Reclaiming the common sense foundations of knowledge:
The mathematics of Einstein's Special Relativity (SR), and Cantor's diagonal.
Abstract: The regression of mathematics to mathematical logic has blinded mathematicians to the vital role of the "meaning" of symbolic expressions. As a result, the fundamental difference between the co-ordinate expressions ("equations") of analytic geometry and kinematics, on the one hand, and the equations of algebra and calculus, on the other, is no longer seen. The orthodox defence of the mathematics of SR is not, as claimed by some critics, a conspiracy, but evidence of a truly tragic stultification which, to put it mildly, has serious implications for the entirety of mathematical physics.
Kinematics, in mathematical physics meanwhile replaced by vector algebra, is the special branch of analytic geometry devoted to the study of moving points. Where, in analytic geometry, we have the position vector r of a fixed point, with its scalar components x, y, z, in kinematics the position vector r(t) is that of a moving point, with the scalar components x(t), y(t), z(t). There is here no fourth time-co-ordinate.; the t-"variable" serves only as a scalar measure on the three space-axes. Hence, in Einstein's 1905 kinematic transformation, the position vector is that for a (any) point on the surface of an expanding sphere, with its three scalar components x(t), y(t), z(t), the magnitude of which depends on the direction of the position vector r = ct. Einstein needs a separate equation for the time-"variable" in the second system because of his attempt to introduce system-dependent units of time measurement; such a change of the unit of measurement does not turn his time-"variable" into a fourth dimension.It is mainly for two reasons that mathematicians have mistaken Einstein's transformation in the kinematic part of his 1905 paper for a case of a 4D linear transformation.
First, linear transformations under which the properties of certain geometric structures are invariant had, from ca. 1870 until well into the 20th century, been a major research area, with Klein and Poincaré as leading participants; see, e.g. Kline, Vol.3, Ch. 38 and 39. Einstein's transformation equations had exactly the form of such a linear "invariant" transformation. In the excitement over an application in frontline physics one would have paid no attention to the banal fact that, in kinematics, the time-variable is not at all a variable of the status of the x, y, z, and that therefore, Einstein's kinematic transformation differs fundamentally from the linear "invariant" transformations on which mathematicians were working at the time. The fundamental role of the boundary (the ct of SR) had also been shown to be an essential part of a certain kind of geometric structure, in that, in hyperbolic geometry, "the points of the geometry are the points interior to this circle" (Kline, Ch. 38), a mathematical principle which found its expression in Minkowski's "postulate of the absolute world" ("Space and Time" of 1908) and, more crudely, in the principle of physics that "nothing can move faster than light".
Second, in consequence of the triumph of logicism, mathematicians had been led to disregard the meaning of mathematical expressions and operations; geometrical reference had been denounced as particularly fallacious. Among mathematicians leading at the time of the rise of SR, the geometric meaning of the unsophisticated symbolic forms and operations of kinematics would have been completely disregarded. Nobody appears to have tried to read the expressions used in Einstein's 1905 kinematic transformation as describing a perfectly conventional geometric operation; had one done so, the mismatch between his symbolic expressions and their geometric referents could not have escaped attention. Because of the great diversity of the symbolic forms of analytical geometry (the equations for lines, planes, curves, surfaces, including the equation for the line in "space-time" representing the x(t) of kinematics as a function - see Lamb), the vital distinction between, on the one hand, the x, y, z as components of the position vector of a static point (analytical geometry) or of the displacement in 3D of a moving point (kinematics), and on the other, the x, y, z (and t) as algebraic variables would have been lost. A typical expression of kinematics for a displacement on the x-axis, such as x' = x - vt, would have been misread as a linear algebraic equation. Although the x-, y-, z-components, through the Pythagorean relation, are interdependent with the position vector of analytical geometry, and the 3d displacement of kinematics, these x, y, z do not behave like the variables of algebraic equations (linear or quadratic). Except in the case of rotation of the co-ordinate axes, the x, y, z never appear together in a linear equation; in the kinematics of SR we may combine x and t, as in x' = x - vt, only because the element vt, like the x, is a displacement on the x-axis.
It is this misreading of the symbolic expressions of kinematics as a type of 4D linear transformation of invariant theory or as "ordinary" algebraic equations, and the failure to refer these expressions to their defined geometric-kinematic meaning, which is responsible for the failure to recognize that Einstein's LT, the outcome of his transformation of co-ordinates (for translation of origin through vt) is absurdly false. This debacle lies at the root of the persistent unease in regard of real or supposed paradoxes. Among those unhappy with SR there is an unfortunate consensus that rejection of SR as a physical theory would be a solution: a dangerous illusion. The bulk of objections (especially the triumphalist "Kritische Stimmen" of the Germans) is overconfident and largely misconceived (or worse). (In view of our complete ignorance of the precise nature of the actual physical processes at the stages of emission, transmission, absorption, the return to concepts like the ether, for instance, solves nothing whatever.) The only relevance of SR in the evolution of thinking in physics is in its role as an illustration of the rise of phantastically nonsensical mathematical formulations, as the foundation for "theory", because the geometric meaning of the mathematical operations of physics is no longer understood. (On the importance of the geometric meaning see also Prof. Kanarev.)
Once we recognize the misreading of the expressions of analytical geometry and kinematics as 4D linear "invariant" transformations or as "ordinary" algebraic equations, the diversity of symbolic forms similar to, or identical with, the LT of SR kinematics is irrelevant for physics; on Einstein's 1905 kinematical derivation, see 1.2. below. Most particularly, the various geometric interpretations of 4D algebra, e.g. Minkowski's hypercone, are not applicable to the LT of kinematics.Comprehension of the distinction between the "meaning" of different kinds of the "LT" found in the literature inevitably requires attention to some apparently esoteric formal expressions; however, this is not intended as a learned exposition or discussion. In addition to the pitfalls intrinsic to the interpretation of formal expressions, we should be aware of the conceptual problems created by trends in philosophy. I have already mentioned logicism; equally lethal for the comprehension of SR kinematics is operationalism (see below). In 1905, the LT, in its various forms, had been a topic for learned exposition.
I do not discuss here the ad hoc equations proposed by Lorentz-FitzGerald-Larmor; these are not "transformations" in the sense as understood by mathematicians. Because of their purely formal similarity, forms such as these tend to confuse the investigation of the "problem" of the various strictly mathematical solutions.Einstein's 1905 transformation is unique in that he attempts to derive a mathematically acceptable form by purely kinematical considerations. His t-equation, usually taken to refer to a fourth dimension, is deceptive. Despite its significance for the "meaning" of the LT in physics, we must postpone discussion of the 3D case because of the infatuation of the learned community with 4D "space-time"; 4D forms must be examined first.
