Mathematics Education: The case for the urgent reversal of curricula changes that have been fatal not only for science but for all of mathematics.

1. Dicovery on an unrecognised, farcically elementary error which eludes mathematics at the height of its development to the present.

In 1952, then undergraduate assistant and tutor at one of the mathematics institutes at the Technische Hochschule Aachen, I had decided to abandon studies after my second year for these reasons: changes in curricula already in progress could be seen to substitute for the labour of visual geometric scrutiny mere rote learning by way of "analytic" methods (e.g. determinatns1). After the years of authoritarian rule (1933-1945), there were ethically more worthy tasks to which one would devote one's life.

I reencountered mathematics in 1989. After the death of my husband in 1985, I sought distraction in "general" studies on the German model. A paper in Philosophy discussed contradictions in Einstein (the case of the train in Relativity: The Special and the General Theory, originally published in 1916). Source texts in affordable paperbacks were easily available, and it was clear that I would have to update my knowledge of mathematics. To be able to assess developments, I joined the Mathematical Association (MA) and subscribed to some of its journals. To my delight, members were selling "outdated" textbooks (including the crucially important Dynamics by Horace Lamb). The reason for "contradictions" in Special Relativity (SR) was easy to see, go to 2.

I learnt of other critics and of ever growing academic protests groups2. I joined a new international group, and published for its members the "Special Relativity Letter" with the aim of bringing mathematical difficulties out into the open. From 1999 I published a website "sapere aude. - Reclaiming the common sense foundations of knowledge", devoting one of its pages to the many papers critical of Einstein's mathematics. Ever since, I have corresponded with hundreds of critics, and read and re-read the many thousands of papers in the evergrowing archives of the various organisations. Because of a spine problem which, for instance, makes is difficult for me to use a computer, in 2019 I closed my website and retired from the work. But the mystery how it could have been possible for mathematicians to fail to recognise the grotesque error (and instead, from Poincaré to Klein, to turn the geometric "problem" into an algebraic fantasy) led me to renewed study (history of maths, especially the problem with "analysis" after Descartes and Newton), and to my report, in 2022, to the government and a submission to the MA.

2. The Nature of the Error.

The case of the train in Relativity resembles closely Einstein's argument in his 1905 paper "The Electrodynamics of Moving Bodies" (with other papers available in "The Principle of Relativity"). On close examination, Einstein presumes that "synchronised" clocks are able to go at different rates depending on the direction of a light signal, a physical impossibility. (In 1993, "Philosophy" accepted for publication a paper where I point this out, without mentioning the paradoxical Lorentz Factor caused by a mathematical error.) Briefly, there are two units of time measurements that differ, t and t', but Einstein assumes that vt=vt' (the displacement between the origins of systems in relative motion). The same error is evident in Poincaré (see Whittaker's History of Theories of the Aether and Electricity).
In consequence, one and the same length, depending on the choice of the unit of time measurement, turns out to be both shorter and longer than itself (the reciprocal contraction of length, by the reciprocal Lorentz Factor). That there are contradictions is unavoidable: hence the longstanding biennial London Conferences "Physical Interpretations of Relativity Theory" (PIRT), and the agonised prefaces to some secondary expositions. (General relativity is built on the apparent outcome of Einstein's 1905 paper, with the same "clock" problem as in SR.) Among the mathematicians closely associated with SR is Felix Klein, who admired Minkowski and shows how his linear transformations add meaning to the significance of SR. Misreading the pathlength equations in Einstein, mathematicians would immediately be tricked into the analytic (algebraic) solution where the error vt=vt' is invisible; is was therefore thought that the outcome of Einstein's 1905, despite its paradoxicality, is irrefutable. This particular paradox came to be hailed as a triumph because mathematics can here be seen to reveal to us truths that transcend human understanding. I'll discuss the error, with its origin in analysis, in section 6.

