Because without Cause

If years ago Wesley Salmon tried to put ‘cause’ back into the ‘because’ (of explanation), in this book Marc Lange aims to take it out.^[1] More precisely, he sets out to show that while causal explanations are important in science, there is a sensible way of conceiving of scientific explanation in non-causal terms. He describes this non-causal class of explanations as further divided into sub-categories, of which the most important one is ‘explanation by constraint’. Only Parts 1 and 2 deal with scientific explanation per se; Part 3 discusses explanation in mathematics, and Part 4 considers these two types of explanation together. Lange draws on some of his recent articles (as acknowledged), but there is good reason to read the book even if you are familiar with the articles. They appear in various chapters, but extended and reworked. Moreover, the volume as a whole articulates a general view of non-causal scientific explanation that can’t be grasped from the articles alone.

The book has plenty to recommend it: broadness of vision and ambition—he covers a lot of ground (both scientific and philosophical)—as well as a wealth of examples (several original, all worked out in detail). However, at the end of the day (so to speak; the book is 489 pages long, and it takes many days to read), I am afraid that I finished it bothered by several persistent puzzles. I will go over a few of them below, from a longer list. Careful articulations of these points are obviously not possible here, so it as an open question as to whether they amount to (serious) flaws, or merely reflect my own misunderstanding.

Here is a brief account of what one can find in the volume. Part 1 will receive more attention, since I will only have space to comment on it; but it is the longest, comprising the first four chapters (out of eleven in total). It spells out the abovementioned type of non-causal scientific explanation—‘by constraint’. Chapter 1 identifies what Lange calls ‘distinctively mathematical’ scientific explanations (DMSEs hereafter), where the constraints are of a mathematical nature. But they need not be so, and Chapter 2 identifies other types—for example, those provided by the conservation laws and symmetry principles. They act as constraints insofar as they are more necessary than the various force laws. What puts these constraints on a higher rung on the ladder of necessity is explicated in counterfactual terms: presumably, the conservation of energy would hold even if the force laws (typically understood as physical necessary) had been different (or there had been additional force laws). Chapter 3 is mostly devoted to special relativity. It examines whether the coordinate transformations can be taken as constraints and features an analysis of Einstein’s much-discussed distinction between ‘theories of principle’ and ‘constructive theories’. The next chapter is about some rather ignored composition laws (such as the composition of forces, that is, the parallelogram law) and the claim assessed here is whether they act as constraints (as opposed to ‘coincidences’—where, again, the distinction is spelled in counterfactual terms).

Part 2 (Chapters 5 and 6) takes up non-causal scientific explanations that are not explanations by constraint. Chapter 5 deals with ‘really statistical’ explanations (in population biology), and Chapter 6 investigates ‘dimensional explanations’ in physics. Both come out in Lange’s analysis as non-causal. The very rich Part 3 (Chapters 7, 8, and 9) could count as a philosophy of mathematics book in itself. It discusses, against the background of many examples, explanation in mathematics, examining explanatory proofs (as opposed to mere derivations) of purely mathematical propositions. Here Lange offers an original account of what such an explanatory proof is, and contrasts it with the accounts of Steiner and Kitcher. Part 4 (Chapters 10 and 11) adds new thoughts about non-causal explanation in science and mathematics when considered together.

All in all, Lange aims at a mean between extremes: he doesn’t wish to offer a unique model of explanation; on the other hand, he argues that explanations have more in common than just a label. Inside these two groups (the causal and non-causal), he maintains that the similarities are genuine, so they are naturally characterized this way.

Now let me mention some concerns. First, it is not clear to me what Lange actually means by ‘causal explanation’ and, more generally, by ‘causal’. His topic is of course non-causal explanation, and this is presented (unsurprisingly) as explanation that is not causal. But, what is causal explanation, then? Here is what one finds (p. 5):

[…] I am not appealing to some account of what makes an explanation ‘causal’ that aims to fit either some pretheoretic intuitions about which explanations are ‘causal’ or some scientific practice of labeling certain explanations ‘causal’. Rather, I am trying to elaborate a notion of ‘causal’ explanation that not only motivates many philosophers to contend that all scientific explanations are causal, but also helps us to understand how scientific explanations work.

This is helpful but not enough to answer the question above. The paragraph continues as follows:

Distinctively mathemati­cal explanations are ‘non-causal’ because they do not work by supplying infor­mation about a given event’s causal history or, more broadly, about the world’s network of causal relations.

