This month’s Insights column was an attempt to use simple puzzles to highlight the consequences of the infinity assumption in the physical world. The idea was sparked by an article by the physicist Max Tegmark that was written for the book, “This Idea Must Die.” Tegmark’s article is excerpted in a blog at Discover magazine under the title, “Infinity Is a Beautiful Concept — and It’s Ruining Physics.” Tegmark is exceptionally critical of the assumption of the continuum that is used in the spectacularly successful theory of cosmic inflation, giving density fluctuation predictions that are in beautiful agreement with precision measurements from experiments such as the Planck and the BICEP2 experiments. Yet inflation has given rise to the measure problem, which according to Tegmark is “the greatest crisis facing modern physics. When we try to predict the probability that something particular will happen, inflation always gives the same useless answer: infinity divided by infinity… This means that today’s best theories need a major shakeup by retiring an incorrect assumption. Which one? Here’s my prime suspect: ∞.” The infinity assumption also gives initially nonsensical answers in quantum field theory, but physicists have found workarounds using the “mathematically ugly” process of subtraction of infinities, much criticized by many prominent physicists including Dirac, Feynman and Salam.

Tegmark does not dispute that infinity is useful in mathematical models applied to physics. He gives the example of the air:

Keeping track of the positions and speeds of octillions of atoms (in air) would be hopelessly complicated. But if you ignore the fact that air is made of atoms and instead approximate it as a continuum — a smooth substance that has a density, pressure, and velocity at each point — you’ll find that this idealized air obeys a beautifully simple equation explaining almost everything we care about: how to build airplanes, how we hear them with sound waves, how to make weather forecasts, and so forth. Yet despite all that convenience, air of course isn’t truly continuous. I think it’s the same way for space, time, and all the other building blocks of our physical world.

He continues:

Let’s face it. Despite their seductive allure, we have no direct observational evidence for either the infinitely big or the infinitely small. We speak of infinite volumes with infinitely many planets, but our observable universe contains only about 10

^{89}objects (mostly photons). If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information, specified by a number with infinitely many decimal places. In practice, we physicists have never managed to measure anything to more than about seventeen decimal places.

Tegmark’s justification for ditching infinity is practical, but there are convincing philosophical reasons as well. The Insights questions this month, though simple, were an attempt to explore in depth what pitfalls are created when we make the assumption of infinity in our mathematical models. Before we look at the solutions, let us examine the fundamental relationship between mathematics and reality, which I will explore under the following four points:

- The map is not the territory.
- Infinity is valid in mathematical models and can be very useful.
- In the physical world, there are compelling practical and philosophical reasons to reject infinity as a default assumption.
- There will be limiting cases where the mathematical infinity assumption and the physical absence of infinity result in different answers.

** The map is not the territory**: This is a most important principle, never to be forgotten whenever mathematics is applied! Mathematics is essentially a mental creation, an invented conceptual model that abstracts and idealizes certain patterns and relations from a given real or imagined situation, and allows us to reason quantitatively with these aspects only. In making a mathematical model we start from implicit or explicit assumptions and rules. The process of making these assumptions and rules automatically creates a complex virtual conceptual structure, sometimes incredibly rich, such as the field of complex numbers, within which mathematical discoveries such as Euler’s beautiful formula

*e*can be made. But these discoveries are of things that were implicit in our assumptions to begin with, and completely contingent on them. Under modern “embodied mind” views of the nature of mathematics that are most consistent with our scientific worldview, the idea of mathematical objects existing in some Platonic world that has an ontological status beyond our brain-based cognitive apparatus is simply ancient mysticism, an instance of unwarranted romanticization of mathematics.