[1.2.1 - 1]
[1.2.1 - 1] is "generalized" to
[1.2.1 - 2]
[1.2.1 - 1] and [1.2.1 - 2] are instances of the general 4D algebraic forms
[1.2.1 - 3]
Note that the geometic "meaning" of [1.2.1 - 1] cannot be decided upon purely formal grounds. Mathematicians involved in the evolution of SR mathematics (e.g. Poincaré and Minkowski) interpret both [1.2.1 - 1] and [1.2.1 - 2] as configurations in hyperspace (the 4D "space-time" of Minkowski). In the SR of physics and Einstein (1905), as an application of conventional 3D analytic geometry, [1.2.1 - 1] is unproblematical. If we have a sphere with radius ct, the quadratic equation is the conventional equation for a 3D sphere. For the translated system, ct' is merely the position vector - conventionally measured from the translated origin; the quadratic equation expresses that the x'-, y'-, z'-components must obey the Pythagorean rule.All these equations are instances of the general quadratic form
Note that, even in an explicitly geometric interpretation (as distinct from abstract algeba), the distinction between 3D and 4D cannot be decided on purely formal grounds, but depends on our assumed "meaning" (or reference: what is the geometric configuration these equations are meant to represent?)
These forms reflect trends in mathematical thinking, in that formal
operations are independent of interpretation: they allow any kind of
application, for instance, to the purely numerical problems of
conventional algebra, or to geometric configurations in nD space. A
linear solution would be assumed to have the form as found, for
instance, in Bôcher's 1907 exposition for algebra, namely
[1.2.1. - 6]
[1.2.1 - 6] is similar to the form in Klein's geometric interpretation (Vorlesungen ...,
Teil II, Kap. 2), in his exposition of Minkowski's theory, namely the
general case of 4D transformation for translation of origin and rotation
[1.2.1 - 7]
where the zi are the (constant) magnitudes of translation, and the ai, ..., ei the equivalents of the direction cosines in analytic geometry for the rotation of the co-ordinate axes.
Voigt's general form is similar, except that he assumes the elements corresponding, in a geometric interpretation, to the constants of translation to be zero; in a geometric interpretation (as distinct from an uninterpreted algebraic form), Voigt's approach would solve the case of 4D rotation.
The quadratic equations [1.2.1 - 1] to [1.2.1. - 5] are indeterminate; their solution requires restrictions by appeal to geometric and/or formal considerations. At this 4D stage of SR mathematics, it was resonable to be guided by the forms proposed in the early literature: x' was to be a linear function f(x, t), y' and z' linear functions f(y) and f(z); on formal grounds t' as a linear function f(x, t) would have appeared suggestive.
It is at this crucial point that Einstein mounts the mathematical bulldozer: he appears to show that a solution for t' linear in (x, t) is, indeed, mathematically necessary, and that furthermore, by consideration of the conventional case of 3D physics, a new form of the LT not only arises on purely mathematical grounds but reveals certain "counter-intuitive" properties of space and time as such. Once discovered, this new form was found to fulfill all the requirements of the 4D case of the mathematicians; read as an application to 4D geometry, it would denote rotation in 4D space ("space-time") - no translation. Since the 4D case is indeterminate, different kinds of solution, depending on the choice of restrictions, would have been possible.
Einstein's LT, in its formal simplicity, would have appealed on aesthetic grounds. Infatuation with the 4D case has prevented attention to the invalidity of Einstein's 3D treatment; some perfectly reasonable algebraic restrictions are inadmissible in the 3D case of physics and Einstein (1905).
Two essential restrictions are not applicable in the 3D case of physics and Einstein (1905):In consequence of the failure to pay attention to the crucial difference between 3D and 4D interpretations of formally identical symbolic expressions, it has become customary, even among physicists, to derive the LT as an algebraic solution of sets of 4D quadratic equations as in [1.2.1 - 1] or [1.2.1 - 2] (see, for instance, Bergmann, Chapter IV).
1. the condition that t' must be linear in (x, t) (in Einstein an error typical of his shoddy logic);
2. the condition that v must have the same value in either transformation (S to S'; S' to S) (in Einstein an unreflected assumption typical of his uncritical way of thinking).
Among the derivations of the LT, Einstein's 1905 procedure is unique: he alone tries to proceed in accordance with the traditional methods of kinematics; his operationalist notions reflect a philosophical trend of his time.
Operationalism:Einstein's writings elsewhere (e.g. the notorious "Simple Derivation") suggest that his thinking is foggy; to call his grasp of logic weak would be a euphemism. By purely kinematic considerations, faulty at virtually every step, he appears to show that the much debated Lorentz Factor is intrinsic to mathematical space and time as such. The enthusiastic acceptance of Einstein's 1905 solution by the group about Minkowski, then in Goettingen, would have led Einstein to follow the 4D interpretation; Einstein's later derivations do no longer adhere to the procedure of 3D kinematics. However, the "Simple Derivation" is instructive: a diagram shows the origin of S' moving along the x-axis of S - this is impossible in 4D because the 4D transformation rotates the axes about a common origin; the diagram therefore shows that Einstein never understood the difference between 3D and 4D.
Thinking in physics had pre-empted the explicit operationalism of the philosophers of science:
the "inertial systems", instead of co-ordinate systems in mathematical space;
physical objects (bodies, mass-points, light-flashes), instead of mathematical points indicating position in mathematical space;
or the clocks - at rest or moving, situated at all points in co-ordinate space - instead of the magnitude of the t-variable for a given point;
the "light path" of SR, instead of the displacement in the mathematical space of kinematics.
The very term "thought experiment" indicates that the nature of geometric operations is no longer understood.
It is instructive to compare the "operationalists" language in contemporaneous German texts (e.g. Hertz's Mechanics, also his teacher's Helmholtz) with more critical British texts (e.g. Lamb's Dynamics of 1914 and Maxwell's Matter and Motion of 1877); but Kirchhoff's language, in Mechanik, 1877, is also rather more critical and precise. Minkowski, Einstein's teacher, was for several years a colleague of Hertz at Bonn university.
Before any examination of the detail of Einstein's 1905 transformation, we need to remind ourselves of the status of the time-variable in classical analytic geometry (with kinematics and dynamcis) and calculus.220.127.116.11. Einstein's 1905 derivation of the LT
Since the SR transformation assumes velocities to be constant, we may confine ourselves to the simple case of kinematics.If we have an expression like x = vt, this can "mean" (represent, refer to) different "things":
1. x may be the displacement of a point P moving along the x-axis. The figurative representation, as in classical analytic geometry and kinematics (see, e.g., Sommerville and Lamb), is
Fig. 18.104.22.168 - 1:
For a point moving in 3D an equation like [1.2.1 - 1] makes perfect sense: it expresses the Pythagorean relation between the position vector and its x-, y-, z-components.(In mathematical jargon, t is here an auxiliary or supplementary variable. There is here no t-co-ordinate or time-"dimension"; t "works" here as a multiplier of quantities on the space-axes.) Note that the "moving point" is a standard concept even in modern textbooks of geometry. Where velocities, as here, are constant, ratios in the geometric configuration are independent of the magnitude of t, much as the properties of a triangle are independent of its assumed size.
2. In calculus, we use configuration in function space ("space-time") to express the dependence of x on t. In this case we interpret the expression x = vt strictly in its sense as a function x = f(t) (in SR, a linear function). The typical figurative representation in function space would be
(x) | | | . . . . . . . . . * | * . | * . | * . | * . | * . | * . O________________________(t) Fig.22.214.171.124 - 2For a point moving in 3D, the function space for the 3D displacement is necessarily 4D.
