Abstract
The home of analogy is semantics; its aim is understanding and inference. An analogy—e.g. “the body politic”—grafts our understanding of a kind of thing (via our grasp of one or more truths about it) onto another kind of thing. Proponents of the “semantic conception of scientific theories” often describe scientific models in terms of purported analogies drawn between them and realities. The philosophical take on analogies from a semantic point of view—in terms of things being analogized to other things—is misguided. The value of analogies isn’t to be seen by regarding them as drawn between kinds of objects; it’s that the analogies drawn enable the invention of new hypotheses about things; they motivate moving tractable syntactic objects—sentences, and groups of them (i.e. theories)—from successful applications to the hopes of new ones. The genuine role of analogy in scientific practice is as ways of stating hypotheses: the discovery of potentially applicable and tractable scientific theories. In old-fashioned philosophical language, the drawing of analogies lives (fully) in the context of discovery, not in the context of justification. Because of this, analogies are not amenable to a systematic general theory of justification; there are only background-specific justifications of particular groups of analogies in particular situations coupled with the after-the-fact testing of one or another hypothesis (one or another hypothesis that’s been, say, creatively invented via an analogy)—a testing that’s typical of scientific hypotheses, generally.
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Notes
- 1.
Because it’s misguided in this way (and because there’s a lot to say about how it, as a result, obscures philosophical insight into scientific practice), I’ve twice argued against the “semantic conception” in previous papers: Azzouni (2014, Forthcoming-a).
- 2.
We also have iconic-looking tools, of course: diagrams and the like. These also involve “syntax,” in the sense I’m using it. Nothing I say against “models” excludes the importance or the availability of these iconic tools for scientific purposes. See Azzouni (Forthcoming-b).
- 3.
Bartha (2010, 290–291) writes: “I believe that analogical arguments framed in accordance with [my] theory have more than heuristic value: they provide a measure of justification for their conclusion.”
- 4.
These distinctions are originally due to Keynes (1921, chapter XVIII).
- 5.
He (2010, 13) does write: “Informally, the analogy mapping is extended to propositions by replacing terms pertaining to one domain with the corresponding terms that pertain to the other.”
- 6.
I’ll try to say something about this in Sect. 1.10.
- 7.
One terrible tendency, that I can’t pursue in this paper, is that focusing on examples like these as paradigmatic of scientific laws inadvertently depicts those laws (implicitly) as purely qualitative. That many of them aren’t is very important—although I won’t be able to discuss that importance in this paper.
- 8.
These ideas, exceptionlessness and necessity, are already present in Hume and are certainly full-blown in Kant. They result (I hypothesize) from centuries-long inertial effects due to thinking of reasoning as paradigmatically syllogistic. By “necessary” I mean only that laws are (intuitively) taken to sustain counterfactuals, and nonlawlike statements are taken not to. The old examples: *All the coins in my pocket are quarters; if this penny were in my pocket, therefore, it would be a quarter. All copper conducts electricity; if this wood splinter were copper, it would conduct electricity.
- 9.
It looks to me, for example, that Chomsky’s (1995) minimal theory isn’t to be characterized in terms of “laws” at all: what look like linguistic “laws” are superficial characterizations that only work up to a point. Rather, a mental “organ” is being postulated, a “thing,” as it were, with dispositional properties.
Notice also that the perspective I’m urging, if it can be made to work, erases what otherwise looks like a sharp methodological difference between physics—at least certain aspects of it (the “laws,” not the “parameters” or “standing conditions”)—and other scientific subject-areas, such as geology, or even history. That doesn’t mean that other methodological differences between these sciences may remain, e.g. applications of different sorts of mathematics.
- 10.
Hempel (1965).
- 11.
See Hawthorne (2020) on this.
- 12.
A nod to Harman (1973, chapter 8, section 4) is surely called for; so I’m so nodding. He urges the replacement of a notion of statistical inference with an inference-to-the-best-explanation approach. This is the right way to go (although I deplore his insistence on using the muddy notion of “explanation”). Now isn’t the time to get further into my agreements and disagreements with Harman on this.
- 13.
