Studies in Applied Philosophy, Epistemology and Rational Ethics Emiliano Ippoliti E ditor Heuristic Reasoning Studies in Applied Philosophy, Epistemology and Rational Ethics Volume 16 Series editor Lorenzo Magnani, University of Pavia, Pavia, Italy e-mail: [email protected] Editorial Board Atocha Aliseda Universidad Nacional Autónoma de México (UNAM), Coyoacan, Mexico Giuseppe Longo Centre Cavaillès, CNRS - Ecole Normale Supérieure, Paris, France Chris Sinha Lund University, Lund, Sweden Paul Thagard Waterloo University, Waterloo, ON, Canada John Woods University of British Columbia, Vancouver, BC, Canada About this Series Studies in Applied Philosophy, Epistemology and Rational Ethics (SAPERE) pub- lishes new developments and advances in all the fields of philosophy, epistemology, and ethics, bringing them together with a cluster of scientific disciplines and tech- nological outcomes: from computer science to life sciences, from economics, law, and education to engineering, logic, and mathematics, from medicine to physics, human sciences, and politics. 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Vorms, Paris, France More information about this series at http://www.springer.com/series/10087 Emiliano Ippoliti Editor Heuristic Reasoning 1 3 Editor Emiliano Ippoliti Department of Philosophy Sapienza University of Rome Rome Italy ISSN 2192-6255 ISSN 2192-6263 (electronic) ISBN 978-3-319-09158-7 ISBN 978-3-319-09159-4 (eBook) DOI 10.1007/978-3-319-09159-4 Library of Congress Control Number: 2014948769 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Contents Reasoning at the Frontier of Knowledge: Introductory Essay .......... 1 Emiliano Ippoliti Why Should the Logic of Discovery Be Revived? A Reappraisal ........ 11 Carlo Cellucci Are Heuristics Knowledge–Enhancing? Abduction, Models, and Fictions in Science .......................................... 29 Lorenzo Magnani Heuristic Appraisal at the Frontier of Research ..................... 57 Thomas Nickles Why Do Scientific Revolutions Begin? ............................. 89 Donald Gillies Withstanding Tensions: Scientific Disagreement and Epistemic Tolerance ......................................... 113 Christian Straßer, Dunja Šešelja and Jan Willem Wieland Heuristics as Methods: Validity, Reliability and Velocity .............. 147 Anna Grandori Dynamic Generation of Hypotheses: Mandelbrot, Soros and Far-from-Equilibrium ....................................... 163 Emiliano Ippoliti v Reasoning at the Frontier of Knowledge: Introductory Essay Emiliano Ippoliti The advancement of knowledge is the big goal in human understanding. To get it, we often have to push beyond the frontier of knowledge, where our understanding dissolves and where new, strange entities appear. These require bold explorations and the consequent discoveries are not idle mind games, but crucial tools for our future life. And to have a method for carrying out these explorations is essential. Tellingly, in his famous documentary Cosmos and the homonymous book, Carl Sagan spent some of his most inspired words to stress this point: In the last few millennia we have made the most astonishing and unexpected discoveries about the Cosmos and our place within it […]. They remind us that humans have evolved to wonder, that understanding is a joy, that knowledge is prerequisite to survival. I believe our future depends on how well we know this Cosmos in which we float like a mote of dust in the morning sky. Those explorations required skepticism and imagination both. Imagination will often carry us to worlds that never were. But without it, we go nowhere. Skepticism enables us to distinguish fancy from fact, to test our speculations [1, p. 7]. Sagan’s contrasting of imagination and skepticism evokes the two main roots of logic and reasoning: ampliative reasoning, heuristics and methods for discovering on one hand, and non-ampliative reasoning, deduction, and methods for justify- ing and grounding our findings on the other. From these two roots have grown, branched out and borne fruit the two main traditions in logic and philosophy of science and philosophy of mathematics in particular. These traditions have seen several conflicts during the history of western scientific and philosophical thought especially on the battleground of the role of logic, reasoning and philosophy in human understanding. The latest clash was generated by the birth of mathematical logic following Frege’s works. The battle is hardest fought between the orthodox view and the maverick view of philosophy of mathematics. E. Ippoliti (*) Sapienza University of Rome, Rome, Italy e-mail: [email protected] © Springer International Publishing Switzerland 2015 1 E. Ippoliti (ed.), Heuristic Reasoning, Studies in Applied Philosophy, Epistemology and Rational Ethics 16, DOI 10.1007/978-3-319-09159-4_1 2 E. Ippoliti The orthodox view is that philosophy is a meta-activity, a thinking about think- ing that exists to clarify concepts, remove flaws and eradicate misunderstanding. Hence, reasoning teaches us to prevent