Notes....

1. ^ No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid).
2. ^ Steen, L.A. (April 29, 1988). The Science of Patterns Science, 240: 611–616. and summarized at Association for Supervision and Curriculum Development, ascd.org
3. ^ Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 978-0-7167-5047-5
4. ^ Jourdain.
5. ^ Peirce, p. 97.
6. ^ ^{a b} Einstein, p. 28. The quote is Einstein's answer to the question: "how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" He, too, is concerned with The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
7. ^ Eves
8. ^ Peterson
9. ^ Both senses can be found in Plato. Liddell and Scott, s.voceμαθηματικός
10. ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub "mathematics", "mathematic", "mathematics"
11. ^ S. Dehaene; G. Dehaene-Lambertz; L. Cohen (Aug 1998). "Abstract representations of numbers in the animal and human brain". Trends in Neuroscience 21 (8): 355–361. doi:10.1016/S0166-2236(98)01263-6. 
12. ^ See, for example, Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim
13. ^ Kline 1990, Chapter 1.
14. ^ "A History of Greek Mathematics: From Thales to Euclid". Thomas Little Heath (1981). ISBN 0-486-24073-8
15. ^ Sevryuk
16. ^ Johnson, Gerald W.; Lapidus, Michel L. (2002). The Feynman Integral and Feynman's Operational Calculus. Oxford University Press. ISBN 0821824139. 
17. ^ Eugene Wigner, 1960, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Communications on Pure and Applied Mathematics 13(1): 1–14.
18. ^ Mathematics Subject Classification 2010
19. ^ Hardy, G. H. (1940). A Mathematician's Apology. Cambridge University Press. ISBN 0521427061. 
20. ^ Gold, Bonnie; Simons, Rogers A. (2008). Proof and Other Dilemmas: Mathematics and Philosophy. MAA. 
21. ^ Aigner, Martin; Ziegler, Gunter M. (2001). Proofs from the Book. Springer. ISBN 3540404600. 
22. ^ Earliest Uses of Various Mathematical Symbols (Contains many further references).
23. ^ Kline, p. 140, on Diophantus; p.261, on Vieta.
24. ^ See false proof for simple examples of what can go wrong in a formal proof. The history of the Four Color Theorem contains examples of false proofs accidentally accepted by other mathematicians at the time.
25. ^ Ivars Peterson, The Mathematical Tourist, Freeman, 1988, ISBN 0-7167-1953-3. p. 4 "A few complain that the computer program can't be verified properly", (in reference to the Haken-Apple proof of the Four Color Theorem).
26. ^ Patrick Suppes, Axiomatic Set Theory, Dover, 1972, ISBN 0-486-61630-4. p. 1, "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects."
27. ^ Zeidler, Eberhard (2004). Oxford User's Guide to Mathematics. Oxford, UK: Oxford University Press. p. 1188. ISBN 0198507631. 
28. ^ Waltershausen
29. ^ Shasha, Dennis Elliot; Lazere, Cathy A. (1998). Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Springer. p. 228. 
30. ^ Popper 1995, p. 56
31. ^ Ziman
32. ^ "The Fields Medal is now indisputably the best known and most influential award in mathematics." Monastyrsky
33. ^ Riehm
34. ^ Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics, Oxford University Press, 2005.
35. ^ Clay Mathematics Institute, P=NP, claymath.org
36. ^ Like other mathematical sciences such as physics and computer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians