The physics of infinity
nature.com
nature.com
0 = nothing = non-existence.
There's something about nihilism. It's being a self defeating argument, but educating. Then there's constructivism in various senses.
There's something about finitism, too, computable numbers and what not. Infinity is rather ... monotonous.
If you're going to look at more than just integers, infinity is everywhere, in between every number you can count, and then some! 0 is as meaningless but so is every other number!
If you limit to integers or even just whole numbers, 0 is meaningful, just like all the numbers...
https://en.wikipedia.org/wiki/Boltzmann_brain
I'm not sure if a functioning elephant is more or less likely to be honest....
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.
Not really, deterministic interpretations of QM don't require such mental contortions.
The field theory covering "quantum noise", the existence of virtual particles, and the vacuum is also completely different.
I'm not sure how well established Virtual Quantum Noise is, but saying that an empty place contains this is just wrong. If it has an effect in the propagation of a real particle in to said space, it wouldn't be empty anymore. That's confusing, and it might be helpful to note that these quantum fluctuations happen on the boundaries of the space, not just everywhere (and if they did, they wouldn't matter any more than my imaginary friend the pink elephant Safr riding the fluctuations).
You are talking of black holes, by the way, aren't you?
As far as I know this is not physically conceivable, if physically conceivable means physical laws as we understand them.
Quantum vacuum state, or zero-point field is not completely void empty.
Then we found things smaller. Sub-atomic particles. Literally makes no sense because it literally means smaller than the smallest thing.
The universe is apparently fractal af.
> 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?
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.
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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(- ∞) = + ∞
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.
lim (x->0⁺) 1/x = ∞
There are several places in physics where you encounter infinities, the most prominent case probably is the aforementioned uncertainty principle, which leads to 'loops' in QED and need to be countered by renormalization[0].
Physicists have very good intuition but sometimes making their statements mathematically rigorous requires extra work. I think a good example is in the history of field theory. Over time the intuition was made into something more formal. The article in the OP discusses renormalisation.
[0] https://en.wikipedia.org/wiki/Projectively_extended_real_lin...
> 1/0
Infinity
Q.E.D.
unless you want to claim that javascript is occasionaly illdefined.
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.
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?
>"...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.
How does that follow? Mathematics is perfectly capable of modeling continuous quantities.
What, to you, would not qualify as a human abstraction?
> How do you define quantity, and can you do it without the notion of counting and discrete objects?
Absolutely. See Tarski's axiomatization of the real numbers, for instance.
Nothing insofar as "what" implies a load of human abstractions. But it's clear that there is a lot of "stuff" out there that no human has ever experienced. It is extremely improbable that humans exist at a "Goldilock's scale" wherein we are even physically capable of experiencing "everything" and of finding boundaries on the universe.
> Absolutely. See Tarski's axiomatization of the real numbers, for instance.
I meant counting in the broadest sense, which is where numbers themselves emerge from. It seems plausible that there are alien modes of cognition that don't rely on the notion of object and can approach continuous "stuff" more directly. Maybe even some terrestrial organisms work this way.
Are you asking how define quantity without quantity?
Also, I think it's fair to say there are quantities that no human has ever "experienced".
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.
"Physicists have found it convenient to use the concept as a mathematical idealization where infinity occurs as a limit of large numbers, even though in physical reality there is no infinity of anything. An example is the Fourier series: while mathematically an infinite number of terms are needed to give an exact representation of the function representing an arbitrary waveform, this does not in fact mean that there are infinite physical frequencies occurring in the real physical system."
There's some philosophers who'd argue that, but I think the arguments are pretty weak.
But really, this discussion is too large to satisfactorily deal with here.
We characterize all sorts of things in all sorts of ways, trying to model the reality we perceive, but actually those models are also reality and they are not the same as the thing being modelled.
From this standpoint everything fictional exists but whether there is an 'infinite' amount of it is a super annoying question that just keeps going in circles so I don't really care so much what the answer is.
But if it's infinite then there is a god, right? :^)
No they don't. The mental representations exist, the thought processes exist. That doesn't mean the fictional entities exist above and beyond those.
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.
Oppositely, the variable X of a polynomial behaves much like an infinity, but is not invertible.
Their argument depends crucially on whether infinitesimals/infinities are invertible.
It is the application of p-adic analysis to quantum mechanics.
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.
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.
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.
The 4 turnips do have physical reality but on the other hand, the 4 in this scenario is not a realized concept, there is nothing physical about the 4 in this scenario since all the things in it are made of particles made of mostly nothing.
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.
But sure, since the days of William of Occam pragmatic metaphysics have been popular, but again popularity does not necessary prove something is right. Just explaining both sides of the argument here and trying to resolve the potential missunderstandings
Of course "they" do - just like anything you care to use a word for, e.g. "humans." All words are, in fact, abstractions, and these abstractions do exist in a meaningful sense.
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.)