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Aristotle made a distinction between infinities that were in potential (dunamis) and in actuality (energia); and stated that actual infinities did not obtain in the physical world. This is the basis of Kants antinomies of time and space.

It has also indicated in physics were theories 'breakdown'; for example black holes were discovered when matter was squeezed to an infinite point; and the quanta of radiation when the theoretical explanation showed that the blackbody radiative spectrum would be infinite.

Is it possible to argue with Aristotle and consider that there are actually physically real infinities in Nature?

This leads to a separate question - can one argue that theoretically infinities aren't physically obtainable; or is it an empirical notion?

note

From an instrumental perspective it appears no classical direct macroscopic quantity purely by definition can be real; what would it mean for say 'velocity' or 'energy' to be infinite?

Mozibur Ullah
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  • Sadly it is no longer possible to argue with Aristotle, for obvious reasons. But surely one must distinguish between the limitations of our theories, and the nature of reality itself. There is no contemporary physical theory that posits any infinite or infinitesimal quantities in nature. Note that I exclude multiverse theories, since those are beyond experimentation hence beyond science itself. – user4894 Sep 24 '14 at 02:35
  • If one takes the multiverse seriously then surely one has an infinity of particles? Aren't physical theories one aspect of *Being*? Spinoza has a section on physics in his *Ethics*; whats wrong with marking this *Ontology*? – Mozibur Ullah Sep 25 '14 at 13:40
  • Its worth pointing out that physics is rooted in Milesian cosmology and materialism. – Mozibur Ullah Sep 25 '14 at 13:50
  • [related](http://philosophy.stackexchange.com/questions/16166/to-use-the-limited-to-pursue-the-unlimited-is-foolish) :) – Drux Sep 26 '14 at 07:53
  • I don't know if Balck Hole has infinity, but my room is sure infinite. Continuity is the answer (curse you Leibniz!). Moreover we everyday live with infinity which we dont want to notice. Time. Do you think(feel) that time is infinite? Or you think that there is something non infinite in time? – Asphir Dom Oct 06 '14 at 10:27
  • @user4894 The multiverse doesn't go beyond science itself. Modern physics is all about positing unobservable entities and processes to explain that which is observable. You might argue that we ought not be ontologically committed to these things but a multiverse is by no means a 'special' entity to be excluded from development of physics. – Joe Lee-Doktor Aug 22 '18 at 11:10
  • @JoeLee-Doktor You don't seem to be distinguishing between theories that can be verified (up to a good number of decimal places) by experiment; and ideas such as the multiverse, which can never be confirmed by experiment. – user4894 Aug 23 '18 at 18:21
  • Consult this paper for some writing on the subject: https://arxiv.org/abs/1801.05016 – Joe Lee-Doktor Aug 23 '18 at 18:32
  • But @user4894 could you tell me more about this distinction? What do you mean by 'confirmed by experiment'? – Joe Lee-Doktor Aug 23 '18 at 18:32
  • @JoeLee-Doktor Are you joking? Science 101. Take Newtonian gravity. It predicts bowling balls fall down. We drop a bowling ball, it falls down. That's a confirmation. We can confirm quantitative predictions too, to a good degree of accuracy. Nobody has an experiment that could confirm or falsify multiverse theory. If you're seriously asking me to explain experimental science, I can't take your comments seriously. – user4894 Aug 23 '18 at 18:41
  • Wow. You should do some reading on philosophy of science. There are many different conceptions of how experimental science should be/is performed. But you seem to be talking about a fairly standard confirmation theory there. Simply, you derive consequences from the existence of a multiverse for our observable universe. Even if the multiverse itself cannot be observed, its effects on our own may be. There are other skeptical worries you could raise with this methodology but this is standardly how scientific discoveries in fundamental physics work. – Joe Lee-Doktor Aug 23 '18 at 18:47
  • @JoeLee-Doktor: I have done some reading on the philosophy of science - enough to recognise what you are talking about. Can you mention five experimentally verifiable tests of the multiverse apart from the famous one by Weinberg on the value of the cosmological constant? – Mozibur Ullah Aug 29 '18 at 13:16
  • @MoziburUllah "Experimentally verifiable test" is an interesting phrase. Anyway, your expectation that someone should be able to deduce predictions of a multiverse hypothesis on the spot is absurd. I'm not a theoretical physicist and the idea of a multiverse emerges from many different complex theories of modern physics. The point is, the multiverse is not something which is uniquely unscientific in physics. People have a problem with it because you cannot 'observe' it. News flash: You can't observe particles. – Joe Lee-Doktor Aug 29 '18 at 15:07
  • @JoeLee-Doktor: I didn't bring up the multiverse in this question. The question is about the distinction between actual and potential infinities in physics. I'm merely referring to your phrase 'you derive consequences from the existence of a multiverse for our observable universe.' If you think that is absurd to then ask 'what are the physically observable consequences' then you have an absurd relationship to reason. I was trained as a theoretical physicist and I've yet to hear of any significant observable consequences apart from the one I mentioned by Weinberg. – Mozibur Ullah Aug 29 '18 at 15:11
  • @JoeLee-Doktor: You can observe atoms through electron-scanning microscopes. News-Flash: you're wrong - badly. – Mozibur Ullah Aug 29 '18 at 15:15
  • @JoeLee-Doktor: Had there been any significant predictions by the multiverse hypothesis I think by now we would have heard of them. The science news that focuses on this highly speculative idea is sensationalist, because cheap news is usually sensationalist - like Fox News - and usually bears a superficial relationship to anything important. Real physics is pretty boring which is why cheap news outlets won't touch it with a barge-pole. – Mozibur Ullah Aug 29 '18 at 16:14
  • News flash: Atoms =/= sub-atomic particles. Even so, there is considerable controversy in the field of philosophy of science as to whether or not electron microscope observation actually counts as "observation" since the data collected from an electron microscope requires such detailed theoretical interpretation. – Joe Lee-Doktor Aug 29 '18 at 19:34
  • @joe Lee-doktor: Not really, it just means there are different classes and kinds of observations. – Mozibur Ullah Aug 29 '18 at 21:29
  • @MoziburUllah Time to read some van Fraassen. – Joe Lee-Doktor Aug 29 '18 at 22:21
  • @MoziburUllah https://plato.stanford.edu/entries/science-theory-observation/ might be a starting point. – Joe Lee-Doktor Aug 29 '18 at 22:23
  • Oh and https://plato.stanford.edu/entries/constructive-empiricism/. – Joe Lee-Doktor Aug 29 '18 at 22:30
  • @Joe Lee-Doktor: Yawn. Perhaps it's time for you to actually crack open a physics textbook and learn the physics part of philosophy of physics? As for constructive empiricism - do you recall where I asked you for 'five significant predictions of the highly speculative multiverse hypothesis?' A question which you rather clumsily avoided answering? That's so constructive... – Mozibur Ullah Aug 29 '18 at 22:46
  • @MoziburUllah No worries dude I'm starting a degree in theoretical physics next year. And it's pretty clear you still haven't read anything about constructive empiricism so sweeeet -and whether or not *I* can personally muster predictions made by the existence of a multiverse changes nothing about whether or not it's the case. Google is your friend. – Joe Lee-Doktor Aug 29 '18 at 23:19
  • @Joe Lee-Doktor: Well, it's one thing to begin a course in theoretical physics and it's another to complete it with distinction. As for constructive empiricism - I'll point out that yet again you avoided the question I asked empirical evidence for the multiverse. That sort of manoeuvre won't help when you come to taking your exams... – Mozibur Ullah Aug 29 '18 at 23:23
  • @Joe Lee-Doktor: The first quotation from your link to the SEP expands on the notion of constructive empiricism as being an 'empirically adequate' theory; hence my questions about concrete predictions of the multiverse. As it is, it's a highly speculative theory with inadequate empirical and theoretical support - it's basically kept in the public eye through the sensational picture it conjures up. That ain't so sweet. Yeah, Google is your friend if you're looking for a mish-mash of superficial and ill-digested ideas. – Mozibur Ullah Aug 29 '18 at 23:30
  • That won't help when it comes to passing your exams - in fact too great an exposure to Google will only hinder the learning process. – Mozibur Ullah Aug 29 '18 at 23:32

