Infinity

Abstract: Whoever trusts in nothing but the material not only doesn’t trust in the universal witness of the  church, but also doesn’t trust in calculus, upon which modern technology is based, and on which mathematicians and physicists the world around take for granted.

Men and women are made in the image of God. While the most critical part of this image is our free will to choose God over ourselves, the good Lord has also given us bodies and minds. These faculties are limited, unlike God’s, and thus we cannot know anything for sure. Instead, we put faith in good assumptions (not necessarily our own) to run our lives and, more specifically, to reason soundly about our world. Of course Christians ostensibly first put all our hope, faith, and love in Lord Jesus Christ.

We begin to believe in the existence of the unseen from an early age—around eight to twelve months old, according to Piaget, a renowned developmental psychologist. But much of what we all, child and adult, put faith in is not available at all to our senses; ergo, much of what we all effectively believe ¹ in is not material. For instance, as children we have faith that our parents will provide us with food, shelter, love, and other basic needs. As adults we have faith that our dates will show, that medicine will help to heal us, and that the sun will rise tomorrow. Although some of these assumptions are well-founded, we still cannot know the future until we are there, and then we don’t believe, but know. More interesting examples include the following: corporations, small businesses, and consumers believe their money exists, whether in banks’ computers or in five-cent strips of paper and cloth. Mathematicians believe in the concepts of set, number, order, pattern, and several other things besides. And those who own hot red sports cars believe that “speed” is “distance” “divided by” “time”—and so do police officers. But these are all ideas that, although they have representatives in reality, aren’t themselves objects.²

A most interesting example of something all physicists and mathematicians effectively believe in is infinity. Infinity pervades physics because calculus pervades physics, and much, if not most, of modern mathematics is concerned, directly or indirectly, with analysis of the infinite and infinitesimal, a term usually shortened to just “analysis.” Mathematical analysts, since the rigor revival of the 19th century, have been wary of infinity³  and avoid the subject by saying that a theorem gets closer and closer to being true as x increases more and more—but never actually declaring what happens if x reaches the end, so to speak.⁴  Physics has, in general, been less concerned if reaching the end can be justified logically, and more with whether experiment confirms a theory. Insofar as Newton’s infinitesimals fairly accurately predict what occurs in experiments, physicists accept his theory. In any case, both disciplines rely heavily on belief in infinity to support their theories.

Physics, for example, often refers to a “very small” (read: infinitesimal) bit of, say, fluid, calculates a property of this bit of fluid, and then “adds up” all these bits to determine a property of the whole fluid. Physics will also talk about the “very small” distance a car travels over a “very small” bit of time, and uses this to determine the speed of said car at an “instant” in time. A typical proof of one of the critical laws of electricity, Gauss’s law, involves adding up “little bits,” and the celebrated laws of Maxwell each describe properties dealing with these small pieces of whatever. I just want to make the point that most, if not all, of everyday physics uses these infinitesimals expressly, without qualms.

Mathematicians aren’t always so forthcoming about infinity. In arithmetic, for example, we learn that there are an infinity of prime numbers. The usual proof for this essentially says that supposing that we’ve found the last prime number is always a wrong supposition, so that we can always find another prime in addition to the ones we’ve already found. This is a typical way of showing the infinity of a set of objects with a given property: we can always find another such object. But the definition doesn’t really involve infinity itself per se, but only draws nearer and nearer to infinity. Math sometimes does this dance around infinity, never actually saying outright, “Here’s infinity, and here’s what happens when you get there.”

However, another way mathematics deals with infinity is flat-out to assume that it exists, and impose properties upon it. For instance, whereas in Euclidean geometry parallel lines are lines that never cross, as far as you extend them, in projective geometry parallel lines are lines that cross at “infinity,” and we actually have a line called “infinity,” where the horizon is, so to speak. In complex analysis one lets the so-called Riemann sphere represent the complex plane, and on this sphere the north pole usually represents infinity. And in upper-level infinitesimal real analysis, one starts exploring spaces like the plane, space, four dimensional space, . . . , and eventually we get to Hilbert space, which explicitly has infinite dimensions. Moreover, one sees the same principle underlying all of infinitesimal analysis of real⁵  numbers: for any two non-overlapping sets of real numbers with every element of one of the sets greater than every element of the other set, there is a real number lying between them⁶. There’s no waffling here—“there is,” and that is that, no bones about it. This already implies that given two distinct real numbers, say 0 and 1, there are infinitely many real numbers⁷. In fact, a very famous mathematician, Cantor, actually showed that there are “more” real numbers than there are integers⁸! Thus several theories in mathematics depend upon assuming the existence of infinity.

In this way we see that critical theories in both mathematics (e.g. calculus) and physics (e.g. basic mechanics) depend upon implicitly or explicitly assuming that infinity exists in some sense. But by definition, mathematical infinity is not something we can experience materially; we can’t count an infinity of apples, nor can we divvy up space into all its pieces, not even theoretically. The concept, then, of infinity that the materialists will accept (and, considering the evidence above for at least some concept of infinity, must accept) is ultimately an admission of limitation, the admission that there are thoughts we cannot think, places we cannot go, and sizes we cannot see. It seems to me that this reasonable assumption, though, is contradictory for materialism, which, as I understand, only accepts as real those things available to the senses. Materialism strikes me either as presumptuous, in assuming that we mortals can exhaustively perceive all the universe by ourselves, or else as too restrictive, saying that although we are finite and mortal, the only things we can legitimately accept as extant are the things which readily present themselves to our physical senses. Even babies know better than that.

Notes

¹Which is to say we act as though we believe.

²Now, when we Christians say that we believe in Christ, this is in the sense of a father saying to his son, “I believe in you, son.†It's more trust, sense of duty, and admiration than an intellectual admission to a possible fact or falsehood. But, of course, if you are to believe in someone, you first have to believe that such a person might exist, or you're being intellectually dishonest.

³They have been wary primarily since Fourier's groundbreaking but slightly incorrect work on heat theory and trigonometric functions. Mathematicians have had to be extra careful sifting the wrong from the right in his theory, because carelessness with infinity leads often to paradoxes, and those are unacceptable in mathematics.

⁴Abraham Robinson demonstrated in the early 1960s that the 18th century concept of the infinitesimal and the infinite could be mathematically defined with rigor comparable to that of the 19th century definition. However, his justifications were based on fairly technical results in second-order logic, and so the 19th century style of analysis, being easier to establish rigorously, has remained the standard.

⁵And complex numbers. For an explanation of complex numbers, see http://www.clarku.edu/~djoyce/complex.

⁶So between the set of complex numbers whose square is bigger than two, and the set of positive numbers whose square is less than two, there is a number that we call √2.

⁷For between 0 and 1 lies a number, say ½, and between 0 and ½ lies a number, say ¼, and between ½ and 1 lies a number, say ¾¦.

⁸ Many philosophers, theologians, and mathematicians took issue with such an odd notion; Cantor himself worried especially about the theological implications of his discoveries, holding correspondence with theologians and even sending a letter to the Pope. Cantor suffered from bouts of depression because of lack of support in the mathematical community for his theories and at last declared that God Himself had revealed these things to him.

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