Chapter 1: On Infinity
Matthew: Concepts of the infinite variety can be found throughout mathematics. First of all, there is the endless supply of natural numbers. There is the real number line that extends to infinity in both the positive and negative directions. The Euclidean and hyperbolic planes are infinite.[1] We also have infinite series, infinite integrals, and countless other examples. Now, I know that philosophy is often concerned with the definitions of important terms. I would like to ask you to consider “infinity.”
Philip:I have often heard it defined as “that which is without bound.” In mathematics I would assume it is usually referring to boundless quantity, but in philosophy this could mean boundless duration, boundless existence, or boundless power.[2] A problem that I have with this definition, however, is that it is not falsifiable. Since it is being defined in the negative sense – based on what it doesn’t have – it is impossible to prove that infinity as a concept does not exist. You would have to find the beginning and end, the bounds, of everything in existence.
Matthew: That would be what James Thomson calls a “supertask.”[3]
Philip: So that may be a problem with the definition I mentioned, because then the concept of infinity is invincible from attack.[4]
Matthew: True, but in mathematics the definition can be salvaged and the problem resolved from the other direction – we can directly prove the existence of an infinite quantity, thus affirming the existence of the concept itself. Suppose there was a finite set containing all of the natural numbers in ascending order.[5]If we consider we see that it is a natural number through closure under addition, but it is not included in our original set because it is larger than the set’s greatest element. Thus, the complete finite set cannot exist and there must be infinitely many natural numbers. That is the simplest and most intuitive example I can think of, though many other things can be proven to be infinite in the same fashion.[6]And in modern set theory, this notion of the infinite successiveness of the integers is a foundational axiom. In fact, it is so fundamental that Peano’s first axiom declares “each integer has a unique successor.”
Philip: Okay, the proof makes sense to me, but I’m still not sure if I’m completely comfortable with the definition we started with. Maybe we’ll hit upon a better one as we continue.
Matthew: At this point I’m not as concerned with the definition as I am with some seeming contradictions that exist. For instance, it can be shown that the set of natural numbers is larger than the set of even numbers, but it can also be shown that the two sets are the same size. Galileo Galilei actually stumbled across this logical oddity near the turn of the17th Century.[7] First, we can consider the vastly familiar even numbers – 2, 4, 6, etc. It is certainly true that the set of even numbers is entirely contained within the natural numbers – 1, 2, 3, 4, etc. We would say the evens are a subset of the natural numbers. But observe that the set of natural numbers contains elements other than the evens, namely, the odd numbers, and there are a lot of them. So we can say that the evens are a proper subset of the natural numbers, and the quantity of natural numbers must be greater than that of the evens as a result.
Philip:I’m with you so far.
Matthew: On the other hand, let’s imagine the entire set of natural numbers being laid out in a single row, and likewise for the entire set of even numbers. We can place the elements of these two infinite sets into
one-to-one correspondence with each other. Now we can evaluate their sizes by determining which row runs out first. But it becomes clear that for every natural number there is an even number associated with it through the one-to-one correspondence. And for every even there will always be a natural number. In this way, it becomes clear that the two sets are the same infinite “size.”[8]
Philip:Okay, but isn’t that a contradiction. How can something be strictly larger than something else and simultaneously be the same size?
Matthew: Even though it smells like a contradiction, it’s not. The first argument is based on invalid logic, and it is the second that is currently accepted. The problem with claiming that an infinite set is larger than another infinite set because it contains additional elements is that finite comparisons such as “greater than” and “less than” are exactly that – finite. It is fallible to assume that finite conventions extend directly into the infinite realm without the need for modifications or reinterpretations. As Galileo realized, “the values of ‘equal, ‘greater,’ and ‘less’ are not applicable to infinite, but only finite, quantities.”Indeed, during Galileo’s time, those concepts did not carry into infinity. It took an entire rethinking in the form of the one-to-one correspondence test, known as Hume’s Principle, which is utilized now for comparison purposes now.
Philip: So under Hume’s Principle, an infinite set can have a proper subset that is also infinite.
Matthew: Exactly. And that idea can even be used to form a new definition, as suggested by Dedekind – a set has infinitely many elements when it contains a proper subset that can be placed in a one-to-one correspondence with itself.
Philip: With this definition, we can again say that the natural numbers are infinite because we found a one-to-one subset with the evens. You also mentioned the real number line. Can we now say that the real numbers are infinite because they contain the natural numbers, which are obviously infinite?