^{iπ}=-1Mathematics is, no doubt, one of the best toolkits we have. We are animals that fill a cognitive evolutionary niche, and therefore it is not surprising that many of us have minds that find quantitative reasoning and mathematical concepts beautiful. This has often resulted in hype among enthusiasts who are enamored of mathematics, as for example the 20^{th} century physicist Eugene Wigner and his romantic idea of the “The Unreasonable Effectiveness of Mathematics in the Natural Sciences. But this romanticism needs to be tempered by a healthy skepticism. Mathematics is actually a very poor tool in modeling complex phenomena with chaotic solutions. A case in point is that even the “three-body problem” — the problem of exactly calculating the motions of just three bodies interacting through gravitation — cannot be tractably solved analytically. If you didn’t know that, let it sink in completely. We cannot solve motion under gravitation for even three bodies, let alone 25 or a thousand or 10^{89}! We are fortunate to live in a universe that does not have a handful of forces of comparable strength operating among everyday objects. We are lucky that on human scales, we can neglect the interaction of more than two bodies in most problems: If it were otherwise, we would not be able to do dynamics analytically. (I recommend reading Isaac Asimov’s classic science fiction story “Nightfall,” in which a civilization living on a planet that orbits a system of six suns finds it hard to discover the laws of gravitation.) Most of our physics is basically a simple science of binary interactions, unlike messy sciences such as biology, sociology and psychology where there is the complex interplay of many variables, and where our mathematical models become extremely approximate and error-prone.

** The validity of infinity in mathematics**: The concepts of the infinitely small and the infinitely large in mathematics are assumptions with the help of which we can construct our conceptual models of virtual universes which may be applicable to the real world (Euclidean geometry) or fictional worlds (Cantor’s tower of infinities). All that is required for mathematical validity is that our assumptions, taken together, are consistent, that we can reason with them and draw unambiguous conclusions. Our conclusions represent the theoretical limits that can be reached once all practical problems are solved. Thus the Hilbert Hotel problem is about a fictional world that illustrates the principle of a countable infinity (Cantor’s aleph-null) using the procedure of one-to-one correspondence, and showing that the cardinality of an infinite subset is equal to that of an infinite set that contains it, under the assumptions and definitions given. It is meaningless to invoke practical objections like the speed of light or the time required to fill the rooms in the Hilbert Hotel problem — it’s about a fictional world. On the other hand, the Euclidean dimensionless point assumption is tremendously useful in making predictions in the real world, and conclusions based on it represent the limit to which practical measurements can approach. Note however, that this is not the only assumption that can give correct answers — an assumption of discrete points that are, say, a thousand times or a million times smaller than the best precision of our current measurements will generally give the same answers in practice. The dimensionless assumption is just more convenient mathematically, and can be applied almost universally without a second thought.

** Reasons to reject infinity in the physical world**: Just because we can navigate successfully in most cases on the basis of a convenient map does not mean that all the assumptions we make in it are actually realized in the physical world. As an example, the Mercator projection, commonly used in cartography, shows Greenland to be far larger than it actually is. Similarly, at limiting cases, the physical world may be quite different from what is predicted by our mathematical assumptions. We have already seen Max Tegmark’s example of inflation and the problem of renormalization in quantum field theory. There is also the practical philosophical principle that there are costs associated with the real world: In thermodynamics as in practice, there is no such thing as a “free lunch” — or else there would be potentially inexhaustible supply of energy, infinitely strong forces, singularities that swallow everything, and all manners of other catastrophes. Thus Jeff’s comment that an infinite number of photons can coexist in the same space, which quantum theory allows, is negated by energy considerations, as explained very nicely by K. Kuromori. To paraphrase Tegmark, it is hubris to extrapolate from 17 decimal places to an infinite number of them, or from 10

^{89}to aleph-null just because we find it mathematically convenient — recall the three-body problem. Infinity is an extraordinary concept to claim in the real universe, and to apply Carl Sagan’s dictum, it should require extraordinary evidence for us to accept it. It is somewhat ironic that it is Max Tegmark who has publicized this argument, considering that the other idea for which he is famous, “The Mathematical Universe,” is in effect a claim that the map is indeed the territory!

So if the physical universe cannot contain infinity (which is a wise default position to have in the absence of extraordinary evidence), there will be limits at which our infinity-laden mathematical models will fail. Our simple problems were meant to explore these limits. We may find that the infinity-based predicted outcome can still hold, but for different reasons (as in the first problem); or that the infinity-based predicted outcome is wrong, and we have to reason differently to reach qualitatively different conclusions (second and third problems).

**Solutions:**

- Hilbert’s Hotel

This first question is just a warm-up to show how we can replace infinitistic thinking with finitistic thinking. It concerns the famous Hilbert’s Hotel, an idea introduced by David Hilbert in 1924.

Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. Can it accommodate 1,000 new guests without increasing the number of guests in any of the occupied rooms? If you had a finite number of rooms, the pigeonhole principle would apply. In this context, this common sense principle says that you cannot have n+1 pigeons in n holes if there is only room for one pigeon in each hole. But in an infinite hotel, it’s easy! We just move every resident from his or her room n to room n + 1,000. Voilà! Rooms one to 1,000 are now empty!

Notice the sleight of hand involved in using infinity in this way. This solution cannot work with a finite number of rooms, no matter how large. Let us restrict ourselves to the notion that the number of rooms can be as large as the size of the universe allows, but must be finite. Can the question still be answered positively? Well, it turns out that you can easily accommodate 1,000 new guests in a finite physical hotel that is currently full. And the arrangement will take less time than moving a single person from one room to another. All it takes is the reasonable assumption that there is a tiny, nonzero probability of a person checking out within a given time. Let’s assume, conservatively, that the probability of a guest checking out on a given day is one in a hundred. Can you see how the hotel can put up its additional guests?

As mentioned above, the Hilbert’s Hotel solution is not to be taken seriously as a realworld problem: It was devised by Hilbert to illustrate the conclusion that there can exist a one-to-one correspondence between a countably infinite set and a subset of it that is also infinite. It therefore follows from the conceptual existence of an infinity of counting numbers, and finds an application in measure theory (which is also just a conceptual model). However, there is a physical limit to the number of rooms that a hotel in the physical universe can have. Under the given assumption though, approximately 1/100^{th} of the rooms will empty in a day, 1/2400^{th} in an hour and 144,000^{th} in a minute. So in a hotel with 144 million rooms we can expect a thousand rooms to be available every minute. Such a large hotel is well within the size limitations of the universe! The difference between such a humongous hotel and Hilbert’s hotel is that Hilbert’s Hotel can accommodate an infinite number of guests when full. A huge hotel can indeed accommodate a large number of guests in a short time based on a finite turnover probability, but the number must be finite.

- The 100, 200, 300 Triangle

Imagine that the laws of physics prevent you from measuring anything smaller than 0.001 micron. Can you have a triangle on the plane that has sides measuring 100, 200 and 300 microns? Can such a triangle, which you would expect to be impossibly flat, have a measurable area? Can you go further and have a triangle that has a sum of two sides that measures smaller than the length of the third side? The answers may surprise you.

Using a triangle calculator, it is easy to analyze a triangle with sides of “true length” 100.0004, 200.0004 and 299.9996 microns that would yield measurements of 100, 200 and 300 micron under the smallest measure. Its area would be 60 square microns, which indicates a height of 1.2 microns using the smallest side as base. This height is 1,200 times the smallest measure. So far from being impossibly flat, this triangle would appear unmistakably triangular. We can indeed go further: A triangle with sides of true length 100.0004, 200.0004 and 300.0006 would be measured as 100, 200 and 300.001 microns. Even this “impossible” triangle has an easily detectable height 480 times the size of the smallest measure.

Notice that we are using the standard Euclidean assumptions, which include the idea of dimensionless points as a backdrop to physical reality to compute these answers and assign a true length measure. But we needn’t use a measure that included the idea of infinitely small points. Any conceptual measure that relied on discrete finite sized points with dimensions much smaller than our hypothesized smallest physical measure would have given the same answers. But the assumption of the infinitely small, precisely because it is idealized, gives us convenient and elegant formulas with which we can compute conveniently.

Lee made the interesting comment that “if space has discrete units, then there is no such thing as a triangle. The mathematical triangle becomes an abstract approximation.” Actually this is true in the real world today — triangles are really abstract approximations, because no real world line has zero width. This problem concerns a 100, 200, 300 micron figure that we would recognize as a triangle, using the same criteria that we would apply to any real world triangle.

Bee points out the interesting observation that “any such minimum to space…is inevitably in conflict with special relativity.” Since special relativity is based on standard mathematical assumptions of continuity, this is not surprising. I don’t know what the resolution of this, but I’d like to point out that there is nothing sacrosanct about special relativity at these length scales. We also know that special relativity has to be violated in the “interior communication” of entangled quantum particles in any realist model, even though we cannot make use of this.