9 Answers9

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Most physicists don't accept infinities for a very obvious reason: such infinite physical objects are not quantifiable! That is, we can't measure them or even prove that they are infinite.

Through the history of physics, infinities were raised in formulas, and usually in these cases the formulas were thrown away, considered as incomplete, or they kept searching for mathematical tricks to avoid them. That is, they were considered as mathematical artifacts. Those approaches until now have been very successful.

As an example, when physicists tried to apply Maxwell's equations of electromagnetism to electron self-energy infinities were raised. That was actually a huge problem, because in other areas those equations was extremely successful in describing reality. Later on we understood that they were incomplete and quantum electrodynamics solved those infinities.

Infinities were also raised in general relativity with the singularity of the black hole. In line with our previous practice, physicists considered it to be incomplete, because so far we have not been able to successfully unify gravity with quantum physics, which we hope would put a "limit" on the kind of singularity that can exist and "patch" the law's break down as you mentioned.

There are other examples that could be mentioned, but they are perhaps harder for non-physicists to understand. As I recall, currently there are two major problems left with infinity in physics: gravitational singularities and vacuum energy.

P.S. (1)

The example mentioned by "Niel de Beaudrap" is totally misleading in my opinion, because there is actually no infinite temperature due to relativity, and Plank Temperature is the most that we can get. And the negative infinite temperature that he mentioned is a mathematical artifact, because in this case the physical meaning of temperature breaks down and it becomes just some abstract mathematical parameter that holds no physical meaning by itself. Even so it takes the same place in formulas like the usual temperature, so it is just an analogy.

P.S. (2)

Some modern theories of cosmology admit the existence of an infinite amount of different universes. That is, they admit infinities. Anyway, those are just theories, and it seems (till now) that there is no way to prove them.

