The items in part 2 have been drastically cut; this leaves the page unbalanced. The sketch is unsatisfactory; but the elementary nature of the muddle does not warrant excessive expenditure of time and effort. The custom of 'formal papers' on SR maths is irresponsible because it impedes scrutiny of available renditions. All relevant topics and terms (in particular the distinction between the x, y, z of 'algebra' and x, y, z as components of displacement vectors) are treated in exhaustive detail in the standard maths literature for engineers (e.g. Anton, Berberian, Kreider, Roe). Had physicists and empiricists paid attention, these grotesque confusions in mathematical physics should never have been able to take root.I present here evidence of loss of comprehension of the meaning of mathematical operations. Because of the simple coordinate geometry involved, nothing could be more suitable than the 'space-time' analysis of Special Relativity (keywords include Lorentz Transformation, anti-relativity, time dilation, clock paradox, twin paradox). The item on Cantor's diagonal is included in order to show that it is not only mathematical physics which is affected.For references, literature, and on the philosophical background of developments in mathematics and physics as expressions of irrationalism, see sapere aude.
Much of the literature of physics and of the debate among critics centres on the role of 4D 'space-time'. Mystagogues unacquainted with classical mathematical physics are convinced that the term 'space-time' denotes the totality of existence in eternity, including possible worlds past, present and future. As a matter of fact, there is here absolutely nothing new; the widespread confusion concerning 4D 'space-time' and 3D 'space' is merely the result of the failure to observe that in the analysis of movement we conventionally use two different kinds of equation and (mathematical) space: vector equations in reference to 3D figures (diagrams) in vector space (t an auxiliary variable, not a coordinate), and scalar equations in reference to 4D figures (graphs) in configuration (function) space (all variables indiscriminately referred to coordinates). (The choice between partial differentiation and differentiation by the chain rule, for instance, presupposes recognition of this fundamental difference.) If we use the wrong figure, we easily overlook the mistake responsible for the paradoxical SR gamma ("Lorentz factor", g). Note that all SR effects involve the gamma; if it cancels in the kinematics, special relativity in its entirety, including the debate about the clock or twin paradox, is left without foundation.
The 3D/4D distinction is forgotten for a number of reasons. First, overwhelmed by the vast mass and complexity of the mathematics to be mastered, physicists untrained in espistemology have come to rely blindly on the judgment of mathematicians. Second, at the time of the rise of the new physics, new interests (set theory, symbolic logic) had led pure mathematics far beyond topics applicable to physics. In addition, unfortunate developments (formalism; the Cantor-Peano 'mathematics of uncertainty') had eliminated figures as unreliable and restricted coordinate algebra to dimensionless numbers (n-tuples) only. (The vital importance of 'visualization' is today recognized by some educators; see Tall, Pt. 2.2.3 below. Placing formal considerations first puts the cart before the horse; since algebraic equations merely describe a scenario in coordinate space, it is vital that we start with our figure.) Third, the philosophy of mathematics, from its anti-empiricist orientation traditionally hostile to the intuitive mathematical practices needed in physics, instituted as normative some of the weird mathematical notions then in fashion; to date, despite the criticism by mathematicians (e.g. Freudenthal, Gray, Pulte, or Tall; Pt.2.2.3. below), this wholly useless definition of the meaning and scope of mathematics remains normative for the foundation of physics.
The difference between the 3D vector diagrams and the 4D function graphs of mathematical physics is this.
1. 3D vector diagrams
In the study of motion we use the 'time' as an auxiliary (supplementary, parametric) variable. If we say that a point moves with the velocity v in the direction of increasing x we have x = vt; both x and vt denote one and the same (vector) displacement on the x-axis of the coordinate system we use to make our quantitative analysis more easy to follow. For a point moving in 3D the 'diagram' is 3D.
2. 4D graphs of functions
There are advantages in using a figure to show the functional dependence between x and t; function graphs are essential as soon as the relation is no longer linear. The graph for our x = vt becomes 2D (the 'space-time' graph beloved by mystics inverts the velocity graph where, following the convention for function graphs, t = horizontal, x = vertical); for a point moving in 3D space the 'graph' (and the mathematical space) would be 4D. Hence Lagrange's assertion that analysis is 4D, and the insistence of relativists that the 4D space-time graphs for points moving in 3D space cannot be presented because our intuition is limited to 3D. But observe that, even if we use this graph, the displacement remains 'on' the x-axis; the t-axis, whether orthogonal or (as preferred by mathematical mystagogues) not orthogonal, shows merely the numerical ratio between x and t; using ct or ict for the time axis makes no difference. (Note that the Minkowski generalization of the non-zero 4D interval confuses two different algebras: the 4D algebra of numbers where such a non-zero 'interval' makes sense, and the requisite 3D algebra of vectors where such a 4D 'interval' is necessarily zero.) Once we start thinking about this we may laugh about worldlines in the 4D function space of pure mathematics; see also the comment on Thiele in the introductory reading list.
This simple difference helps us to understand why the proofs of generations of the most brilliant mathematicians keep arriving at the paradox of reciprocal 'contraction' (like its predecessor, the Lorentz-FitzGerald contraction, a mathematical impossibility; see briefly below). For without the correct figure we easily overlook a farcically silly mistake. This is, that, if we change the time scale, the relative velocity is no longer numerically the same in the different scales. Assuming that it would be (see Fig.1 for c and v having the same direction) puts OO':OP = OO':O'P, where OO' = vt, OP = ct, O'P = ct'.
