Causal Finitude Defended

At the heart of the Kalam Cosmological argument is the question of a past infinite universe is coherent or actual. Over the last 20-30 years, most of the development of the argument has centered on the growing body of evidence that our universe had a beginning approximately 13.7 billion years ago. While the research is promising and fruitful, it faces a notoriously petty objection of “we just don’t know”. Science primarily practices inductive reason, making certain knowledge, especially about the distant past, difficult to erect a case on. To be fair, this objection has been used on both sides – atheists writing IOUs for some future model that will extend the Universe into the past eternally and Young Earth Creationists writing IOUs that further examination will yield a young Earth. Thus, while this may be fertile grounds for exploration, it will never provide the certainty that math and logic can provide.

Causal Finitism

Causal Finitism is the principle that causal chains cannot be past infinite. It has always had a certain intuitive appeal, but it has been difficult to draw out the exact logical incoherence. Drs. Pruss and Koons may have effectively exposed this incoherence once and for all. 

Some Historical Objections

First, let’s talk about the objections to causal finitism through the years. The original formulations against causal finitism have aimed at the idiosyncrasies of the thought experiments that accompanied the argument. Perhaps the most famous of these arguments is the grim reaper experiment. 

A grim reaper is assigned to kill Jack at 12:30 AM. However, another is assigned to kill Jack at 12:15. Yet another at 12:07:50 and so on and so forth, each time halving the time between Midnight and the last reaper. The paradox is this: there is no way for Jack to live past midnight, but there is also no grim reaper who can kill Jack, as there are always an infinite number of reapers between any reaper and Jack.

The historical objections to this normally take the form of attacking the bizarre treatment of time.

  1. Perhaps time cannot be divided in half infinitely. Perhaps there are discrete units of time.
  2. Perhaps the reaper could complete the task under normal temporal conditions.
  3. How does this make sense with the B theory of time?
  4. What exactly is the logical contradiction?

The Pruss/Koons Causal Paradox

With the goal of addressing these concerns and drawing out the actual logical contradiction, the philosophers created a novel thought experiment with remarkable qualities.

  1. It does not require the halving of time
  2. It doesn’t require the interval between each event be the same
  3. It doesn’t even require any time be between events, time is meaningless to the analogy, it could all be simultaneous causation
  4. A clear logical contradiction can be extracted from the thought experiment in the form of A!=A

Here is a rough formulation of the thought experiment.

Imagine Jack is sitting in a chair in a line of chairs with others to the left and right. Everyone has a number. Let’s say Jack has number N, the person to the left N-1 and to the right N+1, so on and so forth.

Each person in the sequence is given a simple task. They will receive a sheet of paper from the person on the left. If the sheet is blank, the person should write their number on it. If a number is already on it, they should make no mark on the paper. The person should then pass the paper to the person on their right. 

When the paper arrives in Jack’s hand, what happens? There is no way that Jack receives a blank paper because there have been infinitely many people before him with a chance to fill in their number. Let’s call this Set A. Set A consists of N-1 through N-∞.

What property do all members of Set A have in common? All members of Set A can put their number on the sheet if blank.

However, we can also think of Set B. Set B also consists of N-1 through N-∞.

What property do all members of Set B have in common? All members of Set B can’t put their number on the sheet because there are an infinite number of prior opportunities for another member to write their number.

Ok, so here is the logical contradiction: We can imagine a third set, which we will call Set C. Set C consists of the sheet of paper in each person’s lap N-1 through N- ∞. But, we know that each paper has two properties… (1) it must have a number on it and (2) it can’t have a number on it.

Thus, we have true logical contradiction. If X is the property of having a number on the paper, then each member of Set C both has property X and does not have property X. Ergo, X and !X.

Concluding Remarks

Here we have a logical contradiction entailed by a simple infinite causal series. The task of writing a number on a sheet of paper and passing it is coherent in any finite series, but is incoherent in an infinite series.  I highly recommend a much more thorough and technically sophisticated proof found in Pruss’s book Causation, Infinity and Paradox. This book masterfully builds through the history, the objections, and finally the rigorous logical proof behind this story version of the Grim Reaper paradox.

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