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Foundations of Geometry

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David Hilbert's "Grundlagen der Geometrie" is a work of great significance for anyone interested in mathematical foundationalism, the history of geometry, and intellectual history and philosophy in general. Sadly, however, the translation of this edition is extremely poor --- not simply akward, or rough, but careless to the point of making the text unreadable. If I did not have access to the German original, I would have long ago given up on making sense of the translation. In Theorem 7, for example, it speaks of "points which are not on the plane alpha." The German is extremely ambiguous, but mathematically it only makes sense if you interpret the sentence as referring to the "line a." On page 31, the translator commits the unpardonable error of mistaking "nun" (now) for "nur" (only). At the end of theorem 34, and entire equation was left out, and the meaning of the sentence completely bungled. Most extraordinary is Theorem 35, where what should be translated as "It follows from Theorem 22 that the sum of two angles of a triangle is less than two right angles" becomes "the sum of the angles of a triangle is less than two right triangles." In the very next sentence, "mithin" is interpreted as "hence," implying a direct logical entailment where there is none. It should have been rendered simply as "of course." Finally, in the next paragraph, it reads "where epsilon denotes any angles." The German has "irgendeinen Winkel" --- unambiguously singular.

Given the tremendous importance of Hilbert's Foundations, it is quite sad that there is not a quality translation available.

This historic book is available for free from Project Gutenberg http://www.gutenberg.org. Search for Geometry. This book is one of a few books available. This is the complete Open Court text. It is available both as a pdf file and a TeX file.

Hilbert gives his new system of axioms and studies their consistency, independence and necessity. Consider for example the theorem that the angle sum in any triangle cannot be greater than two right angles. We can prove it as follows. Consider a triangle ABC with the angles labelled so that ABC<=ACB. Let D be the midpoint of BC. Draw AD and extend it to E so that AD=DE. By SAS, ACD=BDE, so that angle CAD=angle BAE and angle DBE=angle ACB. Thus ABC has the same angle sum as ABE. ABC<=ACB means that AC=BE<=AB, so angle BAE<=angle AEB, so angle BAE<=angle BAC/2. In other words: for any angle A in any triangle we can construct a new triangle with equal angle sum that has as one of its angles A/2. By repeating this process we can make the angle A as small as we like. Thus, if the angle sum of some triangle was greater than two right angles, and we applied this procedure, we would get a new triangle where two of the angles are greater than two right angles, which is impossible. The "as small as we like" part gives away the fact that we are relying on Archimedes' axiom, which is necessary. "The investigation of this matter which [Max] Dehn has undertaken at my urging led to a complete clarification of this problem. ... If Archimedes' axiom is dropped then from the assumption of infinitely many parallels through a point it does not follow that the sum of the angles in a triangle is less than two right angles. Moreover, there exists a geometry (the non-Legendrian geometry) in which it is possible to draw through a point infinitely many parallels to a line and in which nevertheless the theorems of Riemannian (elliptic) geometry hold. On the other hand there exists a geometry (the semi-Euclidean geometry) in which there exists infinitely many parallels to line through a point and in which the theorems of Euclidean geometry still hold. From the assumption that there exist no parallels it always follows that the sum of the angles in a triangle is greater than two right angles." Another interesting topic is the connection between laws of algebra and the theorems of Pappus (which Hilbert calls Pascal's) and Desargues. Geometrically, we can multiply two numbers a and b using only the axioms of projective geometry as follows. We choose a line to be the "x-axis" and call one of its points the origin O and another of its points the unit 1. Mark Oa and Ob on this line. Draw another line, the "y-axis", through O. Pick some point i on the y-axis. Connect 1 and i, and draw the parallel to this line through b, meeting the y-axis at b' (as usual, "parallel to l" means: meets l at an arbitrarily designated line called the line at infinity). Connect a and 1 and draw the parallel to this line through b'. In Euclidean geometry this line cuts the x-axis at ab. In general, then, we may define multiplication in this way. The algebraic identity ab=ba now becomes a geometric theorem. This is the beautiful part: ab=ba is not just any old geometric theorem, it is in fact equivalent to Pappus's theorem: the construction of ab consisted of the line connecting 1 and i and three more lines, the construction of ba consists of the line connecting 1 and i and three more lines, each of which is parallel to one of the lines from the ab construction. Therefore, deleting the line connecting 1 and i, Pappus applies and says ab=ba. Similarly, Desargues is equivalent to a(bc)=(ab)c.

Unlike other books of geometry , the author of this book constructed geometry in a axiomatic method . This is the feature which differ from other books of geometry and the way I like . Let's see how the author constructed axiomization geometry . Intuition and deduction are two powerful ways to knowledge . The axioms are the intuitive principles which are needless to be proved . The theorems are the demonstrated propositions which are deduced from axioms . Although axioms are intuitive , they may have the demonstrated propositions called theorems which contradict . If they do , the system of the axiomization geometry would break down . Because it has some false propositions if you think the contradictory ones as truth , and vice versa . There are all the discussions of the problems above in chapter 2 called consistency which is very important in an axiomatic system .