If, as in SR, we have two points moving along the x-axis (that representing the position of a light signal for y, z = 0, and the origin O' of the second system), we have two different functions, namely x = ct and x = vt, with their lines at their appropriate angles. Although the configuration in function space would still enable us to determine the relative displacement between the two moving points, it does not help us to understand what happens in an SR-type transformation.In this case, namely when the symbols x, y, z denote functions, equations like [1.2.1 - 1] of [1.2.1 -2] are nonsensical; they may be "meaningful" in other mathematical applications, but not in reference to the "space-time" configurations of the SR of physics.The failure to attend to conventional mathematical usage and "meaning" is already evident in that SR texts (including Minkowski) have x as the abscissa and t as the ordinate.
Note that, although the function-lines lie in the x-t-plane, the respective moving points P do not, like the train in some SR texts, ascend the slope in "space-time". Movement, in the given case, proceeds along the x-axis only; the ascending line merely expresses that the magnitude of the displacement on the x-axis increases with increasing values of t.
Einstein's 1905 derivation (like any kind of mathematical argument) cannot be understood by reading discussions in the secondary literature; insight requires attention to Einstein's own argument. I use the translation of the 1905 paper in the Dover edition, Einstein et al., The Principle of Relativity, pp. 37 - 65.Einstein tries to obtain the co-ordinates and time-equation for points on a sphere with the radius ct about the origin O of a co-ordinate system S, in relation to the origin O', of a second system S', moving in the positive direction of the x-axis with the velocity v. A figurative representation for the imagined geometric scenario (point P in the first octant) might be (simplified for z = 0)
I retain here parts of an earlier draft, to which I make merely a few brief additions (the case does not merit scholarly treatment). For the sake of brevity, I retain some of Einstein's operationalist expressions.
(y) (h) | | |. . | . . . . . . . . . . P | | .* . | | . * . | | . * . | | . * . | | . * . | *. | * . _O_____O'____________________)_ (x, x) Fig.126.96.36.199 - 1
To remind ouselves of the asymmetry of the figure, I include a figure that shows points P and Q in different octants of the sphere about O:
(y) (h) | | Q . . . . . . . . . . . . |. . | . . . . . . . . . . P . .* | | .* . . . * | | . * . . . * | | . * . . . * | | . * . . . * | | . * . . . | *. | * . _(___________________________O_____O'____________________)_ (x, x) Fig.188.8.131.52 - 2p. 44-45 (A striking illustration of Einstein as an apprentice mathematical houdini.):
Instead of obtaining the three co-ordinates and the time equation for one single point, he obtains the co-ordinates of three different points; the time equations for these points differ; he retains only the time equation for points on the x-axis (y, z = 0; applicable to the two points when x = ct, x = -ct; x = ct, x = -ct) ("must be linear").
The equation on top of p.45, is the solution of [(x - vt)/(c-v) + (x - vt)/(c = v)]/2 (p.44, nonsensically construed for (x - vt) "infinitesimally small"), i.e. Einstein superimposes the speed of a two-way signal on that of a one-way signal.
The resulting equation can be tidied up to t = ab2(t - vx/c2).
The time equations for the other other points, obtained for x = vt, thus retain the two-way & one-way combination of the light speed for signals in other directions. Observe how the "elegant", completely arbitrary scaling factor a, by substituting F for ab, can be used to get rid of the surplus factor b in all four equations.
Note that the retained time equation, even though restricted to signals moving along the x-axis only, is already direction-dependent and thus completely useless. (In its final form, it reduces, on the right, to bt( 1 - v/c), and on the left, to bt(1 + v/c). Interpreted as a clock rate, for signals to the right clocks would go fast, and go slow for signals to the left. The b, as an ad hoc quantity perfectly acceptable in the physical theories of Lorentz-FitzGerald-Larmor, emerges in the strictly mathematical transformation of SR by mistake and reduces there to 1.)
The time equation is redundant; in Einstein's LT we merely have t = x/c and t = x/c. (Note that the complete uselessness of the enterprise, namely the anisotropy of the scenario, is already expressed in the equation for x: bx(1 - v/c) to the right, bx(1 + v/c) to the left.)
p. 46 (the quadratic equations):
Note that the figure, spherical about O, is not also spherical about O'; the direction-dependence of the time-equation obscures that there is no isotropy in S'. The direction-dependent quantity ct is merely the position vector; the quadratic equation for c2t2 is not that of a sphere but expresses a mere Pythagorean relation.
p. 47 (the inverse transformation):
The reciprocal b emerges here by mistake. Hoist with his own petard, Einstein, who is to teach mankind how to think correctly about time, typically forgets that all velocities in S' must be corrected for the changed time measurement; instead of (c - v) and (c + v) we had got c (the very purpose of the transformation); the relative velocity, similarly, is no longer the "same", namely v, in both systems.
B. J. Tonkinson (see his NPA profile) quotes Torretti, according to whom reciprocity requires the relative velocity to have the same value, namely v, in the time of either system. Note the fallaciousness of abstract principles!Since the time equation (valid only for y, z = 0) is direction-dependent, the magnitude for the relative velocity in S', say v', is also direction-dependent. We can easily obtain it from a simple figure.
In the case of a signal moving to the right, We have
Fig.184.108.40.206 - 3:
where OO' = vt, OP = ct, O'P = (c - v)t = ct.
If OO' = v't, then here vt/(c-v)t = v't/ct and therefore v' = vc/(c - v). We need to apply a paradoxically reciprocal factor b in the inverse transformation only because the assumption that OO' = vt measures OO' too short. If we insert the correct value for v' we find, instead, that x = (x + v't)/b. That is to say, if quantities in S' are shorter than corresponding quantities in S, as common sense logic should expect, these quantities in S must, in that precise ratio, be longer: if a length s = as, then s = s/a.
Similarly, for a signal moving to the left, we have
Fig. 220.127.116.11 - 4:
where OO' = vt, OP = -ct, O'P = -(c + v)t = -ct.
If OO' = v't, then here OO'/O'P = vt/[-(c + v)]t = v't/(-ct) and therefore v' = vc/(c + v). In this case, the assumption that OO' = vt measures OO' too long; since x = OP = |O'P - OO'|, x turns out to be too small because we have subtracted a too large value for OO'. Again, inserting the correct magnitude v't in the inverse transformation gives us x = (x + v't)/b. (As above, if quantities in S' are shorter than corresponding quantities in S, these quantities in S must be longer in the same ratio.)
If we assume the systems to be equivalent, and any factor to be reciprocal, a correct mathematical argument would show that such a factor would be 1.
After running out of steam in October 2011, I continue in January 2012, but restrict myself to some brief points.On p.48, Einstein proves that the b (actually stretching!) means contraction.
On p. 49, he "derives" the second equation for t. The full equation in the LT had been derived for points on the x-axis; the equation for the points of intersection of the h-z-plane with the sphere about O (when x = vt) had then tacitly been dropped under the table. Now that equation, for the case when x = vt, turns up again, and we get t = t/b: the inconvenient direction-dependence of the full time equation has been rendered invisible.
