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1 of 1 people found the following review helpful:
5.0 out of 5 stars
Plato's Ghost: The Modernist Transformation of Mathematics
Gray's thesis, subdued through much of the book, is that the rise of modern mathematics not only coincided with the rise of what historians call Modernism in the arts, but that mathematics in its own way shared with Modernism an analogous change in viewpoints, values, and intellectual concerns. He doesn't propose any specific influences from the arts upon...
Published 11 months ago by Sam Adams
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29 of 41 people found the following review helpful:
3.0 out of 5 stars
Party-line modernism
This book has nothing in particular to say. It fills its pages with unimaginative, thoroughly neutral, semi-encyclopaedic surveys of one branch of mathematics after another, one philosophical debate after another, and so on, while offering next to nothing by way of synthesis or interpretation. I shall criticise Gray for being rather more uncritical than...
Published on November 25, 2009 by Viktor Blasjo
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29 of 41 people found the following review helpful:
3.0 out of 5 stars
Party-line modernism,
This review is from: Plato's Ghost: The Modernist Transformation of Mathematics (Hardcover)
This book has nothing in particular to say. It fills its pages with unimaginative, thoroughly neutral, semi-encyclopaedic surveys of one branch of mathematics after another, one philosophical debate after another, and so on, while offering next to nothing by way of synthesis or interpretation.
I shall criticise Gray for being rather more uncritical than befits a historian in his acceptance of party-line modernism. The tenet of party-line modernism that I shall focus on is the myth that history shows that intuition must be abandoned since it leads to "false" results. It is an important task for historians to reject such propaganda abuses of history as the fabrications that they are; but unfortunately Gray is somewhat rubbing the back of the establishment in this case. A typical statement of the myth in question is the following passage, where Gray is supposedly quasi-paraphrasing Perron: "Spatial intuition is a very frequent source of error, especially when it is used to supplant proofs, as, for example, in proofs of the intermediate value theorem. 'Intuition is a crude instrument that lets us make out true relationships only imprecisely' (p. 204), and this is particularly so of our understanding of curves, which may fail in all sorts of ways to have the intuitive properties one suspects." (p. 275) The propaganda myth is that intuition leads one to suspect that curves should have certain properties while they really don't. Rather, the problem is that the intuitive notion of "curve" does not correspond precisely to the formal mathematical notion. So the "error" referred to above is not at all an error of intuition; it is the error of stupidly taking intuition to apply to formal objects. All of this is spelled out explicitly by Perron himself on the very page that Gray is referring to above (204). But you will have to go read the original article to find that out, for Gray omits it, thus skewing Perron's point to agree with party-line modernism. (In other contexts, however, Gray does quote people making the exact same point (almost verbatim) as Perron; namely Pierpont on p. 229 and Felix Klein on p. 197.) Gray's discussion of the Dirichlet principle is similarly skewed. Weierstass's "decisive" criticism of this principle constituted, according to Gray, "evidence, it would seem, that a mixture of physical intuition and mathematical naivety was capable of leading mathematicians astray" (p. 75). But again intuition is being blamed for something that was not its fault. In fact, the Dirichlet principle is perfectly true, and it was only by extrapolating a particular formal generalisation of it (which no one claimed was intuitively obvious) that Weierstass was able to construct a so-called "counterexample." But perhaps the clearest example of Gray's party-line tendencies is his enthusiastic approval of the ludicrous propaganda history of arch-establishmentarian axe-grinder Ernest Nagel: "Nagel argued [that] the use of duality in projective geometry plays havoc with intuition and, he argued, opens the door to purely logical reasoning. ... I think [this claim] is on the mark" (p. 19). It is baffling how Gray can accept such nonsense. The leading systematiser of duality in projective geometry was Jakob Steiner, the most intuitively inclined mathematician of all time. And Gray himself quotes Enriques as saying that "projective geometry refers to intuitive concepts, psychologically well defined" (pp. 122, 363) and Klein agreeing that it is "always intuitive" (p. 123). Another illustration of Gray's underhand attack on intuition concerns Euclidean geometry. Pasch wrote accurately that "Elementary geometry cannot only be reproached for its difficulties, but also for its incompleteness and obscurities ..." From here Gray concludes: "[Pasch's] criticism of elementary, intuitive geometry from the standpoint of late nineteenth century criteria of rigor was typical." (p. 118). Note Gray's sneaky insertion of the word "intuitive." Pasch did not use this word, and he had good reason not to. Sure enough, Euclid's Elements contains numerous flaws from a formal point of view; for example, the triangle congruence "theorems" should really be axioms and so on. But it makes no sense to blame intuition for this. It is plainly a flaw of the Elements qua formal system. Now perhaps some party-liners might object that it was intuition that tricked Euclid into making this mistake. To this I have two replies. First, I would say that we have intuitions about geometry, not about axiomatic structures. Secondly, I would ask if, in the opinion of this person, there could ever be any mistake in mathematics that he would attribute to formalism rather than intuition. Because if this is not a mistake of formalism then I do not know what such a mistake would look like, whence it would appear that one runs the risk of simply defining "intuition" as "that thing that causes all errors whatever in mathematical reasoning."
