sapere aude: page 2

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sapere aude: page two

The loss of meaning

Last revised: 23 August 2009
url: http://www.btinternet.com/~sapere.aude/page2.html

1. The Lorentz Transformation and the common sense foundations of knowledge

ฉ G. Walton 2009

The Lorentz Transformation (LT) of Special Relativity (SR) appears much too trivial to be of relevance for the debate about the foundations of knowledge. There are believed to be foundational aspects of SR, but the dismal quality of objections suggests ignorance and incompetence rather a than a problem with foundations. The LT is of significance because it strikingly illustrates the danger of developments in the foundations of mathematics.

Logicism has become dominant in the philosophy of mathematics; according to Russell, mathematics has no content, only form (Russell, Russell & Whitehead; cf. Kline History, Ch. 51: The Foundations of Mathematics). In consequence, "counter-intuitive", that is to say blind, symbol-pushing has been elevated to an ultimate art of reasoning. Physicists rightly insist on physical/intuitive models: but it is just these that modern mathematics forbids as a matter of principle. The LT warrants attention because it reveals the futility of the logicist programme: it is identical in form with operations elsewhere in the standard mathematical literature, yet the proper solution can be seen to depend on content, that is to say, on the (abstract!) mathematical subject matter.

Philosophers (Maziarz, for instance) lament the neglect of "critical metaphysics" with its vital role of judging the nature and function of abstraction in all the sciences, and most particularly in mathematics. But even here anti-empirist trends in the theory of knowledge come to the fore, in that no attention whatsoever is paid to the role of visual logic (the abstract images of geometry) as an indispensable part of mathematics (including the mathematics of number).

Prominent critics of SR (e.g. Prof. McCausland) would have it that SR is quite independent of the LT, and that therefore an investigation of the LT misses the point. But it is by means of the LT that Einstein appears to have arrived at the much debated effects (clock retardation; contraction of spatial extension by the reciprocal Lorentz factor, subsequently generalized in GR); without these effects there is nothing left of SR. In fact, Einstein's derivations of the LT reveal just that farcical illogicality and weakness of thought which render his verbal expositions so infuriatingly baffling. This lamentable state of affairs is evident upon the briefest glimpse at Einstein's quantitative treatments; the failure to recognize the significance of the LT has therefore alarming implications for thinking in physics.

In my discussion I need to adduce sets of equations; before that unpleasant task I insert here a sketch of the likely historical origin of the Einstein's 1905 LT as a mathematical form. My page 3 lists critics' objections; see below for a brief discussion.

In 1905, the forms adduced in Einstein's sources would have reminded of forms elsewhere in mathematics (quadratic forms, linear transformations); indeed, his teacher Minkowski was an acknowledged authority on quadratic forms with a special interest in geometric interpretations and extension to nD. nD geometry and manifolds, likewise, though lacking physical applications, had been familiar for decades (Grassmann 1844, Riemann 1854, Cayley 1843, and many others; see Kline, History, Ch.43). To Einstein, the quantities in the physical theories of, e.g., Lorentz or Larmor, would have appeared to invite interpretation as coefficients in a transformation. His verbal justification and discussion (งง1, 2, 4) merely reflect his ignorance of the mathematical meaning of such an operation, and of the mathematical meaning of the specific steps in his first derivation of the LT.
I might briefly mention Einstein's apparent conception of dimensionality: 3D or 4D? Although the 1905 paper uses the equations for the four variables x, y, z, t (x, h, z, t), there is here no suggestion of a 4D geometry. His 1907 paper on SR, in fact, adduces only equations for the three space-variables ("we conclude after a simple calculation that the transformation equations must be of the form..." - there follow the four equations of the complete LT): an unlikely approach for a 4D treatment. Similarly, the "Simple Derivation" of 1917, refers to a sketch that has the quantity vt on the x-axis, which would not be the case in 4D. The 1922 Princeton Lectures, however, present the LT as 4D.

The LT and other forms of that type

Einstein (1905) appeared to have solved the following problem: to find the correlation between ct'(x', y', z') and ct(x, y, z), for co-ordinates and times of systems S(x, y, z, t) and S'(x', y', z', t') in relative motion (displacement of origins OO' = vt):
[1] c²t² = x² + y² + z²
c²t'² = x'² + y'² + z'².

