Infinitesimals and Epistemic Panic: Why Classical Probabilities Aren’t Enough
(Or: I Swear This Isn’t Just Math Trauma)
Cold Open: The Panic
Let’s say someone offers you a bet. Flip a fair coin: if it lands heads, you win $1. If it lands tails, nothing happens. Seems harmless, right?
Now let’s make things more interesting. Same coin, same deal—but we add a catch: before you flip, a billion-sided die is rolled. If it lands on face #7, you lose your entire savings account. Everything. Gone. The chance of that happening? 1 in a billion. Practically zero. But not quite.
Statistically, this is a “good” bet. The expected value is still in your favor. Classical probability shrugs and says, “Go for it.” But your stomach drops. You start thinking about that one weird time you won a Twitter giveaway with 20,000 entries. You remember the feeling of technically safe going horribly wrong.
This is what I call epistemic panic—the moment when your brain tells you everything is fine, but some deeper part of you goes, “Are we sure? Like, cosmically sure?” It’s that uncanny valley between zero and not-zero, where classical probability gives up, and your decision theory starts to sweat.
And honestly? That panic is valid. Our models assume perfect rationality, infinite resources, bounded utilities [3]. Reality assumes you have rent to pay and an anxiety disorder.
Classical probability isn’t built for this kind of fragility. It can’t distinguish between the merely improbable and the existentially unbearable. Which is why I think it’s time we admit something radical:
Sometimes, we need math that’s a little more weird—and a lot more honest.
Soooooo....What’s Wrong with Classical Probabilities?
Classical probability theory is neat. Too neat.
It assigns every proposition a number between 0 and 1. You’re supposed to keep your beliefs in line with the axioms—normalize to 1, never assign a negative probability, update via Bayes, and so on. If you follow the rules, you’re safe from certain kinds of irrationality. You won’t fall for a Dutch Book [1]. You’re a good Bayesian citizen.
Until, of course, the edge cases show up with knives.
Enter Pascal’s Mugging, Ex Ante Pareto violations, and the creeping sense that tiny probabilities can have massive consequences. A one-in-a-trillion chance of a trillion-dollar payoff still breaks expected utility theory [3]. A one-in-a-billion risk of catastrophic loss still makes your heart rate spike. Classical probability is supposed to protect us from these situations—but often, it just shrugs and says, “Well, the math checks out.”
And then there’s the zero problem. In standard probabilism, a proposition with probability zero is treated as impossible. Not just unlikely—impossible. But that’s a pretty strong stance for a discipline that models uncertainty. In continuous spaces (like physics or infinite lotteries), lots of meaningful outcomes have probability zero. That doesn’t mean they can’t happen—it just means the math doesn’t know how to talk about them properly [2].
So we get weird. We hedge. We say, “It’s practically impossible,” or “Let’s treat it as negligible,” or “You’d be irrational not to ignore it.”
But ignoring it still feels wrong. Because the risk, however small, doesn’t disappear. It just hides in the margins of your model. And when you’re operating under pressure—when the stakes are existential, or the utilities are unbounded—those margins matter.
What we need isn’t just tighter reasoning. We need a framework that can actually express the fact that some things are very, very unlikely—but not ignorable. That’s where infinitesimals come in.
Enter: Infinitesimal Credences
So what do you do when “zero” is too confident, but anything above zero feels like a lie?
You call in the infinitesimals.
Infinitesimals are numbers that are greater than zero but smaller than any real number you can name. They belong to a mathematical universe called nonstandard analysis, where the real number line is extended to include infinitely small and infinitely large values [4]. It’s like adding a secret annex to your number system—quiet, weird, and game-changing.
In epistemic terms, infinitesimal credences let us say:
“This outcome is not impossible. It’s just barely possible. But not so barely that we pretend it’s not there.”
They’re tiny little epistemic flags, waving from the far edge of your belief system, saying: I still matter. Not enough to rearrange your life over. But enough to show up in your model when it counts.
