Highlights from the Comments on "A Measured Response"

For those who haven’t seen, this is a follow-up to my recent article (below).

First of all, I am overwhelmed by the positive response to my article—I'm still a little confused why what is functionally a measure theory lecture is my most popular post. Y'all are nuts, thank you so much.

Secondly, Bentham's Bulldog has replied to me in many places! Scott Alexander too, and doesn't that make me feel like I met a celebrity? Obviously they think I'm wrong, but among other things, BB has challenged me to engage with the actual philosophical literature. So I have! Or more accurately, I used Gemini’s Deep Research tool (which is unbelievable by the way) to give me a comprehensive summary of the state of the art in "the measure problem" of fine-tuning. It produced a 32-page document with 55 references, and frankly I read it in one sitting because it was really fascinating. The LLMs lack a true voice (unless “fun police” is a voice, looking at you Gemini), but they are extraordinarily clear writers.

Anyway, since the discussion surrounding my post was so interesting, I thought I would capitalize on the mom— sorry, I meant to say summarize the discussion for all the interested readers! I am going to shamelessly steal Scott Alexander’s style of “Highlights on X” follow-ups, because blatant theft is the highest form of flattery.

This article will be roughly three parts, I’ll summarize some interesting points from the literature review, I’ll respond directly to Bentham’s Bulldog’s article/comments, and then I’ll highlight some particularly interesting/insightful comments from the discussion.

Note/TL;DR: This post will be less didactic (though there IS some stat mech!), and thus possibly less interesting than my previous post, but since the previous post rapidly became my most popular ever, by FAR, I figured I owed a summary/response.

The essence is that Bentham’s Bulldog missed the point a little with some of his objections, and underestimated the ability to do math on alien-taco situations in others. The literature, however, does NOT miss the point and basically agrees that without solving the measure problem somehow you’re sorta screwed, and proposed various contentious ways of dealing with it. My critique stands.

0.5. Brief Recap and Bentham’s Challenge

Briefly, the fine-tuning argument holds that due to the true fact that the universal constants we observe in physics (speed of light, cosmological constant, Higgs boson mass, etc.) fall in a very narrow life-permitting range, that this is strong evidence of an outside “fine-tuner,” i.e. God. Bentham’s Bulldog made this argument in, among others, his article The Fine-Tuning Argument Simply Works.

In my previous post, linked above, I argue that the fine-tuning argument that God exists is ill-posed and the mathematics behind it meaningless. In essence, when Bentham’s Bulldog does a calculation of the Bayesian probability of the creator, he uses a whole bunch of undefined terms and the result one gets could be literally any number. Bentham’s Bulldog naturally disagrees:

This is an interesting article--basically raising the measure problem--but I find it weirdly triumphalist. In that original article I did not address the measure problem but I have elsewhere (here for example). You shouldn't be totally confident that the measure problem wrecks my view if you haven't read what I—or people on my side—say about the measure problem!

I give a lot of more specific objections in the linked post. But crucially we can *obviously* say that things are improbable even from an infinite set. So, for instance, there's an infinite set of things that the nearest aliens might be thinking about presently. An infinite number of them involve tacos. But still, I would guess with high confidence that the nearest alien to me is not thinking about tacos

Let’s tackle (partially, anyway)¹ the first of his objections, that I don’t engage with him or the philosophical literature on this.

1. The Literature

Physics and Cosmology

So the first thing that really jumped out at me is that the measure problem is way more broad than I thought. Forget God for a moment, the problem of measure is just generally a problem for theories of the multiverse. As Gemini writes:

Eternal inflation posits that our observable universe is but one "pocket universe" or region within a vastly larger, perhaps infinite, multiverse, where new universes are constantly budding off. These pocket universes may possess different physical constants or even different effective laws of physics.

The measure problem is the challenge of defining a consistent and unambiguous way to calculate the relative probabilities or frequencies of different types of universes or events within such an infinite ensemble.

I hadn’t really even considered this, but obviously this problem exists in pure physics!

Alan Guth gives a memorable example of why this is a problem—what’s the ratio of two-headed to one-headed cows in the multiverse?

“In a single universe, cows born with two heads are rarer than cows born with one head,” he said. But in an infinitely branching multiverse, “there are an infinite number of one-headed cows and an infinite number of two-headed cows. What happens to the ratio?”

This is a really good question, the answer, of course, is that it depends how you define your universe-generator, i.e. your probability space/measure.

But defining a “typical” or “probable” universe poses similar challenges for the anthropic principle.

