sapere aude: Archive.

First published: 2 October 2016
Last updated: 18 July 2017


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1. Bibliography.

2. Notes to current argument on my homepage (under construction).
2.1. Role of physics in debacle
2.2. Preface: Are pathslengths in kinematics fourdimensional?
2.3. A sketch of relevant ideas in mathematical development.

3. Earlier versions (excerpts) of page2 - Section 1 (Lorentz Transformation):
4. Cantor's diagonal: an instance of the absurd falliousness of abstract procedure.

1. Bibliography.

1.1. 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.

Møller, C., The Theory of Relativity, 2nd ed., OUP 1972.

Nunn, T.P., Relativity and Gravitation, University of London Press, 1923.

Oppenheimer, J.R., Lectures on Electrodynamics, Gordon & Breach, New York, 1970.

Pauli, W., Theory of Relativity (1921), Dover 1981.

(Poincaré, see 2)

Rindler, W., Introduction to Special Relativity, 2nd ed., Clarendon, Oxford, 1991.

Rogers, E.M., Physics for the Inquiring Mind, Princeton U. P. 1960.

Rosser, W.G.V., Introductory Relativity, Butterworths, London, 1967.

Russell, B., ABC of Relativity, Fourth revised Edition, Unwin Hyman, London, 1985.

Schwartz, M., Principles of Electrodynamics, McGraw Hill, New York, 1972.

Schwinger, J., Einstein's Legacy, Scientific American Library, New York, 1986.

Shadowitz, Albert, Special Relativity (W.B. Saunders, Philadelphia, 1968), Dover 1988. (4D).

Silberstein, L., The Theory of Relativity, MacMillan, London, 1914.

Stephenson, G., & Kilmister, C.W., Special Relativity for Physicists (1958), Dover, 1987.

Taylor, E.F., & Wheeler, J.A., Spacetime Physics: Introduction to Special Relativity, 2nd ed., W.H. Freeman, New York, 1992.

Tolman, R.C., Relativity Thermodynamics and Cosmology (1934), Dover, 1987.

Voigt, W., Ueber das Doppler'sche Prinzip. Nachrichten v. d. Königl. Ges. d. Wissenschaften, Göttingen: 1887.

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

1.2. General bibliography: philosophical, mathematical background.

Under revision.

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.)

Aristotle. Unjustly derided, by physicists and anti-empiricists alike, for teaching, like everybody before Galilei, a physics that does not measure up to modern standards. Kline's and Heath's extensive treatment does justice to his crucial role in the development of mathematics.

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 fêting of Einstein's genius.)

Bunge, Mario, Causality and Modern Science. New York: Dover reprint (3rd ed.), 1979 (orig. Harvard U.P., 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'.)

Ferguson, E.S., Engineering and the Mind's Eye. Cambr.: MIT Press, 1982. (On the debilitation of essential engineering skills by counter-intuitive mathematics.)

Freudenthal, H., Mathematics as an Educational Task. Dordrecht: Reidel, 1973.
(See p.114 for a criticsm of the Russell & Whitehead program: "as dead as a doornail" yet "seductive for mathematicians"; no questions, no problems: problems cannot even be formulated.)
id., Revisiting mathematics education. Dordrecht: Kluwer, 1991.

Grassmann, H.G., Die lineare Ausdehnungslehre, ein neuer Zweig der Mathematik, 1844. and
id., Die Ausdehnungslehre, Vollständig und in strenger Form bearbeitet, Berlin: 1862. Excerpt of this in D.E. Smith, 1959, 684-696.

Gray, Jeremy & Moore, Gregory H. (dispute about the relevance of logicism & formalism), Historia Mathematica 23 no 4 (Nov. 1996) and 24 no 2 (May 1997).

Heath, T.L. (ed.), Euclid: The thirteen books of the Elements, 3 vols (1908). Dover reprint, 1956.
id., A History of Greek Mathematics, 2 vols (1921). Dover reprint, 1981.
id., Mathematics in Aristotle. Oxford: Clarendon 1949. Compulsory reading.

Helmholtz, H., Dissertation Ueber die Tatsachen, welche der Geometrie zugrunde liegen, Nachr.d.K.Gesellschaft d.Wissenschaften zu Gottingen, math.-physik.Kl.,1868.
id. "The Origin and Meaning of Geometrical Axioms", Mind (1876).
id., Epistemological Writings, Hertz/Schlick Centenary Edition 1921, reprint. Dordrecht: Reidel, 1977.
id., Popular Scientific Papers, ed. Kline, M.. New York: Dover, 1962.

Hertz, Heinrich: Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt (1894); Engl. transl.: The Principles of Mechanics Presented in a New Form (Introduction by Helmholtz). 1899 (London: MacMillan).
Traditionally, mechanics had been one of the most important branches of mathematics, a tool for empiricist analysis. Hertz's text reflects the new counter-intuitive spirit: exposition of subject matter and method in the form of a logical treatise with abstract mathematical formalisms. Not surprisingly, admired by Russell; as seen by Mach (Die Mechanik ..., Kap. 2.9), beautiful but not recommended for application.

Johnson-Laird, Philip N.: see 1.3. below.

Jordan, D. W., and Smith, P.: Mathematical Techniques - An Introduction for the Engineering, Physical and Mathematical Sciences. OUP, (my edition 1994).

Klein, F., Vorlesungen über die Entwicklung der Mathematic im 19. Jahrhundert.
I. Teil (pure mathematics), 1926, Berlin: Springer.
II. Teil (mathematical physics), 1927, id. (including 4D SR)

id., Elementary mathematics from an advanced standpoint.
Pt.1: Arithmetic, Algebra, Analysis. 3rd ed., 1924. NY Dover (undated).
Pt.2: Geometry. 3rd ed. 1939, London: MacMillan.

Kline, M., Mathematical Thought from Ancient to Modern Times. OUP: 1972. (Compulsory reference for all critics.)
id., Mathematics: The Loss of Certainty. OUP: 1980. (See especially Ch. IX - XI on the rise of logicism.)

Lamb, Horace, Dynamics. CUP: 2009 reissue of the 1961 edition (first edition of 1914). (Especially valuable as modern textbooks omit, perhaps as obvious, the simple case of constant velocity with its displacement graph - no t-co-ordinate.)

Liebeck, H., Algebra for Scientists and Engineers. London: John Wiley & Sons, 1969. MacFarlane Smith, I., Spatial Ability: Its Educational and Social Significance. London University Press: 1964. (On the the danger to the nurture of skills of non-verbal reflection by the rise to dominance of the "Western culture of articulacy".)

Maziarz, E.A., The Philosophy of Mathematics, New York: Philosophical Library, 1950. (On failure of philosophy in its role as critic; comprehensive bibliography.)

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

Passmore, John, A Hundred Years of Philosophy. (Duckworth, 1957) 2nd ed. Harmondsworth: Penguin, 1968.

Poincaré, Henri, items from the collected works:
Sur la Dynamique de l'Électron (Académie des Sciences, t. 140, p.1504-1508; 5 juin 1905); (Oeuvres, La Section de Géométrie, Vol. IX, pp.489-493).
Sur la Dynamique de l'Électron (submitted July 1905 to Rendiconti del Circolo matematico di Palermo, t. 21, p. 129-176; published 1906); (Oeuvres, La Section de Géométrie, Vol. IX, pp.494-550).

Price, M., Mathematics for the Multitude? London: The Mathematical Association, 1994. (See Ch.3 for the dispute among proponents of "pure" vs. "hands-on" mathematics; note Russell's influence.)

Pyenson, L., The Young Einstein. Bristol: A. Hilger, 1985. (Detailed discussion of Einstein's sources in 1905.)

Riemann, B., Ueber die Hypothesen, welche der Geometrie zugrunde liegen. (1867). Darmstadt: 1959.
id., On the Hypotheses Which Lie at the Foundations of Geometry. Tr. H.S.White.
In D.E.Smith, ed., A Source Book in Mathematics, Dover, New York, 1959.

Roe, J., Elementary Geometry. OUP: 1993.

Russell, B.

For a useful summary of his work and influence, especially the influence of his early work, see Audi, Robert (ed.): The Cambridge Dictionary of Philosophy, Cambridge, New York, etc.: CUP (any ed., 1st ed. 1995).
Russell (1903) spells out admirably the task of the philosopher of mathematics, namely as critic and arbiter of the premisses, and of the principles of deduction. But to the great detriment of mathematics and science, he is thwarted by his own beliefs, for instance:
1. that mathematical space is constituted of unextended atoms of mathematical matter (no continuity until proven by Cantor, of all people!), hence his objection to Euclid that he fails to prove that lines have no gaps, e.g. at a point of intersection;
2. that the properties of actual space can only be determined by experiment and observation (false: what we observe are the interactions of physical things; "space" is a concept, an abstract construct as in mathematics).
Important arguments are found widely distributed in R.'s Works; I list a small selection by their year of publication.
1897: An Essay on the Foundations of Geometry. London: Routledge, 1996.
(1902: "The Teaching of Euclid", publ. in London: Mathematical Gazette; see Price, op. cit, Ch. 3, note 60.)
1903: The Principles of Mathematics. London: Routledge, 1992.
(1910, &Whitehead: Principia Mathematica - Vols. 2&3 are online: worth looking at for the sheer madness of this weird creation - see Freudenthal.)
1914: Our Knowledge of the External World.
1919: Introduction to Mathematical Philosophy. London: Allen and Unwin, 1919.
1927: The Analysis of Matter. London: Routledge, 1992.

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.)

Thwaites, Bryan, (Director) The School Mathematics Project. CUP, revised edition, 1967 (see Price).

Torretti, R., Philosophy of Geometry from Riemann to Poincaré. Dordrecht: Reidel, 1978.

Trigg, Roger, Beyond Matter: Why Science needs Metaphysics. West Conshohocken: Templeton Press, 2015.
Unaware of the existence of the objections and explicit refutations by dissidents, T. bases his argument upon orthodox teaching and claims to verification. He raises valid objections, on metaphysical grounds, to problems of existence and verification of the objects of orthodox theoretical physics. Unfortunately, the desastrous lacuna as to the metaphysical status of legitimate abstractions is not brought to the fore.

Weiskrantz, L. (ed.), Thought Without Language. New York: Oxford University Press, 1988.

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.

Whitrow, G. J. (ed.), Einstein - the man and his achievement. Dover reprint 1967 of BBC publication of 1967. (Revealing quotations, from Einstein admirers, for instance, on his astounding naivety and uncritical thought.)

1.3. Bibliography: Surveys of "state of art".

(Under construction.)

Audi, Robert (ed.), The Cambridge Dictionary of Philosophy. Cambridge, New York, Cambrridge, et al.: CUP, 2nd ed. 1999.

Gazzaniga, Michael S. (Gen. Ed.), The Cognitive Neurosciences. Cambridge (Mass.): MIT Press, (any recent new edition; mine is of 1995).
Most important here Section VIII: "Thought and Imagery", Introduction by Stephen S. Kosslyn.

Johnson-Laird, Philip: An immensely important author (exhilarating to read), with a long list of titles, mostly out of print.

Kline, M., Mathematical Thought from Ancient to Modern Times. OUP: 1972. (Compulsory reference for all critics. Awesone achievement. Superb as a refesher course for familiar topics, as well as an introduction to new ones, including the philosophical background to developments.)

Kosslyn, Stephen M., Thompson, William L., and Ganis, Giorgio, The Case for Mental Imagery. OUP: 2006. (Comprehensive review of the empirical evidence for the largely non-propositional structure of spatial thought. See also above: Introduction to Section VIII in Gazzaniga.)

Passmore, John, A Hundred Years of Philosophy. (Duckworth, 1957) 2nd ed. Harmondsworth: Penguin, 1968. (Comprehensive discussion of theories of mind.)

Russell, Bertrand:

Russell is listed here because in these works, especially, the anti-empiricist theory of mind turns up with a vengeance. Single-handedly, he succeeded in persuading the mathematics profession to remove the classical study of geometry from the curriculum. (His ABC of Relativity, listed above in 2.1., would have us believe that, before Einstein, the geometry of physicists had been based on the logic of the sense of touch of our pre-human ancestors!)

2. Notes to current argument on my homepage (under construction).

In view of arguments in the cognitive sciences (role of logic, visualization and concepts of space) the list can be shortened to a few relevant points, rendering my earlier over-elaborate completely redundant. Urgent re-writing of this section is in progress.
Content of this section (under construction):
2.1. Role of physics in debacle
2.2. Preface: 2.2.1. Are pathslengths in kinematics fourdimensional?
2.2.2. The problem of t as a parametric variable
2.3.A sketch of relevant ideas in mathematical development.

