Mark Steiner ז”ל | The Hebrew University of Jerusalem (original) (raw)
In Memory of Professor Mark Steiner ז״ל by Mark Steiner ז”ל
This volume is dedicated to the memory of Mark Steiner, a brilliant and influential philosopher o... more This volume is dedicated to the memory of Mark Steiner, a brilliant and influential philosopher of mathematics. Steiner was an early victim of the Covid 19 pandemic that put an abrupt ending to his highly creative life. That his intense intellectual activity continued virtually to his last day is perhaps some consolation, but at the same time it intensifies the feeling of loss and untimeliness. The essays collected here (all published here for the first time) address a battery of central issues in the philosophy of mathematics. Each of the papers engages in one way or another with Steiner's work and the problems central to it. But each is no less a contribution in its own right, and this attests to the profound impact of Steiner's thought on the community of philosophers of science and mathematics. Steiner's work in the philosophy of mathematics revolved around two large themes: Platonism and Applicability. But dealing with these grand questions led him to address more specific issues, most prominently issues of mathematical explanation and of the nature of numbers. And he spent considerable effort in interpreting the related work of Hume and of Wittgenstein. Steiner's devout Platonism (nowadays "mathematical realism") underlay his first book, Mathematical Knowledge, and remained a constant factor throughout his career. This Platonism had an ontological and an epistemological face, and it came through in Steiner's persistent anti-reductionism. Ontologically, one wonders how the abstract objects and structures that constitute the subject matter of mathematics can exist? Can numbers, infinite sets, or the continuum be real? A related semantic worry is, can we give content to the notion of mathematical truth? Steiner himself held that mathematical objects do existsimpliciter. They exist, if for no other reason than that "scientific theories are committed to the existence of mathematical entities, and since we regard some of (those theories) as true, we must regard mathematical entities as existent" (1983c). In Quine's famous phrase, mathematical entities are scientifically "indispensable." Yes, these entities are abstract, and they may well be infinitary; but they are not fictions, façons de parler, or ephemeral chimera. Mathematics, for Steiner, is a science, among all the sciences that document and explore reality. vii
Jerusalem Studies in Philosophy and History of Science sets out to present state of the art resea... more Jerusalem Studies in Philosophy and History of Science sets out to present state of the art research in a variety of thematic issues related to the fields of Philosophy of Science, History of Science, and Philosophy of Language and Linguistics in their relation to science, stemming from research activities in Israel and the near region and especially the fruits of collaborations between Israeli, regional and visiting scholars.
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Articles and Reviews by Mark Steiner ז”ל
This heretofore-unpublished lecture focusses on the phenomenon of explanation by redescription in... more This heretofore-unpublished lecture focusses on the phenomenon of explanation by redescription in mathematics. In such cases, the explanandum is fitted into a new mathematical framework from which it can be easily derived or even seen to be self-evident. Keywords Mark Steiner • Explanation • Explaining away in physics • Explaining away in mathematics • Fitness • Redescription Editors' Introduction The following text is based on the reading version of Steiner's lecture, "Explaining and Explaining Away in Mathematics: the Role of Fitness". The lecture was delivered on March 11, 2014 at the "Workshop on Beauty and Explanation in Mathematics" organized by Professor Manya Raman Sundstrom at Umeå University in Sweden. Professor Sundstrom is referred to in the text as Manya Raman and Manya. We have made the following editorial changes: [1.] We corrected obvious typographical missteps. [2.] We have made some clarificatory emendations. These are enclosed in square brackets. [3.] While mostly using the figures used by Steiner in his PowerPoint Presentation, we have deleted repetition or redundant ones that were added in the talk for emphasis. [4.] For copyright reasons we have substituted the original Figs. 7 and 8 with figures drawn by Moshe Shapiro.