Felix Klein, following Minkowski's mathematical treatment (Vorlesungen ..., Teil II, Kap. 2), generalizes [1] to
[2]

ds² = dx² + dy² + dz² - c²dt²
= dx'² + dy'² + dz'² - c²dt'².

The generalization immediately explains the confusion as to the meaning of "invariance": invariant c in treatments of [1], as against invariant ds in [2] (here the norm of the 4d position vector).

I will return later to [1] as a 3D problem. In consequence of Einstein's inadequate grasp of the meaning of symbolic operations, the subsequent treatment of the LT has followed [2], for mathematicians a welcome application of 4D geometry. I will therefore present the LT normative for the physics literature as formulated by Klein.

Klein starts with the orthogonal transformation for the general case of translation of origin and rotation in 4d (co-ordinates x₁, x₂, x₃, x₄, subsequently renamed x, y, z, l for ict, simarly for x'₁, ... x₄, renamed x', ..., l'), namely
[3]

x' = a₁x + b₁y + g₁z + e₁l + z₁,
y' = a₂x + b₂y + g₂z + e₂l + z₂,
z' = a₃x + b₃y + g₃z + e₃l + z₃,
l' = a₄x + b₄y + g₄z + e₄l + z₄,

where the a_i, ..., e_i satisfying [2] have to be found.

The transformation extends to 4D the orthogonal transformation familiar from 3D analytical geometry (e.g. Sommerville); Klein's 4D coefficients a_i, ..., e_i (as also the coefficients in [4] and [5] below) are the equivalents of the direction cosines in 3D.

Klein subsequently simplifies the case as follows. First, z_i = 0; the case is thereby effectively restricted to rotation only; it corresponds now to the operation "change of basis" for orthogonal systems in contemporary linear algebra. Second, y' = y, z' = z; that is to say, rotation is restricted to the x-t-plane. (There follows the conventional solution: the LT as first presented in Einstein (1905)).

The mathematical literature follows Klein's normative exposition, as, for instance in Liebeck:
[4]

x'= g(x - vt),
y' = y, z' = z,
t' = ax + bt,

to be solved for g, a and b,

or, in Bergmann as more typical of the physics literature:
[5]

x' = a(x - vt),
y' = y, z' = z,
t' = bt + gx,

to be solved for a, b and g.

Klein's transformation [3], restricted to the case of rotation, represents a geometric interpretation for the case of 4D of the general nD linear transformation of Higher Algebra, as for instance in B๔cher, namely
[6]

x'₁ = a₁₁x₁ + ... + a_1nx_n
...
x'_n = a_n1x₁ + ... + a_nnx_n;

because of the frequent change of symbolic usage, the date of publication, namely 1907, is of particular interest.

So far, so good. [3] to [6] are forms that appear to yield their solution independent of content; [6] is of a higher degree of abstraction because its (abstract!) geometric meaning is left out of consideration. But now compare [6] with a further form also presented in B๔cher:
[7]

x' = a₁x + b₁y + c₁z + d₁t,
y' = a₂x + b₂y + c₂z + d₂t,
z' = a₃x + b₃y + c₃z +d₃t,
t' = a₄x + b₄y + c₄z + d₄t.

This is the so-called projective transformation. The t-variable does here not denote the time as in the equations of mechanics; it does not represent a coordinate; the transformation is 3D. Despite the essential formal identity with [3] to [6], [7] yields a fundamentally different solution because it differs in "content". The solutions in [3] to [6] had appeared independent of content because these forms represent one and the same content (4D rotation). No doubt, those more familiar with the mathematical literature than I am would be able to adduce other examples where the reliance on form alone can be seen to be misleading.

The LT, as first formulated by Einstein, is one such example; it is of interest because of its role in the evolution of modern mathematical physics.

Einstein's (1905) LT as a form

I commence with purely formal aspects. I postpone the discussion of "content", namely the reasons why two identical forms yield different solutions.

As a form, Einstein's exposition of the linear equations for the solution of [1] is identical with [4] or [5], the 4D approach to the solution of [2]. But [1], unlike [2], is 3D. However, a form identical with [4] or [5] would also be applicable to the 3D case, albeit restricted to points on the x-axis only. Yet the conventional solution, correct in 4D, does not follow. Contrary to the logicist doctrine, identical forms, namely 3D and 4D [4] or [5], have different solutions. The solution depends on content, namely the geometric meaning. I proceed now to the discussion of "content".

