What Is Geometry?
Geometry is perhaps the most elementary of the sciences that enable man, by purely intellectual processes, to make predictions (based on observation) about physical world. The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry. |
H. M. S. Coxeter (1907-2003) |
Perhaps it may be asserted, that there are no difficulties in geometry which are likely to place a serious obstacle in the way of an intelligent beginner, except the temporary embarrassment which always attends the commencement of a new study ... |
A. De Morgan (1806-1871) |
And, for geometry, till of very late times it had no place at all (at universities), as being subservient to nothing but rigid truth. And if any man by the ingenuity of his own nature had attained to any degree of perfection therein, he was commonly thought of a magician and his art diabolical. |
Thomas Hobbes (1588-1679) |
Geometry is a branch of mathematics that is concerned with the properties of configurations of geometric objects - points, (straight) lines, and circles being the most basic of these. Although the word geometry derives from the Greek geo (earth) and metron (measure) [Words], which points to its practical roots, Plato already knew to differentiate between the art of mensuration which is used in building and philosophical geometry [Philebus (57)]. Earlier in the dialogue [Philebus (51)], Socrates mentions the matter of beauty:
I do not mean by beauty of form such beauty as that of animals or pictures, which many would suppose to be my meaning; but, says the argument, understand me to mean straight lines and circles, and the plane or solid figures which are formed out of them by turning-lathes and rulers and measures of angles; for these I affirm to be not only relatively beautiful, like other things, but they are eternally and absolutely beautiful, and they have peculiar pleasure, quite unlike the pleasures of scratching. |
In another dialogue - Phaedrus (274) - Socrates ascribes creation of geometry, albeit in a company of other arts, to the god Theuth who resided in the Egyptian city of Naucratis. Truth be told, Phaedrus questions Socrates' account right away: "Yes, Socrates, you can easily invent tales of Egypt, or of any other country." But even if not of divine origin, the objects of geometry are not to be found in the physical world. They are pure abstractions, creations of the human mind.
Around 300 BC, Euclid gave the definitions of points and lines that withstood two millennia of diligent study. The mathematicians of the 19^{th} found them lacking. According to Euclid, A point is that which has no part. As F. Klein [Klein, p. 196] notes "a point is by no means determined by this property alone." According to Euclid, A line is length without breadth. Even if length and breadth are accepted as the basic notions, Euclid's definition conflicts with the existence of curves that cover a surface [Klein, p. 196]. According to Euclid, A straight line is a line which lies evenly with respect to its points, which Klein [ibid] finds completely obscure. Klein goes to considerable length to uncover and explain the deficiencies in Euclid's Elements. A less benevolent but still very accessible critique, was given by B. Russell and can be found in C. Pritchard's The Changing Shape of Geometry [Pritchard, pp. 486-488]. Klein, for example, notes that such a simple proposition as the statement that two circles each passing through the center of the other meet in two points is not derivable from Euclid's postulates without a leap of faith.
Modern mathematics found two ways to remedy the deficiencies and place geometry on a sound foundation. First, mathematicians have perfected the axiomatic approach of Euclid's Elements. They came to a realization that it's impossible and in fact futile to attempt to define such basic notions as points and lines. In analytic geometry, on the other hand, both points and lines are perfectly definable. However, analytic geometry contains no "geometric axioms" and is built on top of the theory of sets and numbers.
The most influential work on the axiomatization of geometry is due to D. Hilbert (1862-1943). In Foundations of Geometry, that appeared in 1899, he listed 21 axioms and analyzed their significance. Hilbert's axioms for plane geometry can be found in an appendix to [Cederberg, pp. 205-207] along with an unorthodox, but short, axiomatization by G. D. Birkhof [Birkhof, Cederberg, pp. 208-209] and a later one, influenced by that of Birkhof, by the S.M.S.G. (School Mathematics Study Group) [Cederberg, pp. 210-213]. (The School Mathematics Study Group has been set up in the 1960s as a response to the success of the Soviet space program and the perceived need to improve on math education in the US. The effort led to the now defunct New Math program.)
