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| '''Albert Harold Lightstone''' (1926–1976)<ref>{{cite web |title= ''Mathematical Concepts and Methods in Science and Engineering'' |location= www.faqs.org |publisher= Plenum |url= http://www.faqs.org/copyright/masters-theses-in-the-pure-and-applied-sciences-accepted-by-3/ |accessdate= March 31, 2011}}</ref> was a Canadian mathematician. He was one of the pioneers of [[non-standard analysis]], a doctoral student of [[Abraham Robinson]], and later a co-author with Robinson of the book ''Nonarchimedean Fields and Asymptotic Expansions''.<ref>''[[Nonarchimedean field]]s and [[asymptotic expansion]]s.'' Lightstone, A. H. and Robinson, Abraham. North-Holland Pub. Co. (Amsterdam and New York), 1975.</ref>
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| ==Biography==
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| Lightstone earned his PhD from the [[University of Toronto]] in 1955, under the supervision of Abraham Robinson; his thesis was entitled ''Contributions To The Theory Of Quantification''.<ref>{{mathgenealogy|id=16086|name=Albert Harold Lightstone}}</ref> He was a professor of mathematics at [[Carleton University]]<ref name="trisect"/> and [[Queen's University]].<ref name="cal">[http://www.queensu.ca/calendars/artsci/MATHEMATICS_AND_STATISTICS.html Queen's University Academic Calendar, Mathematics and Statistics], retrieved 2011-03-31.</ref>
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| ==Research==
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| ===Decimal hyperreals===
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| In his article "Infinitesimals" in the ''[[American Mathematical Monthly]]'' in 1972,<ref>{{citation |last=Lightstone |first=A. H. |title=Infinitesimals |jstor=2316619 |journal=[[American Mathematical Monthly]] |date=March 1972 |volume=79 |number=3 |pages=242–251 |doi=10.2307/2316619 |mr=0300889 }}</ref> Lightstone described an extended decimal notation for the [[Hyperreal number|hyperreals]]. Here there is a digit at every [[hypernatural]] rank rather than merely a digit for every rank given by a natural number. Such a hyperreal decimal is written as
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| :<math>a.a_1 a_2 \ldots ; \ldots a_{H-1} a_H a_{H+1} \ldots\,.</math>
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| Here the digit <math>a_H</math> appears at rank <math>H</math>, which is a typical infinite hypernatural. The semicolon separates the digits at finite ranks from the digits at infinite ranks. Thus, the number 0.000...;...01, with digit "1" at infinite rank ''H'', corresponds to the infinitesimal <math>10^{-H}</math>.
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| The difference 1 - 0.000...;...01 is 0.999...;...9, with an infinite [[hypernatural]]'s worth of digits 9. An alternative notation for the latter is | |
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| :<math>0.\underbrace{999\ldots9 }_H \, </math>
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| where ''H'' is an infinite hypernatural. The extended decimal notation provides a rigorous mathematical implementation of student intuitions of an infinitesimal of the form 0.000...01. Such student intuitions and their usefulness in the learning of [[infinitesimal calculus]] were analyzed in a 2010 study by Robert Ely in the [[Journal for Research in Mathematics Education]].<ref>{{Citation |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146|url=http://u.cs.biu.ac.il/~katzmik/sgtdirectory/ely10.pdf}}. This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|"0.999..."}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}</ref>
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| ===Other research===
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| Lightstone's main research contributions were in [[non-standard analysis]]. He also wrote papers on [[angle trisection]],<ref name="trisect">{{citation
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| | last = Lightstone | first = A. H.
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| | mr = 1571175
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| | issue = 2
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| | journal = [[Mathematics Magazine]]
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| | pages = 99–102
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| | title = A Construction for Trisecting the Angle
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| | jstor = 2688331
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| | volume = 35
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| | year = 1962}}</ref> [[matrix inversion]],<ref>{{citation
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| | last = Lightstone | first = A. H.
