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References 225
34. L. Henkin, On mathematical i nduction, Amer. Math. Monthly 67,
no. 4 (19 60), 323–338.
35. E. Hewitt, Rings of real - valued contin uous functions. I. Tran s. Amer.
Math. Soc. 64 (1 948), 45–99. Errata: Trans. Amer. Math. Soc. 64
(1948), 596.
36. K. E. Iverson, Notation as a tool of thought, Comm. ACM 23 (Augu st
1980), 444–465.
37. V. Kanovei and M. Reeken, Nonstandard Analysis, Axiomati-
cally. Springer-Verlag, 2004.
38. V. Kanovei and S. Shelah, A definable nonstandard model of the reals.
J. Symbolic Logic 69 n o. 1 (2004), 1 59–16 4.
39. K. U. Katz and M . G. Katz, When is .999 . . . less than 1? The Montana
Mathematics Enthusiast 7 no. 1 (201 0), 3–30. ISSN 1551–3440.
40. H. J. Keisler, Elementary Calculus: An Approach Using In-
finitesimals (Experi mental v ersion). Bodgen & Quigley, 1971.
41. H. J. Keisler, Elementary Calculus: An Approach Using
Infinitesimals. Prindle Weber & Schmidt, 1976. ISBN 978-
0871509116.
42. H. J. Keisl er, Elementary Calculus: An Infinitesimal Approach.
Second Edition. Prindle, Weber and Schimidt, Boston, 1986.
43. H. J. Keisler, Elementary Calculus: An Infinitesimal Ap-
proach. On-line edition licensed for non-commercial use un-
der a Creative Commons License. http://www.math.wisc.edu/
~keisler/calc.html.
44. H. J. Keisler, Founda tions of Infinitesimal Calculus. On-line
edition, http://www.math.wisc.edu/~keisler/foundations.
html, retrieved 29 June 2009 .
45. A. P. Kirman and D. Sondermann, Arrows theorem, many agents,
and invis ible dictators, J. E conomic Theory 5 (1972), 267–277.
46. D. Knuth, The Art of Computer Programming. Vol. 2: Seminu-
merical Algorithms. 3 rd edition, Addison-Wesley, 1998.
47. D. Knuth, Letter to the Editor, Notices Amer. Math. Soc. no. 3 (M arch
1998).
48. N. Koblitz, Algebraic Aspects of Cryptography. Springer-Verlag,
1998.
49. P. C. Kocher, Timing attacks on implementations of Diffie-Hellman,
RSA, DSS, and other sys tems, in Advances in Cryp tography: Pro-
ceedings of Crypto ’96, Sprin ger-Verlag, 1996, pp. 104– 113.
50. P. C. Kocher, J. Jaffe and B. Jun, Differential p ower analysis, in Ad-
vances in Cryptography: Proc. of Crypto ’99, Springer-Verlag,
Lect. Notes Comp. Sci. 1666 (1999), pp. 388–397.
51. Synthetic Differential Geometry. Cambridge University P ress,
2nd ed., 2006 (London Math. Society Lect. Notes Ser. vol. 333). ISBN-
10: 0521687381. ISBN-13: 978-052168738 6.
52. A. N. K ol mogorov, Local structure of turbulence in an incompressible
fluid for very large Reynolds numbers, Doklady Acad Sci. USSR 31
(1941), 301–305.
53. E. Landau, Foundations of Analysis. Chelsea, 1966.
54. R. P. Langlan ds, Mathematical retrospections, http://
publications.ias.edu/rpl_works/L12/shaw/photoshell.
pdf. Accessed 5 September 2009.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
226 References
55. A. H. Lightstone, Infinitesimals. Amer. Math. Monthly 79 (1972),
242–251.
56. M. Livshits, You could simplify calculus, http://arxiv.org/abs/
0905.3611. Accessed 18 Janaury 2010.
57. J. Kemeny, Mathematics without numbers, Daedalus 88 (1959), 577–
591.
58. P. Maddy, How applied mathematics became pure, The Review Of
Symbolic Logic 1, no. 1 (2008), 16– 41.
