Elementary Calculus: An Infinitesimal Approach

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, 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).

File:Standard part function with two continua.svg

Thus the microscope is used as a device in explaining the derivative.

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.{{sfn|Davis|Hausner|1978}}{{sfn|Blass|1978}}{{sfn|Madison|Stroyan|1977}} Keisler's student K. Sullivan,{{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 }} 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."{{sfn|Sullivan|1976}}

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 microcontinuity in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".{{citation

| last1 = Błaszczyk | first1 = Piotr

| last2 = Katz | first2 = Mikhail

| author2-link = Mikhail Katz

| last3 = Sherry | first3 = David

| arxiv = 1202.4153

| doi = 10.1007/s10699-012-9285-8

| journal = Foundations of Science

| pages = 43–74

| title = Ten misconceptions from the history of analysis and their debunking

| volume = 18

| year = 2012| s2cid = 119134151

}}

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: $(x-y)^2\; =\; x^2-2xy+y^2$.
• A trigonometric identity: $\backslash sin^2\; x\; +\; \backslash cos^2\; x\; =\; 1$.
• A rule for logarithms: If x > 0 and y > 0, then $\backslash log\_\{10\}\; (xy)\; =\; \backslash log\_\{10\}x\; +\; \backslash log\_\{10\}y$.

• Criticism of nonstandard analysis
• Influence of nonstandard analysis
• Nonstandard calculus
• Increment theorem

{{reflist}}

• {{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

|author1-link = Andreas Blass

|title=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|journal=Bull. Amer. Math. Soc.

|volume=84

|number=1

|year=1978

|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 $\backslash int\; \backslash sqrt\{(dx)^2\; +\; (dy)^2\}$ 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).

• {{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|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|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 = Katz | first1 = Karin Usadi | last2 = Katz | first2 = Mikhail G. | author2-link = Mikhail Katz | arxiv = 1007.3018 | issue = 1 | journal = The Montana Mathematics Enthusiast | 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= 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=Dover Publications | 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

|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.|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.|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|

journal=The American Mathematical Monthly|volume= 83|year=1976|pages= 370–375|issue=5|publisher=Mathematical Association of America|jstor=2318657}}

• {{citation

|first=David

|last=Tall

| author1-link = David O. Tall

|url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1980c-intuitive-infls.pdf|title=Intuitive infinitesimals in the calculus (poster)|publisher=Fourth International Congress on Mathematics Education, Berkeley|year=1980}}

• [http://www.math.wisc.edu/~keisler/calc.html Book in PDF format]

{{Infinitesimals}}

Category:Calculus

Category:Nonstandard analysis

Category:Mathematics of infinitesimals