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⚫ | {{Use dmy dates|date=October 2020}} | ||
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⚫ | '''''Elementary Calculus: An Infinitesimal approach''''' is a textbook by ]. The subtitle alludes to the ] numbers of the ] system of ] and is sometimes given as '''''An approach using infinitesimals |
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| name = Elementary Calculus: An Infinitesimal Approach | |||
| image = File:Elementary_Calculus,_An_Infinitesimal_Approach.png | |||
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| caption = Second edition | |||
| author = ] | |||
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| language = English | |||
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| subject = ] | |||
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| publisher = Dover | |||
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| pub_date = 1976 | |||
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⚫ | '''''Elementary Calculus: An Infinitesimal approach''''' is a textbook by ]. The subtitle alludes to the ] numbers of the ] system of ] and is sometimes given as '''''An approach using infinitesimals'''''. The book is available freely online and is currently published by Dover.{{sfn|Keisler|2011}} | ||
==Textbook== | ==Textbook== | ||
Keisler's textbook is based on Robinson's construction of the ]. Keisler also published a companion book, ''Foundations of Infinitesimal Calculus'', for instructors which covers the foundational material in more depth. | Keisler's textbook is based on Robinson's construction of the ]. Keisler also published a companion book, ''Foundations of Infinitesimal Calculus'', for instructors, which covers the foundational material in more depth. | ||
Keisler defines all basic notions of the calculus such as continuity, derivative, and integral using infinitesimals. The usual definitions in terms of |
Keisler defines all basic notions of the calculus such as ], ], and ] using infinitesimals. The usual definitions in terms of ε–δ techniques are provided at the end of Chapter 5 to enable a transition to a standard sequence. | ||
In his textbook, Keisler used the pedagogical technique of an infinite-magnification microscope, so as to represent graphically, distinct ]s infinitely close to each other. Similarly, an infinite-resolution telescope is used to represent infinite numbers. | In his textbook, Keisler used the pedagogical technique of an infinite-magnification microscope, so as to represent graphically, distinct ]s infinitely close to each other. Similarly, an infinite-resolution telescope is used to represent infinite numbers. | ||
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==Reception== | ==Reception== | ||
⚫ | The book was first reviewed by ], noted for his work in constructive mathematics. Bishop's review was harshly critical; see ]. Shortly after, ] and Hausner published a detailed favorable review, as did ] and ].{{sfn|Davis|Hausner|1978}}{{sfn|Blass|1978}}{{sfn|Madison|Stroyan|1977}} Keisler's student K. Sullivan,<ref>{{Cite web |url=http://www.math.wisc.edu/oldhome/directories/alumni/1974.htm |title=UW Math PhD Alumni (1974) |access-date=29 November 2011 |archive-url=https://web.archive.org/web/20120607220243/http://www.math.wisc.edu/oldhome/directories/alumni/1974.htm |archive-date=7 June 2012 |url-status=dead }}</ref> as part of her PhD thesis, performed a controlled experiment involving 5 schools, which found ''Elementary Calculus'' to have advantages over the standard method of teaching calculus.{{sfn|Keisler|2011}}{{sfn|Sullivan|1976}} Despite the benefits described by Sullivan, the vast majority of mathematicians have not adopted infinitesimal methods in their teaching.{{sfn|Tall|1980}} Recently, Katz & Katz{{sfn|Katz|Katz|2010}} give a positive account of a calculus course based on Keisler's book. O'Donovan also described his experience teaching calculus using infinitesimals. His initial point of view was positive, {{sfn|O'Donovan|Kimber|2006}} but later he found pedagogical difficulties with the approach to nonstandard calculus taken by this text and others.{{sfn|O'Donovan|2007}} | ||
