Kirk Durston Does Mathematics!

Kirk Durston is a local evangelical Christian who likes to construct unconvincing arguments for his faith. Every few years he trots them out at my university.

Here is one of his more recent attempts, a discussion of infinity. Not surprisingly, it is a confused mess.

Durston's argument is based, in part, on a distinction that does not really exist: between "potential infinity" and "completed infinity" or "actual infinity". This is a distinction that some philosophers love to talk about, but mathematicians generally do not.* You can open any contemporary mathematical textbook about set theory, for example, and not find these terms mentioned anywhere. Why is this? It's because mathematicians understand the subject well, but -- as usual -- many philosophers are extremely muddled thinkers when it comes to infinity.

Here is how Durston defines "potential infinity": "a procedure that gets closer and closer to, but never quite reaches, an infinite end". So, according to Durston, a "potential infinity" is not a set but a "procedure". Yet the very first example that Durston gives is "the sequence of numbers 1,2,3, ... gets higher and higher but it has no end". The problem should be clear: a "sequence" is not a "procedure"; a (one-sided infinite) sequence over a set S is a mapping from the non-negative integers to S. From the beginning, Durston is quite confused. His next example is "the limit of a function as x approaches infinity". But a "limit" is not a "procedure", either. Durston also doesn't seem to understand that limits involving the symbol ∞ can be restated to avoid it entirely; the ∞ in a limit is a shorthand that has little to do with infinite sets at all.

He defines "actual infinity" or "completed infinity" as "an infinity that one actually reaches", which doesn't seem to have any actual meaning that I can divine. But then he says that "actual infinity" or "completed infinity" is "just one object, a set". Fair enough. Now we know that for Durston, an "actual" or "completed" infinity is a set. But what does it mean for a set to "reach" something? And if we consider the set of natural numbers, for example, what does it mean to say that it "reaches" infinity? After all, the set of natural numbers N contains no number called "infinity", so if anything, we should say that N does not "reach" infinity.

But then he goes on to say "First, a completed countable infinity must be treated as a single ‘object’." This is evidently wrong. For Durston, a "completed infinity" is a set, but that doesn't prevent us from discussing, treating, or thinking about its members, and there are infinitely many of them.

Next, he says "it is impossible to count to a completed infinity". That is true, but not for the reason that Durston thinks. It is because the phrase "to count to a set" is not defined. We never speak about "counting to a set" in mathematics. We might speak about enumerating the elements of a set, but then the claim that if we begin at a specific time and enumerate the elements of a countably infinite set at, say, once a second, we will never finish, is completely obvious and not of any interest.

Next, Durston claims "one can count towards a potential infinity". But since he defined a "potential infinity" as a procedure, this is clearly meaningless. What could it mean to "count towards a procedure"?

He then goes on to discuss four requirements of an infinite past history. He first asserts that "the number of seconds in the past is a completed countable infinity". Once again, Durston bumps up against his own claims. The number of seconds is not a set, and hence it cannot be a "completed infinity" by Durston's own definition. Here he is confusing the cardinality of a set with the set itself.

Next, he claims that "The number of elapsed seconds in the future is a potential infinity". But earlier he claimed that a potential infinity is a "procedure". Here he is confusing a cardinality with a procedure!

Later, Durston shows that he does not understand the difference between finite and infinite quantities: he claims that "the size of past history is equal to the absolute value of the smallest negative integer value in past history". This would only be true for finite pasts. If the past is infinite, there is no smallest negative integer, so the claim becomes meaningless. So his Argument A is wrong from the start.

At this point I think we can stop. Durston's claims are evidently so confused that one cannot take them seriously. If one wants to understand infinity well, one should read a basic text on infinity and set theory by mathematicians, not agenda-driven religionists with little advanced training in mathematics.

* There are certainly some exceptions to this general rule. The "actual"/"potential" discussion started with Aristotle and hence continues to wield influence, even though mathematicians have had a really good understanding of the infinite since Cantor. Cantor met with resistance from some mathematicians like Poincaré, but today these objections are generally regarded as groundless.