Cosmology View
My views on Cosmology and Physics
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Book by David Michalets
An (original) line precedes original content from the source.
A (remark) line precedes my remark from my review of the preceding original content.
My remark applies to only this section of the original.
Section I of 35
(original)
IN your schooldays most of you who read this book made acquaintance with the noble build ing of Euclid's geometry, and you remember — perhaps with more respect than love — the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. By reason of your past experience, you would certainly regard every one with disdain who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you: "What, then, do you mean by the assertion that these propositions are true?" Let us proceed to give this question a little consideration.
Geometry sets out from certain conceptions such as "plane," "point," and "straight line," with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as "true." Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. they are proven. A proposition is then correct ("true") when it has been derived in the recognised manner from the axioms. The question of the "truth" of the individual geometrical propositions is thus reduced to one of the "truth" of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called "straight lines,"
to each of which is ascribed the property of being uniquely determined by two points situated on it.
The concept "true" does not tally with the assertions of pure geometry, because by the word "true" we are eventually in the habit of designating always the correspondence with a "real" object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical onnection of these ideas among themselves.
It is not difficult to understand why, in spite of this, we feel constrained to call the propositions of geometry "true." Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas. Geometry ought to refrain from such a course, in order to give to its structure the largest possible logical unity. The practice, for example, of seeing in a "distance" two marked positions on a practically rigid body is something which is lodged deeply in our habit of thought.
We are accustomed further to regard three points as being situated on a straight line, if their apparent positions can be made to coincide for observation with one eye, under suitable choice of our place of observation.
If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.
It follows that a natural object is associated also with a straight line. Three points A, B and C on a rigid body thus lie in a straight line when, the points A and C being given, B is chosen such that the sum of the distances AB and BC is as short as possible. This incomplete suggestion will suffice for our present purpose.
Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the "truth" of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the "truth" of a geometrical proposition in this sense we understand its validity for a construction with ruler and compasses.
Of course the conviction of the "truth" of geometrical propositions in this sense is founded exclusively on rather incomplete experience. For the present we shall assume the "truth" of the geometrical propositions, then at a later stage (in the general theory of relativity) we shall see that this "truth" is limited, and we shall consider the extent of its limitation.
(remark)
Einstein finishes this first section with a reference to a "later stage" because more will follow about geometry. This is clear, just by reading the table of contents of the book. I am literally alarmed when he questions the "truth" of measuring distances between 3 points on a rigid body.
Geometry sets the foundation for measurements by defining the basic relationships between a plane, line, point, angle,and distance. Trigonometry further studies angles and lengths of sides. Algebra unifies these disciplines by enabling the use of symbols in a compact formula to represent a mathematical relationship between items.
Results of a formula can be plotted by each variable having an axis perpendicular to another. their independence is shown by their axes being perpendicular.
Parallel axes indicate the ratio between associated variables.
2 variables are easy to plot on a sheet of graph paper, having many evenly spaced parallel and perpendicular lines. A 3rd axis can be mannually added to the graph paper by adding a line at 45 degrees from the point of origin, X0, Y0, Z0, which is at a convenient point on the paper. Adding a 4th axis on a flat sheet of paper is difficult.
Geometry describes the fundamental assumptions of whatever coordinate system we use for our measurements. Physics is the science of energy, matter and motion. Measurements are required for analysis.
Geometry and defined coordinate systems are the foundation of measuring locations, distances and motion, which is just a change in measured position, or a distance, over a measured time duration.
A coordinate system is defined for the observer's environment and the measurements being made.
Experiments are useful only when they can be repeated for confirmation of initial results.
To compare an verify results, the measurements must be independent of observer.
Suggesting a measured distance is not the "truth" will require extraordinary evidence from Einstein.
Go to Table of Contents, to read a specific section.
last change 05/07/2022