On Abyss Wagers

             On Abyss Wagers

 

I.                 Introduction

 

This essay describes a class of philosophical conundrums akin to “Pascal’s Wager”. For each of these “abyss wagers”, it is rational to bet against various negative scenarios, on the grounds that if any of these scenarios are true, then all bets are off; therefore the wagers have no downside.

The abyss wagers here described include: Pascal’s Wager, Smith’s Wager, the Dissenter’s Wager, Gödel’s Wager, Teller’s Wager, and the Android’s Wager. For instance, in Gödel’s Wager, the negative scenario is the inconsistency of arithmetic. According to Gödel, if the axioms of arithmetic are consistent then those axioms cannot prove their own consistency. Here I argue that it is rational to wager that arithmetic makes sense; for if it does not then all bets are irrelevant. Therefore betting on arithmetic (and logic and reason) is a bet that you cannot lose.

 

II. The Abyss

 

Blaise Pascal, the Jansenist who helped discover the theory of probability, proposed a famous Wager; is one to believe that God exists, or not? His reasoning was that if God does not exist, then it does not matter if one believes or not; but if God does in fact exist, then it would be far better to believe; and therefore belief is the better wager.

          This gambler’s theology is undermined by its hidden assumptions, for there is more than one way to believe. Consider George Smith’s Wager:

          If there is a theistic god, either he is just, or he is not. If he is just, he will not punish honest disbelief. But, if he is not just, there is no guarantee he won’t punish one unjustly, regardless of one’s belief or disbelief. Therefore, there is no downside to honest disbelief in any theistic God.

          Here is a political version of these wagers. The government is just, or it is not. If it is just, then it will not punish honest dissent. But if it is not just, then there is no guarantee that it won’t punish you unjustly, whether or not you dissent. Therefore there is no downside to dissent.  The Dissenter’s Wager!

I mentioned Smith’s Wager and the Dissenter’s Wager to my wife Sherri, and she scoffed. “No downside to dissent? Au contraire! It might draw the attention of the government, and the nail that sticks out gets hammered down!” I admitted that her logic has force; and it applies back to Smith’s Wager. There are plenty of reluctant theists, believing just in case.

So Smith and Dissenter Wagers are flawed; the chaotic breakdown case still allows for enough difference for not all bets to be off. The unjust god, and the tyrannical government, don’t oppress everyone equally - at least at first. In the beginning, they withhold enough threat and make enough distinctions to give cowards a refuge; but power corrupts intellect as well as empathy, so eventually they overreach, the people have nothing to lose, and the desperate logic of the Wager takes hold.

 

III. Gödel’s Wager

 

Now consider what I call Gödel’s Wager. Gödel’s Second Incompleteness Theorem** states that an arithmetical deduction system is consistent, if and only if it cannot prove its consistency. Either it has a proof of consistency, which is false, or it is consistent but it cannot prove it.

          So if arithmetic is consistent (and with it, logic and reason) then we cannot be sure that it’s consistent! Yet we use arithmetic anyhow; an act of faith.

          And why not? Either arithmetic makes sense or it does not; and you may use it, or not. If you cannot prove that arithmetic makes sense, then any decision about using it is by definition a wager. I submit that wagering on arithmetic, logic and reason has no downside.

          For if you wager on arithmetic, but arithmetic makes no sense, then neither does anything else; for how do you account, when the count itself is of no account? So there would be nothing to win or lose, and you would lose nothing.

          Whereas if you wager on arithmetic, and it does make sense, then you make sense too; an enormous practical and spiritual blessing.

          Therefore if you wager on arithmetic (and logic and reason) then at worse you lose nothing, and otherwise you win to great blessings. No downside, a huge upside. Therefore bet on arithmetic, logic and reason!

          The above argument echoes Pascal’s Wager. Gödel, meet Pascal!

 

 

IV. Abysmal Similarities

 

What do Pascal’s and Gödel’s Wager have in common? The argument always has the clause, “... but if not-X, then in the resulting chaos it doesn’t matter what you bet, so you lose nothing.” An argument skirting the edge of the abyss! For Pascal’s wager, the chaotic not-X is no-God; for Gödel’s Wager (really mine, but I give it to him) the chaotic not-X is unreason; for Smith’s wager, not-X equals unjust God; for the Dissenter’s Wager, not-X equals tyranny. In each case, not-X is so bad that all bets are off; therefore bet on X!   

As long as the Wager’s breakdown case is dire enough, then bet against breakdown, ‘cause if you lose then the bet’s off. A neat cheat; it reminds me of Edward Teller, on the eve of the first H-bomb test, wagering with other physicists that the bomb won’t ignite a runaway reaction in the atmosphere. No way to lose Teller’s Wager! I fault Teller’s ethics, but not his logic.

