20:
426:
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
469:. 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. As well as covering the title topics, it also contains a long section on the axioms for several algebraic structures:
578:
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. The scholarship was established by Lightstone's wife after his death.
344:
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
195:
247:, 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
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782:. 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
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515:. It was based on an initial draft by Robinson, and finished posthumously by Lightstone, who himself died soon after. It begins with an introduction to
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396:(Prentice Hall, 1964). This introductory textbook is divided into two parts, one providing an informal introduction to
482:
569:. However, it was criticized for the slow pace of its first section and for its overall lack of mathematical rigor.
93:
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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1026:
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503:. This is an introductory textbook that attempts to make the material from Robinson's 1966 monograph
630:
Lightstone, A. H. and
Robinson, Abraham. North-Holland Pub. Co. (Amsterdam and New York), 1975.
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470:
59:
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same domain as the function itself, and its "dull and lengthy discussions of elementary topics".
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Symbolic Logic and the Real Number System: an
Introduction to the Foundations of Number Systems
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541:(Mathematical Concepts and Methods in Science and Engineering, vol. 9, Plenum Press, 1978,
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and the second using formal methods to prove the consistency and completeness of the
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485:. One idiosyncrasy is that, rather than axiomatizing the real numbers using
54:
in 1955, under the supervision of
Abraham Robinson; his thesis was entitled
857:
30:(1926–1976) was a Canadian mathematician. He was one of the pioneers of
19:
419:
389:
Lightstone was the author or co-author of several books on mathematics:
190:{\displaystyle a.a_{1}a_{2}\ldots ;\ldots a_{H-1}a_{H}a_{H+1}\ldots \,.}
1091:
1046:
993:
956:
928:
841:
Lightstone, A. H. (1968), "Group theory and the principle of duality",
820:
744:
681:
658:
Lightstone, A. H. (1962), "A Construction for
Trisecting the Angle",
284:
The difference 1 - 0.000...;...01 is 0.999...;...9, with an infinite
1083:
985:
972:
Hunt, Burrowes (1969). "Symbolic Logic and the Real Number System".
812:
736:
85:
in 1972, Lightstone described an extended decimal notation for the
18:
404:. It is aimed at students who already have some familiarity with
288:'s worth of digits 9. An alternative notation for the latter is
702:
Queen's
University Academic Calendar, Mathematics and Statistics
799:
Lightstone, A. H. (1968), "Two methods of inverting matrices",
519:
with many helpful examples, brings in the necessary tools from
493:, it bases its axiomatization on sequences of decimal numbers.
561:, and a third part on applications of model theory including
598:
Mathematical
Concepts and Methods in Science and Engineering
535:, and finishes with three chapters on asymptotic expansions.
461:(Harper and Row, 1965). This book provides a course in the
394:
The
Axiomatic Method: An Introduction to Mathematical Logic
767:"Nonstandard student conceptions about infinitesimals"
507:
more accessible, and to demonstrate the usefulness of
297:
257:
233:
206:
102:
1311:. www.queensu.ca: Queen's University. Archived from
1281:. www.queensu.ca: Queen's University. Archived from
549:). This book was published posthumously, edited by
539:
437:
Solutions to the exercises for
Concepts of Calculus
349:were analyzed in a 2010 study by Robert Ely in the
721:Lightstone, A. H. (March 1972), "Infinitesimals",
418:(Harper and Row, 1965). This is a textbook on the
330:{\displaystyle 0.\underbrace {999\ldots 9} _{H}\,}
329:
273:
239:
219:
189:
38:, and later a co-author with Robinson of the book
361:Lightstone's main research contributions were in
1360:Academic staff of Queen's University at Kingston
408:, and one of its themes is an algebraic view of
553:. It is organized into three parts, one on the
497:Nonarchimedean Fields and Asymptotic Expansions
40:Nonarchimedean Fields and Asymptotic Expansions
499:(with Abraham Robinson, North-Holland, 1975).
