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.
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 1089 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 eiπ=-1 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.
Mathematics 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 20th 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 1089! 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 1089 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).
- 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 1/100th of the rooms will empty in a day, 1/2400th in an hour and 144,000th 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.