Of interest is the composition of velocities (p.50). The required formula can be read without difficulty directly from a simple diagram. Blind to the meaning of his symbolic operations, Einstein obtains his hilariously nonsensical result by shuffling together x = ct and x = wt; the resulting formula has been parrotted in the entire SR literature (the "critical" Dingle included).
For a point W on the x-axis with the coordinate x = wt, we have, for x = ct (P) on the right,
O'W:O'P = w't:ct = (w - v)t:(c - v)t etc., and for x = -ct (Q), on the left
O'W:O'Q = w't : ct = (w - v)t : (c + v)t etc.
A further remark on Einstein's comprehensive applications: since his formulae never follow there can be no application of SR to physics; there is no relativistic Doppler effect. (We surely have worthier tasks, or topics to exercise our brains, than speculations about non-existent SR applications.)
Note: Checking on a source revealed a discrepancy which, apart from deleting the offending item, I have solved by the wholesale demotion of all quotations to the status of summaries.The reciprocal effect of length contraction and time dilation, which appears by logical necessity to emerge from the kinematic part of the special theory of relativity, has been variously explained as
1. true but not really true (guess who)
3. not real
5. the result of the relativity of simultaneity
6. determined by measurement
7. a perspective effect
Here is a small selection from the literature; for references see below. Unless placed in quotation marks, authors' assessments are summarized.
1. Effects are true but not really true:
Pride of place goes to Eddington [1928, 33-34]:
"The shortening of the moving rod is true , but it is not really true."
(Thanks to Prof. I. McCausland, Toronto, for contributing this gem.)
2. Effects are real:
Arzelies [1966, 120-121]:
The Lorentz Contraction is a Real Phenomenon. ...
Several authors have stated that the Lorentz contraction only seems to occur, and is not real. This idea is false. So far as relativistic theory is concerned, this contraction is just as real as any other phenomenon. Admittedly ... it is not absolute, but depends upon the system employed for the measurement; it seems that we might call it an apparent contraction which varies with the system. This is merely playing with the words, however. We must not confuse the reality of a phenomenon with the independence of this phenomenon of a change of system. ... The difficulty arises because we have become accustomed to the geometrical concept of a rigid body with a definite shape, whatever the measuring system. This idea must be abandoned. ... We must use the term "real" for every phenomenon which can be measured ... The Lorentz Contraction is an Objective Phenomenon. ...
We often encounter the following remark: The length of a ruler depends upon its motion with respect to the observer. ... From this, it is concluded once again that the contraction is only apparent, a subjective phenomenon. ... such remarks ought to be forbidden.
Krane [1983, 23-25]:
It must be pointed out that time dilation is a real effect that applies
not only to clocks based on light beams but to time itself. All clocks
will run more slowly as observed from the moving frame of reference. ...
The length measured by the moving observer is shorter. It must be emphasized that this is a real effect.
Matveyev [1966, 305]:
The dimensions of bodies suffer contraction in the direction of motion ... A body is, therefore, "flattened" in the direction of motion. This effect is a real effect ...
Møller [1972, 44]:
Contraction is a real effect observable in principle by experiment. It expresses, however, not so much a quality of the moving stick itself as rather a reciprocal relation between measuring-sticks in motion relative to each other. ... According to relativistic conception, the notion of the length of a stick has an unambiguous meaning only in relation to a given inertial frame. ... This means that the concept of length has lost its absolute meaning.
Pauli [1981, 12-13]:
We have seen that this contraction is connected with the relativity of simultaneity, and for this reason the argument has been put forward that it is only an "apparent" contraction, in other words, that it is only simulated by our space-time measurements. If a state is called real only if it can be determined in the same way in all Galilean reference systems, then the Lorentz contraction is indeed only apparent, since an observer at rest in K' will see the rod without contraction. But we do not consider such a point of view as appropriate, and in any case the Lorentz contraction is in principle observable. ... It therefore follows that the Lorentz contraction is not a property of a single rod taken by itself, but a reciprocal relation between two such rods moving relatively to each other, and this relation is in principle observable.
Schwinger [1986, 52]:
Each will observe the other clock to be running more slowly. This is an objective fact. It is not a property of clocks but of time itself.
Tolman [1987, 23-24]:
Entirely real but symmetrical.
3. Relativistic effects are not physically real:
Taylor & Wheeler [1992, 76]:
Does something about a clock really change when it moves, resulting in the observed change in the tick rate? Absolutely not! Here is why: Whether a clock is at rest or in motion ... is controlled by the observer. You want the clock to be at rest? Move along with it. ... How can your change of motion affect the inner mechanism of a distant clock? It cannot and it does not.
4. Relativistic effects are apparent:
Aharoni [1985, 21]:
The moving rod appears shorter. The moving clock appears to go slow.
Cullwick [1959, 65, 68]:
[A] rod which is at rest in S' ... appears to the observer O to be contracted ... Similarly, a rod at rest in S will appear in S' to be contracted....
Jackson [1975, 520]:
The time as seen in the rest system is dilated.
Joos [1958, 243-244]:
The interval appears to the moving observer to be lengthened. A body which appears to be spherical to an observer at rest will appear to a moving observer to be an oblate spheroid.
McCrea [1954, 15-16]:
The apparent length is reduced. Time intervals appear to be lengthened; clocks appear to go slow.
Nunn [1923, 43-44]:
A moving rod would appear to be shortened. An interval is always less than measured by the other observer.
Whitrow [1980, 255]:
Instead of assuming that there are real, i.e. structural, changes in length and duration owing to motion, Einstein's theory involves only apparent changes, and these are independent of the microscopic constitution and hidden mechanisms controlling the structure of matter. [Unlike]... real changes, these apparent phenomena are reciprocal.
5. Relativistic effects are the result of the relativity of simultaneity:
Bohm [1965, 59]:
When measuring lengths and intervals, observers are not referring to the same events.
French [1968, 97],
Rosser [1967, 37],
Stephenson & Kilmister [1987, 38-39]:
Measurements of lengths involve simultaneity and yield different numerical values.
6. Relativistic effects are determined by measurements:
Schwartz [1972, 113]:
Each observer determines distances to be foreshortened.
7. Relativistic effects are comparable to perspective effects: Rindler [1991, 25-29]:
Moving lengths are reduced, a kind of perspective effect. But of course nothing has happened to the rod itself. Nevertheless, contraction is no illusion, it is real. Moving clocks go slow, a 'velocity-perspective' effect. Nothing at all happens to the clock itself. Like contraction, this effect is real.
8. Relativistic effects are mathematical:
Eddington [1924, 16-18]:
The connection between lengths and intervals are problems of pure mathematics. A travelling clock gives a low reading.
Minkowski [1908, 81]:
[The] contraction is not to be looked upon as a consequence of resistances in the ether, or anything of that kind, but simply as a gift from above, - as an accompanying circumstance of the circumstance of motion.
Rogers [1960, 496]:
Thus we have devised a new geometry, with our clocks and scales conspiring, by their changes, to present us with a universally constant speed of light.