59. Yu. I. Mani n, Mathematics and Physics. Birkhäuser, 1981.
60. J. E. Marsden and A. Weinstein, Calculus Unlimited. Benjamin-
Cummings Publishing Co., Longman, US, 1981. ISBN/ASIN:
0805369325. ISBN-13: 978080536932 8.
61. J. Mas on, L. Burton, and K. Stacey, T hinking Mathemat ically.
London: Addison-Wesley, 199 2.
62. B. Mazur, When is one thing equal to some other thing?
4
http://www.math.harvard.edu/~mazur/preprints/when_
is_one.pdf.
63. B. Maz ur, How d id Theaetetus prove his theorem? http://
www.math.harvard.edu/~mazur/preprints/Eva.pdf Accessed
6 March 201 0.
64. G. Michaelson, An Introduction to Functi onal Prog ramming
through Lambda Calculus. Addison-Wesley, 198 9.
65. J. Michell and C. Ernst, The axioms of quantity and the theory of
measurement. Translated from Part I of Otto Hölder’s German Text
“Die Axiome der Quantita
¨
t und die Lehre vom Mass”.
5
J. Math. Psy-
chology 40 no. 3 ( 1996), 235–252; no. 4 , 345–356.
66. M. Mi nsky, Automated proof of Pons Asinorum, newsgroup
post of 24 December 199 9, http://www.math.niu.edu/~rusin/
known-math/99/minsky, accessed 12 July 2009.
4
I cannot resist temptation to quote at length from The Onion,
http://www.theonion.com/content/news_briefs/professor_
sees_parallels:
University of Texas professor Thom Windham once again furthered
the cau se of human inquiry in a clas s lecture Monday, as he continued
his longtime practice of finding connections between things and other
things, pointing out these parallels, and then ela borating on them in
detail, campus sources reported.
“By drawing parallels between things and other, entirely different
things, I not only further my own studies, but als o encourage young
minds to develop this comparative methodology in their own work,”
said Windham, holding his left hand up to represent one thing, then
holding his right hand up to represent a separate thing, then bringing
his hands together in simulation of a hypothetical synthesis of the
two th ings. “It’s not just similarities that are important, though—the
differences between thin gs are also worth exploring at length.”
Fifteen years ago, Windham was awarded tenure for doing this.
5
Originally published i n 1901, Berichte über die Verhandlungen der
Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig,
Mathematisch- Physikaliche Classe 53 (1901), 1–64.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
References 227
67. R. Osserman, Mathematics of the Gateway Arch, Notices Amer.
Math. Soc. 57 no. 2 (2010), 220–229. http://www.ams.org/
notices/201002/rtx100200220p.pdf.
68. M. Nakamura and S. Kakutani, Banach limits and the C(ech com-
pactification of a countable discrete set Proc. Imp. Acad. 19 no. 5
(1943), 224–229.
69. E. Nelson, Internal s et theory: A new approach to nonstandard anal-
ysis, Bull. Amer. Math. Soc. 83 (1977), 1 165–1198.
70. E. Nel son, The syntax of non standard a nalysis, Ann. Pure and Appl.
Logic 38 (1988), 123–134 .
71. O. Ore, Number Theory and Its History. Dover Publications,
1976.
72. G. Peano, The principles of arithmetic, p resented by a new method,
in From Frege to Gödel (Jean van Heijenoort, ed.). Harvard, 1967.
73. D. Pierce, Induction a nd recursion. A talk at Logic Colloquium 2008—
ASL European Summer Meeting, Bern, Switzerland, 3–8 July 2008.
74. D. Pierce, Numbers. http://metu.edu.tr/~dpierce/
Mathematics/Numbers/. Accessed 16 April 2010.
75. G. Polya, Mathematics and Plausible Reasoning, vol. 1 (reprint
of the 1954 original) Princeton University Press, Princeton, 1990.
76. M. O. Rabin, Probabilistic algorithm for testing primality, J. Number
Theory 12 (1980), 1 28–13 8.
77. T. C. R atliff, A comparison of Dodgson’s method and the Borda count,
Economic Theory 20 (2002), 3 57–372.
78. P. Ribenboim, 13 Lectures on Fermat’s Last Theorem. Springer,
1979. ISBN-10: 0387904328; ISBN-13: 978-0387904320.
79. A. M. Robert, Nonstandard Analysis. Dover Publications, 2003.
ISBN0-486-43279-3.