Let’s say we’re just one of many universes, but ours is the only one that supports life. Well then of course we find ourselves in this universe, we literally can’t exist in the others! That’s the weak anthropic principle. There’s a strong anthropic principle, which posits that the universe must exist this way, that sounds a lot like a design argument to me.

This is all highly related to Nick Bostrom’s “Strong Self-Sampling Assumption,” which states that we should reason as if we are randomly selected from the class of all observers. But… what class? The measure problem again!

Let’s give some more interesting examples of how different measures affect your cosmology.

So you’ve got some infinite collection of universes, generated according to some process, and you need to calculate relative probabilities. The problem is that in an eternally inflating spacetime, any event that is physically possible will occur an infinite number of times, rendering simple frequency counting useless. Thus, you “regularize” by taking a finite subset of the multiverse, then taking the limit to infinity. There are several choices of cutoff:

• Proper-time cutoff: Counts events or observers up to a fixed hypersurface of proper time, then lets this time go to infinity.

• Scale-factor cutoff: Uses the expansion scale factor as the time variable for the cutoff.

• Stationary measure: Compares processes based on the time since each process individually reached a stationary state.

• Causal diamond measure: Considers the finite four-volume accessible to a single observer from the beginning to the end of their possible observations.²

I’ll just give one example: The proper time cutoff yields an overwhelming bias towards “young” universes. This implies we should all be “Boltzmann Brains,” random thermodynamic fluctuations that simulate life and memory, because these are more probable in a young universe than an old one. Sean Carroll doesn’t like this, because any theory that implies you’re a Boltzmann brain should make you skeptical of your ability to reason coherently about theories. Other cutoffs give you other biases.

Chris Smeenk calls this a "predictability crisis"—eternal inflation literally cannot make testable predictions without solving the measure problem.

I want to raise two more interesting physical arguments. One of them is Max Tegmark’s Mathematical Universe,³ and another is in a paper, which I need to actually read, not just skim, called Algorithmic Theories of Everything that Onid sent me. This proposes that universe histories are sampled from probability distribution based on their algorithmic complexity—universes with shorter descriptions are more probable, and weakens the requirement for universe histories from Turing computable to formally describable.

Okay, well, what do philosophers say?

Philosophers

Pretty much everyone agrees that the universe sure looks fine-tuned to us. To explain this, philosophers like to use either abduction (extrapolating from evidence to hypothesis, sometimes called Inference to Best Explanation), or more formally, Bayesian reasoning, which I argued is ill-posed. Gemini highlighted six explanations

• Design/Theism: It was fine-tuned by God.

• Multiverse: There are many possible universes and we live in one that permits it because of the:

• Anthropic Principle: We observe only what supports our existence.

• Deeper Laws: A more general principle makes fine-tuning seem natural.

• Simulation: We live in a simulation fine-tuned by someone not God

• Brute Fact: Look man, we live where we live, get over it.

I love the way Gemini put this: “The philosophical debate surrounding these explanations often reveals underlying differences in epistemic values and criteria for what constitutes a satisfactory explanation.” Oh my god, yes.

In one sense, the “Brute Fact” explanation is extremely attractive—posit nothing, we live where we live. This is a non-problem. You can see the appeal, in a way. I still like “deeper laws,” personally. Or multiverse, in some way, but isn’t that just an instance of a deeper law?

Now it gets a little wacky, but I love the attempts at formalism! Philosophers propose a few alternate probabilities that differ from standard (Kolmogorov axiomatic) formulation of probability.

• Finitely-Additive Probability (FAP)⁴, of which the first article I found explaining it is called “LOST CAUSES IN STATISTICS I: Finite Additivity.” This version of probability drops “countable additivity” in favor of “finite additivity,” essentially arguing that P(A or B or C or …) = P(A) + P(B) + P(C) + … can’t apply if there are infinitely many terms.

• Non-Archimedean Probability (NAP) Theory: Extends probabilities to include infinitesimals.

• Non-Normalizable Quasi-Probability: Drops the normalization requirement.

• Truncation: This one is the simplest—instead of relaxing the axioms of probability, just truncate the space you work on. This is subject to the “but why that truncation” problem.

I think this is really interesting philosophically, though you’d have to give me a really good justification why it’s a better idea to drop very reasonable axioms of probability theory just so that fine-tuning arguments for God work, when in my opinion the “deeper laws” objection works better anyway. Either way, one can have a debate on this, because, and this is key, every philosopher in the literature summary confronts the measure problem as though it’s fatal without some sort of workaround. I’m actually fairly happy I was challenged to do this, because it makes me pretty happy with my critique. My core objection is physics, not philosophy, and it is correct per se. Bentham’s Bulldog was pointing out that philosophers have various arguments why we can get around it and still reason with solid foundations.

