Approaching Infinity

M. Huemer

• Publisher: Palgrave Macmillan
• Year: 2016
• ISBN:
  9781137560872
• Topic: Philosophie
• Subtopic: Philosophische Metaphysik

Approaching Infinity addresses seventeen paradoxes of the infinite, most of which have no generally accepted solutions. The book addresses these paradoxes using a new theory of infinity, which entails that an infinite series is uncompletable when it requires something to possess an infinite intensive magnitude. Along the way, the author addresses the nature of numbers, sets, geometric points, and related matters.The book addresses the need for a theory of infinity, and reviews both old and new theories of infinity. It discussing the purposes of studying infinity and the troubles with traditional approaches to the problem, and concludes by offering a solution to some existing paradoxes.

Part I

The Need for a Theory of Infinity

1

The Prevalence of the Infinite

Two things are infinite: the universe and human stupidity.

– Albert Einstein/Frederick S. Perls¹

1.1 The concept of infinity and the infinite

What is the infinite? The infinite is sometimes described as that which is unlimited, boundless, or so great that it is impossible for anything to be greater. For instance, an infinitely heavy stone would be a stone so heavy that it would be conceptually impossible for anything to be heavier. (Whether there could be such a stone is another matter.) An infinite area would be an area such that no area could be larger.

What about the noun ‘infinity’ – what does this refer to? Infinity is commonly thought of as a very large number – a number larger than all the other numbers; or, a number so large that if you add one to it, the result is no larger than you started with. Another way to get at the idea is to define infinity by reference to some particular thing that is infinite. For instance, we might say: infinity is the number that describes how many natural numbers there are. Or: infinity is the number that describes how big all of space is.

Much later in this book, we see why the above characterizations are inadequate and what a more adequate characterization looks like. But that requires much more work, so for now, I leave the reader with the above characterizations, which at least point at the intuitive conception of infinity and the infinite.

Why is infinity interesting? Three reasons: first, the infinite is an important part of reality. Second, many important philosophical arguments invoke the concept of infinity, infinite regresses in particular. Third, infinity is extremely puzzling, and we have yet to attain a clear grasp of it. We begin in this chapter with the first of these three points; the second and third points are elaborated in Chapter 2 and 3, respectively.

1.2 The infinite in mathematics

Pace Einstein, more than two things are infinite. In mathematics, there are a variety of infinite classes of abstract objects. The most familiar of these is the set of counting numbers (the numbers 1, 2, 3, and so on). There is no last member of this set; no matter how high you count, there are always more counting numbers. And the point is of course not one about human counting capacities; the main point of interest is one that would hold even if there were no conscious beings: for any natural number n, there is another greater than it.

Similarly, the real numbers (which include whole numbers in addition to fractions and irrational numbers) are infinitely numerous, since again, for any real number r, there are real numbers larger than r. Indeed, if you pick any two real numbers, there are infinitely many others sandwiched between them. For example, between the numbers 2 and 2.1, there are the numbers 2.01, 2.011, 2.0111, and so on.

These numbers are the least controversial; anyone who believes in numbers at all believes in the reals. Negative numbers used to be controversial since it seemed odd that there should be a quantity less than nothing. Potentially controversial classes of numbers include imaginary numbers, infinitesimals, hyperreals, and transfinite cardinals. Each of these kinds of number, if they exist at all, are also infinitely numerous. If there are imaginary numbers, then there are infinitely many of them (i, −i, 2i, −2i, ... ); if there are infinitesimals, then there are infinitely many of them; and so on.

That is just to speak of numbers. Other mathematical objects, if they exist at all, are also infinitely numerous. For instance, if sets exist, then there are the sets {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, and so on. Or, if you prefer sets built from concrete elements, there are {the moon}, {{the moon}}, {{the moon}, the moon}, {{{the moon}}}, and so on. Almost any other type of mathematical object will also be infinitely numerous, if that type of object exists at all (for example, vectors, groups, functions, points, spaces).