The idea of the last sentence appears again (p. 18):

Rather, what makes an explanation ‘causal’ is how it works: that it derives its explanatory power by virtue of supplying relevant infor­mation about the explanandum’s causes or, more broadly, about the world’s net­work of causal relations. In other words, in a causal explanation, an explainer’s explanatory credentials derive partly from the information it supplies regarding the world’s network of causal relations.

It is of course up to the reader to decide how illuminating this is. Recall, Lange relates his project to Salmon’s; yet Salmon spills a lot of ink trying to pin down what causation, and causal explanation, are. So, there is no comparison here with what Lange has to offer. It may well be the case that he decided that such a clarification was not needed (other philosophers’ theories of causation are in fact mentioned in the volume). But then a motivation for this decision would have been welcome in a (very long) book having ‘cause’ in its title.

I was motivated to look for a detailed answer to this question because it seems as though his examples of DMSEs, if they are not to be dismissed as pseudo-explanations (see below), could be construed as causal explanations. Also, this clarification would have been helpful for someone reflecting on an important subtlety Lange introduces, namely, that even though an explanation doesn’t ‘cite causes’ (Lange’s phrase), it may still qualify as causal. The converse also holds: some explanations, despite citing causes, are not causal—since they are not used in the explanation ‘as causes’. (What deserves more analysis is, of course, the ‘as’.) Moreover, his agreement (I take it) with Steiner (p. 407), that the law of conservation of energy is not ‘causal’, depends on the details of what one means by this.

Another concern has to do with Lange’s position on the role of mathematics in scientific explanations. This discussion was initiated by Steiner in the 1970s and reopened by several other philosophers more recently.^[2] A premise of this debate was that one should look for, and examine, explanations in which mathematics features prominently and, importantly, whose explanandum is a physical fact.^[3] For instance, the explanandum in Alan Baker’s well-known example—that ‘Cicadas’ life-cycle measured in years is prime’—is meant as a physical fact. In virtue of this, it is also (and this goes without saying) a contingent fact, since other life-cycles are possible. Initially, Lange (p. 4) seems aware of this premise and says the following:

‘Distinctively mathematical’ explanations are scientific explanations, as dis­tinct from explanations in mathematics—the subject of part III […] That is, part III is concerned with explanations in which the facts being explained (the ‘explananda’) are theorems of mathematics, whereas the explana­tions with which I will be concerned in this chapter (and in the rest of parts I and II) take as their targets various facts about the natural, spatiotemporal world.

If the explananda in the DMSEs are ‘facts about the natural, spatiotemporal world’, then, one may surmise, they are physical-contingent. And they may perhaps be physical-necessary too, like the ordinary laws of nature (assuming, as is customary, that they are necessary). Yet there are surprises waiting for us:

Ultimately, I will argue that roughly speaking, these explanations [DMSEs] explain not by describ­ing the world’s causal structure, but rather by revealing that the explanandum is necessary—in particular, more necessary than ordinary laws of nature are. (p. 9)

Let’s call the explanandum of a DMSE a ‘physical-super-necessary’ proposition, as a shortcut for ‘more necessary than an ordinary law of nature’. An illustration is MOTHER: ‘Mother cannot evenly divide twenty-three uncut strawberries to her three children’. Note that such physical-super-necessary propositions should not be confused with (logico-)mathematical ones—recall, they are meant to state ‘facts about the natural, spatiotemporal world’.

It is the modal status of the explanandum of a DMSE that raises questions about Lange’s engagement in the debate on the role of mathematics in scientific explanation, as I mentioned above. As noted, the authors in this literature were typically after explanations featuring mathematics and physical-contingent explananda. So if Lange’s DMSEs have only physical-super-necessary explananda, this means that they were not after his kind of explanations to begin with. It is thus not clear why he seems to criticizes them (Baker in particular). On the other hand, if he is not criticizing them, is then his aim just to appropriate the term ‘distinctively mathematical explanation’? Then the issue becomes purely terminological—what he means by ‘distinctively mathematical’ or ‘non-causal’ is not what these authors mean by these terms^[4]—and there is still no engagement.