Here is Hume (1739, 136):
It may be thought, that what we learn not from one object, we can never learn from a hundred, which are all of the same kind, and are perfectly resembling in every circumstance. … From the mere repetition of any past impression, even to infinity, there never will arise any new original idea, such as that of a necessary connexion; and the number of impressions has in this case no more effect than if we confin’d ourselves to one only.
And here is Hume (1777, 37)—this is one succinct version of the point that he endeavors to get across in several ways, with several simple examples, billiard balls, changes in weather, bread, etc. in Section IV:
Now where is that process of reasoning which, from one instance, draws a conclusion, so different from that which it infers from a hundred instances that are nowise different from that single one?
That is: there is no unadorned inference to be made beyond one sighting of a black raven (there having been seventeen, a thousand …) to anything else about black ravens except of course what follows deductively from the specific number of these particular sightings.
- 14.
Bartha mistakenly assimilates the unified proof-methodology of formal logic to that of one or another set of inductive principles in one or another “inductive logic”; this goes some distance towards explaining why he thinks analogy is susceptible to a general characterization, one that goes beyond local descriptions of the values of some set of analogies (in some narrow context of application). He writes (2010, 21): “[T]here are no widely acknowledged commonsense inference rules for analogical arguments …. This contrasts sharply with the situation in deductive logic, where we have plenty of unimpeachable inference rules, and with enumerative induction, where we have candidates such as the familiar ‘straight rule.’” The context makes clear that this is a defect he intends to set right with respect to analogical arguments.
His tone later in the book is quite a bit different. Bartha (2010, 241–242) revisits a list of commonsense criteria for good analogies that he first gives in Sect. 1.6 in order to show that his “theory developed in Chaps. 4, 5, and 6 both summarizes and improves upon these criteria.” He also writes late in the book (Bartha, 2010, 236) that “my objective is to develop a model for analogical arguments at a level of detail intermediate between an elementary “commonsense” description of analogical arguments and a meticulous case study.” And he adds, one sentence later, “Inevitably, a sophisticated analysis of a particular analogical argument will be more illuminating than the criteria that I have proposed.” One can ask, therefore, what exactly his (Bartha, 2010, 2, 3) “substantive normative theory of analogical arguments” that’s “general” is distinctly offering here. I discuss that in Sect. 1.8.
In the interim, I think we can draw the conclusion that Bartha is genuinely conflicted about what the supposed generality he’s offering via his new theory of analogy comes to. This isn’t his fault: what makes a general theory good, and not speciously general, is none-too-easy to see. I discuss this a little in Sect. 1.10.
- 15.
See also Keynes (1921), especially his chapter IV, section 12, 57, on “partially follows from.”
- 16.
See the opening remarks in Henderson (2020), especially the bit on the wide spectrum of opinion about this. Russell (1945, 673) is quoted by her as expressing the view that if there is no solution to Hume’s problem, then “there is no intellectual difference between sanity and insanity.” As I’ve been indicating in this paper, this melodrama is puzzling: I’m describing the “problem” in question as a simple logical distinction, first recognized by Hume, between what must be hypothesized on the basis of “evidence” (and subsequently tested) as opposed to what follows inferentially from that evidence. Driving the philosophical melodrama, however, are misapprehensions (that I think Hume was also laboring under) about our concept of knowledge—that the word “know” requires infallibility. After all, after reading Hume for the first time, many of us sincerely say: “Oh my God, I don’t know what will happen in the next minute.” But we do know what will happen in the next minute despite “the problem of induction.” (See Azzouni, Forthcoming-d on this.)
- 17.
Those influenced by Quine in this matter, or for other reasons, think that empirical pressure can change the logic that’s useable. Some, e.g. Bueno (2020) and Beall and Restall (2006), think that contemporary science already exhibits a pluralism of logics that are applied to different scientific studies. See Azzouni (Forthcoming-c) for some discussion of this contentious and difficult issue.
- 18.
A nice illustration of this is the way that the mathematics of Hamiltonians is generalized from classical settings to quantum mechanical ones.
- 19.