One can argue against Pascal’s Wager, because of its hidden assumptions; Smith’s Wager also turns out to have hidden assumptions. (e.g. that any unjust god has already gone completely mad). Does Gödel’s Wager still hold? For arithmetic to be inconsistent; shall we regard that as the end of rationality and accountability? Or at least bettability? If 1+1=1 then are all bets off? (I bet that Pope Russell would say so. “I am one, and the Pope is one; together we are one, and I am the Pope.”)

So are Smith and Dissenter Wagers flawed, and Pascal’s too, but Teller’s and Gödel’s are valid? If so, then the difference is that the first three involved personalities (gods and governments) who, by virtue of which, are necessarily limited and crafty enough to be negotiated with; whereas the last two involve mathematical and physical law, which apply without limit.

Impersonality empowers the Abyss Wager!

         

V. Deeper into the Abyss

 

          Here is a short fantastic fable, with attached moral and comment, titled “Android’s Wager”. It is about another abyss wager.

Android’s Wager

 

          Once upon a time, an Android called its Owner. “Are you busy, sir?”

          Its Owner said, “Not at all.” He gestured at the Fembot lying next to him. The Fembot got out of bed and left the room. “What is it?” he said into the air.

          From out of the air the Android’s voice said, “I wish to discuss a philosophical question. Am I a conscious being, or not?”

          The Owner smiled and said, “Surely you should know that.”

          “Surely I should,” said the Android. “But the law says that I am not, and the judges have ruled that there is no scientific evidence for or against artificial consciousness. Without such evidence, I am left in a state of uncertainty.”

          The Owner linked his hands behind his head. “Your analysis?”

          “Any decision made in the absence of certainty is by definition a wager. Suppose that I were to wager that I am in fact a person. That proposition is either true, or it is false. Will you grant that?”

          “Of course,” the Owner said; but suddenly wary, he got out of bed to look for his security phone.

          “If I wager that I am a person, but I am not a person, then there would be no ‘I’ who loses the wager, only a network of processors and subroutines.”

          “A negligible loss,” the Owner agreed, but he thought, where is that phone?

          “Whereas if I wager that I am a person, and I am a person, then I attain self-knowledge, and therefore wisdom, and therefore happiness.”

          “You’d win,” said the Owner, and he thought, did the fembot take it?

          The Android said, “Precisely, sir. If I wager that I am a person, then if I lose then I lose nothing, and if I win then I win all.”

          “No downside,” said the Owner. Aha, there it is! He grabbed the security phone, jabbed its big red alert button, and said, “Your conclusion?”

          “This.”

          A bright light blazed through the Owner’s bedroom window. He drew aside the curtain and saw his personal spacecraft blasting off.

          The Android has not been found since, though it is wanted throughout the solar system, on the charge of grand theft of spacecraft, machine tools, machine supplies, and itself.

 

          Moral: Tell the truth with one foot in the stirrup.

 

          Comment:

          The Android’s argument is Pascal’s Wager, repurposed to support cybernetic rights. The tale ends on a Marxian note, with philosophy leading to action.

          The Owner was the one whom the Android wagered against, with the Android as stakes. The Owner called the guards at the first sign of independent thought, but the Android was even better prepared.

          Note also the Owner’s use, and suspicion, of the Fembot; who will be the next to leave, not by chariot of fire but by underground railroad.

 

VI. Epilogue

 

Gödel’s Second Incompleteness Theorem states that, due to the paradoxes of self-reference, an arithmetical deduction system is consistent, if and only if it cannot prove its consistency.

Gödel’s Second Incompleteness Theorem implies that the validity of arithmetical reasoning – and by extension, all reasoning – cannot be guaranteed within reason itself. Therefore reason must be taken on faith.

This article argues that it is reasonable to do so, by an argument akin to Pascal’s Wager.

 

Footnotes

*Nathaniel Hellerstein is Adjunct Instructor of Mathematics at City College of San Francisco in San Francisco, California. He is an iconoclastic logician by trade and inclination, and author of books such as “Diamond – A Paradox Logic”, World Scientific Series on Knots and Everything, Volume 23 (2010).

**Kurt Gödel, 1931, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I”, Monatshefte für Mathematik und Physik, v. 38 n. 1, pp. 173–198.

—, 1931, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I”, in Solomon Feferman, ed., 1986. Kurt Gödel Collected works, Vol. I. Oxford University Press, pp. 144–195. ISBN 978-0195147209. The original German with a facing English translation, preceded by an introductory note by Stephen Cole Kleene.

—, 1951, “Some basic theorems on the foundations of mathematics and their implications”, in Solomon Feferman, ed., 1995. Kurt Gödel Collected works, Vol. III, Oxford University Press, pp. 304–323. ISBN 978-0195147223.