2. Directly Responding to Bentham’s Bulldog

First, a mistake from me.

Three commenters, from the article and the reddit thread⁵ pointed out that one of my arguments was flawed. This one, specifically.

“But by construction, you get P(fine-tuning∣¬God) = 0 due to selecting a value from a set of zero measure (in this case, selecting a value from a continuous range). By the premise of this whole argument, then God too must select, whether randomly or not, from a set of zero measure. The probability of God selecting any particular value, P(fine-tuning∣God), is therefore also zero.”

Why does defining the God hypothesis more precisely not end up with that "1/0 \to \infty" situation, as opposed to undefined?

—u/[Anonymous] (reddit)

This remark and the associated note 17 is wrong; by assumption God is not choosing randomly.

—u/Curates (reddit)

Also, it’s a minor point, but you mention that God could not select from a non-measurable set, but this isn’t true. God would, presumably, act non-randomly, and one can definitely non-randomly select from a non-measurable set (for instance, selecting 0 from the Cantor set). Since God is presumably defined as omnipotent, omniscient, and wanting to create life, P(fine-tuning | God) would be 1.

—Onid (Substack)

These three are right and I am wrong.

The point I was trying to highlight was that even if you manage to define naturalism on some continuous interval, P(God selecting our constants from any continuous interval) is still 0, because he might just as well have chosen something else within the life-permitting interval.

While this is technically true for a God who selects randomly, it’s quite simple to counter this argument by simply stating that God doesn’t choose randomly.

I'd have been better off not even attempting to posit a God selection mechanism at all. That’s getting into theology which is (clearly) not my strong point. Onid, however, delivers my rejoinder for me:

Obviously P(fine-tuning | not God) is still completely undefined, so it’s not like this changes anything.

—Onid

Yes, this is right. I'd pretty happily accept a version of the fine-tuning argument that defined both of these things somehow and then made a Bayesian argument. It would have to look like:

Assumption 1: God is this way, here's my justification.

Assumption 2: Naturalism works this way, multiverse budding, whatever, and here's P(our constants | my multiverse).

This is 100% valid and I have no (math) problems with it. I would probably quibble about your justifications or argue with you about why that specific God, but you'd be good on the math.

As an aside, Onid also said something very kind that made me very happy.

Anyway, your point about the monster group was brilliant. I hadn’t heard that one before. —Onid

Thank you!! I was very proud of this analogy, because it’s authentically mine and I feel like it works really well to demonstrate why something really odd can be true but derived from a really simple and uncontroversial principle. Looking forward to your article on SIA!

Bentham’s Bulldog’s Specific Objections

His Comments

Ok let’s look at his specific objections to my article.

But crucially we can *obviously* say that things are improbable even from an infinite set. So, for instance, there's an infinite set of things that the nearest aliens might be thinking about presently. An infinite number of them involve tacos. But still, I would guess with high confidence that the nearest alien to me is not thinking about tacos

The whole point of my article was no we cannot obviously say things are improbable from an undefined set. He’s eliding infinite with undefined. Some infinite sets are totally fine, most are not and you must pick one or I will yell at you. This objection misses the point entirely.

The second objection is better. Let’s talk tacos, because that one generated a ton of discussion and I think I can contribute.

Well-Defined: Aliens Holding Tacos

I originally tried to answer this question with aliens-thinking-tacos, please see the footnote for my formalization attempt,⁶ but I want to thank u/kaa-the-wise on reddit for pointing out that aliens thinking about tacos has the “what is thinking, anyway” question embedded in it, and runs into the hard problem of consciousness.

So let me modify: aliens holding tacos. Same essence of improbability, no consciousness. Hopefully Bentham’s Bulldog is okay with this.

My response to this is quite straightforward: When we ask "what's the probability that the nearest alien is holding a taco?" we're operating entirely within known physics, and we can quite reasonably formulate a probability space to operate on. We need some statistical mechanics background though.

Very Brief Stat Mech

Statistical mechanics is a branch of physics concerned with the problem that there are just too many atoms. We’re sure that the physics of atoms ultimately gives rise to the physics of e.g. baseballs, but there’s ~10^23 atoms and we can’t write down the equations of motion. So statistical mechanics provides a rigorous framework for taking ensemble averages of many, many atoms, and pulling out useful average quantities, like temperature, entropy, and so on.