Matthew: We can say the real numbers are infinite, but we have to be careful. If we use the Dedekind-type definition, we have to find a subset within the real numbers that is the same “size” as the real numbers – meaning a one-to-one correspondence exists between the two. In more specific terms, the subset would have to have the same cardinality as the real numbers. Cantor proved that no such correspondence exists between the naturals and the reals, implying they have different cardinalities.[9]
Philip: So we have to find some other subset that has the correct correspondence.
Matthew: Yeah, and the simplest example is just the interval of real numbers from 0 to 1. It’s plain to see that it’s a subset, and even though it’s not plain to see, there actually is a one-to-one correspondence between the two sets.[10]
Philip: You said Hume’s Principle is currently the standard practice, but it seems to lead to some fairly counter-intuitive results. I’m not saying that it invalidates the principle, though, because human intuition can often lead us astray.
Matthew: I believe Galileo was realizing the same thing when he compared the naturals and the evens. He said that difficulties arise when we try to comprehend infinity “with our finite minds.”[11]
Philip: I’ve been reading about a similar thing in Plato’s Parmenides. The problem there is that people try to conceptualize the Forms by comparing them to material objects, but as soon as you try to materialize the Forms you’ve already missed the point. Even Socrates was guilty of this. At first he said that the Forms were as the day, because day can be in many places at once without losing its day-ness. Then he went a step further and said that it must also be like a sail, laid out over everything as the day lays out over the Earth. But this was his big error because different parts of the sail would be over different parts of the Earth, and if you removed a particular section of the sail you would have altered the original. This materialization, which the human mind is constantly tempted to perform, is not an accurate way to think about the Forms. A Form can give out shares of itself without altering itself.
Matthew: Infinity is difficult to grasp for the same reason – it can lose a part of itself but remain infinite. However, that is not true for things we come in contact with in everyday life.
Philip: Which is why philosophers are leery of sensory data. We can constantly see parts being removed from the whole, and consequently, the whole is altered. But this does not extend to the Forms.
Matthew: And we can constantly see numbers being subtracted from or divided in half, and consequently, the original number is reduced. But this does not extend to infinity, because infinity minus any finite value is still infinity, and infinity divided in half is still infinity.
Philip: Have we stumbled upon another definition of infinity?
Matthew: Let’s hear it.
Philip:Maybe we can say that if a quantity has a part of itself removed and remains the samequantity, then it is infinite.
Matthew: Interesting. We should think about that one a bit more.[12] But before we do, I wanted to bring up the fact that infinity is not just this side-concept in mathematics where weird wild stuff happens. It was actually the careful consideration of the infinite and its reciprocal, the infinitesimal,[13] that allowed mathematics to rigorize itself to the point they have today. In particular, it was the limit concept that increased the logical merit of our continuous functions and our calculus, among other things.
Philip:What are some examples of this limit notion?
Matthew: Well, we’ve seen how dangerous it can be to try and deal directly with infinity. So mathematicians will instead introduce a limit. They will consider what happens as a variable approaches infinity. This keeps them on firmer logical ground and it allows them to do some pretty powerful mathematics. For instance, we can determine some identities with infinity, and we can nearly divide by zero.We can also work with functions going to infinity at different rates using l’Hopital’s rule.[14]
Philip: This reminds me of a famous paradox in the history of philosophy.
Matthew: Let’s hear it.
Philip:They are known as Zeno’s paradoxes and there are three of them. But I know that calculus supposedly answers one, so I’ll see if I can remember it.
Consider an arrow that has just been released from a bow. We can imagine viewing the arrow’s movement within a certain portion of time, say the arrow’s first second of flight. We can further divide this time and look at only the first half second, or the first quarter second. We can even imagine viewing this arrow in the window of a single moment, where it is absolutely motionless because it has no time in which to move. Now, since time is a collection of moments, the arrow’s flight comprises many moments just like the one we isolated. But the arrow’s movement in each moment is zero, and the sum of many zeros is zero. Therefore, motion of the arrow, or anything else for that matter, should be impossible.[15]
Matthew: I think I see how calculus addresses this problem, because the notion of isolating a single moment of time is analogous to “zooming in” on a distance function. This can be done with the limit of secant lines, which gives us the tangent line. And the reason the tangent line idea works is because smooth, continuous functions are what we call locally linear – when you zoom in on them enough, they lose any curvature that they may have and appear as a straight line.From there, it is easy to find the slope of a straight line and subsequently, the slope of the function.