Edit(1)

In response to "shane's" answer, I would like to emphasize that in physics it is basically possible (at least theoretically) to move from point A to B in 0 time, and that is not only due to entanglement in quantum physics as mentioned in comments (and which really depends on the interpretation you use), but even due to more "classical" reasons, which is general relativity, because it has the ability to bend the space-time sheet to connect two points on different sides of it. It should be mentioned that time here is a relative thing, so we should be really cautious about "relative to which observer" it will be 0 time. Anyway there is no "infinite" speeds here.

Frank Hubeny
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TMS
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  • I notice that the article you link to mentons 'infinite tempretures' but they describe them as artficial ie formal. – Mozibur Ullah Sep 25 '14 at 22:42
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    @MoziburUllah, yes and that what I mentioned too, it is just formal analogy, not physical thing. – TMS Sep 25 '14 at 22:47
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    I like this answer. The mention of vacuum energy is plausible, but subject to my naive understanding of physical theory. If infinity is to be found, vacuum energy looks like a good place to look. – nwr Sep 26 '14 at 00:32
  • @NickR: sure; I think gravitatonal singularities are another plausible candidate; Hawkings solution was to hide them behind a horizon; like quark confinement - except you can scatter a quark; in string theory (or hypothesis) there are microstate calculations that match black hole entropy; of course once it was conjectured that they had entropy its worth looking for microstates - which has the additional benefit of removing the singularity. – Mozibur Ullah Sep 26 '14 at 07:00
  • @TMS: yes, agreed; I tend to think that 'voids' and 'infinities' make no sense physically... – Mozibur Ullah Sep 26 '14 at 07:04
  • "[negative temperature is] some abstract mathematical parameter that holds no physical meaning by it self" ... what can it mean, for negative temperature to play a role in a physical theory and yet have no physical meaning? Is temperature only 'real' if it is a possible reading of a length of a column of mercury, or a similar system, set against a properly calibrated linear scale? – Niel de Beaudrap Oct 03 '14 at 10:49
  • Your comment about infinite and negative temperature ignores the unifying trend of physical theory. We should not propose meaningless algebraic equivalencies, but we ought recognise algebraic equivalencies where they exist. Be careful, because your same line of reasoning would suggest that 'quarks' and 'potential energy' are merely fables used as accounting devices, and that the Einstein equation, E=mc^2, is a meaningless muddle. These, too, are possible positions to take in the interpretation of physics, but very heterodox, and in the end more fictionalist than the realism you seem to prefer. – Niel de Beaudrap Oct 03 '14 at 10:55
  • did you hear of Renormalization? Problem of infinity is not matter of physics it is the matter of mathematics and reason (human mind). It is clear that there are no infinities in physics because time of any experiment is finite. – Asphir Dom Oct 06 '14 at 10:35
  • @NieldeBeaudrap Quarks are exitations of qunatum fields, not potential energy, and here precisly no analogy, but a deep physical relation, it's totaly different of what I mentioned. – TMS Oct 07 '14 at 15:54
  • @AsphirDom I mentioned Vacuum Energy, and renormalization kinda related to this, but it is harder to grasp so I didn't mention it in detail. – TMS Oct 07 '14 at 15:54
  • @TMS: I'm aware that quarks are not potential energy. What they have in common is that they do not correspond to directly observable phenomena. Quarks do not occur in isolation: their presence is only inferred (and more indirectly than e.g. electrons). Similarly, potential energy originates as the idea of there being somewhere that kinetic energy has to go to. (Kinetic energy, of course, is also not directly measurable; similarly for the baryons that quarks form.) The analogy is this: like negative temperatures, they are formal explanatory devices for phenomena beyond our direct experience. – Niel de Beaudrap Oct 07 '14 at 16:23
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Physical infinities lead to impossibilities pretty quickly. For instance, suppose it were possible for something to move at an infinite speed, then the time it would take for the object to pass from point A to point B would be 0. But then there is some instant t such that the object is at A at t, and at B at t. Hence the same object is in two different places.

These are the kind of considerations that Aristotle advances and it's not clear to me how he could be wrong. If anything I'd think that the discovery that light moves at a fixed speed is a pretty impressive confirmation of Aristotle's basic insight.

