Abstract
English translation of Moritz Pasch,“Physikalische und mathematische Geometrie,” Annalen der Philosophie mit besonderer Rücksicht auf die Probleme der Als-Ob-Betrachtung 3 (1922), pp. 362–374. Pasch discusses “the fundamental differences that separate physical from mathematical geometry.” In the former, points are understood to be physical objects. In the latter, the term “point” is an undefined expression in a system of axioms. Physical points can move. Mathematical points cannot, although, in mathematical geometry, “intuitive” motions have “purely intellectual” counterparts: transformations. Because our instruments are not infinitely precise, one physical point can be represented equally well by distinct coordinates. In mathematical geometry, distinct coordinates always represent distinct points.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
To make the connection with projective coordinates, we need to use a result from p. 146 of my Lectures [1, 4]: a proper point on a proper line always determines an “associated” point. The origin O of our rectangular coordinate system determines an associated point for each axis OE 1 E 2 E 3: say \(A,B,\varGamma \). The planes \(B\varGamma E_1, \varGamma AE_2,ABE_3\) intersect at a point E. If we now give the point O the additional name Δ, we have five points \(A,B,\varGamma, \Delta, E\) that, according to p. 190 of the Lectures [1, 4], determine a system of projective coordinates. The homogeneous coordinates x 1,x 2,x 3,x 4 are related to our rectangular coordinates x,y,z as follows.
$$x=\frac{x_1}{x_4}, y=\frac{x_2}{x_4}, z=\frac{x_3}{x_4}.$$ - 2.
If, instead of S, one chooses a system S ′ of projective coordinates, then this yields a notation
$$x_1'\;|\; x_2'\;|\; x_3'\;|\;x_4'\;||\; S'\;|\;W$$where \(x_1^{\prime},x_2^{\prime},x_3^{\prime},x_4^{\prime}\) are homogeneous linear functions of \(x_1,x_2,x_3,x_4\).
- 3.
In homogeneous coordinates (see the note to the first paragraph of no. 2):
$$x_1\;|\; x_2\;|\; x_3\;|\;x_4\;||\; S\;|\;W$$More generally:
$$x_1'\;|\; x_2'\;|\; x_3'\;|\;x_4'\;||\; S'\;|\;W$$for a system S ′ as in the note to the second paragraph of no. 2.
- 4.
References
Pasch, Moritz. 1882. Lectures on modern geometry. Leipzig: B.G. Teubner.
Pasch, Moritz. 1904. On the introduction of imaginary numbers. Archiv der Mathematik und Physik 7:102–108.
Pasch, Moritz. 1909. Foundations of analysis. Leipzig: B.G. Teubner.
Pasch, Moritz. 1912. Lectures on modern geometry, 2nd edn. Leipzig: B.G. Teubner.
Pasch, Moritz. 1914. Variable and function. Leipzig: B.G. Teubner.
Pasch, Moritz. 1921. Implicit definition and the proper grounding of mathematics. Annalen der Philosophie 2:145–162.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Pollard, S. (2010). Physical and Mathematical Geometry. In: Pollard, S. (eds) Essays on the Foundations of Mathematics by Moritz Pasch. The Western Ontario Series in Philosophy of Science, vol 83. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9416-2_8
Download citation
DOI: https://doi.org/10.1007/978-90-481-9416-2_8
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-9415-5
Online ISBN: 978-90-481-9416-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)