And the best part? This isn’t just a vibes-based patch. Infinitesimals preserve coherence. You can build a consistent, deductively valid probability system that includes them [5]. Think of it like Bayesian reasoning with better emotional range. You still get expected values, conditional probabilities, and rational updating—you just gain the ability to register nuance where classical theory taps out.
Nonstandard analysis sneaks past classical limits without forcing you to give up structure or go full mystic. It just asks you to admit that maybe not every important belief fits neatly between 0 and 1.
And honestly? That feels like progress.
Why This Isn’t Just a "Vibe"
At this point, it’s fair to ask:
Are infinitesimals just… an emotional support number line?
And look, I get it. It’s tempting to treat them like math for people who panic at parties (of possible worlds). But infinitesimals aren’t just a poetic flourish. They solve real problems that classical models can’t touch—without giving up coherence, precision, or rational justification [5].
Take accuracy-first epistemology. The idea is that good beliefs are ones that track the truth—and you can evaluate them using scoring rules [6]. In this framework, infinitesimals give you finer control. They let you distinguish between “literally impossible” and “basically impossible, but still real enough to haunt me.” That’s a meaningful distinction, especially when you’re dealing with vast hypothesis spaces or uncertain priors.
Then there’s Dutch Book avoidance. If you believe something has probability zero, you’re functionally saying it can’t happen. But if it does, you're toast—and worse, you're incoherent [1]. Infinitesimals let you hedge just enough to avoid guaranteed loss while still treating the belief as negligible in day-to-day reasoning. It’s a kind of epistemic modesty, [5] an admission that your models might be wrong in precisely the ways you didn’t plan for.
This also plays beautifully with bounded utility theory. If your decision model assumes payoffs are finite (as they usually are), even an infinitesimal risk can matter [3]. Infinitesimals help you navigate those thin spaces between paralysis and recklessness—like the mathematical equivalent of carrying a tiny umbrella “just in case.”
They’re not magical. They won’t solve every paradox. But they’re not vibes either. They’re a precision tool for edge-case sanity, a way to make rational space for possibilities too small for classical models, but too real to ignore
Wrap-up to a Very Long™ Post
I used to think infinitesimals were just math’s little freaks—cute, niche, maybe dangerous. Like a formalism you keep in the basement in case you need to prove something weird. I respected them the way you respect a volatile genius: from a safe distance, preferably with snacks.
But the more I thought about how we actually live with uncertainty—the kind that makes you spiral at 2am over a one-in-a-billion scenario—the more I realized: classical probability doesn’t always cut it. Not because it’s wrong, but because it’s too confident. Too clean. Too willing to ignore what your gut insists is real.
Infinitesimals don’t solve that entirely. But they give me language for it. They let me model the difference between “won’t happen” and “might, but please God no.” They let me make peace with the fact that technically zero and emotionally catastrophic aren’t the same thing.
Sometimes you don’t need certainty. You just need a little more nuance.
And a protest sign.
Cold Open: The Panic
Let’s say someone offers you a bet. Flip a fair coin: if it lands heads, you win $1. If it lands tails, nothing happens. Seems harmless, right?
Now let’s make things more interesting. Same coin, same deal—but we add a catch: before you flip, a billion-sided die is rolled. If it lands on face #7, you lose your entire savings account. Everything. Gone. The chance of that happening? 1 in a billion. Practically zero. But not quite.
Statistically, this is a “good” bet. The expected value is still in your favor. Classical probability shrugs and says, “Go for it.” But your stomach drops. You start thinking about that one weird time you won a Twitter giveaway with 20,000 entries. You remember the feeling of technically safe going horribly wrong.
This is what I call epistemic panic—the moment when your brain tells you everything is fine, but some deeper part of you goes, “Are we sure? Like, cosmically sure?” It’s that uncanny valley between zero and not-zero, where classical probability gives up, and your decision theory starts to sweat.
