Book Reviews: Richard L. Epstein, Reasoning in Science and Mathematics: Essayson Logic as The Art of Reasoning Well, Advanced Reasoning Forum,2011, 134 pp., ISBN-13: 978-0983452126, ISBN-10: 0983452121

Rush, Penelope A.

English

Akademicka Platforma Czasopism

Logic and Logical PhilosophyVolume 22 (2013), 365–371DOI: 10.12775/LLP.2013.017Book ReviewsRichard L. Epstein, Reasoning in Science and Mathematics: Essayson Logic as The Art of Reasoning Well, Advanced Reasoning Forum,2011, 134 pp., ISBN-13: 978-0983452126, ISBN-10: 0983452121.This is one of a series of small handbooks: good as practical guides whichwill no doubt help students and early career scholars navigate their wayaround notions they may encounter in applied courses. Notions suchas theory, model, explanation, prediction, falsification and assumptionare explained within the context of science and mathematics. Taken assuch practical guides, the books are informative, useful and often timesstimulating  offering incentive for further study where the complexityof the issues or concepts covered goes beyond their representation in thebooks (the endnotes serve this particular feature well, suggesting furtherreading and going into some of the missing detail and subtleties that themain text sometimes glosses).This particular work is a highly practical approach to the considera-tion of mathematics and science. Although it does describe a philosophyof these areas, it is perhaps best not viewed primarily as a philosophicalpiece but rather as a work seeking ways we can utilize theories, exper-iment and mathematics in scientific and everyday knowledge. Indeed,we are directly cautioned not to take the work as seeking to provide‘ultimate explanations’ for some of the key questions in the philosophiesof mathematics and science, but to take the work as “grounding math-ematics” with “minimal metaphysics” in order to stimulate better workin the field: i.e. “to help students grasp new concepts and make use ofthem, not only as a subject to be learned but as a tool for modellingfurther” (p. 93).© Nicolaus Copernicus University (Toruń) 2013 ISSN: 1425-3305366 Book ReviewsBy and large, the book’s perspective comes from a particular stanceon human reason, set out in more detail elsewhere, but briefly sum-marized in the opening chapters. The basic concepts laid out thereare then applied to reason about the particular domains of science andmathematics. It is probably worth noting that the view of human ‘rea-son’ outlined in the introductory background is set out along classical‘critical thinking’ lines: it comes from consideration of what underpinspractical argument skills, and is based on classical rules of inference andwell known rules and principles establishing the strength or validity ofarguments.In line with the series of which it is part, this particular work isa useful introduction to the discourse and presuppositions of scienceand mathematics and so could be particularly helpful for the studentunacquainted with what lies beneath many of the core notions in thesefields and unsure of how to approach and interpret these notions in a waythat works  i.e. the work could help to open the door to understandingthe larger themes and theories in these fields.Not only does it go a long way toward explaining and demystifyingthe scientific approach, it also provides various examples translating anduntangling the cross-currents; outright disagreements; and developmentswithin the fields, as well as within some of their different sub-fields.The use of cartoons help to facilitate the practical aims of the booknicely  highlighting assumptions we might make on visual or prima facieanalyses of data. The examples, though, are also interesting in their ownright: e.g. examples of ‘duplicable and replicable experiments’ (pp. 55–62) range over a recipe for vegetarian Chile; an ironic experiment ongrass growth; the feeding behaviour of primates; the refraction of lightrays; and the growth of living nerve cells in vitro.The ‘outside in’ view of scientific reasoning offered leads to the pro-vision of ‘rules of thumb’ again particularly applicable to the studentor novice finding their way through the field. Notable among these arethose presented at the close of the chapter on experiments: “Never trustthe first or even the first few experiments that claim to establish a signif-icant correlation” and “to reason well, imagine the possibilities” (p. 65).Wonderful advice!Indeed, the emphasis throughout the work is continually on useful-ness, both to the reader and of the theories within the fields examined.This emphasis is particularly apparent in the essay on mathematics,which begins with the premise that a “good story of mathematics” shouldBook Reviews 367not only answer the host of issues dubbed the ‘usual suspects’ in the phi-losophy of mathematics (e.g. ‘what is mathematical truth?’ and ‘what ismathematical intuition?’) but also “be consistent with how we actuallydo mathematics” and “be useful to mathematicians, leading to new andinteresting work in mathematics” (p. 68).The story then offered could thus perhaps best be understood froma largely instrumental perspective: prioritising the utility of the accountover a more traditional philosophical defence of its more speculativeaspects.The story given is primarily that of “mathematics as the art of ab-straction”. That mathematics can be understood as abstracted fromexperience is an idea that those in the field will have encountered before,but the instrumental flavour of the account here offers a fresh look at thenotion. At times the story given is reminiscent of Shapiro’s structural-ism, at times of Maddy’s ‘Second philosophy’, though the naturalismhere is more Millian than Quinean.One of the interesting ideas put forward can be seen as arising natu-rally from the introductory chapters on reason and from a key emphasisthat the work places on the similarities between science and reason. Thisidea is that mathematical abstractions are akin to scientific abstractionsand so should be seen as schematic and not the sort of thing able tobe true or false until their application is made clear. The idea, flippingHilbert’s Formalism almost on its head, deserves more attention thanI can give it here, but a small discussion follows nonetheless.The central claim is that mathematical proofs should be seen asarguments, specifically arguments that a given mathematical inferenceis valid. The sense, shared by most mathematicians and philosophersthereof, that mathematical claims are the sorts of thing which can betrue or false (if any claim can be) is accounted for by embedding math-ematical claims in their theories: the suggestion is that we imagine thateach mathematical claim has an implicit caveat; ‘2 + 2 = 4’ then reallymeans “ ‘2+2 = 4’ follows from the Peano’s axioms”. This, Epstein says,accounts for the conviction held by many that the truths of mathematicsare ‘necessary’: the necessity comes not from there being “no possibleway such [. . . ] claim[s] could be false” but from “the necessity that theclaim must follow from the assumptions of the theory” (p. 77).But the sense that the truths of mathematics are ‘really true’ is alsoaccounted for in the book by the suggestion that mathematics holds onlyin those select domains in which it applies and that it is only applied to368 Book Reviewsdomains in which it holds: the idea here is that the final claim of a proofis ‘preserved to be true’ by “applying it only in cases in which it is true”(p. 70).The two notions are in some tension with one another  to the extentthat (apparent) mathematical truth is linked both to internal mathemat-ical inference and to external applicability. But the interest here, for meanyway, is more in the central idea: that mathematical claims can bethought of as true of particular domains or of specific inferences, but notin themselves.I’m not sure the picture Epstein paints can account for interest inmathematics for its own sake and the way in which mathematics proceedsoften without regard for any of its potential applications, even appar-ently without regard for the possibility that whole swathes of it maynever have any application. Equally, mathematical truth is commonlythought of as (at least) sturdier somehow than empirical truth, not tomention as objective. The idea that mathematics is true only in so faras its inferences are internally valid and only insofar as the domainsto which it is applied are sufficiently analogous to its theories seems toundercut some of the most persistent of our convictions about the natureof mathematics: e.g. on the one hand that the assumptions from whichwe draw our inferences are themselves true and, on the other, that expe-rience cant falsify mathematics, not because we selectively ignore thoseelements of experience to which mathematics does not seem to apply, butbecause mathematics is not dependent on any empirical state of affairsor contingent truth.I personally find other accounts more appealing: e.g. Mark Steiner’sargument in his [1998] that mathematics is an anthropomorphic en-deavour whose application to science suggests a ‘user friendly’ world.Steiner’s account allows mathematical claims to have genuine truth val-ues and at the same time, provides a convincing account of the applica-bility of mathematics.Steiner also provides some novel and compelling reasons as to whywe should proceed with caution when propounding most versions of nat-uralism  specifically those versions that can fairly be described as ‘anti-anthropocentric’. That the assumptions underpinning the tale offeredin this book are naturalistic is clear throughout the text. The philos-ophy that emerges could perhaps, in the final analysis, be describedas a hybrid between empiricism and naturalism. I think it meets theessential elements of Steiner’s definition of naturalism: “naturalism de-Book Reviews 369mands that philosophy be part of, or continuous with, natural science”(Steiner [1998, p. 55] and so can fairly be described as belonging in the‘anti-anthropocentric’ camp, although I leave it to readers to judge thisfor themselves.Leaving aside these weightier philosophical issues for a moment (as,in fairness we should, when reading the book), it is worth