  • Yes, agreed; I would have been surprised to find his basic insight into this being confounded; and of course you're about the speed of light; and it puts into context the debate about the instantaneous 'transmission' of both light and the force of gravity. Your paradoxical example is intriguing; particularly in relation to QM where sometimes it is argued that objects can be in more than one position at the same time; this isn't to say I'm disputing your reasoning as of course the same reasoning holds for the wave-function. – Mozibur Ullah Sep 24 '14 at 14:45
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    I'm not competent to speak about QM, i'm afraid. I've heard the claim advanced that QM show us that one thing, an electron say, can be in two places. But it's not clear to me why it is this *ontological* claim is the thesis we should hold, and not just the weaker *epistemological* claim "we can't know which of these places the electron is in." As I said, I'm not competent to argue about this though, unfortunately. –  Sep 24 '14 at 14:49
  • The standard (Copenhagen) interpretation of QM is epistemological.. – Mozibur Ullah Sep 24 '14 at 19:41
  • @shane "Objects" of quantum mechanics are not objects to which Aristotle's reasoning even applies. It's not that an electron can be in two places at once, but that for quantum objects the question of what place they are at is meaningless as well as most other questions that come from imagining them as little balls whose location we just don't exactly know. – Conifold Sep 25 '14 at 19:17
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    @Conifold: the question of position and momenta for an electron in QM isn't *meaningless*; the quantum wave-function is precisely deterministic so its specified exactly; its a probability wave and phenomenologically speaking it manifests itself, if one takes the standard interpretation, as a 'wave' or 'particle'; its more that the question of interpretation becomes difficult - as exemplified by ongoing debates; as for Aristotle: what does an 'infinite' momenta, position or energy *mean* here? For a value to be measurable it must be finite; notably the dirac function takes an infinite – Mozibur Ullah Sep 26 '14 at 00:00
  • value on point support; but it is interpreted solely under an integral so its rendered finite; particles per se may not exist but that doesn't them being useful:in scattering calculations via Feynman diagrams or as strings in string theory where they are essentially particles with extension like the original Democritean atom; personally speaking, QM with the standard Bohrian interpretation looks Kantian to me ie an epistemological rather than an ontological theory; where is the *ding an sich*? – Mozibur Ullah Sep 26 '14 at 00:14
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    Wave function is not a probability wave, the square of its absolute value is probabilistic, but there is no closed equation of motion for it to turn it into a wave. Wave/particle is Bohr's metaphor, not the modern interpretation, they don't behave like either or even as a combination of both. Future position and momenta as computed in quantum mechanics are not even expected values in the classical sense, and certain effects can be transferred instantaneously in a meaningful way (quantum nonlocality) although Aristotle's "reasoning" seemingly applies to them too. – Conifold Sep 26 '14 at 00:18
  • Sure; its exactly because its amplitude is probabilistic that I called it a probability wave/function; Not only Bohr but Feynman too; and its still very useful; which *modern* interpretation? Which is why [Quantum non-locality](http://en.wikipedia.org/wiki/Principle_of_locality) is controversial: ' To date, no test has simultaneously closed all loopholes to the idea that entangled particles violate the principle of locality or engage in superluminal communication'. – Mozibur Ullah Sep 26 '14 at 00:42
  • The ding an sich is the quantum state, we just don't have an analog in our life experience to correspond to it informally. Waves and particles don't fit, but we got nothing else, so a qualitatively new intuition has to be developed. Questions like how did the electron get from A to B shouldn't be asked, just like one doesn't ask about the shape of water. "Position", "momentum", "particle" only sound familiar, they behave very differently from the classical concepts under those names. – Conifold Sep 26 '14 at 00:45
  • As the article I linked to explains Newtons theory violated non-locality and it took Lorentz, Poincare and Einstein to return to it; one doesn't give up such a simple philosophical principle easily; sure, but its also true that there is a'family resemblence' to use Wittgensteins term for the lack of a better one. – Mozibur Ullah Sep 26 '14 at 00:58
  • The majority interpretation is still Copenhagen, which "rejects questions like "where was the particle before I measured its position?" as meaningless" http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics#The_Copenhagen_interpretation. Actually, quantum non-locality is consistent with locality in the sense of relativity (energy doesn't travel faster than light, only corellations might), the uncertainty principle does allow violations even of that. But it's not controversial, math just works out that way, only interpretations are. – Conifold Sep 26 '14 at 01:06
  • You speak of speaking of impossibilities too quickly too quickly. –  Oct 08 '14 at 22:53
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Yes. Infinities in physical theories are possible.

In principle, a physical model is just a mental device for reasoning about experiences. "Infinity" is also a device for reasoning about things; and for all a physicist might protest that there are no actual infinities, it certainly is used as a convenient tool for approximations in lower-year physics exercises in university. As such, all one demands of an "actual infinity" in a physical theory is that some quantity is assigned an infinite value to describe a physical situation, and that the future behaviour of that same physical system is predictable by the laws of physics in that model.

I assert that, even without examples, it is unclear how you could ever exclude a physical theory from giving useful predictions despite admitting infinite quantities. Given enough mathematical sophistication, you can readily "squeeze in" (non-canceling) infinities and still have a consistent model of physics. Perhaps this would not be an infinity of the form "infinite number of apples" or "infinite speed", but why can more exotic infinite values be ruled out?

In fact, there is an example: temperature can be infinite. This is not a state of affairs which one would expect of a thermodynamic system (an "infinitely hot" black body would contain an infinite amount of thermal energy). But thermodynamics is formulated in such a way that one can speak of infinite temperatures of other systems, e.g. a row of magnets (or spins) in a surrounding magnetic field. It is possible to have the magnets almost all pointing opposite their neighbours (a high-energy configuration), so that there are few ways to add any more energy to the system (while preserving the constraint of being a row of magnets). This is a configuration of negative temperature, which is more energetic than any configuration of positive temperature, and also unstable. If disturbed, it will quickly relax to a configuration where most magnets align with their neighbours, which has positive temperature — passing momentarily through a configuration with infinite temperature as it does so, not unlike a ball thrown into the air momentarily being at rest as it accelerates downwards.