And honestly? That panic is valid. Our models assume perfect rationality, infinite resources, bounded utilities [3]. Reality assumes you have rent to pay and an anxiety disorder.
Classical probability isn’t built for this kind of fragility. It can’t distinguish between the merely improbable and the existentially unbearable. Which is why I think it’s time we admit something radical:
Sometimes, we need math that’s a little more weird—and a lot more honest.
Soooooo....What’s Wrong with Classical Probabilities?
Classical probability theory is neat. Too neat.
It assigns every proposition a number between 0 and 1. You’re supposed to keep your beliefs in line with the axioms—normalize to 1, never assign a negative probability, update via Bayes, and so on. If you follow the rules, you’re safe from certain kinds of irrationality. You won’t fall for a Dutch Book [1]. You’re a good Bayesian citizen.
Until, of course, the edge cases show up with knives.
Enter Pascal’s Mugging, Ex Ante Pareto violations, and the creeping sense that tiny probabilities can have massive consequences. A one-in-a-trillion chance of a trillion-dollar payoff still breaks expected utility theory [3]. A one-in-a-billion risk of catastrophic loss still makes your heart rate spike. Classical probability is supposed to protect us from these situations—but often, it just shrugs and says, “Well, the math checks out.”
And then there’s the zero problem. In standard probabilism, a proposition with probability zero is treated as impossible. Not just unlikely—impossible. But that’s a pretty strong stance for a discipline that models uncertainty. In continuous spaces (like physics or infinite lotteries), lots of meaningful outcomes have probability zero. That doesn’t mean they can’t happen—it just means the math doesn’t know how to talk about them properly [2].
So we get weird. We hedge. We say, “It’s practically impossible,” or “Let’s treat it as negligible,” or “You’d be irrational not to ignore it.”
But ignoring it still feels wrong. Because the risk, however small, doesn’t disappear. It just hides in the margins of your model. And when you’re operating under pressure—when the stakes are existential, or the utilities are unbounded—those margins matter.
What we need isn’t just tighter reasoning. We need a framework that can actually express the fact that some things are very, very unlikely—but not ignorable. That’s where infinitesimals come in.
Enter: Infinitesimal Credences
So what do you do when “zero” is too confident, but anything above zero feels like a lie?
You call in the infinitesimals.
Infinitesimals are numbers that are greater than zero but smaller than any real number you can name. They belong to a mathematical universe called nonstandard analysis, where the real number line is extended to include infinitely small and infinitely large values [4]. It’s like adding a secret annex to your number system—quiet, weird, and game-changing.
In epistemic terms, infinitesimal credences let us say:
“This outcome is not impossible. It’s just barely possible. But not so barely that we pretend it’s not there.”
They’re tiny little epistemic flags, waving from the far edge of your belief system, saying: I still matter. Not enough to rearrange your life over. But enough to show up in your model when it counts.
And the best part? This isn’t just a vibes-based patch. Infinitesimals preserve coherence. You can build a consistent, deductively valid probability system that includes them [5]. Think of it like Bayesian reasoning with better emotional range. You still get expected values, conditional probabilities, and rational updating—you just gain the ability to register nuance where classical theory taps out.
Nonstandard analysis sneaks past classical limits without forcing you to give up structure or go full mystic. It just asks you to admit that maybe not every important belief fits neatly between 0 and 1.
And honestly? That feels like progress.
Why This Isn’t Just a "Vibe"
At this point, it’s fair to ask:
Are infinitesimals just… an emotional support number line?
And look, I get it. It’s tempting to treat them like math for people who panic at parties (of possible worlds). But infinitesimals aren’t just a poetic flourish. They solve real problems that classical models can’t touch—without giving up coherence, precision, or rational justification [5].
