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And if so, are there any interesting implications? According to the storyline, Galileo launched modern science by declaring the necessity of rendering physical events countable. What is countable must be "defined" or literally translated into finite units.

Newton's great leap followed this maxim, notoriously placing a countable limit on the infinite Zeno-like regress of divisions arriving at "points" on a curve. Rendering motion countable. It worked! It worked so marvelously that all the metaphysical debates about it at the time were happily allowed to expire.

But what is the status today of the old philosophical bugbear of "the infinite"? Cantor's set theory produced a kind of "countable" definition of "infinity." But this was originally a disturbing turn for many and, as far as I know, does not have many if any applications in physics.

So, what is the status of "infinity" in philosophy and science now? Is it more or less accepted that science can only get going by performing the (I am tempted to say castrating) act of "defining" to enable counting? And Cantor sealed the deal by defining infinity itself in terms of counting?

(I ask in part because I am always a bit uneasy with modern cosmology and statements like "countable" hydrogen atoms in the "universe.") In any case, are there interesting current controversies about infinity in physics, math, and philosophy these days? Preferably understandable to the amateur.

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    See; 'The Influence of Spinoza's Concept of Infinity on Cantor's Set Theory' (Science Direct, Vol. 40 Issue 1, March, 2009, l.p.. 25-35. Your presence is a welcome addition to the SEP. Caute, – Charles M Saunders Aug 30 at 3:55
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    You start off by presenting a completely incorrect discussion of the history, mathematics, and physics, so I don't see how anyone can answer this, other than by correcting all your misconceptions. I suggest you start by posting a question on hsm.SE to clear up your confusions. Facts matter. – Ben Crowell Aug 30 at 14:34
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    @Ben Crowell. Well, okay. Not sure about you, but the reason I ask questions is precisely to correct my misconceptions. Since apparently every single statement in my "storyline" is wrong, I'll just take that as your answer. – Nelson Alexander Aug 30 at 16:13
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    Frankly, I do not see how either Galileo or Newton followed this maxim, calculus isn't countable in finite units. What was once disturbing in Cantor is the acceptance of actual infinity, but mathematics that came out of it is now ubiquitous in physics and everywhere else. I suppose one could interpret current intuitionists and predicativists as proposing alternative schemes with only potential infinity, but the conclusion then would be that the status of infinity is irrelevant for the kind of math that science needs. I am not even sure what "the act of defining to enable counting" refers to. – Conifold Aug 30 at 18:01
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    I do not know anything about Science, but modern mathematics has a pretty good understanding about "infinity". To clarify, mathematicians today regularly deal with infinite objects and they have a rigorous definition and clear understanding of what this means. This isn't very advanced mathematics either; students are taught this in their first year of an undergraduate mathematics program. – Agnishom Chattopadhyay Aug 31 at 22:08
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In the world of physics, things can get very very large, but not infinite. For example, if a physical model of some phenomenon predicts an infinite result in some circumstance, it signals a hard limit on that model's applicability, and it means there are physics that the model does not contain which are important in that particular case. It is then the job of the physicist to uncover what is missing, and add it in.

In addition, the amount of matter in the universe in the form of particles like protons and neutrons (called baryons) is not infinite; the baryon count is known to be of order ~10^80 which is an enormous number- but not infinite.