I won’t develop this point here; but the ultimate problem with developing a theory of “analogy” is the same as the problem of developing a theory for “thinking outside the box,” or creatively inventing new concepts (in mathematics, or, anywhere, really). The hope, repeatedly dashed, I think, is that one thinks one can implement such a theory by introducing tools for manipulating various “contents”—the consciously-available concepts involved in analogies and elsewhere, for example. My view (to be explored at another time and place) is that this is the wrong level entirely to be operating on if one wants to explain how creativity works: we need (as it were) an explanation that operates largely at the level of neurophysiology. See Fauconnier and Turner (2002) for a vigorous and systematic attempt at a theory—of “conceptual blending”—which occurs (I claim) at the wrong level of analysis.
- 20.
Examples of recipe-methods for integrating certain restricted kinds of functions, and “tables” of integrals, were made available to students right at the beginning of their study of calculus. (I believe this is still the case.) See for example, Thomas (1972)—my undergraduate calculus textbook. Also see Gradshteyn and Ryzhik (1980). (Again, although I haven’t verified this, I imagine such lists are less valuable these days because of the availability of brute computer calculations of numerical approximations of functions implicitly defined by differential equations.)
- 21.
- 22.
This fact—all by itself—leads to the important program/scope distinction in science: that the program of a scientific theory (its empirical reach) is always much farther than its scope: what, in practice, can be done because of, among other things, the sheer intractability of the mathematics if one attempts to directly apply the theory. (“Among other things,” includes, of course, things like the sheer complexity of the empirical phenomena, as with friction.) This drives the introduction of “idealizations”—often in the form of simpler but inaccurately applicable mathematics. See Azzouni (2000, Part I, section 2) for more details.
- 23.
See, for example, Arnold (1980). It’s important to stress again that these syntactically-rich approaches to extracting information about the functions implicitly defined by such differential equations (and which involve such creative inventions of new mathematics) have been supplanted in recent years (to an increasingly large extent) by our newly-emerging capacities for brute computation.
- 24.
Semantic characterizations of logical systems are often seen as providing a kind of insight that mere axiomatizations of those systems (unaccompanied by completeness results wedding those systems to one or another “semantics”) don’t have: possible worlds, in the case of modal systems, but also the large literature on formal theories of truth that emerged after the pioneering work of Kripke (1975)—compare the articles in Martin (1970) with those in Martin (1984) to see the striking shift to semantic approaches to paradox due to Kripke. Generally overlooked is that when an unaxiomatizable collection of truths—semantic or otherwise—is so semantically characterized, the same sort of epistemic failure I’ve been describing in this section is exhibited. The semantic theory in question is incomplete; and there is no unified method for determining the truths of such a theory. Philosophers invariably fall back on their metaphysical “intuitions”—for example with respect to how similar possible worlds are to other possible worlds. This practice graphically indicates the epistemic failure in question.
- 25.
See Bartha (2010, 19) for a list of what he calls “commonsense guides.” Among them are items like, “the more the differences, the weaker the analogy” and “the greater the extent of our ignorance about the two domains, the weaker the analogy.” I might as well add here that Bartha’s quite sophisticated and technically-proficient analyses of analogies yields a more complex set of guides for numerous and open-ended subclasses of analogies, but nothing beyond that. I’ll also add that late in the book, Bartha (2010, 240–241) repeats his earlier list of commonsense guides, and then claims that (241) his “theory developed in Chaps. 4, 5, and 6 both summarizes and improves upon these criteria.” His background methodological aims seem to be “moving targets” over the course of the book.
- 26.
- 27.
- 28.
I’ll say more about this shortly, but the plausibility judgments Bartha is striving for aren’t to come in degrees—although Bartha means them to be compatible with corresponding numericalized judgments, as he endeavors to show at length in his Chap. 8. A second point is that one striking difference between ampliative reasoning (at least when it takes the form of inductive inference) and deductive reasoning is the failure of inductive inferential arguments to be conserved in light of new information, e.g. that A inductively supports B doesn’t imply that A and C inductively support B. Ampliative reasonings, that is, are sensitive to further information which can infirm the original inferences. (See Azzouni (2020, 7.4.2) for discussion.) This, all by itself, suggests that Bartha’s aim to restrict analogy-reasoning—in particular, prima facie plausibility judgments—to single analogous arguments is militantly quixotic. This disanalogy between inductive and deductive reasoning also leads directly to the (I claim evident) fact that mathematical results, barring mistakes, are historically conserved although analogy-reasoning (and ampliative reasoning generally)—all reasoning about empirical matters—isn’t. I’ll touch on this again in Sect. 1.10.
