Abstract
This chapter contains a brief summary of several types of mathematical knowledge needed to read this book, including the elements of logic, set theory, mapping theory, and algebraic structures such as number systems and vector spaces. Definitions of basis, dimension, linear mappings, isomorphism, matrices and determinants are given; there is also discussion of the roles of axioms, theorems, and definitions in a mathematical theory. The main development of the book begins here with the statement of eight incidence axioms and proof of a few theorems including one from Desargues.
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Notes
- 1.
Square bracketed numbers refer to entries in References, just before the Index.
- 2.
After the French mathematician and philosopher René Descartes (1596–1650), inventor of the Cartesian coordinate system, which we will explore in the later chapters of this work. He has been called “the father of modern philosophy.”
- 3.
You might prefer to think of a mapping as a rule which associates a single second element f(x) with each first element x. Our definition is a bit more formal, as it does not depend on the undefined notion of a “rule.”
- 4.
Technically, the condition (G1) is redundant here because the definition of an operation requires that the result be a member of the same set.
- 5.
After the Norwegian mathematician Niels Henrik Abel (1802–1829).
- 6.
More generally, an algebraic number is a complex number that is a root of a polynomial equation with rational coefficients. (A treatment of complex numbers is available in a supplement accessible from the home page of this book at www.springer.com.) The set of all algebraic numbers is a subfield of the complex numbers, as is the set of real numbers, and their intersection is the subfield of real algebraic numbers described here. The field of algebraic numbers is algebraically closed, meaning that any root of a polynomial with coefficients from the field is also a member of the field. The field of real algebraic numbers is not algebraically closed, as is readily seen from the fact that the polynomial equation \(x^{2} = -1\) has no real solution.
- 7.
The proof is essentially that found in Halmos, Finite Dimensional Vector Spaces [9], pp. 9–14.
- 8.
There are a number of good books on vector space theory; Finite Dimensional Vector Spaces [9], by Paul Halmos (1916–2006), remains a classic. Originally published by Van Nostrand in 1958, it is still in print from Springer.
- 9.
We use the words “term” and “word” interchangeably.
- 10.
“When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable; the expressions in this group we call PRIMITIVE TERMS or UNDEFINED TERMS, and we employ them without explaining their meanings. At the same time we adopt the principle: not to employ any of the other expression[s] of the discipline under consideration, unless its meaning has first been determined with the help of primitive terms and of such expressions of the discipline whose meanings have been explained previously…” Alfred Tarski, Introduction to Logic: and to the Methodology of Deductive Sciences, 4th ed., page 118, Dover (1995) [21].
- 11.
Except for the third named author who, living in Michigan, routinely walks on (solidified) water.
- 12.
The following has been attributed to David Hilbert, as a way of saying that in proving geometric theorems we must use only the axioms, rather than any “real” interpretation of geometric objects: “One must be able to say at all times—instead of points, straight lines, and planes—tables, beer mugs, and chairs.”
References
Bachmann, F.: Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edn. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1973). ISBN 978-3642655388
Behnke, H., Bachmann, F., Fladt, K., Kunle, H. (eds.): Fundamentals of Mathematics, vol. 2. MIT, Cambridge (1974). ISBN 978-0262020695. Chapter 3, Affine and Projective Planes, R. Lingenberg and A. Bauer; Chapter 4, Euclidean Planes, J. Diller and J. Boczec; also Chapter 5, Absolute Geometry, F. Bachmann, W. Pejas, H. Wolff, and A. Bauer; tr. by S. Gould
Birkhoff, G.D., Beatley, R.: Basic Geometry, 2nd edn. AMS Chelsea Publishing Co., New York (2000, Reprint). ISBN 978-0821821015
Burton, D.M.: The History of Mathematics: An Introduction, 7th edn. McGraw Hill, New York (2010). ISBN 978-0073383156
Doneddu, A.: Étude de géométries planes ordonnées. Rend. Circ. Mat. Palermo Ser. II 19(1), 27–67 (1970). ISSN 0009-725X
Euclid of Alexandria: Elements. The Thirteen Books of Euclid’s Elements (3 vols., tr. by Thomas L. Heath, Cambridge University Press, 1925 edn.) (facsimile). Dover, New York (1956). ISBN 978-0486600888 (vol. 1), ISBN 978-0486600895 (vol. 2), ISBN 978-0486600901 (vol. 3); also available in a single volume from Digireads.com. ISBN 978-1420934762
Ewald, G.: Geometry: An Introduction. Ishi Press (2013). ISBN 978-4871877183
Greenberg, M.J.: Euclidean and Non-Euclidean Geometries, Development and History, 4th edn. W.H. Freeman and Co., New York (2008). ISBN 978-0716799481
Halmos, P.R.: Finite Dimensional Vector Spaces, 2nd edn. Springer, New York (1974, 1987). ISBN 978-0387900933
Hilbert, D.: Foundations of Geometry, 2nd edn. (tr. by P. Bernays). Open Court, Chicago (1971). Also available via Project Gutenberg at http://www.gutenberg.org/ebooks/17384 in a translation by E.J. Townsend, Open Court (1950). ISBN 978-0875481647
Holme, A.: Geometry, Our Cultural Heritage, 2nd edn. Springer, Heidelberg (2010). ISBN 978-3642144400
Karzel, H., Sperner, E.: Begründer einer neuen Ordnungstheorie. Mitteilungen der Mathematischen Gesellschaft in Hamburg, Hamburg 25, 33–44 (2006)
Lumiste, U.: Foundations of Geometry. Estonian Mathematical Society, Tartu (2009)
Martin, G.E.: Transformation Geometry: An Introduction to Symmetry, Chapter 14. Springer, New York (1982). ISBN 978-0387906362
Millman, R.S., Parker, G.D.: Geometry, a Metric Approach with Models, 2nd edn, Chapter 10. Springer, New York (1993). ISBN 978-0387974125
Moulton, F.R.: A simple non-desarguesian plane geometry. Trans. Am. Math. Soc. 3(2), 192-195 (1902). American Mathematical Society, Providence. ISSN 0002-9947, JSTOR 1986419
Pasch, M.: Vorlesungen Über Neuere Geometrie (Lectures Over More Recent Geometry), Ulan Press (2012, English reprint) ASIN: B009ZJXQ3A
Rosen, K.: Discrete Mathematics and Its Applications, 7th edn. McGraw Hill, Boston (2011). ISBN 978-0073383095
Sperner, E.: Die Ordnungsfunktionen einer Geometrie. Mathematische Annalen (Springer) 121, 107–130 (1949)
Tarski, A.: What is elementary geometry? In: Henkin, L., Suppes, P., Tarski, A. (eds.) The Axiomatic Method, with Special Reference to Geometry and Physics. Proceedings of an International Symposium at the Univ. of Calif., Berkeley, 26 December 1957–4 January 1958. Studies in Logic and the Foundations of Mathematics, pp. 16-29. North-Holland, Amsterdam. Brouwer Press (2007, Reprint). ISBN 978-1443728126. MR0106185
Tarski, A.: In: Tarski, J. (ed.) Introduction to Logic: And to the Methodology of Deductive Sciences, 4th edn. Dover, New York (1995). ISBN 978-0486284620
van der Waerden, B.L.: De logische grondslagen der Euklidische meetkunde (Dutch). Chr. Huygens 13, 65–84, 257–274 (1934)
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Specht, E.J., Jones, H.T., Calkins, K.G., Rhoads, D.H. (2015). Preliminaries and Incidence Geometry (I). In: Euclidean Geometry and its Subgeometries. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-23775-6_1
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