errors, and logic is its main tool. Here logic is purely deductive, that is, a closed set of sound mechanical rules. The maverick view claims that philosophy contributes to the hunt for new knowledge, by providing a logic and a method for its generation. Here logic is an open set of fallible rules for the generation of hypotheses from a set of data, and method is a framework for solving problems. A recent example of this view is Cellucci’s revised version of the analytic method (see [2]). Truth be told, there is a branch of the orthodox view that maintains that phi- losophy contributes to the advancement of knowledge, but it comes with the criti- cal thesis that deduction and axiomatization can extend our knowledge. This is a crucial point on which the maverick view challenges the orthodoxy. The mavericks think that deductive logic cannot genuinely extend our knowl- edge. They argue that no deductive rule is ampliative since the content of the con- clusion is already present in its premises. According to this view, a deduction only makes explicit the information that is implicit in its premises: a deduction allows us to unfold and rewrite the information embedded in the axioms in a way that is much more understandable and testable but, logically, it cannot extend them. Axiomatic-deductive systems establish relations of logical dependence between known findings but cannot produce new findings. Moreover, the relation between hypotheses and consequences, axioms and the- orems, is radically different in these two views. According to a very radical mav- erick view, the starting point of an enquiry is not the axioms, but the consequences and the theorems. This point is expressed nicely by Hamming, who points out that this is true even in mathematics: The idea that theorems follow from the postulates does not correspond to simple observa- tion. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid’s postulates came from the Pythagorean theorem, not the other way. For over thirty years I have been mak- ing the remark that if you came into my office and showed me a proof that Cauchy’s theo- rem was false I would be very interested, but I believe that in the final analysis we would alter the assumptions until the theorem was true. Thus there are many results in math- ematics that are independent of the assumptions and the proof [3, pp. 86–7]. The bottom line: in the hunt for new knowledge we cannot employ axioms and deduction. Axioms are the pawns, not the queens, of our understanding, and they can be sacrificed on the chessboard of knowledge. Deductions are conservative moves: they protect your pieces and strengthen your position but do not offer ways to create lines of attack to win a match. The orthodox view replies that axioms are the rough diamonds of our knowl- edge, and that deduction is the tool used to cut them. In this view the cut diamond is a new product, with new properties and relations between its parts, in other words new knowledge, so deduction is ampliative. The orthodoxy supports this claim with several arguments, such as the semi-decidability of the theories, the sur- prise of unexpected consequences, the need of new individuals in deduction, and Reasoning at the Frontier of Knowledge: Introductory Essay 3 the epistemic aspect of conclusions (see e.g. [4–6]). In a nutshell, these arguments set out to show that by deducing consequences we gain genuine new knowledge since the consequences (theorems) are to axioms as plants to their seeds (using a Fregean metaphor). The seeds in itself are not enough to obtain the plants, the truth of our postulates is not enough to foresee the truth of their consequences. We need an effort to obtain a deduction from given axioms—to choose and combine the premises in the appropriate way—in the same way we need a work to get a plant from its seeds. A plant is something new with respect to the seeds, so deductive consequences are new knowledge.1 Moreover the orthodox view states that you get new plants or new properties of a plant just working on the seeds, that is on their combination and modicfi ations. In other words, drawing deduction by relaxations, changes and combinations of axioms are ways to produce new knowledge. The maverick view, in turn, argues that these arguments miss the big point: there is no way to logically extend our knowledge by means of deductions from axioms. The axioms are the only things needed in deductions: an axiomatic-deduc- tive system is a closed world, unlike a plant that is an open-world that needs to interact with an environment to grow from seeds. Moreover, you don’t need any kind of work or effort to get deductive consequences from a set of axioms since this task can be done mechanically by the British Museum Algorithm. The real issue here is the definition of new , or novelty, that is what can be con- sidered as new knowledge. On the orthodox side, establishing new logical rela- tions between known components is regarded as new knowledge, on the maverick side only the production of an unknown component is regarded as