1220:Bulletin of the American Mathematical Society
1151:Bulletin of the American Mathematical Society
774:Journal for Research in Mathematics Education
351:Journal for Research in Mathematics Education
56:Contributions To The Theory Of Quantification
8:
1277:Mathematics & Statistics Specific Awards
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527:, spends two chapters describing how to do
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1307:The Albert Harold Lightstone Scholarship
1251:The Albert Harold Lightstone Scholarship
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1103:
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1132:
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1011:Webber, G. Cuthbert (1966). "Review of
967:
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786:falling short of 1 by an infinitesimal
587:
79:In his article "Infinitesimals" in the
58:. He was a professor of mathematics at
1006:
1004:
1355:Academic staff of Carleton University
1255:. www.canadian-universities.net. 2010
7:
1340:20th-century Canadian mathematicians
807:(1), University of Wisconsin: 1–7,
251:, corresponds to the infinitesimal
50:Lightstone earned his PhD from the
89:. Here there is a digit at every
14:
974:The American Mathematical Monthly
463:construction of the real numbers
447:(Appleton-Century-Crofts, 1969,
1234:10.1090/S0273-0979-1979-14718-9
1165:10.1090/S0002-9904-1977-14277-8
844:Canadian Mathematical Bulletin
674:10.1080/0025570X.1962.11975312
445:Fundamentals of Linear Algebra
1:
724:American Mathematical Monthly
644:Mathematics Genealogy Project
82:American Mathematical Monthly
1350:University of Toronto alumni
1039:10.1126/science.153.3735.519
431:Concepts of Calculus, vol. 2
944:by D. R. Dickinson (1966),
365:. He also wrote papers on
1386:
1185:by J. M. Plotkin (1980),
917:Journal of Symbolic Logic
640:Albert Harold Lightstone
34:, a doctoral student of
28:Albert Harold Lightstone
23:A.H. Lightstone at chess
711:, retrieved 2011-03-31.
707:March 27, 2010, at the
274:{\displaystyle 10^{-H}}
1345:Mathematical logicians
858:10.4153/cmb-1968-006-9
602:. www.faqs.org: Plenum
555:propositional calculus
517:non-Archimedean fields
439:(Harper and Row, 1966)
433:(Harper and Row, 1966)
373:, and applications of
347:infinitesimal calculus
331:
275:
241:
221:
191:
24:
16:Canadian mathematician
1144:Nonarchimedean Fields
1111:Nonarchimedean Fields
626:asymptotic expansions
622:Nonarchimedean fields
529:non-standard analysis
513:asymptotic expansions
509:non-standard analysis
505:Non-Standard Analysis
363:non-standard analysis
332:
276:
242:
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220:{\displaystyle a_{H}}
192:
52:University of Toronto
32:non-standard analysis
22:
1315:on December 24, 2010
1187:Mathematical Reviews
1115:Mathematical Reviews
1072:Mathematical Gazette
946:Mathematical Gazette
942:Concepts of Calculus
908:The Axiomatic Method
888:Mathematical Reviews
880:The Axiomatic Method
765:Ely, Robert (2010),
661:Mathematics Magazine
563:nonstandard analysis
416:Concepts of Calculus
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1066:(1967). "Review of
1031:1966Sci...153..519L
557:, a second part on
410:mathematical proofs
60:Carleton University
1213:Mathematical Logic
1183:Mathematical Logic
574:Awards and honours
521:mathematical logic
402:predicate calculus
327:
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75:Decimal hyperreals
64:Queen's University
25:
1285:on March 29, 2012
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1283:the original
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511:in studying
504:
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467:formal logic
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379:formal logic
375:group theory
360:
341:
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286:hypernatural
283:
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91:hypernatural
80:
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27:
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1370:1976 deaths
1365:1926 births
1211:"Review of
1142:"Review of
1078:(375): 78.
525:ultrapowers
1334:Categories
1181:Review of
1109:Review of
980:(6): 716.
940:Review of
906:Review of
878:Review of
788:0.000...1.
784:"0.999..."
583:References
567:set theory
531:using the
523:including
87:hyperreals
1319:March 31,
1289:March 31,
1259:March 31,
851:: 43–50,
606:March 31,
465:based on
412:in logic.
316:⏟
309:…
264:−
181:…
147:−
136:…
130:…
46:Biography
1209:(1979).
1140:(1977).
914:(1966),
705:Archived
420:calculus
70:Research
1194:0497355
1122:0414354
1092:3613660
1047:1719891
1027:Bibcode
1018:Science
994:2316722
957:3614713
929:2270630
895:0163834
867:0229507
829:0231832
821:2687951
753:0300889
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471:groups
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1088:JSTOR
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953:JSTOR
925:JSTOR
817:JSTOR
801:Delta
770:(PDF)
741:JSTOR
678:JSTOR
475:rings
385:Books
1321:2011
1291:2011
1261:2011
624:and
608:2011
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144:H
140:a
133:;
125:2
121:a
115:1
111:a
107:.
104:a
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