How sharp is a geometric focus in the real world?Ĭonsider the case of an elliptical billiard or pool table. 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.ģ. Imagine that the laws of physics prevent you from measuring anything smaller than 0.001 micron. Let us substitute a less daunting but still microscopic length for the Planck length. But what if there is a physical limit to the smallest length of space that is measurable, say somewhere close to the Planck limit of 1.6 x 10 -35 meters? Will this theorem of geometry still hold near that length? It is a theorem of plane geometry that the sum of the lengths of any two sides of a triangle is greater than the length of the third. What if there are physical limits to the smallest measurable amount of space? Can you see how the hotel can put up its additional guests?Ģ. Let’s assume, conservatively, that the probability of a guest checking out on a given day is one in a hundred. All it takes is the reasonable assumption that there is a tiny, nonzero probability of a person checking out within a given time. And the arrangement will take less time than moving a single person from one room to another. 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.
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. This solution cannot work with a finite number of rooms, no matter how large. Notice the sleight of hand involved in using infinity in this way. But in an infinite hotel, it’s easy! We just move every resident from his or her room n to room n + 1,000. 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. 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. It concerns the famous Hilbert’s Hotel, an idea introduced by David Hilbert in 1924.Ĭonsider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. This first question is just a warm-up to show how we can replace infinitistic thinking with finitistic thinking. Can a number that is finite but very large substitute for infinity? Focus on how the theoretical answers change when you discard the notion of infinity.ġ. In these examples, do not get stuck on practical details.
Here are three puzzles that illustrate this. In many cases, better or at least more useful answers can be obtained if we just stick to very large or very small quantities. We don’t have to examine the foundations of physics to see examples of how the infinity assumption can give qualitative answers that are not quite correct in the real world. While “most physicists and mathematicians have become so enamored with infinity that they rarely question it,” Tegmark writes, infinity is just “an extremely convenient approximation for which we haven’t discovered convenient alternatives.” Tegmark believes that we need to discover the infinity-free equations describing the true laws of physics. But can infinities truly exist in any aspect of the physical world? Is space truly infinite, as some inflationary models of the universe assert, or is it in some way “pixelated” at the lowest level? In an extremely interesting book, This Idea Must Die, in which many eminent thinkers describe scientific ideas they consider wrong-headed, the physicist Max Tegmark of the Massachusetts Institute of Technology argues that it is time to banish infinity from physics. Infinities implicitly pervade many familiar mathematical concepts, such as the idea of points as mentioned above, the idea of the continuum, and the concept of infinitesimals in calculus. Mathematicians have developed the theory of infinity to an exquisite degree - Georg Cantor’s concept of transfinite numbers is notable for its beauty, “a tower of infinities with no connection to physical reality,” as Natalie Wolchover put it in a recent Quanta article on the finite-infinite divide in mathematics.
0 kommentar(er)
