What Does Analysis Teach?

My favorite essay so far.

Florence Bacus, aka Moral Law Within on substack, writes great stuff and you should read her. (I’m very happy someone else is talking about Functional Decision Theory on substack!)

Among other things, she has a mathy-er background than some of the other substack philosophers, and that excites me at the same time it makes me a little more concerned that some readers will get Eulered. I am intrigued by her attempts to prove moral laws from no moral assumptions, and I don’t think it is doomed to fail. However, I think she’s a little too sanguine about what analysis of stipulated terms alone can teach us.¹ Hence, this essay: about what we can and can’t learn, and where I think Florence’s writing so far may go wrong.

1. What Analysis Can’t Teach

Florence’s chief example of analysis comes from continuity, so we’ll start there.² Suppose Thea is thinking about the nature of continuous functions. She thinks of a continuous function as a function you can draw in a single smooth line without picking up your pen:

A continuous and a discontinuous function.

Of course, Thea will have a lot of questions about continuity:

What about the function g that outputs x² for x less than 0 and x²+1 for x [greater] than or equal to 0? There’s certainly a discontinuity at 0, but it seems like the function is continuous with only that exception, i.e. g continuous at every point except 0. What does that mean? The only time you have to pick your pencil up is at 0? But what does it mean in general to say that a function is continuous or not at a point? Perhaps it means that the point lies in a region on the graph that you can draw without picking your pencil up. So, should be conclude that if a function f is continuous at x, then there is some region of nonzero size containing x that f is continuous on? How do we even prove anything about continuous functions? Do we appeal to intuitions about what you can do without picking a pencil up? Should we fund empirical studies to investigate drawing without picking a pencil up?

Fortunately, mathematicians don’t have to worry about this, because we have a different idea of what it means to be continuous—the ε-δ (epsilon-delta) definition of continuity.³ In pure mathematical syntax, f is continuous at c iff

\(\forall\varepsilon > 0,(\exists\delta>0:y\in\mathbb R, |y - c| <\delta\implies |f(y)-f(c)| <\varepsilon)\)

Frightened? Don’t be. If you think about a graph, you can visualize this as saying that for any arbitrarily narrow horizontal band of space, centered on f(c) and stretching horizontally forever, there is some narrow band of vertical space centered on c such that all the values of f(x) within the vertical band are also within the horizontal band.

A visual for ε-δ continuity. Left: no matter how narrow we make the red band (how small we make ε), we can narrow the green band (δ) to keep all green band values in the red band. Right: there are some red bands (some values of ε) where no matter how narrow we make the green band (δ), there will be some values of f in the green band but outside the red band.

Of course, ε-δ can also extend to cases where there aren’t nice graphs and visuals, and that’s a major advantage to using ε-δ for mathematics. Florence’s example, the crazy Thomae’s function, is actually ε-δ continuous at every irrational number:

Thomae’s function. Good luck drawing nice red and green bands on that one!

Now imagine Thea, who’s been thinking about continuity in terms of drawing graphs, runs into Alex, who teaches her about ε-δ continuity. NB: the dialogue that follows is meant to be broadly illustrative about the limits of analysis. It is not meant to represent Florence’s beliefs, strawman or steelman; Florence has addressed this problem and is much more sophisticated than Alex.

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Alex: …and so, using only analysis, we can show that Thomae’s function is indeed continuous at every irrational! What do you say to that? I hope you have abandoned your silly intuitions about pen-drawing and whatnot!

Thea: Well, you’ve certainly taught me something new about ε-δ. But I don’t see how this answers any of my questions about continuity.

Alex: I don’t think you understand. ε-δ is continuity.

Thea: Well, ε-δ is certainly a well-defined concept, and that’s probably what mathematicians mean when they talk about continuity. But I was thinking about the set of functions that you can draw without picking up your pen. At first, I thought your ε-δ might be able to teach me something about them. But since Thomae’s function is one of your ε-δ functions, clearly not.

Alex: Didn’t you hear a word I said? You can’t rely on intuitions to tell you what functions are and aren’t continuous. I can prove that Thomae’s function is continuous, no matter how much your intuitions rebel, and there’s no escaping that.