- The Elliptical Pool Table

Consider the case of an elliptical billiard or pool table. An ellipse is a geometric figure that has two foci. Any straight line drawn from one focus to the circumference of the ellipse is reflected to the other focus. Now assume you have a pool table with a pocket at one focus.

Let us assume that the table is perfectly manufactured. There is still the problem that the mathematical focus is a dimensionless point, whereas the ball, being a physical object, has a finite size. How does this finite size affect how accurately the ball goes to the other focus when hit? Given this, and the fact that no pool player is perfect, will you get equally good results no matter which direction you hit the ball as long as it is at a focus initially? If the major axis of the table is 2 meters in length and the minor axis is 1 meter, what is the best direction to hit the ball from one focus so that it bounces and rolls into the pocket at the other focus? Assume the pocket is about 1.5 times the diameter of the ball. Will your conclusion change if the ball and pocket are made as small as physically possible without changing their relative sizes?

The same Euclidean assumption of the dimensionless point also implies that there is a zero probability that a finite sized ball can be placed or hit exactly at the point focus (probabilities under the infinity assumption can only be assigned to positioning inside length segments within larger segments, never for an exact position). So the ball’s path will always start from a point a finite distance away from the true focus. Assume this is *x* millimeters (mm). After reflection from the side of the table the ball will be travel through a point a finite distance away from the second focus. (The spin imparted by not hitting the ball at the absolute center will compound this problem, but let’s neglect that for now.) Assume that the length of the initial path is *a* mm and that of the reflected path is *b* mm. Then, if we use the principle of similar triangles as seen in the figure, the ball’s distance from the second pocket after reflection will be approximately *x* mm times *b/a*.

The path ratio *b/a* either magnifies or diminishes the original error depending on whether it is greater or smaller than 1. So the best results will be obtained if you make the initial path as long as possible and the reflected path as short as possible. It is easy to see that this happens when the ball is hit directly along the major axis to the other end of the table. Of course, this will result in the path passing through the pocket, so the best course would be to hit it at a small angle so that it clears the pocket and bounces off the far wall toward the pocket. Using an ellipse calculator we can see that the original error will be diminished by a factor of 12.9 by hitting it close to the far end and magnified 12.9 times by hitting it toward the near end. So it is not true that all directions are equally good. Making the ball as small as possible so that all points on it are closer to the focus doesn’t make an appreciable difference at all since the magnification factor remains the same. So even on Bellos’ elliptical pool table, you will need some skill: It will help you only when you bounce the ball on the side far away from you. If you do the opposite, it will magnify your mistakes.

Sand pointed out another way in which a finite ball will behave differently from a point. It will collide with the border a little earlier than in the case of the “ideal trajectory.” Therefore, as Sand correctly states: The actual trajectory after the collision will then be parallel to the ideal trajectory. The distance or deviation between the two paths, though, will be* r* times *sin(a)* where* r* is the radius of the ball and *a* is the angle of incidence with the table wall (the deviation cannot be higher than *r*). This deviation will also tend toward zero for a ball hit to the far end, so this factor is also optimal for the solution stated above.

Notice that here too, we used the similar triangles calculation from Euclidean geometry with some approximations to arrive at our conclusion, even as we explicitly disavowed the Euclidean assumption of the point focus. Incidentally as Wylie pointed out, there is a wonderful problem called the “Ellipsoid Paradox” which uses the shape of the ellipsoid to “prove” that you can construct a battery that can continuously charge itself and therefore give power forever. The reasoning is completely sound, except for the implicit assumption that a finite object can be placed at the focus, which is the source of the fallacy. In actual fact, the focus is infinitesimal and therefore impossible to find in the real world. I have discussed this fallacy and its solution in detail, showing that unfortunately there is no free lunch allowed by thermodynamics in the universe.

So the bottom line is: Infinity is permissible in mathematics applied to physics because it makes things convenient and tractable in most cases. However, we must be alert for limiting cases where our models are bound to fail, and we will then need to apply different methods. As for dimensionless points, as John Merryman put it, “[A dimensionless point] is a tradeoff, a self-negating concept, versus a fuzzy concept. Since it is an abstraction in the first place, having it self-negating is the cleanest solution.”

The *Quanta* T-shirt for this month goes to John Merryman for the quote above. Thanks to everyone for the great discussions. See you for another round of Insights next month.

Very interesting article, thank you for sharing! Tegmark could quote J.L.Borges as exergis ("There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite."). Or even Renè Guenon, for his idea to substitute "infinite" with "indefinite". Of course the infinite is a dangerous concept, but also negating his existence could lead to some paradox, like thinking about a finite "building block" of everything; it's quite hard to draw conclusions… I'd say that the infinite we deal with is most of all a "rule" to extract (create?) something, usually a pattern of sort.

Just for fun: Let's imagine an infinite list of pictograms like these:

♤°~¤☆#▪¿♢…

We subtract one of them, let's say the poor ♤. The list is still numerically infinite, but also incomplete.

(The pictograms are an infinite bigger than the one of numbers, like Cantor would say, but to decide that we can "subtract" or that the list is "infinite but incomplete" depends only on our rules. Here for example "all the pictograms" and "infinite pictograms" can't coincide.)

The idea of infinity was invented this way: if n is the largest number, (n+1) is larger. However, this assumes that, having used n symbols, we can find still another. So what are the "symbols" going to be? If they are fermions, there is clearly a finite number within the event horizon. Thus there is a number n so that (n+1) does not exist.

https://patriceayme.wordpress.com/2011/10/10/largest-number/

Update 7-1-2016We have made the following two updates in this article since it was originally published yesterday (6-30-2013):

1. The figure has corrected to show the correct direction of deviation (the original figure had a mirror image). A line showing the ideal error minimizing path has been added.

2. The explanation of the Ellipsoid Paradox has been clarified as follows: 'The reasoning [behind the paradox] is completely sound, except for the implicit assumption that a finite object can be placed at the focus, which is the source of the fallacy. In actual fact, the focus is infinitesimal and therefore impossible to find in the real world.'

The original language was ambiguous.

It should to be noted that this article and the previous one use

toy problemsto highlight philosophical issues with the assumption of infinity in mathematics and physics (the Insights column is a puzzle column after all). There are, however, several links to excellent expositions of more serious physics in the text for readers who wish to go deeper. I'd also like to recommend the following excellent links provided by readers:Bee cited a detailed investigation of the Planck length in Relativity and Quantum Gravity, by Sabine Hossenfelder.

Pete Attkins: An explanation of the point spread function

Māris Ozols: Struggles with the continuum in physics by John Baez

Finally, James Franklin, a philosopher of mathematics, referred to his book on the subject, whose conclusions about infinity appear to be similar to what we have discussed here.

Thank you for sharing these. They all these are worth exploring more deeply, and I'd certainly love to. Now if only there were infinite time 🙂

PS: Also note that Tegmark’s comments were made a couple of years ago, and the BICEP-2 result has been deflated since then. No definitive experimental evidence for cosmic inflation has been found, but support for it remains strong, as this article explains.

@Patrice Ayme

It seems that you are making the exact same conflation of "the map" and "the territory" that I've recommended should be avoided. There is no such thing as the largest number in our conceptual model of numbers, but there is at any given point, a limit on the number of particles in the physical universe. If tomorrow we find that each fermion consists of a million vibrating strings, we can easily accommodate the new limit because of the flexible conceptual structure provided by the infinite assumption in our mathematics.

I really enjoyed this article, since I think it gets at some very important questions about infinity which, physicists are still grappling with and need to address. However I am not sure I agree with all the answers the author has proposed. In particular I think the author is dead wrong in the following statement:

" the idea of mathematical objects existing in some Platonic world that has an ontological status beyond our brain-based cognitive apparatus is simply ancient mysticism, an instance of unwarranted romanticization of mathematics."

In my opinion there are so many avenues of modern physics which are driving us into the arms of Plato, from quantum mechanics to the relationship between information and order. I understand that the author is re-coiling in fear from the all the potential pseudo science hidden when Plato and infinities begin to conspire to generate metaphysics, parallel universes and evil goatee sporting Stocks. However I take issue with an attack on Plato, and I think Plato's point is becoming increasingly worthy of consideration. The fact that our fundamental "units of matter" to quote Heisenberg are "not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language" I think is the big reason why math is increasingly viewed in such a radical light.

One can disagree on this point and hope that after 80 years of searching we would discover the 'real' physical objects hiding behind the math, but I think we should be a little careful in dismissing Mr. Plato so flippantly.