80. A. Robins on, Non-sta ndard Analysis. Amsterdam, North-Holland,
1966. Later edition: A. Robinson, Non-standard Analysis (Prince-
ton Landmarks in Mathematics and Physics). Prin ceton University
Press, Princeton, 1996.
81. G. J. Simmons, How to insure that data acquired to verify treaty com-
pliance are trustworthy, Proc. IEEE 76, no. 5 (May 1988) , 621–6 27.
82. V. I. Smirnov, Course of Higher Mathematics. Vol. 1. Elsevier,
1964.
83. R. P. Stanley, E numerative Combinatorics. Vol. 2, Cambridge
University Press, 1999.
84. R. P. Stanley, Solu tions to exercises on Catalan and related num-
bers, http://www-math.mit.edu/~rstan/ec/catsol.pdf, ac-
cessed 12 July 20 09.
85. K. Sullivan, The teaching of elementary calcu lus u sing the nonstan-
dard analysis approach, Amer. Math. Monthly 83 (1976), 370–375.
86. G. H. Suter, An elementary exampl e of a transcendenta l p-adic num-
ber, Math. Magazine 49, no. 1 (1976), 4 2. Stable URL: http://www.
jstor.org/stable/2689885. A ccess ed: 25/12/200 8 07:22.
87. D. Taimina, Crocheting Adventures with Hyperbolic Planes. A
K Peters, 2009. ISBN-13: 978-15688-145-20.
88. A. Tarski, A Decision Method for Elementary Algebra and Ge-
ometry. Rand, 1948.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
228 References
89. L. van den Dries, Tame Topology and o-Minima l Structures,
London Math. Soc. Lect. Notes Ser., vol. 248. Cambridge University
Press, Cambridge, 1998.
90. P. Walker, A lemma on divisibility, Amer. Math. Monthl y 115, no. 4
(2008), 338.
91. D. Wells, The Penguin Book of Curious and Interesting Puz-
zles, Penguin Books, 1992.
92. D. Wright, Mathematics and Music. Amer. Ma th. Soc., Providence
R.I., 2009. ISBN 978-0-8218-487 3-9.
93. D. Zagier, Problems posed at the St Andrews Colloquium,
1996. http://www-groups.dcs.st-and.ac.uk/~john/Zagier/
Problems.html.
94. D. Zeilberger, “Real” analysis is a degenerate ca se of discrete analy-
sis, 2001 . http://www.math.rutgers.edu/~zeilberg/ .
Philosophy and Methodology of Mathematics
95. A. Arana, On formally measuring and eliminating extraneous no-
tions in proofs. Philosophia Mathematica 17 (2009), 208–219.
96. V. I. Arnold, What is Mathematics? Moscow, MTsNMO, 2004.
97. S. Awodey, An answer to Hellman’s question: Does Category theory
provide a framework for mathematical structuralism? Philosophia
Mathematica 12 (2004), 54–64.
98. J. D. Barrow, Pi in the Ski: Counting, Thinki ng and Being. Ox-
ford, Oxford University Press, 1987.
99. J. P. van Bendegem, Alternative mathematics: the vague way, Syn-
these 125 (2000), 19–31.
100. G. Birkhoff, Mathematics and psychology, SIAM Review 11 no. 4
(1969), 429–469.
101. R. P. Boas, Can we make mathematics intelligible? Amer. Math.
Monthly 88 no. 10 (1981), 727 –731.
102. A. V. Borovik, Is mathematics sp ecial? In PhiMSAMP. Philosophy
of Mathematics: Sociological Aspects and Mathematical Prac-
tice (Benedikt Löwe and Thomas Müller, eds.). College Publications,
London, 2010. Texts in Philosoph y 11, pp. 1–27.
103. A. V. Borovik, Mathematics under the Microscope, Amer. Math.
Soc., Providence, R.I., 2010.
104. J. A. Bracken, A new look at time a nd eternity, Theology and Science
2, no. 1 (2004), 77–8 8. DOI: 10.1080/147467 0042000196621.
105. P. Cameron, Autobiography, http://www.maths.qmul.ac.uk/
~pjc/diaries/biog.pdf. Accessed 7 September 2009.
106. J. Carter, Individuation of objects—A problem for structuralism?
Synthese 143 (2005) 291–307.