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    "In the world of physics, things can get very very large, but not infinite.". The observable universe may be finite, but much speaks for the universe itself to not be finite. Same for the baryon count - that applies to the observable universe. – Polygnome Aug 30 at 11:53
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    @nick012000 And that is excatly where our current models are not applicable anymore and we don't know what is actually going on – Christian Aug 30 at 17:35
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    Sets don't have to contain large numbers to be infinite, e.g. the interval $[0,1]$. – usul Aug 31 at 0:31
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    "It is known". Seems to me it's more like "It is believed". – J D Aug 31 at 5:41
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    @J D, I am a physicist, not a philosopher. if you want a complete explanation you should post a question on the physics stack exchange. But prepare yourself for a torrent of mathematics if you do. – niels nielsen Aug 31 at 16:10
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First, let's concede there are two conceptions of the infinite. One is the potential and the other is the actual. As for excluding the infinite, I think it's fair to say that the answer is a resounding no. One of the greatest advancements of science was Galileo's quantization of science; of course, one often then mentions the great leap of Newton and Leibnitz who introduced the calculus and infinitely small quantities. Any scientist worth his salt has taken calculus-based mechanics and E&M as entry-level science courses either in secondary or higher education. So, how can one champion Newton's laws of physical motion which are subject to the calculus and then reject the infinite? What about the use of extended real numbers? Reject those too? Very metaphysically unwieldy if not outright contradictory. The fact is that cosmology is a highly-mathematically centric pursuit and therefore is subject to mathematical suppositions. The age of the universe, the size of the universe, and other aspects of the universe lend themselves quite nicely to infinite quantities of one sort or another.

In fact, according to James Robert Brown in his entry, 'Mathematics, Role in Science' included in Blackwell's Companion to the Philosophy of Science (p.257), he writes "do space-time and the quantum state exist in their own right, separate from their mathematical representations: or are they nothing but mathematical entries." Think about it. Under the general theory of relativity, space-time is inherently curved and forces operate according to the curvature of the fields that determine them. And with the curvature of space comes the derivative and an infinite number of points. So, if one's metaphysics predisposed one to believe that the curvature of space-time is real, infinity is an inescapable aspect of physical reality.

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    I don't know much math, which is why I ask such questions. I had thought of Newton and Leibniz as deriving a "limit" to a potentially infinite calculation "to any degree of precision." But what these infinitesimals "really" meant was controversial and only formalized later on. So I had thought of them as (de)fining the problem to in a way to prevent infinite regress, of the sort we get in Zeno or Richardson's coastline. As for space time, I believe the current math derives a "beginning," so not sure how that squares with infinity. Since "points" are zero dimensional can we call them "real"? – Nelson Alexander Aug 30 at 16:34
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    @NelsonAlexander: Most "real" numbers can't even be represented - are the real numbers real? "Most" integers would not fit into the observable universe - are they real? Mathematical models are only models. The important thing (which this answer fails miserably to convey) is not whether they are an accurate description of the universe, but whether they allow us to make predictions about experiments. – BlueRaja - Danny Pflughoeft Aug 30 at 22:48
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    @NelsonAlexander Inifinity is still metaphysically controversial, as are the nature of reality and existence. What you need to understand is that there is a philosophical debate that separates along a dichotomy between realists and instrumentalists, roughly. An instrumentalist rejects that it's worth even talking about "true" reality, and instead believes that theory that predicts a phenomenon is only useful or not; in this case, it's irrelevant to discuss noumenon. – J D Aug 31 at 5:49
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    > So, how can one champion Newton's laws of physical motion which are subject to the calculus and then reject the infinite? What about the use of extended real numbers? Reject those too? -- Calculus tries to avoid infinite real hard, it's about approaching infinite. Hence, infinite(s) aren't considered numbers in the strictest sense; most I know mostly consider extended reals as notation that allows to write x = ∞ when others would perhaps write x → ∞. So I'd argue that you can indeed reject infinite in some ways, especially after taking calculus. – ljrk Aug 31 at 10:52
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    Agreed, hence my opening with potential and actualy infinity. One can reject actual infinity (I tend to, in fact), but still accept potential infinity. But rejecting some members of a class is not rejecting the class in it's entirety. Hence, by rejecting actual infinity, one can still accept potential infinity. Don't tell me you don't believe that infinite loops on computers don't actually exist. You'd argue, of course, they're not acually infinite, to which I would reply, what empirical evidence do you have that time on a go-forward basis isn't infinite? – J D Aug 31 at 16:44
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The set-theoretic definition of infinity is not "countable or uncountable," i.e. is not read off the concept of countability. Afaik the definition is, "A set is infinite if and only if it can be put into a one-to-one correspondence with one of its proper subsets." Or even, "A set is finite if it is not infinite," that's somewhere in the SEP articles on set theory.