noting againthat a key explanatory tool employed throughout is that of mathemati-cal analogy. The explanatory work to which Epstein puts this particularphenomenon or practice relies heavily on the understanding of physicalor scientific analogy outlined earlier in the text. That is, earlier in thetext, reasoning by analogy is described in tandem with abstraction fromexperience: models and theories are described as constructed in ser-vice to various practical or physical applications of analogical reasoning(models and theories are seen as good to the extent that they ‘work’ inan application of analogical reasoning to some domain or phenomena towhich they apply).The earlier work on analogies and abstraction in science is recappedin the section on mathematics: “A comparison becomes reasoning byanalogy when a claim is being argued for: on one side of the comparisonwe draw a conclusion, so on the other side we can draw a similar one [. . . ]Science sometimes proceeds by analogy. But scientists always proceedby abstracting: choosing some aspect(s) of experience to pay attentionto and claiming, perhaps implicitly, that all other aspects of experiencein these kinds of situations don’t matter. What we pay attention to givesus the constraints for saying whether a claim is true or false” (pp. 68–69).The understanding of mathematics presented in the book could, Ithink, be viewed as informed directly by these ideas. The story of math-ematics as schematic, for example, draws from the above analysis of theway in which science can proceed, beginning the particular analogy herewith: “Mathematics abstracts from experience, too, only much morethan any science” (p. 69).It is, Epstein says, when we can see “a clear path of abstraction’to mathematical theories and entities that we can make sense of theapplication of these theories and entities to experience or the ‘real world.Epstein also says that abstractions of abstractions, or analogies be-tween mathematical theories are “merely aesthetic until applications arefound” (p. 75)Returning for a minute to the weightier philosophical issues, thisseems to me not to meet some of the requirements laid out earlier:370 Book Reviewsincluding those of explaining the practice of mathematics and aidingmathematical discovery. To press the earlier point: it is hard to see howmathematicians proceed and mathematics progresses as if the theoriestherein will be true or false or able to be tested and thus established astrue or false upon discovery of an application of those theories  rather,the truth of the theory seems to rely on mathematical standards of proofalone. Applicability and truth do not seem, to me at least, to be quiteso linked in mathematics as they appear in Epstein’s description of thefield, at least not from the perspective of mathematicians themselves.Steiner’s [1998] can help clarify some the issues here too: he pointsout that often when analogies are (successfully) drawn between math-ematical theories or between a mathematical and physical theory theremay simply be no path of abstraction from any material or natural prop-erty to the mathematical theory. Indeed, in many cases, there’s another(at times quite mysterious, even magical) path leading us in quite theopposite direction: from the theory to the natural world (for just one ex-ample, see the discussion of Hilbert spaces in quantum mechanics pages36–44): indeed Steiner argues that two central types of mathematicalanalogies: ‘Pythagorean’ and ‘formalist’, are both absolutely indispensi-ble to contemporary physics and “deeply antinaturalistic” (p. 54).These observations are not meant to suggest Epstein’s story is anythe less engaging or useful to enquiring minds. It is, rather, meant asa perspective on the story he offers from metaphysics and the realm of‘ultimate explanations’.That is, from that perspective, I’m not sure Epstein’s account ofmathematics as best understood as abstractions from experience is con-vincing. There are competing theories (e.g. realism: that the structuresmathematics describes exist in their own right, as well as Steiner’s an-thropomorphism) that, to me at least, more thoroughly address some ofthe ‘usual suspects’ Epstein lines up at the beginning of the book.Nonetheless, I enjoyed reading the work, and I would recommend itto anyone with an interest in the philosophy of science or mathemat-ics, as well as to teachers and researchers interested in the question ofwhat constitutes good reasoning and critical thinking skills across theseparticular domains.Book Reviews 371References[Steiner 1998] Steiner, M., The Applicability of Mathematics as a PhilosophicalProblem, Harvard University Press, London, 1998.Penelope A RushThe University of TasmaniaSandy Bay7001, Australiaprush@westnet.com.au

Richard L. Epstein, “Reasoning in Science and Mathematics: Essays on Logic as The Art of Reasoning Well”

https://apcz.umk.pl/LLP/article/download/LLP.2013.017/781

Richard L. Epstein, “Reasoning in Science and Mathematics: Essays on Logic as The Art of Reasoning Well”

Abstract

Similar works

Full text

Available Versions

Akademicka Platforma Czasopism