Einstein's 'Simple Derivation of the Lorentz Transformation' forms Appendix I to Relativity. First published in German in 1917, the book was written for the amateur reader (English tranlation published in 1920 by Methuen). According to the bibliography in A.P. Schilpp (Albert Einstein: Philosopher - Scientist, Open Court, 1949 & 1969; 706), the book constitutes the only comprehensive survey by Einstein of his theory, and is his most widely known work. (One gathers from Schilpp that, even in 1949, the debate about simultaneity had disintegrated into irreconcilable philosophical factions: clearly a waste of time.)Here is Einstein's text:
Einstein's great mathematical fame rests on his work on the General Theory (one might add that the mathematics for that theory was provided by Marcel Grossmann, and that, as usual, one looks in vain for an acknowledgement). The simple derivation of 1917 was therefore written at the time of Einsteins greatest mathematical mastery. (The 'mastery' is immediately evident if one compares the 'simple' derivation with the inchoate transformation of 1905). The derivation uses only the most elementary "algebra", and should present no difficulty whatsoever even to the amateur reader. Yet it has been entirely ignored; if one draws the attention of expert philosophers to it, they refer such supposedly technical stuff to mathematical experts.
The "Simple Derivation" is a typical illustration of Einstein's capacity to turn a farcically elementary problem into mathematical esoterics and thus to render it completely unintelligible - a capacity which has earned him the appellation of genius and the admiration of mystagogues. The problem can be solved without any difficulty whatsoever; see below.
For the relative orientation of the coordinate systems indicated in [an earlier figure for 3 space axes], the x-axes of both systems permanently coincide. In the present case we can divide the problem into parts by considering first only events which are localised on the x-axis. Any such event is represented with respect to the coordinate system K by the abscissa x and the time t, and with respect to the system K' by the abscissa x' and the time t'.Comment: Let's pause at this point and look at an appropriate figurative representation. Care is here needed because any figure would necessarily reflect whether we assume propagation to be isotropic in K or in K'; it cannot be isotropic in both. To be on the safe side I use two versions:
We require to find x' and t' when x and t are given.
A light signal, which is proceeding along the positive axis of x, is transmitted according to the equation
x = ct or x - ct = 0 .
Since the same light signal has to be transmitted relative to K' with the velocity c, the propagation relative to K' will be represented by the analogous formula
x' - ct' = 0 .
Those ... events ... which satisfy  must also satisfy . Obviously, this will be the case when the relation
(x' - ct') = l(x - ct) 
is fulfilled in general; where l indicates a constant; for, according to , the disappearance of x - ct involves the disappearance of x' - ct'.
If we apply quite similar considerations to light rays along the negative x-axis, we obtain the condition
(x' + ct') = m(x + ct) .
By adding (or subtracting) equations  and , and introducing for convenience the constants a and b in place of l and m, where
a = (l + m)/2, b = (l - m)/2,
we obtain the equations
x' = ax - bct, ct' = act - bx .
We should thus have the solution of our problem, if the constants a and b were known. These result from the following discussion.
Fig. 3.a: Isotropy in K
Fig. 3.a: QO = OP
Fig. 3.b: Isotropy in K'
Fig. 3.b: QO' = O'P
So far, so good. Although not 'false', the indeterminate zero equation  warns of trouble ahead. The derivation derails fully with equation . As here explicitly defined, the symbols x and x'  denote quantities which differ from those previously used in  and . We have, in , x = -ct, x' = -ct', whereas, in  and , x = ct, x' = ct'.
But that is not all. There is, first, the problem that symbols like x and x' may appear ambiguous, in that it is not immediately evident whether they represent positive or negative values, that is to say, in the case of geometry, the displacements of points moving to the right or left. Second, the question of isotropy must be now be faced. Although equations  and  are compatible with either version of the figure, careless symbol use here leads us to assume that QO = OP as well as QO' = O'P; addition and subtraction can only result in mathematical nonsense. Note that the presence of l and m serves to assure the negligent reader that the difference between the ratios QO/OP and QO'/O'P is properly being taken into account. For the quantities l and m, and presumably therefore the ratios QO/OP and QO'/O'P, are assumed to differ, for otherwise b = 0. But Einstein's actual treatment of the symbols x, x', ct and ct' is at variance with the assumption that these ratios differ. The vague assumption that the ratios QO/OP and QO'/O'P are equal as well as different is a typical instance of Einstein's logic.
Let's first sort out the ambiguity of symbols like x or x'. In the case of a 3D displacement OP(x,y,z), the variables x, y, z denote the components of ct. For a point on the x-axis such that x = ct, the expressions x and ct (x' and ct' respectively) are alternative names for one and the same displacement. Of these alternatives ct (ct' respectively) is preferable, for the direction of movement is clearly indicated by the sign. In contrast, shoddy thinkers like Einstein easily forget a definition like x = -ct (movement to the left). To avoid this kind of confusion, let's eliminate x and x' in favour of their safer alternatives.
Einstein's equations  and  should then read:
(ct' - ct') = l(ct - ct)
(-ct' + ct') = m(-ct + ct).
We could stop here, for all operations can already be seen to cancel. But let's continue.
In order to distinguish between the symbols used in the different equations let's re-write  and  using subscripts:
(x'3 - ct'3) = l(x3 - ct3), [3*]
(x'4 + ct'4) = m(x4 + ct4). [4*]
If we now eliminate the ambiguous x and x' in favour of the safer alternatives ct, ct', -ct and -ct', these equations become
(ct'3 - ct'3) = l(ct3 - ct3), [3*]
(-ct'4 + ct'4) = m(-ct4 + ct4). [4*]
Clearly, addition and subtraction cannot lead to Einstein's equation  because all operations cancel. This is the case regardless whether movement is to be isotropic in K or K'. Even though, with the invalidity of , the 'Simple Derivation' has lost its foundation, we may look in passing at some of the subsequent equally brilliant considerations adduced to conjure up the LT. The main lines of the argument are these:
The coordinates of O' are x' = 0 and x = vt. From , we find avt = bct. Further progress can be made by evaluating  for t = 0 and t' = 0, when we find x' = ax and x' = a(1 - v2/c2)x. From the Principle of Relativity we have x'/x = x/x', therefore a = (1 - v2/c2)-1/2. Q.E.D. Some Q.E.D.
To conclude: After the revealing start of the derivation, namely from  to , it should be clear that nothing of value is to be expected of Einstein's mathematical brilliance. Need one wonder if admirers like Reichenbach believed Einstein (of EPR) to have proven the insufficiency of classical logic? Yet academic physics would persuade us to purchase from this "genius" the claim that, by recourse to non-Euclidean geometry and tensor calculus, he has obtained results that transcend the powers of the Newtonian metric.
A simple solution of the problem of Einstein's "Simple Derivation"
The given case restricts points to the x-axes, namely
---------O---O'--------------Pwhere OP=x=ct, O'P=x'=ct' (t=x/c, t'=x'/c) and OO'=vt.
Despite the absence of dynamic effects, this appears to lead directly to the Lorentz Transformation, as follows. The entire literature assumes that, despite change of the time unit, the relative velocity is "the same" in both systems, so that OO'=vt=vt'. It is this uncritital assumption that leads to the mysterious effect, Minkowski's "gift from above". (Correctly we should put OO'=vt=v't', where the magnitude of v' can easily be obtained from the figure. If we use v't' instead of vt' the "gift from above" vanishes, for then k=1; but let's proceed as the relativists do.)