This chapter is about imagination and its historical role in the philosophy of mathematics of the... more This chapter is about imagination and its historical role in the philosophy of mathematics of the eighteenth-century Scottish philosopher David Hume, and the great twelfth-century Jewish philosopher and legalist Moses Maimonides (1138-1204). We will demonstrate some surprising connections between philosophers who would seem to have had nothing in common. Nor should we be surprised to discover the continuing relevance of this subject in recent philosophy. Accordingly, we shall conclude the chapter with a short discussion of the work of a contemporary philosopher of mathematics, Charles Parsons, who stresses the role of the imagination as providing a foundation for mathematical intuition.¹ Maimonides, in his Eight Chapters, defines the imagination as "the power that preserves the impressions of sensibly perceived objects after they vanish from the immediacy of the senses that perceived them. Some impressions are combined with others, and some are separated from others" (Maimonides 1983: 63). He concludes that the imagination can present what is impossible to perceive. David Hume's definition of the imagination is not much different,² though his epistemological attitude towards the imagination is radically different from that of Maimonides. As we will see shortly, Hume holds that what ¹ In fact the present chapter originated in a talk I gave at a conference in honor of Parsons, at the Van Leer Jerusalem Institute, January 2014. ² "WE find by experience, that when any impression has been present with the mind, it again makes its appearance there as an idea; and this it may do after two different ways: either when in its new appearance it retains a considerable degree of its first vivacity, and is somewhat intermediate betwixt an impression and an idea; or when it entirely loses that vivacity, and is a perfect idea. The faculty, by which we repeat our impressions in the first manner, is called the MEMORY, and the other the IMAGINATION" (Treatise 1.1.3; SBN 8ff.). References to the Treatise are to Hume, A Treatise of Human Nature, ed. Norton and Norton, hereafter cited in the text as Treatise, followed by Book, part, and section number, and to Hume, A Treatise of Human Nature, ed. Selby-Bigge, rev. by Nidditch, cited in the text as "SBN" followed by the page number.
A Note on Maimonides and Al-Ghazali, Leibniz and Clarke Although it is regarded as highly probabl... more A Note on Maimonides and Al-Ghazali, Leibniz and Clarke Although it is regarded as highly probable 1 that Maimonides read at least the Tahafut of Al-Ghazali, 2 it is not easy to find a "smoking gun." 3 Maimonides does not mention Al-Ghazali by name-but this means little, since he doesn't even mention R. Saadia Gaon by name (in the Guide), nor does he mention the names of any of the Muslim theologians, the "mutakallimun," a discussion of whose doctrines was supposed to be, and is, one of the aims of the Guide. 4 Maimonides' description of the Mutakallimun, Guide 1:71-74, is contradicted by a number of doctrines of Al-Ghazali. For example, the mature Al-Ghazali expresses skepticism regarding the doctrine of atomism, 5 which was once the dominant doctrine of the Mutakallimun.
The 12 paragraphs or halakhot of Maimonides' first chapter of Yesodei Hatorah are an astonishing ... more The 12 paragraphs or halakhot of Maimonides' first chapter of Yesodei Hatorah are an astonishing tour de force of law, philosophical exposition, Hebrew style, and literary excellence. Maimonides sets forth what he sees as the basic principles of Jewish belief (or "knowledge" as he puts it): the existence of God (1-2), the necessary existence of God (3), 1 His immateriality (7), 27
I use the following abbreviations: PI = Philosophical investigations (Wittgenstein 2009), RFM = R... more I use the following abbreviations: PI = Philosophical investigations (Wittgenstein 2009), RFM = Remarks on the foundations of mathematics (Wittgenstein 1978), LFM = Wittgenstein's lectures on the foundations of mathematics, Cambridge, 1939 (Wittgenstein 1976), PG = Philosophical grammar (Wittgenstein and Rhees 1974).)
During the course of about ten years, Wittgenstein revised some of his most basic views in philos... more During the course of about ten years, Wittgenstein revised some of his most basic views in philosophy of mathematics, for example that a mathematical theorem can have only one proof. This essay argues that these changes are rooted in his growing belief that mathematical theorems are 'internally' connected to their canonical applications, i.e., that mathematical theorems are 'hardened' empirical regularities, upon which the former are supervenient. The central role Wittgenstein increasingly assigns to empirical regularities had profound implications for all of his later philosophy; some of these implications (particularly to rule following) are addressed in the essay.