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| | mr = 0231832
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| | journal = Delta
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| | pages = 1–7
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| | publisher = University of Wisconsin
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| | title = Two methods of inverting matrices
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| | volume = 41
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| | year = 1968}}</ref> and applications of [[group theory]] to [[formal logic]].<ref>{{citation
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| | last = Lightstone | first = A. H.
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| | mr = 0229507
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| | journal = Canadian Mathematical Bulletin
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| | pages = 43–50
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| | title = Group theory and the principle of duality
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| | url = http://www.math.ca/10.4153/CMB-1968-006-9
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| | volume = 11
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| | year = 1968}}</ref>
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| ==Books==
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| Lightstone was the author or co-author of several books on mathematics:
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| *''The Axiomatic Method: An Introduction to Mathematical Logic'' (Prentice Hall, 1964). This introductory textbook is divided into two parts, one providing an informal introduction to [[Boolean logic]] and the second using formal methods to prove the consistency and completeness of the [[predicate calculus]].<ref>Review of ''The Axiomatic Method'' by [[Reuben Goodstein|R. L. Goodstein]], ''Mathematical Reviews'', {{MR|0163834}}.</ref> It is aimed at students who already have some familiarity with [[abstract algebra]], and one of its themes is an algebraic view of [[mathematical proof]]s in logic.<ref>Review of ''The Axiomatic Method'' by [[Peter B. Andrews (mathematician)|Peter Andrews]] (1966), ''[[Journal of Symbolic Logic]]'' '''31''' (1): 106–108, {{JSTOR|2270630}}.</ref>
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| *''Concepts of Calculus'' (Harper and Row, 1965). This is a textbook on the [[calculus]] of [[real function]]s of a single variable. Reviewer D. R. Dickinson wrote that it "contains much novel and interesting material"; however, he also complained of its pedantic avoidance of variables (using identity functions in their place), its unnecessary insistence on considering only functions whose derivative has the same domain as the function itself, and its "dull and lengthy discussions of elementary topics".<ref>Review of ''Concepts of Calculus'' by D. R. Dickinson (1966), ''Mathematical Gazette'' '''50''' (373): 329–330, {{JSTOR|3614713}}.</ref>
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| **''Concepts of Calculus, vol. 2'' (Harper and Row, 1966)
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| **''Solutions to the exercises for Concepts of Calculus'' (Harper and Row, 1966)
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| *''Fundamentals of Linear Algebra'' (Appleton-Century-Crofts, 1969, ISBN 0-390-56050-2)
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| *''Symbolic Logic and the Real Number System: an Introduction to the Foundations of Number Systems'' (Harper and Row, 1965). This book provides a course in the [[construction of the real numbers]] based on [[formal logic]].<ref name="hunt">Review of ''Symbolic Logic'' by Burrowes Hunt (1969), ''[[American Mathematical Monthly]]'' '''76''' (6): 716–717, {{DOI|10.2307/2316722}}.</ref> Its goal is both to show how the real numbers can be developed from simpler concepts in arithmetic, and to demonstrate the impact of logic on the rest of mathematics.<ref name="gcw">Review of ''Symbolic Logic'' by G. Cuthbert Webber (1966), ''[[Science (journal)|Science]]'' (New Ser.) '''153''' (3735): 519, {{DOI|10.1126/science.153.3735.519}}, {{JSTOR|1719891}}, {{Bibcode|1966Sci...153..519L}}.</ref> As well as covering the title topics, it also contains a long section on the axioms for several algebraic structures: [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[Boolean algebra (structure)|Boolean algebras]].<ref name="goodstein">Review of ''Symbolic Logic'' by [[Reuben Goodstein|R. L. Goodstein]] (1967), ''Mathematical Gazette'' '''51''' (375): 78, {{JSTOR|3613660}}.</ref> One idiosyncrasy is that, rather than axiomatizing the real numbers using [[Dedekind cut]]s or [[Cauchy sequence]]s, it bases its axiomatization on sequences of decimal numbers.<ref name="hunt"/><ref name="gcw"/><ref name="goodstein"/>