107. J. Carter, Categories for the working mathematician: making the im-
possible pos sible. Synthese 162 (2008), 1–13. DOI 10.1007/s11229–
007–9166–9.
108. J. van Cleve and R. E. Frederick, eds. The Philosophy of Right
and Left: Incongr uent Counterparts and the Nature of Space.
Kluwer, 1 991. ISBN-10: 079 23084 41. ISBN-13: 978- 07923 08447.
109. A. Comte, Catéchisme positivist. 1852.
110. D. Corfield, Towards a Philosophy of Real Mathematics. Cam-
bridge University Press, 2003. ISBN 0521817226.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
References 229
111. P. J. Davis, Fidelity in mathematical discourse: is one and one really
two? A mer. Math. Monthly 79 no. 3 1972, 252– 263.
112. J. W. Dawson, Jr. Why do mathematicians re-prove theorems?
Philosophia Mathematica 14 no. 3 (2006), 269–286.
113. S. Della Sala and R. Cubelli.
´
S“Directional apraxia”: a unitary ac-
count of mirror writing foll owing brain injury or as found in nor-
mal young children. J. Neuropsychol. 1 (2007), 3–2 6. DOI:10.1348/
174866407X180783.
114. U. Dudley, What is mathematics for? Notices of the AMS 57 no. 5
(2010), 608–613.
115. D. Fallis, Intenti onal gaps in mathematical proofs, Synthese 134
(2003), 45–69.
116. L. Feigenson, S. Dehaene, and L. Spelke, Core systems of number,
Trends in Cognitive Sciences 8 (2004), 307– 314.
117. D. Gale, Tracking The Automatic Ant: And Other Mathemati-
cal Explorations. Sprin ger, 1998. ISBN-10: 0387982728. ISBN-13:
978-0387982724.
118. M. Gardner, Is Nature ambidextrous? Philosophy and Phenomeno-
logical Resea rch 13 (December 1952), 200–211.
119. M. Gardner, The Ambidextrous Universe. W. H. Freeman, New
York, 1989.
120. J. T. Gatto, Why schools donŠt educate, Nexus 4 no. 4
(1997). http://www.seniormanagementservices.com/Images/
JohnTaylorGattoTeacherAward.pdf.
121. J. T. Gatto, Dumbing Us Down: The Hidden Curriculum of
Compulsory Schooling. New Society Publishers, 2002. ISBN-10:
0865714487. ISBN-13: 978-0865714489.
122. T. Gowers, Mathematics: A Very Short Introduction. Oxford Pa-
perbacks, 200 2. ISBN-10: 0192853619. ISBN-13: 978-0192853615.
123. A. Grothendieck, Récoltes et Semailles. http://acm.math.
spbu.ru/RS/. Accessed 17 April 2010.
124. J. W. Hadamard, Essay on the Psychology of Invention in the
Mathematical Field. Princeton, NJ: Princeton University Press,
1945.
125. M. Harris, “Why mathematics?” you might ask, Princeton Com-
panion of Mathematics (T. Gowers, ed.), Princeton University
Press, 2009, pp. 966–976.
126. R. Hersh, What is Mathematics, Really?. Vintage, London, 1998.
127. R. Hersh and V. John-Steiner, Loving and Hating Mathematics:
Challenging the Myths of Mat hematical Life. Princeton Univer-
sity Press, 2010. ISBN-13 : 978-0691142470.
128. I. Kant, On the fi rst groun d of the distinction of regions in space
(1768), in The Philosophy of Right and Left: Incongruent
Counterparts and the Nature of Space (J. van Cleve’s and R. E.
Frederick, eds.). Kluwer, 1991, pp. 27–33.
129. E. C. W. Krabbe, Strategic maneuvering in mathematical proofs, Ar-
gumentation 22 (2008), 453 –468. DOI: 10.1007/s10503-008-9098-7.
130. I. Lakatos, Proofs and Refutations. Cambridge University Press,
1976.
131. I. Lakatos, Cauchy and the continuum: the significance of nonstan-
dard analysis for the history and philosophy of mathematics. Math.
Intelligencer 1 no. 3 ( 1978), 151 –161.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
230 References
132. G. Lakoff and R. Núnez, Where Mathematics Comes From: How
the Embodied Mind Brings Mathematics Into Being. Basic
Books, New York, 2000.