This volume is dedicated to the memory of Mark Steiner, a brilliant and influential philosopher o... more This volume is dedicated to the memory of Mark Steiner, a brilliant and influential philosopher of mathematics. Steiner was an early victim of the Covid 19 pandemic that put an abrupt ending to his highly creative life. That his intense intellectual activity continued virtually to his last day is perhaps some consolation, but at the same time it intensifies the feeling of loss and untimeliness. The essays collected here (all published here for the first time) address a battery of central issues in the philosophy of mathematics. Each of the papers engages in one way or another with Steiner's work and the problems central to it. But each is no less a contribution in its own right, and this attests to the profound impact of Steiner's thought on the community of philosophers of science and mathematics. Steiner's work in the philosophy of mathematics revolved around two large themes: Platonism and Applicability. But dealing with these grand questions led him to address more specific issues, most prominently issues of mathematical explanation and of the nature of numbers. And he spent considerable effort in interpreting the related work of Hume and of Wittgenstein. Steiner's devout Platonism (nowadays "mathematical realism") underlay his first book, Mathematical Knowledge, and remained a constant factor throughout his career. This Platonism had an ontological and an epistemological face, and it came through in Steiner's persistent anti-reductionism. Ontologically, one wonders how the abstract objects and structures that constitute the subject matter of mathematics can exist? Can numbers, infinite sets, or the continuum be real? A related semantic worry is, can we give content to the notion of mathematical truth? Steiner himself held that mathematical objects do existsimpliciter. They exist, if for no other reason than that "scientific theories are committed to the existence of mathematical entities, and since we regard some of (those theories) as true, we must regard mathematical entities as existent" (1983c). In Quine's famous phrase, mathematical entities are scientifically "indispensable." Yes, these entities are abstract, and they may well be infinitary; but they are not fictions, façons de parler, or ephemeral chimera. Mathematics, for Steiner, is a science, among all the sciences that document and explore reality. vii
Jerusalem Studies in Philosophy and History of Science sets out to present state of the art resea... more Jerusalem Studies in Philosophy and History of Science sets out to present state of the art research in a variety of thematic issues related to the fields of Philosophy of Science, History of Science, and Philosophy of Language and Linguistics in their relation to science, stemming from research activities in Israel and the near region and especially the fruits of collaborations between Israeli, regional and visiting scholars.
This heretofore-unpublished lecture focusses on the phenomenon of explanation by redescription in... more This heretofore-unpublished lecture focusses on the phenomenon of explanation by redescription in mathematics. In such cases, the explanandum is fitted into a new mathematical framework from which it can be easily derived or even seen to be self-evident. Keywords Mark Steiner • Explanation • Explaining away in physics • Explaining away in mathematics • Fitness • Redescription Editors' Introduction The following text is based on the reading version of Steiner's lecture, "Explaining and Explaining Away in Mathematics: the Role of Fitness". The lecture was delivered on March 11, 2014 at the "Workshop on Beauty and Explanation in Mathematics" organized by Professor Manya Raman Sundstrom at Umeå University in Sweden. Professor Sundstrom is referred to in the text as Manya Raman and Manya. We have made the following editorial changes: [1.] We corrected obvious typographical missteps. [2.] We have made some clarificatory emendations. These are enclosed in square brackets. [3.] While mostly using the figures used by Steiner in his PowerPoint Presentation, we have deleted repetition or redundant ones that were added in the talk for emphasis. [4.] For copyright reasons we have substituted the original Figs. 7 and 8 with figures drawn by Moshe Shapiro.