Is the negative, and infinite, temperature "real"? Only as real as "temperature" ever is, which is a parameter in our description of the world — a subtler one than position and momentum perhaps, but one which we accept readily enough. Our model of physics tells us that this is one way that temperature can be. If you side with Aristotle and have read your Popper, you might say that this falsifies our theories of thermodynamics. But why would it not instead falsify the principle "there are no actual infinities"? Perhaps you would prefer to consider the thermodynamic parameter β = -1/T instead of the temperature T; indeed, this is common practise in physics research, and clarifies the first law of thermodynamics (absolute zero corresponds to β being negative infinity). You'll have to cope with every day's weather having a forecast of sub-zero "negative inverse temperature", but that's possibly just the price to pay for having your life free of the spectre of the infinite. However, society has standardised on using temperature as we know it; and so until further notice, our best physical theories do allow some actual infinities.

Niel de Beaudrap
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    Aristotle distinguishes between things we can measure as infinite (i.e., by cutting into small pieces) and actual infinities. Negative infinite temperature seems to be an artifact of how we work with temperatures rather than an actual infinite (I found this helpful for deciphering how it works -- http://physicscentral.com/explore/action/negative-temperature.cfm ). But I don't actually have anything resting on the resolution of this question. – virmaior Sep 25 '14 at 11:22
  • @viramor: As a materialist, I find distinguishing between an "actual" infinity versus "something we can measure" strange. In principle, even such classic intangibles as 'love' and 'hope' may have multidimensional measurable values via a suitably subtle theory, given that they have observable impact on our behaviour. What is the difference between reality and that which we measure? Kant has some things to say on the matter after Aristotle, but it's not clear that we can get any clearer idea by positing that the true answer is impossible to actually know except filtered through our senses. – Niel de Beaudrap Sep 25 '14 at 13:58
  • @viramor: Dividing things into smaller pieces only yields potential infinities anyway: I do not think this is a 'measured infinity'. (Any such measurement, essentially by fiddling with units, would be finite.) That leaves your objection: who is to say that infinite temperatures are not merely artefacts of our theories? That's a fair objection, of course; but that amounts to asking whether our notion of temperature is ill-founded. It subverts not only the one possible value, but the entire framework of measurement. So either infinite temperature is acceptable, or the entire theory is flawed. – Niel de Beaudrap Sep 25 '14 at 14:00
  • ' it is unclear how you could ever exclude a physical theory from giving useful predictions despite admitting infinite quantities. '; yes - of course; but that isn't my question; undoubtably infinities are useful; but the question is whether they are *real*; 'thermodynamics is formulated in such a way that one can speak of infinite temperatures of other systems' - has an *experimental* setup demonstrated 'infinite temperature'? – Mozibur Ullah Sep 25 '14 at 14:22
  • I'm not asking about 'toy' theories that have theoretical import only or to demonstrate purely *logical* possibilities - important as they undoubtably are; and nor a mathematical artifact such as *1/a* as *a* approaches zero. – Mozibur Ullah Sep 25 '14 at 14:24
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    @MoziburUllah: "*I'm not asking about 'toy' theories that have theoretical import only or to demonstrate purely logical possibilities*" — Nor is the spin-system described above a toy theory: it is a physical system, to which the concept of infinite temperature is pertinent, using the existing broad-ranging theory of statistical mechanics. "*has an *experimental* setup demonstrated 'infinite temperature'*" — Do you mean 'has it read out infinity on a thermometer'? No. But from turning off to turning on, lasers achieve infinite (and negative) temperatures routinely. – Niel de Beaudrap Sep 25 '14 at 14:59
  • @MoziburUllah your questions indicate why I can't grok this question: there are useful/confirmed physical theories that formally involve infinite quantities (theory of 2nd order phase transitions come to mind) and which clearly describe what is going on in real, finite, physical systems but may not have ontological impact: is the correlation length in a ferromagnetic system _real_ enough that when it diverges it is an _actual_ infinity? Does the fact that, so far, this has only occurred in finite systems mean it is only a potential infinity? – Dave Sep 25 '14 at 15:02
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    @MoziburUllah: but crucially... "*undoubtably infinities are useful; but the question is whether they are* real" — Unless you commit to a specific philosophy of mathematics and of physics for the question, it is unclear how I could better answer you. I have answered from a formalist and naturalist point of view. If you are sufficiently a mathematical realist to believe that five is "real", that the square root of five might *not* be "real", and that there is only one true theory of physics (all others being clumsy imitations or derivatives), the answer might be 'no'. Otherwise, how to answer? – Niel de Beaudrap Sep 25 '14 at 15:02
  • @Dave: I think we can at least say that correlation lengths in finite systems are always finite. They may be potentially unbounded in the sense that by extending the system you can extend the longest distance over which there are correlations, but the system in which the correlations exist is finite. I was fretting similarly whether to use the example of wavelength of a constant field (having zero frequency), but decided that this was a notion of 'wave' that only a mathematician could love. Temperature seems to stand out as *bona fide* realisable 'physical' infinity. – Niel de Beaudrap Sep 25 '14 at 15:10
  • Do you mean 'has it read out infinity on a thermometer'? Partly, yes; 'I have answered from a formalist and naturalist point of view' - the formalist viewpoint isn't the one I'm asking about here; naturalist, is this the 'thermometer'? To which you answer 'no'; or the 'spin-system' to which you answer yes? When addressing theoretical constructs that are *inferred* I'd suggest that one ought to be cautious about their ontological import; it isn't a commitment to an philosophy of mathematics that I see as being important here – Mozibur Ullah Sep 25 '14 at 15:37