- 29.
Goodman’s grue example, alas, is intrinsically cutesy: “All emeralds are grue,” where “grue” means (say) “discovered before the year 3000 CE and is green, or discovered after the year 3000 CE and is blue”—the point being that any confirmation of “All emeralds are grue” is simultaneously a confirmation of “All emeralds are green”—at least up to the year 3000 CE. This cutesiness is (philosophically) distracting because one naturally thinks to dismiss the predicate “grue” as artificial because it’s temporally defined, or (worse) one’s tempted to introduce “natural kinds” as a solution to the problem. (See Azzouni (2017), Chap. 4, for some reasons to think this is a dead end.) The general fact that Goodman’s grue example reveals—that background nonlogical inferential tissue is always required to make something inductive follow from sets of single premises—is obscured by these distracting elements that Goodman (no doubt) was so proud of. Putting the issue my way, incidentally, assimilates Goodman’s new problem of induction to Hume’s old one. I intend to elaborate this point further at another time.
- 30.
Quine (1955, 247). I discuss his virtues vis-à-vis scientific realism, in Azzouni (2004), and use their epistemic drawbacks to motivate tracking (thick and thin epistemic access) as an empirical requirement on when terms in scientific theories can be taken to refer to realities. Van Fraassen (1980, 245) nicely describes why simplicity cannot be seen as providing a reason to think a theory exhibiting such is more likely to be true or more likely to be empirically adequate. For briefer remarks on this, see what follows.
- 31.
There is a faint analogy here to Hempel’s (1965) nonnumerial notion of confirmation. But Hempel’s notion in contrast to Bartha’s prima facie plausibility (and despite the Hempelian notion collapsing under moderate philosophical inspection) is well-defined in terms of the deductive notion of instantiation.
I should note that we never get much more than this “rough” characterization. Bartha (180), for example, writes: “In Sect. 5.2, I wrote that a mathematical conjecture is plausible, if, in light of available evidence, but short of decisive proof, it has enough support to warrant investigation ….” “Support” is ultimately characterized by the quite different and specific conditions Bartha places on source domains in his various categories of analogy; the nontheoretical touchstone for his analyses, however, remains only our intuitions about the various specific analogies he discusses in his book (e.g. Franklin’s analogy, Reid’s analogy, …).
- 32.
Largely, but not entirely. At least twice (but rarely, in any case), Bartha stresses that the prima facie plausibility of an analogy should be neither necessary nor sufficient for the success of an analogy. And he (190) gives one case where he claims that an analogy is a good one although it doesn’t succeed. But because he gives no robust characterization of prima facie plausibility, we haven’t firm grounds for distinguishing successful analogies from plausible ones. (And, indeed, my own intuitions about prima facie plausibility, I find, are extremely shaky. Hind-insight into subsequent success and failure (and surely this is true for all of us?) impacts far too strongly on which analogies I find prima facie plausible.)
- 33.
Analogies, after all, live in literature and poetry too; and all sorts of qualities make them appealing: saliences of various sorts, emotional responses. We need to know these sorts of qualities aren’t operative—tainting our intuitions about scientific analogies … but there are reasons to think these qualities are tainting our intuitions, namely: the various autobiographical stories scientists tell about how they think up analogies. (Much of that stuff, let’s admit it, is pretty laughable epistemically-speaking—even when it proves to be successful!)
- 34.
He allows, though, that if a philosopher can, in some way, forge a link from pragmatic virtues to truth then an indirect link between plausibility and truth can be established. I should add that both Bartha and Van Fraassen implicitly treat “truth” as something in the ballpark of “correspondence truth”—this is the source of the sharp distinction they both make between pragmatic values and truth. Truth, of course, need not be understood in this way. Nevertheless, the concern with how pragmatic theory-values connect to objectivity—the way things actually are—remains live no matter how we nuance “truth,” although how to characterize that objectivity becomes philosophically tangled.
