to correct me that they read what I said
carefully. For instance, "reasonable basis" != "undeniable proof", which
experience leads me to believe is a difference a hypothetical replier might,
in their rush to reply, overlook. Also note the _two_ clauses in the question;
if it is simply impossible to create a correct model of reality at all, we may
never get to the "infinity" portion of the question.)

~~~
simonh
If you could do that - create a correct model of reality - at all infinities
or not, there's a Nobel Prize with your name on it. :)