Einstein (1905) attempts to solve [1] which, unlike [2], is 3D. In [1], ct and ct' are the position vectors with the components x, y, z, and x', y', z'. ct is here the radius of a sphere with its centre at the origin O; the abstract geometric case would be applicable to the case of isotropic propagation of light, regardless whether wave-like or corpuscular. Points on the sphere about O are to be referred to a second coordinate system (say S'); by definition, the origin O' of S' is to be displaced through vt on the (coincident) x- and x'-axes. Fig. 1 shows a simplified case (z, z' = 0); because of the inevitable asymmetry about O' I use P for points to the right of O and O', and Q for points to the left. In [1], ct' is the position vector from O' of points on the sphere about O. The equation for ct' in [1] expresses merely the Pythagorean relation between the position vector and its 3D-components.


                            (y)  (y')
                             |     |
 Q  . . . . . . . . . . . .  |. .  | . . . . . . . . . . P
 .   .*                      |     |                 .*  .
 .       . *                 |     |             . *     .
 .           .  *            |     |         .  *        .
 .               .   *       |     |     .   *           .
 .                   .    *  |     | .    *              .
 .                       .   | *.  |   *                 .
_(___________________________O_____O'____________________)_ (x, x')  


                           Fig.1

The t- and t'-variables in [1] are not a coordinates but auxiliary or supplementary variables. The components of ct and ct' are functions of t and t': x(t), y(t), z(t) and x'(t'), y'(t'), z(t'). For greater clarity, this should be rendered explicit, namely, depending on direction,
x = c_xt (positive direction) and x = -c_xt (negative direction), as also for y and z,
y = c_yt (positive) and y = -c_yt (negative),
z = c_zt (positive) and z = -c_zt (negative); similarly for x'(t'), y'(t'), z'(t').

[1] becomes
[1a]

c²t² = t²(c_x² + c_y² + c_z²)
c²t'² = t'²(c_x'² + c_y'² + c_z'²).

Einstein had postulated a linear solution. (The later [2], as a rotation about a common origin, has indeed a linear solution.) The form [1a] makes that an unlikely assumption. A linear solution can be seen to be ruled out in 3D; it restricts the case to 1D (y, z, y', z' = 0).

But even though the solution for 1D is linear (and follows from an exposition identical in form with [4] or [5]), the LT is not the correct solution. The accepted derivation of the LT, blindly following the 4D case, had apparently proven the necessity, on purely mathematical grounds, of the reciprocal Lorentz factor. Its emergence is most elegantly shown in a fallacious argument sometimes used in the literature (I forget where: it is not my invention). A "straight" transformation, e.g. x' = x -vt, etc., does not appear to succeed. Since the movement of systems is strictly relative, the strange purely mathematical effect must apply to both equally; let's therefore assume that

x' = k(x - vt), t' = k(t - vx/c²),
x = k(x' + vt'), t = k(t' + vx'/c²),
where k turns out to be the Lorentz factor.

The error in the argument is most easily seen if we restrict the case to points on the x-axis only (Fig. 2).

Q------------------------O------O'----------------P,

Fig.2

On the right of O, we have OP = ct, O'P = ct' = (c-v)t, and on the left, QO = ct, QO' = ct' = (c + v)t.

Common sense should have warned us that any change of the time-variable in S' implicates all velocity measurements, except of c for which the "invariance" transformation has been set up. Furthermore, OO' cannot be both vt as well as vt'; caution suggests OO' = v't'. Now the transformation depends on the direction. A brief look at Fig. 2 shows that this is also true for v'. v' can be seen to equal vc/(c-v) for points to the right of O, and vc/(c+v) for points to the left. If we use these corrected values for v', the "straight" transformation succeeds: k = 1.

A linear solution, although correct for the 4D case [2], is not applicable to the 3D case [1]. That is to say, the LT is not a valid solution of the case as set up by Einstein in 1905 (and much debated and reformulated in the subsequent literature of physics). It seems to me disturbing that such a simple matter, as the difference between [1] and [2], should have been disregarded. Objections to SR and its implications have been confined to conclusions that never follow, and to concepts of space: an alarming state of affairs. The acceptance of the LT, as a solution of the geometric case as defined by physicists, has grave implications for the debate about common sense and the foundations of knowledge. For it reveals the danger of the regression to symbolic abstraction (effectively "blind symbol pushing") which has become normative in the entirety of mathematical physics. The absurdly simple case of the LT should remind philosophers that scrutiny of the role of abstraction, in science generally as in mathematics, is overdue.