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| *''Nonarchimedean Fields and Asymptotic Expansions'' (with Abraham Robinson, North-Holland, 1975). This is an introductory textbook that attempts to make the material from Robinson's 1966 monograph ''Non-Standard Analysis'' more accessible,<ref name="fenyo">Review of ''Nonarchimedean Fields'' by I. Fenyo, ''Mathematical Reviews'', {{MR|0414354}}.</ref> and to demonstrate the usefulness of [[non-standard analysis]] in studying [[asymptotic expansion]]s.<ref name="loeb">Review of ''Nonarchimedean Fields'' by [[Peter A. Loeb]] (1977), ''[[Bulletin of the American Mathematical Society]]'' [http://projecteuclid.org/euclid.bams/1183538676 '''83''' (2): 231–235], {{DOI|10.1090/S0002-9904-1977-14277-8}}.</ref> It was based on an initial draft by Robinson, and finished posthumously by Lightstone, who himself died soon after.<ref name="fenyo"/><ref name="loeb"/> It begins with an introduction to [[non-Archimedean field]]s with many helpful examples, brings in the necessary tools from [[mathematical logic]] including [[ultrapower]]s, spends two chapters describing how to do [[non-standard analysis]] using the [[Levi-Civita field]], and finishes with three chapters on asymptotic expansions.<ref name="fenyo"/>
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| *''Mathematical Logic: An Introduction to Model Theory'' (Mathematical Concepts and Methods in Science and Engineering, vol. 9, Plenum Press, 1978, ISBN 0-306-30894-0). This book was published posthumously, edited by [[Herbert Enderton]]. It is organized into three parts, one on the [[propositional calculus]], a second part on [[formal semantics (logic)|formal semantics]], and a third part on applications of model theory including [[nonstandard analysis]] and [[set theory]].<ref name="plotkin">Review of ''Mathematical Logic'' by J. M. Plotkin (1980), ''Mathematical Reviews'', {{MR|0497355}})</ref> However, it was criticized for the slow pace of its first section and for its overall lack of mathematical rigor.<ref name="plotkin"/><ref>Review of ''Mathematical Logic'' by [[John Crossley (mathematician)|J. N. Crossley]] (1979), ''[[Bulletin of the American Mathematical Society]]'' '''1''' (6): 1003–1005, {{DOI|10.1090/S0273-0979-1979-14718-9}}.</ref>
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| ==Awards and honours==
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| Queen's University annually awards the Albert Harold Lightstone Scholarship, named for Lightstone, to a fourth year honors undergraduate student majoring in mathematics or statistics.<ref>{{cite web |year= 2010 |title= ''The Albert Harold Lightstone Scholarship'' |location= www.canadian-universities.net |url= http://www.canadian-universities.net/Scholarships/A/Albert-Harold-Lightstone-Scholarship.html |accessdate= March 31, 2011}}</ref><ref>{{cite web |title= ''Mathematics & Statistics Specific Awards'' |location= www.queensu.ca |publisher= Queen's University |url= http://www.queensu.ca/studentawards/completeawardslist/automatic/upperyear/artsci.html |accessdate= March 31, 2011}}</ref> The scholarship was established by Lightstone's wife after his death.<ref>{{cite web |title= ''The Albert Harold Lightstone Scholarship'' |location= www.queensu.ca |publisher= Queen's University |url= http://www.queensu.ca/giving/recognition/donorstories/fundatog.html#albert |accessdate= March 31, 2011}}</ref>
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| ==References==
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| {{reflist|colwidth=40em}}
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME = Lightstone, AH
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = Canadian mathematician
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| | DATE OF BIRTH = 1926
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| | PLACE OF BIRTH =
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| | DATE OF DEATH = 1976
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| | PLACE OF DEATH =
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| }}
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| {{DEFAULTSORT:Lightstone, AH}}
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| [[Category:20th-century mathematicians]]
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| [[Category:Canadian mathematicians]]
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| [[Category:Mathematical logicians]]
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| [[Category:University of Toronto alumni]]
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| [[Category:Carleton University faculty]]
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| [[Category:Queen's University faculty]]
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| [[Category:1926 births]]
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| [[Category:1976 deaths]]
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