___O_____O'____________________P__Fig.1
In short, although one need not question the existence of a truly dynamic effect involving a quantity g = (1 - v2/c2)-1/2, the relativistic argument, when corrected, would give us a g which equals 1. All the much debated problems and paradoxes - from time dilation to the twin paradox - disappear. (Not entirely; but there are other reasons why the Lorentz transformation, even if corrected, is completely useless. Note that the transformation seeks to introduce a novelty, namely 'relativistic' unit vectors; this has two consequences. First, ordinary vector algebra does no longer work. Second, the meaning of velocity changes; for instance, attempts to confirm or refute relativity by velocity measurements forget that the light velocity is constant not absolutely but only under the requisite Lorentz type transformation.)
Incidentally, the original Lorentz-FitzGerald theory had introduced the novel but hardly defensible idea that we may 'solve' problems by assigning to one and the same 'thing' two different quantities: it is this (mathematically inadmissible!) 'idea' which opens the way to the 'new physics'. (The mathematical novelty is defended even by critics because the effect is believed to be real. But it is nonsense to say that the contracted body may present as not contracted in some other system of coordinates, namely a mythical frame of reference at rest with the universe. If the body is really contracted it has the same extension in any frame of reference.) Mathematics is helpless against such abuses: it can only confirm that the two quantities l and l' - entered for one and the same extensional object - differ. SR must have been attractive to opponents and followers of Lorentz alike. For it seemed to show that the difference is mathematically necessary and that Lorentz, in failing to discover its reciprocity, had simply not gone far enough.
I preface this part with a figure (simplified to the case when z = 0) used later to facilitate the examination of classical derivations:
(y) (y')
| |
Q . . . . . . . . . . . . |. . | . . . . . . . . . . P
. .* | | .* .
. . * | | . * .
. . * | | . * .
. . * | | . * .
. . * | | . * .
. . | *. | * .
_(___________________________O_____O'____________________)_ (x, x')
Fig.2
Comment on Fig.2: To draw attention to the asymmetry of the
figure, not evident in the Lorentz transformation, I include a point Q to the
left of the origins. (I do not show the double image needed to represent the
effect of 'contraction'; the main reason for this omission is that the
transformation actually 'stretches' the figure in the direction of the x-axis;
we should therefore need, for P as well as Q, a pair of apparent images, one
shifted inward and the other shifted outward. The reader is asked to complement
the figure in his imagination.) For ease of reference I have connected the
points O and O' with P and Q, and placed the marks ( and ) on the x-axis.
Observe that the transformation uses the symbolism of analytical geometry which
was the norm before universal adoption of vector algebra. All extensive
quantities ('displacements') are there directed, i.e. vectors ; the only
difference from vector algebra is that we do not yet write, e.g., x + y + z = r,
but must still use the norm x2 + y2 +
z2 = r2 . Note in particular that early vector
algebra (see e.g. Maxwell, Whitehead, or Dickson), like the more rigorous
modern treatments (e.g. Berberian), does NOT use bold letters (see also Crowe's
history); it is always evident from the general form of an equation whether the
elements are scalar or vector. (Note further that elementary vector algebra
simplifies the symbolism because the universal unit vector is implicitly taken
for granted. As soon as we consider scale transformations it is therefore
vital to observe more advanced practice where unit vectors are rendered
explicit. In addition, to prevent falsification by deep-rooted
operationalist preconceptions, it is advisable to substitute for v, c, t
innocuous symbols like a, b, n; the former may be reinstated after solution of
the problem x2 + y2 + z2 - (ct)2 = x'2 + y'2 + z'2 - (ct')2 = 0. Further, to track down more easily the terrible
mistake responsible for the preposterous 'paradoxical' gamma, be
specific: try displacements on the x-axis only, rather than overambitious 3D
transforms 'in general'.)
In earlier versions I had presented lengthy examinations of the 'algebra'. But this is superfluous for readers with the most elementary algebraic competence, and useless for those who, lacking such competence, must place their blind trust in gurus. I restrict myself therefore to a very brief discussion.Because of difficulty of access to the original paper I base my discussion on the quotation in Ch. IX, A. O'Rahilli, Electromagnetic Theory, Dover reprint, 1965, 325. Voigt presents the coordinate differences for displacements referred to two coordinate systems in relative motion along their x-axes. (Mathematicians who object that coordinate systems 'cannot move' might look at the transformations of Klein and Noether where such movement is included as a matter of course; see, e.g., van der Waerden.) Since it is only the differences that matter, the argument is easier to follow if we assume that the origins O and O' of the systems S and S' coincide when t,t'=0. The constant kappa, introduced by mathematical convention, turns out to be unity; to keep things as simple as possible we may therefore here ignore it. (As a matter of fact, such constants - including the gamma - make sense only in the algebra of numbers; in reference to coextensive geometric elements any such constants are necessarily unity.) We start with our figure which is central to clarification (z=0).
(y) (y')
| |
Q . . . . . . . . . . . . |. . | . . . . . . . . . . P
. .* | | .* .
. . * | | . * .
. . * | | . * .
. . * | | . * .
. . * | | . * .
. . | *. | * .
_(___________________________O_____O'____________________)_ (x, x')
Fig.1
In the simplified form for the case when O and O' coincide at the time t,
t' = 0 Voigt's coordinate transformation becomes
It is then easy to verify that
Before we proceed to a cursory discussion, note in particular that [1] explicitly assumes x' to depend upon direction; although the 'algebra' 'works', this is incompatible with the equivalence of frames of reference (isotropy in S as well as S').As is immediately evident, Voigt commits errors such as the following:
1. The Michelson-Morley result had shown that propagation in the vicinity of moving bodies cannot be assumed to be homogeneous; Voigt might have been expected to understand that recourse to linear algebra is therefore ruled out. (Because of the lack of isotropy in S' the inadmissible reatment would, in any case, not solve the Michelson-Morley paradox.)