1 of 1 people found the following review helpful:
5.0 out of 5 stars
Plato's Ghost: The Modernist Transformation of Mathematics,
By Sam Adams (Minnesota. USA) - See all my reviews
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This review is from: Plato's Ghost: The Modernist Transformation of Mathematics (Hardcover)
Gray's thesis, subdued through much of the book, is that the rise of modern mathematics not only coincided with the rise of what historians call Modernism in the arts, but that mathematics in its own way shared with Modernism an analogous change in viewpoints, values, and intellectual concerns. He doesn't propose any specific influences from the arts upon mathematics or particular mathematicians, although he does briefly note influences going the other way. In philosophy, however, the stature of Immanuel Kant (1724-1804) and therefore of Kant's views on cognition and intuition in mathematics, especially in the light of the later discoveries of non-Euclidean geometry, stimulated mathematicians' thinking about mathematics as being within the purview of cognition and about mathematicians' own notions of the cognitive status and role of so-called mathematical intuition in mathematical knowledge. This scrutiny of epistemological concepts in relation to mathematics included, and indeed required, a critical examination of the pivotal notions of logic, definition, and proof. Here, the philosophical convictions of Gottfried Leibniz (1646-1716) on the nature of logic and on the relation of mathematics to logic stimulated mathematicians' thinking. The development of mathematics had reached a point where mathematicians were concerned to find the unequivocal and comprehensive epistemological basis of mathematics. That basis, if it was not found in some form external to mathematics, be it as Kant traced it or otherwise, would be found within the anatomy of mathematics itself. This left open the question of the cognitive relations of mathematics to the physical world. The pursuit of clarity and accuracy within mathematics had, independently of the concerns mentioned earlier, led to considerations of mathematics as a form of language, and soon involved the question of the cognitive relations of any specific language (upon which the language of mathematics seemed to depend) or, more abstractly even the cognitive relations of language in general, to the physical world. The linguistic distinction between syntax and semantics became especially important with the increased study of axiomatic systems, and with that distinction came, as understanding increased, the distinction between the provable and the true. The additional distinction between a consistent axiomatic system and a true axiomatic system raised further questions. Wherein, then, is found truth? Truth proposed as a relation raises questions of ontology, including the ontology of relations, and if a relation is an abstract structure, then questions are raised about the ontology of abstract structures, which abound in mathematics (the concept of structure itself being abstract). From these considerations, more questions concerning cognition