Reading Notes: August 29th, 2024

“A surface is commonly defined as the boundary of a space. Thus, the surface of a metal sphere is the boundary between the metal and the air; it is not part either of the metal or of the air; two dimensions only are ascribed to it. Analogously, the one-dimensional line is the boundary of a surface; for example, the equator is the boundary of the surface of a hemisphere. The dimensionless point is the boundary of a line; for example, of the arc of a circle. A point, by its motion, generates a one-dimensional line, a line a two-dimensional surface, and a surface a three-dimensional solid space….No difficulties are presented by this concept to minds at all skilled in abstraction. It suffers, however, from the drawback that it does not exhibit, but on the contrary, artificially conceals, the natural and actual way in which the abstractions have been reached…A more homogeneous conception is reached if every measurement be regarded as a counting of space by means of immediately adjacent, spatially identical, or at least hypothetically identical, bodies, whether we be concerned with volumes, with surfaces, or with lines. Surfaces may be regarded as corporeal sheets, having everywhere the same constant thickness which we may make small at will, vanishingly small; lines, as strings or threads of constant, vanishingly small thickness. A point then becomes a small corporeal space from the extension of which we purposely abstract, whether it be part of another space, of a surface, or of a line. (Mach, Space and Geometry, 48-49)

“As we see, every geometrical measurement is at bottom reducible to measurements of volumes, to the enumeration of bodies. Measurements of lengths, like measurements of areas, repose on the comparison of the volumes of very thin strings, sticks, and leaves of constant thickness.” (Mach, Space and Geometry, 81)

“If we were to ask an unbiased, candid person under what form he pictured space, referred, for example, to the Cartesian system of coordinates, he would doubtless say: I have the image of a system of rigid (form-fixed), transparent, penetrable, contiguous cubes, having their bounding surfaces marked only by nebulous visual and tactual percepts,—a species of phantom cubes. Over and through these phantom constructions the real bodies or their phantom counterparts move, conserving their spatial permanency.” (Mach, Space and Geometry, 83-84)

“In geometry one sometimes starts with the concept of the measure of length without properly having established this concept. In doing so it is taken for granted that all the lengths occurring in a figure have a determinate numerical ratio to one of them chosen by us. The length chosen is then called the unit of length. The number indicating how often the unit is contained in a line segment is called the measure-number of this line segment. The line segment to be measured can be either commensurable or incommensurable to the unit; in the latter case the measure number will be irrational [i.e., cannot be represented by a ratio of two integers]. The measure-number of a square’s diagonal for example is irrational if the side of the square is chosen as the unit.” (Hölder, Intuition and Reasoning in Geometry, §24)

“The concept of content of plane figures and bodies has not been constructed deductively by Euclid, but it is possible to do so. In the case of bodies, one may regard the content as an empirical concept extracted from the usage of measures of capacity [e.g., ounces, cups, pints, quarts and gallons; or liters, centiliters, milliliters and kiloliters] in measuring fluids, and in this way one may also get to the content of plane figures. Euclid takes it for granted that figures have content, and that to them also the axioms apply that equals added to equals yield equals and the greater the added to the greater yields the greater. If however one does not want to presuppose the concept of content, but to establish it geometrically, one first and foremost has to show that figures, in particular figures of different form, can be compared as to their magnitude. As a first step congruent figures will be declared to be equal; the figure that completely comprises the other figure will be called the greater one. In order to compare any two figures, they have however to be cut through, and it must be proved that the result of the comparison does not depend on the actual way this is done. The construction of the proportions of line segments as well as of the concept of content presupposes that certain geometrical operations are repeated an indefinite number of times. Indeed, in the consideration of proportion, we had to take the nth multiple of a line segment, n being an indeterminate number.” (Hölder, Intuition and Reasoning in Geometry, §30-§31)