2.1. Role of physics in debacle

Whatever the confusion in mathematics and its philosophical background, we should never have got into our difficulty if physicists had only paid attention to the queer quantitative/mathematical logic in Einstein 1905. For instance, the mathematical procedure is based on the definition of synchronicity (par.1) and the relativity of lengths and times (par.2); the transformation (par.3) states that the clocks have been synchronized. That is to say, the clocks of the moving system are capable of giving a reading conformable with t' = l/(c-v) as well as t' = l/((c+v). The self-contradictory implications of the now inevitable time-equations (one and the same clock running fast as well as slow) arise because the "theory" rests on a logical impossibility (there can be no such clocks). The kinematic argument in par.3, apart from this foundational fallacy, is furthermore invalid on purely mathematical/geometric grounds (e.g. the fallacious reciprocal factor needed for the inverse). Physicists cannot be excused from sharing a great part of the guilt for the rise of the grotesque nonsense and irrationality of post-Einsteinian theoretical physics (and the waste of resources in trying to find a way out).
To date, objections are being raised on physical grounds (role of, appeals for, experiments; role of aether - see, for instance, the completely irrelevant error list in the GOM documentation); any attention to the mathematical impossibility, as well as invalidity, of the theory is being rejected as misconceived.

2.2. Preface:
2.2.1. Are pathlengths (displacements) in kinematics four-dimensional?

Mathematical treatments of SR assume ct, ct' to be entities in 4D "space-time". In Einstein's 1905 derivation, ct, ct' are pathlengths. To date, "general" textbooks of mathematics, as had been the case before Einstein, explicitly show (by figures) that the path of an object, moving with some constant or variable speed in 3D space, is a curve in 3D coordinate space. (For the classical treatment, see e.g. Lamb, Dynamics, 1908, still in print.) A modern treatment is found in the acclaimed The School Mathematics Project - Revised Advanced Mathematics, Thwaites, B. (Director), (Cambridge, New York, ..., Sydney: CUP, Revised Ed., 1967), Book 1, Chapter 12, "Kinematics". pp.227f. There is here no time-coordinate; as the text puts it: "In this example the time (t) acts as a parameter." The symbolic expressions of the path [with components x(t), y(t), z(t), for a given value of t] are "parametric equations".
A graph of the function of the 3D-position vector, or its one-dimensional x-, y-, z-components, however, does require a time-coordinate, or time-dimension. There are cases where we take recourse to such a graph with a t-coordinate, for instance, a 2D graph for movement along one coordinate axis only, to answer the question whether, or when and where, two points, having started at different times, meet. (This does not apply to SR because, by definition, we there consider the pathlengths of two moving points (lightsignal and origin of second system of coordinates) the starting points of which coincide at the time (times) t, t'=0.
The mathematical treatment of pathlengths in kinematics applies generally to any moving point or "body"; the physical nature of the "body" (light, e.g.) or its speed (whether low or some ultimate value, like c) are irrelevant. Nor does it matter that, in the case of spherical propagation, the position vector may lie in any direction; what we attend to is always the pathlength for a point moving in a particular direction, with its x-, y, z-components for that particular point. (If propagation is spherical, x, y, z obey the equation of a sphere, with ct as the radius.) There are here no complications whatsoever that would change the logic of the mathematical procedure. A complication appears to arrive because Einstein introduces a "relativistic" unit of time measurement (t'). The position vector ct', for a point on the surface of the sphere with radius ct for a given time t, measured in reference to the origin of the second system (N.B.: not at the centre of the sphere) is evidently asymmetrical. The ratio t'/t therefore depends on the direction of the lightray. That is to say, for a given time t, the value of the parameter t' varies, and the "parametric" equation for ct' now is a function of x, y, z. That the parameter t' varies for a given time t does not turn it into a "variable" in the sense of algebra or mathematical logic. (See Section 1 of my page2.)

Why then the assumption, in the mathematics of SR, that Einstein's set of equations describes an entity in four-dimensional space-time?

2.2.2. The role of t as a parametric variable.

I insert here a sketch how one might start to find a way out of the debacle. (The sketch had been put into a previous version of my homepage.)
The acceptance of Einstein's 1905 solution ("Lorentz Transformation") as mathematically valid is the consequence of a quite minor oversight, but one that caused a massive derailment for the entirety of theoretical physics. This is because the acceptance opened the way for the notion of "mathematical truths" that transcend the limit of the natural human intelligence ("common sense" as understood in classical philosophy). Henceforth, "paradox", instead of symptom of some error somewhere in an argument, was to count as a revelation of a higher kind of truth; errors elsewhere in theoretical physics, in themselves perhaps as "trivial" as in SR, would be less likely even to be suspected.

As to Special Relativity, a major bone of contention, I would set the direction narrowly:

To pay attention to the role, or status, of the t in SR. Its difference in geometry as compared to other branches of mathematics is the clue to the SR debacle:

What, then, is the problem concerning the role, or status, of the t in SR?

As is made clear in many textbooks of coordinate geometry, if we consider the pathlength attained by a moving point, the t in an equation such as x = vt is a parametric variable (not a coordinate or dimension). If in doubt, critics are urged to verify this. In my archive I briefly quote from one textbook in reference to usage in kinematics. Einstein 1905 explicitly refers to points ("rays") moving along some line (e.g. the x-coordinate); the quantities x, x', y' z' adduced by Einstein are pathlengths. We need a separate equation for t' only because, in SR, t' and t differ.

However, if the explicit geometric reference is ignored and the set is read in an abstract algebraic sense, there are here four variables (coordinates, dimensions): x, y, z, t.

In addition, mathematicians would be likely to have ignored the special status of the t in Einstein's 1905 purely geometric set because many leading exponents of the theory would already have been seeking solutions from within the Maxwell symbolism where, as in algebra, we do, in fact, need t as a fourth variable (time-dependent field fluctuations, etc.).

As soon as the special role of the t symbol is understood, as there is now no fourth dimension, we are able to take recourse to a sketch of Einstein's lightsphere, and thus follow every single one of the steps in the 1905 transformation. (I try to set this out in my page 2.)

Recent correspondence reveals difficulties concerning the geometric meaning of, say, some symbol x denoting a geometric distance. Textbooks of geometry, wisely, always qualify the distance in reference to some figure, say for a point P on the x-axis: OP = x = ct. In SR, for the distance between origins O and O', we would have OO' = x = vt. In geometric shorthand, with a figure present, the OP and suchlike are often omitted; much of the alarm over the occurence in SR of both x = ct and x = ct is unjustified for that reason. Unfortunately Einstein is at best unreliable. Only if we always laboriously spell out the entire geometric expression - say, OP = x = ct or whatever - are we able to see where, overconfident and without close reference to the figure he clearly has in mind, he is in serious error, and why the derivation in its entirety is not only invalid but alarmingly so. (Critics might try the OP = x = ... form in his par. 5: absolutely hilarious!)

With hindsight, that mathematicians have paid no attention to the evidently geometric nature of Einstein 1905 set of equations, might be dismissed as a regrettable mistake, which, unfortunately, established Einstein as a "great" scientist with his equally "questionable" General Relativity yet to come. At the very least, we should take note that our blind trust in the reliability of professional mathematical judgment has been shown to be unwarranted.

Furthermore, modern mathematical developments have been driven by arguments in philosophy (theories of logic; philosophy of mathematics, e.g. Russell on geometry). Physicists would be strongly advised to familiarize themselves with at least some historical surveys: Kline (mentioned by several critics) and Passmore are admirable.

After all, classical mathematical physics had been built by and for physicists;
the tragedy of SR give us a case to reclaim our inheritance.

2.3. A sketch of relevant ideas in mathematical development.

2.3.1. Historical Accident - schools, personnel where and when - clearly plays a role.

Kline (Preface, p. IX): "Every Branch of mathematics bears the stamp of its founders, and great men have played decisive roles in determining the course of mathematics. But it is their ideas that have featured; biography is entirely subordinate. In this respect, I have followed the advice of Pascal: 'When we cite authors we cite their demonstrations, not their names.'"
One of the reasons why Einstein's equations seem weird, or, worse, are misread as functions, is that today we us vector notation instead of coordinate geometry for his problem; many physicists are not at all familiar with the older symbolism. As observed by Kline, "Aristotle knew that forces can be represented by vectors". Instead, 19th century mathematical physics, as e.g. in Kirchhoff's Mechanics (1877) normative in 1905, is still using the logic of classical coordinate geometry where direction is indicated by the +/- sign. I insert here the dates of important publications in this development (Kline, ch.32, pp.772f.)
No surprise then that Einstein, 1905, adheres to the older practice of coordinate geometry for his displacements in 3D space.
  • 1879 Frege, Begriffsschrift (audi p. 328)
  • 1897 Russell, Essay on the Foundations of Geometry ("metric" geometry not part of mathematics; various arguments in later works)
  • 1898 Whitehead, Universal Algebra (comprehensive, authoritative exposition of formal calculus)
  • 1903 Russell, The Principles of Mathematics
  • 1904 Hilbert, Ueber die Grundlagen der Logik und Arithmetik
  • 1910-13: Russell & Whitehead, Principia Mathematica
  • 1919 Russell, Introduction to Mathematical Philosophy

    2.3.2. Mathematical ideas

    Kline records in detail the rise of mathematical ideas and their role in developments. Crucial for problems in physics after the rise of "counter-intuitive" mathematics (mathematical logic, set theory) are

    ideas of space.

    All the rest (whether the human brain "thinks" in terms of "images" or of an exclusively propositional calculus) is now receiving attention in neuroscientific research, with spectacular results: the ghosts of a Hilbert (mathematical formalism in terms of uninterpreted symbols), of the Frege-Russell school (algebraic calculus free from the transience of reality of a metric geometry), or a Russell (denial of the existence of a visualized idealized distance as in Euclid) can be assumed to have been laid to rest.

    What is important though is the concept of space.

    As already emphasized by Aristotle, we dont't see "space" but objects: the notion of "space" is generalized from the distance between objects. So far, so good. What non-physicists tend to forget is that the "empty space" between objects is not empty, but permeated by all manner of physical entities such as fields. Hence the fundamental mistake of imposing a special metric on the geometry merely because one such force, namely gravitation, is "universal", with the consequence that the dynamic effects of all other forces must now be transformed into a non-Euclidean metric taken to be basic. The effect of any such different forces had presented no problem as long as a Euclidean metric had been taken as the basis (though mistakenly believed to be "true"). Why, then, this silly innovation driven by mathematicians and philosophers ignorant of physical reality? As Kline discusses at length, influential ideas more often than not arise as mere fads. In the present case, one particular anti-empiricist headache could thus be got rid of: the supposedly "mystical", "superstituous" belief in "force" or "cause"; for an excellent discussion see Bunge.

    As to our capacity to analyze geometric configurations by visual attention and purely mental imagery, one would expect this to have evolved as animals adapt to the requirements of survival in a world of moving "things" (prey or predator). Modern neuroscientific tests show that, in the monkey, "space on the retina" reappears as "space on the cortex" (i.e. perceived spatial arrays visible with relative distances between points precisely preserved).

    Mathematicians and philosophers, furthermore, would have us believe that, even if we imagine we are able to "see with the eye of the mind", a proper quantitative analysis can only be achieved by applying a purely formal propositional calculcus. It had already been shown (Fermelo) that such a calculus is unable to arrive at a solution; the case has now long been settled by computer studies, and in particular studies in conjunction with artifical intelligence systems.
    The upshot is: as conceded by Riemann, the local metric is Euclidean. It is not "physical" but, as clearly grasped in antiquity, an abstraction, an idealization. This basic Euclidean metric permits the application of any other, non-Euclidean, metric; that it is easy to operate is not entirely irrelevant.

    3. Earlier versions of page2: Introduction and treatment of Lorentz Transformation

    Under construction.

    Because of the difficult background (history and philosophy of mathematics, role of logicism) my texts had required frequent updating. To make a start here, I select at random a version from a time where I include stuff on the 4D treatment, now omitted because (as we now understand what is the matter with the ludicrous t-"variable") we may ignore all this as not really relevant.
    Version of January 2012:


    The purpose of my effort, in response to the difficulties exposed in the arguments of critics (see page3), is the removal of a conceptual-epistemological blockage that prevents recognition that and why the LT of SR as understood by physicists including Einstein (1905) is absurdly false. The acceptance of the LT has implications for mathematical physics in general.
    In consequence of the triumph of logicism, a philosophical movement spearheaded by Russell, mathematicians had been led to disregard the meaning of mathematical expressions and operations; geometrical reference had been denounced as particularly fallacious. Among mathematicians leading at the time of the rise of SR, the geometric meaning of the symbolic forms and operations of kinematics would have been completely disregarded. Because of the great diversity of the symbolic forms of analytical geometry (the equations for lines, planes, curves, surfaces, including the equation for the line in "space-time" representing x(t) as a function - see Lamb), the vital distinction between, on the one hand, the x, y, z as components of the position vector of a static point (analytical geometry) or of the displacement in 3D of a moving point (kinematics), and on the other, the x, y, z (and t) as algebraic variables would have been lost. A typical expression of kinematics for a displacement on the x-axis such as x' = x - vt would have been interpreted as a linear algebraic equation. Although the x-, y-, z-components, through the Pythagorean relation, are interdependent with the position vector of analytical geometry, and the 3d displacement of kinematics, these x, y, z do not behave like the variables of algebraic equations (linear or quadratic). Except in the case of rotation of the co-ordinate axes, the x, y, z never appear together in a linear equation; in the kinematics of SR we may combine x and t, as in x' = x - vt, only because the element vt, like the x, is a displacement on the x-axis.

    It is this misreading of the symbolic expressions of kinematics as algebraic equations, and the failure to refer these expressions to their defined geometric meaning, which is responsible for the failure to recognize that Einstein's LT, the outcome of his transformation of co-ordinates (for translation of origin through vt) is absurdly false. This debacle lies at the root of the persistent unease in regard of real or supposed paradoxes. Among those unhappy with SR there is an unfortunate consensus that rejection of SR would be a solution: a dangerous illusion. The bulk of objections (especially the triumphalist "Kritische Stimmen" of the Germans) is overconfident and largely misconceived (or worse). (In view of our complete ignorance of the precise nature of the actual physical processes at the stages of emission, transmission, absorption, the return to concepts like the ether solves nothing whatever.) The only relevance of SR in the evolution of thinking in physics is its role as an illustration of the rise of phantastically nonsensical mathematical formulations, as the foundation for "theory", because the geometric meaning of the mathematical operations of physics is no longer understood. (On the importance of the geometric meaning see also Prof. Kanarev.)

    In its own sphere, philosophy has long recognized that formal logic is completely useless for the solution of actual logical problems; formal logic flourishes merely as a speciality for experts. When will the philosophy of mathematics comprehend that its relentless drive to reduce mathematics to mathematical logic is as fatal for mathematics as the reduction to formal logic would be for philosophy? Mathematicians are unlikely to comprehend the metaphysical-epistemological implications of the current catastrophic cognitive impairment.

    1. The fundamentally different versions of the Lorentz Transformation (LT) and the significance Einstein's 1905 transformation.

    Once we recognize the misreading of the expressions of analytical geometry and kinematics as algebraic equations, the section on the forms of the LT is redundant because the diversity of algebraic equations, formally similar to, or like the LT, is irrelevant for the debacle of kinematics. This entire section 1. is in need of revision.
    Draft first uploaded on 30 September 2011.

    As emphasized by Einstein in his exposition for a lay readership, Relativity (Section XI), a definite transformation for the space-time magnitudes of an event removes the apparent disagreement between our fundamental results of experience in regard to light transmission. Despite the generally vague logic of his verbal statements, Einstein here correctly diagnoses the central role of the mathematical treatment (the LT). Critics who dismiss mathematical nitpicking as a distraction from the physical, logical, and metaphysical problems generated by SR appear unaware that all SR insights and effects are mere phantoms conjured up by invalid mathematical operations. Not only would nitpicking rid us of these phantoms; it would vastly advance our insight into the nature of our conceptual constructs and operations: a prerequisite for "thinking" in physics.

    Comprehension of the distinction between the "meaning" of different kinds of the "LT" found in the literature inevitably requires attention to some apparently esoteric formal expressions; however, this is not intended as a learned exposition or discussion. In addition to the pitfalls intrinsic to the interpretation of formal expressions, we should be aware of the conceptual problems created by trends in philosophy. This would require lengthy discussions of "operationalism" and "logicism", both tendencies lethal for comprehension of the mathematics of SR.

    Operationalism: Thinking in physics had pre-empted the explicit operationalism of the philosophers of science:
    the "inertial systems", instead of co-ordinate systems in mathematical space;
    physical objects (bodies, mass-points, light-flashes), instead of geometric points, at rest or moving, indicating position in mathematical space;
    the "light path" of SR, instead of the displacement in the mathematical space of kinematics.
    The very term "thought experiment" indicates that the nature of geometric operations is no longer understood. We thus get nonsense like the "oberserver" in the SR transformation, the "rod" representing a section of a co-ordinate-axis, the clocks - at rest or moving, situated at all points in co-ordinate space - instead of the magnitude of the t-variable for a given point.

    Logicism (today the bane of mathematics: see, e.g. Freudenthal or Gray), long before its triumph at the time of Russell, had rejected any appeal to figurative representation. Visualization of the geometric "meaning" of symbolic expressions of any kind - arguably the most difficult aspect of mathematical comprehension - was thus neglected. Unsurprisingly, even at the time of Einstein, the ability to "read" equations, i.e. to "see" their geometric referents, had already been lost.
    (After the wholesale logicist dismissal of visualization as an integral part of mathematical cognition, operationalism must have seemed attractive because it allowed some degree of insight into the meaning of operations, supposedly subject to purely formal rules, upon supposedly abstract and uninterpreted symbolic expressions.)

  • In 1905, the LT, in its various forms, had been a topic for learned exposition.
    I do not discuss here the ad hoc equations proposed by Lorentz-FitzGerald-Larmor; these are not "transformations" in the sense as understood by mathematicians. Because of their purely formal similarity, forms such as these tend to confuse the investigation of the "problem" of the various strictly mathematical solutions.
    Einstein's 1905 transformation is unique in that he attempts to derive a mathematically acceptable form by a consideration of light propagation in 3D space; his t-equation, usually taken to refer to a fourth dimension, is deceptive. Despite its significance for the "meaning" of the LT in physics, we must postpone discussion of the 3D case because of the infatuation of the learned community with 4D "space-time"; 4D forms must be examined first.

    1.1 The quadratic equations of SR and their 4D solution.

    The SR literature uses the following sets of quadratic equations:

    [1.1 - 1]

    x2 + y2 + z2 - c2t2 = 0,
    x'2 + y'2 + z'2 - c2t'2 = 0.

    [1.1 - 1] is "generalized" to
    [1.1 - 2]

    x2 + y2 + z2 - c2t2 = k2,
    x'2 + y'2 + z'2 - c2t'2 = k2.