2. The transformation, even if it were valid, would be useless for physics. The time scale ('clock rate') depends upon direction: the presence of the x-component implies variation between |ct'| = ß|(+c - v)t| and |ct'| = ß|(-c - v)t|. Clocks that adapt their rate to the direction of signals do not exist. (It is this uselessness of a relativistic time scale which compels physics to adhere to some kind of 'universal' scale unit.)
3. More seriously, deluded by the mathematics of formalism, Voigt forgets the difference between two completely different 'algebras': the algebra of number and that of extension. Oblivious of the geometry defined by [1] and [3] he blindly applies the algebra of number. This 'works' because in the algebra of number [3] is indeterminate. [1] is merely one of an infinity of purely arithmetical solutions, one arrived at by arbritary constraints: y',z' = y,z; t' linear in x and t only; presence of the ß. A transformation of this kind should have used the algebra of extension (analytic or coordinate geometry); equations reflect the geometry of the case and Voigt's solution is evidently false.
4. In his fanatical pursuit of 'generality' Voigt fails to attend to two different kinds of functional dependence between variables:
a) the dependence of ratios on the direction of OP (and O'P);
b) the dependence of extensions upon t. However, ratios are independent of t which (unlike t') divides out.
5. If P, in S as well as S', refers to one and the same geometric point, the x-component of O'P is coextensive with (x - vt); it follows that ß = 1. (If P refers to different geometric points the argument becomes nonsensical.)
6. Voigt fails to state that equation [3] must be zero; as rendered explicit by Poincaré and Minkowski, [3] is compatible with the geometrically nonsensical 'generalized' 4D 'interval'. But even in the case of the zero 'interval' we are misled to assume that both sides of [3] refer to spheres; this is not the case because t' depends upon direction.
To arrive at the (useless) relativistic ratio t':t one might rewrite [3] more correctly (ß = 1) in terms of the components of ct
As already suggested, a detailed discussion is superfluous or useless (e.g. why a linear dependence of t' on t and x only implies y, z = 0); readers so inclined should write this out for themselves.
Here one may only take note of the presentation of such mathematical nonsense by a leading member of one of the most prestigious German schools in mathematical physics - preparing the ground for 'advances' elsewhere (Princeton; Cambridge/UK). The fanatical pursuit of abstraction is now known to have caused havoc throughout mathematics, there exists a large educational literature; see, e.g., ed. Tall; unfortunately, mathematicians continue to believe in SR as one of the greatest triumphs of their profession. Let's leave Voigt and turn to Poincaré.
(This item would benefit from editorial cutting for which I don't have the time. I should add that the repellent nature of the subject matter acts as a disincentive. Apology to the reader.)Our concern is the reciprocal effect associated with special relativity; it suffices to focus attention upon a small number of formalisms in Poincaré's huge oeuvre on this topic.
The conceptual and formal groundwork for the algebraic treatment of the Lorentz Transformation (LT) has been laid in Poincaré's comprehensive discussions of the literature, e.g. his vast papers on the topic since 1887 (Oeuvres, Vol. IX). There we find him raising, and returning to, themes like the isotropy of propagation in systems in relative motion, the principle of the relativity of physical effects, the method of clock synchronization, the algebraic rescaling of 'time', the promise of a mathematical solution of contradictions, and the demonstration of an apparent dynamic effect inherent in the relativity of movement. The final algebraic exposition and development of the LT are found in P.'s papers of 1905 and 1906 (Oeuvres, Vol. IX, pp. 489-493, 494-550); in papers of 1908 and 1912 (pp.551-586, 587-619) P. returns to logical and philosophical questions. Formal considerations had led him to simplify the transformation; as this makes his arguments difficult to follow it seems best to commence with brief excerpts followed by a translation into conventional symbolism. To emphasize the priority of the geometric scenario described by the 'algebra' a copy of our figure accompanies both parts.
Excerpt from Poincaré's mathematical treatment
Note: References, by page, are to Oeuvres, Vol. IX. Because of ASCII restrictions I use the letter e instead of P.'s Greek letter eta.In his brief paper of 1905, Poincaré (p. 490) presents the transformation as derived by Lorentz, and proposes to name it after him, namely
[1] x' = kl(x + et), y' = ly, z' = lz, t' = kl(t + ex),
where x, y, z are the coordinates and t is the time before the
transformation, x', y', z' and t' after the transformation, and e a constant
which defines the transformation, namely
Before we go further, some observations are in order. First, Poincaré's expositions bring to light an anomaly inherent in the vague language of electromagnetism, one aggravated by developments in mathematics. This is the notion that the vectors representing magnetic effects are orthogonal to the electric 'field', vaguely thought of as a 3D plane. Early geometry had treated orthogonal elements as imaginary; thus there arises the assumption (orthodox even among influential critics to date) that magnetism requires a fourth ('imaginary') dimension of space, believed to be denoted by elements containing it. Second, the contraction hypothesis had tried to overcome the Michelson-Morley paradox by the assumption that we need to correct the measurement of lengths; in Lorentz's theory to the contraction of moving bodies corresponds a contraction of electrons. Competing theories (Abraham, Langevin) similarly ignored that there is more to 'bodies' than electrons. Since Poincaré is concerned only with the mathematical implications of the Lorentz Transformation this question need not trouble him; he treats the electron exactly as Russell does the mathematical point (an 'atom of mathematical matter'); empty space, without further ado, is thus treated as a continuum of electrons. In addition, P.'s usage of symbols appears ambiguous. On the one hand, x,y,z denote extensions as functions of the time t (1905, 492; 1906, 511 & 543); on the other, they denote quantities independent of the time (1906, 499).