Thea: You can prove that Thomae’s function is ε-δ, certainly. I would be wrong to deny that! But it seems to me that our respective usages of “continuity” pick out different clusters in thingspace. When you say continuity, you mean ε-δ; when I say continuity, I mean pen-paper. I’m not disputing which is the “true” definition of continuity. In fact, we should probably Taboo the word “continuity” for the rest of the conversation, for clarity. All I mean is that I’ve been thinking about pen-paper. Your ε-δ sounds interesting, but as you’ve just shown, it is not equivalent to pen-paper.

Alex: But ε-δ can capture everything that is pen-paper, and more.

Thea: That’s not necessarily a strength in a definition. The cluster of mammals in thingspace includes all cats, but that doesn’t tell me anything new about cats, and it would be a mistake to conflate the two and conclude that bears are cats. It seems to me that if the set of ε-δ functions really includes every pen-paper function, I am still talking about cats and you are talking about mammals.

Alex: But if I can prove properties that all ε-δ functions have, then those properties will apply to pen-paper functions too. Just as if I were somehow able to rigorously prove statements about mammals, they would apply to cats too.

Thea: True, so long as you can show me that all pen-paper functions are ε-δ functions. Now that would really interest me!

Alex: Well, give me a rigorous mathematical definition of pen-paper functions and I can give you a proof, right now.

Thea: But I don’t have a rigorous mathematical definition of pen-paper functions.

Alex: Then how can I be sure you are talking about anything coherent at all? How do I know that pen-paper functions refer to any well-defined set of functions?

Thea: It probably doesn’t refer to a set, where objects are in or out. I can’t give you a rigorous mathematical definition of what a cat is, either. No matter what criteria I give, I am sure you can find some borderline case where it seems unclear whether or not some creature is a cat. But it seem to me that if I am interested in cats, I can still learn things about a bunch of individuals clustered in thingspace, even if the border is fuzzy.

Alex: This is the definitive advantage of ε-δ over pen-paper, then. There are no fuzzy borders for ε-δ; it is perfectly precise. True statements about ε-δ can be proven, not just argued. Mathematical analysis makes no assumptions, and it proves only definite truths.

Thea: I could equally say: this is the definitive weakness of mathematical analysis, in that you begin by restricting yourself to only perfectly precise concepts. It can tell you a lot about those concepts. It can even tell you about precisifications of vague concepts. But it can’t tell you directly about vague things, like cats and paper-drawing and families and countries, because they are excluded a priori—or worse, a precise concept like a measure of cat-ness scaling with Bayesian probability is defined with the name “cat” to trick you into thinking the vague concept and the scalar precise concept are the same thing.

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I think Thea is right here.⁴ I might question why Thea cares so much about pen-paper. After all, ε-δ lets you do calculus, which is quite empirically useful for physics, but pen-paper probably doesn’t. But:

• This argument is not a proof that pen-paper is wrong and ε-δ is correct—it’s a pragmatic argument about use, and relies on judgment, observation, and intuition.

• If Thea is really intrinsically interested in the properties of pen-paper, ε-δ doesn’t have anything to teach her about that question.

I am also not trying to say silly relativistic things like a proof by vibes is just as good as a rigorous proof for mathematical statements. Mathematics has its own rules, and it is precisely defined to exclude vibes. That means that mathematics doesn’t tell you anything about your vibes, but it also tells you your vibes don’t tell you a damn thing about mathematics. Analysis can tell us things about mathematics with absolute certainty, and it’s important not to get this twisted. Hence, the next section:

2. What Analysis Can Teach

Lewis Carroll’s “What the Tortoise Said to Achilles” is brilliant, and the source of the argument I’m about to investigate. Unfortunately, it is not easily quotable because there are a lot of interruptions in the dialogue, the language is a bit antiquated, and I have other arguments I want to explore in the same dialogue. So I’m going to reprise our characters Alex and Thea in place of Achilles and the Tortoise:

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Alex has written 3 statements on a whiteboard:

1. If x = z and y = z, then x = y.

2. A = C and B = C.

3. Therefore, A = B.

Alex: …and that’s the first component of Euclid’s first proposition. The third statement follows from the first two. Therefore—

Thea: I don’t see how that’s the case.

Alex: …what? Do you deny one of the premises?

Thea: No, they both make sense to me. Even if I did deny one, that wouldn’t affect the conclusion.

Alex: Then… are you denying modus ponens?

Thea: What’s modus ponens? You haven’t written that down.

Alex: Ah, maybe this is your confusion. Modus ponens is a rule of inference. It’s nothing more than the fact that if φ implies q, and φ is the case, then q is the case. 3 follows from 1 and 2 via modus ponens.

Thea: That feels like an additional assumption you’re making, isn’t it? I don’t see anything that says “if φ implies ψ, and φ is the case, then ψ is the case” written up there on the whiteboard. Couldn’t someone deny modus ponens?

Alex: …no? That’s just what it means to say that φ implies ψ. There isn’t an extra assumption on top of φ implies ψ.

Thea: So if I were to accept modus ponens, then 3 would follow from 1 and 2.

Alex: Yes.

Thea: Then it seems to me you should have written:

1. If x = z and y = z, then x = y.

2. A = C and B = C.

2.1. If 1 and 2 are true, then 3 is true.

3. Therefore, A = B.

Alex: …okay, sure, I guess if writing it out explicitly like that helps you. I suppose someone could hypothetically deny modus ponens, though they’d be wrong. Anyhow, we can now see that 3 follows from 1, 2, and 2.1. So, Euclid’s first proposition—

Thea: Whoa, slow down there. We included the assumption that “if 1 and 2 are true, then 3 is true,” but now you’re sneaking in this extra assumption that “if 1, 2, and 2.1 are true, then 3 is true.” None of our assumptions say anything about what the consequence for 3 is if those premises are true.

Alex: Yes, they do. The content of 2.1 just is what is the case if 2.1 is true. If you accept 2.1, then if 1 and 2 are true, 3 is true.

Thea: Oh, I accept that.

Alex: Then what could you possibly be confused about?

Thea: I’m not confused. I’m just saying that you’re adding this extra baggage that if 1, 2, and 2.1 together are true, then 3 is true. I see two of your assumptions are if-then conditionals, and one of them is that A = C and B = C. But you are making this implicit assumption “if 1, 2, and 2.1 are true, then 3 is true.” That isn’t on the board, and I’d like it made explicit. You should really say:

1. If x = z and y = z, then x = y.

2. A = C and B = C.

2.1. If 1 and 2 are true, then 3 is true.

2.2. If 1, 2, and 2.1 are true, then 3 is true.

3. Therefore, A = B.

Alex: There’s no extra baggage! You don’t understand what it means for one proposition to imply another. If you understood that, you would see that there is nothing else going on.

Thea: Well, you said you were going to show me a rigorous proof, and so far it sounds like you’re only appealing to my intuitions about what it means that one proposition implies another. If you can prove that this is indeed a consequence of implication, then I’ll be willing to listen to you. Otherwise, it sounds like you’re agreeing with me that intuitions can’t be discarded and analysis only takes you so far.

Alex: …I can’t give you a rigorous proof.

Thea: Well, then, it sounds like I win.

Alex: No, you misunderstand. I can’t give you a rigorous proof, because you can’t actually understand what it means for one proposition to imply another. I can’t give a rigorous proof to a cat, either—I can write all the steps out perfectly, and that won’t be enough to make the cat understand. Whether or not you can comprehend what I’m saying depends on your brain chemistry. It does not speak ill of mathematical truth to say that speaking it out loud doesn’t give me wizardly power to reach in and rewire your neurons to agree with me. Nor do I really care if you use the words “if/then” “condition/result”, or “zerg/blortch.” If you accepted—really accepted—what I mean when I use the words “if/then”, there would be no room for debate. I can’t make you understand what logic means by logic alone. But that is not a shortcoming of logic, it’s a shortcoming of you.

Doug’s Law: how to interpret a language cannot be communicated in that language alone.

Decrypt This Message

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I hope it’s uncontroversial that Alex is right, for exactly the reasons laid out in his last paragraph. It’s important to contrast this with the first case, though, to see what’s different. In the continuity argument, Thea did not deny any of the properties of ε-δ continuity, or claim it was an ill-defined concept, or that it had unspoken assumptions. She was just personally concerned with pen-paper continuity, so truths about ε-δ didn’t apply to the cluster of things she was curious about to begin with. Here, Thea is claiming that Alex’s own definitions are wrong or insufficient internally. She thinks that Alex’s argument is invalid despite claiming to accept his assumptions.⁵ There is a difference between believing that you accept modus ponens and actually accepting modus ponens.

I used to feel pretty impressive in philosophical arguments by telling people that mathematics requires axioms and can only prove what is consistent with the axioms, holding them as assumptions above and beyond everything else. Now I think that’s the wrong way to think about it. I think what mathematical axioms really do is identify the class of things that we are talking about, from which we can prove absolutely true statements about that class of things.⁶

3. Speaking of: Florence Bacus

I worry that sometimes Florence starts with good analysis—determining the implications of a stipulative definition—and then adds assumptions, implicitly or explicitly, such that I am no longer sure the stipulative definition is what we care about. In “Against intuitionism,” for example, we start with this:

Sometimes, when I do something, I do not decide in a way that has implications for how I am to act if I find myself with different desires. When I decide to take a walk because it seems intrinsically desirable, I do not thereby commit myself to taking a walk even in cases where I find something else more desirable. Other times, I decide to do something in a way motivated independent of brute desire. When I decide not to steal, I am motivated in a way that commits me to not stealing whatever my desires may be, just as long as circumstances are relevantly similar. I hold that the question of what to do and the question of what I morally ought to do are the same. … But what if the above is not what actually characterize the concepts of moral obligation and permission as people generally use them? My answer: “I do not care; I stipulate my subject matter to be answering the question of what to do, which is in itself an important an important question. If people mean something else by their words, what I am doing would remain indispensable.”⁷

Which is all fine, although I would love it if Florence’s definition of ought were a little more explicitly described. Nevertheless, we can do our best to pull out from this definition implications of how one ought* to act But then a little later in the essay, we get:

Why does ought imply can? Simple: one cannot rationally judge that an agent ought to do something they judge they cannot do.* Specifically, if I judge I ought to do something, then I settle on doing it (with a desire-independent motivation, but that assumption is not needed here). But I cannot rationally settle on doing something that I take myself to be incapable of; capability just is the concept we attach to things that we can think about whether to do. Therefore, if I make the judgment of obligation, I have to take myself to be able to do the obligatory thing.

* footnote: In other words, I conclude that ought implies can because an agent who judges they ought to do something is committed to judging they can do it. Reasoning at the level of judgments and commitments is necessary here, since morality here is defined by what it means to make a moral judgment (namely, to come to a certain kind of conclusion about what to do).⁸

And this simply does not follow from the stipulative definition of ought that we agreed on, that you ought to act in a certain way iff you are committed to that course of action by choosing some end that is independent of your desires.

• “Capability just is the concept we attach to things that we can think about whether to do” is not necessarily true.

  • For one, I can have a justified belief that I can go to the store and buy bread tomorrow, when in fact the store will be closed due to an unexpected flood. I am incapable of buying bread, but I can think about whether to go buy them.

  • For another, suppose that I am trapped on a rickety bridge. I don’t believe I am capable of jumping to safety. But I learn that people who settle on making the jump—not just trying, but actually executing it successfully—are slightly more likely to get across. Thus, it is best for me to settle myself on jumping to safety, even though I don’t believe I am capable. So I can think about whether to jump to safety without attaching the “capability”

• “One cannot rationally judge that an agent ought to do something they judge they cannot do.” Also not necessarily true. Randy believes he is possessed by the devil and must burn my house to the ground; he judges that he cannot do otherwise. This is in fact false. But he still ought not to burn my house down! So what is it about my assertion that is “irrational”?

  • Maybe I’m using “ought” colloquially, rather than talking about the stipulative “ought*” that Florence is discussing. But Florence’s stipulation isn’t exactly clear to begin with, and I am not totally clear how it would apply in this situation. That makes me less confident

  • Perhaps Randy is not obligated to not burn down my house, but is obligated to test whether the devil can actually control him by trying to resist. But Randy might still believe that even his resistance was caused by the devil—perhaps the devil wants to trick him into thinking he’s doing it of his own free will, to corrupt Randy further. So resistance doesn’t change Randy’s judgment of whether he’s capable of doing otherwise. But Randy perfectly is capable of staying home and cowering, even if he thinks the devil will eventually force him to burn your house down; he ought to do that.

To rescue this, you need to add a bunch of definitions about what you mean by rationality, capability, commitment, judgment, etc. Then you can prove that ought—in that stipulated sense—implies can. But then we’ve abandoned the simple stipulation that “ought” is just about what to do—which is a problem we all care about, no matter what we call it. We started by caring about pen-paper continuity, and we were promised we could learn a bunch of truths about it from no additional assumptions—and then halfway through we realize that the proofs given only apply if you’re really just talking about ε-δ, and we have no assurances that ε-δ is equivalent to the concept we originally cared about.