Let’s say I have N particles in a 1-dimensional box of side length L, the positions and momenta of these particles is then (x1, x2, x3, ..., p1, p2, p3...). This is a 2N-dimensional “phase space.” The particle states at any point in time occupy some configuration in this phase space. Phase space is continuous, and so there are an infinite number of states, yet we can very reasonably and meaningfully discuss densities of those states, because we can attach a sigma-algebra! This sigma-algebra is the Borel Set, which I briefly mentioned in my article, but essentially consists of countable unions, intersections, and complements of open intervals. It’s a measurable set, and thus we can assign a measure to it (usually Lebesgue). For example, we can discuss the "density" of particles over some interval dx or the system being in some region d^N x d^N p of phase space.

How does this help us with aliens holding tacos?

Well, this is nothing more than a configuration of atoms on the phase space of the universe! Aliens-holding-tacos is some subset of the possible configurations of the Borel set on universe phase space, and (probably) a tiny tiny tiny fraction of it. We can perfectly well use Bayesian reasoning on this and get answers that align with our intuition. The nearest alien is very probably not holding a taco, though if they were, first contact is going to be amazing.

I want to emphasize this, because there was a lot of confusion on this point. Infinite or continuous vs. discrete and/or finite is not the appropriate distinction for having sets of things that we can reasonably talk about—it's describability, it’s the sigma-algebra, it’s the measure.

I should note that this is basically also my response to Scott Alexander’s challenge as well, and I’ll get to that later.

Bentham’s Bulldog’s Article

A third objection is called the measure problem.

…

Now, this objection has the problem of stupendously misstating the physics behind much of fine-tuning. Regarding the cosmological constant, for instance, physicists don’t assume that each value of the constant is equally likely, but instead note that many different values affect the cosmological constant. These values are determined at small scales, but almost exactly cancel out at large scales, so that the universe doesn’t expand either too quickly or too slowly. This is a coincidence on the order of the number of seconds humans have spent happy minus the number of seconds humans have spent unhappy equaling just twelve seconds. The values going into the fine-tuning argument are about 10^120 times greater than the value of the cosmological constant, and when one allows these values to cancel out, they get a slight positive cosmological constant.

This objection is simple to dispense with, it does not actually refute the measure problem, instead it restates that fine-tuning is surprising. This is true! It being true does not give you a well-defined space to work in. That’s what I mean about slightly missing the point, though it’s unfair of me to say that this article, written prior to mine, “missed the point” of my article. Nevertheless.

It also misstates probability. The value of the constants can only be calculated across scales consistent with the basic standard model, and only a tiny slice of values consistent with the standard model produce complex structures. This would be like, to use an analogy from Robin Collins, hitting a tiny red dot on a dartboard surrounded by a white area. Even if most of the dart board is shrouded in darkness, this is very improbable! It doesn’t matter the total percentage of the dartboard that is occupied by red, because the surrounding region is almost all white—hitting the tiny red dot is thus astronomically improbable.

Okay here this is wrong. “The value of [the constants for a hypothetical set of universes]” can absolutely not be calculated “consistent with the basic standard model” without a whole boatload of assumptions that are by no means assured. Well-defined space. Note again the analogy used is a well-defined space. Arguments by analogy hurt me.

One can also get their probabilities to nicely sum to 1 by taking a ~log prior across the values of the constants! When they do that, once again, they get a ridiculous degree of fine-tuning.

Here is a little more subtle, he is trying to pick a space/measure! This isn’t a critique of the measure problem so much as an attempt to show it’s invariant under different ones, but that is just not correct, unfortunately. Pretty much every physicist/philosopher in the Deep Research review acknowledged this is hard, and you have to make some very questionable assumptions (though potentially justifiable!). See the literature review

But once again, we don’t need to bother with all that. The objection simply proves too much. If it were right, then the initial conditions spelling out “made by God, with love” in every language wouldn’t be evidence. After all, there are an infinite number of shapes that the initial conditions could conceivably take on. In fact, the math behind the actual fine-tuning is considerably less messy than the math behind figuring out the odds of the initial conditions spelling out “made by God, with love,” as there are probably infinite different ways the initial conditions could spell that out, and no particularly elegant measured over the values that would render that improbable.

Aha, this is the same objection as aliens-holding-tacos above, and has the same justification. This is well-defined, just completely intractable to exactly calculate. Infinite ways does not equal ill-defined, as the very brief stat mech overview showed us.

He also has some objection to the “deeper physics” postulate that I personally like, but that’s handled well enough in my previous article, I think, where I attempt to prime your intuition that actually, really really weird unintuitive stuff comes from basic obvious seeming principles sometimes.