Dear Pradeep Mutalik:

Thanks for the answer. You say:”It seems that you are making the exact same conflation of "the map" and "the territory" that I've recommended should be avoided. There is no such thing as the largest number in our conceptual model of numbers, but there is at any given point, a limit on the number of particles in the physical universe. If tomorrow we find that each fermion consists of a million vibrating strings, we can easily accommodate the new limit because of the flexible conceptual structure provided by the infinite assumption in our mathematics.”

What limits the number of particles in a (small enough) neighborhood is density: if mass-energy density gets too high, according to (generally admitted) gravity theory, not even a graviton could come out (that’s even worse than having a Black Hole!)

According to Quantum Theory, to each particle is associated a wave, itself computed from, and expressing, the momentum-energy of said particle.

Each neighborhood could be of Planck radius. Tessellate the entire visible universe this way. If too each distinct wave one attaches an integer, it is clear that one will run out of waves, at some point, to label integers with. My view does not depend upon strings, super or not: I just incorporated the simplest model of strings.

Another mathematician just told me: ‘Ah, but the idea of infinity is like that of God’. Well, sure. But mathematics, ultimately, is abstract physics. We don’t need god in physics, as Laplace pointed out to Napoleon (“Sire, je n’ai pas besoin de cette hypothese”).

The presence of God, aka infinity, in mathematics, is not innocuous. Many mathematical brain teasers become easier, or solvable if one assumes only a largest number (this is also how computers compute, nota bene). Assuming infinity, aka God, has diverted mathematical innovation away from the real world (say fluid flow, plasma physics, nonlinear PDEs, nonlinear waves, etc.) and into questions akin to assuming that an infinity of angels can hold on a pinhead. Well, sorry, but modern physics has an answer: only a finite number.

I never believed in the completed infinity. And yet here are papers which purport to start out using all finite mathematics, but need infinity anyway,

http://www.personal.psu.edu/t20/papers/mathtext

"UNPROVABLE THEOREMS AND FAST-GROWING FUNCTIONS"

"So in a hotel with 144 million rooms we can expect a thousand rooms to be available every minute."

Yep, Queuing Theory 101.

@Vahid Ranjbar,

There are certainly beautiful idealized forms in mathematics as Plato suggested. However the question is, where do they come from, and where do they reside? For Plato, they exist independent of the world in some ethereal realm, they are the essences on which the physical universe is based. Physical reality is therefore an imperfect imitation of mathematical forms.

The embodied mind view of mathematics turns this around. The universe has embedded regularities, which our imaginations and our cognitive ability turn into perfect, beautiful forms. These mathematical forms (like infinity), therefore exist in our minds, which create them by idealizing reality.

What appeals to you depends on your predilections.

To a certain extent this question is similar to another: Did God make man in his image (a poor imitation!) or did man make God in his image (a perfect idealization)? The former answer, like Plato’s, requires the invention of a special ontological category and realm, necessitates a mental ability to directly apprehend this realm and raises unanswerable questions about where God and the forms came from. The latter answer does not have these consequences, and is far more in tune with the scientific worldview.

Does the latter answer decrease the magic of mathematics? No, as Feynman would say, ” It does not do harm to the mystery to know a little about it. For far more marvelous is the truth than any artists of the past imagined it.” I’m in agreement with Feynman — far more marvelous is the truth than Plato, or the philosophers of ages past, imagined it. The magic becomes even deeper when you know a little about how it came about, because now you also know that your concepts, building on the ideas of the past, are deeper and closer to the truth!

What happens to the assumptions regarding infinity if one assumes nothing can be created or destroyed, including spacetime itself? In other words, how do the current notions of infinity vibe with the idea of percentages, which are just relative measures?

Didn't the ideas for current infinite set theory get developed back when we still thought the world was governed by Newtonian Mechanics, and thus reflect that worldview?

Just some fun thoughts 🙂

Also, if I took a sphere and expanded it from the center of the Milky Way, counting the contents in an ordered fashion in the order they contacted the surface of the sphere, could I count everything in it in an ordered fashion in a single line sense? If so, what would keep me from continuing this process until I counted the whole Universe? Time?