page 3 lists critics' objections to the 1905 derivation. Unfortunately, these tend to confine themselves to the symbolism and to isolated irregularities in Einstein's procedure. While "mistakes" may puncture the Einstein myth, they do not advance understanding: why do mathematicians support SR, and why, on purely mathematical grounds, do we seem to be condemned to the paradoxically reciprocal Lorentz factor? For comparison, here are my own brief notes:
Einstein's 1905 exposition is a chain of unwarranted presuppositions, non sequitur's and invalid conclusions. (It is salutary, when reading Einstein's mathematical arguments, to bear in mind his reputation, among his friends, for ambiguity, illogicality and uncritical adoption of popular notions.) For instance (page refs are to the Dover edition; I use the letters S and O for system and origin):
1. p.44: (From definition of the geometric scenario on p.43 - points on spherical surface referred to S and S' with translation of origin. Elementary coordinate geometry, except for change of the unit of measurement, with the purpose of rendering different (!) displacements in S_i equal to ct_i.) Average velocity of a two-way signal along the x-axes, leading to a partial differential equation (discontinued).
2. p.45: Equation for t from the presupposition that it must be linear, warranted for signals along the x-axis but not, as subsequently uncritically taken for granted, for signals in other directions. (The coefficient a indicates already Einstein's failure to distinguish between geometric quantities - coincident, that is to say, identical displacements - and the referents of other algebras: number; linear algebras as in the later 4D rotation of mathematicians where such a coefficient denotes the equivalent of a direction cosine). Note that, because of the assumption of linearity, the transformation implies y, z=0; these symbols have become mere dummies.
3. p.46: A deus ex machina: but for the (nonsensical) coefficient j instead of the previous a, the conventional LT, with the tacit substitution of the square root for c²/(c² - v²). (N.B.: Deliberate, not a "mistake", for without it the later inverse transformation would not "work".)
4. p.47: The inverse transformation, and the supposed proof of the mathematical necessity of the reciprocal Lorentz factor: a third system S" moving, in the time t of the second system, with the velocity -v is assumed to be "at rest" with the first system S, because we arrive at quantitative identities (but for the j, subsequently to be ironed out). But the stretching by the reciprocal Lorentz factor is here needed mathematically precisely because the quantities under examinations are not identical (the origin O" of system S" moving, in the time t, with the velocity -v would not coincide with the origin of the first system). So much for the derivation proper.
5. pp.48-49: there follow the typically Einsteinian "proofs":
a) The figure, assymetric about O', stretched by the (nonexistent because erroneous!) Lorentz factor is an ellipsoid contracted along the x-axis;
b) The inconvenient asymmetry in the equation for t can be eliminated. (We are left with the so-called proper time, used in arguments about clocks and the twin paradox.)
6. p.50: Composition of velocities, blind symbol pushing again, of a truly mind-boggling idiocy: any velocity other than v, c, e.g. any w' (w' < c), can, of course, easily be read from a geometric diagram.
Mistake No.4, namely the assumption that, despite the change in the time scale (unit of measurement, basis), the displacement between origins is both vt as well as vt', is made also in much of the critical literature. This mistake is like the one often made with percentages, namely the assumption that, if a = b + 20%, then b = a - 20%. On the whole, the symbolic gibberish of SR clearly shows that, as often suspected, mathematics makes us stupid.

2. Tower of Babel

This is the edited version of an item distributed among critics in 1997. Once we begin to understand what has gone wrong, we may marvel at the ingenuity of generations of thinkers in rationalizing the outcome of an utterly simple but faulty geometric argument.

On the nature of relativistic effects

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)
2. real
3. not real
4. apparent
5. the result of the relativity of simultaneity
6. determined by measurement
7. a perspective effect
8. mathematical.

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 ...

M๘ller [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.

3. Einstein's "Simple derivation"

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.)
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.

Here is Einstein's text:

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'.
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 [1].
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 [2].
Those ... events ... which satisfy [1] must also satisfy [2]. Obviously, this will be the case when the relation
(x' - ct') = l(x - ct) [3]
is fulfilled in general; where l indicates a constant; for, according to [3], 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) [4].
By adding (or subtracting) equations [3] and [4], 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 [5].
We should thus have the solution of our problem, if the constants a and b were known. These result from the following discussion.