133. B. Larvor, What is dialectical philosophy of mathemat-
ics? Philosophia Mathematica 9 no.2 (2001), 212–229. DOI:
10.1093/philmat/9.2.212.
134. M. Leng, Phenomenology and mathematical practice, Philosophia
Mathematica 10 no. 1 (2002), 3–2 5.
135. D. Lomas, Some initial steps toward a much-needed critical epis-
temological realis m in mathematics and science education. In:
Philosophy of Educa tion 2001 (S. Rice, ed.). University of Illi-
nois at Urbana-Champaign, 2001. http://www.ed.uiuc.edu/
EPS/PES-yearbook/2001/lomas%2001.pdf.
136. D. Lomas, What perception i s doing, and what it is not doing, in math-
ematical reasoning. British J. Philosophy of Science 53 no. 2 (2002) ,
205–223.
137. Yu. I. Manin, Mathematics as Metaphor. A mer. Math. Soc., P rovi-
dence, R.I., 2007. ISBN-13 978-0-8218-4331- 4.
138. M. Otte, Does mathematics have objects? In what sense?, Synthese
134 (2003) , 181–216.
139. J. Pérez La raudogoitia, A beautiful su p ertask. Mind 105 (417) (1996),
81–83.
140. M. Potter, Reason’S Nearest Kin: Philosophy of Arithmetic
from Kant to Carnap. Oxford University Press, 2000.
141. Y. Rav, Why do we prove theorems? Philosophia Mathematics (3) 7
(1999), 5–41.
142. G.-C. Rota, The pernicious influence of mathematics on ph ilosophy,
Synthese 88 (1991), 165–178.
143. G.-C. Rota, The phenomenology of mathematical beauty, Synthese
111 (1997) , 171–182.
144. G.-C. Rota, The phenomenology of ma thematical proof, Synthese 111
(1997), 183-196.
145. B. Rotman, Signifying Nothing : the Semiotics of Zero. Stanford
University Press, 1993.
146. S. Shapiro, Philosophy of Mathematics: Struc ture and Ontol-
ogy. Oxford University Press, 1997. ISBN 0-19- 51393 0-5.
147. D. Sherry, The wake of Berkeley’s Analyst: rigor mathematicae? Stud.
Hist. Philos. Sci. 18 no. 4 (1987), 455–48 0.
148. B. Shulman, What if we change our axioms? A feminist inquiry into
the foundations of mathematics. Configurations 4 no. 3 (1996), 427–
451. DOI: 10.1 353/con.1996.0022.
149. Z. K. Silagad z e, Zeno meets modern science, Acta Physica Polonica B
36 no. 10 (2005) , 2887–2929.
150. A. Sloman, The Computer Revolution in Philoso-
phy:Philosophy, Science and Models of Mind. Humanities
Press, 1978. ISBN 0-391-00830-7.
151. R. Sorensen, The aesthetics of mirror reversal, Philosophical Studies
100 no. 2 (2000) , 175–191.
152. B. Sriraman, The influence of Platonism on mathematics research
and theological beliefs, Theology and Science 2, no. 1 (2004), 131–
147.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
References 231
153. D. Struik, The sociology of mathematics revisited: A personal note.
Sci. and Society 5 0 (1986), 280–299.
154. N. Tennant, On the n ecessary existence of numbers. Nous 31 (1997),
307–336.
155. W. Thurston, On proof and progress in mathematics. Bulletin of the
American Mathemati cal Society 30 no. 2 (1994), 161–177.
156. R. Tieszen,Phenomenology and mathematical knowledge, Synthese
75 (1988), 373–403.
157. R. Tragesser, Gian-Carlo Rota a nd the phenomenological philosophy
of mathematics, Philosophia Mathematica (3) 8 (2000), 3–8.
158. C. Werndl, Justifying definitions i n mathematics—going beyond
Lakatos Philosophia Mathematica 17 no. 3 (2009) 313–3 40.
159. E. Wigner, The unreasonable effectiveness of mathematics in natu ral
sciences. Communicati ons in Pure and Applied Mathematics 13, no.
1 (1960), 1–14.
160. R. L. Wilder, Mathematics as a Cultural System. Pergamon, Ox-
ford, 1981. ISBN-10: 0080257968. ISBN-13: 978-00 80257 969.