So in summary, none of Bentham’s Bulldog’s defenses refute the core critique. He convincingly argues that fine-tuning is surprising, but offers us no way to do probabilistic calculations with it. He further offers an intuition pump about aliens and tacos, or equivalently unlikely scenarios that are well-defined, once again demonstrating the peril of arguing by analogy.

The core ambiguity and the problem of measure remain intact. If I were him (taking some charitable liberties with his position) I could posit something like a Borel set on finite-parameter-range multiverses where our universe is a tiny slice, and thus improbable, then use this to argue for theism. That’s emphatically not the only valid choice, we could easily have deeper laws, or a wacky underlying set (Vitali universe?), or literally anything else, but this would at least be a (somewhat arbitrary, but nevertheless) valid mathematical statement.

3. Highlights from Comments

Attempts to Well-Pose Fine-Tuning

Scott Alexander wanted some clarification on whether the fine-tuning argument works if you pose it a little more formally.

Suppose that handwavy/intuitively, the constant could be anywhere between -infinity and infinity. But in order for hydrogen to exist, the constant must be between 0 and 10. And in order for life to exist, the constant must be between 3.790 and 3.792.

Now you could say something like "of all the universes where hydrogen exists, only 1/5000 have life, this is Bayesian evidence for fine-tuning" and this would be fine. As far as I know, Bulldog could say this - the hydrogen example is made up, but probably something like it is true.

Would this rescue fine-tuning? And if so, can't you then say "and in fact, since the set of universes with hydrogen is a tiny subset of all possible universes, the real value must be incomparably larger"? -u/ScottAlexander

Now I wrote my own response, but I, u/kaa-the-wise, and u/Express-Smoke1820 basically said the same thing, and they were less long-winded, so I cede the floor.

No, this very much would not be fine. It is inconceivable what the range of "all the universes where hydrogen exists" is, and what the probability measure on them is. 1/5000 is pulled out of thin air and is just a shiny mathematical wrapper for a feeling.

…

To give this proper grounding, you have to define both the space and the probability measure, also inevitably defining the limits to the model you are offering, because these can serve as the basis for potential disagreement.

I further suspect that one motivation of not doing so is to make one less vulnerable to such disagreement. —u/kaa-the-wise

A sharp response in both senses. Perhaps a bit harsh at the end, the mistake is subtle, after all, but absolutely correct. u/Express-Smoke1820 concurs, with a slightly different flavor:

The dangerous part here is “intuitively, it’s obvious that at the very least it could be anywhere between 0 and 10.”

“Anywhere between 0 and 10” is a well-defined probability space, but there is no non-arbitrary reason to draw the line at 10, or 100, or 30 trillion, or to restrict the set to rational or even to real numbers. So, to the question “what are the odds we get a value between 3.790 and 3.792 out of all possible values,” it might seem like the answer is a hard 0 for any finite interval, but it’s actually not even that - “values” simply doesn’t refer to a measurable set (from the article: “Real numbers? Positive only? Complex? We have no idea, and exactly one example of a universe to guess from.”). This is a problem, not just because it sets out no discernible sample space, but also because it defines no σ-algebra we could use to work with any such space. —u/Express-Smoke1820

They also raise a further objection, which I found thoughtful and interesting.

And (though this goes beyond the scope of MEMR’s article) I’d say there is an additional dimension of uncertainty - the set of natural laws other than the ones we’ve got which God could have created, but didn’t. These should also be considered in any explanation of why God chose these laws and tuned them. But the space of alternative sets of laws of physics is so far beyond quantifiable that I wouldn’t even know where to begin. —u/Express-Smoke1820

Me again now:

I think the valid version would have to look something like “according to this theory of the multiverse, we have universes generated with constants in this range. Life only exists in this tiny subset! Wow, how lucky did we get to end up in this one? Hmm…”

That’s a valid argument, conditioned on the theory of the multiverse being right. But we don’t have that.

Requiring This Much Math Stifles Thought

This objection was touched on earlier, but raised by both Bentham’s Bulldog and Scott Alexander. It deserves an answer as a legitimate concern.

[A]s a less legible objection, I'm nervous that work like this is taking something that's obviously true - it's really weird that these constants are in the tiny realm suitable for life - and merely raising enough objections that you're not allowed to talk about it unless you have ten years of postgraduate math. I think in order to convince me, you would need to find a way to intuitively show me how this actually defuses the surprisingness rather than just lets you pounce on anyone who tries to discuss it rigorously. —u/ScottAlexander

I do slightly feel like I'm getting Eulered. —Bentham’s Bulldog

I had to look this one up. From context, it sounds like “Eulering” someone is throwing math at them they don’t understand until they give up, but I my guess is it probably comes from Scott’s Slate Star Codex article Getting Eulered.

