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).

The request to exclude mathematical experts might seem strange. The danger of experts in mathematics is great for these reasons:
• Mathematical abstraction, as distinct from symbolic abstraction, is a subject of metaphysics¹.
• The evidence must be tracked down in histories.
• "Special Relativity": is a speciality even in mathematics, see.

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.

This is a corrected version, 02/09/22, of the original text, 21/08/22: equations in discussion of Lorentz Transformation had been mistyped. The corrected passages are marked thus: ** [corrected passage] ** .



Abstract

A comparison of school textbooks of different periods reveals changes that closely mirror the reasons why, in 1952, I had decided to abandon university study after my second year. An accidental encounter with academic protest groups, forty years later, suggested a study of the reasons for these changes. A brief chronology of my journey is therefore best suited to discuss in detail what might "have gone wrong".

Until the end of the eighteenth century visualisation (geometry) was regarded as an alternative to analytic methods. The current consensus that algebra/analysis is always preferable rests on a misunderstanding.

Developments in the eighteenth and nineteenth centuries saw the rise of new "algebras" while the importance of Newton's "geometry of moving points" escaped attention (especially the fundamental difference between the variables of algebra and those of mechanics).

Almost contemporaneous with the rise of the new "algebras" and "formal logic" were developments in mathematical physics which culminated in "special relativity" as the springboard for "theoretical physics". "Special relativity" rests on a coordinate transformation (using Newton's pathlength equation) and exists in two different forms:

1. Einstein's 1905 transformation;
2. all other transformations.
Both forms present as a mixture of misconception, confusion, oversight, and plain errors:
1. misconception: the belief that mathematics is able to reconcile two geometric models that contradict one another;
2. confusion: SR tries to solve a nonexistent problem (the first model was already known to be untenable; the second understood to make sense as well as supported by experimental evidence);
3. oversight: the special role of the time "variable" which in pathlength equations is NOT a "dimension";
4. plain errors: many and some evident even on purely logical grounds, prior to scrutiny of the mathematical arguments, and especially the astounding error responsible for the "paradoxical" "Lorentz Factor".
Einstein's 1905 transformation appeared to demonstrate that mathematical space as such possesses dynamic properties. This supposedly proven physical significance of SR has been a serious hindrance for progress in experimental physics.

Further developments in mathematics and "special relativity" supplemented one another, with the result that the teaching of classical three-dimensional geometry (including the "geometry of moving points" as the basis for mechanics), always an annoyance for logicians, was eventually completely discontinued.

Classical coordinate geometry (including the "geometry of moving points") had been an essential tool for physicists who, to a large extent, think in terms of visual models. Younger generations, lacking any acquaintance with the symbolic expressions of SR, are unable to reconstrue a sound mathematical basis for experimental work in physics.

Developments which have led to this grotesque impasse have been no less detrimental for mathematics, in that, despite the amazing growth of entirely new forms and structures, it has come to be seriously impoverished. An investigation is indicated.

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).

In order to avoid having to return to the topic later, I will here briefly mention changes noticed in textbooks on my return to mathematics forty years later, now living in the UK. (In a housemove fifteen years ago I disposed of great masses of publications, including most of such "new" books.)

"Visual" methods had almost completely disappeared. Instead of geometric analysis (including the compound angles of trigonometry) students are expected to memorise long lists of names of esoteric geometric shapes.
Generally, arithmetic and algebra had been replaced by
set-theoretical exercises (sorting objects into boxes)
and the mystagoguery of

• "infinity",
• "number" and
• the gaps on the "numberline"
The "gaps" are created by the arbitrary classification of numbers by logicians obsessed with order. Ever since Descarte's linking of algebra and geometry, the solution of quantitative problems, not a concern of logicians, is arrived at by geometric ratios.

Whereas the old curricula were confined to the (abstract) mathematical foundation for methods needed in practical applications, even Prof. Thwaites' renowned School Mathematics Project, with its huge visual input, includes material present for no other reason than that it has been created and therefore constitutes important "mathematical knowledge", thus crowding out stuff desperately needed by students of engineering and physics.

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.