Otoh things like Skolem's paradox are some kind of evidence for an absolute countability thesis, here, depending on how far they are taken.

As for physics, I believe Feynman at least wrote of "subtracting infinities" in renormalization. This might mean just a translation of the math into a finite form, or it might suggest operations involving infinity. This is the crux of the QM/GR gap, as renormalization doesn't work for gravity.

And then there are infinite multiverses to consider, infinity of history or of the future or of space or... Infinite-dimensional space even shows up (in a Hilbert or Minkowski form, if not both, but I don't recall clearly).

Ultimately, infinity can be hypothesized for something, is hard to prove, might be implied by a model, inductively corroborated, or vitiated by falsification survival rates; saying, "Infinity did it," is not quite like, "God did it," in a scientific context, so to say.

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  • Thanks, I'm pretty mathless, I'll have to look at the idea of "countability." I vaguely recall that definition of the infinite sets but I thought that was a new way of defining "counting." Subtracting infinities? Oof! That sounds dangerous! – Nelson Alexander Aug 30 at 16:48
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    A set is countable iff it can be put into a one-to-one correspondence with the set of natural numbers or a finite set. Now, think of clouds of irrational numbers between natural ones. Irrationals have countably many digits, but you wouldn't "count" to 2.6558643798643357... and then from there to... What? Any of those digits after the heading can vary in the cloud. You can continuously stream over these clouds but they're not discrete enough to be counted. – Kristian Berry Aug 30 at 17:04
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    @NelsonAlexander not exactly subtracting, but L'Hôpital's rule is an example of meaningfully dividing infinities. – kylefinn Aug 31 at 14:10
  • Renormalization isn't really "subtracting infinities". That's the pop-sci version of what's happening, and as with most pop-sci explanations, it's wrong if you take it too seriously. In fact, the real story is almost the opposite: the infinities in the theory are handled by imposing finite-energy "cutoffs", beyond which the predictions of the theory are no longer expected to hold. Renormalization theory tells you what happens when you move these finite-energy cutoffs around. – probably_someone Aug 31 at 20:25
  • Also, renormalization isn't really the crux of the QM/GR gap. There are well-known quantum theories which are clearly non-renormalizable (QCD, the theory of quarks, gluons, and color charge, is probably the most famous example), but that doesn't stop them from interacting with other quantum theories in an intelligible way. – probably_someone Aug 31 at 20:29
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No, there's no need whatsoever to exclude the infinite from science.

The gold standard for a scientific hypothesis is that the hypothesis

  • is consistent with all known observations,
  • successfully predicts observations made after the hypothesis was formulated, and
  • is the simplest hypothesis that fulfills the first two criteria.

If a hypothesis meets these criteria, then it will be generally accepted as a reliable tool for making predictions about the world. In other words, it will probably be accepted as being true, or at least as being true enough for all practical purposes.

And if a successful hypothesis invokes infinity, then so be it! Scientists don't throw away useful tools simply because infinity shows up somewhere.

Perhaps the most notable example of an infinity showing up in physics is time. In almost every model that exists in physics, from Newton's laws to general relativity to quantum mechanics, time is modeled as being infinitely divisible. In other words, every period of time, no matter how brief, is composed of even briefer periods of time. So, according to these models, during any period of time, infinitely many things happen.

The reason that these hypotheses model time as being infinitely divisible is that if they modeled time as being only finitely divisible, then they would (perhaps counterintuitively) be much more complicated, while still not explaining observations any better than they do currently.

(A common misconception is that the Planck time is the smallest unit of time. As far as I know, there is no evidence for this, and few if any theoretical scientists believe this.)