  • but a materialist one; I haven't previously come across the concept of negative temperatures; but [this](http://arxiv.org/pdf/1304.2066v1.pdf) paper by Dunkel and Hilbert claim: – Mozibur Ullah Sep 25 '14 at 15:44
  • ''A considerable body of experimental and theoretical work claims the existence of negative absolute temperatures in spin systems and ultra-cold quantum gases. Here, we clarify that such findings can be attributed to the use of a *popular* yet *inconsistent entropy* definition, which violates fundamental thermodynamic relations and fails to produce sensible results for simple analytically tractable classical and quantum systems. Within a mathematically consistent thermodynamic formalism, based on an entropy concept originally derived by Gibbs,... – Mozibur Ullah Sep 25 '14 at 15:46
  • absolute temperature remains positive even for systems with bounded spectrum. We address spurious arguments against the Gibbs formalism and comment briefly on heat engines with efficiencies greater than one'. – Mozibur Ullah Sep 25 '14 at 15:47
  • Putting aside the claim of 'inconsistency' and the violations of ' fundamental thermodynamic relations' the theory you allude to appears controversial; in which case making serious ontological claims for it, at least for me, is problematic; its worth pointing out that the paper states that 'negative absolute temperature' was introduced by Purcell and Pound in '56; and that Ramsey discussed some of their ramifications including the possibility of 'Carnot engines of efficiency greater than 1'; ie getting more energy than you put in... – Mozibur Ullah Sep 25 '14 at 16:03
  • @MoziburUllah: What devices do you deem acceptable for reading temperature? How sophisticated are they allowed to be? — Your caution as to ontological import presupposes that "infinity" is an ontologically contentious idea. Why so? It is a formal concept. What makes infinity intrinsically more extravagant than, say, pi? Are we obligated to force "infinity" into a role where it is represents a notion of excess of 'stuff', like mass or energy or time? If so you will have to formulate some notion of 'amount' which is general enough to encompass any concievable theory involving 'stuff'. – Niel de Beaudrap Sep 25 '14 at 18:26
  • @MoziburUllah: I cannot comment just now on the paper you have found, but if sound, it would not simply undermine the notion of negative temperature but the entire way in which statistical mechanics is discussed. Where is the boundary of topics, then, on which you are prepared to discuss things ontologically? Gravity and QM are inconsistent with each other; can we not then discuss those either? Are we restricted to our subjective experiences, and our personal notions of the infinite? – Niel de Beaudrap Sep 25 '14 at 18:37
  • I was simply asking what you mean by 'naturalism'; I take it that we would both agree that a thermometer reading is 'natural'; more extravagant devices need to be examined on their merits; which obviously occurs in the discourse of physical science; I would say not that infinity per se is 'suspect' but 'difficult'; and there are differing accounts of it - theological, physical, formal and philosophical; we aren't obligated to 'force' infinity into a 'role'; but rather examine carefully how infinity expresses itself in each context; the point of contention I see is this: – Mozibur Ullah Sep 25 '14 at 19:04
  • that you see infinity as a formal concept; whilst I accepting the role of the formal in mathematics struggle to reconcile it, for lack of a better notion, with physical intuition. My question was pointing towards the problematic of the infinite in physics; I tend to take it as one does as Occams razor; as a principle. – Mozibur Ullah Sep 25 '14 at 19:15
  • @MoziburUllah: naturalism meaning simply that 'nature', with measurable and with regularly acting phenomena, is all that there is. Why is measuring "temperature" most reasonable when performed by trying to bring mercury or alcohol in thermodynamic equilibrium with a system? A thermometer is a device which reacts under certain environmental influences, and is useful for 'measurement' because of the controllability of that reaction; any other device which reacts controllably is also a good measurement device — of whatever phenomenon prompts that reaction. (e.g. Geiger counters of radiation). – Niel de Beaudrap Sep 25 '14 at 21:31
  • @MoziburUllah: I take your point with "the role of the formal in mathematics struggle to reconcile [infinity]"; it is analogous to the question of ignosticism versus (a)gnosticism, where the latter is happy to contemplate the *sort* of being one *wants* to talk about, while the former demands first a solid definition. But in the case of the infinite, this struggle is over, *because* of formal investigation. The notion itself of the infinite is no longer much more mysterious than the integers. The only question is whether you wish only to consider infinities-in-place-of-integers, to count. – Niel de Beaudrap Sep 25 '14 at 21:39
  • For roughly the same reason one wouldn't accept Russells theory of types if it showed that 1+1 is indeed 2; sure and clearly any classical device is natural; but can the same be said for the configuration manifold of a single particle - is it *real*? Is it natural? It is certainly useful; so are statistics; that the average family in the uk has 2.2 children is certainly useful; but not in any sense real; it is of course the wrong question; its use in a coherent set of questions about population demographics is the right question – Mozibur Ullah Sep 25 '14 at 21:56
  • (That should be 'indeed not 2'); I can't say I agree; at least in physics one wants or looks for *correspondance*; there is, in my toy example, for example no (direct) correspondance with 2.2 children with actually existing children; No, not particularly; and neither Kant nor Aristotle made that comparison - perhaps you were confused by my use of the word 'struggle'; the question is about the infinite 'real', and not a logical one; Aristotle admitted the logical one but not the real – Mozibur Ullah Sep 25 '14 at 22:19
  • @dave:yes to both. – Mozibur Ullah Sep 25 '14 at 22:52
  • There is a simpler answer, isn't there? Aristotle argued against Zeno's appeal to infinite divisibility, but not conclusively. If I cross the room, haven't I actually traversed infinity? I haven't cut up the distance into smaller parts, because then I could not have crossed the room. Yet every attempt afterward to make the distance densely packed, such that _I_ am continuous from A to B, can be subdivided. What happens to this argument when it's subjected to quantum physics? – Dave L. Oct 01 '14 at 03:30
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Actual infinities are neither possible in physics nor in mathematics. The reason is so simple that it is generally overlooked.

