29 of 41 people found the following review helpful:
3.0 out of 5 stars
Party-line modernism, November 25, 2009
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."
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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, February 15, 2011
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...
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