The 1906 paper commences with an enunciation of the Postulate of Relativity as a law of nature (495). In accordance with mathematical practice, for reasons of simplicity we put c = 1 (498), so that, in the case of spherical propagation, the radius of the sphere of propagation simplifies to r = t. Having quoted the LT [1](p.499), P. observes that we easily deduce
[2] x = k(x' - et')/l, t = k(t' - ex')/l, y = y'/l, z = z'/l.
The original transformation had already shown that the moving electron is ellipsoidal, where change in the ratio ß:1 is taken to signify contraction. P. observes that the inverse transformation, likewise, changes a spherical electron into an ellipsoid, one 'contracted' in the ratio ß:1. Although the inverse transformation 'proves' the Principle of Relativity, Poincaré does not further comment on this world-shaking finding; we shall return to a brief discussion. In view of the treatment up to this point we should not be surprised by Poincaré's complete blindness to the mathematical paradox of reciprocal contraction and to the mistake which causes it. The rest of P.'s treatment is for us useless; it suffices to list its major stages.
From [2], which P. treats like an alternative equation for 'the electron', he proceeds to calculate the volume of 'the electron' at the instant when t'=0. Not surprisingly, this results in a 'divergence' from Lorentz (pp.500ff., 523ff.). (I have already mentioned P.'s ambiguous use of the symbols x,y,z; on the one hand, x,y,z are the time-dependent components of the displacement r = t; on the other, they denote positions which are not necessarily time-dependent, such as a position not at the origin when t'=0. Lorentz takes the light-path in moving systems to be measured in terms of bodies constituted of electrons, and the time-dependent light-path (x,y,z) therefore to consist of time-dependent sequences of electrons as elements. In contrast, P.'s ambiguity in regard of x,y,z leads him to ignore their time-dependence, and to base his extensive formal evaluation on the time-independence of space-elements which may be electrons or more generally differentials.)
Insertion of this new electron volume, in brilliant chains of 'functional determinants', leads to new equations for the potential (501) and for the force before and after the transformation (502-503).
In part 4 of the paper (pp.513-515) P. returns to the mathematical platitude already enunciated in his Introduction (p.496), namely that this transformation, like any other, forms a group. The mind boggles that this divertissement, concerning a physically useless transformation, has given rise to the advanced physics of the Poincaré Group.
P. observes that, if we rotate the systems through 180o around the y-axis, we need to change the signs in [1] and [2]; the geometrical or physical meaning of such a rotation is not considered.
Finally, his ambiguity in regard of the correlation between x,y,z and t, leads P. to introduce a space where t is a fourth dimension (542), and where the LT constitutes a rotation around the origin. Note that nD spaces, and algebraic forms interpretated as rotations, are unobjectionable in pure mathematics; but we must be clear that, because of the vectorial character of the elements in the equations of physics, the 4D case is there not applicable.
Of interest for us are only P.'s equations [1] and [2]; I have listed some
other items of his papers only because of their suggestiveness. It is best to
postpone [2] until after translation into conventional symbolism; so let's
briefly relate [1] to the figure. The original LT has S' moving to the right;
since c = 1, we have -1 < e < 0.
(y) (y')
| |
Q . . . . . . . . . . . . |. . | . . . . . . . . . . P
. .* | | .* .
. . * | | . * .
. . * | | . * .
. . * | | . * .
. . * | | . * .
. . | *. | * .
_(___________________________O_____O'____________________)_ (x, x')
Fig.1
Since c = 1, we have OP = t, O'P = t'. To see that then t = r and t' = r'
it is unnecessary to write out the full
x2 + ... = t2 (similarly for x').
The simplification c = 1, r = t, one regularly found in mathematical treatments, should help to dispel the delusion that, in special relativity, we have a 'vector' c which is 'the same' for all observers in relative motion. As already mentioned, all elements in this treatment are displacements, and the classical literature (like more demanding texts today) does not use bold letters; the vectors are r and r' (ct, ct', or here t, t').
Translation into conventional symbolism
P.'s equation [1] is intended to represent the LT which, in its conventional form, reads
x' = ß(x - vt), y' = y, z' = z, t' = ß(t - vx/c2) [1*.a-d],
where ß = 1/(1 - v2/c2)1/2.
We have examined this transformation already in the section on Voigt and need here look only at the difficulties introduced by Poincaré's highly abstract rendition. Note that putting -v=e immediately puts an obstacle in the way of geometric comprehension; the beauty of mathematical generality can here be seen to get into the way of doing meaningful physics. But be comforted. Common sense has a way of turning the tables: in [2], namely
x = k(x' - et')/l, t = k(t' - ex')/l, y = y'/l, z = z'/l,
we see Poincaré hoist by his own petard. For the substitution of the beautifully 'general' e for the ugly particular -v leads him into a mistake exactly like the one often made in conjunction with percentages. Equation [2] is exactly like concluding that, if A = B + e%, then B = A - e%. Here too, as in P.'s physics, the equation 'works' only if we introduce a factor k, such that A = k(B + e%), B = k(A - e%), where k = [1 - (e%)2]-1/2.
To see what has gone wrong here let's recall the geometric scenario, and let's simplify it to the case of points on the x-axis only. That is to say, instead of our previous figure (Fig.1), namely
(y) (y')
| |
Q . . . . . . . . . . . . |. . | . . . . . . . . . . P
. .* | | .* .
. . * | | . * .
. . * | | . * .
. . * | | . * .
. . * | | . * .
. . | *. | * .
_(___________________________O_____O'____________________)_ (x, x')
Fig.1
we now assume that Q and P lie on the x-axis, so that, for some given t,
t', we have OP = ct, O'P = ct', OQ = -ct, O'Q = -ct', where OO' = vt, as in
Fig.2, namely
_Q___________________________O_____O'____________________P_ (x, x')
Fig.2
But we must go further. Even though the very notion of 'contraction'
neither makes sense nor solves the supposed problems of physics, we need a
figurative representation which allows comparison of lengths before and after
contraction. Fig.2 is insufficient; Fig. 3 places lengths 'contracted' in the
ratio ß:1 at an angle.
(P*)
.
.
(O'*) .
.
_Q___________________________O_____O'____________________P_ (x, x')
.
.
.
.
(Q*)
Fig.3
Note to Fig. 3: Since 'contraction' is believed to be associated with movement at the constant velocity v, all parts of a system are equally afflicted. For ease of comparison, I let the inclined line pass through O and mark points on this line by an asterisk. Note that this representation is merely meant to facilitate comparison between parts of a line before and after contraction; although the ease of visualization may have suggested the idea of 'worldlines' at an angle, our figure has nothing to do with rotation of systems of coordinates round the fixed origin of 4D space-time.Before we proceed, we may note that Poincaré's Principle of Relativity is attractive because it seems to agree with the law on action and reaction. It assumes that lengths in both systems are necessarily affected by movement relatively to one another; the physical explanation might lie in a resistance to relative movement. That is to say, in either system the rest-lengths l0 and l'0 have changed to l = kl0 or =ßl0 and l' = kl'0 or = ßl'0. Unfortunately, in that case, a comparison would show that the ratio between the respective coextensive lengths is actually unaltered. We find x and x' in equations [1] and [2]; are they the rest- or moving lengths? If both change equally, why don't we compare rest and moving lengths in the same system instead of lengths in different systems? (The answer here is that a simpler approach should immediately reveal the error concerning e.)
Let's examine the case as it would be understood by mathematicians familiar with historical developments. That is to say, let's ignore the hopeless mass of objections which result from the ill-digested arguments of skeptics and from 'operationalism' ('How do we know ...?'; 'In physics, by x etc. we mean rods, or conductors, which may shrink or bend in many ways ...'). Let's start with the error concerning e, by a comparison of parts of Fig.3.
(P*)
.
.
(O'*) .
.
_Q___________________________O_____O'____________________P_ (x, x')
.
.
.
.
(Q*)
Fig.3
The conventional LT assumes that OO', or OO'*, equals vt' or perhaps ßvt'.
Depending on what we mean by contraction, we had assumed that OO' or OO'* = vt
or perhaps ßvt. Regardless of the meaning of contraction, we might agree that
vt:ct = ßvt:ßct = OO':OP = OO'*:OP*, so that v = c(OO':OP) = c(OO'*:OP*). If we
translate quantities into the new scale t' in S', what becomes of OO' or OO'*?
The new time scale had given us for O'P or O'*P* ct' = ß(c-v)t. Use of vt' in
reference to OO' or OO'* implies the ratio v:c = OO':O'P, or v = c(OO':O'P)
instead of v = c(OO':OP). Since OP>O'P, regardless of what we mean by
contraction, vt' measures OO' too short. The simplest way forward is to
introduce a new symbol v', such that v't':ct' = OO':O'P, and to find the ratio
v':v.
Since v't':ct' = OO':O'P and OO':O'P = vt:(c-v)t, we have
v' = c(OO':O'P)
and OO':O'P = v:(c-v),
so that v' = vc/(c-v).
Hence, regardless of what we mean by contraction, vt' measures the ratio OO':O'P too short. Not surprisingly, since (v'+c)t'/ß = ß(v+c)t', the paradox of 'reciprocal' contraction vanishes. That is to say, if we assume lengths in S' to be contracted in the ratio ß:1, the mathematics confirms that, as we should expect from 'ordinary' logic, when compared with lengths in S', lengths in S are 'contracted' in the opposite ratio, namely 1:ß. We should here not be misled by the allure of the beautiful ß. The mathematics works for any ratio k:1; it merely shows that, if A = kB, then necessarily B = A/k.
What is left of Poincaré? Even without the ghastly mistake in [2], obscured by the lofty abstractions c = 1 and e = -v/c, his equation [1] would have been useless for the reasons briefly described in the section on Voigt.
As already mentioned, P. subsequently states that we need to reverse the signs in the transformation if we rotate the systems by 180o round the y-axis. He does not explain the geometrical or physical meaning and seems to have taken for granted that this would be understood. He is evidently unaware that, if we pursue symbolic operations in abstraction from geometric representation, the problem of isotropy never arises because such symbolic expositions give us only half the picture.
Physicists, in contrast to mathematicians, need to assess the complete geometric scenario (e.g. spherical propagation); that is to say, one and the same coordinate representation must include movement in positive as well as negative directions of space. It is true that the symbolism of analytic geometry reflects thinking in physics; as one should expect, it provides for the anomaly of 'forward' movement in a 'backward' direction (and vice versa). Nevertheless, one should have expected Poincaré to be sensitive to the strangeness of 'negative' displacements; the following item by Einstein illustrates the havoc caused by negligent treatment.
