Pascal’s Wager is an argument proposed by the French philosopher and mathematician Blaise Pascal in his posthumously published work Pensées (1670).

Pascal proposed the wager as a method of helping a person torn between belief and unbelief in God when they don’t feel able to settle the question based on evidence.

As such, the wager is not an evidential or “cognitive” argument for belief in God. It is an example of practical or “non-cognitive” reasoning.

In essence, Pascal seeks to show that—whether or not God exists—it is in the interests of a person who is unable to decide between these options to go ahead and believe.

It thus offers practical reasons to believe rather than new evidence to believe.

I won’t go into the details of Pascal’s Wager, because I’ve written about it elsewhere (for example, here).

However, the wager relies on insights that can be useful in other situations, and I’d like to explore some of those here.

Context

First, we need to understand the context in which the wager was proposed and what its limitations are.

We’ve already mentioned that it is not designed to give new evidence. That’s the point of the wager. It’s meant to help someone who has reviewed the evidence and still feels unable to decide.

As a result, the wager turns to look at matters besides evidence—that is, what is in the person’s interest.

There is nothing wrong with interest-based, practical reason. Humans constantly make prudential judgments about what to do based on their interests: Is it in my interest to take this job or that? To marry this person or that? To watch this movie or that?

Making decisions that maximize our interests is a fundamental part of the human experience. Such reasoning is built into us.

An Objection: Proportion to Evidence

Some question whether it is legitimate to apply practical reason to matters of belief.

Some have claimed that we have a moral duty to proportion our beliefs strictly to the evidence we have supporting them.

It is difficult to know what advocates of this claim are envisioning, because this is not how humans work. We do not constantly review our beliefs and assign numerical probabilities to them.

Much less do we proportion the beliefs themselves, so that we would say, “I 75% believe this, but I 25% disbelieve it.”

Beliefs are binary. In the typical human experience, we either believe something or we don’t.

We may have different degrees of confidence about our belief, but the belief itself is either there or it isn’t.

How Things Work in Science

It is readily admitted by scientists that the results of science are always provisional.

No matter how much evidence has been accumulated for a scientific theory, it’s always possible that new evidence will emerge that indicates the theory must be modified or rejected in favor of a better one.

But that doesn’t stop scientists from believing particular scientific claims.

Based on the evidence so far accumulated, they accept—let’s say—the existence of electrons. They believe in them, and then they proceed about their business on the premise that electrons exist, without doubting this.

If someone asks them how sure they are that electrons exist, they may stop and mentally review the evidence and say something like, “Well, the results of science are always provisional, so I can’t say with infallible certainty that they do. But the evidence is so strong that I can’t imagine a scenario where sufficient evidence would emerge to overturn their existence. So, I believe that electrons do exist, and I don’t worry about the tiny chance that they don’t.”

In saying something like this, a scientist would be acknowledging that:

• 1. There is always a gap between the evidence at hand and total certainty, and
  2. That this gap is sufficiently small that the scientist doesn’t worry about it.

In other words, the scientist has made a leap of faith to overcome the evidential gap. He then adopts the belief that electrons exist, and he doesn’t deem it worthwhile to worry about the possibility that he is wrong unless something happens to cause him to reflect on the question.

Everyday Life

Such leaps of scientific faith are omnipresent in the sciences, but the same applies in all areas of human life.

For example, most people believe that their spouses are not secretly trying to kill them. The evidence for this proposition is significantly less than the evidence for the existence of electrons.

In fact—among a population of billions—any number of people do try to kill their spouses. But—absent evidence that this is the case in a particular instance—the odds are so low that it is not worth worrying about.

People thus accumulate a certain amount of evidence—e.g., that someone loves them and will not kill them—they adopt the belief, “I am safe with this person,” they marry them, and then they don’t worry about it until significant evidence emerges to the contrary.

This is simply how human belief works.

And so, the idea that we should proportion our belief to the evidence does not describe the human experience.

Instead, we see enough evidence that we deem it rational to adopt a belief, we adopt it, and then we don’t worry about the chance we are wrong until something happens that causes us to question the belief.

In other words, we make a leap of faith to overcome the gap between the evidence we have and the position of belief (i.e., acceptance of a proposition without worrying about it) that we need to achieve in order to move on with life.

Paranoia and Self-Interest

We even have a word for people who fail to do this and who continue to worry about the possibility they are wrong: We call them paranoid.

If—despite the evidence a person has that they are safe with their spouse—they continue to worry about the idea that their spouse is going to kill them, that person is paranoid, and we tell them so.

“Look,” we may say, “it is hypothetically possible that your spouse is plotting your murder. But the evidence for that is so small that you shouldn’t be worrying about it. You are only hurting yourself by doing so—and you may be dooming your marriage to failure.”

By making an argument like this, we are appealing to the person’s interests.

They are currently hurting themselves with unnecessary worry—which is contrary to their interests.

And they may in the future hurt the interests of both themselves and their spouse by dooming a marriage that can otherwise benefit both.

In appealing to them to stop worrying, we urge them to use practical reason to overcome the evidential gap between what they’ve seen and the subjective certitude they need to move on with their life on the belief that they are safe with their spouse.

In other words, we are counseling them to make a leap of faith in their own self-interest.

Back to Science

This is the same thing every scientist does when they make a leap of scientific faith between the evidence that electrons exist and the belief that they do—or any other scientific belief they may entertain.

At some point, it would become scientific paranoia to continue to have doubts or anxiety about the existence of electrons (or whatever).

We would thus counsel a paranoid scientist to set aside his doubts and move on—given that he lacks compelling evidence to the contrary.

Is it rational—in terms of self-interest—for the scientist to worry about the reality of electrons, or is it better to believe that they do and move on—being willing to reconsider this if contrary evidence emerges in the future?

If the scientist continues to devote time and energy to the non-existence of electrons—in spite of the current evidence—he is hurting himself and his career.

He is harming his quest for greater scientific understanding by wasting time on an exceedingly unlikely hypothesis, and also hurting society at large by denying others the discoveries he could otherwise make.