Is the Planck scale the scale at which everything is finite? Why not at 10^{-500} meters, or at progressively smaller scales, as we discover and engineer using more refined physics? At what point can we be completely certain there is no more refined physics to discover? Equally, just because we cannot currently see anything beyond, say, 15 gigaparsecs using light does not mean that there is an end to what might be detected using progressively more refined physics. Feel free to rule out infinity outwards or inwards in your own universe because of metaphysical preferences, but I would offer that it is better to avoid claiming that anything is ultimately either infinite or finite, either uncountably or not, insofar as empirical evidence is always thin.

What if the speed of light is not constant, but instead depends on the density of the space it is traversing? What if it acts like a limit? What if the things you experience are just light slowed way, way down?

https://www.sciencenews.org/article/speed-light-not-so-constant-after-all

-"Generally if light is not traveling at c it is because it is moving through a material. For example, light slows down when passing through glass or water."

Also, what do SI units refer to? Is that some kind of absolute quantity?

Lastly, what is the density, in an absolute sense, of a piece of the earth compared to a piece of space somewhere in between the earth and the sun?

How does the classic view of space and time stand up to the ideas presented in the more than 2000 year old Zeno's Paradox?

@Peter Morgan,

Sure, the limits for the very small and the very large could be many orders of magnitude smaller and larger, respectively, than we suspect today. We could certainly leave that open pending advances in physics, but what about infinity?

While it is fine to be agnostic about the presence or absence of infinity in the physical world, this should not be taken to imply that the probabilities of these two scenarios is about equal. This is not just for metaphysical reasons, but in fact there are a constellation of reasons from logic, mathematics, physics and psychology to support the idea that both varieties of infinity are very, very unlikely to exist in the physical world and this may even be impossible. You could say that the probability of infinity is infinitesimally small 😀

Some of the reasons have been discussed here but let me make an attempt to list them.

Logic:

As John Merryman put it in his comment, “[A dimensionless point] is a tradeoff, a self-negating concept.” And so is the idea of parallel lines ever meeting at infinity in Euclidean geometry, as Mike S. pointed out — after all, the distance between them remains the same always. We just overlook the logical inconsistencies of foundational assumptions like these because we are so used to the successes of geometry in so many real world applications.

Mathematics:

As I pointed out, there are an infinite number of possible discrete theories that can give the same real world predictions as our classical mathematical theories based on infinity. It’s just that they are difficult to work with analytically, although they are exactly what are used in computer models.

Measure theory which is the standard mathematical theory based on continuity is likewise very successful, but it cannot assign a probability for an object to be in any position whatsoever, as we saw above in problem 3. What this means is that it is impossible to find any specified point in space, and all exactly specified positions are impossible! In fact, measure theory can only assign probabilities to discrete positions —which seems to indicate that even within the theory the infinitesimal assumption, while helpful, is a contradiction. In most real world cases, this does not matter, but in case of the repeated reflection from a focus, it makes a qualitative difference as in the Ellipsoid Paradox, and in the billiard table. Discrete models give the right answer here —our infinity-laden classical answers are simply wrong.

There are more sophisticated paradoxes caused by the assumption of infinitely small points such as the Banach-Tarski paradox which states that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using rigid motions only, form two solid balls, each the same size and shape as the original.

https://www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml

Physics:

There is no doubt that physical theories based on the continuum are creaking in the joints at small and large scales as we saw in the problems encountered in cosmologic inflation and renormalization in quantum field theories. In the case of quantum field theories, there is a “cutoff” distance that is assumed below which the calculations are not pursued. This distance is unknown in a given problem, but by plugging in measured quantities and using some dubious “cancelling of infinities” you can calculate results that agree well with experiment. It may be that at small scales there is discreteness that is variable—it may vary with the energy of the system. So there is definitely new physics that happens on these scales, which is exactly what is to be expected if the continuity assumption is wrong with respect to space and time.

There is a beautiful six-part (so far) discussion of the problems with the continuum hypothesis in the theories of physics by the mathematical physicist John Baez, which commenter Māris Ozols provided a link for. It lists all the paradoxes and problems caused by the assumption of the continuum in physics including some that are mentioned above. My favorite one is that in a 5 body scenario, it is possible to “prove’ that you can have a body that can cover an infinite distance in a finite time!

https://www.physicsforums.com/insights/struggles-continuum-part-1/

All the above are arguments against the infinitely small. Similarly, there are logical and mathematical arguments against the infinitely large as well such as the differences between a real number and infinity as seen in the Hilbert’s Hotel example. The infinitely big requires infinite energies, and this would generate all kinds of paradoxes and catastrophes. The universe would be filled with light of all wavelengths, and as William Dobni, commented, "if there an infinite universe then there would be an infinity of almost vanishingly small nearly massless neutrinos such that there would be so many neutrinos clumped everywhere that we would all be literally suffocating under their ubiquitous presence."