This chapter is about imagination and its historical role in the philosophy of mathematics of the... more This chapter is about imagination and its historical role in the philosophy of mathematics of the eighteenth-century Scottish philosopher David Hume, and the great twelfth-century Jewish philosopher and legalist Moses Maimonides (1138-1204). We will demonstrate some surprising connections between philosophers who would seem to have had nothing in common. Nor should we be surprised to discover the continuing relevance of this subject in recent philosophy. Accordingly, we shall conclude the chapter with a short discussion of the work of a contemporary philosopher of mathematics, Charles Parsons, who stresses the role of the imagination as providing a foundation for mathematical intuition.¹ Maimonides, in his Eight Chapters, defines the imagination as "the power that preserves the impressions of sensibly perceived objects after they vanish from the immediacy of the senses that perceived them. Some impressions are combined with others, and some are separated from others" (Maimonides 1983: 63). He concludes that the imagination can present what is impossible to perceive. David Hume's definition of the imagination is not much different,² though his epistemological attitude towards the imagination is radically different from that of Maimonides. As we will see shortly, Hume holds that what ¹ In fact the present chapter originated in a talk I gave at a conference in honor of Parsons, at the Van Leer Jerusalem Institute, January 2014. ² "WE find by experience, that when any impression has been present with the mind, it again makes its appearance there as an idea; and this it may do after two different ways: either when in its new appearance it retains a considerable degree of its first vivacity, and is somewhat intermediate betwixt an impression and an idea; or when it entirely loses that vivacity, and is a perfect idea. The faculty, by which we repeat our impressions in the first manner, is called the MEMORY, and the other the IMAGINATION" (Treatise 1.1.3; SBN 8ff.). References to the Treatise are to Hume, A Treatise of Human Nature, ed. Norton and Norton, hereafter cited in the text as Treatise, followed by Book, part, and section number, and to Hume, A Treatise of Human Nature, ed. Selby-Bigge, rev. by Nidditch, cited in the text as "SBN" followed by the page number.
A Note on Maimonides and Al-Ghazali, Leibniz and Clarke Although it is regarded as highly probabl... more A Note on Maimonides and Al-Ghazali, Leibniz and Clarke Although it is regarded as highly probable 1 that Maimonides read at least the Tahafut of Al-Ghazali, 2 it is not easy to find a "smoking gun." 3 Maimonides does not mention Al-Ghazali by name-but this means little, since he doesn't even mention R. Saadia Gaon by name (in the Guide), nor does he mention the names of any of the Muslim theologians, the "mutakallimun," a discussion of whose doctrines was supposed to be, and is, one of the aims of the Guide. 4 Maimonides' description of the Mutakallimun, Guide 1:71-74, is contradicted by a number of doctrines of Al-Ghazali. For example, the mature Al-Ghazali expresses skepticism regarding the doctrine of atomism, 5 which was once the dominant doctrine of the Mutakallimun.
The 12 paragraphs or halakhot of Maimonides' first chapter of Yesodei Hatorah are an astonishing ... more The 12 paragraphs or halakhot of Maimonides' first chapter of Yesodei Hatorah are an astonishing tour de force of law, philosophical exposition, Hebrew style, and literary excellence. Maimonides sets forth what he sees as the basic principles of Jewish belief (or "knowledge" as he puts it): the existence of God (1-2), the necessary existence of God (3), 1 His immateriality (7), 27
I use the following abbreviations: PI = Philosophical investigations (Wittgenstein 2009), RFM = R... more I use the following abbreviations: PI = Philosophical investigations (Wittgenstein 2009), RFM = Remarks on the foundations of mathematics (Wittgenstein 1978), LFM = Wittgenstein's lectures on the foundations of mathematics, Cambridge, 1939 (Wittgenstein 1976), PG = Philosophical grammar (Wittgenstein and Rhees 1974).)
During the course of about ten years, Wittgenstein revised some of his most basic views in philos... more During the course of about ten years, Wittgenstein revised some of his most basic views in philosophy of mathematics, for example that a mathematical theorem can have only one proof. This essay argues that these changes are rooted in his growing belief that mathematical theorems are 'internally' connected to their canonical applications, i.e., that mathematical theorems are 'hardened' empirical regularities, upon which the former are supervenient. The central role Wittgenstein increasingly assigns to empirical regularities had profound implications for all of his later philosophy; some of these implications (particularly to rule following) are addressed in the essay.