such. The clash between mavericks and orthodoxy, not only on the issue of new knowledge, has come to various attempts of reconciliation. For instance, recently Paolo Mancosu has tried to harmonize the two views within his ‘philosophy of mathematical practice’ framework (see [9]). The above Sagan’s quote suggests a fruitful way to look at this problem, and also a way out to the clash. In effect, we need both ampliative and non-ampliative reasonings in the advancement of knowl- edge. They serve different purposes and have different roles within the same pro- cess. The ampliative reasoning offers means to produce new hypotheses capable of enlarging our understanding. The non-ampliative reasoning provides means to test and assess these hypotheses by confronting them with existing knowledge, strengthening the process of generation of hypotheses itself. In this sense, non- ampliative reasoning is useful and even necessary for the advancement of knowl- edge also from a maverick view point. Moreover, the work on axioms, from time to time, can produce new knowledge, since the relaxations or the changes of our postulate can be an effective heuristic move—even though it does not require to endorse a structuralist view. In particular working deductively from axioms is a means of control, a means of discovering errors in our postulates and knowledge, and learning from them—this is a big lesson from the history of set theory. The relation between ampliative and non-ampliative reasoning can be expressed also in terms of risk-management, that is cost-benetfi ratio. Basically, the non-ampliative 1 See [7, pp. 326–332] and [8, pp. 16–72] for a detailed discussion of this point. 4 E. Ippoliti reasoning is a risk-aversion strategy: it aims at minimizing as much as possible the possibility of doing mistakes, but in order to reach this goal it pays a cost, that is the fact that the novel epistemic gain it offers is small or negligible. The ampliative reason- ing is a risk-taking strategy: it has a potentially high cost—namely the possibility of doing bad mistakes by means of its set of fallible inferences—, which is balanced by the benetfis of deep epistemic gains. This follows from the paradox of inference, which remind us that the tension between soundness and ampliativity in our reasoning cannot be dissolved. The point is that while non-ampliative reasoning has been developed exten- sively in the history of philosophical and scientific thought, the same cannot be said for ampliative reasoning. One obvious reason for this is the intrinsic difficulty of producing risk-taking strategies, that is ways of reasoning at the frontier of knowledge and research. At this stage of knowledge, most of our tools for manag- ing knowledge and solving problems vanish: the hypotheses and concepts we rely on become more and more tentative and uncertain, our knowledge-base about the objects under investigation becomes poorer and poorer, the problem-state and problem-goal can be ill-defined, the allowed ‘moves’ on the entities of our inquiry can be unknown or only partially known, as are the constraints on them. We really have feeble light, and most of our steps are made in darkness. Ampliative reason- ing provides a way of increasing this light and so the recent resurgence of interest in it is hardly a surprise.2 This volume sets out to contribute to this increase and to offer ways of obtain the advancement of knowledge in this continually expanding land, populated by moving targets. But, in a sense, this difficulty is just the lesson from the ‘maver - icks’ tradition.3 In effect the very origin of the term ‘maverick’ recalls this point. It is an eponym that derives from the eccentric Texan rancher Samuel Maverick. One of his unusual traits was that he did not brand his cattle, and the noun ‘maverick’ was first used in 1867 to denote his unbranded cattle. Accordingly Maverick’s cows turned out to be considered as outsiders, impossible to categorise by usual labels—as they were. In ampliative reasoning this feature is amplified by the fact that, quoting Bacon [22, pp. I–CXXX], “the art of discovery may improve with discoveries” (“artem inveniendi cum inventis adolescere posse”). That is, the intrinsically dynamic nature of ampliative reasoning. In effect, on one side there is the ongoing inquiry into methods for discovering, and on the other side we have that cases of discovery can be rationally evaluated, reconstructed and offered as a means of improving the ‘method’ of discovery itself. The papers in the volume focus on a set of issues that are at the center of the development of ways of reasoning at the frontier of knowledge and of constructing ‘methods’ of discovery, such as models for revolutions and changes in paradigm, ways of treating scienticfi disagreement in a rational way—crucial when revolutions happen and strong disagreement can emerge inside the scienticfi community—, the 2 See in particular [2, 10–21]. 3 On the role of the term and concept of “mavericks” I would like to thank David Nicholson for his valuable advice.