Unlike Euclid's Elements, modern axiomatic theories do not attempt to define their most fundamental objects, points and lines in case of geometry. The reason is nowadays obvious: all possible definitions would apparently include even more fundamental terms, which would require definitions of their own, and so on ad infinitum. Instead, the comprehension of the fundamental, i.e. undefined, terms builds on their use in the axioms and their properties as emerge from subsequently proved theorems. For example, the claim of existence of a straight line through any two points, that of the uniqueness of such a line or the assertion that two lines meet in at most one point, tell us something about the points and the lines without actually defining what these are. (The first two are Hilbert's axioms I.1 and I.2, while the last one is a consequence of the first two.)
The usage of the undefined terms, in the above paragraph, certainly meets our expectation and intuition of the meaning of the terms points and lines. However, depending on intuition may be misleading, as, for example, in projective geometry, according to the Duality Principle, all occurrences of the two terms in the axioms and theorems are interchangeable.
Modern geometry is thus a complete abstraction that crystallizes our ideas of the physical world, i.e., to start with. I say "to start with", because most of the edifice built on top of the chosen axioms, does not reflect our common experiences. Mathematicians who work with the abstract objects develop an intuition and insights into a separate world of abstraction inhabited by mathematical objects. Still, their intuition and the need to communicate their ideas are often fostered by pictorial representation of geometric configurations wherein points are usually represented by dots and straight lines are drawn using straightedge and pencil. It must be understood that, however sharpened a pencil may be, a drawing is only a representation of an abstract configuration. Under a magnifying glass, the lines in the drawing will appear less thin, and their intersection won't look even like a dot thought to represent an abstract point.
If it were at all possible, placing a magnifying glass in front of our mind's eye would not change the appearance of points and lines, regardless of how strong the magnification could be. This is probably not very different from the meaning Euclid might have meant to impute to the objects he had tried to define. The difference is not in the imaging of the geometric objects, but in the late realization that the definition is not only not always possible, it may not even be necessary for a construction of a theory.
As a word of precaution, the diagrams supply an important tool in geometric investigations, but may suggest wrongful facts if not accompanied by deductive reasoning. (Worse yet, faulty deductive reasoning may accidently lead to correct facts in which case you may be left oblivious of the frivolous ways in which a correct fact had been obtained.)
The second approach to resolving inconsistencies in the Elements came with the advent of analytic geometry, a great invention of Descartes and Fermat. In plane analytic geometry, e.g., points are defined as ordered pairs of numbers, say,
There are many geometries. All of these share some basic elements and properties. Even finite geometries deal with points and lines and universally just a single line may pass through two given points. Thus I believe that a frequently used term "Taxicab Geometry" is a misnomer. The taxicab metric is a useful mathematical concept that turns the plane into a metric space - in one way of many. Which, still, does not make it a geometry.
References
- G. D. Birkhoff and R. Beatley, Basic Geometry, AMS Chelsea Publ., 2000, 3^{rd} edition
- D. A. Brannan, M. F. Esplen, J. J. Gray, Geometry, Cambridge University Press, 2002
- J. N. Cederberg, A Course in Modern Geometries, Springer, 1989
- A. De Morgan, On the Study and Difficulties of Mathematics, Dover, 2005
- D. Hilbert, Foundations of Geometry, Open Court, 1999
- T. Hobbes, Leviathan, ch. 46, Penguin Classics, 1982
- B. Jowett, The Dialogues of Plato, Random House, 1982
- F. Klein, Elementary Mathematics from an Advanced Standpoint: Geometry, Dover, 2004
- D. Pedoe, Geometry: A Comprehensive Course, Dover, 1988
- C. Pritchard (ed.), The Changing Shape of Geometry, Cambridge University Press, 2003
- S. Roberts, King of Infinite Space, Walker & Company, 2006
- S. Schwartzman, The Words of Mathematics, MAA, 1994
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