Assume there exists the simplest actual infinity, the complete set of natural numbers. It cannot have any connection to the real world and it cannot be applied in mathematics.

What means to apply a natural number? It means to identify it by a name or to abbreviate it by digits (for instance to connect it with other numbers).

If you try this with any available natural number n, then you can easily see that 100*n* is also a natural number. Hence n belongs to the first percent of the complete set. Alas same is true for 100*n* and any multiple of n. Therefore you will not be able to identify any natural number beyond the first percent of the complete set.

Of course instead of 100 every larger factor can be used. Therefore all natural numbers that can be applied or identified belong to a vanishing initial segment of the complete set --- if such a set exists somewhere. Its existence would not have any consequences since almost all of its elements are inaccessible.

Of course we cannot describe anything actually infinite in physics because of the restrictions just mentioned. Physics however is the description and analytical treatment of reality. Therefore nothing can be actually infinite in physics.

What about the universal quantifier applied to actually infinite sets in set theory? It is simply ignorance of facts.

Hilbert7
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    There's mathematics about transfinite numbers. Therefore, they are possible in mathematics. Reasoning about infinite sets doesn't involve considering all the individual members, which would be impossible. – David Thornley Aug 22 '18 at 18:34
  • There are names of so-called transfinite numbers and there are names for angels and devils and spaghetti monsters. None of these delusions has any real contents, and none is mathematics. The proof above shows that there is no universal quantification possible over infinities. But without that there is no transfinity. – Hilbert7 Aug 23 '18 at 11:49
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    That's not really a proof, particularly when contrasted with the large number of mathematicians who work with infinities, and the fact that they have no problem with quantifications. When all the experts are against you, consider that you may be mistaken. – David Thornley Aug 23 '18 at 20:32
  • "The large number of mathematicians who work with infinities" is as conclusive as the large number of astrologians who work with astrology. That is in no way a convincing argument. In "fact that the experts have no problem with quantifications" because they are not aware of the limitations, perhaps by Freudian suppression. (They would no longer be "experts".) What I have proved is mathematical fact: You cannot use natural numbers beyond the first percent! – Hilbert7 Aug 24 '18 at 11:33
  • What you're saying is that we cannot name all natural numbers in finite time. Thus you confirm that there must be infinitely many of them! Otherwise we could name them all in finite time. – Ingo Aug 25 '18 at 15:42
  • @ Ingo: No, nothing about finite time! What I said is clearly visible above. Every natural number of the infinite set that can be used belongs to the first percent. This holds for all infinitely many named in infinite time. – Hilbert7 Aug 25 '18 at 16:02
  • Congratulatipns! You've discovered the infinitely many infinite subsets of the natural numbers! – Ingo Aug 26 '18 at 17:15
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Infinity, as a concept, exists in mathematics. Not only are there different types of infinities, but there is even an infinity of infinities! Physics also makes use of mathematical infinities (integrals have to be evaluated from - and + infinity), but in keeping with the intent of the question, there is only one physical object that comes "close" - the universe. The universe is discontinuous at the edge, and therefore, (by definition) it is infinite.

