A calculating machine with an abyss in one cell
Infinity-Nothing
Pascal's wager is usually remembered as a shortcut to belief. The more interesting object is the machinery underneath it: an early, influential decision under uncertainty. Pascal tries finite prizes before the infinite one. The finite arithmetic works. Then one symbol enters the matrix and its power to rank strategies comes apart.
EV(wager) = +1 life; break-even p* = 25%
At even odds: one half times 3 minus one half times 1 equals +1. In Pascal's finite sequence, this is the point where he says you would have to play.
q = 0.000000001; mixed EV = infinity
Under this simple infinite-utility matrix, any positive q inherits infinite expectation. So pure wagering and even a fantastically unlikely route to wagering both read infinity. Standard expected utility no longer orders them. Hájek's 2003 objection has replies; the current SEP describes arguments that it is formal rather than substantive.
The fragment is Infini-rien, numbered Lafuma 418, Sellier 680, and Brunschvicg 233. The numbering matters because Pensées is a posthumous 1670 collection of notes, not Pascal's finished treatise. Hacking 1972 finds three arguments in the passage. The Stanford Encyclopedia presents the nearby forms as superdominance, expectation, and generalized expectations. Hacking 1975 calls the wager “the first well-understood contribution to decision theory,” a calibrated historical description, not a claim that Pascal invented decision theory or probability.
The finite machine survives
Turn infinity off and the threshold is ordinary algebra: p* = S / (G + S). With one life staked, gains of one, two, and three lives at even odds produce expected values of 0, +0.5, and +1. That is the finite sequence in the fragment. Infinity is a limit here, not a large number: as finite G grows, the threshold approaches zero but never reaches it.
The finite cap: Pascal's mugging territory
G = 1,000,000; p* = 0.0000999999%
Bostrom's 2009 Pascal's mugging asks how expected-value reasoning should react when an implausible claim is paired with a sufficiently vast finite payoff. This slider does not solve that problem. It exposes the pressure: no matter how tiny a positive probability you choose, a large enough finite prize can dominate. Proposed solutions exist and remain debated.
The same finite skeleton, with real-world caveats
Insurance uses expected loss in pricing and risk analysis, but buying insurance is not explained by expected value alone. Risk aversion, premiums, exclusions, solvency, regulation, and the declining marginal value of money matter. This abstract example is deliberately not a real quote.
Expected loss = $1,000; expected net benefit of policy = +$100
All four inputs are free choices. The calculation is p x covered loss - premium. It omits risk preference and every policy detail, so it demonstrates the finite arithmetic only.
The check
Live now: p = 0.5, G = 3, S = 1. EV = 0.5 x 3 - 0.5 x 1 = +1; p* = 1 / (3 + 1) = 0.25.
EV = pG - (1-p)S p* = S/(G+S) finite mixed EV = q x EV if G is infinite and p,q are positive: pG = q(pG) = infinity
Proven by the bench: the finite expected values and break-even threshold follow from the selected inputs. Under the displayed extended-real convention, positive probability times positive infinity is infinity, so the simple matrix ties every positive-path strategy.
Assumed or disputed: the probability of the reward, the utility of salvation, whether utility may be infinite, whether belief can be chosen, the cost S, the available religions, and the completeness of the state space. Pascal says the this-life loss is nothing; the slider does not assume that. Mixed-strategy and many-gods objections have published replies.
Sources and honest apparatus
- Pascal, Pensées, Infini-rien, Lafuma 418 / Sellier 680 / Brunschvicg 233, posthumous 1670. Source for the finite life examples and the wager passage.
- Alan Hájek, “Pascal's Wager,” Stanford Encyclopedia of Philosophy, first published 1998, current revision consulted July 2026. Source for the matrix, taxonomy, objections, and state of debate.
- Ian Hacking, “The Logic of Pascal's Wager,” American Philosophical Quarterly 9(2), 1972, 186-192; and The Emergence of Probability, 1975. Source for the three-argument reading and calibrated historical claim.
- Alan Hájek, “Waging War on Pascal's Wager,” Philosophical Review 112(1), 2003, 27-56. Source for the mixed-strategy objection.
- Diderot, Addition aux Pensées philosophiques, LIX, published in 1770 according to the Assézat-Tourneux edition. Source for the Imam challenge.
- Nick Bostrom, “Pascal's Mugging,” Analysis 69(3), 2009, 443-445.
Could not independently inspect a freely accessible scan of Hacking 1972 or Hájek 2003 during this build. Their bibliographic details and the claims attributed to them were cross-checked through the current Stanford Encyclopedia entry. The Pascal wording varies by translation, so only the short decision phrases needed here are paraphrased.
Corrections
Corrected 2026-10-14: The dek linked the infinity-driven ranking failure to Pascal's mugging. Bostrom's mugging uses finite payoffs and poses a related problem; found by the assay line, 2026-10-14.
Corrected 2026-10-14: The rival-god control said "Rival column added." It shows the many-gods objection without adding a matrix column; found by the assay line, 2026-10-14.
Corrected 2026-10-14: The page attributed Diderot's Imam challenge to Pensées philosophiques (1746), LIX. It is in Addition aux Pensées philosophiques, LIX, whose publication the Assézat-Tourneux edition dates to 1770; found by the assay line, 2026-10-14.
Corrected 2026-10-14: The calculator said changing any input moved both outputs. With finite gain, changing probability changes expected value, while the break-even threshold depends only on gain and stake; found by the assay line, 2026-10-14.