We thus counsel him to set aside his worries and make the scientific leap of faith needed to overcome the evidential gap between what experiments have shown and belief (acceptance without worry) that electrons exist.

Preliminary Lessons

From the preceding, I take it that there is simply a difference between the degree of confidence that the evidence alone would warrant and the belief that corresponds to this.

It is rational to make leaps of faith between the two—and it is rational to do so on practical (prudential) grounds.

At some point, the evidential chance of being wrong is low enough that it simply is not worth worrying about the idea one is wrong.

Instead, it is in one’s interest—and the interests of others—to set aside doubts and proceed on the basis of belief.

At some point, we judge it impractical to continue to worry above the evidential gap and choose to embrace a belief on practical grounds. That’s just how humans work.

I thus take it as established—at least from this point forward—that there is a difference between:

• • Whether we believe a proposition (which is binary; we either believe a proposition or we don’t), and
  • What degree of confidence we feel regarding the proposition when we review the evidence for it.

I further take it as established that:

• • It can be rational to believe a proposition even if the confidence level we feel based on the evidence is less than what would be required for infallible certainty,
  • We all do this constantly; we all wager, all the time, and
  • There is nothing wrong with this; it is how human cognition works.

This puts us in a position to consider interesting aspects of the reasoning involved in Pascal’s Wager.

A Limit of Pascal’s Wager

Pascal’s Wager was formulated to help a person in a specific situation—being torn between belief in the Christian God and a western form of skepticism that would involve agnosticism or materialistic atheism. As a result, it does not deal with other religious options.

Many have pointed out that there are other options, and the wager doesn’t address them. This is true, but it does not deprive the wager of its utility for those who are in this situation.

In his 1896 lecture “The Will to Believe” (later published as an essay), William James provided helpful discussion of this subject, noting that—for various people—some hypotheses are “live” while others are “dead.”

James defined a live hypothesis as “one which appeals as a real possibility to him to whom it is proposed,” whereas a dead hypothesis is one that does not strike the hearer as a real possibility.

James referred to the decision between two hypotheses as an “option” and stated:

A living option is one in which both hypotheses are live ones.

If I say to you: “Be a theosophist or be a Mohammedan,” it is probably a dead option, because for you neither hypothesis is likely to be alive.

But if I say: “Be an agnostic or be Christian,” it is otherwise: trained as you are, each hypothesis makes some appeal, however small, to your belief.

Pascal’s Wager, then, is designed to help a person for whom both Christianity and western skepticism are live hypotheses.

Other Wagers

However, wager-style reasoning can be applied to other situations. To cite a simple example that I’ve discussed before, one can construct a kind of “reincarnation wager.”

Suppose a person’s live option is whether to believe in reincarnation or whether to believe that this life is the only one we have.

How we spend our time has consequences—whether it is achieving goals with respect to this life or with respect to the afterlife.

Consequently, if a person feels unable to decide the issue of reincarnation based on evidence, it will be in his interest to believe the latter so as to make the most of the time he has. If it turns out he is wrong and he reincarnates, he will simply get more time to pursue his goals and “get it right.”

There also can be a similar “afterlife wager” for those who have a live option between believing that there is no afterlife and the possibility that there is an afterlife in which we experience positive or negative consequences based on what we do in this one.

If one is unable to decide this question based on evidence, it will be prudent to assume that there is such an afterlife so as to take reasonable steps to ensure a good afterlife.

If it turned out that the person were wrong and there was no afterlife, the person would not experience a negative one and would only have wasted reasonable efforts in pursuit of a good one.

Wagering, Materialism, and Morals

In light of the applicability of wager-style arguments to other situations, I’d like to address one involving materialism and morals.

Despite the fact we all constantly wager and adopt beliefs based partly on practical rather than evidential reasons, one of the concerns limiting the use of wager-like reasoning is a nagging anxiety people have about whether they are doing something “wrong” by adopting beliefs on these grounds.

I concede that people have a moral intuition that there needs to be some kind of relationship between belief and evidence.

For example, we have the intuition that we would be violating what philosophers call our “epistemic duties” if we chose to believe something that had a massive amount of evidence against it and no evidence for it.

This is true. However, it is not applicable to the situation that Pascal’s Wager is designed to address.

The wager is specifically intended to address a situation in which a person has considered the evidence and still feels unable to make an evidence-based decision.

Further, as William James points out, we may be forced to make a choice, for to refuse to adopt belief in a proposition is to adopt the alternative of non-belief in it. James discusses this in terms of a decision between adopting a religious view or not doing so:

[W]e see, first that religion offers itself as a momentous option. We are supposed to gain, even now, by our belief, and to lose by our nonbelief, a certain vital good.

Secondly, religion is a forced option, so far as that good goes. We cannot escape the issue by remaining sceptical and waiting for more light, because, although we do avoid error in that way if religion be untrue, we lose the good, if it be true, just as certainly as if we positively chose to disbelieve.

It is as if a man should hesitate indefinitely to ask a certain woman to marry him because he was not perfectly sure that she would prove an angel after he brought her home. Would he not cut himself off from that particular angel-possibility as decisively as if he went and married some one else?

Scepticism, then, is not avoidance of option; it is option of a certain particular kind of risk. Better risk loss of truth than chance of error—that is your faith-vetoer’s exact position. He is actively playing his stake as much as the believer is; he is backing the field against the religious hypothesis, just as the believer is backing the religious hypothesis against the field.

One thus does not escape the trap of making a choice in this situation. It is simply a choice between belief and non-belief.

And—in the absence of evidence that decides the matter—it is made on non-evidential grounds no matter which choice is made.

Religion vs. Scientific Materialism

The above illustrates the difficulties with the idea that it is somehow immoral—a violation of epistemic duties—to adopt a belief based partly on pragmatic rather than evidential concerns.

However, there is more that can be said about this when a particular situation is considered—that is, one like Pascal’s original situation of a person torn between Christianity and skepticism.

Today in the West, skepticism typically entails a form of materialism in which science is given a primary place (i.e., scientific materialism).