History of M athematics
161. W. F. A nderson, Arithmetical computations in Roman numerals.
Classical Philol ogy 51 no. 3 (1956), 145–150.
162. Archimedes, The Works of Archimedes (Sir Thomas Heath,
transl.). Dover Publications, 20 02. (ISBN 048642084 1)
163. E. L. Bennett, Fractional quantities in Mi noan bookkeeping. Ameri-
can J. Archaeology 54 (1950), 204–222.
164. J. L. Berggren, Episodes in the Mathematics of Medieval Islam.
Springer, 2 003. ISBN 0-387-40605-0.
165. H. J. M. Bos, Differentials, higher-order differentials and the deriva-
tive in the Leibnizian calculus. Arch. History Exact Sci. 14 (1974),
1–90.
166. C. B. Boyer, Fundamental steps in the development of numeration.
Isis 35 (1944), 153–165.
167. C. B. Boyer, Note on Egyptian numeration. Mathematics Teacher 52
(1959), 127–129.
168. W. H. Brock and M. H. Price, Squared paper in the ni neteenth cen-
tury: Instrument of science and engineering, and symbol of reform in
mathematical education. Ed. Stud. Math. 1 1 no. 4 (1980), 365–381.
DOI: 10.1007/BF00231211.
169. G. Cantor, Contributions to the Founding of the Theor y of
Transfinite Numbers(P. E. B. Jourdain, transl.). Dover Publi ca-
tions, 1915.
170. P. Cartier, A mad day’s work: from Grothendieck to Connes and Kont-
sevich. The evolution of concepts of space and symmetry. Bull. Amer.
Math. Soc. 38, no. 4 (2001), 389–408.
171. J. C leave, Cauchy, convergence and continuity, Britis h. J. P hilos. Sci.
22 (1971), 27–37.
172. P. C. Cohen, A Calculating People: the History of Numeracy in
Early America. Uni versity of Chicago Press, 1982.
173. R. Cooke, The History of Mat hematics: A Brief Course.
Wiley-Interscience, 2005. ISBN-10: 0471444596. ISBN-13: 978-
0471444596.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
232 References
174. R. Cooke, Classical Algebra: Its Nature, Origins, and Uses.
Wiley-Interscience, 2008. ISBN-10: 0470259523. ISBN-13: 978-
0470259528.
175. B. Datta, Anc ient Hindu Geometry: The Science of the Sulba.
Cosmo Publications, 2003. ISBN-10: 8170204607. ISBN-13: 978-
8170204602.
176. J. W. Dauben, Georg Cantor: His Mathematics and Philoso-
phy of the Infinite. Princeton University Press, 1990. ISBN-10:
0691024472. ISBN-13: 978-0691024479.
177. J. W. Dauben, Abraham Robinson. The Creatio n o f Nonstan-
dard Analysis. The Personal and Mathematic al Odyssey.
Princeton University Press, 1995.
178. H. M. Edwards, Eu ler’s definition of the derivative, Bull. Amer. Math.
Soc. 44 ( 2007), 575–580.
179. P. Ehrli ch, The rise of non-Archimedean mathematics and the roots
of a mis conception. I. The emergence of non-Archimedean systems of
magnitudes. Arch. Hist. Exact Sci. 60 no. 1 (2006) , 1–121.
180. Theorematum quorandumad numeros primos spectantium demon-
stratio. Comment. acad. scient. Petrop. 8 (1736), 1741, p. 141–146 .
181. W. Felscher, Bolzano, Cauchy, epsilon, delta. Amer. Math. Monthly
107 no. 9 (2000) , 844–862.
182. J. V. Field, Rediscovering the Archimedean polyhedra: Piero della
Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele
Barbaro, and Johannes Kepler. Arch. Hi st. Exact Sci. 50, nos. 3–4
(1997), 241–289. DOI: 10.1007/BF00374595.
183. J. Friberg, Numbers and measures in the earliest written records.
Sci. Amer. 250 no. 2 (1984), 110–118.
184. Gal ileo Galilei, Two New Sciences, translated by Henry Crew and
Alfonso de Salvio (Dover Publications, New York, NY, 19 54). Origi-
nally publi shed by Elzevir, 1638.
185. K. F. Gauss, Disquisitiones Arithmeticae, 1801. Available at
http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=
PPN235993352, accessed 19 december 2010.