Assuming reciprocity of any effect, "we" put x'=k(x-vt), t'=k(t-vx/c2) [Ex.1a,1b],
x=k(x'+vt'), t=k(t'+vx'/c2) [Ex.2a,2b].
Entering x' and t' [Ex.1a,1b] in equations [Ex.2a,2b] we have x=k2x(1-v2/c2) etc. and thus k=(1-v2/c2)-1.
I had removed my argument on this topic in 2011 because I wanted to avoid distraction from the scandal of the mathematics of special relativity. Re-reading, in January 2012, several of Russell's key texts with his appraisal of Cantor, as well as that of others as reported by Kline, persuades me that, in view our concern with the consequences of pernicious developments in mathematics, the item merits re-instatement. Cantor's proof resembles the fallacy at the root of of special relativity, both in the supposed "triumph of mathematical reason" as well as in the elementary nature of the error, namely one which, in both cases, is easy to see.In earlier versions of this webpage I had included a longer argument to show that Cantor's diagonal procedure fails. The fallaciousness of Cantor's abstract procedure, on careful inspection, is immediately evident; the simplest possible presentation should therefore suffice.
Cantor's proof is widely quoted in the literature on the foundations and history of mathematics. One of many of his proofs using similarly abstract procedures, it is to establish the uncountability of the real numbers, and, by implication, the continuum hypothesis which is believed to be vital not only for analysis but for geometry.
According to Russell, before Cantor, any argument about curves or surfaces having (e.g. at supposed intersections) points in common would have lacked foundation: this invalidated Euclid as well as classical analytical geometry.Cantor's thought is believed to have laid the foundation for the development of modern mathematics (see, e.g. Kline, Vol.3; or B. Russell, The Principles of Mathematics, 2nd ed. of 1937, London: Routledge paperback ed., 1992).
My concern here is with the fallaciousness of the procedure itself, not with any implications.
I use Cantor's formalism and part of his own text as quoted, in translation, in John Fauvel and Jeremy Gray (eds), The History of Mathematics - A Reader. Basingstoke and London: The MacMillan Press Ltd., 1987 (pp. 579-580).
Briefly, Cantor considers the collection M of elements E = (x1, x2, ..., xn,
...), with infinitely many coordinates each of which is either m or w.
He proceeds to prove that M "does not have the power of the series 1, 2,
..., n, ...."
"If E1, E2, ..., En, ... is any simply infinite sequence of the manifold M, then there is always an element E0 of M which does not agree with any E.
To prove this let
To recognize that a vital premiss, tacitly taken for granted, is false, we need to consider what the procedure actually tries to do.
Wittgenstein, in his Remarks on the Foundations of Mathematics (Part II), rather vaguely thinking in terms of an arbitrarily ordered collection of elements such asThe easiest way to see what is wrong with Cantor's conclusion is by ordering the elements in the way one would actually build such a collection. Cantor's m and w are difficult to distinguish; we may choose any other more easily readable letters (say l and q); there should be no objection to opting for 0 and 1. 0 and 1 are easy on the eye; the argument is actually meant to be significant for the "continuum hypothesis" (e.g. points on the number line) and generally for numbers, and we are familiar with such collections. The argument works in the same way if we adhere to Cantor's own m and w, except that the result is more difficult to read. So lets start building the collection envisaged by Cantor, and observe what happens if we substitute b for a as indicated by Cantor. I underline the "diagonal" coordinates where 0 is to be substituted for 1, or 1 for 0.
0.1246798...(II.18) had guessed the error by asking whether the diagonal procedure might not work because there might be more rows than columns.
The most reasonable way of building the collection is by proceeding from (0, 0, 0, ...) to (1, 1, 1, ...), that is to say, the start of the collection should look like this:
We see "without further ado" (to quote Cantor against himself) that the new element E0 formed by the diagonal procedure is merely (1, 1, ..., 1, ...). It is only the abstractness of the procedure which blinds us to the fact that, for any n, the number of permutations (of this kind which allows repetitions) is 2n. The front edge of the progression of permutations proceeds like a curve with the power 2n. The diagonal has the linear power of n only; with increasing n the diagonal therefore recedes ever further away from the front edge of the progression of permutations. For a collection of classical mathematics (potential infinite only) we should say that the collection has n columns and 2n rows. But even ignoring this classical restriction, it is clear that, for an "infinite" collection also, the diagonal traverses only a small part of the collection.
The procedure, in effect even if not explicitly, relies on the one-one relation between columns and rows. Elsewhere in this branch of mathematics, as also in Cantor's own work, we find appeals to one-one relations between points in space and points on a line, supposedly proving that their numbers have the same power. The diagonal procedure allows us to see what may be amiss with set-theoretical proofs of this kind.While it is true that the E0 disagees with all elements encountered by the procedure, the procedure never reaches that part of the collection where the element E with which E0 does agree, namely here (1, 1, ..., 1,...), is located.
Readers have objected that "my" collection is useless because applicable to the rationals only, whereas Cantor's proof includes the irrationals. This objection is mistaken. As we build our collection, proceeding to infinity, we would, in their proper places, put the elements with non-repeating binary decimals. The fault is not in "my" collection (merely using 0 and 1 instead of Cantor's m and w) but intrinsic to Cantor's conception of a diagonal procedure in reference to such a collection. Certainly, the diagonal, in its linear progression, can never leave that part of the collection where the rational numbers are located. The collection itself, however, ("mine" as well as Cantor's own), if proceeding to infinity, by the way it would most reasonably be built, does of necessity include the irrationals, except that these can never be met by the "diagonal".Having shown that the diagonal cannot be formed in the way as envisaged by Cantor, one would be wasting words if one were to enter into a discussion about implications.
Angel, R.B., Relativity: The Theory and its Philosophy, Oxford: Pergamon, 1980. (Highly recommended.)
Arzelies, H., Relativistic Kinematics, Pergamon, Oxford, 1966.
Bergmann, P. G., Introduction to the Theory of Relativity, (1942), Dover, 1976.
Bohm, D., The Special Theory of Relativity, W.A. Benjamin, New York, 1965.
Cullwick, E.G., Electromagnetism and Relativity, 2nd ed., Longmans, London, 1959.
Durrell, C.V., Readable Relativity, Bell, London, 1931. (By a leading British mathematician; standard text for older British mathematics teachers.)
Eddington, A.S. The Mathematical Theory of Relativity, 2nd ed., CUP 1924.
Eddington, A. S., The Nature of the Physical World, 1928, CUP / MacMillan (NY).
Einstein, A., "On the Relativity Principle and the Conclusions Drawn
from it", (1907), Collected Papers, Princeton U.P., 1989, Vol.2 (Ppb),
id., Ether and the Theory of Relativity (1920), in Sidelights on Relativity, Dover, 1983, 3-24.
id., The Meaning of Relativity, (1921), Chapman & Hall, London, 1967 or Routledge, London, 2002.
id. Relativity: The Special and the General Theory, 15th Ed. (Methuen 1960) Routledge, London, 1993.