Diderot was quite the clever debater, and soon this scandalous new atheism thing was the talk of St. Petersburg. This offended reigning monarch Catherine the Great…

[S]o she asked legendary mathematician Leonhard Euler to publicly debunk and humiliate Diderot. Euler said, in a tone of absolute conviction: “Monsieur, (a+b^n)/n = x, therefore, God exists! What is your response to that?” and Diderot, “for whom algebra was like Chinese”, had no response. Thus was he publicly humiliated, all the Russian Christians got an excuse to believe what they had wanted to believe anyway, and Diderot left in a huff. —Scott Alexander

Scott has some very nuanced views about this in his article. He tries to thread the needle of not letting people throw math at you to shut you up, vs. acknowledging that math is sometimes very important to make good arguments.

I suppose I have two thoughts on this. The first is that this is a valid criticism of my article! I did use a lot of math, and I did so intentionally. Now I did the best I could to make the article didactic, to really teach people something important and nontrivial about measure theory. While of course still mixing in the Latex and not writing a textbook. But nevertheless, if you have to read a lecture on measure theory to understand it, is it still a good argument, or just Eulering?

I am sympathetic to the notion of being Eulered, especially when it happens to me, but except in egregious cases, I think I reject this concept outright unless used in bad faith. Sometimes arguments just are complicated and technical, and the “problem of measure” is not coherently debatable without some understanding of measure. Especially when you invoke math in your argument, you must, at some point, be ready to do the math, and do it right.

It is, however, incumbent upon those who know more math to attempt to teach it to those who don’t. Especially when making a complicated technical argument.

Just as a note, I’m trying not to come off too elitist here so I’ll say this: I love teaching! I sincerely try to do a good job at explaining the math, and you all have been very kind! Please, ask me for clarification, I will do my level best to explain any point I make at any level of sophistication you ask: five-year-old, postgraduate, whatever. I reserve the right to do much worse on the philosophy points, though, I’m already a little outside my lane here.

Is the Measure Problem a Fully General Dismissal?

Scott worries more along the same lines.

Also, isn't "there could be some unknown physical law causing this" a fully general dismissal for all possible evidence? Suppose that SETI detected a message saying "HI WE ARE ALIENS" in perfect English. It's always possible that there could be some unknown physical law that causes this to happen. Bayes' Theorem just says that the it's-really-aliens hypothesis called exactly this outcome, whereas "there are unknown physical laws" could cause approximately anything - and therefore we should greatly increase our confidence in aliens. In the same way, "someone is optimizing for life" predicts fine-tuning within this range, and "there are unknown physical laws", while they could explain this, could equally well explain any other value of constants.

This is more or less the same as Bentham’s Bulldog’s argument regarding aliens, tacos, etc. and has the same answer. We have computationally intractable but nevertheless well-defined answers to questions that work with known physics. It’s perfectly fine to use Bayes’ rule.

Other Nice Comments

There’s a whole field of philosophy called formal epistemology that studies how to apply rigorous probabilistic methods to philosophical questions like those raised by fine tuning arguments. They have developed tools to address problems you’re concerned with, they’re more sophisticated than you might think. —u/Curates

This is honestly just really interesting, and I knew nothing about this, but the Stanford Encyclopedia of Philosophy has an article on it.

Too much math for me here but I wonder if it boils down to what I think: When people say something like "If such and such constant was 0.0001 different, reality wouldn't work!" There's a buried premise that 0.0001 is 'small'. Small compared to what? The decimal point feels completely arbitrary here. We can dive in infinitely and stick the decimal point way further back, now the number can be hugely different and have no discernible impact.

I call this the Rough-Tuning Rebuttal.

But my physics isn't so hot either so maybe I'm wrong. —u/Ontheflodown

This is maybe not exactly right but it’s very close and very concise.

I’ve raised these same concerns in comments before as well, and I think there are even larger concerns.

1) Bertrand’s paradox. Even if you did have a well defined sigma algebra, its not clear how to pick a uniform prior (in the language of your post, you can have the sigma algebra, and the event space, but picking P is not necessarily easy)

2) Universes with different laws of physics, or uncomputable laws of physics, or individual universes which are too large to be measured make assigning a probability space impossible.