⚫ | G. R. Blackley remarked in a letter to Prindle, Weber & Schmidt, concerning ''Elementary Calculus: An Approach Using Infinitesimals'', "Such problems as might arise with the book will be political. It is revolutionary. Revolutions are seldom welcomed by the established party, although revolutionaries often are." |
||
⚫ | G. R. Blackley remarked in a letter to Prindle, Weber & Schmidt, concerning ''Elementary Calculus: An Approach Using Infinitesimals'', "Such problems as might arise with the book will be political. It is revolutionary. Revolutions are seldom welcomed by the established party, although revolutionaries often are."{{sfn|Sullivan|1976}} | ||
The book was reviewed by ], noted for his work in constructive mathematics. Bishop's review was vitriolic<ref>] (1996) Arguments, logic and proof: mathematics, logic and the infinite. History of mathematics and education: ideas and experiences (Essen, 1992), 113–148, Stud. Wiss. Soz. Bildungsgesch. Math., 11, Vandenhoeck & Ruprecht, G"ottingen.</ref> and was the subject of a recent dramatic skit;<ref>{{citation | |||
| last1 = Kanovei | first1 = Vladimir | |||
| author1-link = Vladimir Kanovei | |||
| last2 = Katz | first2 = Karin U. | |||
| last3 = Katz | first3 = Mikhail G. | |||
| author3-link = Mikhail Katz | |||
| last4 = Schaps | first4 = Mary | |||
| author4-link = Mary Schaps | |||
| doi = 10.1007/s10699-013-9340-0 | |||
| issue = | |||
| journal = ] | |||
⚫ | |||
| title = Proofs and Retributions, Or: Why Sarah Can't ''Take'' Limits | |||
| volume = | |||
⚫ | |||
Hrbacek writes that the definitions of ''continuity'', ''derivative'', and ''integral'' implicitly must be grounded in the |
Hrbacek writes that the definitions of ''continuity'', ''derivative'', and ''integral'' implicitly must be grounded in the ε–δ method in Robinson's theoretical framework, in order to extend definitions to include nonstandard values of the inputs, claiming that the hope that nonstandard calculus could be done without ε–δ methods could not be realized in full.{{sfn|Hrbacek|2007}} Błaszczyk et al. detail the usefulness of ] in developing a transparent definition of ], and characterize Hrbacek's criticism as a "dubious lament".<ref>{{citation | ||
| last1 = Błaszczyk | first1 = Piotr | | last1 = Błaszczyk | first1 = Piotr | ||
| author1-link = | |||
| last2 = Katz | first2 = Mikhail | | last2 = Katz | first2 = Mikhail | ||
| author2-link = Mikhail Katz | | author2-link = Mikhail Katz | ||
| last3 = Sherry | first3 = David | | last3 = Sherry | first3 = David | ||
| author3-link = | |||
| arxiv = 1202.4153 | | arxiv = 1202.4153 | ||
| doi = 10.1007/s10699-012-9285-8 | | doi = 10.1007/s10699-012-9285-8 | ||
| issue = | |||
| journal = ] | | journal = ] | ||
| pages = | | pages = 43–74 | ||
| title = Ten misconceptions from the history of analysis and their debunking | | title = Ten misconceptions from the history of analysis and their debunking | ||
| volume = | | volume = 18 | ||
| year = 2012 |
| year = 2012| s2cid = 119134151 | ||
}}</ref> | |||
==Transfer principle== | ==Transfer principle== | ||
Between the first and second edition of the ''Elementary Calculus'', much of the theoretical material that was in the first chapter was moved to the epilogue at the end of the book, including the theoretical groundwork of |
Between the first and second edition of the ''Elementary Calculus'', much of the theoretical material that was in the first chapter was moved to the epilogue at the end of the book, including the theoretical groundwork of nonstandard analysis. | ||
In the second edition Keisler introduces the extension principle and the transfer principle in the following form: | In the second edition Keisler introduces the extension principle and the transfer principle in the following form: | ||
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== See also == | == See also == | ||
*] | *] | ||
*] | *] | ||
* ] | * ] | ||
* ] | * ] | ||
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==References== | ==References== | ||
*{{citation|last=Bishop|first=Errett| |
*{{citation|last=Bishop|first=Errett|author-link=Errett Bishop|title=Review: H. Jerome Keisler, Elementary calculus|journal=Bull. Amer. Math. Soc.|url=http://projecteuclid.org/euclid.bams/1183538669|volume= 83|year=1977|pages= 205–208|doi=10.1090/s0002-9904-1977-14264-x|doi-access=free}} | ||
