Grundlagen der Geometrie Classic Reprint German Edition David Hilbert 9780259542384 Books
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Excerpt from Grundlagen der Geometrie
Die vorliegende Untersuchung ist ein neuer Versuch, für die Geometrie ein vollständiges und möglichst ein f aches System von Axiomen aufzustellen und aus denselben die wichtigsten geometrischen Sätze in der Weise abzuleiten, daß dabei die Bedeutung der verschiedenen Axiomgruppen und die Tragweite der aus den einzelnen Axiomen zu zie henden Folgerungen möglichst klar zutage tritt.
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Grundlagen der Geometrie Classic Reprint German Edition David Hilbert 9780259542384 Books
Part of the reason I bought this book was to employ this as my companion to reading Euclid in Greek: Volume 1: Book I.Although the commentaries of Heath on the Greek text by Euclid are very instructive on the whole, not all can be accepted without reservation.
As an example, the comment on the Definition 7 (όρος ζ’) etymologically decomposes the word επίπεδος into επι and πεδ, where πεδ is claimed as the root of πους in view of the conclusion that the word conveys the idea of ‘something that you can stand on’. My French literature on this language however lists the root of πους as ποδ (not πεδ), shows no declination in the form of πεδ for the word, and decomposes επίπεδος into επι and πέδον (i.e., on earth).
As another example, the commentator affirms that the fact that there is only one straight line connecting two points is implicit in the first postulate (αίτημα α’) so that the axiom to that effect ‘which was interpolated among the genuine Euclidean axioms is unnecessary.’
One however wonders if the implicit assumption that Heath detects in the first postulate was as obvious to the man who took all the pains to define the straight line (ευθεία).
As a matter of fact, the geometers of almost two and a half millennia later, whom David Hilbert represents in the form of this book, thought it necessary to make this an axiom, right after the axiom that corresponds to the first Euclidean postulate.
In reality, the pair of books turned out each to be an excellent companion to the other, Sancho Panza to Don Quixote so to speak, or Sam Gamgee to Frodo Baggins if you prefer, shedding light not only on each other, but also in front as a pair of torch lights to illuminate the ever so steep and treacherous a path across the magnum opus of Emmanuel Kant, one of the reasons that I began to read Euclid.
Indeed, while Euclid takes pains to define points, lines, planes, angles and figures before going on to deduce the relations between them, the geometers of the early 20th century go the other way around and axiomatize the relations without ever bothering to define those fundamental parts of Space. It is then fascinating to observe that out of those abstract relations emerge theorems that mirror the familiar concepts that we instinctively associate with those parts, a fact that invites one to adopt a critical view toward what the German philosopher terms Die transscendentale Aesthetik, and in particular toward Die Mathematik der Ausdehnung (of extension, cf. Critique of Pure Reason (Penguin Classics)).
Time consuming to read, but time worth consuming to read.
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Grundlagen der Geometrie Classic Reprint German Edition David Hilbert 9780259542384 Books Reviews
Part of the reason I bought this book was to employ this as my companion to reading Euclid in Greek Volume 1 Book I.
Although the commentaries of Heath on the Greek text by Euclid are very instructive on the whole, not all can be accepted without reservation.
As an example, the comment on the Definition 7 (όρος ζ’) etymologically decomposes the word επίπεδος into επι and πεδ, where πεδ is claimed as the root of πους in view of the conclusion that the word conveys the idea of ‘something that you can stand on’. My French literature on this language however lists the root of πους as ποδ (not πεδ), shows no declination in the form of πεδ for the word, and decomposes επίπεδος into επι and πέδον (i.e., on earth).
As another example, the commentator affirms that the fact that there is only one straight line connecting two points is implicit in the first postulate (αίτημα α’) so that the axiom to that effect ‘which was interpolated among the genuine Euclidean axioms is unnecessary.’
One however wonders if the implicit assumption that Heath detects in the first postulate was as obvious to the man who took all the pains to define the straight line (ευθεία).
As a matter of fact, the geometers of almost two and a half millennia later, whom David Hilbert represents in the form of this book, thought it necessary to make this an axiom, right after the axiom that corresponds to the first Euclidean postulate.
In reality, the pair of books turned out each to be an excellent companion to the other, Sancho Panza to Don Quixote so to speak, or Sam Gamgee to Frodo Baggins if you prefer, shedding light not only on each other, but also in front as a pair of torch lights to illuminate the ever so steep and treacherous a path across the magnum opus of Emmanuel Kant, one of the reasons that I began to read Euclid.
Indeed, while Euclid takes pains to define points, lines, planes, angles and figures before going on to deduce the relations between them, the geometers of the early 20th century go the other way around and axiomatize the relations without ever bothering to define those fundamental parts of Space. It is then fascinating to observe that out of those abstract relations emerge theorems that mirror the familiar concepts that we instinctively associate with those parts, a fact that invites one to adopt a critical view toward what the German philosopher terms Die transscendentale Aesthetik, and in particular toward Die Mathematik der Ausdehnung (of extension, cf. Critique of Pure Reason (Penguin Classics)).
Time consuming to read, but time worth consuming to read.
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