Conventional science is driven by empirical phenomena—things that can be observed and measured using the conventional senses (sight, hearing, etc.) and their technological extensions (microscopes, telescopes, spectrometers, gas chromatographs, etc.).

Science is held to be incapable of investigating non-empirical phenomena (souls, spirits, God), and so these are deemed outside the realm of science.

Indeed, for scientific materialism, it is the non-empirical quality of these entities that drives rejection of their existence in the first place.

However, it isn’t only souls, spirits, and God that are not subject to empirical investigation. It is also morality.

Moral properties like good and evil, right and wrong, cannot be detected with the senses or their technological extensions. As a result, it is difficult to see how morality could be real if scientific materialism were true.

Another Wager

This leads us to another wager—this time between a religious worldview and scientific materialism:

1) Suppose a person adopts a religious worldview, and it turned out that scientific materialism were right and that there are no non-empirical things.

In that case, the person would not be violating their epistemic duties because morality would be a fiction, and the person had done nothing wrong by being religious.

2) On the other hand, suppose that a person adopts a worldview of scientific materialism, and it turns out the religious worldview is correct.

On the religious worldview, morality is real, and one should be a moral person. The person then has a choice:

a) In keeping with their scientific materialism, they could reject the real existence on the grounds that it is non-empirical. In this case, they would be doing something wrong because the religious worldview is true and morality is real.

b) Or, despite their scientific materialism, they could continue to accept the real existence of morality. In this case they also would be doing something wrong, because they are violating their own principles, and violating your own principles is morally wrong.

We thus see that (1) if a person incorrectly adopts a religious viewpoint, he does nothing wrong, while (2) if he incorrectly adopts scientific materialism, he inescapably does something wrong.

Given these facts, the logical thing to do is to accept the religious worldview since—whether it is correct or not—one avoids doing something wrong.

Testing the Wager

One way of testing this wager is to ask, “If the religious worldview is true, could I still be doing something wrong by adopting it? Not in the sense of being religious, because we’re assuming this view is true. But perhaps by violating my epistemic duties in some way from within a religious perspective?”

At this point, we are speaking purely from within a religious perspective. We are taking it that religion is true and asking whether one can violate one’s epistemic duties and thus do something morally wrong.

The answer, of course, is yes. From a religious perspective, people of any stripe—religious or not—need to be moral people, and that includes honoring their epistemic duties.

If a person—religious or non-religious—stifles his conscience to convince himself that murder is an okay thing to do, then he is violating his epistemic duties.

So, yes, religious people can violate their epistemic duties. But what does this have to do with the question of being religious itself?

We can infer from this that one should not violate one’s epistemic duties by adopting beliefs that one should not, so don’t join a religion that teaches them.

If you have a functioning conscience, don’t become a member of the Manson Family and participate in its murder sprees. And if you have good evidence that evolution is true, don’t join a church that insists on Young Earth Creationism.

The Religious View in General

But how would one be violating one’s religious duties merely by adopting a religious point of view?

This returns us to the question of evidence and what relationship it has with belief adoption.

If a person thought that he had conclusive evidence against religion, then he should not adopt a religious point of view.

And if a person thought he had conclusive evidence for religion, then he should adopt it.

However, neither of these situations is what wager-style arguments are designed to address (or at least the kind that we are considering). They are for people who don’t think that they can settle the matter based on their review of the evidence.

But there is still a need to settle it, and so wagers appeal to practical reason to overcome the evidential gap—just as we do in science and in everyday life.

Given the omnipresence of pragmatic leaps of faith in every field of human endeavor—indeed, in virtually every belief we adopt except as the result of a mathematical demonstration—it is hard to see how using practical reason to overcome an evidential deadlock could be seen as violating our epistemic duties.

We use practical reason to overcome evidential gaps all the time. It is built into human nature, and so we are simply acting in accord with our nature when we do so. There is nothing wrong with this.

Intellectual and Moral Coherence

Further, adopting a religious perspective provides a greater degree of intellectual and moral coherence than adopting scientific materialism.

Whether or not one is religious, we have an inbuilt moral sense that tells us that we have moral duties, including the epistemic ones that the person torn between religion and materialism is concerned about.

On a materialist view, these may have an evolutionary explanation, but they do not objectively bind, and—as non-empirical—they should not be given credence.

Nevertheless—unless they are psychopaths—materialists find themselves inescapably falling back into thinking and acting as if morality is objectively real. They are as horrified by murder, bigotry, and oppression as anyone—even though their worldview would imply that there is nothing objectively wrong with any of these.

Materialists thus have a lived experience that is inconsistent with their belief system, resulting in a lack of coherence between the two.

By contrast, on the religious view, non-empirical entities are real, and this provides an intellectual framework that allows our in-built moral sense to be what we take it to be—a reflection of reality and something that is objectively binding on us.

The religious view thus provides a form of coherence between the intellectual and the moral that scientific materialism does not.

Coherence between belief and lived experience is a desirable feature of worldviews, and the religious worldview offers this regarding moral realism, whereas materialism does not.

This is one more reason—in addition to the evidential and pragmatic reasons—to prefer the religious worldview.

In Reasonable Faith, William Lane Craig seeks to show the absurdity of an actually infinity of things by appealing to surprises that await us at Hilbert’s Grand Hotel.

This is a thought experiment proposed by the German mathematician David Hilbert (1862-1943). In the thought experiment, Hilbert envisioned a Grand Hotel with an infinite number of rooms, all of which are full.

Hilbert then posed a series of scenarios that would allow the hotel to accept additional guests—both any finite number of guests, and even an infinite number of guests!

Craig poses four scenarios involving Hilbert’s Hotel, which I will summarize this way:

1. 1. A Single New Guest: The hotel is full and a new guest arrives. The manager has each current guest move to the next higher room, freeing up the first room for the new arrival.
   2. Infinite New Guests: The hotel is full and an infinite number of new guests arrive. The manager has each current guest move to the room whose number is double that of their current room. This puts all the existing guests in the even numbered rooms, leaving the odd numbered rooms free to accept the infinite number of new arrivals.
   3. Odd Numbers Check Out: The hotel is full and all the odd numbered guests decide to check out, leaving their rooms vacant. The manager then has the guests in the even numbered rooms move to the room whose number is half that of their current room. This fills up both the odd and the even numbered rooms, and the hotel is completely full again.
   4. Guests Above 3 Check Out: The hotel is full and all of the guests with room numbers above 3 check out, leaving only three guests in the hotel. Yet since the set {4, 5, 6, . . . } has the same “size” as the set {1, 3, 5 . . . }, it would seem that the same number of guests checked out in this example as in the previous one, but the resulting number of guests in the hotel was different.

