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The physics of infinity
- deleted 8y ago[deleted]
- techno_modus 8y ago> 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.
- subroutine 8y agoWhat aspect of waves as they exist in our universe is infinite?
- whatshisface 8y agoIf waves are continuous then they are over the real numbers, and there are an infinite number of reals between any two points.
- subroutine 8y agoThis is specifically addressed in the article - this is a function how our maths represent phenomena: "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."
- jamesrcole 8y agoIt 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 8y agoIf 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 8y agoThe 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 8y agoZeus 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."
- jamesrcole 8y ago> Zeus and Sherlock Holmes could both be argued to have had transitive causal influences upon us There's some philosophers who'd argue that, but I think the arguments are pretty weak.
- simonh 8y agoThe fictional character Sherlock Holmes certainly exists.
- jamesrcole 8y ago
- nobody_nowhere 8y agoStatements like this always bring me back to Goedel's incompleteness theorem...
- mannykannot 8y agoI 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.
- techno_modus 8y agoYour room and elephants are different objects. Your room has some size and it is not empty but it has zero elephants or a finite number of zero size elephants. 0 = nothing = non-existence.
- whatshisface 8y agoThe complete wave function of your room contains some elephant-like fluctuations in the distant decimal places.
- qubex 8y agoI have met the Queen of England exactly zero times.
- whatshisface 8y agoThe complete wavefunction of you and the Queen of England involves a handshake-like fluctuation in the distant decimal places.
- qubex 8y agoBut once either of us dies there won't. Zero can be meaningful in describing the world. I agree infinity, in its truest sense, can't... but zero definitely can.
- NihilumExNil 8y agoNo, no, null is not meaningful. That's the whole point. It is meaningful as neutral element of a ring, as center of a coordinate system or as limit of some function, sure, but only in comparison to other values and the other values are all that matters. Not having the other values, having literally nothing, the word ''having'' would be meaningless. There is always something. 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.
- misja111 8y agoI 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.
- jacobmoe 8y ago"4" is a concept, but you don't deny that the 4 turnips on your table do have physical reality, do you? Is the notion that there are an infinite number of turnips spread out through an infinite universe "just a concept" whose physical reality we don't have to bother ourselves with, in a way that those 4 turnips aren't?
- zaarn 8y agoI will not deny that a collection of atoms, mostly carbon, hydrogen and oxygen, with the rough weight of 4 turnips, is on the table and looks like 4 turnips. 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.
- simonh 8y agoThe 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.
- cjohansson 8y agousefulness is a pragmatic attribute, it does not by itself say that something really exist in reality, unless you believe that everything that is useful exists. I think the OP do not believe that everything that is useful exists in physical reality, therefore seeing four tables does not mean that fours exists in physical reality separated from tables or other objects. So you are probably using different metaphysics.. 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
- andyjohnson0 8y agoFrom 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 8y agoDivision 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 8y agoI 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 8y agoCan 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.
- deleted 8y ago[deleted]
- foobar1962 8y ago1/0 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 8y agoWhile 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.
- Koshkin 8y agoSure there is an infinity without a sign.
- j1vms 8y agoAFAIK, 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
- curlcntr 8y ago"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 8y agoI 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.
- jkabrg 8y agoThe dual numbers contain infinitesimal numbers but not infinite numbers. 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.
- monster_group 8y agoThe article doesn't load for me in Firefox or Chrome. It's stuck at 'Loading enhanced PDF...'.
- maaark 8y agoDoesn't load for me either. Stuck at 'pay us money'.
- everdev 8y agoInfinity 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 8y agoIf you had read the article, you'd learn that the problem is much richer and more complicated than the way you stated it.
- pphysch 8y agoThe 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 8y agoEven 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?
- swerveonem 8y agoWell said, I wish I knew you in real life.
- pphysch 8y agoI 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.
- throwaway37585 8y ago> If the universe is not truly quantized... mathematics is not related to physical reality How does that follow? Mathematics is perfectly capable of modeling continuous quantities.