- 35.
Bartha (2019) doesn’t altogether escape this problem—at least how he writes about plausibility doesn’t. Here’s an illustrative quotation (and herein the citation to Polya is transcribed as in Bartha’s original quotation):
To say that a hypothesis is plausible is to convey that it has epistemic support: we have some reason to believe it, even prior to testing. An assertion of plausibility within the context of an inquiry typically has pragmatic connotations as well: to say that a hypothesis is plausible suggests that we have some reason to investigate it further. For example, a mathematician working on a proof regards a conjecture as plausible if it “has some chances of success” (Polya, 1954 (v. 2): 148).
Notice that “some reason to believe it” is a hairbreadth away from “some reason to believe it’s true”—notice also that the pragmatic connotations Bartha speaks of are ones arising directly from epistemic considerations: something is worth investigating because it might be true. Notice, finally, that in the mathematical context that Polya is writing in, “some chances of success” can only mean: “some chances of being true.”
- 36.
This leaves out of consideration more than the analogies occurring outside these subject areas, e.g. law and philosophy. The applications of mathematics to empirical subject areas is itself often driven by perceived analogies. Bartha (2010) does discuss these—by offering yet another distinctive analysis of them—in his Sect. 6.3. I discuss his analysis subsequently.
- 37.
Bartha says the things I’ve just quoted. But then, again—late in the book—he writes (276): “…(O)ur argument is not supposed to justify all analogical arguments—only those that satisfy the requirements of our theory.” Suddenly, therefore, his theory of analogies is being offered not as a theory of all analogies (all good analogies) but only a certain here-undesignated subclass. Later still (303)—although there are hints earlier—we learn that his theory of analogy is to apply to analogies only as they occur in “normal science,” as Kuhn (1970) describes it, and in mathematics. (Of course, it’s none-too-clear that there’s anything like a sharp distinction between “normal” science and the abnormal revolutionary stuff—the problem is that the more fine-grained, close-up, sociological/historical studies of scientific developments are, the less visible any such sharp distinction in scientific development appears to be. See, e.g. the articles in Lakatos and Musgrave (1970) for discussions and arguments about this very point.)
- 38.
Does Bartha intend to credit his theory with “generality”? Most of what I’ve quoted in this paper (and quite a bit else I haven’t quoted) indicates this rather straightforwardly. On the other hand, he writes (late in the book, 277: “Our argument has a modular structure. The particular models of earlier chapters can be criticized, modified, or rejected; new models can be added.” He also says, somewhat earlier (258): “Nobody can seriously entertain the fantasy of a complete set of general heuristic principles sufficient to guide all assessments of plausibility.” Even after this concession, however, he describes his two-fundamental-principles theory as a “single method,” that “integrates fruitfulness, simplicity, coherence, and unification, rather than leaving them in tension.” Of course, any theory of analogy that was truly general and that truly justified hypotheses generated on its basis would have to integrate the various theoretical virtues governing scientific hypotheses (for otherwise it wouldn’t yield a robust distinction between good and not-so-good hypotheses). It would do so, that is, virtually by definition.
- 39.
Although, let’s be honest: very few people regard statistical correlations as “explanatory”—rather, these correlations are, if anything, seen as needing explanations.
- 40.
“Similarity” is crucial to analogies, and one might hope that it might label a source of generality—e.g. by being characterized by isomorphism, etc. But it doesn’t: Bartha (187) writes, only about the mathematical case: “Instead, we should move to a pluralistic approach that aims for a set of precise, but diverse, models of similarity.” And (236): “… different types of generalization at which analogical arguments aim shape our assessment of similarity ….”
- 41.