3. Classical mathematics in its development: from Plato and Aristotle to Descartes and Newton.

It is clear that geometry in Plato and Aristotle means visual study; see the Heath's A HISTORY OF GREEK MATHEMATICS. His chapters IX (Plato) and X (From Plato to Euclid) are essential reading for anyone studying the case. It has been said that philosophy is thought becoming self-concious. In Descartes we see mathematical thought becoming conscious of mental operations performed at unconscious level by all moving animals (see 4. The significance of research in cognitive science for classical mathematics). Descartes's pure mathematical space, while applicable to any number of dimensions, in the three-dimensional case, as distinct from the space of physical reality, concurs precisely with findings in cognitive science: an inferential visual system that permits the detection of deviations and irregularities: its rectangularity and perfect symmetry are essential.

Two further "problems" are solved:
1. "number", in all its supposed mystery (rational, irrational, algebraic, transcendental, complex) is seen to be nothing but geometric ratio.
2. Questions like those concerning parallels or superposition, in the light of reason, lose their substance like night fogs in the light of the sun.

In order not to have to return to the topic later, I may here mention the difference between Descartes' space and that of non-Euclidean geometry: Newton had recognised that his higher power curves could be seen as traced by mathematical points moving in the three-dimensional space of Descartes; all that was needed for the study of astronomy was to find the shape of the path, say of planets as they move around the sun, that approximates some higher power equation.

The mechanics developed from his idea starts with the "pathlength" equation; I'll discuss its ambiguity in section 6.

To return to the space of Descartes:
His LA GEOMETRIE (tr. D.E.Smith and L.M.Latham) opens thus:
"Any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain lines is sufficient for its construction." (p. 297 because the first edition appeared as a kind of appendix to the Discours de la Méthode.)
On p.304, after an initial study of geometrical problems, he writes: "This is one thing which I believe the ancient mathematicians did not observe, for otherwise they would not have put so much labor in writing so many books in which the very sequence of the propositions shows that they did not have a sure method of finding all, but rather gathered together those propositions on which they had happened by accident."

D.'s awesome treatment of geometrical problems merits close attention; his Geometry should be essential reading for teachers.

Discussions in the UK of the teaching of geometry show that the fundamental significance of Descartes had escaped attention; hence the astonishment of observers from Germany and France. From my own learning experience in Germany (coordinate geometry in 1941 for age 11) geometry meant Descartes.

4. The significance of research in cognitive science for classical mathematics.

German pedagogy aims at a "balanced" personality and therefore prescribes, beyond examination subjects, acqaintance with philosophy and all the sciences, such as the life sciences including cognitive science. Neglect of this in strictly technological education may have disadvantaged mathematicians, especially in Britain.

Since Darwin's theory of evolution, the physiology of mental operations, including spatial thought, rapidly became a major resarch area. By 1905, cortical maps were in print. Today, to quote from the entry for this topic in Audi: "cognitive science, an interdisciplinary research cluster ... seeks to account for intelligent activity whether exhibited by living organisms (especially adult humans) or machines. Hence cognitive psychology and artifical intelligence constitute its core." It is estimated that 90% of sense data enter by way of the eye; hence the visual system is a major subject of study, especially as, to date, it has been impossible to construe an AI model that works like the cognitive processing in the human brain.

A teaching of mathematics that is ignorant of the actual cognitive capacities of learners is in danger
1. not only to hinder and prevent growth in knowlege and understanding,
2. but instead to waste precious time (available for teaching as well in the lives of learners)
3. and to impose nothing but the purely formal-analytical artifices that had been construed after Descartes and Newton.

It will be difficult to remedy this catastrophic deficiency, initially in the teaching of teachers, before one could even think of re-thinking teaching at primary and secondary school level: thus condemning yet another generation to irreparable cognitive damage. The hugely sophisticated specialised literature is far too large to master. One would have to find one's way in by way of dictionaries, such as the one edited by Audi, or the one by L.R. Gregory. Any textbook for A-level psychology discusses relevant points (in preparation for this report, I have found Gross hugely informative.)