and so-called mathematical intuition in mathematics arose. Mathematics, it seemed, was entangled with the very core of fundamental philosophical questions, notably prominent in Kant's philosophy, on how we know, on what we know, on what there is to know, and on whether what there is to know is all there is. The rise of modern mathematics had lead to complex questions of philosophy that mathematicians began now to debate. What was the relation of logic to mathematics and of logic to knowledge? What was the relation of language? What was the relation of experience? What was the relation of mind? With a basis of mathematics eventually established (although not proven) within set theory (an axiom system within mathematics itself) in conjunction with formalized mathematical logic, and with the heuristic success of associated formalist views of mathematics, which carefully maintain the distinction between syntax and semantics and thus between proof and truth, unanswered philosophical questions on mathematics began to seem less forceful and germane to mathematicians. Because set theory is itself a mathematical structure, a basis found within the anatomy of mathematics itself, questions concerning the ontology of mathematics, especially that of sets and natural numbers, held the attention of philosophers, whereas working mathematicians, if pressed to offer an ontological theory to outsiders, settled upon a relatively indifferent, imaginative realism. The crisis was over. [Gray is not always this specific but all of this is implicit in his discussion.] = CONTENTS = [fully expanded > contents page = a.b] * Introduction _ I.1 Opening Remarks __ I.1.1 Mathematical Modernism ____ I.1.1.1 What is in this Book ____ I.1.1.2 The Spread of Mathematical Modernism ____ I.1.1.3 A First Overview ____ I.1.1.4 Modernisms __ I.1.2 Mehrten's Moderne Sprache Mathematik __ I.1.3 Disclaimers ____ I.1.3.1 Modernity ____ I.1.3.2 What this Book is Not ____ I.1.3.3 Plato's Ghost ____ I.1.4 Acknowledgments ____ I.1.4.1 Permissions _ I.2 Some Mathematical Concepts * 1. Modernism and Mathematics _ 1.1 Modernism and Branches of Mathematics __ 1.1.1 Ontology and Epistemology __ 1.1.2 Psychology and Language _ 1.2 Changes in Philosophy __ 1.2.1 The Path Out of Kant __ 1.2.2 The Path to Logic and Logicism __ 1.2.3 Formalism __ 1.2.4 Science, Mathematics, and Philosophy _ 1.3 The Modernization of Mathematics __ 1.3.1 Experts and Audiences __ 1.3.2 Professionalization * 2. Before Modernism _ 2.1 Geometry __ 2.1.1 Projective Geometry ____ 2.1.1.1 Pole and Polar ____ 2.1.1.2 Duality __ 2.1.2 Non-Euclidean Geometry ____ 2.1.2.1 Lobachevskii ____ 2.1.2.2 Bolyai ____ 2.1.2.3 The Significance ____ 2.1.2.4 Geometry __ 2.1.3 Acceptance: Riemann and Beltrami ____ 2.1.3.1 Beltrami __ 2.1.4 Professional Aspects _ 2.2 Analysis __ 2.2.1 What to Look For in a History of Mathematical Analysis __ 2.2.2 Cauchy ____ 2.2.2.1 Cauchy's Definition of the Integral ____ 2.2.2.2 First Responses __ 2.2.3 Weierstrass __ 2.2.4 George Green and Potential Theory _ 2.3 Algebra __ 2.3.1 Algebraic Number Theory _ 2.4 Philosophy __ 2.4.1 Kant __ 2.4.2 Two Post-Kantians: Herbart and Fries ____ 2.4.2.1 Herbart ____ 2.4.2.2 Fries __ 2.4.3 Mathematicians and Scientists as Philosophers of Mathematics ____ 2.4.3.1 Grassmann ____ 2.4.3.2 Riemann __ 2.4.4 Kronecker's Foundations for Arithmetic __ 2.4.5 Helmholtz's Foundations for Arithmetic and Geometry ____ 2.4.5.1 Arithmetic ____ 2.4.5.2 Geometry __ 2.4.6 Erdmann and Tobias ____ 2.4.6.1 Erdmann ____ 2.4.6.2 Tobias _ 2.5 British Algebra and Logic __ 2.5.1 Boole __ 2.5.2 The Americans: Pierce and Ladd ____ 2.5.2.1 Algebraic Logic by 1880 _ 2.6 The Consensus in 1880 * 3. Mathematical Modernism Arrives _ 3.1 Modern Geometry: Piecemeal Abstraction __ 3.1.1 Projective Geometry: The Kleinian View __ 3.1.2 Projective Geometry: Rigor, Duality, Novel Spaces, Novel Ingredients __ 3.1.3 Non-Euclidean Geometry __ 3.1.4 The Helmholtz-Lie Space Problem ____ 3.1.4.1 Space Forms _ 3.2 Modern Analysis __ 3.2.1 What are the Real Numbers? __ 3.2.2 Cantor's Introduction of the Transfinite ____ 3.2.2.1 Ordinal Numbers ____ 3.2.2.2 Catholic Modernism ____ 3.2.2.3 Cardinal Numbers ____ 3.2.2.4 The Continuum Hypothesis __ 3.2.3 The Philosophy of Paul Du Bois-Reymond _ 3.3 Algebra __ 3.3.1 Dedekind __ 3.3.2 The Unity of Nineteenth-Century Mathematics __ 3.3.3 Kronecker _ 3.4 Modern Logic and Set Theory __ 3.4.1 Some German Philosophers __ 3.4.2 Frege ____ 3.4.2.1 Frege on Number ____ 3.4.2.2 Frege's Grundgesetze __ 3.4.3 Dedekind __ 3.4.4 Peano _ 3.5 The View from Paris and St. Louis * 4. Modernism Avowed _ 4.1 Geometry __ 4.1.1 Abstract Italian Geometry __ 4.1.2 Hilbert ____ 4.1.2.1 Straightness and Shortest Distance ____ 4.1.2.2 Poincaré ____ 4.1.2.3 Enriques __ 4.1.3 Implicit Definitions __ 4.1.4 The Nagel-Enriques Thesis __ 4.1.5 Non-Euclidean Geometry __ 4.1.1 Poincaré's Geometric Conventionalism ____ 4.1.6.1 Calinon and Lechalas _ 4.2 Philosophy and Mathematics in Germany __ 4.2.1 Geometry and Intuition ____ 4.2.1.1 Klein ____ 4.2.1.2 Hölder ____ 4.2.1.3 Borel __ 4.2.2 Hilbert, Husserl, Frege __ 4.2.3 Hilbert, Nelson, and the Neo-Friesians _ 4.3 Algebra __ 4.3.1 Group Theory __ 4.3.2 Vector Spaces _ 4.4 Modern Analysis __ 4.4.1 The French Modernists ____ 4.4.1.1 Measure Theory __ 4.4.2 Dimension __ 4.4.3 Continuous Curves __ 4.4.4 Riesz on Space and Topology __ 4.4.4 Modernism and Modern Analysis _ 4.5 Modernist Objects __ 4.5.1 Hensel's New Numbers __ 4.5.2 Knots and Topology _ 4.6 American Philosophers and Logicians __ 4.6.1 Pierce ____ 4.6.1.1 Russell versus Pierce __ 4.6.2 Royce __ 4.6.3 American Axiomatizers _ 4.7 The Paradoxes of Set Theory __ 4.7.1 Thinking about Sets __ 4.7.2 Paradox __ 4.7.3 Hilbert's First Thoughts __ 4.7.4 Zermelo and Well-Ordering __ 4.7.5 The "Five Letters" __ 4.7.6 Zermelo's Axiomatization __ 4.7.7 Poincaré: Impredicativity __ 4.7.8 The Schoenflies-Korselt Exchange _ 4.8 Anxiety __ 4.8.1 The Appreciation of Error __ 4.8.2 Anxiety: Kronecker and Enriques __ 4.8.3 Perron's Inaugural Address _ 4.9 Coming to Terms with Kant __ 4.9.1 The Leibnizian Revival ____ 4.9.1.1 Russell __ 4.9.2 Poincaré Replies __ 4.9.3 Russell and Whitehead ____ 4.9.3.1 Hausdorff __ 4.9.4 Around 1910: Weyl, Winter, Study, and Cassirer ____ 4.9.4.1 Weyl ____ 4.9.4.2 Winter ____ 4.9.4.3 Study ____ 4.9.4.4 Cassirer __ 4.9.5 Brouwer * 5. Faces of Mathematics _ 5.1 Introduction _ 5.2 Mathematics and Physics __ 5.2.1 On the Roles of Mathematics in Physics __ 5.2.2 Maxwell __ 5.2.3 Riemann __ 5.2.4 Poincaré contra Duhem ____ 5.2.4.1 Le Roy and Duhem __ 5.2.5 Hertz __ 5.2.6 Hilbert __ 5.2.7 Minkowski __ 5.2.8 Einstein _ 5.3 Measurement __ 5.3.1 Classical and Representational Theories __ 5.3.2 Poincaré __ 5.3.3 Measuring the Infintesimal ____ 5.3.3.1 Du Bois-Reymond's and Stolz's Numbers __ 5.3.4 Bettazzi ____ 5.3.4.1 The Bettazzi-Vivanti Debate ___5.3.5 Veronese ___5.3.6 Hölder __ 5.3.7 Frege __ 5.3.8 Russell: Measurement as Ordering __ 5.3.9 Campbell on Measurement in Physics _ 5.4 Popularizing Mathematics around 1900 __ 5.4.1 Introduction __ 5.4.2 Paul Carus __ 5.4.3 Poincaré __ 5.4.4 Enriques _ 5.5 Writing the History of Mathematics __ 5.5.1 History and Historians in Germany ____ 5.5.1.1 German Historians of Mathematics __ 5.5.2 History and Historians in France and Italy ____ 5.5.2.1 French Historians of Mathematics ____ 5.5.2.2 Italian Historians of Mathematics __ 5.5.3 Why the History of Mathematics was Written ____ 5.5.3.1 Non-Euclidean Geometry ____ 5.5.3.2 A Connection to Mathematical Modernism? * 6. Mathematics, Language, and Psychology _ 6.1 Languages Natural and Aritificial __ 6.1.1 National Languages in Mathematics around 1900 __ 6.1.2 An International Language __ 6.1.3 Mathematics as a Language __ 6.1.4 An Ideal Language __ 6.1.5 Nineteenth-Century Linguistics __ 6.1.6 Semantics __ 6.1.7 Hilbert and Semantics _ 6.2 Mathematical Modernism and Psychology __ 6.2.1 Poincaré __ 6.2.2 Intuition and Psychology in a German Setting __ 6.2.3 Helmholtz on Knowledge and Visual Perception __ 6.2.4 Wilhelm Wundt __ 6.2.5 Cognitive Foundations of Mathematics ____ 6.2.5.1 Wundt ____ 6.2.5.2 Lipps ____ 6.2.5.3 Santerre * 7. After the War _ 7.1 The Foundations of Mathematics __ 7.1.1 Introduction and Overview __ 7.1.2 Hilbert and Proof Theory __ 7.1.3 Brouwer and Weyl __ 7.1.4 Axiomatic Set Theory __ 7.1.5 Gödel ____ 7.1.5.1 Coda _ 7.2 Mathematics and the Mechanization of Thought __ 7.2.1 Can Computers Think? __ 7.2.2 Hilbert __ 7.2.3 Turing and the Turing Test __ 7.2.4 Von Neumann and Neural Networks _ 7.3 The Rise of Mathematical Platonism __ 7.3.1 Working Platonists? __ 7.3.2 Schlick's Anti-Platonism __ 7.3.3 Bernay's Formulation __ 7.3.4 Platonism, Nominalism, and Fictionalism __ 7.3.5 Carnap's Linguistic Frameworks __ 7.3.6 Challenges to Philosophy __ 7.3.7 Alternatives to Platonism . . . __ 7.3.8 . . . and Gödel's Platonism __ 7.3.9 Hilbert's Garden _ 7.4 Did Modernism "Win"? __ 7.4.1 Objects __ 7.4.2 Proofs __ 7.4.3 The Philosophy of Mathematics _ 7.5 The Work Is Done * Appendix: Four Theorems in Projective Geometry __ Pappus, Desargues, Uniqueness of the Fourth Harmonic Point, Pascal * Glossary * Bibliography * Index |
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Plato's Ghost: The Modernist Transformation of Mathematics by Jeremy Gray (Hardcover - September 2, 2008)
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