186. A. Givental, The Pythagorean Theorem: What is it about? Amer.
Math. Monthly 113 no. 3 (2006), 261–265.
187. A. Glaser, Histo ry of Binary and Other Nondecimal Numera-
tion. 1971, out of print, but available at http://www.eipiphiny.
org/books/history-of-binary.pdf.
188. I. Grattan-Guinness, Nu mbers, magnitudes, ratios and proportions
in Euclid’s Elements: how did he han d le them? Hist. Math. 23 (1996),
355–375. [Printing correction: 24 (1997) 213.]
189. I. Grattan-Guinnes s, Th e mathematics of the past: distinguishi ng its
history from its herita ge, Historia Mathematica 31 (2004), 163–185.
190. I. Grattan-Guinness, On certain somewhat neglected featu res of the
history of mathematics, Arch. Int. Hist. Sci. 55 (2005), 383—402.
191. I. Ha cking, The Emergence of P robability: A Philosophical
Study of Early Ideas About Probability, Induction and Sta-
tistical Inference. Cambrid ge University Press, 19 75.
192. A. Heeffer, Sources in the history of algebra before 1600, http://
logica.ugent.be/albrecht/math.php. Accessed 9 July 2010.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
References 233
193. L. Howard, On a periodical variation of the ba rometer, apparently
due to the influence of the sun and moon on the atmosphere, Philo-
sophical Magazine 7 (1800), 355–363.
194. J. Høyrup, Lengths, Widths, Surfaces. A Portrait of Old Babylo-
nian Algebra and Its Kin. Springer, 2002. ISBN-10: 03 87953035.
ISBN-13: 978- 03879 53038.
195. J. Høyrup, Jacopo Da Firenze’s Tractatus A lgorismi and Ea rly
Italian Abbacus Culture. Birkhäu ser, 2007. ISBN-10: 376438390 9.
ISBN-13: 978- 37643 83909.
196. V. J. M. Huguet and A. C. Hernán d ez, Euler y la teoría de números,
Cap. 6 in El Legado Matemático de Le onhard Euler a Tre-
scieñtos Anos de su Nacimiento (A. A. Meneses, J. Delgado,
and F. M. Pérez, Eds.). Universidad Autónoma Metropolitana, Mex-
ico, 2007, pp. 147–179. Available at http://mx.oocities.com/
alfons_physikvier/EulerBuch.pdf, accessed on 19 December
2010.
197. D. Jesseph, Lei bniz on the foun d ations of the calculus: the question
of the reality of infinitesimal magnitudes. Leibniz and the sciences.
Perspect. Sci. 6 no. 1–2 (1998), 6–40.
198. G. G. Joseph, The Crest of the Pe acock: Non-European Roots of
Mathematics. Tauris, London, 1991. ISBN-10: 1850432856. ISBN-
13: 978-1850432852.
199. G. G. Joseph, A Passage to Infinity: Medieva l Indian Mathemat-
ics from Kerala and Its Impact. Sage Publications Pvt. Ltd, 2009.
ISBN-10: 81321016 85. ISBN-13: 978-8132101680.
200. I. Kleiner, Hi story of the infinitely small and the infinitely large in
calculus, Ed. Stud. Math. 48 (2001), 137–174.
201. J. Kozhamthadam, Jesuit contribution to the origin and development
of modern science and mathematics. Indian J. Hist. Sci. 42 (2007),
13–30.
202. L. Kvasz, History of geometry and the development of the form of its
language, Synthese 116 (1998), 141–186.
203. F. Lasserre, The Birth of Mathematics in the Age of Plato.
Hutchinson, London, 1964.
204. D. Laugwitz, Definite values of infinite sums: aspects of the founda-
tions of infinitesimal analysis around 18 20. Arch. Hist. Exact Sci. 39
no. 3 (19 89), 195–245.
205. D. Laugwitz, Cauchy-Zahlen als Grundlage der Infinitesimalmathe-
matik. Mathematische Semesterberichte 38 (1991 ), 175–213.
206. D. Laugwi tz, On the historical development of infinitesimal mathe-
matics. Amer. Math. Monthly 10 4 no. 5 (1997), 447–455 .
207. D. Laugwitz, On the historical development of infinitesimal math-
ematics. Parti II. the conceptual thinking of Cauchy. Amer. Math.