5. The misunderstanding concerning the superiority of "analysis" over visual geometry.

The misunderstanding arises because Newton, Laplace and Lagrange assert the superiority of analysis over geometry; but this superiority exists only in those parts of mechanics which can no longer be visualised (calculus and many-body problems). Mechanics starts with the pathlength equation where the time "variable" is a parameter, not a coordinate or dimension.

Laplace's Traité de mécanique céleste was published 1791-1804. Playfair in his 1808 review of the translation warns that only a small minority of mathematicians are able to read these equations. By that time, mainstream mathematics was already entering "analysis", ironically started by Lagrange whose Mécanique analytique had been published in 1788. Mechanics, a small mathematical speciality, would not have been of interest for mainstream mathematics. Mechanics flourished, untroubled by developments elsewhere in mathematics. Mainstream mathematics never notices the fundamental difference between a geometry of static points and structures and Newton's geometry of moving points. Mainstream mathematicians, from now on, studied exclusively problems in static systems. In static geometry, spatial ratios can be expressed algebraically (more recent textbooks of three-dimensional geometry, e.g. E.A.Maxwell, Elementary Coordinate Geometry, OUP: 1951, introduce determinants at an early stage). Curiously, mathematics next returned to "synthetic" geometry (figures - always static) which Cayley (Kline, Ch.38) "approached ... from the standpoint of algebra". Cayley's ideas were adopted by Klein (Kline, Ch.38) and he succeded in "subsuming the various metric geometries under projective geometry"; "his basic idea is that every geometry can be characterised by a group of transformations and that a geometry is really concerned with invariants under this group of transformations". Mathematics is now fully set up for the SR debacle. But antinomies turn up elsewhere. Kline (Ch.41) next addresses the foundations of real and transfinite numbers, which is followed by mathematical logic which Kline, in a superb overview, discusses in his very last chapter (53). Next comes Hilbert's formalism, which adopts the logical calculus; according to Hilbert, "mathematics proper is a collection of symbolic systems, each building its own logic along with its mathematics, each having its own concepts, its own axioms, its own rules for deducing theorems such as rules about equality or substitution, and its own theorems. The developments of each of these deductive systems is the task of mathematics."

6. The downfall of analysis.

 In view of the fact that mathematicians, and those physicists who have been victims of mathematics teaching since the time when three-dimensional geometry and mechanics disappeared from curricula and textbooks, do no longer know what geometry looks like, it is necessary to quote passages from an important survival of a sane past: the Dynamics by Horace Lamb. The book was first published in 1914, with a second edition in 1923. After Lamb's death in 1934, the book was reprinted six times until 1960. What is significant is that Minkowski's space-time paper had been published in 1908; the full relativity furore started after the 1919 eclipse expedition. Somewhat earlier, there had been the confusions of "Perryism" (that the usage of pure mathematics and mechanics is unintelligble for students); Lamb in his text, clearly addresses this misinterpretation of what is meant by pure mathematics. From the text, and from the oddity of the frequent reprinting, one may speculate that there was, among mathematicians more widely, an attempt to confront the disaster of relativity.

The difference between the two readings of one and the same equation (a one-dimensional pathlength or a two-dimensional "function") became relevant at the time of the rise of "relativity". Poincaré and others (see Whittaker), when trying to resolve the conflict between the old aether theory and the interferometer experiments, had proposed to introduce the notion of "local time". P. and Minkowski were analysts, both had worked on invariants. Einstein tried to show that an equation by Poincaré follows directly from the equations in Newton's mechanics. Poincaré applies the logic of analysis: his t is a fourth coordinate or dimension. Einstein was a physicist; like others, in his 1905 paper he does not seem to have paid attention to the notion of time as a fourth dimension. That x and x' are pathlengths emerges clearly from his text. Minkowski, in his 1908 "Space-time" paper "generalised" Einstein's formalism. Einstein is reported to have complained that he did no longer understand his own theory, but in his 1921 Princeton Lectures (The Meaning of Relativity) he adopts Minkowski's four-dimensional "generalisation". (Critiques of the 1905 paper were published as early as 1908.)

Einstein's 1905 paper "On the electrodynamics of moving bodies".