I have a similar problem when I hear Florence talk about the ultimate role of intuition in morality:

For the mathematician, intuition is indispensable: it is needed to know what lines of inquiry are worth putting effort into, making guesses as to what is true so that you know what to try to prove, and looking for flaws in others’ proofs. … Likewise, in ethical philosophy, we need intuitions for the same reasons: if the Categorical Imperative turns out to imply it’s immoral to wear a green shirt on a Tuesday, I need to go back and check my work; if I find myself having to choose whether to kill one man to harvest his organs and save five, I will refrain from doing so because it is intuitively wrong, as I do not yet have an Ethical Theory of Everything. But as long as I am relying on intuition, my judgments are merely provisional, and I need to continue the search for actual justifications for them.

The problem being that while analysis can tell you the implications of any particular definition, it cannot tell you which definitions to care about unless there is something internally inconsistent with an alternative definition. If the Categorical Imperative implies it’s immoral to wear a green shirt on a Tuesday, you should not just “check your work” in the sense of making sure that the conclusions follow from the premises. You should wonder whether you really care about the original, stipulative definition you gave of “morality” that makes it immoral to wear a green shirt on a Tuesday. Someone who starts interested in pen-paper continuity, but with only a fuzzy conception of what they are actually interested in, adopts an interest in ε-δ because they think that might just be equivalent to this fuzzy concept in their head they care about. When they discover Thomae’s function is ε-δ continuous, no amount of “checking their work” will reveal an inconsistency in the ε-δ logic. Their mistake was right at the beginning believing that ε-δ continuity was the thing that they cared about in the first place.

Maybe this is just because I think more like Yudkowsky than like Bacus on morality:

“I should X” means that X answers the question, “What will save my people? How can we all have more fun? How can we get more control over our own lives? What’s the funniest jokes we can tell? …”

And I may not know what this question is, actually; I may not be able to print out my current guess nor my surrounding framework; but I know, as all non-moral-relativists instinctively know, that the question surely is not just “How can I do whatever I want?”⁹

I may not be able to give a stipulative definition of morality, from which all the correct things can be deduced. If that’s the case, then any analytic argument from stipulation has the possibility of being wrong even if the internal logic is totally consistent, because the stipulation may point out a different cluster of things than what I actually do and should care about.

[Hell, if you’re an epistemologist, just look at Gettier cases. “Justified true belief” is an internally consistent definition of knowledge, but almost everyone rejects it now despite nobody having a great account of what knowledge is. They just have reasons to think that the stuff they are pointing at isn’t the Gettier cluster!]

Perhaps there is a 2,000 IQ move that can discover, irrefutably, what it means to care about something in the first place, and derive from that some truth, but if so I haven’t found it yet. Doubtless many have tried. Maybe some have succeeded! It’s definitely worth further reading, but until then, I am unconvinced.

1. Amusingly for someone who’s all about proceeding from stipulative definitions, Florence never defines precisely what she means by “analysis.” Looking at the SEP, it seems like her usage is close to Russell’s

2. Bacus, Florence. “What Action Is.” Substack newsletter. Moral Law Within, July 15, 2025. https://morallawwithin.substack.com/p/what-intentional-action-is.

3. For brevity I will refer to ε-δ continuity simply as ε-δ from here on out, though there are ε-δ definitions of other concepts, like differentiability.

4. Similar arguments are made in Yudkowsky’s sequence A Human Guide to Words, most notably “Empty Labels” and “The Parable of Hemlock.”

5. Among other things, she is confusing her Hofstadterian stack depth. Readers of Gödel, Escher, Bach will understand.

6. Does that make me a Platonist? …Maybe? My philosophy and my mathematics are decent; my philosophy of mathematics, not so much. I have more reading to do.

7. Bacus, Florence. “Against Intuitionism.” Substack newsletter. Moral Law Within, July 11, 2025. https://morallawwithin.substack.com/p/against-intuitionism.

8. Ibid.

9. Yudkowsky, Eliezer. Morality as Fixed Computation. August 8, 2008. https://www.lesswrong.com/posts/FnJPa8E9ZG5xiLLp5/morality-as-fixed-computation.