What Zeno was doing with the arrow was similar, because he was zooming in to a single moment. By imagining the change in time being zero – what he called a moment – he saw in his mind’s eye a motionless arrow, and then he concluded that its velocity at this moment was zero. But this was his error, because the arrow can have an instantaneous velocity even when the change in time is zero. Going back to our tangent line analogy, this process of zooming does result in a line, but it does not have to be a horizontal line. Sending the change in x to zero does not necessarily send the slope to zero as well. Thus, Zeno’s arrow can have a velocity at a particular moment, you just are not able to see it within that moment.
Philip: So this is yet another example of problems arising through a lack of care when dealing with infinite values, in this case, infinite divisions of time.
Matthew: I would say so, yes.
Philip: Well, I think we’ve learned a lesson of infinite value in that regard.
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[1] The spherical plane is also infinitely large, though in a different manner than the Euclidean and hyperbolic planes. The Euclidean and hyperbolic planes are both infinite and unbounded, while the spherical plane is bounded due to its positive curvature. If, on an infinite time scale, you were to travel in a straight line in elliptic geometry, you would eventually arrive back at your starting position. This is not true of Euclidean or hyperbolic geometry. For more, see Appendix D.
[2] The theistic god of Judaism, Christianity, and Islam is often ascribed all of these characteristics – boundless duration, boundless existence, and boundless power. In this way, the concept of god is an embodiment of infinity. Ironically, philosophers have utilized a rejection of infinity in arguments for god’s existence. For example, Thomas Aquinas claimed that the chain of causes could not go infinitely backward so there must be a “first cause,” namely god. His argument is essentially as follows: infinity cannot exist, therefore infinity exists.
[3] A supertask involves infinitely many steps and therefore can never be completed. One such task arises in Russell’s “Grand Hotel” paradox which is discussed in this chapter. Supertasks are also found in Zeno’s paradoxes, discussed in note 14.
[4] The existence of infinity is rejected by Leopold Kronecker in his philosophy of finitism. In this strict branch of constructivism, it must be possible to construct all mathematical concepts in a finite number of steps. Kronecker and his critical view of mathematical conventions are best known through his words, “God created the natural numbers, all else is the work of man.”
[5] This is possible due to the well-ordering principle of the natural numbers.
[6]Euclid of Alexandria used a similar argument over 2,000 years ago to prove that the set of prime numbers is infinite.His argument can be paraphrased thusly: Suppose the finite set contains every prime number. By considering the number it can be shown that either or its divisor is a prime that was excluded from the set, which implies that a finite set containing all primes does not exist and there is an infinitude of primes.
[7] For more from Galileo, see Dialogues Concerning Two New Sciences, translated by Crew and del Salvio.
[8] For another seeming contradiction, see Appendix A.
[9]The natural numbers are countably infinite. Cantor proved that the real numbers are uncountably infinite and therefore strictly larger than the natural numbers. For a sketch of this proof, see Appendix B.
[10]The function can serve as a correspondence between the real numbers and the open interval from 0 to 1. Utilizing the one-to-one conceptualization of infinite size, it has also been shown that the interval of real numbers from 0 to 1 is the same “size” as a square built upon that interval. This counter-intuition is similar to the fact that the natural numbers are the same “size” as the rational numbers.
[11] A perfect example of our minds failing to comprehend infinity resides in Bertrand Russell’s Grand Hotel. Russell begins by asking us to think about a hotel with a finite number of rooms. When these rooms are full it is obvious that no other guests can be accommodated. Russell then beckons us to consider a hotel with an infinite number of rooms, and suppose again that the rooms are full. The unexpected result is that this Grand Hotel can still accept another guest. You simply ask the person in Room 1 to move to Room 2, the person in Room 2 to move to Room 3, and so forth. This opens up Room 1 for the new occupant. Furthermore, you could accommodate infinitely many more guests by asking the person in Room 1 to move to Room 2, the person in Room 2 to move to Room 4, the person in Room 3 to move to Room 6, and so on, opening up an infinite number of rooms – the odds. Though this may seem like a paradox, it is actually just another example of infinity behaving differently than we would expect from our familiar knowledge of the finite.
[12] Johann Bernoulli relied on a version of this definition when considering the sum of the harmonic series. He found that while simultaneously He then concluded that the sum must be infinite (the series diverges) because it remains the same whether 1 is added or removed.
[13] An infinitesimal is a conceptual value that is greater than zero but less than any positive real number. Initially, the calculus developed by Leibniz used dx to represent this concept as an infinitely small change in x. This can lead to some pleasing results. For example, consider Then, using this infinitesimal, we have