Guill
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  • 1) Infinite integrals means that we evaluate them up to infinity, but it never reach it! this is basics of limits and real analysis. 2) The universe doesn't have an "edge" nigher it is discontinuous there.. – TMS Oct 07 '14 at 15:56
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It is easier to argue that infinitessimals are part of almost all of our physics, and to be infinitely small, one still needs real notions of infinity. Even when we do not accept that measurements can be arbitrarily accurate, we still think of the spatial coordinate system as continuous. For that to happen requires infinite subdivision, and therefore infinity.

  • Infinite subdivisions are abstract not physical things, and most probably space is not continuous, but has small divisions, called "Plank Length". – TMS Oct 07 '14 at 15:59
  • People modelling what goes on at the Planck Length are still using continuous coordinates. I understand the notion of 'quantum foam' suggests space may be quantized, but we do not yet have a physics that works without continuous space. –  Oct 07 '14 at 18:44
  • @ jobermark: Nobody uses continuous coordinates because nobody *can* use them. Note that almost all "real" numbers cannot be used in computing because the set of computable numbers is countable. And not even this set is available, because real mathematics (in contrast to set theory) is restricted to numbers that can be finitely defined with a Kolmogov-complexity available within the computing system - that is far less than infinity! – Hilbert7 Aug 24 '18 at 11:39
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Nice. In order to assess if infinities are physically possible, we should first ask if (physical) finities are actually possible. The question assumes such fact, but that's debatable.

Finite numbers are a systemic mental construct: they are defined by ideal boundaries. For example, we take a continuous line, and split it in similar parts. That is our representation of the physical nature. If we find boundaries, we can count things, or formally systems. Although systems are just mental constructs.

For example, if there are few clouds in the sky, you can count them, and you can say that there is not an infinite number of clouds. But what happens if the sky is completely covered? Would you say that there's an infinite number of clouds? Or would you say there's just one cloud? Or that there's zero clouds?

Time and space are similar entities: sometimes we perceive them as discrete chunks, other times as continuous stuff. So we have learned to enumerate them. For example, a minute is a discretization of time. The equivalence with clouds in a rainy day would be to draw a grid on the sky and count clouds if cells are occupied.

There's an idea I enjoy exploring: physical nature would be like a number, but without the decimal point. What is the meaning of that? What would be a number without integer and fractional parts? But in fact, the problem comes from the other side: why have we chosen to make integers out of nature? Why did we created the decimal point? What is the point of numbering things? That's because our mind needs to define borders, limits, boundaries, frontiers in order to interact with nature. A cloud or a rainbow exist as an integer unit... depending on my subjective physical location, my perception, my memory, the scale of my existence. The same happens with a river. Or a tree. Or a rock. It seems that the borders of a rock are much more defined than the borders of a rainbow, but it's just a matter of scales. Things don't exist physically.

So, everything happens in our subjective perceptions. So, perhaps the real question is... are finities actually possible? And my personal answer is no. We have discretized matter in our minfs, but physically, everything is just energy, has no boundaries. In consequence, finities are not possible. Ergo, infinities cannot be physically possible. It is enough to count the clouds in the sky. Perhaps you and me can agree on the number, but does that physically mean something?

Update: two metallic objects can be put together and they keep being two objects. But the only reason they keep being two separate entities is because there's air between the surfaces. If two objects are joined in space, they become one; "there is no way for the atoms to ‘know’ that they are in different pieces of [metal]" (Richard Feynman). So, the apparent "number" of parts is just a subjective appreciation. Finite entities are apparent to our perception, but that has no physical meaning.

Or perhaps you mean that if an eternally-living person were able to count the number if clouds in all stars (after defining a precise taxonomy of clouds), it will never finish. That is out of our current knowledge. Perhaps she will only be able to count a finite number of clouds... forever.

RodolfoAP
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This question should be posted on the Physics SE and/or the Mathematics SE, so that the professionals can have a whack at it. But in the meantime, here is my take:

To begin with: although he may have been the leading mathematical physicist of his time, Aristotle by modern standards was neither a physicist nor a mathematician. Both fields have progressed so far in the meantime that whatever he might have thought or believed in his time is not even irrelevant today.

In real-world physics, there are no infinite physical quantities: no infinite forces, no immovable objects, no infinite field strengths. The appearance of an infinity in an equation is a signal of your having hit the applicability bound of that equation, and of the need for a new set of physical concepts if you wish to go further. The renormalization process in quantum electrodynamics is a perfect example of this. If you wish to argue otherwise, go over to the Physics SE and make your case with the professionals.

On the other hand, since mathematics is under no obligation to represent the real world, infinities can and do exist in that realm- and the practitioners of transfinite mathematics are neither deluded fools nor charlatans. Again, if you wish to assert the opposite, the Mathematics SE awaits.

niels nielsen
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In the Universe (as whole) there is only one infinite and absolute physical structure - it is Minkowski 4-D (spacetime) This so called ''Minkowski light cone'' is not abstract concept

The real image of ''light cone or 4-D'' is Zero Vacuum: T=0K.

Israel Sadovnik
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  • If you have any references that take this same view they would help support your answer and give the reader a place to go for more information. How do we know the Minkowski 4-D spacetime is not an abstract concept? Regardless, welcome. – Frank Hubeny Aug 01 '19 at 16:55