Craig regards all of these as absurd and concludes both that such a hotel could not exist and that an actual infinity of things cannot exist.

How might we respond to this?

Yes, They’re Strange

My first response would be to say, “Yes, these are strange, surprising, and counter-intuitive. If you want, you can even call them absurd.”

But that’s not what I’m interested in. There are lots of true things that fit those descriptions.

What I’m concerned with is whether any of these situations entail logical contradictions that God could not actualize in some possible world, and it’s not obvious that they do.

What Infinity Means

It seems to me that part of the problem is that we have a persistent tendency to slip into thinking of infinity as if it is a particular, concrete, limited number. It’s not. By definition, it is unlimited. That’s what the term “infinite” means. But as long as we use labels like “infinity,” we tend to slip into thinking of this as if it were an ordinary, limited number rather than something of unlimited magnitude.

It can help to strip away the label and repose the question without it.

• • Suppose that I had an unlimited number of apples. Then you bring me a new apple, and I add it to my collection. How many apples do I have now? Well, it’s still an unlimited number. Adding a new item to an already unlimited collection isn’t going to change the status of the collection from unlimited to limited, so it’s still unlimited.
  • Suppose that you bring me a whole bunch of apples—an unlimited number of them—and I dump them into my collection. How many are there now? Again, adding even an unlimited number of apples to my already unlimited collection won’t change the status of my collection from unlimited to limited, so it’s still unlimited.
  • Suppose that you go through my apple collection and pull out all the odd numbered apples (let’s suppose that I’ve conveniently numbered them so I can always find the apple I want). How many are left in the collection? Pulling out every other apple from an unlimited number of apples would still leave an unlimited number there, so my collection is still of unlimited size. And I can re-number the apples I have left if I want.
  • Finally, suppose that you pull all of the apples out of my collection except the first three. How many do I have left? Three. “Why is the number different than when I pulled out the odd numbered ones?” you might ask. “Because of which ones you pulled out,” I reply. “The first time, you pulled out every other apple, but the second time you pulled out all of the apples but the first three. Of course, you’re going to get a different number left over at the end.”

What we’ve just done here is run through the same scenarios that Craig uses, only we’ve done it with apples instead of hotel rooms and we’ve done it using the intuitive term “unlimited” rather than less intuitive terms like “infinity” or “aleph-null,” both of which suggest a definite amount to our ears.

When we keep the fact that the amount is unlimited clearly in focus, the situation sounds quite a bit less counter-intuitive. The reason is that we aren’t losing sight of the not-limited amounts we are playing with.

But Could All This Really Exist?

Certainly not in our universe—not without God suspending the physical laws he has set up to govern it.

If a hotelier decided to build a hotel with an infinite number of rooms, he would be immediately confronted with the fact that there isn’t enough room on earth to house such a hotel, there aren’t enough construction workers to build it, and he doesn’t have an infinite amount of money.

Similarly, I can’t go to a store, or an orchard, or any number of stores and orchards, and buy an infinite number of apples. (Nor do I have an infinite amount of money—quite the opposite, in fact!)

So, both the hotelier and I would quickly discover that our projects are practically impossible (meaning: not possible given the practical restrictions we operate under).

Yet suppose the hotelier and I were to make go of our projects, somehow pulling in resources from off-world once earth’s supplies started to go low. At some point the hotel and the apple collection would collapse under their own mass.

If we kept loading material into them, they would become so massive that they would begin to fuse, and they would both turn into stars.

If we still kept loading material into them, they would become more massive yet and, eventually, millions of years later, go supernova and turn into black holes.

Such is the way of things, given the physical laws that operate in our universe.

Even if God were to intervene and miraculously allow the things to be built and not become black holes, we still wouldn’t be able to do the infinite amount of guest and apple shuffling involved in the thought experiments. We wouldn’t have the time!

So, I’m quite prepared to concede that, in addition to being practically impossible, the infinite hotel and the infinite apple collection are also physically impossible (meaning: not possible given the physical laws of our universe).

But that’s not our question. We’re not asking about practical possibility or even physical possibility. We’re asking about logical possibility.

How useful is a hotel metaphor?

Here I would like to introduce what I think is the main problem with Craig’s use of Hilbert’s Hotel to disprove the possibility of an infinite history: I just don’t think it’s relevant.

Even if you grant that Hilbert’s Hotel involves a logical impossibility, that doesn’t show that all actual infinities involve a logical impossibility.

We’ve already shown that there is an actually infinite set of mathematical truths, so that is logically possible.

And that actual infinity does not seem to be subject to the same kind of difficulties that Hilbert’s Hotel raises. Even if we were to start numbering the different mathematical truths the way the rooms in Hilbert’s Hotel are numbered, we couldn’t perform the kind of manipulations on them that the hotelier does.

We couldn’t have the mathematical truths move around the way the guests in the hotel do. We couldn’t make new mathematical truths appear or existing mathematical truths vanish.

We could change the numbers we apply to these truths, but that’s not the same thing. That’s just our labeling. It doesn’t affect the truths themselves.

The apple collection fares no better. We can’t move, add, or delete mathematical truths the way we can apples. Mathematical truths are just there.

And so it seems that the kind of puzzles we can generate with Hilbert’s Hotel simply do not arise with the actual infinity of mathematical truths.

This means, if we grant that Hilbert’s Hotel involves a logical contradiction, that we have to ask the question: How relevant is it to the idea of God creating an infinite history for the universe? Is the idea of an infinite history in the same category as Hilbert’s Hotel or in the same category as the set of mathematical truths?