Why should Einstein's expositions of special relativity merit attention? It is a common error, among admirers and critics alike, to assume that this is the case because the theory plays an important role, good or bad, in modern physics. The real reason why they merit attention is that they are early instances of 'mathemagics', namely the increasing reliance of modern physics on so-called counter-intuitive mathematical operations. The simple derivation is of particular importance, because its use of simple algebra allows insight into the grossly invalid symbolic usages which have become the norm in modern mathematical physics.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; the English tranlation was 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 derivation uses only the most elementary algebra, and should present no difficulty whatsoever even to the amateur reader. Yet it has been entirely ignored. More seriously, if one draws the attention of expert philosopher-mathematicians to it, they profess inability to read it, and refer such supposedly technical stuff to mathematics departments. Physicists see no reason to use such an amateur exposition; the prejudice against Einstein's critics persuades them that no time need be wasted on attention.
The reaction of leading mathematicians is interesting. The majority reassure the critic that Einstein's reputed sophistication places his work absolutely beyond doubt, full stop. A minority base their well-meaning explanations on the conviction that protesters are half-wits in search of elementary mathematical enlightenment. A major hurdle is here the neglect of geometry, with its important but forgotten distinction between 3D and 4D. Without exception, the outcome of any such correspondence is silence, blacklisting, or ridicule in front of a public which takes Einstein's supposed genius for granted.
To do Einstein's supposed genius full justice, I quote verbatim. Recourse to our figure is best postponed. I shall try to restrict my comment to purely analytical considerations.
Note: Because of ASCII restrictions, some symbols used by Einstein are not available. I use S instead of K. In my comment, inequalities need to be circumscribed.
Here is Einstein's text:
For the relative orientation of the co-ordinate systems indicated in [the previous 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 co-ordinate system S by the abscissa x and the time t, and with respect to the system S' by the abscissa x' and the time t'.Comment: On the difference between space-time graphs and diagrams of points moving on space axes, see the Preface to Part 2. The case as here outlined by Einstein is represented by the two co-incident x-axes only. Space-time graphs are used only in order to visualize the rate of change dx/dt. In the case of constant speeds this ratio is independent of the time; a space-time graph is therefore not needed. Note that such distinctions, first noticed by Maxwell, were fairly new at the time of the origin of the new physics and (though praised by Boltzmann) completely ignored by German and French mathematicians.
We require to find x' and t' when x and t are given.Comment: Let's pause at this point and use our Fig.2 (as in section 2.2.). Care is here needed because Fig.2 necessarily reflects whether propagation is to be isotropic in S or in S'. To be on the safe side, and to take nothing for granted, we should use two versions of Fig.2: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 S' with the velocity c, the propagation relative to S' 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') = µ(x - ct) [3]
is fulfilled in general; where µ 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') = ø(x + ct) [4].
By adding (or subtracting) equations [3] and [4], and introducing for convenience the constants a and b in place of µ and ø, where
a = (µ+ø)/2, b = (µ-ø)/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.
Fig.2a: Isotropy in S
_Q___________________________O_____O'____________________P_ (x, x')
Fig.2a: QO = OP
Fig.2b: Isotropy in S'
_Q_____________________O_____O'__________________________P_ (x, x')
Fig.2b: 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 Fig.2. For regardless whether QP = OP or QO' = O'P, we may say that OP = ct
and O'P = ct'. On the 'meaning' of t and t', see the following reminder.
Reminder: In conventional analytical geometry, so-called auxiliaries like t define the unit of measurement; in scenarios such as the present one where ratios are constant, modern algebras dispense with the auxiliary variable and use unit vectors instead. Physics uses the auxiliary t to denote the time scale to which clock rates are to be set. There can exist no objection to the introduction of two different scales t and t', such that the displacements in S and S' are quantified as OP = ct and O'P = ct'. This is like using meters in S and yards in S', or like measuring temperatures in different scales. The notion that the units of time measurement cannot be freely determined by humans is a gross operationalist fallacy. SR has led into trouble not because of its (failed) attempt to use different units of time measurement, but because, in consequence of an unbelievably stupid mistake, we seem to be led into the paradox of reciprocal contraction. Musings about 'time' are counter-productive because they distract attention from the false geometric argument.
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. But for the illicit symbol use in [4], [1], [2] as well [4] are compatible with either version of Fig.2. 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. (A brief note here. Although the zero-equations [3] and [4] would 'work' for any multiplier, the presence of µ and ø serves to assure the negligent reader that the difference between the ratios QO'/QO and O'P/OP is properly being taken into account. Note also also that µ and ø, and presumably therefore QO'/QO and O'P/OP, are assumed to differ, for otherwise b = 0. But Einstein's 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'/QO and O'P/OP 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) = vt or ct, the variables x, y, z denote the components of vt or ct. If movement is along the x-axis only, say x = vt or x = ct, then the expressions x as well as vt, or x as well as ct, respectively, are alternative names for one and the same displacement. Of the two alternatives, vt or ct are preferable. The direction of movement is here clearly indicated by the sign; +v or +c means movement to the right (in direction of increasing x), and -v or -c to the left; t denotes the number of steps, to be multiplied by v or c. In contrast, shoddy thinkers like Einstein easily forget that they had defined x so as to denote movement to the left. To avoid this kind of confusion, let's eliminate x in favour of its safer alternative.
Einstein's equations [3] and [4] should then read:
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'4 + ct'4) = ø(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'4 + ct'4) = ø(-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 S or S'. 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.
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? Let's leave Einstein and turn to Bergmann.
