No claim I've made has been criticized by the religious more than the claim that there are actual infinities. On the face of it, this is odd, given that God is supposedly infinite. But what motivates these critics is their belief that if no actual infinities exist, then the past cannot be infinite, and thus the universe had a beginning – which of course allows them to then more easily argue for the necessity of a creator.
These same people admit the infinity of the future, however – it's just that they don't see it as problematic. They believe that it doesn't represent an actual infinity, but only a potential one.
To understand this distinction, think of the process of counting. This process is potentially infinite – meaning it can in principle go on forever. However, no matter how long one counts, one will never actually reach infinity. Similarly, the future, it is said, is only potentially infinite: there will always be more tomorrows, but for all that the totality will always remain finite (reality never arrives at infinity), and so there never will be an actually infinite future.
An infinite past, on the other hand, is seen as problematic, because all of its yesterdays have already occurred – which means there have already been an actually infinite number of them.
This difference between the past and the future is highly intuitive – though interestingly, Aristotle, who rejected actual infinities and who came up with the actual/potential distinction, disagreed with it. Mathematician James A. Lindsay is another infinity denier who disagrees (see his book Dot, Dot, Dot, pp. 52-53).
It's also true that the above difference cannot be maintained if one thinks of all of temporal reality as equally real, so that the future is every bit as much “there” as the present and past; the difference only works on the common sense view that time involves an actual coming into being of moments. But let's accept this common sense view (as maybe we should), and also leave aside the views of Aristotle and Lindsay (both of which I think can reasonably be denied). Let's grant, in other words, the difference between the past and the future in the above argument. Even in that case, I maintain, the infinite future presents us with an actual infinity, and therefore with a problem for infinity deniers.
Here's why. Although the future hasn't happened yet, it remains the case that there is a matter of fact regarding each and every single future event. Consider: if I say that tomorrow, x will happen, then my statement is either true or false. After all, either x will happen tomorrow or it will not. Even if the future is not predetermined, so that it has not yet been “decided” whether x will happen or not, it remains the case that it will either happen or not happen. Thus, my statement is either true or false. Moreover, if, as it turns out, x does happen, then my statement is true. (After all, in that case it certainly isn't false.) And the same thing applies to every tomorrow: no matter what future time you care to consider, there is a matter of fact regarding what will happen at that time.
But this means that if the future is infinite, then there is an actual infinity of facts regarding what will happen. And what's worse – at least in that it makes the point more palpable – is that if you believe God knows all the facts about the future, then you must believe he knows an actual infinity of facts. (For more on the future and God's supposed knowledge of it, see my paper Is God's Foreknowledge Compatible with Free Will?, especially the addendum.)
This actual infinity of facts should be every bit as problematic as any other for infinity deniers. I conclude that infinity deniers who make the above argument are actually wrong.
These same people admit the infinity of the future, however – it's just that they don't see it as problematic. They believe that it doesn't represent an actual infinity, but only a potential one.
To understand this distinction, think of the process of counting. This process is potentially infinite – meaning it can in principle go on forever. However, no matter how long one counts, one will never actually reach infinity. Similarly, the future, it is said, is only potentially infinite: there will always be more tomorrows, but for all that the totality will always remain finite (reality never arrives at infinity), and so there never will be an actually infinite future.
An infinite past, on the other hand, is seen as problematic, because all of its yesterdays have already occurred – which means there have already been an actually infinite number of them.
This difference between the past and the future is highly intuitive – though interestingly, Aristotle, who rejected actual infinities and who came up with the actual/potential distinction, disagreed with it. Mathematician James A. Lindsay is another infinity denier who disagrees (see his book Dot, Dot, Dot, pp. 52-53).
It's also true that the above difference cannot be maintained if one thinks of all of temporal reality as equally real, so that the future is every bit as much “there” as the present and past; the difference only works on the common sense view that time involves an actual coming into being of moments. But let's accept this common sense view (as maybe we should), and also leave aside the views of Aristotle and Lindsay (both of which I think can reasonably be denied). Let's grant, in other words, the difference between the past and the future in the above argument. Even in that case, I maintain, the infinite future presents us with an actual infinity, and therefore with a problem for infinity deniers.
Here's why. Although the future hasn't happened yet, it remains the case that there is a matter of fact regarding each and every single future event. Consider: if I say that tomorrow, x will happen, then my statement is either true or false. After all, either x will happen tomorrow or it will not. Even if the future is not predetermined, so that it has not yet been “decided” whether x will happen or not, it remains the case that it will either happen or not happen. Thus, my statement is either true or false. Moreover, if, as it turns out, x does happen, then my statement is true. (After all, in that case it certainly isn't false.) And the same thing applies to every tomorrow: no matter what future time you care to consider, there is a matter of fact regarding what will happen at that time.
But this means that if the future is infinite, then there is an actual infinity of facts regarding what will happen. And what's worse – at least in that it makes the point more palpable – is that if you believe God knows all the facts about the future, then you must believe he knows an actual infinity of facts. (For more on the future and God's supposed knowledge of it, see my paper Is God's Foreknowledge Compatible with Free Will?, especially the addendum.)
This actual infinity of facts should be every bit as problematic as any other for infinity deniers. I conclude that infinity deniers who make the above argument are actually wrong.
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