Some Similarities

At first glance, you might think that an infinite history would go in the same category as Hilbert’s Hotel.

For one thing, Hilbert’s Hotel involves an infinite amount of space (the infinite rooms it contains), and an infinite history would involve an infinite amount of time. We could map the rooms of Hilbert’s Hotel onto the individual moments of time in the universe’s history.

Furthermore, the rooms in Hilbert’s Hotel have content (guests and the things they are doing in their rooms), and the moments of history have content (people and the things they are doing—or at least whatever God would have chosen to put in them).

But these similarities turn out to be rather superficial. In fact, you could map any countably infinite set onto any other countably infinite set. You could, thus, map the rooms of Hilbert’s Hotel onto something totally non-physical, like the mathematical truths that fit the form X + 1 = Y, like this:

• • Room 1 –> (1 + 1 = 2)
  • Room 2 –> (2 + 1 = 3)
  • Room 3 –> (3 + 1 = 4)
  • Room 4 –> (4 + 1 = 5)

And just as each room in the hotel has contents (guests), each slot in our sequence of mathematical truths has contents (the particular truth in question).

The real issue is whether the moments of history and their contents can be manipulated the way that the rooms and guests in a hotel can.

And they can’t be.

Time Can Be Rewritten?

Despite Doctor Who’s repeated assurances that time can be rewritten, this question has proved a real puzzler for philosophers and scientists.

Scientists are of different opinions about whether time travel to the past is physically possible. Einstein was startled when, in honor of his 70th birthday, Kurt Gödel gave him a proof showing that Einstein’s own equations would allow for the possibility of time travel if we are living in a certain kind of universe.[1]

So what would happen if you really were able to go back in time? Could you change history? Could you carry out the famous “grandfather paradox,” where you kill your own grandfather before the conception of your father, thus preventing your own birth? (And, by extension, preventing you from coming back in time to kill your grandfather.)

According to much of the thinking on the issue, there would be two possibilities:

1) You simply would not be able to kill your grandfather. According to this view, time has an as-yet undiscovered principle that prevents time paradoxes from happening. This was proposed in the 1980s by the Russian physicist Igor Novikov (b. 1935), and the proposed principle is known as the Novikov self-consistency principle.

2) You would be able to kill your grandfather, but in so doing you would create a new timeline. According to this view, you would be able to kill your grandfather, but this would not change your history. The timeline in which your father and you were born would still exist. That timeline has to still exist, because you need to leave at some point to come back in the past. What would happen if you kill your grandfather is you would cause a new timeline to branch off from the original one, and in the new timeline you would never be born. You would still be around, however, because your original timeline is still out there, undamaged.

If the first of these possibilities is correct then the Doctor is simply wrong. Time can’t be rewritten.

If the second of these possibilities is correct then the Doctor is right. –ish. Up to a point.

You could “change history” in the sense of causing a new timeline to branch, one in which you are never born. But you didn’t change your history. The timeline where you were born, grew up, and travelled back in time is still out there. It’s still real. It didn’t un-happen from an external perspective looking at the timelines.

How does this help us answer our question about an infinite history?

Forward, into the Past!

The ancients really don’t get enough credit, because since at least the time of the Greeks, people have been asking whether God could change the past.

The poet Agathon (c. 480-440 B.C.) wrote that:

For this alone is lacking even to God,
To make undone things that have once been done.

This statement is quoted and endorsed by Aristotle (Nichomachean Ethics 6:2).

The same view is taken up and endorsed by St. Augustine (c. 354-430), who wrote:

Accordingly, to say, if God is almighty, let Him make what has been done to be undone, is in fact to say, if God is almighty, let Him make a thing to be in the same sense both true and false [Reply to Faustus the Manichean 26:5].

And the same view is endorsed by St. Thomas Aquinas, who wrote:

[T]here does not fall under the scope of God’s omnipotence anything that implies a contradiction. Now that the past should not have been implies a contradiction. For as it implies a contradiction to say that Socrates is sitting, and is not sitting, so does it to say that he sat, and did not sit. But to say that he did sit is to say that it happened in the past. To say that he did not sit, is to say that it did not happen. Whence, that the past should not have been, does not come under the scope of divine power [Summa Theologiae I:25:4].

Notice that both Augustine and Aquinas identify the same reason for God not being able to change the past: It involves a logical contradiction.

Does it?

From Here to Eternity

The view that I’ve advocated in this paper—that God is outside of time and that, consequently, the B-Theory of time is true—indicates that God cannot change the past—at least not in the way relevant to our discussion.

To illustrate this, let’s think about a particular moment in my own past, which I have already referred to: the moment when I was five years old and, in my grandmother’s kitchen, reached for a broom, causing a Coke bottle to explode and injure my knee.

That moment is present to God in eternity.

In the eternal now, he is creating that moment in history, and, by my free will, I am grabbing the broom and causing the Coke bottle to explode.

Could God change history so that this doesn’t happen? Could he, for example, give my grandmother a sudden inspiration a few moments earlier so that she can intervene and prevent me from grabbing the broom?

No.

Why not?

Because that moment is present to him in the eternal now. It’s real. He can’t undo it, because there is no time where he is. He can’t first let it happen and then undo it. That would imply the passage of time for God. It would imply more than one moment in the eternal now, in which case God would not be in eternity but in time.

What God could do (I assume) is create a second timeline,[2] in which my grandmother does stop me from breaking the Coke bottle, but that would leave intact the original moment in which I did break it. That moment is still there, still present to God in eternity, and thus still real.

If Akin is right about this then so were Agathon, Aristotle, Augustine, and Aquinas: God cannot undo history.

Once an event has happened in time, it’s set in eternity.

This has important implications for the question of whether he could make an infinite history.

Not Like Hilbert’s Hotel

If God can’t change history then time is not like Hilbert’s Hotel.

God cannot, in the eternal now, cut up, rearrange, and reshuffle the moments of history the way Hilbert’s hotelier moved guests around. If Moment 2 follows Moment 1 in time, then that is the way they are arranged before God in eternity. He can’t swap their order because in eternity there is no time for him to swap them in.