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  • Thanks, I shared that misconception about Planck time. I guess you are right that in reality science goes on without worrying about "infinity" in an informal sense. But I thought there were cases where it becomes a problem in preventing infinite regress in measurement and calculations. – Nelson Alexander Aug 30 at 16:56
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    Nevertheless, when a theory predicts an infinity, physicists have learned to suspect that this prediction does not hold. Take for instance the electric field. Classically, the charge density of the electron is infinite. However, quantum mechanics tells us that you need an infinite amount of energy to localize an electron in an infinitely small volume, and thus the charge density is limited by the available energy. Likewise, general relativity predicts a singularity inside a black hole, but physicists believe that this will vanish once we have a unified theory of gravity and quantum mechanics. – cmaster - reinstate monica Aug 31 at 21:37
  • I have seen many physicists write that the Planck time is the smallest unit of time "that has any meaning". – Guy Inchbald Sep 1 at 9:34
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Mathematically, infinities fall into two distinct classes; countable and uncountable. For example the set or rational numbers is countable, the set of real numbers is not.

Note that Newtonian countability is still a countable infinity, being merely the sequential terms in a convergent infinite series.

Another answer has explained that infinities in a physical model are believed to indicate a breakdown of the theory. This is called a boundary condition to the theory.

Nevertheless, speculative cosmologists love to bandy "infinity" around as if they meant it. For example in the theory of eternal inflation, the process has been going on for ever and will continue to do so, constantly spawning new universes such as ours. Another such eternal model is the conformal cyclic universe of Roger Penrose. These solve one chosen mathematical problem at the expense of introducing the ultimate no-no of infinity.

Others will talk of "infinity" but when pressed they either avoid answering or retreat to "arbitrarily large" or, to approximate Douglas Adams, "so humunguously vast that they are unmeasurably more vast than the vastest thing you can possibly imagine, which is really and absolutely indistinguishable from the real thing and therefore is to all intents and purposes and therefore factually the real thing". Or some such garbage.

So while one can say that mainstream science has no time for infinities, speculative science can lack a certain self-consistency.

Nevertheless, for purposes of argument, infinities may be tolerated in speculative theories on the assumption that the refined version will iron them out. The singularity at the heart of a black hole provides an example; since we cannot look inside a black hole to see what is in there, all theories about it are pretty much speculative, but nevertheless it can be instructive to try.

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  • Thanks, helpful. And helpful to know that, as suspected, the term isn't used very consistently in cosmology or talk about "the universe." – Nelson Alexander Aug 30 at 19:20
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The belief that the infinite does not really exist goes back at least to Aristotle. Parmenides even questioned the reality of plurality and change. (Einstein's vision has much in common with Parmenides). Towards the end of the nineteenth century an acrimonious exchange took place between Kronecker and Cantor regarding the reality of the actual (as opposed to potential) infinite. Kronecker claimed that only the finite integers really exist and all else is merely the work of man. Cantor countered that the essence of mathematics was its freedom and that he had attained a larger vision than Kronecker had who could not see the infinite. Most mathematicians have followed Cantor and found his paradise a more beautiful and alluring universe.

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Infinity is a useful concept. Using infinities like in mathematics, we can get answers which are measurable, meaningful. The question may be whether there is really anything that exists and indeed an infinite. The infinities just as a mathematical concept can be explained and may be argued to exist eg integer series 1,2,3,... so on. But it's just logical or theoretical existence. In the physical universe, it appears that the universe may be infinite or a black hole may be infinitely dense at its core. Science may resolve these questions with measurable answers, maybe in the future. As of now, its useful to describe such entities by using infinities. The universe is expanding into the "nothing". The "nothing" as an entity in this sense can really be said to be infinite.

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This is just a formal fallacy.

It doesn't follow that given the necessity of accountable numbers, the infinity should be excluded by science. If the need exists, it is nevertheless compatible with the formality of the infinity.

Science does not require of the exclusion of a concept that is valid in any formal context, for operations where it is not needed.

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