Take accuracy-first epistemology. The idea is that good beliefs are ones that track the truth—and you can evaluate them using scoring rules [6]. In this framework, infinitesimals give you finer control. They let you distinguish between “literally impossible” and “basically impossible, but still real enough to haunt me.” That’s a meaningful distinction, especially when you’re dealing with vast hypothesis spaces or uncertain priors.
Then there’s Dutch Book avoidance. If you believe something has probability zero, you’re functionally saying it can’t happen. But if it does, you're toast—and worse, you're incoherent [1]. Infinitesimals let you hedge just enough to avoid guaranteed loss while still treating the belief as negligible in day-to-day reasoning. It’s a kind of epistemic modesty, [5] an admission that your models might be wrong in precisely the ways you didn’t plan for.
This also plays beautifully with bounded utility theory. If your decision model assumes payoffs are finite (as they usually are), even an infinitesimal risk can matter [3]. Infinitesimals help you navigate those thin spaces between paralysis and recklessness—like the mathematical equivalent of carrying a tiny umbrella “just in case.”
They’re not magical. They won’t solve every paradox. But they’re not vibes either. They’re a precision tool for edge-case sanity, a way to make rational space for possibilities too small for classical models, but too real to ignore
Wrap-up to a Very Long™ Post
I used to think infinitesimals were just math’s little freaks—cute, niche, maybe dangerous. Like a formalism you keep in the basement in case you need to prove something weird. I respected them the way you respect a volatile genius: from a safe distance, preferably with snacks.
But the more I thought about how we actually live with uncertainty—the kind that makes you spiral at 2am over a one-in-a-billion scenario—the more I realized: classical probability doesn’t always cut it. Not because it’s wrong, but because it’s too confident. Too clean. Too willing to ignore what your gut insists is real.
Infinitesimals don’t solve that entirely. But they give me language for it. They let me model the difference between “won’t happen” and “might, but please God no.” They let me make peace with the fact that technically zero and emotionally catastrophic aren’t the same thing.
Sometimes you don’t need certainty. You just need a little more nuance.
And a protest sign.
Sources
[1] Bruno de Finetti, Theory of Probability (1974). One of the foundational sources for subjective probability and the Dutch Book argument.
[2] Michael Caie, “Rational Probabilistic Incoherence,” Philosophical Review, 2013. Discusses failures of classical coherence and alternatives.
[3] Brian Weatherson, “Decision Theory,” Stanford Encyclopedia of Philosophy (2020). Overview of bounded utility, infinite payoffs, and decision-theoretic edge cases.
[4] H. Jerome Keisler, Foundations of Infinitesimal Calculus (1976). Classic introduction to nonstandard analysis and the hyperreal number system.
[5] Kenny Easwaran, “Regularity and Hyperreal Credences,” Philosophical Studies, 2014. Defends the rational use of infinitesimals in epistemic modeling.
[6] Jim Joyce, “Accuracy and Coherence: Prospects for an Alethic Epistemology of Partial Belief,” in Realism and the Aim of Science (1998). Central text in accuracy-first epistemology.
[1] Bruno de Finetti, Theory of Probability (1974). One of the foundational sources for subjective probability and the Dutch Book argument.
[2] Michael Caie, “Rational Probabilistic Incoherence,” Philosophical Review, 2013. Discusses failures of classical coherence and alternatives.
[3] Brian Weatherson, “Decision Theory,” Stanford Encyclopedia of Philosophy (2020). Overview of bounded utility, infinite payoffs, and decision-theoretic edge cases.
[4] H. Jerome Keisler, Foundations of Infinitesimal Calculus (1976). Classic introduction to nonstandard analysis and the hyperreal number system.
[5] Kenny Easwaran, “Regularity and Hyperreal Credences,” Philosophical Studies, 2014. Defends the rational use of infinitesimals in epistemic modeling.
[6] Jim Joyce, “Accuracy and Coherence: Prospects for an Alethic Epistemology of Partial Belief,” in Realism and the Aim of Science (1998). Central text in accuracy-first epistemology.
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