God cannot, in the eternal now, create or delete additional moments in time beyond those he has already created, the way I could add or subtract apples from my collection.

There is no time in which God could do these things, because that is the way eternity works by definition. It is thus logically impossible.

That means that the kind of puzzles that arise with Hilbert’s Hotel simply do not arise with time from God’s perspective.

An actually infinite history for the universe thus would go in the same category as an actually infinite number of mathematical truths. It does not give rise to Hilbert paradoxes.

Neither does an actually infinite future for the universe, which is why God is able to create one of those—as he has.

Therefore, even if one grants that Hilbert’s Hotel involves a logical contradiction and could not be realized by an omnipotent God, it does not matter. Viewed from the eternal perspective, an infinite amount of time (past or future) does not generate the problems of Hilbert’s Hotel and so is not in the same category.

[1] This solution is known as the “Gödel metric.” For a good popular introduction to the subject of time travel from the perspective of contemporary physics, see The Physics of the Impossible by Michio Kaku, who is a professor of theoretical physics at City University of New York.

[2] Because God is in eternity, this second timeline, and any others he might create, would all exist simultaneously for him in the eternal now, just as our timeline does. From the eternal now, he would be simultaneously creating all of the histories that exist.

We know that God created the universe a finite amount of time ago, but defenders of the Kalaam cosmological argument say that God had to do it this way. He had no other choice.

William Lane Craig has proposed the following argument to support this claim:

1) An actually infinite number of things cannot exist.

2) A beginningless series of events in time entails an actually infinite number of things.

3) Therefore, a beginningless series of events in time cannot exist.

Depending on your view of time, there are potential problems with the second premise—as I’ve written elsewhere.

However, here I’d like to consider the first premise.

Is it true that an actually infinite number of things can’t exist?

What’s an “Actual” Infinite?

We need to be aware of the difference between what philosophers and mathematicians call “potential” infinities and “actual” infinities.

• • Something is potentially infinite if it goes on endlessly, but there is no frame of reference in which all of its elements exist.
  • Something is actually infinite if it goes on endlessly and there is a frame of reference in which all its elements exist.

For example, suppose that you have a machine that makes cubes. Today it makes a cube, tomorrow it makes a cube, and it keeps on like that endlessly. No matter how many days you go into the future, there will still be a finite (limited) number of cubes.

From within the perspective of time, there is no day where the “infinity-eth” cube pops out of the machine, because “infinity” is not a number on the number line. There is no “number just before infinity,” and so you can’t count to infinity.

Yet the series of cubes that the machine will make is endless, which is what “infinite” means—unlimited or unending (Latin, in- “not” + finis = “limit, end”).

So, while the number of cubes you have grows toward infinity, it never gets there. That’s why the cubes would be said to be potentially infinite rather than actually infinite—because they don’t all exist at the same time.

But suppose you didn’t have a cube-making machine. Suppose instead that God decided to create an infinite number of cubes all at once. Bam! It’s done. All in a flash.

In this case, you would have an infinite number of cubes—that all exist at once—and so that would be an actually infinite set of cubes.

The key point to remember is that both potential infinites and actual infinites involve limitless numbers of things. The difference is that in a potential infinity these elements don’t all exist at once, while in an actual infinity, they do.

Can Actual Infinities Exist?

While many authors assert that actual infinities can’t exist, we should test this. Can we think of anything actually infinite?

How about numbers?

We often represent the set of natural numbers like this: {0, 1, 2, 3 . . . }. The reason we put the ellipsis (the three dots) at the end is to say that this series of numbers goes on forever in the same way it began, with one number after another, with no end to them.

That’s an infinite set!

And all those numbers already exist. It’s not like there’s a mathematician somewhere inventing new numbers in his workshop.

We may make up names for new numbers—like the number googol, which is 10-to-the-power-of-100, or googolplex, which is 10-to-the-power-of-googol—but we didn’t create these numbers. We only named them.

Even before they were thought of or named, it would remain true that googol multiplied by 2 is 2 googol, that googol minus google is 0, and that googol divided by google is 1.

So, it appears that an infinite quantity of natural numbers exists, whether we’ve thought of or named them or not.

Since the set of numbers does not grow with time—only our knowledge of them does—the set of natural numbers is an actually infinite set, as mathematicians commonly acknowledge.

The Truth of the Matter

Numbers are not the only actual infinity we can think of. There also are truths (facts), as we can easily see:

• • It is true that 1 + 1 = 2.
  • It is true that 2 + 1 = 3.
  • It is true that 3 + 1 = 4.
  • And so on.

So, not only does an actually infinite quantity of numbers exist, an actually infinite quantity of truths also does.

How might a defender of the Kalaam argument respond to this?

The Nature of Numbers and Truths

It’s easy to point out that there is a difference between numbers and truths and the kind of objects we see in the world around us.

For example, if I have two cubes that I’m holding in my hands, I’m physically touching them. But I don’t seem to be touching the number 2. Numbers aren’t physical objects you can see, hear, or touch.

Neither are truths. It may be a truth that Abraham Lincoln died in 1865, but I can’t hold this truth in my hands like an apple or put a ruler beside it and measure how long it is.

Things like numbers and truths seem to be in a different category than things like cubes and apples. Things in the first category are often called abstract objects, while those in the latter are often called concrete objects. We also might call them physical objects.

The sciences have given us a lot of information about how physical objects work, but we can’t use science to investigate abstract ones. They lie in the realm of philosophy, and among philosophers there are different views about their nature.

Some philosophers (known as anti-realists) deny that abstract objects are real and propose other ways of understanding them. Other philosophers (known as realists) hold that they do exist, but again there are different understandings (e.g., do abstract objects exist in an abstract realm of some kind? do they only exist in physical objects? are they based in the mind of God?).

Options for the Kalaam Defender

A Kalaam defender could adopt an anti-realist position and say that things like numbers and truths simply do not exist, which would mean that there aren’t actual infinities of these—because numbers and truths don’t exist in the first place!

But this seems hard for many to imagine, including various supporters of the Kalaam argument.