Since the logic of the case depends on the 1905 derivation, I exploited the similarity between the clock at the "midpoint" of the train and the clock at the "midpoint" in the derivation. As the derivation proceeds, it turns out that the "halves" either side of the "midpoint" differ from one another, with the result that the clock at the "midpoint" is expected to go slow as well as fast. But the best bit of my paper is a quotation from an article in Nature [Vol. 151, No. 6321 (9 May 1991), 114] - unfortunately I do not name the author and do not remember whether I tried to communicate:
Current theory ... does not even dare to mention ... the notion of a 'real physical situation'. Defenders ... say that this notion is philosophically naive ... and that recognition of this constitutes a deep new wisdom about the nature of human knowledge. I say that it constitutes a violent irrationality, that ... the distinction between reality and our knowledge of reality has become lost, and the result has more the character of medieval necromancy than of science."

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, C.B. & Merzbach, U.C., A History of Mathematics, Wiley: 1968 & 1989,
• Kline, M., Mathematical Thought from Ancient to Modern Times, OUP: 1972.

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:

• 1637 Descartes' "Geometry"
• 1671 Newtons' Method of Fluxions
• 1676 Newton's "Enumeratio linearum tertii ordinis"
• 1742 McLaurin Treatise of fluxions
• 1797 Lagrange's Mécanique Analytique
• 1799-1825 Laplace's Mécanique Céleste.

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.

(The Dynamics of Horace Lamb, on the very first page, draws attention to this and present the diagram for the pathlength.)

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.

Scientists contributing to this branch of physics follow Newtonian usage; but authors use terms that are difficult to translate into mathematical statements. In electrodynamics, the nature of the moving thing is also unclear: photon, ray, wave packet?
Maxwell, in Matter and Motion, p. 5, writes "The configuration of a material system may be represented in models, plans, or diagrams. The model or diagram is supposed to resemble the material system only in form, not necessarily in any other respect.We shall use the term Diagram to signify any geometrical figure ... by means of which we study the properties of the material system. ... A body so small that, for the purposes of our investigation, the distances between its different parts may be neglected, is called a material particle. ... The diagram of a material particle is of course a mathematical point, which has no configuration."
Maxwell himself, in his writings on the aether theory, calls the moving "thing" a ray. For the sake of brevity, instead of referring to a "moving mathematical point", I follow usage in the texts of physicists: a moving "ray".

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.

Here are some "philosophical" statements by Poincaré that resemble statements in Einstein's 1905 paper.
Poincaré, in 1904, enunciated the "principle of relativity" : "The laws of physical phenomena must be the same for a 'fixed' observer as for an observer who has uniform motion of translation relative to him." "There must arise a new kind of dynamics, which will be characterised above all by the rule, that no velocity can exceed the velocity of light."
Einstein writes: "It is known that Maxwell's electrodynamics ... when applied to moving bodies, leads to asymmetries which do not seem to be inherent in the phenomena. ... Examples of this sort ... suggest that the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the "Principle of Relativity") to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body."

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'.

Theorists, from the start, tried to generalise the finding to geometric models for a three-dimensional space, where, for the moving system, the x'-component of the position vector O'P would be x'=x-vt.

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.

For those who likes rules, the rule would be: "If a mechanics equation contains a space- as well as a time-variable, that equation is a pathlength equation where the time is not a coordinate or "dimension" but a parameter".

The Lorentz Transformation.

The Lorentz transformation of SR exists in two different forms:

• Derivations other than Einstein's follow the interferometer experiments of Michelson & Morley: a two-way pathlength.
  
  The 1908 paper by Minkowski is based on the outcome of Einstein's 1905 one-way transformation, but he alters the mathematics: instead of the position vectors accepted by all, obeying the expressions OP²= (ct)² = x² + y² + z² and O'P²= (ct')² = x'² + y'² + z'², he writes (ct)² - x² - y² - z² = (non-zero constant) and (ct')² - x'² - y'² - z'² = (non-zero constant), thus turning the case into a mathematical construct no longer related to the problem of physics.
  
  
• Einstein's 1905 transformation begins with the the case of the one-way pathlengths of signals moving in opposite directions of the X-axis. The pathlength equation for the rest-system had been assumed to be x=+/-ct, depending on direction. For the moving system Einstein adduces the form x'=x-vt; as we have x=+/-ct, the correct parametric form of x'=x-vt is x'=(c +/- v)t, depending on the direction of the signal. (As will be seen in the discussion of Einstein's 1905 tranformation, Einstein, in the very first use of the x', assumes it to be a fixed length.)

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:

• The transformation rests on the misconception that mathematics is able to reconcile that two geometric models that contracdict one another can exist in the same mathematical space, here the space of the X-axis. In the aether model, the pathlengths are x=(c-v)t and x=(c+vt), as against the case of the experiment where we have x'=+/-ct', depending on direction.
• The attempt exhibits confusion: it tries to solve a problem that does not exist. The model of the aether was already widely understood to be untenable. The aether of the model does not exist, and there is no contradiction. In as far as phenomena are unexpected, all that is needed is a physical explanation.
• There is an oversight: namely that in pathlength equations the time is not a coordinate or "dimension". Whereas in classical mechanics, where an equation like x=ct would imply t=x/c, the SR-solution requires a different kind of time-equation. Several authors, including Poincaré, assume that the time-variable is a coordinate or dimension, and that the "space" of a model for spherical propagation is four-dimensional. Most, like Einstein in 1905, do not appear to have paid attention to this question of a "time dimension".
• Although Einstein introduces new "errors", two errors are present in the work of Lorentz and Poincaré.
  • The x'-component of the position vector in the moving system is asymmetric (in the invalid SR solution, with the invalid Lorentz Factor, we have ** x'=(x-vt)/(1-v²/c²)^1/2 **; the position vector O'P in the moving system is therefore asymmetric. The position vector for a point on the surface of the hypothetical lightsphere in the rest-system, OP= ct, obeys the equation OP²= (ct)² = x² + y² + z². The fact that the equation for the asymmetric position vector O'P=ct' obeys the equation O'P²= (ct')² = x'² + y'² + z'², does not prove that propagation in the moving system is also spherical.
  • The error responsible for the paradoxically reciprocal "Lorentz Factor": the assumption that the relative velocity v must be the same in both systems of coordinates. This velocity occurs in the equation for the pathlength OO' (O and O', the origins of the rest and moving system), defined as OO'=vt. The transformation has changed time-measurement [in order to turn (c-v)t and c+v)t into ct']; the v must be correspondingly corrected for otherwise we should have OO'=vt=vt'. If the error is corrected the paradoxical Lorentz Factor cancels, and we are merely left with a time equation that requires clocks to go slow as well as fast, depending on whether the ray and the moving system move in the same or opposite direction(s). (Einstein, although presenting the same false and useless time-equation in his final set of equations, later tries to remedy the asymmetry; I will attend to this in the discussion of Einstein's 1905 transformation.)

Einstein's 1905 transformation.

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

Among authors of secondary expositions of SR are highly qualified mathematicians; perhaps mathematicians pay no attention to Einstein's tortuous and even weird text, see below my quotation from the start of the "transformation". Mathematicians are likely to confine themselves to the mathematics of the equations. As these have been misread as algebra, ignoring the parametric nature of pathlength equations, the invalidity of Einstein's argument is difficult to see (though the horrible error responsible for the paradoxical "Lorentz Factor" should have been seen: the assumption that "the relative velocity must be the same in all frames of reference", or OO'=vt=vt').

The person examining this part of my present text should be a mathematician acquainted with Euclidean coordinate geometry who understands the difference between the abstract space of mathematics and "physical space". In the books by highly erudite physicists, for instance, one finds the notion that the coordinate system of SR exists in the aether, and that actual lightrays are being sent out into this aether with its coordinate-axes; despite Maxwell's warning, the notion of mathematical points denoting the location of a merely hypothetical ray is rejected as useless for physics.

"The mathematics of special relativity" is a mathematical speciality taught in courses of theoretical physics, following Minkowski's theory. "Experts" in this sense should not be appointed to the task of scrutinising my argument and Einstein's strange text.

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 r_AB 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).

Comment
Critics call this equation "the monster equation"; in my subsequent discussion I shall refer to it by that expression.

---

After many futile attempts to render, or paraphrase, Einstein's statements I had to resign myself, specifically for this submission, to type out at length how Einstein thinks of a co-ordinate transformation. His language and differs from that of other mathematicians: Kirchhoff in his 1876 Vorlesungen ueber mathematische Physik: Mechanik, Lamb in his 1914 Dynamic; even Minkowski's 1908 paper "Die Grundgleichungen fuer die elektromagnetischen Vorgaenge in bewegten Koerpern" uses the terms of "ordinary" mathematical discourse. All other autors discussed in Whittaker use the known expressions of classical mechanics, but for the error of introducing a time-coordinate that does not exist. One may therefore be permitted to read Einstein's equations as they would have been regarded by people familiar with usage at the time.

What of the x'=x-vt?
It plays a role in the first equation, the "monster equation". Einstein here uses 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), see ⁴.
The "monster equation" shows that Einstein assumes the pathlength x' to be a fixed length, similar to his rod r_AB of the earlier "definition of simultaneity".

But Einstein's x' is not a fixed length. The x here is the pathlength in the stationary system: x=+/-ct. For a ray in the positive direction of the X-axis, for x'=x-vt=(c-v)t we therefore have:


O.....O'.............P,
where OO'=vt, O'P=x'=(c-v)t.

Now the pathlengths of mechanics, by convention, are for points moving outward from the origin of a system; the case of a point returning to the origin is not in accordance with mathematical practice. Light propagates isotropically; at the time of emission, a ray would therefore have travelled in the negative direction of the X-axis, its pathlength equation would be x'=x-vt. If OQ=x=-ct, we have

Q....................O.....O'.............P
where O'Q=x'=x-vt=-(c+v)t.

But one can figure out what happens if we insist on reflecting the ray, say at a mirror M coincident with P at the instant of reflection, for return to "the origin".

Critics have objected that Einstein does not specify which of the two origins he means. But the equation is clear, the time t is the same for both rays: x=-ct means that the ray returns to O.

At the instant of the arrival of the reflected ray at the origin O of system K we should have

O...........O'.......P.....(M)
During the further time t O' and M have moved through vt,
and we have MO=MP+PO=x'=x-vt=-(c+v)t.

The mathematician reading my text so far may already be questioning Einstein's grasp of the geometry of the case and of the pathlength equation. It is impossible for me to adduce in full all of Einstein's subsequent text of the "transformation" paragraph; reprints of Einstein's paper are available, and I woud beg to refer to a copy. This is also important because I have difficulty reading texts on screen; despite many exhausting hours spent checking and editing, it is unlikely that my text would be free from typing errors, including in the equations copied. I will confine myself to only a few observations.

From the subsequent treatment it is clear that Einstein seeks to render his equations conformal with the forms at the time regarded as orthodox (Lorentz, Poincaré).

It is difficult to see why, after the "monster equation", he commences by employing partial differentiation for x' infinitesimally small.

The partial differention procedure had given Einstein dt/dx' + [v/(c²-v²)] dt/dt = 0, dt/dh = 0, dt/dz = 0.
Einstein had written that the equations must be linear. For the pathlength of and time required by a ray in direction of the X-axis, the equations are obviously "linear"; but Einstein is trying to proceed to the three-dimensional "light sphere". Since, for the position vector OP we have OP=ct, and the required solution for O'P is O'P=ct', for t' we have O'P/c. O'P has the components x'=x-vt, y', z' (where y'=y, z'=z); the solution of (O'P)² is no longer linear.

Why the emphasis on a linear solution? Earlier authors, ignorant of the parametric nature of the pathlength equation, had regarded the x'=x-vt as a linear function in the two variables x and t; and it had become orthodoxy to write the x' (or x) and t' (or t) in the form x' = ax - bt and t'= dt - ex, where the a, b, d, e are functions f(v); y=y' and z=z' are linear.

If the t and t' are regarded as coordinate variables, we seem to have two sets of four linear equations (solved by recourse to matrices); the SR transformation for translation of origin becomes a transformation for rotation about a common origin, as is the common interpretation of SR, including Einstein's 1905 transformation.

(As emerges from Einstein's subsequent text, the t-equation in the full set is the nothing but the equation derived for the ray in direction of the X-axis, when y, y', z, z' = 0.)

After the partial differential treatment, the first solution for t, is t = a[t - vx'/(c² - v²)].
This equation includes the square of the "Lorentz Factor" b, where b=1/(1-v²/c²)^1/2; what is wanted is b, not b². The substition of f for a has the purpose of remedying this defect.

For the three-dimensional case Einstein needs the equations for the h- and z-components of O'P. For this purpose he has rays travelling in the direction of the y'- and z'-axes, and finds that h=ct, z=ct. It is evident that he argues from an image of the sphere about O, where (O'P)² = (OP)² - (OO')² and x=vt. As in this case O'P=ct, he could have derived the required t-equation from his geometric image.
Instead he exploits that here x'=0, and derives the solution from the t-equation found for the ray along the X-axis. As h=y, his finding that y/(c²-v²)^1/2 = t implies that, when h=ct (and z=ct), t = h/c (= z/c) = t(c²-v²)^1/2.

The t-equations for rays not in direction of theX-axis differ therefore from the equation arrived at for rays along the X-axis;
his text ingeniously makes this difficult to see.
Of course his h and z are useless, they are not the wanted h- and z-components of a three-dimensional position vector.

He is now adduces the initial version of the full set of the "Lorentz Transformation", with the f yet to be eliminated. The t-equation is that arrived at for the ray in direction of the X-axis, where y, y', z, z'=0.

He next tries to prove "that propagation is no less a spherical wave with velocity c when viewed in the moving frame". The position vector OP=ct, where ct is the radius of the "lightsphere", obeys the equation x² + y² + z² = c²t². Einstein writes, "[transforming] this equation with the aid of our equations of transformation we obtain after a simple calculation x² + h² + z² = c²t²." He believes that this shows that propagation is spherical in the moving frame, as well. Now the position vector for every point on any surface whatsoever obeys this pythagorean relation; by SR logic every surface must be spherical.

The error responsible for the "Lorentz Factor" b is evident in the inverse transformation. He introduces a third system of coordinates, which is found to be at rest with the stationary system K; for K' he uses the symbols t', x', y', z'. As both systems are at rest with one another, here t'=t, x'=x, y'=y, z'=z.
The t'-equation here (omitting the f), namely t'=t=b(t+vx/c²) expresses the assumption (in other authors explicit) that the relative velocity must be the same in both systems of reference. That is to say: OO'=vt=vt'. To change the expected light velocity (c-v) for rays in the positive direction of the X-axis, (c+v) for rays in the opposite direction], to the c of the interferometer experiment, SR changes the unit of time measurement. This same change, namely c/(c-v) and c/(c+v) is necessary for any other point moving along the X-axis, such as point O', the origin of k. If we write v't'=O'P, and correct for the changed unit of time measurement, we have v'=vc/(c-v) and v'=vc/(c+v) respectively. If, in the t- and x-equations we use the correct value for v', the "Lorentz Factor" cancels, ** that is to say, b=1 **.

Of importance is the next paragraph: "Physical Meaning...", particularly what Einstein writes about "moving clocks".
The t-equation is asymmetric, because here x=+/-ct. [The equation can be simplified to t=bt(1-v/c) and t= bt(1+v/c), depending on direction.] Now the t-equation is the one derived for rays along the X-axis, when x=+/-c. Einstein remedies the defect of the asymmetry of the t-equation by substituting x=vt for x=+/-ct. (The equation thus obtained is the one he had found for rays along the h- and z-axes). His argument here is, again, false.

In interpretations, it is this t-equation that is used, e.g. in discussions of the twin paradox.

The notion of clocks as supplying "the time" of an event is misleading. The magnitude of the t (or t' or t) in a pathlength-equation is the time needed by a point to complete the given pathlength. In any case, the SR solution is inapplicable, as it requires clocks to go at rates that depend on the direction of a ray. As I have already mentioned, there had long existed physical evidence that the light velocity cannot be constant; the SR-device, of changed time-measurement, exists merely for the purpose of perpetuating the myth of the constancy of c.

Herbert Dingle, author of Science at the Crossroads (1972), had previously written a monograph about SR where he adduces all equations and even expands some of them (e.g. the "composition of velocities"). It is strange that he should not have examined Einstein's derivation. For articles by scientists who agreed with Dingle see⁵.

There should be no need for me to write more; as mentioned, I am 92 years old with serious age-related health problems; typing text of this difficulty is so exhausting as to make me ill.

I might add that it is perfectly possible to "do" the transformation intended by Einstein. The result is:

• There is no Lorentz Factor, and t = t(1 +/- v/c).
• As the t'/t ratio depends on the x'-component of ct', which is largest for x=ct, smallest (below zero) for x=-ct, the clocks would have to be able to adapt to a t'/t ratio that can assume all values from (1-v/c) to (1+v/c).
  
  In my paper in Philosophy I had quoted Horace's quip: parturiunt montes, nascetur ridiculus mus (the mountains labour, a ridiculus mouse is born. Or: all that labour and nothing to show for it.).

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.

Einstein's "General Theory of Relativity" (GR) is based on the findings of his SR; as in SR, in GR there is no time-dimension.

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.

The question is whether abstract objects, such as geometrical figures or numbers, "exist". Of significance here is a contemporary debate in the cognitive neurosciences: the role of "thought without language" in "animal cognition" and whether the capacity for "mental images" is common or not, see Weiskrantz, Johnson-Laird and Kosslyn. Bertrand Russell for instance, publishing from 1897, might have lacked the capacity: he insisted that what geometry analyses is physical objects: ropes, marks on paper. Russell was dominant in debates on changes in curricula, see Price.

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.

That the (non-relativistic) geometry of classical mechanics, as ever needed for astrophysics, is taught in courses of n-dimensional geometry for mathematical specialists is useless for physicists.

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" t_A from A towards B, let it at the "B time" t_B 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 t_B -t_A = t'_A - t_B.

In agreement with experience we further assume the quantity 2AB/(t'_A - t_A) 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 t_A [footnote: time here denotes the time of the stationary system], let it be reflected at B at the time t_B 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

t_B - t_A = r_AB/(c-v) and t'_A - t_B = r_AB/(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. 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)
  • 2.1.1. History of the theory of relativity.
  • 2.1.2.Original expositions of the mathematics of "relativity".
  • 2.1.3. Secondary expositions of "relativity".
  • 2.1.4. Publications especially relevant for "relativity" topics.
• 2.2. General bibliography.
  • 2.2.1. History of mathematics.
  • 2.2.2. Philosophy.
  • 2.2.2.1. Metaphysics.
  • 2.2.2.2. Philosophy of mathematics and science.
  • 2.2.2.3. History of philosophy (logic, science)
  • 2.2.3. Mathematics.
  • 2.2.3.1. Textbooks and discussions of maths topics.
  • 2.2.3.2. Pre-relativistic textbooks of mechanics.
  • 2.2.3.3. Visual logic ("geometry") as the coomon sense foundation of mathematical knowledge.
  • 2.2.3.4. Mathematical logic and related texts.

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).

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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.

"On the electrodynamics of moving bodies", 1905, in The Principle of Relativity, Dover reprint of translation of German original: 1952. (This translation is known to be faulty: the inverse transformation refers to the third frame of reference as moving, relatively to the second one, in the positive direction of the X-axis; private communication by Prof. Ian McCausland, ca. 1992).
• "On the Relativity Principle and the Conclusions Drawn from it", (1907), Collected Papers, Princeton U.P., 1989, Vol.2 (Ppb), 252-311.
• Ether and the Theory of Relativity (1920), in Sidelights on Relativity, Dover, 1983, 3-24.
• The Meaning of Relativity, (1921), Chapman & Hall, London, 1967 or Routledge, London, 2002.<.li> Relativity: The Special and the General Theory, 15th Ed. (Methuen 1960) Routledge, London, 1993.

Minkowski,

"Die Grundgleichungen fuer die elektromagnetischen Vorgaenge in bewegten Koerpern." in
H., Gesammelte Abhandlungen, ed. D. Hilbert, 1911; 1967 reprint: NY: Chelsea.
"Space and Time" (1908), in H.A. Lorentz et al., The Principle of Relativity, Dover, 1952,75-91.

Poincaré, Henri,
Whittaker, in footnotes, lists P.'s main original contributions of the relativity debate. Below I list some 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).

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

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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.

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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.)

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

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.

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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.

A History of Greek Mathematics, 2 vols (1921). Dover reprint, 1981.
• Mathematics in Aristotle. Oxford: Clarendon 1949.

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.

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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).

Science and Wisdom. New York: Charles Scribner's Sons, 1940.
• The Degress of Knowledge. London: The Centenary Press, 1937

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.

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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.,

• 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).
• Epistemological Writings, Hertz/Schlick Centenary Edition 1921, reprint. Dordrecht: Reidel, 1977.
• 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; 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,

• Space, Time, Matter (4th Edition, 1921), Dover (original tr.) 1952.
• Philosophy of mathematics and natural science. Princeton University Press, 1949.

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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".

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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).

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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.)

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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.,

• 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.)
  
• Revisiting mathematics education. Dordrecht: Kluwer, 1991.

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).

• The Computer and the Mind - An Introduction to Cognitive Science. London: Fontana, 2nd ed. 1993. (Extensive discussion of the empirical evidence against propositional accounts of mental processing, and for various types of alternative accounts, across the entire range of human cognition.)
• "Mental Models, Deductive Reasoning, and the Brain", in Gazzaniga, Michael S. (ed.), The Cognitive Neurosciences. Cambridge Mass., London: MIT Press, 1995.

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.

• Translations from the Philosohical Writing of Gottlob Frege, eds. Peter Geach & Max Black. Oxford, Basil Blackwell: 1952.
• Posthumous Writings. Oxford, Basil Blackwell: 1979.

Green, J.A., Sets and Groups. London: Routledge&Kegan Paul Ltd., 1965.
Russell, Bertrand
(a small selection of texts)

• 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 - Freudenthal: "as dead as the Dodo - one can't even formulate a problem".)
• 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.

Stoll, Robert R., Set Theory and Logic