*{{citation|first1=Andreas|last1=Blass | *{{citation|first1=Andreas|last1=Blass | ||
|author1-link = Andreas Blass | |author1-link = Andreas Blass | ||
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|number=1 | |number=1 | ||
|year=1978 | |year=1978 | ||
|pages=34–41|url= |
|pages=34–41|url=https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978-14401-2/home.html |doi=10.1090/S0002-9904-1978-14401-2|doi-access=free | ||
}} | |||
::Blass writes: "I suspect that many mathematicians harbor, somewhere in the back of their minds, the formula <math>\int \sqrt{(dx)^2 + (dy)^2}</math> for arc length (and quickly factor out ''dx'' before writing it down)" (p. 35). | ::Blass writes: "I suspect that many mathematicians harbor, somewhere in the back of their minds, the formula <math>\int \sqrt{(dx)^2 + (dy)^2}</math> for arc length (and quickly factor out ''dx'' before writing it down)" (p. 35). | ||
::"Often, as in the examples above, the nonstandard definition of a concept is simpler than the standard definition (both intuitively simpler and simpler in a technical sense, such as quantifiers over lower types or fewer alternations of quantifiers)" (p. 37). | ::"Often, as in the examples above, the nonstandard definition of a concept is simpler than the standard definition (both intuitively simpler and simpler in a technical sense, such as quantifiers over lower types or fewer alternations of quantifiers)" (p. 37). | ||
::"The relative simplicity of the nonstandard definitions of some concepts of elementary analysis suggests a pedagogical application in freshman calculus. One could make use of the students' intuitive ideas about infinitesimals (which are usually very vague, but so are their ideas about real numbers) to develop calculus on a nonstandard basis" (p. 38). | ::"The relative simplicity of the nonstandard definitions of some concepts of elementary analysis suggests a pedagogical application in freshman calculus. One could make use of the students' intuitive ideas about infinitesimals (which are usually very vague, but so are their ideas about real numbers) to develop calculus on a nonstandard basis" (p. 38). | ||
*{{citation|title= Review: J. Donald Monk, Mathematical logic|first=Martin|last= Davis|journal=Bull. Amer. Math. Soc.| volume =83|year=1977|pages =1007–1011|url=http://projecteuclid.org/euclid.bams/1183539465}} | *{{citation|title= Review: J. Donald Monk, Mathematical logic|first=Martin|last= Davis|journal=Bull. Amer. Math. Soc.| volume =83|year=1977|pages =1007–1011|url=http://projecteuclid.org/euclid.bams/1183539465|doi=10.1090/S0002-9904-1977-14357-7|doi-access=free}} | ||
*{{citation|last1=Davis|first1=M.|last2=Hausner|first2=M|year=1978|title=Book review. The Joy of Infinitesimals. J. Keisler's Elementary Calculus|journal=Mathematical Intelligencer|volume=1|pages=168–170 |
*{{citation|last1=Davis|first1=M.|last2=Hausner|first2=M|year=1978|title=Book review. The Joy of Infinitesimals. J. Keisler's Elementary Calculus|journal=Mathematical Intelligencer|volume=1|pages=168–170|doi=10.1007/bf03023265|s2cid=121679411}}. | ||
*{{citation|last1=Hrbacek|first1=K.|last2=Lessmann|first2=O.| last3=O’Donovan|first3=R.|title=Analysis with Ultrasmall Numbers|journal=American Mathematical Monthly|volume=117|number=9|date=November 2010|pages=801–816|doi=10.4169/000298910x521661}} | *{{citation|last1=Hrbacek|first1=K.|last2=Lessmann|first2=O.| last3=O’Donovan|first3=R.|title=Analysis with Ultrasmall Numbers|journal=American Mathematical Monthly|volume=117|number=9|date=November 2010|pages=801–816|doi=10.4169/000298910x521661|s2cid=5720030}} | ||
*{{citation|last1=Hrbacek|first1=K.|editor1-last=Van Den Berg|editor1-first=I.|editor2-last=Neves|editor2-first=V.| chapter=Stratified Analysis?|title=The Strength of Nonstandard Analysis|publisher=Springer|year=2007}} | *{{citation|last1=Hrbacek|first1=K.|editor1-last=Van Den Berg|editor1-first=I.|editor2-last=Neves|editor2-first=V.| chapter=Stratified Analysis?|title=The Strength of Nonstandard Analysis|publisher=Springer|year=2007}} | ||