3) The space of all *logically possible* universes is way way way way way bigger than universes permissible with ZFC. It is not even clear if it makes sense to apply probability theory, which is built upon ZFC, to the larger universe of *all logically possible universes*. Logical possibility is unfathomably big. Im not even sure if “all logically possible universes” is coherent (modal logic notwithstanding) —SorenJ

To point 3) here, I think SorenJ is touching on a good point. We generally think of math as being unitary and absolute, but actually, our mathematics is formally based on a set of axioms called Zermelo–Fraenkel set theory, and if you include the axiom of choice, that makes ZFC. There are other formulations! ZF, for example (no axiom of choice). I think this is important to remember, perhaps not so much for its utility, but as perhaps a reminder when you stretch your intuition too far that the space of everything possible is unfathomably vast.

To point 1), Bertrand’s paradox is just extremely interesting, a classic paradox in probability, and I can’t believe I didn’t talk about it in my article, so I’ll mention it briefly here.

Let’s say you have an equilateral triangle inscribed in a circle. You choose a random chord from that circle, what’s the probability it’s longer than the side length of the triangle, P(>△)?

As it turns out, this problem has no answer—it depends how you construct the chord.

From Wikipedia, https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

• Random Endpoints: You could choose two random points on a circle and draw a chord between them.

  • If you do this, then P(>△) = 1/3.

• Random Radial Point: You could cast a ray from the center to the edge, and choose a random point on this radial line, drawing the chord perpendicular to the radius at this point.

  • If you do this, P(>△) = 1/2.

• Random Midpoint: Choose a random point in the circle, and draw a chord with that as its midpoint.

  • If you do this, P(>△) = 1/4

Which is right? It depends on your construction! This doesn’t say anything we don’t already know, it’s just the most famous illustration of why defining the problem correctly is very important if your probability is to mean something.

From Wikipedia: https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

On the Self-Indication Assumption

Onid in the comments had some very interesting thoughts about the self-indication assumption. He’s currently writing an article about it, so I won’t preempt him here, but I’m looking forward to it!

On Being a Chatbot

A commenter on reddit, with a since-deleted comment, asked me whether I had prompted a chatbot to imitate the style of Scott Alexander for my previous article. No, I did not, and that’s high praise, thank you!

But in the interests of transparency, since I do frequently use LLMs as a writing aid, I thought I’d paraphrase my response below. Essentially, my strategy with LLMs is to try to leverage their strengths and avoid their weaknesses.

Strengths

• Deep Research for literature review is just amazing and I can’t recommend it enough. For deeply technical subjects it is no substitute for reading the papers, but for something like this article? Excellent substitute.

• Beta Reader: I find LLMs incredibly helpful for essentially beta-reading my posts. I’ll prompt it with “Hey, can you read this for clarity and grammar, and please point out any logical holes in my arguments?” It’s shockingly good at this. In the previous essay, it caught a couple errors in my math, some flaws with how I was arguing, etc.

• General Explanation: Let’s say I’m having trouble with the Vitali set construction (and I absolutely was), asking LLMs to break it down for you is just so helpful, as they tailor the explanation to your level and background (if you tell them to). You should, of course, verify with the formal definition after you understand, but formal definitions are usually light on the intuition, and LLMs really shine here.

• Outlining: My previous post was looooong. I sometimes have a lot of thoughts that I’ll write down in a notepad as a stream of consciousness, then ask the LLM to “organize my thoughts into a coherent and flowing structural outline.”

Weaknesses

• Voice: Except for the occasional really good one-liner, I hate how LLMs write on a sentence-to-sentence level. They often write like an undergraduate five-paragraph essay that’s been sanitized by an HR department—Gemini ends every question I ask it with a section literally entitled “Why This Matters” and must have a profound wrap-up section for every essay. I am so glad I learned to write before these things came out, or I would almost definitely converge on this writing style by sheer inertia, and I hate it.

• Novel math/arguments/ideas: Ask them to regurgitate a proof, or make “the strongest version of standard argument,” and they’re great. Ask them to solve a new math problem, one that’s kinda open-ended? I’ve done this at my work a bunch and they’re just terrible at recognizing when a problem is intractable, stopping, re-evaluating, and trying something else. They have no sense of the ground truth, and are very prone to charge off in a single direction and iterate by random walk. Hopefully this gets fixed in the future, but for now, don’t use them for this.

I have a lot of other thoughts on what LLMs are good for and not good for, but remember not to let them substitute for your own thought! They’re a powerful force multiplier, but they’re also a mirror, and you must be careful not to fall into it.

Narcissus and his reflection

Conclusion

With all this said, I suspect this will be my last post on the measure problem for at least a little while. So to all of you who subscribed to my blog thinking it was a measure theory textbook, I apologize, it was a bait and switch! Instead, you’re likely to get more posts like this one:

EDIT: Addendum, some math errors from me in the original post.