French, A.P., Special Relativity, Chapman & Hall, London, 1968.
Goldstein, H., Classical Mechanics, 2nd ed., Addison-Wesley, Reading: Mass., 1980.
Gray, J., Ideas of space, OUP, 1979.
Jackson J.D., Classical Electrodynamics, 2nd ed., John Wiley, New York, 1975.
Joos, G., Theoretical Physics, (1934), 3rd ed., Blackie, London, 1958.
(Klein, see 2)
Krane, K.S., Modern Physics, J. Wiley, New York, 1983.
Liebeck, H., Algebra for Scientists and Engineers. London: Wiley, 1969. (Relativistic 'proofs' by pure mathematics approach, by distinguished British mathematician.)
McCrea, W.H., Relativity Physics, 4th ed., Methuen, London, 1954.
Matveyev, A., Principles of Electrodynamics, Reinhold, New York, 1966.
Mermin, N.D., Space and Time in Special Relativity, Waveland Press, Prospect Heights: Ill., 1968.
Miller, A.I., Albert Einstein's Special Theory of Relativity, Addison-Wesley, Reading: Mass., 1981.
Minkowski, H., Gesammelte Abhandlungen, ed. D. Hilbert, 1911; 1967 reprint: NY: Chelsea.
id., "Space and Time" (1908), in H.A. Lorentz et al., The Principle of Relativity, Dover, 1952,75-91.
Møller, C., The Theory of Relativity, 2nd ed., OUP 1972.
Nunn, T.P., Relativity and Gravitation, University of London Press, 1923.
Oppenheimer, J.R., Lectures on Electrodynamics, Gordon & Breach, New York, 1970.
Pauli, W., Theory of Relativity (1921), Dover 1981.
(Poincaré, see 2)
Rindler, W., Introduction to Special Relativity, 2nd ed., Clarendon, Oxford, 1991.
Rogers, E.M., Physics for the Inquiring Mind, Princeton U. P. 1960.
Rosser, W.G.V., Introductory Relativity, Butterworths, London, 1967.
Russell, B., ABC of Relativity, Fourth revised Edition, Unwin Hyman, London, 1985.
Schwartz, M., Principles of Electrodynamics, McGraw Hill, New York, 1972.
Schwinger, J., Einstein's Legacy, Scientific American Library, New York, 1986.
Shadowitz, Albert, Special Relativity (W.B. Saunders, Philadelphia, 1968), Dover 1988. (4D).
Silberstein, L., The Theory of Relativity, MacMillan, London, 1914.
Stephenson, G., & Kilmister, C.W., Special Relativity for Physicists (1958), Dover, 1987.
Taylor, E.F., & Wheeler, J.A., Spacetime Physics: Introduction to Special Relativity, 2nd ed., W.H. Freeman, New York, 1992.
Tolman, R.C., Relativity Thermodynamics and Cosmology (1934), Dover, 1987.
Voigt, W., Ueber das Doppler'sche Prinzip. Nachrichten v. d. Königl. Ges. d. Wissenschaften, Göttingen: 1887.
Whitrow, G.J., The Natural Philosophy of Time, 2nd Ed. OUP 1980.(Compulsory reading for critics writing on 'time'.)
Isolated critics (see the entries for Aristotle and Maziarz, below) emphasize the special role of philosophy, rather beyond the conception favored by the literati, namely that, in the words of Wilfrid Sellars, the aim of philosophy is "to understand how things in the broadest possible sense of the term hang together in the broadest possible sense". Such a definition tends to omit just the most critical tasks, namely those concerned with language and thought: epistemology, logic, metaphysics. The transfer of areas of study to subfields such as the philosophy of science appears to leave just those basic and general critical tasks unattended. Prominent critics, for instance Dr. Borchardt, lament the tendency of the philosophy of science to accept as normative the pronouncements of scientists; unfortunately, this state of affairs all too easily results in the consensus that philosophy may be dismissed as altogether redundant. The opposite is the case. Although the philosophy of science is certainly unfit to supply the remedy in our current debacle, it is just the basic and general nature and role of concepts and abstractions that require attention - hardly a task to be left to practising scientists. (On the other hand, if one considers the extraordinary havoc wrought by influential thinkers such as Russell, one hesitates to welcome philosophers.)Anton, H., Calculus with analytic geometry. New York: John Wiley and Sons, 1980. (One typical example of the large standard literature on basic mathematical concepts for engineers.)
The omission, in my list, of texts in the philosophy of science is deliberate. For points of particular importance, see comments to individual entries. Because of the neglect of visualization, changes in the usage and terminology of 'algebra' are particularly important, hence the listing of various texts on mathematical topics.
Early versions of my webpage had tried to draw attention to the importance of Aristotle for the debate on foundations. To quote P. H. Wicksteed's 1929 Introduction to Vol. IV of the Loeb edition: "What ... are we to expect from the Physics? Something that is still of philosophical interest; very much that is of historical interest and that has entered deeply into the texture of our language; much of purely intellectual interest and bracing gymnastic; but also much that is of vital significance in relation to that borderland between physical and metaphysical thought where mathematics and philosophy meet."Arnheim, R., Visual Thinking. London: Faber, 1970. (On the impoverishment of the imagination by the mathematics of number.)
By accident, in my local library, I came across a recent philosophical treatment: Ursula Coope: Time for Aristotle - Physics IV.10-14; Oxford: Clarendon Press, 2005. The supposed difficulties with, and the objections to, Aristotle's treatment elaborated here demonstrate wo things:
First, the astounding agreement of his understanding with thinking in classical as well as modern physics (e.g. "now", "point", continuum, time as measure);
Second, the alarming detachment, or retreat, of philosophy, as a discipline, from thought in science - the complete absence of understanding at a fundamental cognitive level.
Such scholarly labours are unlikely to remedy the abysmal loss of power, from Aristotle's supreme intellect at one extreme end of the spectrum, to modern philosopher-scientist "genius" at the other.
Barzun, J., The House of Intellect. London: Secker & Warburg, 1959. (On the fêting of Einstein's genius.)
Bôcher, Maxime Introduction to Higher Algebra. 1907; Dover reprint of 1964.
Boyer, Carl B., The History of the Calculus and its Conceptual Development (1949). New York: Dover, 1959.
Crowe, M.J., A History of Vector Analysis. Univ. of Notre Dame Press, 1967. (Indispensable for critics because of detailed attention to Grassmann's clarification of concepts such as 'axiom'. To be read in the historical context - e.g. Copleston on Kant, and in complementation of ostensibly richer but anti-physics histories such as Kline, v.d.Waerden or Torretti.)
Ferguson, E.S., Engineering and the Mind's Eye. Cambr.: MIT Press, 1982. (On the debilitation of essential engineering skills by counter-intuitive mathematics.)
Freudenthal, H., Mathematics as an Educational Task. Dordrecht: Reidel, 1973.
(See p.114 for a criticsm of the Russell & Whitehead program: "as dead as a doornail" yet "seductive for mathematicians"; no questions, no problems: problems cannot even be formulated.)
id., Revisiting mathematics education. Dordrecht: Kluwer, 1991.