*{{citation | last1 = Katz | first1 = Karin Usadi | last2 = Katz | first2 = Mikhail G. | author2-link = Mikhail Katz | arxiv = 1007.3018 | issue = 1 | journal = ] | pages = 3–30 | title = When is .999... less than 1? | url = http://www.math.umt.edu/TMME/vol7no1/ | volume = 7 | year = 2010}} | *{{citation | last1 = Katz | first1 = Karin Usadi | last2 = Katz | first2 = Mikhail G. | author2-link = Mikhail Katz | arxiv = 1007.3018 | issue = 1 | journal = ] | pages = 3–30 | title = When is .999... less than 1? | url = http://www.math.umt.edu/TMME/vol7no1/ | volume = 7 | year = 2010 | doi = 10.54870/1551-3440.1381 | url-status = dead | archive-url = https://web.archive.org/web/20110720095125/http://www.math.umt.edu/TMME/vol7no1/ | archive-date = 20 July 2011| bibcode = 2010arXiv1007.3018U | s2cid = 11544878 }} | ||
*{{citation|first=H. Jerome|last= Keisler|title=Elementary Calculus: An Approach Using Infinitesimals|year= 1976|url= http://www.math.wisc.edu/~keisler/calc.html|publisher=Prindle Weber & Schmidt|isbn=978-0871509116}} | *{{citation|first=H. Jerome|last= Keisler|title=Elementary Calculus: An Approach Using Infinitesimals|year= 1976|url= http://www.math.wisc.edu/~keisler/calc.html|publisher=Prindle Weber & Schmidt|isbn=978-0871509116}} | ||
*{{citation|first=H. Jerome |last=Keisler|title= Foundations of Infinitesimal Calculus|url= http://www.math.wisc.edu/~keisler/foundations.html | |
*{{citation|first=H. Jerome |last=Keisler|title= Foundations of Infinitesimal Calculus|url= http://www.math.wisc.edu/~keisler/foundations.html |access-date=10 January 2007|publisher= Prindle Weber & Schmidt |year= 1976|isbn=978-0871502155}} A companion to the textbook ''Elementary Calculus: An Approach Using Infinitesimals''. | ||
*{{Citation | last1=Keisler | first1=H. Jerome | author1-link=Howard Jerome Keisler | title=Elementary Calculus: An Infinitesimal Approach | url=http://www.math.wisc.edu/~keisler/calc.html | publisher=] | location=New York | edition=2nd | isbn=978-0-486-48452-5 | year=2011}} | *{{Citation | last1=Keisler | first1=H. Jerome | author1-link=Howard Jerome Keisler | title=Elementary Calculus: An Infinitesimal Approach | url=http://www.math.wisc.edu/~keisler/calc.html | publisher=] | location=New York | edition=2nd | isbn=978-0-486-48452-5 | year=2011}} | ||
*{{citation|first1=E. W.|last1=Madison|first2=K. D.|last2=Stroyan|title=Elementary Calculus. by H. Jerome Keisler|journal=The American Mathematical Monthly|volume=84|number=6|date=Jun–Jul 1977|pages=496–500 | *{{citation|first1=E. W.|last1=Madison|first2=K. D.|last2=Stroyan|title=Elementary Calculus. by H. Jerome Keisler|journal=The American Mathematical Monthly|volume=84|number=6|date=Jun–Jul 1977|pages=496–500 | ||
|jstor=2321930 | |jstor=2321930|doi=10.2307/2321930 | ||
}} | }} | ||
*{{citation|last1=O'Donovan|first1=R.|editor1-last=Van Den Berg|editor1-first=I.|editor2-last=Neves|editor2-first=V.|chapter=Pre-University Analysis|title=The Strength of Nonstandard Analysis|publisher=Springer|year=2007}} | *{{citation|last1=O'Donovan|first1=R.|editor1-last=Van Den Berg|editor1-first=I.|editor2-last=Neves|editor2-first=V.|chapter=Pre-University Analysis|title=The Strength of Nonstandard Analysis|publisher=Springer|year=2007}} | ||
*{{citation|last1=O'Donovan|first1=R.|last2=Kimber|first2=J.|editor1-first=N|editor1-last=Cultand|editor2-first=M.|editor2-last=Di Nasso|editor-first3=D.|editor3-last=Ross|chapter=Nonstandard analysis at pre-university level: Naive magnitude analysis|title=Nonstandard Methods and Applications in Mathematics|series=Lecture Notes in Logic|volume=25|year=2006}} | *{{citation|last1=O'Donovan|first1=R.|last2=Kimber|first2=J.|editor1-first=N|editor1-last=Cultand|editor2-first=M.|editor2-last=Di Nasso|editor-first3=D.|editor3-last=Ross|chapter=Nonstandard analysis at pre-university level: Naive magnitude analysis|title=Nonstandard Methods and Applications in Mathematics|series=Lecture Notes in Logic|volume=25|year=2006}} | ||
*{{citation|last1=Stolzenberg|first1=G.|journal=Notices of the American Mathematical Society|volume=25|number=4|date=June 1978|pages=242}} | *{{citation|last1=Stolzenberg|first1=G.|title=Letter to the Editor|journal=Notices of the American Mathematical Society|volume=25|number=4|date=June 1978|pages=242}} | ||
*{{citation|doi=10.2307/2318657|title=The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach|first=Kathleen|last= Sullivan| | *{{citation|doi=10.2307/2318657|title=The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach|first=Kathleen|last= Sullivan| | ||