In this chapter, Mark Steiner first describes what makes the applicability of mathematics a thorn... more In this chapter, Mark Steiner first describes what makes the applicability of mathematics a thorny philosophical problem-and explains that it is really two prob/ems, one that the philosophers have been discussing and a quite different one that physicists have been asking. He then concentrates on one aspect of the problem, which at first seems quite unproblematic, the question of addition. It leads to a better understanding of what Wigner might have meant when he asked, many years ago, about the "Unreasonable Effectiveness of Mathematics in the Natural Sciences." Mark Steiner is a Professor of Philosophy in the Faculty of Humanities at the Hebrew University of Jerusalem (socrates.huji.ac.ilIProf-A4ark-Steiner.htm). His interests include philosophy of mathematics, philosophy of science, Wittgenstein, and Hume. He has worked on philosophical issues in contemporary mathematics education. In the first paragraph of this chapter, he has written "to the extent one can talk about a 'consensus' of the philosophical community, I'm not in it." This is because he has consistently worked, independently of the trends in the philosophy of mathematics, on issues involving how mathematics is actually practiced. His work has several times been far ahead of that of his contemporaries. His book, Mathematical Knowledge, published in 1975, was one of the first works to say that none of the traditionalfoundational schools-logicism, formalism, intuitionismdescribe how mathematicians come to have mathematical knowledge. Mathematical knowledge cannot be identified with formal proofs, as mathematicians very rarely give formal proofs. Also, some informal arguments have actually produced mathematical certainty, or near certainty, as in some of the work of Euler. Steiner appears to have been the first philosopher to recognize this. His essay on explanatory versus non-explanatory proofs from thirty years ago has now begun to motivate studies on explanation in the mathematicalfield. His second book, The Application of Mathematics as a Philosophical Problem, published in 1998, makes a very important distinction 15. Mathematics Ayy{jecf: The Case if Adcfitiol1
We have before us a fascinating document, an original philosophical work written in the Yiddish l... more We have before us a fascinating document, an original philosophical work written in the Yiddish language by Rabbi Reuven Agushewitz (1897-1950). By the word "original" I mean that this book contains philosophical arguments and ideas that may never have appeared in print before. I mean that the book is not just an account of other philosophers' thought (of that genre there are many books and articles in Yiddish), but is meant as a contribution by the author. It is also a polemical work, a sustained attack on philosophical materialism, ancient and modem, as well as an attempt to show that religion in general, and Judaism in particular, are in complete harmony with the scientific world view (when both science and Judaism are properly understood). And it was written by an ordained rabbi with no degrees in philosophy, in fact who may never have studied even a day in any university (cf. below). The interest in this book is, therefore, threefold: we have here (a) a work containing original philosophical material; (b) written in Yiddish; (c) by an autodidact rabbi who emigrated from Lithuania to the U. S. All in all, we have before us a unique work; there is, to my knowledge, nothing like it. Of course, there were Eastern European Jews who became philosophers (Solomon Maimon, Morris Raphael Cohen), but they abandoned Orthodox Judaism. There were Eastern European Jews who studied philosophy without abandoning Judaism, such as, of course, Rabbi J. B. Soloveitchik, who knew Agushewitz well and wrote an approbation for his Talmudic novellae on Bava Kamma. As it
It is a pleasure to acknowledge some intellectual debts. I should like first to thank Paul Benace... more It is a pleasure to acknowledge some intellectual debts. I should like first to thank Paul Benacerraf for letting me have the benefit of his keen critical sense throughout the preparation of this manuscript. Working under his guidance was a profound learning experience for me, in both how to think and how to write. How much I benefited is to be found in the difference between this work and earlier versions. Next, my thanks to Richard Grandy, whose influence is felt on every page of what follows. I have gained from Grandy's thoughts and from his pithy criticism of my own. I have discussed the topics of my book with a number of philosophers, particularly Parsons. Credit has been given where their ideas have been used. Thanks, finally, are due Max Black and the mysterious logicist who read the manuscript for Cornell University Press and made valuable criticisms. Much of my research was done in Jerusalem during the year 7