There is another option, which is to draw a line between the two classes of objects and say something like, “Look, whether or not actual infinities of abstract objects exist (maybe they do; maybe they don’t), that’s not what I’m talking about. When I say that actual infinities can’t exist, I mean that they can’t exist concretely, in the physical world.”

A person taking this position could acknowledge that actual infinities of abstract objects can exist. What he disputes is that actual infinities of concrete, physical ones can.

Why Not?

The question would be: Why not? Why can’t actual infinities of physical objects be real?

This question is particularly acute from a Christian perspective, since the Christian faith holds that God exists and that he is omnipotent, which means that he can create anything that does not involve a logical contradiction.

So, let’s do a thought experiment:

Imagine a cubic foot of empty space that has a single hydrogen atom in it. If we can imagine that, so can God.

Now imagine a second cubic foot of empty space sitting right next to it, also with a hydrogen atom in it. God can imagine that, too.

Now put a third cubic foot of space next to that, also with a hydrogen atom, so that we have a row of three.

Then imagine a fourth, a fifth, a sixth, and so on. God can imagine each of these as the line of cubic feet extends off into the distance.

In fact—due to his omniscience—God can imagine any number of such units. Unless there is a logical contradiction involved, God could imagine a cubic foot of space with a hydrogen atom for every natural number.

And so, God can imagine an actually infinite volume of space, with a single hydrogen atom in each cubic foot.

Either this scenario involves a logical contradiction or it doesn’t. If it doesn’t, then God can imagine it, since God can do anything logically possible.

And it does not appear to involve a contradiction. As even many defenders of the Kalaam argument admit, the mathematics of infinity are consistent and do not contain logical contradictions.

This situation isn’t like saying, “Suppose God imagines a four-sided triangle.” Not even God can visualize that, because “four-sided triangle” is a contradiction in terms.

“Four-sided” and “triangular” mean different and contradictory things. This expression is just word salad—not something that is actually meaningful.

But our volume of space isn’t like that. Even our puny minds can imagine a row of cubic feet with hydrogen atoms in them. God’s mind is infinite, and so he can imagine the line of cubic feet extending on endlessly—and there doesn’t seem to be a logical contradiction involved in him doing that.

If there’s not, then we’re ready to add a new element to our thought experiment.

“Let There Be Space!”

Because of his omnipotence, God can do anything that doesn’t involve a logical contradiction. So, if God can imagine something, he can also create it.

As a result, if the idea of an infinite row of cubic feet of space—each with a hydrogen atom—doesn’t involve a logical contradiction, then God can make it real.

God thus could create an infinite number of hydrogen atoms and an infinite volume of space to contain them.

If God chooses, actual infinities of physical things could exist!

The only way to avoid this would be to say that, even though the idea of finite space with finite atoms is logically coherent, a contradiction in terms is generated if we extend this to infinity.

In that case, God couldn’t imagine or create this any more than he could a four-sided triangle.

But in that case, I want to know: What’s the contradiction?

As we’ve discussed, a Christian who understands God’s omnipotence should affirm that God can make something unless it is shown to involve a logical contradiction.

The Kalaam defender thus needs to name the contradiction: Which are the specific terms that contradict?

And it won’t do to change the scenario and pose some other one where a new logically contradictory entity is subtly introduced. That happens all too often.

Changing the scenario is what you do when you can’t deal with the current one.

Neither is it sufficient to say, “Hey, infinities have weird properties.” Yes, they do. That doesn’t mean they’re beyond the reach of God’s omnipotence.

So please, deal with the thought experiment I’ve described, flesh out its terms in detail, and name which terms contradict each other—just like how I pointed out that “four-sided” and “triangular” contradict.

Either that or be prepared to acknowledge that God has the power to create actual infinities of physical objects.

We know from Scripture that God created the world a finite amount of time ago, but was that the only option he had? Could God have created a world with an infinite history?

Defenders of the Kalaam cosmological argument claim that he couldn’t have done so.

To show that, they would need to demonstrate that the idea of an infinite history involves a logical contradiction.

Some recent attempts to do this involve paradoxes that have been proposed by different authors.

Let’s look at a couple and see what we can learn.

The (Squished) Grim Reaper Paradox

The core of what has become known as the Grim Reaper paradox was proposed by Jose Bernardete, and the argument has taken different forms.

Here I’ll present a version that is close to the original. I’ll refer to it as the “squished” version for reasons that will become obvious.

Suppose that a guy named Fred is alive at 12:00 noon.

However, there are an infinite number of grim reapers waiting to kill him. For the sake of convenience, we will give the reapers names based on the negative numbers, with the last reaper being Reaper 0.

If Fred is still alive at 1 p.m., Reaper 0 will encounter him and kill him. However, before that, Reaper -1 will encounter Fred at 12:30 p.m. and kill him. Even before that, Reaper -2 will encounter Fred at 12:15 p.m. and kill him. But Reaper -3 will encounter him at 12:07:30 p.m., and so on, with each reaper set to encounter (and kill) Fred in half the remaining distance back to noon.

Which reaper will kill Fred?

The way the situation has been set up, we have a paradox. Reaper 0 should not kill Fred, because Fred should already have been killed by Reaper -1. But this reaper shouldn’t kill him either, because he should have been killed by Reaper -2. And Reaper -3 should have killed him before that, and so on.

It thus looks like Fred can’t possibly survive past 12:00 noon, but it’s impossible to name which reaper kills him. Paradox.

I called the above version “squished” because it squishes the infinite series of grim reapers into a single hour. But we don’t have to do it that way. We’ll see another version later.

The Problem with the Paradox

The resolution of this paradox is fairly straightforward. It has envisioned a situation where Fred begins alive and then will be killed by the first grim reaper he encounters.

The problem is that—if the series of grim reapers is infinite—then it must have no beginning.

To suppose that an infinite series of whole numbers has both a first and last member involves what I’ve called the First-and-Last Fallacy.

• • Infinite series can have no beginning ( . . . -3, -2, -1, 0)
  • They can have no end (0, 1, 2, 3 . . .)
  • Or they can lack both a beginning and an end ( . . . -3, -2, -1, 0, 1, 2, 3 . . . )

But if a series has both a beginning and an end, then it’s finite.