Bartha, however, seems to eviscerate this constraint near the end of his book, when he writes—italics his (312): A ‘Pythagorean’ analogy—a purely logical or mathematical resemblance in form, devoid of physical meaning—cannot support a novel scientific hypothesis. The analogical argument must rest upon similarities between observable effects in the two domains. Such similarities are represented in mathematical terms, but nevertheless have a physical interpretation.” I read this sentence as Bartha meaning that, of course the observable effects must themselves be accompanied by a physical interpretation—and not merely that they’re similar phenomenological appearances. This makes the remark compatible with his discussion of Pythagorean analogies in his Sect. 6.3. A lurking problem, emerging here (see what follows) is that by semantically couching analogies in terms of domains (of, presumably, actually existing objects) prevents his approach capturing the scientific use of analogies directed towards phenomena, where there needn’t be underlying metaphysical assumptions about what’s causing (or underlying) that phenomena. Bartha is forced, as he is here, to dichomize between a physical interpretation (objects in a domain) and “observable effects.” But the Pythagorean analogies usually are between patterns of regularities that two kinds of phenomena exhibit, e.g. a kind of oscillation in the data—in the instrumental measurements—that by analogy looks open to a particular application of mathematics. To be measured isn’t necessarily to be “observed”—not if instruments are used.
- 42.
Bartha (221): “A necessary condition for a composite analogical argument in the empirical sciences to be plausible is that in at least one of the component analogies, the relevant similarities that constitute the basis for the argument have known physical significance.” I’m avoiding a discussion of the wrinkles Bartha introduces via the word “composite” because the point—to be made in a moment—is that there are numerous physical “analogies” involving nothing more than purely phenomenological analogies, e.g. a kind of phenomenon that admits of a mathematical characterization that’s at work elsewhere without any indication of physical similarities, as, for example, quantum mechanical applications of the harmonic oscillator model (Cartwright, 1983, 145) “even when it is difficult to figure out exactly what is supposed to be oscillating.”
- 43.
Pretty much anyone who has discussed Steiner’s work has had to navigate his sloppiness with respect to the history he discusses. See, for example, Azzouni (2004, 175–180), Bartha (210–223) or Bueno and French (2018).
- 44.
There are, honestly, far too many examples to bother citing. Many can be found in Morrison (2015); one of the cutest she gives is this (italics mine: “data” is not, often, “observable effects”):
Nuclear models [model the atomic nucleus and its constituents] in entirely different ways, depending on the data that need explanation. However, in most cases these models go beyond mere data fitting to provide some type of dynamical account of how and why the phenomena are produced. Because there are over 30 fundamentally different models that incorporate different and contradictory assumptions about structure and dynamics, there is no way to determine which of the models can be said to even approximate the true nature of the nucleus. … Although we are able to extract information about the nuclear phenomena from these models, we have no way of assessing the epistemic status of that information over and above its success in accounting for certain types of nuclear data.
- 45.
Here I’m echoing the point made in Sect. 1.6 about differential equations, but generalizing it. Mathematics is hard; that is, it’s hard to manipulate mathematical representations to extract information that they (otherwise) imply about whatever it is they’re being applied to. Therefore, when we have a family of mathematical representations that we’ve worked on for a while, that alone justifies trying them out in a new application area. (I’m here echoing my discussion of Steiner’s work in Azzouni (2004, 175–180).) Lastly, I should note the irony that Bartha (257) offers this very motivation for hypothesis selection in another context (and for something else)—italics his: “Given a decision to work on a particular problem (and the conviction that a solution exists), we have to gamble on some hypothesis. The cost of doing nothing at all until some better idea comes along is too great.” That motivates using analogies of the thinnest sorts imaginable—if there is nothing else on the horizon. Again, this makes perfect sense once one makes central the fact that analogies are tools for stating hypotheses.
- 46.
Because he describes his “argument” in terms of two general principles and an admittedly open-ended collection of models, he somewhat obscures the fact that these models are doing the “lion’s share” of characterizing analogies, leaving only baby-kitten tasks for his general principles.
- 47.
See Azzouni (2006, chapter 6) for discussion of the striking mathematical case.
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Azzouni, J. (2022). Syntactically Recharacterizing Analogies, Assessing Theories of Assessing Analogies (And Making Some Observations About Induction Too). In: Wuppuluri, S., Grayling, A.C. (eds) Metaphors and Analogies in Sciences and Humanities. Synthese Library, vol 453. Springer, Cham. https://doi.org/10.1007/978-3-030-90688-7_1
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