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:

Fig. 3.a: Isotropy in K




_Q___________________________O_____O'____________________P_ (x, x')  


                           Fig. 3.a: QO = OP

Fig. 3.b: Isotropy in K'




_Q_____________________O_____O'__________________________P_ (x, x')  


                           Fig. 3.b: QO' = O'P

Now let's look at Einstein's text. Notice how we are slowly getting into trouble. [1] and [2] are perfectly in order, and compatible with either version of our figure. For regardless whether QO = OP or QO' = O'P, we may say that OP = ct and O'P = ct'.

So far, so good. Although not 'false', the indeterminate zero equation [3] warns of trouble ahead. The derivation derails fully with equation [4]. As here explicitly defined, the symbols x and x' [4] denote quantities which differ from those previously used in [1] and [2]. We have, in [4], x = -ct, x' = -ct', whereas, in [1] and [2], 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 [1] and [2] 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 [3] and [4] should then read:

(ct' - ct') = l(ct - ct)
and
(-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 [3] and [4] 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 [5] 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 [5], 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 [5], we find avt = bct. Further progress can be made by evaluating [5] 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 [1] to [5], 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'--------------P
where 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.

4. The triumph of mathematical unreason: Cantor's diagonal procedure

Briefly, for those not 'in the picture', Cantor's 'diagonal' is THE most celebrated proof that classical notions of truth and reason must be abandoned, and that the abstract, purely formal methods of mathematics are able to establish the existence of truths that transcend ordinary reason. (The literature on the philosophical implications is large and continually growing; it is generally accepted that common sense has been shown to be indefensible.)

Even in a more considered treatment, it would make little sense to present a specialist bibliography; I appenc a selective reading list. Here I may only mention that expositions of the supposedly world-shaking argument are found in virtually all texts on the history or philosophy of mathematics and number theory (e.g. Bell, Boyer, Courant, Hardy, Kline, Russell).

To comprehend its significance it must be kept in mind that new theories of "the infinite" (e.g. since Bolzano) had come to distinguish between two different kinds of "infinity": potential (continuing ad infinitum, so-called denumerable) and actually "completed infinities", the latter represented in terms of strictly formal definitions of supposed totalities which, though non-denumerable, are postulated to include the "infinity" of all possible elements. (The jargon keeps constantly changing, which gives editors a convenient excuse to reject submissions, re-submissions generally being ruled out.)

Cantor's work is exclusively concerned with the paradoxical properties of "completed infinite collections"; the diagonal argument is merely the most celebrated among 'evidence piled upon evidence' of the 'counter-intuitive' properties of this type of collection. Briefly put, his argument is as follows; for his symbols m and w I substitute 0 and 1.

If we consider elements like

{0, 0, ..., 0, ...} {0, 1, 0, 1, ..., 0, 1, ...} {1, 1, ..., 1, ...}

the "completed infinite totality" containing all possible elements, including all possible permutations of the symbols 0 and 1, must have the form

{a₁₁, a₁₂, ..., a_1n, ...} {a₂₁, a₂₂, ..., a_2n, ...} ... {a_n1, a_n2, ..., a_nn, ...}

where it is understood that n signifies the actual "completed infinite". In our present example, the nth element would be {1, 1, ... 1, ...}.

Although, by definition, the totality is to include all possible elements of this type, Cantor proceeds to show that we are able to form an element which must differ from each element of the collection and which therefore is not included in the collection. This paradoxical element is formed by the diagonal procedure, as follows. Remember that the collection contains all possible permutations of the symbols 0 and 1; each a_ik must therefore be either 0 or 1. The first character of this new diagonal element depends on a₁₁; if this is 0 we substitute 1; if it is 1 we substitute 0. Similarly, the second character is 1 if a₂₂ is 0 and vice versa, and so forth for every element of the original collection. We are thus certain that the new element cannot agree with any element included in the original collection.

As I have mentioned, the proof is celebrated, even though a few voices have been raised against it. The major problem is believed to be that of order. As put by Wittgenstein, how is one to be sure that a number differs from

0.1246798... 0.3469876... 0.0127649... 0.3426794... ...(Imagine a long series.)

In fact, he hit the nail on the head by his observation that the diagonal procedure fails if the collection has more elements than each element has characters. (The common objection, e.g. by Russell, is that, "at infinity", this is irrelevant because there are as many units as there are millions, billions, trillions etc.)

