Friday, November 16, 2018

On Abyss Wagers, 5 of 5

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.

*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.

Thursday, November 15, 2018

On Abyss Wagers, 4 of 5

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?”
          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.

          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.

Wednesday, November 14, 2018

On Abyss Wagers, 3 of 5

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!