Monthly 104 no. 7 (1997), 654 –663.
208. G.-G. Leibnitz, Explication de l’arithmétique binaire, qui se sert des
seuls caractéres O et I avec des remarques sur son utilité et sur ce
qu’elle donne le sens des anci ennes figures chinoises de Fohy, Mé-
moires de mathématique et de physique de l’Académie royale des sci-
ences 1703, 85 –89.
209. A. H. Lighthouse, Infinitesemals, Amer. Math. Monthly 79 (1972),
242–251.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
234 References
210. N. N. Luzin, Two letters of N. N. Luzin to M. Ya. Vygodski
˘
i (with
an introduction by S. D. Demidov, translated by A. Shenitzer), Amer.
Math. Monthly 107 (January 2000), 64–82.
211. K. O. May, Historiographic vices. I. Logical attribution. Hist. Math. 2
(1975), 185-187.
212. K. O. M ay, Historiographic vices. II. Priority chasi ng. Hist. Math. 2
(1975), 315–317.
213. K. O. May, What is good history and who shoul d do it? Hist. Math. 3
(1976), 449–455.
214. F. A. Medvedev, No nstandard Analysis and the History of Clas-
sical Analysis. Patterns in the Development of Modern Math-
ematics. “Nauka”, Moscow, 1987, pp. 75–84. (Russian.)
215. F. A. Medvedev, N onstandard analysis and the history of classical
analysis. Amer. Math. Monthly 105 no. 7 (1998), 659–664.
216. R. N etz, The Shapi ng of Deduction in Greek Mathematics: A
Study in Cognitive History. Cambridge University Press, 1999.
ISBN-10: 05216227 94. ISBN-13: 978-0521622790.
217. The Penny Cyclopedia of the Society for the Diffusion of Use-
ful Knowledge, vol. 12. London: Charles Knight and Co., 22,
Ludgate Street, 1838. A scanned image of the original is avail-
able on GOOGLE BOOKS, http://books.google.com/books?
id=DoyETlgz7QYC&printsec=frontcover&dq=editions:
UCM5319406755&lr=#v=onepage&q&f=false.
218. M. A. Peterson, Galileo’s discovery of scaling laws, American
J. Physics 70 no. 6 (June 2 002), 575–580. arXiv:physics/
0110031v3.
219. H. M. Pycior, Augustus De Morgan’s algebraic work: the th ree stages,
Isis 74 no. 2 (1983 ) , 211–226.
220. S. Restivo, Mathematics in Soci e ty and History. Kluwer, 1992.
221. M. Reynolds, The octagon in L eona rdo’s drawings, N exus N etwork J.
10, no. 1 (2008), 51– 76. DOI: 10. 1007/ S0 0004-007-0056-8.
222. J. L. Richards, Augustus De Morgan, the hi story of mathematics, and
the foundations of algebra, Isis 78, no. 1 (198 7), 7– 30. Stable URL:
http://www.jstor.org/stable/232727.
223. S. Rose, Mathematics an d the art of navigation: the ad vance of sci-
entific seamanship in Elizabethan England, Tra ns. Roy. Hist. Soc. 14
(2004) 175 –184.
224. L. A. Sad, M. V. Teixeira, and R. R. Baldi no, Cauchy and the problem
of point-wis e convergence. Archive Internationales d’Histoire des Sci-
ences 51, no. 147 (2001), 27 7–308 .
225. J. Sesia no, An Introduction to the History of Algebra: Solv-
ing Equatio ns from Mesopotamian Times to the Renaissance.
Amer. Math. Soc., Providence, R. I. 2009. ISBN 978-0-8218-4473-1.
226. S. Tanaka, Boltzmann on mathematics, Synthese 119 (1999), 203–
232.
227. I. Thomas, Selections Illustrating the History o f Greek Mathe-
matics with an English Translation by Ivor Thomas. Vol. 1:
From Thales to Euclid. Ha rward U niversity Press, Cambridge,
Ma. 1938 (reprinted 1980). ISBN 0-674-99369-1.
228. C. Vaughan and M. O’Malley, Froude and the contribution of naval
architecture to our understanding of bipedal locomotion, Gait and
Posture 21, no. 3 ( 2005), 350 –362.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK