In a previous post of abstracts, I mentioned philosopher Josh Rosaler’s attempt to clarify the distinction between empirical and formal notions of “theoretical reduction”. *Reduction* is just the idea that one theory reduces to another in some limit, like Galilean kinematics reduces to special relativity in the limit of small velocities.^{a } *Formal reduction* is when this takes the form of some mathematical limiting procedure (e.g., ), whereas *empirical reduction* is an explanatory statement about observations (e.g., “special relativity can explains the empirical usefulness of Galilean kinematics”).

Rosaler’s criticism, which I mostly agree with, is that folks often conflate these two. Usually this isn’t a serious problem since the holes can be patched up on the fly by a competent physicist, but sometimes it leads to serious trouble. The most egregious case, and the one that got me interested in all this, is the quantum-classical transition, and in particular the serious insufficiency of existing limits to explain the appearance of macroscopic classicality. In particular, even though this limiting procedure recovers the classical *equations of motion*, it fails spectacularly to recover the *state space*.^{b }

In this post I’m going to comment Rosaler’s recent elaboration on this idea^{c }:

I was tempted to interpret the thesis of this essay like this:

*The only useful notion of theory reduction is necessarily intertwined with empirical facts about the domain of applicability. A strictly formal notion of reduction is intrinsically broken, and attempts to define one are forced to smuggle in implicit empirical assumptions.*

But a careful reading shows that this is not what Rosaler is claiming. A better claim, and one that is a better summary of the essay, is something like this:

*Theory reduction is a very delicate and fragile idea, often entangled with the empirical question of theory confirmation and the domains of applicability. Whenever formal reduction is used as part of an explanation of our observations — such as explaining why Galilean mechanics is so useful at speeds much smaller than , or why our quantum universe appears classical — one must very carefully check the applicability of the empirical domain being explained. In practice physicists are not careful, and have sometimes offered unsound explanations.*

The need for this discussion arises in the first place because the jump from formal reduction to empirical reduction is subtle, and the folks who need to be chastised quite literally don’t notice when they make the jump.

Some other comments:

- What this essay really needs is several examples from the literature illustrating where people make this mistake, and dissecting them. Obviously, this is very laborious, and such examples are hard to find; the reason this jump is particularly insidious is because it often doesn’t lead to problems except in a small number of cases, such as the quantum-classical transition. Examples not involving the quantum-classical transition would be especially preferable because the quantum-classical transition has all sorts of distracting baggage.
- Relatedly, I think the discussion in the paragraph beginning “A second worry…” on page 10 is correct but liable to be misinterpreted by physicists. My first (and second) instinct when reading this was “Yes, of course there are going to be some nitty gritty details when understanding the reduction between two theories with different mathematical structures (such as general relativity and Newtonian gravity). Obviously you can’t literally take the limit in . I would know what to do in all these circumstances. This philosopher just doesn’t know enough physics.” Remember, Rosaler isn’t arguing that physicists routinely derive wrong conclusions because they mindlessly apply naive mathematical limits. Rather, you just need to notice that drawing conclusions from these limits requires a physicist’s expertise and intuition, and is not included explicitly in the mathematical proof of the limit.
^{d }This framing shares much in common with what I want to argue regarding set selection: physicists say they have a mathematical theory written on paper which explains quantum phenomena, but in fact there are key steps necessary to extract predictions from the theory that are neither written on paper nor obvious to the uninitiated. Instead, these steps are buried in the brains of the physicists, in the form of intuition, and ought to be extracted and explicitly written down.

- I was unhappy with the example in the paragraph starting “As an example, consider the relationship between Newtonian mechanics and special relativity…” on page 13. Perhaps I misunderstand the point, but I think this is arguing against a straw man. Clearly, no one claims (or is misled into thinking) that finding a new theory that reduces to the existing one necessarily implies that the new theory is empirically correct. Everyone understand that the statement “low-level theory Y reduces to high-level theory X” necessarily hinges on whether the data is actually consistent with theory Y. Indeed, physicists often (and correctly and usefully) employ language like “My candidate theories Y1, Y2, and Y3 all reduce to well-accepted theory X; no more than one of them can be true, and in fact they all may be false.” Thus, using the special relativity thought example as a demonstration that reduction is an empirical (rather than formal) notion appears to be attacking a view no one has.
The second example in that section (the alpha particle), of course, I agree strongly with.

- I think it’s easy to get the impression from this paper that Rosaler’s notion of theory reduction would require the inclusion of a giant empirically collected list of all systems to which a reducing theory theory applies. (This is definitely the impression that Rob Spekkens and I got following Rosaler’s talk here at PI.) But this isn’t what Rosaler is saying or, more specifically, this is no different than the uncontroversial situation with theory
*confirmation*. More specifically, Rosaler’s notion of empirical reduction depends on the giant list of all empirical observations only in the same way that, formally, we would consider a theory to be empirically confirmed only if it is consistent with all empirical observations to date. In both cases, we do not actually survey the totality of data collected by humanity, but rather just spot check the data at key places, and use general arguments and induction to convince ourselves that the rest of the data agrees.

### Footnotes

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- Confusingly, philosophers use a reversed convention; they say that Galilean mechanics reduces to special relativity.↵
- There are multiple quantum states that have the
*same*classical analog as , and there are quantum states that have*no*classical analog as .↵ - I thank him for discussion this topic and, full disclosure, we’re drafting a paper about set selection together.↵
- Unfortunately, the only strategy I know of to catch the attention of someone thinking that is to entrap them: find an example where they think they know how to take the limit, trick them to try into applying it naively themselves, and then point out their mistake. But this is a lot harder!↵

Very good post. I am going through some of these issues as

well..