    [1.1 - 1] and [1.1 - 2] are instances of the general 4D algebraic forms
    [1.1 - 3]

    x12 + x22 + x32 + x42 = 0,
    x'12 + x'22 + x'32 + x'42 = 0,

    and [1.1 - 4]
    x12 + x22 + x32 + x42 = k2,
    x'12 + x'22 + x'32 + x'42 = k2.

    Note that the geometic "meaning" of [1.1 - 1] cannot be decided upon purely formal grounds. Mathematicians involved in the evolution of SR mathematics (e.g. Poincaré and Minkowski) interpret both [1.1 - 1] and [1.1 - 2] as configurations in hyperspace (the 4D "space-time" of Minkowski). In the SR of physics and Einstein (1905), as an application of conventional 3D analytic geometry, [1.1 - 1] is unproblematical. If we have a sphere with radius ct, the quadratic equation is the conventional equation for a 3D sphere. For the translated system, ct' is merely the position vector - conventionally measured from the translated origin; the quadratic equation expresses that the x'-, y'-, z'-components must obey the Pythagorean rule.
    Note that, even in an explicitly geometric interpretation (as distinct from abstract algeba), the distinction between 3D and 4D cannot be decided on purely formal grounds, but depends on our assumed "meaning" (or reference: what is the geometric configuration these equations are meant to represent?)
    All these equations are instances of the general quadratic form
    [1.1 - 5]
    Sxik2 = 0, or = k.

    These forms reflect trends in mathematical thinking, in that formal operations are independent of interpretation: they allow any kind of application, for instance, to the purely numerical problems of conventional algebra, or to geometric configurations in nD space. A linear solution would be assumed to have the form as found, for instance, in Bôcher's 1907 exposition for algebra, namely
    [1.1. - 6]

    x'1 = a11x1 + ... + a1nxn
    x'n = an1x1 + ... + annxn.

    [1.1 - 6] is similar to the form in Klein's geometric interpretation (Vorlesungen ..., Teil II, Kap. 2), in his exposition of Minkowski's theory, namely the general case of 4D transformation for translation of origin and rotation
    [1.1 - 7]

    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 zi are the (constant) magnitudes of translation, and the ai, ..., ei the equivalents of the direction cosines in analytic geometry for the rotation of the co-ordinate axes.

    Voigt's general form is similar, except that he assumes the elements corresponding, in a geometric interpretation, to the constants of translation to be zero; in a geometric interpretation (as distinct from an uninterpreted algebraic form), Voigt's approach would solve the case of 4D rotation.

    The quadratic equations [1.1 - 1] to [1.1. - 5] are indeterminate; their solution requires restrictions by appeal to geometric and/or formal considerations. At this 4D stage of SR mathematics, it was resonable to be guided by the forms proposed in the early literature: x' was to be a linear function f(x, t), y' and z' linear functions f(y) and f(z); on formal grounds t' as a linear function f(x, t) would have appeared suggestive.

    It is at this crucial point that Einstein mounts the mathematical bulldozer: he appears to show that a solution for t' linear in (x, t) is, indeed, mathematically necessary, and that furthermore, by consideration of the conventional case of 3D physics, a new form of the LT not only arises on purely mathematical grounds but reveals certain "counter-intuitive" properties of space and time as such. Once discovered, this new form was found to fulfill all the requirements of the 4D case of the mathematicians; read as an application to 4D geometry, it would denote rotation in 4D space ("space-time") - no translation. Since the 4D case is indeterminate, different kinds of solution, depending on the choice of restrictions, would have been possible.

    Einstein's LT, in its formal simplicity, would have appealed on aesthetic grounds. Infatuation with the 4D case has prevented attention to the invalidity of Einstein's 3D treatment; some perfectly reasonable algebraic restrictions are inadmissible in the 3D case of physics and Einstein (1905).

    Two essential restrictions are not applicable in the 3D case of physics and Einstein (1905):
    1. the condition that t' must be linear in (x, t) (in Einstein an error typical of his shoddy logic);
    2. the condition that v must have the same value in either transformation (S to S'; S' to S) (in Einstein an unreflected assumption typical of his uncritical way of thinking).
    In consequence of the failure to pay attention to the crucial difference between 3D and 4D interpretations of formally identical symbolic expressions, it has become customary, even among physicists, to derive the LT as an algebraic solution of sets of 4D quadratic equations as in [1.1 - 1] or [1.1 - 2] (see, for instance, Bergmann, Chapter IV).

    1.2 Einstein's 1905 transformation for translation of origin: the deceptive role of the time "variable".

    Draft version subject to urgent editing and revision, with some additions to follow.

    1.2.1 Preface.

    Before any examination of Einstein's 1905 transformation, we need to remind ourselves of the status of the time-variable in classical analytic geometry and calculus.
    Since the SR transformation assumes velocities to be constant, we may confine ourselves to the simple case of kinematics.
    If we have an expression like x = vt, this can "mean" (represent, refer to) different "things":

    1. x may be the displacement of a point P moving along the x-axis. The figurative representation, as in classical analytic geometry and kinematics (see, e.g., Sommerville and Lamb), is

    Fig. 1.2.1 - 1

    0-------------------------------------P----------> (x-axis)

    For a point moving in 3D an equation like [1.1 - 1] makes perfect sense: it expresses the Pythagorean relation between the position vector and its x-, y-, z-components.
    (In mathematical jargon, t is here an auxiliary or supplementary variable. There is here no t-co-ordinate or time-"dimension"; t "works" here as a multiplier of quantities on the space-axes.) Note that the "moving point" is a standard concept even in modern textbooks of geometry. Where velocities, as here, are constant, ratios in the geometric configuration are independent of the magnitude of t, much as the properties of a triangle are independent of its assumed size.

    2. In calculus, we use configuration in function space ("space-time") to express the dependence of x on t. In this case we interpret the expression x = vt strictly in its sense as a function x = f(t) (in SR, a linear function). The typical figurative representation in function space would be

     | . . . . . . . . .  *
     |                 *  .
     |              *     .
     |           *        .
     |        *           .
     |     *              .
     |  *                 .
       Fig.1.2.1 - 2
    For a point moving in 3D, the function space for the 3D displacement is necessarily 4D.

    If, as in SR, we have two points moving along the x-axis (that representing the position of a light signal for y, z = 0, and the origin O' of the second system), we have two different functions, namely x = ct and x = vt, with their "curves" at their appropriate angles. Although the configuration in function space would still enable us to determine the relative displacement between the two moving points, it does not help us to understand what happens in an SR-type transformation.

    In this case, namely when the symbols x, y, z denote functions, equations like [1.1 - 1] of [1.1 -2] are nonsensical; they may be "meaningful" in other mathematical applications, but not in reference to the "space-time" configurations of the SR of physics.
    The failure to attend to conventional mathematical usage and "meaning" is already evident in that SR texts (including Minkowski) have x as the ordinate and t as the hypothenuse.
    Note that, although the function-"curves" lie in the x-t-plane, the respective moving points P do not, like the train in some SR texts, ascend the slope in "space-time". Movement, in the given case, proceeds along the x-axis only; the curve merely expresses that the magnitude of the displacement on the x-axis increases with increasing values of t.
    1.2.2 Einstein's 1905 derivation of the LT
    Einstein's 1905 derivation (like any kind of mathematical argument) cannot be understood by reading discussions in the secondary literature; insight requires attention to Einstein's own argument. I use the translation of the 1905 paper in the Dover edition, Einstein et al., The Principle of Relativity, pp. 37 - 65.

    The 1905 derivation presents two problems for critics:

    1. The argument is tortuous; errors are extremely difficult to pin down. A discussion might have the purpose of saving other critics time by pointing to errors easily overlooked.

    2. There is the question whether it is morally defensible to spend long hours investigating this mess. On the one hand, we need to liberate physics from the tyranny of the mathematicians; their acceptance of SR, nonsensically transmogrified to 4D, gives us firm ground on which to stand. We should not postpone this unpleasant task, in the pious hope that somebody, sometime, will be mad enough to try to rid physics of this monster. On the other hand, especially for me, it cannot make sense to have to re-formulate and re-write this rubbish again and again (necessarily with attention to the minutiae of Einstein's text). I need no apology for being brief. (Even brevity drags one into cogitations reminiscent of a mathematical Korinthenkacker; the translations "hair-splitter" or "nitpicker" - types equally common in mathematics - do not remotely capture the undoubted Freudian anal-erotic nature of the compulsion.)
    For brevity's sake I retain some of Einstein's operationalist terms.

    Einstein tries to obtain the co-ordinates and time-equation for points on a sphere with the radius ct about the origin O of a co-ordinate system S, in relation to the origin O', of a second system S', moving in the positive direction of the x-axis with the velocity v. A figurative representation for the imagined geometric scenario (point P in the first octant) might be (simplified for z = 0)

    (y) (h) | | |. . | . . . . . . . . . . P | | .* . | | . * . | | . * . | | . * . | | . * . | *. | * . _O_____O'____________________)_ (x, x) Fig.1.2.2 - 1

    To remind ouselves of the asymmetry of the figure, I include a figure that shows points P and Q in different octants of the sphere about O:

    (y) (h) | | Q . . . . . . . . . . . . |. . | . . . . . . . . . . P . .* | | .* . . . * | | . * . . . * | | . * . . . * | | . * . . . * | | . * . . . | *. | * . _(___________________________O_____O'____________________)_ (x, x) Fig.1.2.2 - 2
    p. 44-45 (A striking illustration of Einstein as an apprentice mathematical houdini.):

    Instead of obtaining the three co-ordinates and the time equation for one single point, he obtains the co-ordinates of three different points; the time equations for these points differ; he retains only the time equation for points on the x-axis (y, z = 0; applicable to the two points when x = ct, x = -ct; x = ct, x = -ct) ("must be linear").

    The equation on top of p.45, is the solution of [(x - vt)/(c-v) + (x - vt)/(c = v)]/2 (p.44, nonsensically construed for (x - vt) "infinitesimmally small"), i.e. Einstein superimposes the speed of a two-way signal on that of a one-way signal.

    The resulting equation can be tidied up to t = ab2(t - vx/c2).

    The time equations for the other other points, obtained for x = vt, thus retain the two-way & one-way combination of the light speed for signals in other directions. Observe how the "elegant", completely arbitrary scaling factor a, by substituting F for ab, can be used to get rid of the surplus factor b in all four equations.

    Note that the retained time equation, even though restricted to signals moving along the x-axis only, is already direction-dependent and thus completely useless. (In its final form, it reduces, on the right, to bt( 1 - v/c), and on the left, to bt(1 + v/c). Interpreted as a clock rate, for signals to the right clocks would go fast, and go slow for signals to the left. The b, as an ad hoc quantity perfectly acceptable in the physical theories of Lorentz-FitzGerald-Larmor, emerges in the strictly mathematical transformation of SR by mistake and reduces there to 1.)

    The time equation is redundant; in Einstein's LT we merely have t = x/c and t = x/c.

    p. 46 (the quadratic equations):

    Note that the figure, spherical about O, is not also spherical about O'; the direction-dependence of the time-equation obscures that there is no isotropy in S'. The direction-dependent quantity ct is merely the position vector; the quadratic equation for c2t2 is not that of a sphere but expresses a mere Pythagorean relation.

    p. 47 (the inverse transformation):

    The reciprocal b emerges here by mistake. Hoist with his own petard, Einstein, who is to teach mankind how to think correctly about time, typically forgets that all velocities in S' must be corrected for the changed time measurement; instead of (c - v) and (c = v) we had got c (the very purpose of the transformation); the relative velocity, similarly, is no longer the "same", namely v, in both systems. Since the time equation (valid only for y, z = 0) is direction-dependent, the magnitude for the relative velocity in S', say v', is also direction-dependent. We can easily obtain it from a simple figure.

    In the case of a signal moving to the right, We have

    Fig.1.2.2 - 3


    where OO' = vt, OP = ct, O'P = (c - v)t = ct.
    If OO' = v't, then here vt/(c-v)t = v't/ct and therefore v' = vc/(c - v). We need to apply a paradoxically reciprocal factor b in the inverse transformation only because the assumption that OO' = vt measures OO' too short. If we insert the correct value for v' we find, instead, that x = (x + v't)/b. That is to say, if quantities in S' are shorter than corresponding quantities in S, as common sense logic should expect, these quantities in S must, in that precise ratio, be longer: if a length s = as, then s = s/a.

    Similarly, for a signal moving to the left, we have

    Fig. 1.2.2 - 4


    where OO' = vt, OP = -ct, O'P = -(c + v)t = -ct.
    If OO' = v't, then here OO'/O'P = vt/[-(c + v)]t = v't/(-ct) and therefore v' = vc/(c + v). In this case, the assumption that OO' = vt measures OO' too long; since x = OP = |O'P - OO'|, x turns out to be too small because we have subtracted a too large value for OO'. Again, inserting the correct magnitude v't in the inverse transformation gives us x = (x + v't)/b. (As above, if quantities in S' are shorter than corresponding quantities in S, these quantities in S must be longer in the same ratio.)

