From Gertrud Walton, Winchester,

To the Mathematical Association

For the attention of the Teaching Committee.

In June 2022, in a report on changes in curricula that need revision, forwarded to the Minister for Education, I had argued that the matter is too complex (history, metaphysics&philosophy, mathematics, physics) for the resources of one individual and should require a study group, including members of the empiricist professions but excludding mathematical experts. Since curricula at university level are independent of government guidance I have been advised to submit the case to the Mathematical Association (MA).

As will emerge, my formal education is inferior to that of probably all members of critics' organisations. Because of my age (92), it would not be wise to wait any longer for a better qualified person to rise to the challenge of drawing attention to the disastrous consequences of changes in curricula. Age-related health problems place increasingly severe limits on the work I can do. For some years I have found the simplest computer operations (typing text documents, reading on screen) very difficult and exhausting. I can work only in brief spurts because fatigue causes strange cognitive lapses not noticed by me at the time. In judging the poor style and form of this submission I would beg to take this into account. The appended bibliography, confined to titles of interest for members of critics' organisations, supplements my argument; for easier access I have increased the division into themes.

To identify changes, it is necesssary to compare curricula from different periods: I therefore begin with a chronological account of my maths education at secondary level (1940 to 1950). The German school day lasted 5 hours, followed by a heavy load of homework. From mid-1942 until end 1945 virtually no education was available (air-raids; evacuation; delayed start after the end of the war) nor did we any homework. I might mention that,
  1. my early encounter with Kant's "Wahlspruch der Aufklaerung" (maxim of the enlightenment) - "sapere aude: Have the courage to use your own intellect!" , after the Nazi abuse of our trust, had struck me like a thunderbold;
  2. from early adolescence, it was found that I performed well as a tutor from primary to advanced level (I discovered that the main obstacle is sheer terror) and that classical maths topics are ideally suited to transmit the maxim: students not only are liberated by the discovery that common sense reason (in the sense of classical philosophy) works, but start feeling at home in the world of our experience from which classical mathematics had started. In my final school years, the parents of several girls in my own class called me in and, after an attempted suicide in my class, the school asked me to help (within weeks the girl was able to perform well).
Until the end of my school years, except for tables of logarithms and trigonometry, no printed maths texts were availabe; written information was transmitted via the blackboard. Until 1947, our maths teacher was a superb septagenarian (the majority of younger males were waiting for political clearance). Coordinate geometry was much used for all topics, and a whole term was devoted to spherical geometry, with trigonometry and logarithms (trigonometry including compound angles as in Siddons&Hughes, Trigonometry, CUP: 1928).

Young people from my social background did not consider university study, but I was encouraged to apply for a scholarship at the Technische Hochschule (TH) Aachen. In autumn 1950 I was enrolled as a candidate for a teaching qualification (mathematics, physics, chemistry, philosophy). The large TH had the monopoly for engineering degree courses in North Rhine-Westphalia, by far the most populous Land of West Germany. Mathematics therefore catered mainly for large numbers of students of engineering.
Many courses provided half-yearly performance tests; I did so well in the first maths test that I was invited to join the team of undergraduate assistants&tutors in one of the analysis institutes (largely to assist with the marking of hundreds of written papers, and to be present during hours assigned to tutoring).
Unacquainted with academic convention, I benefitted considerably from time unnecessarily spent listening to discussions among lecturers, and observing their visual work at the blackboard (functions and their derivatives and integrals). Invited to give Referate in seminars, and participation in seminars on advanced themes, made me aware of long-standing disagreements about "foundations" and some of the new "logic" and "infinity" topics.
I found that students found coordinate geometry and the geometric basis of mechanics most difficult; they tended to learn by rote and were encouraged to do so (don't waste time in trying to "understand").
Most alarmingly, I noticed that hugely sophisticated techniques tended to render the subject matter unintelligible: a typical instance of this are determinants/matrices (with their origin in equations for planes in 3D space).
Many years of work as a tutor had enabled me, with astounding practical success, to nurture the natural cognitive capacity of students. The new mathematical ideology does the opposite: it deliberately suppresses such natural capacities (dismissed with contempt as "animal cognition"), and instead trains the young to follow the rules of an abstract "logic": not a moral option after the Nazi disaster. I therefore decided (causing offence to my teachers) to discontinue university study. (Only a few years later the entire mathematics department was turned upside down, with some of the lecturers leaving the TH.) I subsequently worked in non-graduate employment, and, ca. 1990, encountered again the "problem" of the new mathematics only by accident.

After the death of my husband in 1985, I sought distraction in studies. The local reference library, at the time, held rich resources: academic journals, and shelves packed with academic books. In Philosophy, published by the Royal Institute, I found a discussion about contradictions in Einstein's relativity (the case of the train in Relativity). Primary and secondary texts were available in affordable Dover reprints. I thought it best to begin by updating my maths knowledge "in English", bought textbooks available in the trade and secondhand. I applied for membership of the MA, subscribed to its Gazette as well as to two journals on maths education (to see how the contempt for "animal cognition" would be presenting in practice). To my delight, MA-members were selling out-of-date books: just what I needed, including precious classics. Over the following years, I obtained on loan (a fortune spent on fotocopying) all that was available of titles mentioned in "relativity" bibliographies.

I had no difficulty "reading" the equations in Einstein's foundational 1905 paper; I saw that and why his transformation is invalid, and the error responsible for the "paradoxical" "Lorentz Factor". In 1993, Philosophy accepted for publication (in 1995) my contribution to the debate.

I discovered that critics had long been forming professional groups with conferences in many countries2. In correspondence I had noticed a weakness in mathematical objections. From 1997 to 1999 I therefore published a quarterly journal specifically for the purpose of bringing difficulties out into the open. In 1999 I started the website "sapere aude: Reclaiming the common sense foundations of knowledge" which had much feedback even from physicists and mathematicians not connected with any of the protest groups. Over the years, I studied all relevant published critical material and corresponded with hundreds of authors. In 2019, partly for age-related health reasons, I closed my website.

But the question how the known ambiguity of the Newtonian pathlength equation could have been overlooked by even the most erudite and experienced mathematicians (Poincaré, Felix Klein) kept troubling me. When I had abandoned university study in 1952, I had seen that the reliance on abstract symbolic procedure was stultifying students. I would not have dared to believe that such reliance could have the same effect on teachers. After years of reading, checking, and reflection, it became clear that this appears to be the case. Very recently I therefore resumed contact with my friends among the critics to see what can be done.

Geometry, coordinate geometry, and analysis.

To follow the course of developments, I have found two histories of mathematics helpful: Boyer welcomes the modern mathematical disregard of the study of nature; Kline is sensitive to the "uncertainty" as a result of recent developments. Both acknowledge the immense importance of Newton for subsequent developments. The dates of revolutionary changes are suggestive: Descartes allows negative numbers and thus opens up all octants of three-dimensional space, a major mathematical advance. But there exists a fundamental difference between these authors: all except Descartes are engaged in studies of motion.

When John Playfair , in 1808, reviewed the early part of Laplace's Mécanique, he was alarmed to find that only a small minority of British mathematicians would be able to read these equations. The reason for this is that mathematicians are not generally interested in studies of motion. (Mathematicians have told me that "in mathematics, nothing moves", or "in mathematics points move only in the positive direction").

The misunderstanding concerning the consensus on the superiority of analytic methods has arisen because the significance of Newton's new "geometry of moving points" has escaped attention. Newton, Laplace and Lagrange might even be thought to have contributed to the misunderstanding. Newton, while at first preferring geometry, later conceded that analytic methods are indispensable. Kline, p. 615, adduces two quotations from Lagrange and Laplace where they extol the superiority of analytic methods; Lagrange states "no diagrams will be found in this work".
What Newton and his followers would have known that mechanics starts with the equation for the (signed) length

length = product of velocity and time,
where the time-"variable" is not a coordinate or "dimension" but a parameter.

The bulk of mechanics is concerned with derivatives (Newton's and McLaurin's fluxions) and the many-body problem. At this level of the mathematics the time is a coordinate-variable ("dimension"), and geometry is no longer accessible: from the level of the derivative analytic procedures alone are available.

While in algebra all variables are assigned to coordinates ("dimensions"),
in the pathlength equations of mechanics time is a parameter, not a coordinate or "dimension".

This ignored difference between the variables of algebra and geometry lies at the root of the relativity disaster. Correct parametric usage will become clear in my discussion of SR.

The consensus based on a misunderstanding was irrelevant as long as three-dimensional geometry (with mechanics) was included in curricula.

Developments in mathematical physics that culminated in "special relativity" (SR).

For the history of these developments see Whittaker's 2 Vol. History of Theories of the Aether and Electricity.
From Whittaker's text I select only data that show how assumptions and mathematical forms found in Einstein 1905 relate to the work of earlier scientists. Among the earliest proponents of models of the aether had been, for instance, Johann Bernoulli and Euler. Even Cauchy had contributed to the mathematical investigation of its "nature". The model eventually accepted by many was that of a luminiferous medium in which propagation is isotropic, that is at rest relatively to a hypothetical body Alpha, and pervades the substance of all material bodies with little or no resistance. This would allow to verify the velocity with which the earth moves in the aether: experiments culminated in the Michelson-Morley interferometer measurements of 1881 and 1887 which showed that no measurable difference between the times for rays describing equal paths parallel and perpendicular respectively to the earth's motion could be detected. In fact, at the time, it was already known that light, as the carrier of energy with its mass-equivalent, is subject to gravitational acceleration, and that therefore the theory of an aether unaffected by the presence of bodies and pervading them was untenable. The acceleration of light as an energy carrier also implied that its velocity cannot be constant.

The time measurements of the interferometer experiments showed that the light velocity in the aether was too high by a factor of 1/(1-v2/c2), **the square of what later came to be** called the "Lorentz Factor". There followed now attempts to "explain" this mathematically. FitzGerald (June 1892) proposed that the dimensions of material bodies are slightly altered when they are in motion relatively to the aether. Lorentz (November 1892) proposed an enlargement of our concepts of space and time and follows FitzGerald. Over the following years FitzGerald, Larmor, Lorentz and Poincaré proposed versions of a transformation of coordinates, later called the "Lorentz Transformation". Einstein, in his 1905 paper, does not state sources, and much has been written about the question how much of the earlier work was known to him at the time. Pyenson has shown that a comprehensive Encyclopedia article of 1903 by Lorentz would have been available to him.

The pathlength equations of Newtonian mechanics

In the following text I need to introduce mathematical forms that must be rewritten parametrically. Authors either tacitly seem to assume, or state explicitly, that the time-variable is a coordinate variable ("dimension"), whereas in the pathlength equations of Newtonian mechanics this variable is not a "dimension" but a parameter.
One sought to explain the result of the interferometer experiments which had shown that no measurable difference between the times for rays describing equal paths parallel and perpendicular respectively to the earth's motion could be detected, say one arm of the instrument (the "X-arm") aligned with and moving in the positive direction of the "X-axis" of the geometric model representing the hypothetical aether, and the horizontal "Y-arm" of the instrument parallel to the "Y-axis" of the "aether model". For a two-way signal, the velocity over both arms of the instrument was found to be the same, namely c. The pathlength equation for a signal moving in either direction of its "X-arm" would therefore have to be x'=+/-ct'.

For a signal in either direction over a length on the X-axis (representing the X-arm of the instrument), with y'=0, z'=O, we have x'=x-vt=(c-v)t for the signal in the positive direction of the X-axis and x'=x-vt=-(c+v)t for the signal in the opposite direction. The form x'=x-vt, in its similarity to a linear equation in the two variables x and t, is misleading, and makes it difficult to understand the mathematical argument. The Lorentz Transformation.

The Lorentz transformation of SR exists in two different forms:

Einstein's 1905 paper builds on the work of Lorentz and Poincaré. Before a discussion of Einstein's 1905 transformation, one might summarise the preceding work as follows: Einstein's 1905 transformation.

At this point of my submission the question of "experts" becomes critical.

In long passages before the start of the transformation Einstein spells out important premisses; when we come to the symbolic rendition these would be assumed to be known. In the "definition of simultaneity", for instance, Einstein discusses the case of a rod rAB which resembles the setup, result and problem of the interferometer experiment, and which later truns up again in the guise of x'; see 3.

The "theory of the transformation of co-ordinates and times from a stationary system to another system in uniform motion of translation relativiely to the former" commences as follows:
Let us in "stationary" space take two systems of co-ordinates, i.e. two systems , each of three rigid material lines, perpendicular to one another, and issuing from a point. Let the axes of X of the two systems coincide, and their axes of Y and Z respectively be parallel. Let each system be provided with a rigid measuring rod and a number of clocks, and let the two measuring rods, and likewise all the clocks of the two systems, be in all respects alike.
Now to the origin of one of the systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other system (K) and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring rod, and the clocks. ...
We now imagine space to be measured from the stationary system K by means of the measuring rod, and also from the moving system k by means of the measuring rod moving with it; and that we thus obtain the coordinates x,y,t, and x, h, z respectively.
To any system of values x,y,z,t, which completely define the place and time of an event in the stationary system there belong a system of values x,h,z,t, determining that event relatively to the system k, and our task is now to find the system of equations connecting these quantities. In the first place it is clear that the equations must be linear in account of the properties of homogeneity which we attribute to space and time.
If we place x'=x-vt, it is clear that a point at rest in the system k must have a system of values x', y, z independent of time. We first define t as a function of x', y, z, and t. ...
From the origin of system k let a ray be emitted at the time t0, and at the time t1 be reflected thence to the origin of the co-ordinates, arriving there at the time t2; we must then have (1/2)(t0 + t0) = t1, or, by inserting the arguments of the function t and applying the principle of the constancy of the velocity of light in the stationary system:

(1/2)[t(0,0,0,t) + t(0,0,0,t + x'/(c-v) + x'/(c+v)] = t(x',0,0,t+ x'/(c-v).



Allow me to proceed to a brief discussion of the consequences of the acceptance of SR as mathematically valid.

Einstein's 1905 transformation, by the apparent proof of the existence of the "Lorentz Factor", appeared to demonstrate that mathematical space as such possesses dynamic properties.

The supposedly proven physical significance of SR has been a serious hindrance for progress in experimental physics.

Developments in philosophy and mathematics contemporaneous with the rise of special relativity supplemented one another in their influence on maths education. For philosophy, see Passmore. Formal logic, previously a philosophical speciality, becomes a mathematical subject, see Kline, Ch.51 "The Foundations of Mathematics. For reasons discussed by Maziarz, logicians had tended to reject the classical concept of mathematical abstraction; instead they introduced theories of symbolic abstraction.

Special relativity had seemed to prove that the world of our experience is four-dimensional, and that three-dimensional geometry is outdated. Mathematicians not only accepted special relativity as mathematically valid, but hailed its paradox (the reciprocal Lorentz Factor) as a triumph in that it demonstrates the power of mathematics to reveal to us truths that transcend understanding. (Critics, derided as "flat-earthers" and cranks, and put on blacklists, came to refer to the mathematics profession as a mafia.)

Physicists think in terms of visual models. The restriction of (Euclidean coordinate) geometry in curricula, and finally its removal but for rudimentary treatments, is a catastrophe.

Although physicists find it easier to understand Einstein's 1905 exposition of the geometric case than other mathematical treatments of SR, younger generations are completely unable to construe the geometric scenario of the SR problem and to interpret the geometric meaning of the symbolic expressions.

Conclusion

In the course of developments, the vast growth of new mathematical material has crowded out topics that are desperately needed by physicists and engineers. While it is true that mathematics has immensely grown and been enriched by the new geometries and formalisms, the absence in curricula of large parts of classical mathematics (Euclidean three-dimensional coordinate geometry and mechanics with its sophisticated symbolic treatment) is an impoverishment of mathematics itself. An investigation is indicated.

Notes and Bibliography.

1. Notes.

  • 1
    See Maziarz, Ch.I, Section V ("Metaphysical Separation") for a comprehensive discussion.

  • 2
    Germany, Russia, Italy. Go 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

  • 3
    "If at the point A of space there is a clock, an observer can determine the time values of events ... If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events ... But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B ... unless we establish by definition that the "time" required by light to travel from A to B equals the "time" it requires to travel from B to A.
    Let a ray of light start at the "A time" tA from A towards B, let it at the "B time" tB be reflected at B in the direction of A, and arrive again at A at the "A time" t'A.
    ... [The] two clocks synchronize if tB -tA = t'A - tB.

    In agreement with experience we further assume the quantity 2AB/(t'A - tA) to be a universal constant - the velocity of light in empty space. [We call the time thus defined] "the time of the stationary system".

    We now imagine [that a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is imparted to a rigid rod] lying along the axis x of the stationary system of co-ordinates. ... We imagine further that at the two ends A and B of the rod, clocks are placed which synchronise with the clocks of the stationary system ... These clocks are therefore "synchronous in the stationary system".

    We imagine further that with each clock there is a moving observer ... Let a ray of light depart from A at the time tA [footnote: time here denotes the time of the stationary system], let it be reflected at B at the time tB and reach A again at the time t'A. Taking into consideration the princple of the constance of the velocity of light we find that

    tB - tA = rAB/(c-v) and t'A - tB = rAB/(c+v).

    Observers moving with the moving rod would thus find that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous.
    So we see that we cannot attach any absolute significance to the concept of simultaneity ...."

  • 4
    For a comprehensive discussion of this equation, see Bergmann

  • 5
    see the e-book by Prof. Ian McCausland, available at http://www.naturalphilosophy.org/pdf/ebooks/RelativityQuestionMcCausland.pdf

  • 6
    See the "documentation of critical publications", available at https://www.kritik-relativitaetstheorie.de/projekt-go-mueller/
    chap.7: chronology; chap.5: list of publications, az by authors; chap.4: discussion of and excerpts from texts, az by authors.

    2. Bibliography.

    Content of this section:

    2.1. Special Relativity.

    2.1.1. 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).


    2.1.2. 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.


    2.1.3. 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.


    2.1.4. 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, Felix,(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.


    2.2. General bibliography.

    2.2.1. 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.
    Heath, T.L. (ed.), Euclid: The thirteen books of the Elements, 3 vols (1908). Dover reprint, 1956.

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

    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.


    2.2.2. Philosophy.

    2.2.2.1. 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.


    2.2.2.2. 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,

    2.2.2.3. 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.
    See his chapters 6, 7, and 9, "New Developments in Logic", "Some Critics of Formal Logic", and "Moore and Russell".


    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.)


    2.2.3.3. Visual logic ("geometry") 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.

    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