Psychology:

The human mind finds it easy to imagine a process that goes on and on by idealizing from reality, even though we have no direct experience of such things.

The question needs to be asked, why do we even entertain the possibility that the universe can be infinite when it so alien to our experience? The answer, of course, is that our infinity-laden mathematics that has made some spectacularly successful predictions makes infinity seem possible. But as we saw, there can be an infinite number of discrete models that give the same answers as classical mathematical theories, and these discrete models do not suffer from the paradoxes that our idealized theories do. The difference between continuous and discrete models are that the former, precisely because they are idealized, yield expressions and formulas that are easy for us to grasp. The idea that we should follow these theories to their paradoxical end when alternatives that are better but more incomprehensible for us, do exist, is the same kind of ‘humans are at the center of the universe’ thinking that was characteristic of pre-Copernican days. Yes, we want the universe to be easy to understand, but the universe is not obligated to be what we can easily imagine it to be.

Similarly, as I said, it is hubris to extrapolate from the 17 decimal places to an infinite number of them, or from 10^89 to infinity just because we find it mathematically convenient. There is literally an infinite amount of ground to be covered to get from one to another! The successes of our mathematics are no doubt great, but we need to be more humble about our analytical mathematical models that that cannot even exactly solve the three body problem. If we look realistically at our mathematical models, we will see that there is no justification to follow them all the way to infinity.

To quote Tegmark again:

“Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too."

So yes, it is very well to be agnostic about infinity, but I stand by my recommendation that we should not accept infinity in the physical universe without extraordinary evidence.

Unfortunately, like theological questions, the chances are that we will never know for sure. From a practical point of view our capacity to access all of the universe will probably always be bounded in some way. So our effective universe, in all likelihood, will always remain very large but finite, for mundane practical reasons.

The infinity is creation of the human mind.there is not in the cosmos.

Actually special relativity, in regards to entanglement, may not be violated if they're connected via wormholes (or higher dimensional space shortcuts).

Another area where infinity may be absolutely required in the real world is the eternality of the cosmos. This is required since causality presents a contradiction to finite time. It would seem then that the cosmos requires a "bottom" or bedrock that just is and always has been (quantum foam, strings/branes?).

It seems to me that the infinite/continuous is a convenience that makes some of the combinatorial complexity of the finite disappear sometimes. For example in computer science/computability theory, we have provably undecidable problems (such as the halting problem), but the corresponding finite complexity versions of this (e.g. P vs NP) are notoriously difficult and mostly unanswered. Somewhat similarly, in statistics the Central Limit Theorem is really useful if sufficiently large sample are available, but not very helpful in dealing with the details of smaller samples, where much more detailed models are needed. Finally, in arithmetic questions about prime numbers are dealing with the discrete/finite, and surprisingly some of them are answered by a detour to the continuous/infinite (e.g. proof of Prime Number Theorem). Until we figure out a way to deal with the complexity of the finite, the continuous/infinite will be there to smooth things out for us. I have no doubt (neither a proof) that the infinite does not exist outside of this. On the hand, in general it is possible that there really is no way to deal with the complexity of the finite in a practical way (interpreting loosely P<>NP), and in this case the infinite is hear to stay for ever. Which one might say makes it quite real…

The problem of Infinity arises from the concept of "time" used by us scientists to create implausible models. If we remove Newtonian function dx/dt most of Cosmology comes crumbling down.

Science has given us loads of creature comforts. But in the process, concepts of "Time", "Money" and "Mathematics", over the last 250 years, have conditioned the human brain-mind complex in negative ways. As a result we are becoming dumber.

As proof I offer the accomplishments of pre-science civilizations, who had but a copper chisel in their toolbox.

We don't even come to within a lightyear to explain how they achieved what we , today, with all our technology and knowhow, cannot duplicate.