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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?

  • 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 2:35
  • 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 :) – Drux Sep 26 '14 at 7:53
  • 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 6 '14 at 10:27
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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.

  • 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. – Nick R Sep 26 '14 at 0:32
  • @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 7:00
  • @TMS: yes, agreed; I tend to think that 'voids' and 'infinities' make no sense physically... – Mozibur Ullah Sep 26 '14 at 7:04
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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. – shane Sep 24 '14 at 14:49
  • The standard (Copenhagen) interpretation of QM is epistemological.. – Mozibur Ullah Sep 24 '14 at 19:41
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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 0:00
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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 0:18
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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.

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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 -- 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
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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
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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
0

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.

  • 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 7 '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 7 '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. – jobermark Oct 7 '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! – Wilhelm 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.

0

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.

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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. – Wilhelm 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! – Wilhelm 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

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