The series of reapers set to kill Fred has an end—Reaper 0—but if that’s the case, it cannot have a beginning.

This means that there is no first grim reaper that Fred encounters, just as there is no “first negative number.”

The idea of a first negative number involves a logical contradiction, and therefore the (Squished) Grim Reaper paradox is proposing a situation that cannot exist.

The (Spread-Out) Grim Reaper Paradox

To make the Grim Reaper Paradox more relevant to the Kalaam argument, some have proposed a new version that spreads out the grim reapers over an infinite history rather than squishing them into a single hour.

We can put the spread-out version like this:

Suppose that Fred has always been alive, all the way through an infinite past.

Suppose that there is an infinite series of grim reapers set to kill Fred, and they are set to kill him in sequence on New Year’s Day.

On New Year’s Day this year, Reaper 0 is set to kill him if he is still alive. But on New Year’s Day last year, Reaper -1 was set to kill him, and Reaper -2 the year before that, and so on.

Which reaper kills Fred?

Exactly the same paradox results. Each reaper should not be able to kill Fred because a previous reaper should already have done the dirty work.

But the problem is the same: The situation is set up so that the only reaper that could kill Fred is the first one in a beginningless series, and a beginningless series has no first element.

The spread-out version of the paradox thus has the same flaw that the original did: It proposes an entity that involves a logical contradiction and so can’t exist.

Application to the Kalaam Argument

Kalaam defenders use paradoxes like this in an attempt to undermine the idea of an infinite history.

For example, on the rhetorical level, the strategy can work like this:

• • They encourage their audience to imagine a set of circumstances that could exist and that don’t involve a logical contradiction (e.g., Fred exists, a grim reaper is scheduled to kill him at some time).
  • They multiply the circumstance extending finitely into the past (suppose there are some other grim reapers who were also set to kill him previously).
  • They extend this infinitely into the past, generating a contradiction.
  • They point out this contradiction.
  • Finally, they assert that, since the infinite extension into the past caused the contradiction, we must reject the idea of an infinite past.

This reasoning is mistaken, because it isn’t the infinite history that’s the problem. It’s the fact that you’ve proposed a first element in a beginningless series.

Consider the following scenario:

• • Suppose that some guy named Fred exists.
  • Suppose that he existed last year.
  • And the year before that.
  • And that he’s always existed, with no beginning.

That involves an infinite history, but it doesn’t have a contradiction in it—because it does not propose a first element in a beginningless series.

It’s only when you introduce the latter that a contradiction occurs. So, it isn’t the infinite history that’s the problem but the impossible first element. Grim Reaper-like paradoxes thus are disguised forms of the First-and-Last Fallacy.

World Without End

Another way of seeing why the strategy used by Kalaam defenders is problematic is by flipping the arrow of time and considering a mirror image of the paradox that deals with the future:

Suppose that Fred is alive today and will remain alive as long as a grim reaper doesn’t kill him.

Suppose that there is an infinite series of grim reapers to kill him in the future and who we will name using the positive numbers, beginning with 0. Reaper 0 encounters Fred today, Reaper 1 encounters him tomorrow, Reaper 2 the day after, and so on.

But the reapers have agreed that the honor of killing Fred will go to the reaper with the highest number.

Which reaper kills Fred?

The way this scenario has been set up, the only reaper that can kill Fred is the last reaper.

But there can be no last reaper in a series that has no end, just as there can be no first reaper in a series that has no beginning.

The Future Reaper Paradox proposes the same kind of logically contradictory entity that the original paradoxes did.

And the problem is not the fact that the future is infinite in the scenario. The infinite future itself does not involve a contradiction, because it does not propose there being a last element in the series of days stretching into the future.

Indeed! From a Christian point of view, an infinite future is exactly what awaits us. The Christian faith teaches that God will give us endless life and there will be no day on which we pass out of existence. As a result, an orthodox Christian is committed to the idea of an infinite future.

It’s not the endlessness of the future that’s a problem in the above scenario, but the idea of that future containing a logically contradictory entity like the last member of an endless series of reapers.

In the same way, the idea of an infinite past is not a logical contradiction but the idea of a first reaper in a beginningless series.

The Underlying Assumptions

The paradoxes recently proposed in support of the Kalaam argument (and there are many) share a common set of assumptions:

1. 1. Suppose some past scenario (P) that involves a first member.
   2. Suppose some past scenario (P’) that does not have a first member.
   3. Suppose that P and P’ are the same scenario.

By contrast, corresponding future-oriented paradoxes have these assumptions:

1. 1. Suppose some future scenario (F) that involves a last member.
   2. Suppose some future scenario (F’) that does not have a last member.
   3. Suppose that F and F’ are the same scenario.

The way the paradoxes are fleshed out and expressed varies, and this can disguise the fact, but they all share the same set of assumptions.

As a result, they set up logical contradictions, but this does not tell us which of the premises must be rejected to diffuse the contradiction.

This is true of any such paradox. Consider the following geometrical one:

1. 1. Suppose there is some closed geometrical shape (S) that has three sides.
   2. Suppose that there is some closed geometrical shape (S’) that has four sides.
   3. Suppose that S and S’ are the same shape.

This scenario sets up the idea of a four-sided triangle, which is logically impossible. But in resolving the paradox, we don’t have to reject any particular premise.

We could reject the idea that S (the three-sided shape) exists; we could reject the idea that (S’) the four-sided shape exists; or we could reject the idea that S and S’ are the same and then conclude that there must be two shapes. Each of these is a legitimate option.

As a result, when considering past- or future-oriented paradoxes, we don’t have to reject the idea of an infinite past or an infinite future, because neither of these concepts generates a logical contradiction on its own.

It is only when we introduce a logically contradictory entity into these scenarios—like the first member of a beginningless series or the last member in an endless series—that a paradox results.

Those are the entities that need to be rejected, and the recently proposed paradoxes do not disprove either the idea of an infinite past or an infinite future.