Grassmann, H.G., Die lineare Ausdehnungslehre, ein neuer Zweig der Mathematik, 1844. and
id., Die Ausdehnungslehre, Vollständig und in strenger Form bearbeitet, Berlin: 1862. Excerpt of this in D.E. Smith, 1959, 684-696.
Gray, Jeremy & Moore, Gregory H. (dispute about the relevance of logicism & formalism), Historia Mathematica 23 no 4 (Nov. 1996) and 24 no 2 (May 1997).
Heath, T.L. (ed.), Euclid: The thirteen books of the Elements, 3 vols (1908). Dover reprint, 1956.
id., A History of Greek Mathematics, 2 vols (1921). Dover reprint, 1981.
id., Mathematics in Aristotle. Oxford: Clarendon 1949. Compulsory reading.
Helmholtz, H., Dissertation Ueber die Tatsachen, welche der Geometrie zugrunde liegen, Nachr.d.K.Gesellschaft d.Wissenschaften zu Gottingen, math.-physik.Kl.,1868.
id. "The Origin and Meaning of Geometrical Axioms", Mind (1876).
id., Epistemological Writings, Hertz/Schlick Centenary Edition 1921, reprint. Dordrecht: Reidel, 1977.
id., Popular Scientific Papers, ed. Kline, M.. New York: Dover, 1962.
Hertz, Heinrich: Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt (1894); Engl. transl.: The Principles of Mechanics Presented in a New Form (Introduction by Helmholtz). 1899 (London: MacMillan).
Traditionally, mechanics had been one of the most important branches of mathematics, a tool for empiricist analysis. Hertz's text reflects the new counter-intuitive spirit: exposition of subject matter and method in the form of a logical treatise with abstract mathematical formalisms. Not surprisingly, admired by Russell; as seen by Mach (Die Mechanik ..., Kap. 2.9), beautiful but not recommended for application.
Klein, F., Vorlesungen über die Entwicklung der Mathematic im 19. Jahrhundert.
I. Teil (pure mathematics), 1926, Berlin: Springer.
II. Teil (mathematical physics), 1927, id. (including 4D SR)
id., Elementary mathematics from an advanced standpoint.
Pt.1: Arithmetic, Algebra, Analysis. 3rd ed., 1924. Engl.tr.: NY Dover (undated).
Pt.2: Geometry. 3rd ed. 1939, London: MacMillan.
Kline, M., Mathematical Thought from Ancient to Modern Times. OUP: 1972. (Compulsory reference for all critics.)
id., Mathematics: The Loss of Certainty. OUP: 1980. (See especially Ch. IX - XI on the rise of logicism.)
Lamb, Horace, Dynamics. CUP: 2009 reissue of the 1961 edition (first edition of 1914). (Especially valuable as modern textbooks omit, perhaps as obvious, the simple case of constant velocity with its displacement graph - no t-co-ordinate.)
Liebeck, H., Algebra for Scientists and Engineers. London: John Wiley & Sons, 1969. MacFarlane Smith, I., Spatial Ability: Its Educational and Social Significance. London University Press: 1964. (On the the danger to the nurture of skills of non-verbal reflection by the rise to dominance of the "Western culture of articulacy".)
Maziarz, E.A., The Philosophy of Mathematics, New York: Philosophical Library, 1950. (On failure of philosophy in its role as critic; comprehensive bibliography.)
Merz, J.Th., A History of European Thought in the Nineteenth Century. 4 vols. Edinburgh/London: 1907 ff.
Poincaré, Henri, items from the collected works:
Sur la Dynamique de l'Électron (Académie des Sciences, t. 140, p.1504-1508; 5 juin 1905); (Oeuvres, La Section de Géométrie, Vol. IX, pp.489-493).
Sur la Dynamique de l'Électron (submitted July 1905 to Rendiconti del Circolo matematico di Palermo, t. 21, p. 129-176; published 1906); (Oeuvres, La Section de Géométrie, Vol. IX, pp.494-550).
Price, M., Mathematics for the Multitude? London: The Mathematical Association, 1994. (See Ch.3 for the dispute among proponents of "pure" vs. "hands-on" mathematics; note Russell's influence.)
Pyenson, L., The Young Einstein. Bristol: A. Hilger, 1985. (Detailed discussion of Einstein's sources in 1905.)
Riemann, B., Ueber die Hypothesen, welche der Geometrie zugrunde liegen. (1867). Darmstadt: 1959.
id., On the Hypotheses Which Lie at the Foundations of Geometry. Tr. H.S.White.
In D.E.Smith, ed., A Source Book in Mathematics, Dover, New York, 1959.
Roe, J., Elementary Geometry. OUP: 1993.
For a useful summary of his work and influence, especially the influence of his early work, see Audi, Robert (ed.): The Cambridge Dictionary of Philosophy, Cambridge, New York, etc.: CUP (any ed., 1st ed. 1995).1897: An Essay on the Foundations of Geometry. London: Routledge, 1996.
Russell (1903) spells out admirably the task of the philosopher of mathematics, namely as critic and arbiter of the premisses, and of the principles of deduction. But to the great detriment of mathematics and science, he is thwarted by his own beliefs, for instance:
1. that mathematical space is constituted of unextended atoms of mathematical matter (no continuity until proven by Cantor, of all people!), hence his objection to Euclid that he fails to prove that lines have no gaps, e.g. at a point of intersection;
2. that the properties of actual space can only be determined by experiment and observation (false: what we observe are the interactions of physical things; "space" is a concept, an abstract construct as in mathematics).
Important arguments are found widely distributed in R.'s Works; I list a small selection by their year of publication.
Schiemann, G., Wahrheits-Gewissheitsverlust: Hermann von Helmholtz' Mechanismus im Anbruch der Moderne. Darmstadt: Wissenschaftliche Buchgesellschaft, 1997.
Smith, D.E. (ed.), A Source Book in Mathematics, Dover: 1959.
Sommerville, D.M.Y. Analytical Geometry of Three Dimensions,. CUP 1947.
Thiele, Ch., Philosophie und Mathematik (in German). Darmstadt: Wissenschaftliche Buchgesellschaft, 1995.
(Comprehensive survey & bibliography, from an unquestioned dualistic perspective, of trends in the foundations of mathematics, including concepts of space. Typically, Grassmann is not even mentioned. Note the queer outcome of the dualist theory of knowledge where mere abstractions such as mathematical spaces present as mystically co-existing real universes.)
Torretti, R., Philosophy of Geometry from Riemann to Poincaré. Dordrecht: Reidel, 1978.
Weyl, Hermann, Space, Time, Matter (4th Edition, 1921), Dover (original tr.) 1952.
id., Philosophy of mathematics and natural science. Princeton University Press, 1949.
Whitehead, A.N., A Treatise on Universal Algebra. (1898). New York: Hafner,1960. (Linear algebra, fr. Grassmann; important source text on early vector notation.)
id., An Enquiry Concerning the Principles of Natural Knowledge. C.U.P.: 1919.
id., The Concept of Nature. C.U.P.: 1920.
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