journal=The American Mathematical Monthly|volume= 83|year=1976|pages= 370–375|issue=5|publisher=Mathematical Association of America|jstor=2318657}} | journal=The American Mathematical Monthly|volume= 83|year=1976|pages= 370–375|issue=5|publisher=Mathematical Association of America|jstor=2318657}} | ||
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{{Infinitesimals}} | {{Infinitesimals}} | ||
⚫ | {{Use dmy dates|date= |
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] | ] | ||
] | ] | ||
] | ] |
Latest revision as of 18:08, 31 March 2024
Second edition | |
Author | H. Jerome Keisler |
---|---|
Language | English |
Subject | Mathematics |
Publisher | Dover |
Publication date | 1976 |
Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson and is sometimes given as An approach using infinitesimals. The book is available freely online and is currently published by Dover.
Textbook
Keisler's textbook is based on Robinson's construction of the hyperreal numbers. Keisler also published a companion book, Foundations of Infinitesimal Calculus, for instructors, which covers the foundational material in more depth.
Keisler defines all basic notions of the calculus such as continuity (mathematics), derivative, and integral using infinitesimals. The usual definitions in terms of ε–δ techniques are provided at the end of Chapter 5 to enable a transition to a standard sequence.
In his textbook, Keisler used the pedagogical technique of an infinite-magnification microscope, so as to represent graphically, distinct hyperreal numbers infinitely close to each other. Similarly, an infinite-resolution telescope is used to represent infinite numbers.
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 (see figure).
Thus the microscope is used as a device in explaining the derivative.
Reception
The book was first reviewed by Errett Bishop, noted for his work in constructive mathematics. Bishop's review was harshly critical; see Criticism of nonstandard analysis. Shortly after, Martin Davis and Hausner published a detailed favorable review, as did Andreas Blass and Keith Stroyan. Keisler's student K. Sullivan, as part of her PhD thesis, performed a controlled experiment involving 5 schools, which found Elementary Calculus to have advantages over the standard method of teaching calculus. Despite the benefits described by Sullivan, the vast majority of mathematicians have not adopted infinitesimal methods in their teaching. Recently, Katz & Katz give a positive account of a calculus course based on Keisler's book. O'Donovan also described his experience teaching calculus using infinitesimals. His initial point of view was positive, but later he found pedagogical difficulties with the approach to nonstandard calculus taken by this text and others.
G. R. Blackley remarked in a letter to Prindle, Weber & Schmidt, concerning Elementary Calculus: An Approach Using Infinitesimals, "Such problems as might arise with the book will be political. It is revolutionary. Revolutions are seldom welcomed by the established party, although revolutionaries often are."
Hrbacek writes that the definitions of continuity, derivative, and integral implicitly must be grounded in the ε–δ method in Robinson's theoretical framework, in order to extend definitions to include nonstandard values of the inputs, claiming that the hope that nonstandard calculus could be done without ε–δ methods could not be realized in full. Błaszczyk et al. detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".
Transfer principle
Between the first and second edition of the Elementary Calculus, much of the theoretical material that was in the first chapter was moved to the epilogue at the end of the book, including the theoretical groundwork of nonstandard analysis.
In the second edition Keisler introduces the extension principle and the transfer principle in the following form:
- Every real statement that holds for one or more particular real functions holds for the hyperreal natural extensions of these functions.
Keisler then 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 .
See also
- Criticism of nonstandard analysis
- Influence of nonstandard analysis
- Nonstandard calculus
- Increment theorem
Notes
- ^ Keisler 2011.
- Davis & Hausner 1978.
