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The book became the subject of a debate in the 1970s following [[Errett Bishop]]'s negative review (see [[criticism of non-standard analysis]] for a summary of the debate). The book was the subject of a field study (K. Sullivan, 1976) a few years after the publication of the first edition. |
The book became the subject of a debate in the 1970s following [[Errett Bishop]]'s negative review (see [[criticism of non-standard analysis]] for a summary of the debate). The book was the subject of a field study (K. Sullivan, 1976) a few years after the publication of the first edition. |
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In practice, the book is rarely used in a typical calculus classroom, dominated by the traditional approach excluding infinitesimals. |
In practice, the book is rarely used in a typical calculus classroom, dominated by the traditional approach excluding infinitesimals.<ref>Carl M, Wikipedia, 28 nov 2011</ref> |
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==Textbook== |
==Textbook== |
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In his textbook, Keisler pioneered the pedagogical technique of an infinite-magnification microscope, so as to represent graphically, distinct [[hyperreal number]]s infinitely close to each other. |
In his textbook, Keisler pioneered the pedagogical technique of an infinite-magnification microscope, so as to represent graphically, distinct [[hyperreal number]]s infinitely close to each other. |
Revision as of 20:11, 28 November 2011
Elementary Calculus: An Infinitesimal approach (the subtitle is sometimes given as An approach using infinitesimals) is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of Abraham Robinson's non-standard analysis. The book is available online[1] and is due to be re-issued by Dover in 2012.
The book became the subject of a debate in the 1970s following Errett Bishop's negative review (see criticism of non-standard analysis for a summary of the debate). The book was the subject of a field study (K. Sullivan, 1976) a few years after the publication of the first edition.
In practice, the book is rarely used in a typical calculus classroom, dominated by the traditional approach excluding infinitesimals.[2]
Textbook
In his textbook, Keisler pioneered the pedagogical technique of an infinite-magnification microscope, so as to represent graphically, distinct hyperreal numbers infinitely close to each other.
When one examines a curve, say the graph of ƒ, under a magnifying glass, its curvature decreases proportionally to the magnification power of the lens. Similarly, an infinite-magnification microscope will transform an infinitesimal arc of a graph of ƒ, into a straight line, up to an infinitesimal error (only visible by applying a higher-magnification "microscope"). The derivative of ƒ is then the (standard part of the) slope of that line. Thus the microscope is a useful device in explaining the derivative.
Examples of a real statement
To provide a freshman-level explanation of the transfer principle, Keisler first gives a few examples of real statements to which the principle applies:
- Closure law for addition: for any x and y, the sum x + y is defined.
- Commutative law for addition: x + y = y + x.
- A rule for order: if 0 < x < y then 0 < 1/y < 1/x.
- Division by zero is never allowed: x/0 is undefined.
- An algebraic identity: .
- A trigonometric identity: .
- A rule for logarithms: If x > 0 and y > 0, then .
Transfer principle
The transfer principle of non-standard analysis is a mathematical implementation of Leibniz's law of continuity: what succeeds for the finite numbers, succeeds also for the infinite numbers, and vice versa. Mathematically, this takes the following form:
- Every real statement that holds for one or more particular real functions holds for the hyperreal natural extensions of these functions.
See also
- Criticism of non-standard analysis
- Influence of non-standard analysis
- Non-standard calculus
- Increment theorem
References
- Artigue, Michèle (1994), Analysis, Advanced Mathematical Thinking (ed. David Tall), Springer-Verlag, p. 172, ISBN 0792328124 ("The non-standard analysis revival and its weak impact on education".)
- Bishop, Errett (1977), "Review: H. Jerome Keisler, Elementary calculus", Bull. Amer. Math. Soc., 83: 205–208
- Davis, Martin (1977), "Review: J. Donald Monk, Mathematical logic", Bull. Amer. Math. Soc., 83: 1007–1011
- Keisler, H. Jerome (1976), Elementary Calculus: An Approach Using Infinitesimals, Prindle Weber & Schmidt, ISBN 978-0871509116
- Keisler, H. Jerome (1976), Foundations of Infinitesimal Calculus, Prindle Weber & Schmidt, ISBN 978-0871502155, retrieved 10 jan 2007
{{citation}}
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(help) A companion to the textbook Elementary Calculus: An Approach Using Infinitesimals. - Keisler, H. Jerome (2012) [1976], Elementary Calculus: An Infinitesimal Approach (2nd ed.), New York: Dover Publications, ISBN 978-0-486-48452-5
- Nelson, Edward (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society 83(6):1165–1198.
- Schubring, Gert (2005), Conflicts Between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis in 17th–19th Century France and Germany, Springer, p. 153, ISBN 0387228365
- Sullivan, Kathleen (1976), "The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach", The American Mathematical Monthly, 83 (5), Mathematical Association of America: 370–375, doi:10.2307/2318657, JSTOR 2318657, Analysis of experiment to teach freshman calculus from Keisler's book
- Tall, David (1980), Intuitive infinitesimals in the calculus (poster) (PDF), Fourth International Congress on Mathematics Education, Berkeley
References
- ^ Elementary Calculus: An Infinitesimal Approach; On-line Edition
- ^ Carl M, Wikipedia, 28 nov 2011