1.3 The infinite in philosophy

Philosophers often discuss propositions, which are usually thought of as a special kind of abstract object. A proposition is the sort of thing that can be either true or false; however, propositions are to be distinguished both from statements and from beliefs. When you have a belief, the thing that you believe to be the case is a proposition. Multiple people can believe the same thing (the same proposition). A person can also have multiple attitudes toward the same proposition; for instance, you may both believe and regret that you have eaten all the coffee ice cream. The phrase ‘that you have eaten all the coffee ice cream’ denotes a proposition. There can also be multiple sentences that express the same proposition; for instance, ‘It is raining’ and ‘Il pleut’ (the French translation of ‘It is raining’) both express the proposition that it is raining.² If propositions exist, then there are infinitely many of them. There are, for instance, the propositions that 2 is greater than 1, that 3 is greater than 1, that 4 is greater than 1, and so on.

Then there are universals. A ‘universal’ is something that multiple things could have in common, in the sense of something that could be predicated of them. For instance, in my refrigerator I have a tomato and a chili pepper. Both are red; in other words, redness is one of the things they have in common. So redness is a universal. Note, incidentally, that redness would be a universal even if there were fewer than two red things in the universe: redness is a universal because it is the sort of thing that multiple things could have in common, even if in fact multiple things don’t have this in common. (I don’t know why philosophers call these things ‘universals’. The name ‘universal’ makes it sound as though they are shared by everything, but in fact all that is required is that they are capable of being shared by more than one thing.)

How many universals are there? Infinitely many. Every color is a universal – not just the familiar colors such as red, green, and blue, but every shade of color, however specific, including the shades for which English lacks a name. Each of these colors is something that multiple things could have in common. Similarly, every shape is a universal – not just the familiar shapes such as round, square, and triangular, but every irregular shape, including the ones that have no names in English, and including those shapes such that no physical object is actually shaped that way. Each of these shapes is something that multiple objects could have in common. Similarly, every possible size is a universal; every type of emotional state is a universal (for example, being happy, being afraid, being angry); and so on.

Finally, many philosophers have held that there is one particularly interesting entity that is infinite, namely, God. God has been thought to be infinite in a variety of respects – to have unlimited power, unlimited knowledge, to be present everywhere at all times, and so on. According to Descartes, the word ‘infinite’ strictly applies ‘only to that in which no limits of any kind can be found; and in this sense God alone is infinite.’³

1.4 The infinite in the physical world

Physical space – the space that we occupy and move around in – is infinite in two ways. First, space is infinitely extended (it extends infinitely far in all directions). Imagine traveling away from the Earth. No matter how far away you went, it would always be possible to go farther. You would never, so to speak, come to the edge of space, like some giant wall. And the real point here is not about our traveling capabilities; the point of interest is that there are places arbitrarily far from here. That is, for any distance, d, there are places farther from here than d (even if there are no physical objects in those places).

Second, space is infinitely divisible. If you have a line segment, for example, you can divide the segment into a left half and a right half. Then the left half can itself be divided into a left half (the leftmost quarter of the original segment) and a right half. Then that left half (the leftmost eighth of the original segment) can be divided into a left half and a right half. And so on. Again, the point here is not one about human dividing capabilities; the real point is that for any two points, there are other points in between them, which entails that there are infinitely many points, and infinitely many line segments, within any given line segment.

The same observations apply to time. Time appears to extend infinitely in both directions. No matter how far into the past one looks, there were moments earlier than that. No matter how far into the future one looks, there will be moments after that. There may one day come an end of humanity, or an end of the Earth, or even an end of our material universe, but there is not going to come an end of time.

Like space, time is infinitely divisible. Take a one-minute interval of time. It can be divided into the first half and the last half. Then the first half can itself be divided into a first half (the first 15 seconds of the original interval) and a second half. Then that first half can be divided into a first half (the first 7.5 seconds of the original interval) and a second half. And so on. Every time interval contains infinitely many smaller intervals, and infinitely many instants.

1.5 The infinite in modern physics

In modern physics, some infinite quantities have been theorized but remain controversial. A black hole, as described by the general theory of relativity, is a region of infinite density (a positive amount of mass-energy is concentrated in zero volume) and infinite spacetime curvature. The same is true of the Big Bang singularity in which the universe is thought to have originated. However, the appearance of these infinities is generally regarded as signaling a breakdown of the theory (general relativity) under these conditions. To accurately describe a black hole or the conditions...