It was inevitable that an actual mathematician, or at least someone better than I am, show up and find my errors.

Small quibble: I don't understand your mention of AC. If we take on not-AC we can prove that all sets are measurable.

Here I was using the axiom of choice to draw an analogy between unintuitive consequences and simpler principles that imply them. He is right that I am implicitly assuming the axiom of choice in my examples and that I don’t have to do this, though choosing or not choosing the axiom of choice for your universe model is of course a choice. An axiom of axiom of choice?

Bigger quibbles: What do you mean P(God|measurable sets)? The set of measurable sets is not an element of the algebra, it is itself the algebra.

This is sloppy notation on my part and intended as rhetorical flourish, not a formal statement of probability. The set of measurable sets is absolutely just the algebra— this should say something like P(God | “the surprising fact that measurable sets exist and physics occurs on them”), which is completely undefined. I acknowledge the irony of imprecision in a post about making imprecise math statements.

You claim there's no natural choice of algebra. The natural choice of algebra is the Borel algebra over a Euclidean representation of the parameter space.

Elijah suggests the Borel algebra over a Euclidean representation of a fine-tuning parameter is a natural choice. I agree this is true if the sample space (Ω) is well-defined. My core argument, however, is that it isn’t, and so any subsequent choice of σ-algebra is of course also not well-defined.

Then you say that the measure theorist says the probability of picking any particular value from a continuum is zero. This is wrong. Singletons are Borel in R^n. Zero-inflated priors, for example, are bread and butter Bayesian statistics

—Elijah Spiegel

Elijah also points out that the probability of picking a single value from a continuum is not always zero. This is true, you can have “point masses” with nonzero probability and they don’t break your theory and are allowed in Borel sets. The “zero-inflated prior” is essentially having an interval of possible effect sizes and tacking on {0} to that interval to represent “no effect” with nonzero probability. My original statement implicitly assumed an absolutely continuous probability measure. One could mathematically assign a point mass to “our universe's constants,” but of course why would you do that. Nevertheless, he is correct.

Generally speaking, he is right about his small and large quibbles. I think his points show the danger in trying to essentially write a measure theory textbook as a blog post, I have some elisions and errors between the concept I'm pointing to and a fully correct definition, as well as me straight-up forgetting about point masses being allowed in Borel sets, all of which is entirely my fault. This is my bad.

However, I don’t think any of the points he makes are fatal to the overall argument that your sample space of possible physics is just not well-defined. Instead, it just illustrates how hard this problem is in general.

1. Here’s the link to the Gemini 2.5 Deep Research again, it’s really fantastic.

2. Again taken directly from Gemini’s summary.

3. See Scott Alexander’s article where I learned about it and Bentham’s Bulldog’s rebuttal.

4. Yes, that’s the acronym.

5. https://www.reddit.com/r/slatestarcodex/comments/1kzvnzg/a_measured_response_to_benthams_bulldog/

6. Here’s my attempt from the reddit thread when u/[Anonymous] asked how I might go about formalizing this. It was pretty off-the-cuff, but what do you think?

As my favorite textbook says, "When the going gets tough, lower your standards." Let's start with the discrete finite case cause that's definitely easier.

The space of alien thought is tough, but the space of human thought is maybe much easier. What fraction of human thought is about tacos, or taco-like objects? My first instinct is to restrict further, look at human writing, because we have an enormous corpus of writing to look through that includes menus, cookbooks, etc. What fraction of words in the English Corpora are "taco" or taco-adjacent? For example, we could use some sort of similarity metric, Jaccard, some sort of word2vec latent space for LLMs, etc, to define a fuzzy range around the concept of "taco" that might work as a lower bound.

Once I have that number, I need to try to extend this argument, and find some sort of upper bound for how much of human thought can POSSIBLY be about tacos. No one has tacos for every meal, etc. etc. I'd then try to tighten these bounds... somehow.
This would give me an estimate of an upper and lower bound of how much of human thought can possibly be about tacos.

Generalizing this to aliens is... problematic, but maybe we can argue that tacos are a human object, and P(aliens thinking about tacos) < P(humans thinking about tacos)? Perhaps we can try to look at human cultures to see how many of them develop a similar concept of a taco? I'd have to think about that one, and you're right, it posits some sort of alien that can (a) think and (b) likes tacos.

From there, though, the argument is straightforward. I would bring in the 10^{25,888,579,627} possible thoughts and try to normalize.

Might make a great substack article, actually, but I'd never bet anything on my estimate.