But can DMSEs feature other kinds of explananda? For example, explanda such as physical-contingent or not physical-super-necessary (that is, only physical-necessary—for example, an ‘ordinary’ law of nature)? The answer is probably ‘no’, but I cannot locate a principled argument for this. Then, under this assumption, it is fair to ask what good is Lange’s concept of a DMSE (what is its relevance), since it fails to apply to almost all typical explanations in science—that is, to both explanations of physical-contingent facts and explanations of the physical-necessary (‘ordinary’) laws of nature.

Note a related problem here. Lange’s own simple double pendulum example (Section 1.4) shows that even if the explanandum is physical-super-necessary,^[5] and it has a DMSE, it may also have a causal explanation (assuming for the moment that we accept Lange’s use of this term). This amounts to acknowledging that the causal type of explanation can cover even those physical-super-necessary explananda featuring in DMSEs. But does this always happen? Although I was unable to find an answer or a discussion of this issue in the book, it seems one is needed. Because if the answer is affirmative, one’s suspicion that DMSEs may be unimportant is reinforced—especially when coupled with the idea that they may turn out to be pseudo-explanations (see below). So, if it is the case that for any DMSE there is a causal one available, a further interesting question (not addressed in the book) seems to me how DMSEs compare to causal explanations.^[6] Do DMSEs, for instance, provide more understanding? In what sense of ‘understanding’? The proposal is, it seems, that understanding comes from the ‘constraint’: ‘A distinctively mathematical explanation works instead (I will argue) roughly by showing how the fact to be explained could not have been otherwise—indeed, was inevitable to a stronger degree than could result from the action of causal powers’ (pp. 5–6). But then, one wonders, first, what happens with the non-causal explanations that are not ‘by constraint’ (in Part 2)? Second, the proposal becomes dependent on an explication of how the constraints work—and this involves evaluating counterfactuals, which leads to other difficulties, as we will see below.

Another worry is that the physical-super-necessary explananda of DMSEs are actually not ‘facts about the natural, spatiotemporal world’, as Lange presents them, but (purely) mathematical propositions. Consider MOTHER again: Although it does not look like a typical mathematical proposition, the way it is formulated may be irrelevant. After all, ‘two plus two is four’ is a mathematical proposition with no mathematical symbols in it; conversely, ‘Trump has declared bankruptcy 4 times’ is surely not a mathematical proposition. Crucial here is what the proposition says, that is, what the why-question actually asks. Now, it is not that we saw Father dividing the strawberries, so we wonder why Mother, specifically, cannot do it; it is also not the case that blackberries can be so divided, and hence we are not asking about strawberries either. Thus, the reference to mothers and strawberries (and children) seems merely a façade or superfluous, just the linguistic garb of the proposition actually expressed by MOTHER, which then is, one suspects, about numbers—‘A certain number (of uncut strawberries) cannot be divided exactly (by Mother) by a certain number’. This rendering brings MOTHER dangerously close to a mathematical proposition. So, under this hypothesis—that mathematical propositions can sometimes appear in disguise—it turns out that the explananda in Lange’s DMSEs are mathematical propositions.^[7] But they also appear in the explanans (as the ‘constraints’), and then the whole of Chapter 1 becomes very problematic: are these DMSEs circular, or pseudo-explanations?

Let me end by signalling two more issues. First, many philosophers think, and I agree, that any appeal to counterfactuals when attempting to formulate a philosophical view amounts to asking for trouble. Yet this is exactly how Lange explicates the key idea of a constraint as distinct from mere ‘coincidence’. Thus, we may be dealing with a case of obscurum per obscurius here. Moreover, Lange sometimes talks in terms of possible-world semantics for counterfactuals, in which case one may ask, in what possible world is MOTHER false?! If the answer is in none, this puts the proposition on the same footing with the typical logico-mathematical necessary propositions, which is just to be expected if MOTHER is, despite appearances, such a proposition (as speculated above).

The second issue is that although the book is about explanation, and it is customary to discuss explanation together with the notion of understanding, I cannot really say what Lange’s views on the latter are. If the relevant connection is between understanding and the counterfactually explicated constraints, then it is rather hard to fathom—counterfactuals typically muddy the water. Moreover, explanation and understanding seem equated (p. 204), which is definitely too quick. Just like ‘causal explanation’, I am afraid that ‘understanding’ remains somewhat unclear.

Despite all these reservations, my overall assessment is that this is a substantial book well worth studying. It will elicit interesting debates in the years to come.