Reference: Ch.IV of P.G.Bergmann, Introduction to the Theory of Relativity , Dover reprint, 1976, 33-36. Because of ASCII restrictions, substitutions have to be made for some symbols, such as the Greek letters used for coefficients; instead of Bergmann's asterisks I use primed letters for the system moving to the right.Our concern is with derivations of the Lorentz Transformation (LT), and the steps leading to the paradox of reciprocal contraction. Bergmann's classical and beautiful text invites attention. On the one hand, it excels over those by most other physicists because of his successful avoidance of operationalism at its crudest, that is to say, of the inadmissible substitution of physical entities for precisely defined mathematical concepts and procedures. On the other hand, his unquestioned belief that mathematics need never even begin to consider the constraints imposed by the logic of its own explicit geometric definitions plainly reveals the shortcomings of a discipline which has taken leave of its senses ('counter-intuitive', purely symbolic in abstraction from the 'intuition', or from consideration of supposedly misleading figures).
Note that this classical treatment of the space-time formalism is obtained by simplification from the complete (Minkowski) 4D treatment of so-called space-time transformations as recorded by Felix Klein; cf. (in German) Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, II. Teil: Die Grundbegriffe der Invariantentheorie und ihr Eindringen in die mathematische Physik. Berlin: Springer, 1927. In an effort to remedy SR, these latter grandiose and for physics completely nonsensical space-time transformations are also used by some critics, such as Dr. Selleri; see his October 1997 article in Apeiron.Let's start with our familiar Fig.1 in order to remind ourselves of the geometric scenario, once again simplified to 2D (z=0):
(y) (y')
| |
Q . . . . . . . . . . . . |. . | . . . . . . . . . . P
. .* | | .* .
. . * | | . * .
. . * | | . * .
. . * | | . * .
. . * | | . * .
. . | *. | * .
_(___________________________O_____O'____________________)_ (x, x')
Fig.1
Note to Fig.1: OO' = vt, OP(x,y,z) = ct,
O'P(x',y',z') = ct'
At the time of writing (1942), the mathematical necessity of a
relativistic factor was no longer in doubt; Bergmann therefore begins his
derivation by citing, without further comment, the well known equations for the
space axes, namely
't' must depend on t, x, y, and z linearly, because of what we have called the "homogeneity" of space and time. For reasons of symmetry, we assume further that t' does not depend on y and z. Otherwise, two S'-clocks in the Y'Z'-plane would appear to disagree as observed from S. Choosing the point of time origin so that the inhomogeneous (constant) term in the transformation equations vanishes, we have
t' = bt + dx. [4.3]
'Finally, we must evaluate the constants [a] of eq. [4.1] and [b] and [d] of [4.3]. We shall find that they are determined by the two conditions that the speed of light be the same with respect to S and S', and that the new transformation equations go over into the classical equations when v is small compared with the speed of light, c.'
Observe: Bergmann ignores the constraints imposed by the
explicitly defined geometric scenario; the usual assumption that, for small
ratios v/c, the LT would go over into the classical equations is false. Let's
briefly look at these points before diving into Bergmann's treatment. (Note that
we are concerned only with Bergmann's approach to the logical problem. There is
no need to remind the reader of the LT. Familiarity with it is in the following
taken for granted.)
First, as defined, O'P = ct', and therefore O'P/c, are certainly linear in t
(cf. section 2.1. on the treatment by components). But his equations (i.e.
without reformulation into the symbolism of linear vector algebra) must obey
Pythagoras; it is impossible for O'P/c to be linear in x, y, z. His reference to
clocks appears sound but is insufficiently precise. For he should have noticed
that a solution which is linear in x and t only would be equally impossible,
quite apart from the fact that, on purely geometrical grounds, it implies y, z =
0. It would be impossible because movement is bidirectional; x may therefore be
positive or negative. In consequence, [4.3] would give us a time scale which
depends upon direction. Clocks could perform this remarkable feat only if their
front and rear halves went at different rates.
Second, for small ratios v/c the LT does not go over into the
classical equations. To see this, let's look at a point moving in direction of
increasing x, bearing in mind that the LT, in any case, is applicable to points
on the x-axis only. Here the final forms of both [4.1] and [4.3.] reduce to the
banal
To return to Bergmann's treatment: once the linear dependence of t' on x and t only has been set up, the fallacious LT is inevitable. Bearing in mind that Bergmann is blinded by mathematics, his derivation is nevertheless noteworthy because of its formal elegance. The absurdity is increased by the fantastical labour of the algebra needed to obtain a completely nonsensical result.
The speed of moving points is c in S as well as S', namely OP = ct and O'P = ct', their omnidirectional progress is therefore described by either of the two equations
x'2 + y'2 + z'2 = c2t'2. [4.5]
Note: The assumption that t' must be linear in t and x
necessarily implies that y, z = 0. The following derivation is therefore
invalid because [4.5] reduces to
I present the rest only as an instance of elegant and apparently cogent mathematics having turned into an automaton.By applying [4.1-3] we can replace the primed quantities in [4.5] and obtain
The coefficients of x2, t2 and tx must be the same on either side of [4.7], so that we obtain
and finally
From elementary geometry, the reader may work out for himself what is wrong with the inverse LT. Bergmann should have considered that a change of the time scale necessitates the numerical correction of the coefficient denoting the relative speed for S'; the geometry imposes a primed quantity v' which differs numerically from v. The inverse equations, false and useless as they are, should therefore at least be written thus:To conclude, had B. paid the slightest attention to the geometric scenario, his tortuous efforts would have been redundant, for all quantitative correlations had already been set up by explicit definition; this kind of doing mathematics is chasing its own tail. Clearly, in this kind of formal treatment the distinction between geometry and algebra is no longer understood. Since mathematicians hold that every really intelligent person becomes an outstanding mathematician, and that therefore people who are not even 'ordinary' professional mathematicians are to be classed as grossly subintelligent, we may leave it to the mathematical profession whether it values such distinctions. For physics, the distinction is vital because all measurement operations are necessarily geometrical.