Now consider this counter-example; we never need speculate about the supposed properties of "completed infinities" to see that the diagonal procedure merely blinds us. It rapidly runs ahead of itself and merely produces an element further down because the diagonal can never reach it. Here is the counter-example; we order the original collection as follows; I underline the positions from which we are to form the new element.

{0, 0, 0, 0, 0, 0, 0, 0, ..., 0, ...} {1, 0, 0, 0, 0, 0, 0, 0, ..., 0, ...} {0, 1, 0, 0, 0, 0, 0, 0, ..., 0, ...} {1, 1, 0, 0, 0, 0, 0, 0, ..., 0, ...} {0, 0, 1, 0, 0, 0, 0, 0, ..., 0, ...} {0, 1, 1, 0, 0, 0, 0, 0, ..., 0, ...} {1, 0, 1, 0, 0, 0, 0, 0, ..., 0, ...} {1, 1, 1, 0, 0, 0, 0, 0, ..., 0, ...} ...

The character encountered by the diagonal procedure is everywhere zero; it is quite evident that, if, as we may, we so order the elements, it is impossible for any character further to the right to be other than zero. Hence the diagonal procedure merely gives us the banal

{1, 1, 1, 1, 1, 1, 1, 1, ..., 1, ...}

which, by definition, must be included in the collection. Note that Cantor's own method follows the perfectly admissible procedures for collections tending to infinity. The proof fails because it cannot substantiate assumptions as to "infinity"; any such assumption remains a matter of faith.

(There has been the objection that the collection is confined to rational numbers and cannot include the irrationals because "all" elements end in infinitely repeated zeros. But merely substituting 0 and 1 for Cantor's m and w does not alter the character and properties of the collection. An "infinite" collection of all possible permutations would necessarily include elements ending in nonrepeating numerals.)

Conclusion: Cantor's diagonal procedure has misled us by its very abstractness. Far from proving that common sense is shown to be indefensible, it makes untenable the modern conviction that the abstract methods of purely formal mathematics are able to reveal to us truths which transcend ordinary reason.

5. References and reading list:

1. Expositions of and topics associated with special relativity

Aharoni, J., The Special Theory of Relativity, (1965), Dover, 1985.

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), 252-311.
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.

M๘ller, 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.

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.

Whitrow, G.J., The Natural Philosophy of Time, 2nd Ed. OUP 1980.(Compulsory reading for critics writing on 'time'.)

2. The road to perdition: the philosophical and mathematical background

(The philosophy of science has rightly been dismissed as the publicity department of establishment physics; its omission in my lists is deliberate. I include here only a few texts that help to follow the course of pernicious developments. Because of the neglect of visualization, changes in the usage and terminology of 'algebra' are particularly important. Previous versions had listed several metaphysicians from a Roman Catholic background; but their emphasis on common sense - Aristotle, Aquinas - is vitiated by notions of being and truth that reject change and motion as contingent: no help for a physics sunk in epistemological muddles in its search for the structure and causes of change.)

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.)

Arnheim, R., Visual Thinking. London: Faber, 1970. (On the impoverishment of the imagination by the mathematics of number.)

Barzun, J., The House of Intellect. London: Secker & Warburg, 1959. (On the f๊ting of Einstein's genius.)

B๔cher, 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, Vollstไndig 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. Dover reprint, 1956.
id., A History of Greek Mathematics, 2 vols. 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., Epistemological Writings, Hertz/Schlick Centenary Edition 1921, reprint. Dordrecht: Reidel, 1977.
id., Popular Scientific Papers, ed. Kline, M.. New York: Dover, 1962.

Klein, F., Vorlesungen �ber 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. Unfortunately marred by the crude epistemology typical of the entire mathematical profession, resulting in misrepresentations of classical philosophy, based, e.g., on unreflected and nonsensical notions of "physical space".)

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. (Highly recommended; comprehensive bibliography.)

Merz, J.Th., A History of European Thought in the Nineteenth Century. 4 vols. Edinburgh/London: 1907 ff.

Price, M., Mathematics for the Multitude? London: The Mathematical Association, 1994. (See Ch.3 for literature on the history of operationalist mathematics.)

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.

Russell, B., The Principles of Mathematics. London: Routledge, 1992.
id., An Essay on the Foundations of Geometry. London: Routledge, 1996.

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 Poincar้. 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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