Einstein's paper is arranged as follows:
an introductory passage,
a Kinematical Part (I), &1 to &5,
and an Electrodynamical Part (II), &6 to &10.
To evaluate the mathematical argument, in &3 (Theory of the Transformation of Coordinates and Times from a Statioanry System to another System in Uniform motion of Translation Relatively to the Former), to get a sense of Einstein's way of thinking, it is advisable to read through the entire text before &3, especially &2 where he discusses the problem of the moving rodAB, and the method of synchronising moving clocks. My comment commences at the point when Einstein introduces the pathlength equation x'=x-vt, "if we place x'=x-vt it is clear that...". The x' next turns up where he inserts "the arguments of the function t".
Use of x' at this point shows that he has forgotten to add the sign when using x and x' for points ("rays") moving in opposite directions (to the right: x=ct, x'=t(c-v), and to the left: x=-ct, x'=-t(c+v). His long equation for the "function" has the form of the equation for the expected result of the interferometer experiments, the time that light requires to travel forward and backward along the X-arm of the instrument, of length l:
t = l/(c-v) + l/(c+v) = 2l/c(1-v2/c2).

Einstein has not seen that the magnitude of the x' depends on direction; he seems to be thinking of his rodAB in &2. The derivation of the equation for the time t derails in its entirety already at this point. But carry on. The subsequent recourse to differentiation ("if x' be chosen to be infinitesimally small") does not make sense. The first equation for t, namely
t = a(t - vx'/(c2/v2),
follows directly from the false use of the x', for any finite magnitude, taken to be a fixed length.

Einstein seeks to derive the general form of the equation for t in reference to any point representing the light ray moving in any direction. He seems to believe that the three different points, namely of the intersection of the X-, y'-, and z'-axes with the circumference of the sphere at a given time t, can be taken to refer to any one such point. Even for these three different specific points, it is clear that the two new t/t ratios differ from the one valid in the case when y,z=0, although he writes his equations in such a way that this may easily be overlooked. In the full set of equations, Einstein uses the equation derived for "rays" along the X-axes, which is valid only when y,z=0.

He believes to have proven that "The wave under consideration is therefore no less a spherical wave with velocity of propagation c when viewed in the moving system". But the form of his equation

x2 + h2 + z2 = c2t2
does not prove any such thing, because this form would be true for the position vector for points on any surface. In the present instance, we see already that the x depends upon direction: the wave, when viewed in the moving system, is not spherical.

The error responsible for the paradox of the reciprocal Lorentz Factor is seen in the passage where he introduces a third system of coordinates K' with coordinates x', y', z', t'. (The Dover translation of Einstein's original German text is here faulty: K' is described as moving to the right of the moving system instead of to the left. In the correct translation, K' would be at rest with the stationary system.) The error is the indiscriminate use of the v in conjunction with with t as well as t, i.e. vt=vt although t and t differ. The ingenious arguments, throughout the text, about the a and f, have the purpose of getting from the unwanted b2 to the desired b.

&4 in Einstein's paper ("Physical interpretation", the discussion of clocks) should be looked at. The equation in the full set had been derived for the case when y,z=0; it is asymmetric because it depends on the direction of the ray. In his typical mathematical ingenuity, Einstein now uses, instead, the equation valid when x'=0, applicable to all points on the intersection of the plane x'=0 with the lightsphere; point P is to the right of O; O'P is merely shorter than OP in this particular case. Discussions of SR take the equation in &4 to be the "real" meaning of t.

Because of the assumption that relativity has revealed the world of our experience to be a four-dimensional space-time, and that therefore three-dimensional geometry and mechanics are antiquated notions, geometry and mechanics disappeared from curricula and textbooks, thus removing just that which is fundamental to mathematical thought (as distinct from meaningless symbolic operations): visual scrutiny and inference.

Relativity is believed to have unified physics. Instead, as a result of the conviction that the mathematical outcome of Einstein's 1905 paper is valid, today we have two different types of mathematics as well as mechanics:


Textbooks of physics are of particular interest. They are bound to submit to the mathematical diktat. Some do discuss pathlengths, but in vector notation where the problem of the time "variable" never arises. Most start immediately with v=dx/dt, where time is of course a coordinate or "dimension". Discussions of the Lorentz Transformation and its effects never follow Einstein's procedure, but seek to explain the matter as a physically real (non-reciprocal) phenomenon (that bodies moving at high speeds contract, with implications, e.g. for momentum and energy).

7. Conclusion: what needs to be done.

In view of the many topics that require attention (history of mathematics in some detail - Kline and Heath; the difference between visual scrutiny in Greek mathematics and Descartes; Descartes' own examination of Euclidean problems in his book; mechanics as geometry and Einstein's paper; some study of the implications of research in cognitive science) a group is needed to prepare an initial text for teachers for whom Kline's Mathematical Thought from Ancient to Modern Times, the chapters on Plato and Aristotle in Heath's A HISTORY OF GREEK MATHEMATICS, and at least part of Descartes' Geometry are essential reading. For the implications of research in cognitive science, see 4. The importance of the matter and the urgent remedy of a teaching based on currently accepted unjustified beliefs concerning the nature and meaning of mathematics should not be underestimated.

Notes and Bibliography.

1Determinants.
The method had first appealed to Leibniz who envisaged its use for symbolic logic in application to problems of language. He had discovered the "logic" of the equations of analytic geometry, where a set of three equations in three variables denotes three planes. L. writes that, by using determinants, the labour can be delegated to workers ("slaves") ignorant of mathematics (thus liberating "higher" mortals from menial tasks). The set of equations, in its geometric sense, denotes a difficult visual problem that, in physical reality rarely arises in such frequency as to justify the invention of machinery as complicated as determinants. More importantly, "understanding" geometry demands precisely that the capacity for visual scrutiny be exercised for the purpose of mastering it. In as far as we have such a set, what we try to see is whether the planes intersect in some way. With an "algebraic" solution we are visually none the wiser, and our time has been completely wasted.

Used in their "algebraic" sense simple problems may arise in practice more often than in geometry, but still not frequently enough to justify determinants. (Plato dismisses such calculations as not part of pure, abstract arithmetic.)

2Go to
https://www.kritik-relativitaetstheorie.de/projekt-go-mueller/(comprehensive list of publications)

http://db.naturalphilosophy.org/scientists/ http://gsjournal.net/
http://www.antidogma.ru (with links to Russian archives of the internatical St. Petersburgh conferences).

2. Bibliography.

Contents of this section:

  • 2.1. General bibliography. 3.1. Theory of Relativity (Special and General, SR and GR).
    (General relativity builds on the apparent proofs of special relativity. Texts therefore frequently combine the discussion of both theories. Although my focus is on SR, the literature cannot be cleanly separated. There exists, in addition, a confusion in classical physics in regard of the general concept of relativity, as opposed to absolute "space" or "movement"; for a detailed and lucid anlysis of this see Mach)

    2.1. General bibliography.

    2.1.1. History of mathematics.

    (Histories tend to present the case of an advance to excellence, blind to classical achievements and the impact of systemic change. I list here only a small selection.)

    Boyer, Carl B. & Merzbach, Uta C., A History of Mathematics, Second Edition, New York: John Wiley & Sons, 1989.

    Crowe, M.J., A History of Vector Analysis. Univ. of Notre Dame Press, 1967.

    Descartes, René, The Geometry of René Descartes, (tr. D.E. Smith), any edition, e.g. 1954 Dover reprint of the 1925 publication by Open Court.

    Heath, Euclid: The thirteen books of the Elements, 3 vols (1908). Dover reprint, 1956.

    Kline, M., Mathematical Thought from Ancient to Modern Times. OUP: 1972. (especially the Preface, Ch.43, 49 and 51, and on Gauss, Ch.36). 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.)

    Smith, D.E. (ed.), A Source Book in Mathematics, Dover: 1959.

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

    2.1.2. Philosophy.

    Dictionaries of Philosophy, in comparison with detailed specialist treatises, may present surprisingly useful information in a nutshell: e.g.

    ed. Audi, Robert, The Cambridge Dictionary of Philosophy, 2nd edition, Cambridge University Press, 1999. (See, for instance, the entry for "cognitive science".)

    2.1.2.1. Metaphysics.

    Maritain, Jacques, La philosophie de la nature: Essay critique sur les frontiers de son object. Third Ed.. Paris: Pierre Tequi (not dated). Maziarz, E.A., The Philosophy of Mathematics, New York: Philosophical Library, 1950. (Comprehensive bibliography.) Sheen, F.J., Philosophy of Science. Milwaukee: The Bruce Publ. Co., 1934.

    Smith, V. E., The Philosophical Frontiers of Physics. Washington: The Catholic University of America Press, 1947.

    Trigg, Roger, Beyond Matter: Why Science needs Metaphysics. West Conshohocken: Templeton Press, 2015.
    T. raises valid objections, on metaphysical grounds, to problems of existence and verification of the objects of theoretical physics.

    2.1.2.2. History of philosophy (logic, science)

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

    Passmore, John, A Hundred Years of Philosophy. (Duckworth, 1957) 2nd ed. Harmondsworth: Penguin, 1968. On the dubious value of philosophical thought and writings by scientist-philosophers, Ch.9, "Moore and Russell", and Ch.14, "Natural Scientists turn Philosophers".

    2.1.2.3. Philosophy of mathematics and science.

    (The list of publications is large: on my own shelves are numerous texts which, given my age- and health-related restrictions, I do not have the resources to list.)

    Benacerraf Paul & Putnam, Hilary, Philosophy of mathematics - Selected Readings. New York: CUP, 1964.

    Bunge, Mario, Causality and Modern Science. New York: Dover reprint (3rd ed.), 1979 (orig. Harvard U.P., 1959).

    Eddington, A. S., The Nature of the Physical World, 1928, CUP / MacMillan (NY).

    Helmholtz, H.,

    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; unsurprisingly, admired by Russell. As seen by Mach (Die Mechanik ..., Kap. 2.9), beautiful but not recommended for application.

    Hilbert, David, . Translation, La Salle: Open Court, 1971.

    Mermin, N.D., Space and Time in Special Relativity, Waveland Press, Prospect Heights: Ill., 1968.

    Whitrow, G.J., The Natural Philosophy of Time, 2nd Ed. OUP 1980.

    Weyl, Hermann,


    2.2.3. Mathematics.

    2.2.3.1. Textbooks and discussions of maths topics.

    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, including curves traced by "moving points")

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

    Roe, J., Elementary Geometry. OUP: 1993. This is a modern exposition of advanced - n-dimensional - treatment of Newtonian mechanics, which includes the basic 3D mechanics as a matter of course. On the parametric pathlengths equations where the time is not a dimension, see p.91: "the world of our experience is three-dimensional".

    Sommerville, D.M.Y. Analytical Geometry of Three Dimensions,. CUP 1947.

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

    2.2.3.2. Pre-relativistic textbooks of mechanics.

    Kirchhoff, Dr. Gustav, Vorlesungen ueber mathematische Physik: Mechanik. 1876, Leipzig: Teubner.

    Lamb, Horace, Dynamics. CUP: 1960 Reprint of the 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.)

    Mach, Ernst, Die Mechanik in ihrer Entwicklung. Darmstadt: Wissenschaftliche Buchgesellschaft, Nachdruck: 1988.

    Maxwell, James Clark, Matter and Motion. Originally published in 1877. CUP reprint (undated, ed. Larmor). CUP edition reprinted London, SPCK: 1920. Dover reprint: 1991.
    (Maxwell is here included because he affirms the dynamic principles of Newton.)

    2.1.3.3. Visual inference as the common sense foundation of mathematical knowledge.

    (I include here discussions from the cognitive neurosciences, where a dispute, as in mathematics, is raging between supporters of spatial and symbolic processes.)

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

    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.,

    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.

    Gross, Richard D., Psychology - The Science of Mind and Behaviour. London: Hodder&Staughton, 1992.

    Gregory, R.L. (ed.) The Oxford Companion to the Mind, Oxford: OUP. 1st edition 1987, 2nd edition 2004. A rich source of data across the cross-disciplinary research field on the biology of cognition, involving cognitive psychology, cognitive science&AI and philosophy.

    Johnson-Laird, Philip: An immensely important author (exhilarating to read), with a long list of titles, mostly either out of print or available only via "print on demand" (never delivered).

    Kosslyn, Stephen M., and Ganis, Giorgio, The Case for Mental Imagery. OUP: 2006.

    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".)

    Weiskrantz, L. (ed.), Thought Without Language. New York: Oxford University Press, 1988. (Visual "geometry" is one of the most important types of "thought without language, here never even mentioned.)

    2.2.3.4. Mathematical logic and related texts.

    (A small selection that happens to be on my shelves.)

    Cantor, George, Contributions to the Founding of the Theory of Transfinite Numbers. English translation Open Court: 1915. Dover reprint.

    Dedekind, Richard, Essays on the Theory of Numbers. English translation Open Court: 1901. Dover reprint.

    Frege, Gottlieb, The Foundations of Arithmetic. English translation, Oxford, Basic Blackwell: 1950.

    Green, J.A., Sets and Groups. London: Routledge&Kegan Paul Ltd., 1965.

    Russell, Bertrand
    (a small selection of texts)

    Stoll, Robert R., Set Theory and Logic. W.H. Freeman and Company: 1963. Dover reprint 1979.

    3.1. Special Relativity.

    3.1.1. History of the theory of relativity.

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

    Whittaker, Sir Edmund, A History of Theories of the Aether and Electricity, 2 Vols, 1951-1953: T. Nelson, New York (available as a Dover reprint).

    3.1.2. Original expositions of the mathematics of "relativity".

    Einstein, A. "Zur Elektrodynamik bewegter Koerper", in , Nachdruck: Wissenschaftliche Buchgesellschaft Darmstadt (undated) of Original publication by B.G. Teubner, Stuttgart, Fuenfte Auflage 1928.
    Minkowski,
    Poincaré, Henri,
    Whittaker, in footnotes, lists P.'s main original contributions of the relativity debate. Below I list some items from the collected works:
    Voigt, W., Ueber das Doppler'sche Prinzip. Nachrichten v. d. Königl. Ges. d. Wissenschaften, Göttingen: 1887.

    3.1.3. Secondary expositions of "relativity".

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

    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.

    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.

    French, A.P., Special Relativity, Chapman & Hall, London, 1968.

    Gray, J., Ideas of space, OUP, 1979.

    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.

    Miller, A.I., Albert Einstein's Special Theory of Relativity, Addison-Wesley, Reading: Mass., 1981.

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

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

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

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

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

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

    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.

    3.1.4. Publications of interest.

    (Observe how post-relativity expositions of "classical mechanics" accept the SR proofs as a matter of course.)

    Angel, R.B., Relativity: The Theory and its Philosophy, Oxford: Pergamon, 1980.

    Arzelies, H., Relativistic Kinematics, Pergamon, Oxford, 1966.

    Cullwick, E.G., Electromagnetism and Relativity, 2nd ed., Longmans, London, 1959.

    Goldstein, H., Classical Mechanics, 2nd ed., Addison-Wesley, Reading: Mass., 1980.

    Jackson J.D., Classical Electrodynamics, 2nd ed., John Wiley, New York, 1975.

    Joos, G., Theoretical Physics, (1934), 3rd ed., Blackie, London, 1958.

    Klein, F.(Much of the II. Teil of mathematical physics is devoted to Minkowski, and we owe the orthodox symbolism for the Minkowski rotation to Klein.)

    Krane, K.S., Modern Physics, J. Wiley, New York, 1983.

    Matveyev, A., Principles of Electrodynamics, Reinhold, New York, 1966.

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

    (Poincaré, see 2)

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

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

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