x = ß(x' + v't'), t = ß(t' + v'x'/c2)
Unfortunately, while the LT is applicable to points moving on the x-axis only, the time scale t' depends upon direction. In consequence, in the two time scales t' of S', we get two numerically different quantities v'. If these are correctly entered, we obtain the inverse
x = (x' + v't')/ß, t = (t' + v'x'/c2)/ß
which resolves part of the paradox. For, on the one hand, if it made sense to say that lengths in S' are or appear shorter than their equivalents in S, it follows from the inverse that lengths in S are or appear longer than their equivalents in S' (vice versa if lengths in S' are or appear longer). In addition, the quantity ß becomes indeterminate. On the other hand, if we insist on the equivalence of frames in uniform relative motion, it follows that ß = 1.The reason for the paradox is that, in the (two!) time scales of S', a point moving with the speed -v is not the origin of S. The usual treatment, in arrogant abstraction from the supposedly misleading geometric figure, keeps finding everything reciprocally contracted by a factor ß because its proofs rely on a false inverse. Clamour about common sense is mere escapist rhetoric unless we are prepared to attend to piffle such as this. Common sense starts here.
Let the orthodox aping of mathematical inanities, as in Bergmann's example, be a lesson. Classical geometry had made an important contribution to mathematical thinking; the evolution of pure mathematics during more recent centuries has virtually been driven by the geometrical operations of classical physics. The logic of these operations, in their correspondence to a universe ruled by causal laws, has, moreover, formed the educational backbone of a culture of reason. That is why classical physics and its subservient mathematical disciplines are an important part of 'general knowledge', and why the health of thinking in physics is an important indicator of the health of a culture.
Although my discussion falls short of minimal requirements for academic bombast, and is shoddy even by my standards, it suffices to show that Cantor's celebrated proof of the inadequacy of common sense is ridiculously invalid. Since I find the sheer triviality of this new kind of higher mathematics at best depressing, a revision is not intended.
Even in a more considered treatment, it would make little sense to append a specialist bibliography; see the reading list appended to the Introduction to the main webpage. 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
the completed infinite totality containing all possible elements, including all possible permutations of the symbols 0 and 1, must have the form
where it is understood that n signifies the actual completed infinite. In our present example, the nth element would be {1, 1, ... 1, ...}; even doubting mathematicians point out that such actually infinite elements certainly exist (e.g. irrational numbers and endlessly recurring decimal fractions).
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 aik 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
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 is that, at infinity, this is irrelevant because there are as many units as there are millions.)
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.
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
which, by definition, must be included in the collection.
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.
Within the limits dictated by space, I append here (brief) hostile comments.Received January 2002:
My aim had been to shun rhetoric and to make a practical contribution: does the diagonal method issue in a contradiction by yielding an element that could not have been included?
Unfortunately, a discussion is bound to be futile because correspondents typically ignore the literature (for instance: on paradox, reason, common sense, infinity and experience; operations on sets, collections).
One point, though: Re-writing Cantor's binary element {m, w, m, w, ...} more legibly as {0, 1, 0, 1, ...}, and ordering the collection in analogy to the mirror image of the binary system of natural numbers, do not alter the character of elements or of the collection. It is therefore unjustified to query the use of "numbers" in the counter-example.
"I read with some interest your "counter-example" to Cantor's theorem and I must point out that it doesn't actually do the job. In order to prove to be a counter-example the number created by the procedure must actually appear on the given list. But it is quite clear that any number on your list will end in with an indefinite string of 0's while the number given ends in an indefinite string of 1's. Thus the number created by the procedure actually verifies Cantor's theorem. Incidentally, the diagonal prodedure presented (which is usually given in mathematics textbooks) is a dumbed down version of Cantor's original result. Cantor proved that any set A cannot be put into one-to-one correspondence with the power set of A (set of subsets of A) and since the set of reals can be put into one-to-one correspondence with the power set of the natural numbers, the reals possess a higher cardinality. The reason for Cantor's acclaim was that it was both shocking (at the time) and technically correct. Despite your protestations, the proof, even in its dumbed-down form works to prove that the real line possesses a higher cardinality than the natural numbers. How this leads to the abandonment of reason is difficult to see."
Received March 2002:
I suspect that my statement here will have little impact on your opinions,
but I'll point this out anyway.
In your counterexample to Cantor's proof, you
consider the sequence
{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, ...}
Cantor's diagonalization argument gives the
element
{1, 1, 1, 1, 1, 1, 1, 1, ..., 1, ...}
Which you state is on the
list.
But if the all-ones sequence is on the list, then it cannot be the
result of the diagonalization.
If the all-ones sequence is at, say, position
n in the list, then the n-th element in the diagonalization result is a zero, by
definition.
Thus, your counterexample does not work.
A couple of brief
comments on your arguments:
First, I think you have a misunderstanding on the
nature of the Natural numbers. For, while they are infinite in number, any given
natural can be represented by a finitely long sequence of digits. This is the
fundamental difference; Real numbers cannot be represented in that fashion. Your
construction of a sequence is merely an enumeration of the Naturals in binary;
every such value has an infinite "tail" of zeroes, and as such, you will never
reach a sequence consisting entirely of ones.
Second, on more philisophical
grounds (this is more than a bit outside my expertise, but I have to say it):
you appeal to "common sense" in your argument regarding Cantor. I frankly don't
see how you can do this; what *direct* experience of infinity do you have? I
personally have never seen an infinite number of, say, pencils, or an infinitely
long line; infinity is an abstraction, and as such, it is not meaningful to
appeal to "common sense" when discussing it.