    If we assume the systems to be equivalent, and any factor to be reciprocal, a correct mathematical argument would show that such a factor would be 1.

    I pause here in my draft, with brief remarks on some instructive errors at later stages in Einstein argument to follow.
    (End of excerpt from page2, Jan. 2012)

    4. Cantor's diagonal: an instance of the absurd falliousness of abstract procedure.

    I had removed my argument on this topic in 2011 because I wanted to avoid distraction from the scandal of the mathematics of special relativity. Re-reading, in January 2012, several of Russell's key texts with his appraisal of Cantor, as well as that of others as reported by Kline, persuades me that, in view our concern with the consequences of pernicious developments in mathematics, the item merits re-instatement. Cantor's proof resembles the fallacy at the root of of special relativity, both in the supposed "triumph of mathematical reason" as well as in the elementary nature of the error, namely one which, in both cases, is easy to see.
    In earlier versions of this webpage I had included a longer argument to show that Cantor's diagonal procedure fails. The fallaciousness of Cantor's abstract procedure, on careful inspection, is immediately evident; the simplest possible presentation should therefore suffice.

    Cantor's proof is widely quoted in the literature on the foundations and history of mathematics. One of many of his proofs using similarly abstract procedures, it is to establish the uncountability of the real numbers, and, by implication, the continuum hypothesis which is believed to be vital not only for analysis but for geometry.

    According to Russell, before Cantor, any argument about curves or surfaces having (e.g. at supposed intersections) points in common would have lacked foundation: this invalidated Euclid as well as classical analytical geometry.
    Cantor's thought is believed to have laid the foundation for the development of modern mathematics (see, e.g. Kline, Vol.3; or B. Russell, The Principles of Mathematics, 2nd ed. of 1937, London: Routledge paperback ed., 1992).

    My concern here is with the fallaciousness of the procedure itself, not with any implications.

    I use Cantor's formalism and part of his own text as quoted, in translation, in John Fauvel and Jeremy Gray (eds), The History of Mathematics - A Reader. Basingstoke and London: The MacMillan Press Ltd., 1987 (pp. 579-580).

    Briefly, Cantor considers the collection M of elements E = (x1, x2, ..., xn, ...), with infinitely many coordinates each of which is either m or w. He proceeds to prove that M "does not have the power of the series 1, 2, ..., n, ...."
    "If E1, E2, ..., En, ... is any simply infinite sequence of the manifold M, then there is always an element E0 of M which does not agree with any E.
    To prove this let

    E1 = (a11, a12, ..., a1n, ...)
    E2 = (a21, a22, ..., a2n, ...)
    Em = (am1, am2, ..., amn, ...)

    where each amn is a definite m or w.
    Let there be a sequence b1, b2, ..., bn, ... where each b is also equal to m or w. So if ann = m, then bn = w. Then consider the element
    E0 = (b1, b2, b3, ...)
    of M and one sees without further ado that the equation E0 = Em can be satisfied by no integer value of k and all integers n
    bn = amn and also in particular bm = amm
    which is excluded by the definition of bm".

    To recognize that a vital premiss, tacitly taken for granted, is false, we need to consider what the procedure actually tries to do.

    Wittgenstein, in his Remarks on the Foundations of Mathematics (Part II), rather vaguely thinking in terms of an arbitrarily ordered collection of elements such as
    (II.18) had guessed the error by asking whether the diagonal procedure might not work because there might be more rows than columns.
    The easiest way to see what is wrong with Cantor's conclusion is by ordering the elements in the way one would actually build such a collection. Cantor's m and w are difficult to distinguish; we may choose any other more easily readable letters (say l and q); there should be no objection to opting for 0 and 1. 0 and 1 are easy on the eye; the argument is actually meant to be significant for the "continuum hypothesis" (e.g. points on the number line) and generally for numbers, and we are familiar with such collections. The argument works in the same way if we adhere to Cantor's own m and w, except that the result is more difficult to read. So lets start building the collection envisaged by Cantor, and observe what happens if we substitute b for a as indicated by Cantor. I underline the "diagonal" coordinates where 0 is to be substituted for 1, or 1 for 0.

    The most reasonable way of building the collection is by proceeding from (0, 0, 0, ...) to (1, 1, 1, ...), that is to say, the start of the collection should look like this:

    (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, ...)

    We see "without further ado" (to quote Cantor against himself) that the new element E0 formed by the diagonal procedure is merely (1, 1, ..., 1, ...). It is only the abstractness of the procedure which blinds us to the fact that, for any n, the number of permutations (of this kind which allows repetitions) is 2n. The front edge of the progression of permutations proceeds like a curve with the power 2n. The diagonal has the linear power of n only; with increasing n the diagonal therefore recedes ever further away from the front edge of the progression of permutations. For a collection of classical mathematics (potential infinite only) we should say that the collection has n columns and 2n rows. But even ignoring this classical restriction, it is clear that, for an "infinite" collection also, the diagonal traverses only a small part of the collection.

    The procedure, in effect even if not explicitly, relies on the one-one relation between columns and rows. Elsewhere in this branch of mathematics, as also in Cantor's own work, we find appeals to one-one relations between points in space and points on a line, supposedly proving that their numbers have the same power. The diagonal procedure allows us to see what may be amiss with set-theoretical proofs of this kind.
    While it is true that the E0 disagees with all elements encountered by the procedure, the procedure never reaches that part of the collection where the element E with which E0 does agree, namely here (1, 1, ..., 1,...), is located.
    Readers have objected that "my" collection is useless because applicable to the rationals only, whereas Cantor's proof includes the irrationals. This objection is mistaken. As we build our collection, proceeding to infinity, we would, in their proper places, put the elements with non-repeating binary decimals. The fault is not in "my" collection (merely using 0 and 1 instead of Cantor's m and w) but intrinsic to Cantor's conception of a diagonal procedure in reference to such a collection. Certainly, the diagonal, in its linear progression, can never leave that part of the collection where the rational numbers are located. The collection itself, however, ("mine" as well as Cantor's own), if proceeding to infinity, by the way it would most reasonably be built, does of necessity include the irrationals, except that these can never be met by the "diagonal".
    Having shown that the diagonal cannot be formed in the way as envisaged by Cantor, one would be wasting words if one were to enter into a discussion about implications.

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