- Blass 1978.
- Madison & Stroyan 1977.
- "UW Math PhD Alumni (1974)". Archived from the original on 7 June 2012. Retrieved 29 November 2011.
- ^ Sullivan 1976.
- Tall 1980.
- Katz & Katz 2010.
- O'Donovan & Kimber 2006.
- O'Donovan 2007.
- Hrbacek 2007.
- Błaszczyk, Piotr; Katz, Mikhail; Sherry, David (2012), "Ten misconceptions from the history of analysis and their debunking", Foundations of Science, 18: 43–74, arXiv:1202.4153, doi:10.1007/s10699-012-9285-8, S2CID 119134151
References
- Bishop, Errett (1977), "Review: H. Jerome Keisler, Elementary calculus", Bull. Amer. Math. Soc., 83: 205–208, doi:10.1090/s0002-9904-1977-14264-x
- Blass, Andreas (1978), "Review: Martin Davis, Applied nonstandard analysis, and K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, and H. Jerome Keisler, Foundations of infinitesimal calculus", Bull. Amer. Math. Soc., 84 (1): 34–41, doi:10.1090/S0002-9904-1978-14401-2
- Blass writes: "I suspect that many mathematicians harbor, somewhere in the back of their minds, the formula for arc length (and quickly factor out dx before writing it down)" (p. 35).
- "Often, as in the examples above, the nonstandard definition of a concept is simpler than the standard definition (both intuitively simpler and simpler in a technical sense, such as quantifiers over lower types or fewer alternations of quantifiers)" (p. 37).
- "The relative simplicity of the nonstandard definitions of some concepts of elementary analysis suggests a pedagogical application in freshman calculus. One could make use of the students' intuitive ideas about infinitesimals (which are usually very vague, but so are their ideas about real numbers) to develop calculus on a nonstandard basis" (p. 38).
- Davis, Martin (1977), "Review: J. Donald Monk, Mathematical logic", Bull. Amer. Math. Soc., 83: 1007–1011, doi:10.1090/S0002-9904-1977-14357-7
- Davis, M.; Hausner, M (1978), "Book review. The Joy of Infinitesimals. J. Keisler's Elementary Calculus", Mathematical Intelligencer, 1: 168–170, doi:10.1007/bf03023265, S2CID 121679411.
- Hrbacek, K.; Lessmann, O.; O’Donovan, R. (November 2010), "Analysis with Ultrasmall Numbers", American Mathematical Monthly, 117 (9): 801–816, doi:10.4169/000298910x521661, S2CID 5720030
- Hrbacek, K. (2007), "Stratified Analysis?", in Van Den Berg, I.; Neves, V. (eds.), The Strength of Nonstandard Analysis, Springer
- Katz, Karin Usadi; Katz, Mikhail G. (2010), "When is .999... less than 1?", The Montana Mathematics Enthusiast, 7 (1): 3–30, arXiv:1007.3018, Bibcode:2010arXiv1007.3018U, doi:10.54870/1551-3440.1381, S2CID 11544878, archived from the original on 20 July 2011
- 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 January 2007 A companion to the textbook Elementary Calculus: An Approach Using Infinitesimals.
- Keisler, H. Jerome (2011), Elementary Calculus: An Infinitesimal Approach (2nd ed.), New York: Dover Publications, ISBN 978-0-486-48452-5
- Madison, E. W.; Stroyan, K. D. (June–July 1977), "Elementary Calculus. by H. Jerome Keisler", The American Mathematical Monthly, 84 (6): 496–500, doi:10.2307/2321930, JSTOR 2321930
- O'Donovan, R. (2007), "Pre-University Analysis", in Van Den Berg, I.; Neves, V. (eds.), The Strength of Nonstandard Analysis, Springer
- O'Donovan, R.; Kimber, J. (2006), "Nonstandard analysis at pre-university level: Naive magnitude analysis", in Cultand, N; Di Nasso, M.; Ross, D. (eds.), Nonstandard Methods and Applications in Mathematics, Lecture Notes in Logic, vol. 25
- Stolzenberg, G. (June 1978), "Letter to the Editor", Notices of the American Mathematical Society, 25 (4): 242
- 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
- Tall, David (1980), Intuitive infinitesimals in the calculus (poster) (PDF), Fourth International Congress on Mathematics Education, Berkeley
External links
Infinitesimals | ||
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