More on analyzing concepts

This time I am not being coy

I wrote a post about what action is while also attempting to convey some general lessons about analyzing concepts; one of those “Flo thinks Lovers of Wisdom should be quiet until they have mastered real analysis” posts. Stacker of Subs Jack Thompson wrote a post replying to it. Here I do not so much as directly address his reply (since I do not take issue with much of what he initially says), but rather lay out some thoughts on the very interesting considerations in his post. You should read those first, but I’ll also make sure this post makes sense as a self-contained one (except I am not going to re-explain the epsilon-delta definition of continuity; see my linked post what I hope is a pretty okay intuitive gloss of it).

What I claimed

I opened my post talking about real analysis and the standard definition of a continuous function, then jumped to talking about philosophy of action. I was being intentionally coy, and so I cannot complain if various people took various messages.

Basically, the main lesson I wanted to convey was that in the two cases discussed, whether an analysis recovers certain obvious pre-theoretic intuitions, by itself, holds little weight in whether the analysis is correct. I then hoped the reader internalized this message, and then approach my later claims about action without prejudiced judgment, because I think if one does so they will come away thinking I am correct and for correct reasons. The pre-theoretic definition of a continuous function is “one whose graph you can draw without picking your pencil up.” I then conveyed the unnecessary troubles mathematicians would get into if they approached conceptual analysis as Lovers of Wisdom do.

But, wait, why on Earth should we care about correct analyses? I may define words however I want, in order to express whatever truths I want; it is for the reader to judge whether my conclusions, as defined, are significant. Well, the answer is that we expect many of the fundamental concepts we use—action, knowlegde, obligation—to track something really important. Lovers of Wisdom do not spill gallons of ink writing about intuitions about when using the word “knows” is appropriate and then come up with intricate semantic theories that track usage as best they can because they are really interested in how people talk. They do so because they think the word “knows” refers to something really important and that seeing how people use the word will reveal important truths about a concept we care about independently of language. I think this assumption is correct, but also that it is a bad methodological principle for philosophers, but that is a post for another day.

So, if we have what we independently believe to be a super important concept, and it turns out that that concept can actually be defined as such-and-such, then we have a very exciting and useful conclusion.

Further details about what I did and did not intend to convey will be clear in what follows.

Jack on analysis

He begins by portraying a dialogue between two characters, Alex and Thea. Alex thinks the ε-δ definition of continuity is a good analysis of the pre-theoretic concept. This definition implies, for example, that the popcorn (Thomae’s) function is continuous at all and only irrational points. Thea disagrees, holding that ε-δ-continuity might be a useful concept, but it is nevertheless a distinct one from her pre-theoretic concept of continuity. No matter Alex’s protests, Thea takes ε-δ-continuity to be so distinct from what she calls “continuity” that proving things about ε-δ-continuity does not entail anything about continuity. In particular, she denies that the popcorn function is continuous at any point, and she does not take the usefulness of ε-δ-continuity to imply that it is the true analysis of continuity. As an analogy, just because a scientifically useful definition of “vegetable” excludes tomatoes does not mean that my concept of a vegetable excludes potatoes, or even that it ought to. Given this, Jack says the important question remaining is whether Thea should care so much about what she can draw without picking a pencil up; but that is a different matter from whether the ε-δ definition is a correct analysis of her concept.

I do not know if Jack would be suprised that I completely share his verdict about the dialogue (he did not intend Alex as a standin for myself). But, in fact, I stand by my claim that the ε-δ-definition is in fact a correct analysis of continuity, which is all I needed for my post. The important point is that whether I am right depends on whether people generally respond like Thea when they take real analysis. I expect most people who understand the ε-δ definition and then see the proof that the popcorn function is continuous at irrational points will say, “Oh, cool, now it makes sense to me that the popcorn function is continuous at irrational points.” Maybe there are some Theas out there, but that has no implications for what the correct analysis (if any) of continuity is; it merely means Thea has been using the word in a way that is internally valid but which departs from common usage.

Again, why should we care about mere empirical facts about common usage? As before, the answer is because we have good reason to expect that the word, as commonly used, tracks something important, and indirect evidence about something important is important. If I had to give a nominal definition of the common man’s concept of contunity, I would hazard it to be something like “The very important property of functions which includes things you can draw without lifting your pencil and excludes sudden jumps in otherwise continuous functions.” This, I think, is enough to single out the ε-δ definition (or at least something equivalent to it) as a correct analysis of continuity of functions from R to R, given the importance claim supposedly built into the common man’s concept. Were it the case that for most people the exclusion of the popcorn function is a non-negotiable paradigm verdict of “continuity,” then we would be imagining an alternate society¹ in which “continuity” refers to a different concept not correctly analyzed by the ε-δ definition, and their only fault would be a pragmatic one of wedding their words to a relatively useless concept.

One thing to note is that, even given everything above, the mathematician should not ultimately care about whether the ε-δ definition is an analysis of a common concept. The definition should stand or fall depending on whether it’s an analysis of some useful concept. The only thing the relationship between the definition and ordinary language determines is whether we label the definition as an analysis or a useful revision. Appeal to pretheoretic understanding is only useful insofar as you do not have the useful definition yet, and you are looking for indirect clues that will help you find it.

Why, then, do I make such a big deal about analysis, since I claim that questions regarding pretheoretic understanding will be discarded once we find a useful definition? Well, this gets back to why the usual philosophical method is defective. This method places a lot of weight on maintaining the belief that the words “knows” and “action” as ordinarily used refer to the same thing that philosophers want to analyze. So, in the actual world, where (I believe, at least) our pretheoretic notions of continuity and action are not like Thea’s and they really do track the important and independently useful thing, the way philosophers approach analysis of concepts is like a mathematician who says “The ε-δ definition is decisively wrong on the popcorn function” and then decides to throw away the ε-δ definition.

Therefore, my comparison is only apt under the assumption that most people are in fact not like Thea; this was not made explicit, and so it may have looked like I made a general claim that an analysis of an ordinary concept analyzes that very concept just because the exceptional cases are worth tolerating.

I have said that, if you conduct mathematics or philosophy properly, then you may use ordinary intuitions as a provisional guide, but then discard them once you arrive at what is an important, useful, and clearly-defined concept. I would additionally hypothesize that philosophy, in this respect, is special in a way math isn’t. In math, we are free to define whatever mathematical structures we like, but if the definitions you give do not characterize anything useful (like in real analysis) or interesting (like number theory back when it wasn’t obviously useful), no one will care. It is, I think, up to the patterns in nature (which in turn inform what patters we find interesting) to determine which mathematical definitions and theorems are “good” in this way; perhaps there is a possible world where things pop in and out of existence in a manner that makes arithmetic useless, and perhaps the inhabitants of this world would see the prime number theorem as a pointless manipulation of symbols.

In philosophy, however, I believe our main job is to apprehend concepts that are truly and utterly necessary. Just in virtue of being a rational being, I have to come to conclusions about how to act, and my motivations either will or will not motivate me conditional in finding myself with other brute desires. In the case where I am in fact motivated to act a certain way conditional on having any brute desires, I take that to be a judgment of obligation. Now, I do in fact think this view tracks ordinary usage better than any other theory, but also, even if it didn’t, I would still need to figure out what I ought to do in my sense—which, in other words, means deciding whether to act for some reasons independent of my brute desires. If there is some other sense of “ought” that aligns with what the intuitionist says, then they have any intrinsic bearing on how to live.²

1. Unless, of course, my empirical judgment about most people’s considered reactions are wrong, which it may be; what follows should make clear what I would say in that event.

2. I originally intended to respond to the rest of Jack’s post, but ultimately decided it would be unproductive to do so.

I would be remiss to not mention that his following comments made me realize I was not clear about some things, indeed some very important things. Also important is the blatant mistake I made in saying “One cannot rationally judge that an agent ought to do something they judge they cannot do.” All references in that sentence were supposed to be to the same agent, but evidently by the time I got to the word “agent” I forgot I started the sentence with “one,” leaving me with an obvious falsehood.

I will edit my intuitionism post and give him due credit, but know that what he quotes is the original. I thank Jack for his engagement with my post and for making me aware of the corrections that need to be made, and especially for causing me to think about the issues I wrote about above.