Even if simple counting was frequently insufficient to provide convincing evidence, this did not mean that it was opposed in principle: its mathematical limitations were not widely understood, so it had become incorporated as an element of individual clinical judgement rather than being an alternative to it Weisz , p This changed after the mids when, after the introduction of anaesthesia, surgical innovations became more and more frequent.
On the other hand, ulcerated dog larynxes after prolonged intubation were demonstrated to justify tracheotomy rather than intubation despite its low success rate. Now removal of tumours, treatment of ovarian cysts or infectious foci continued to be reported in this simple statistical way — and compared with the presumably fatal issue of conservative therapies.
There were no recognized baselines permitting comparison with other treatments or with no intervention. In the case of systemic therapies, however, few things remained stable during the 19 th century, neither the diagnostic category, nor often the constantly evolving procedures, nor the results. And in doing so they sometimes made use of scientific techniques that were far more sophisticated than counting Weisz , p Medical science versus medical art In , the then young physician Armand Trousseau b.
The translator, Dr. In the present state of science, we must often be content with probability. Louis acknowledges this, whilst he insists that there is a great difference between the probable and the true, for the probable may be false Louis , p XV. In the Introduction he devoted eight pages to the numerical method. This might sometimes be useful, but only secondarily so, for example, when it would lead to new notions in the future. In that way, Trousseau recommended the method and admitted that he had used it himself.
Although Trousseau had never calculated any proportions, let alone any probabilities with all their claimed rigor, he stressed their limitations, saying that they could yield only. I reproach [the method] to count only, […] to stick to the rigorous result like a mathematician…. This [numerical] method is the scourge of intelligence.
It transforms the physician into a clerk, a passive servant of numbers which he has superposed; and the maximal reproach I raise against it is to suffocate medical intelligence Transl. Observing facts, systematizing them by counting, submitting an equal number of cases to two modes of treatment to decide a therapeutic question — these were the characteristics of medical science for Trousseau. But medical science was not to be confounded with medical art.
He stressed. Medicine is more of an art, and the doctor truly worthy of his ministry must above all glorify himself not to be only a learned scientist. And even when the doctor, unfortunately, errs often, one nevertheless finds more charm, more attraction in the study of an art, and [… medicine] needs a bit more intervention of intelligence [understanding and knowing] than the sciences where we are directed by certain and invariable rules Transl.
Trousseau, too, was well aware of therapeutic trials and had done some himself, for example, to evaluate homeopathy in Dean To be sure, his two homeopathy trials for various illnesses were single-blinded, yet he did not report whether symptomatic improvements, if any, were more than transient.
Nevertheless, he believed that these were valid tests. Clearly, Trousseau and therapeutic reasoning remained unconscious, pre-mathematical, complex, messy…and verbose. Another line was probability against certainty. In France, this was articulated particularly forcefully by Claude Bernard. One of the principles underlying his work was that, in nature, every effect was due to a precise cause. This constant relationship of determinism could be discovered through animal experimentation.
This was a typical line of arguments of physiologists, which we shall come across also in Germany see below. Medical science was to look for certainty whereas statistics could only offer probabilities and was therefore inappropriate for physiology. This would be rooted in animal and other preclinical experimentation Morabia In therapeutics — for the time being — one could not do without the probability of statistics; given constant progress, this was an unavoidable concession to pragmatism.
Despite his important contribution based on formal, mathematical probabilistic thinking, Gavarret seems to have had no followers in 19 th century French clinical thinking. This implied unconscious probabilistic thinking. There was much confusion about the notions of statistics, experiment and experience on one level. On other levels, there stood issues of medical science versus medical art and of probability versus deterministic certainty.
Explicit probabilistic thinking was hardly considered. And for various reasons, partly historical, partly out of intellectual curiosity, Gavarret became influential mainly in Germany see below , the USA Warner , Bartlett and, to some extent also in Britain. The long 18 th century Theory. As noted in the introductory sketch of this study, there was some unconscious probabilistic thinking in British clinics during the 18 th century. In this section, I will consider in more detail the motives for, and the modes of, this reasoning, and look further into the 19 th century.
How were they presented? Two traditions were combining to build an indispensable basis for probabilistic reasoning during 18 th century Britain, and these had their origins in the 17 th century with Bacon, Sydenham and Locke Dickersin and Chalmers And there was a growing tendency to report all cases of new treatments observed during a given time period — whether successful or failures, a novelty in itself!
Three modes of probabilistic reasoning by 18 th century clinicians. In other words, there was a transition from seemingly certain knowledge to reliance on relative results based on many observations, successful or otherwise, results that were recognized as partial and evanescent as time went on.
The naval physician James Lind b. A work more perfect and remedies more absolutely certain might perhaps have been expected from an inspection of several thousand … patients. I found a formulation of a conscious, pre-mathematical mode in the same year by the contemporary British physician, John Gregory, and an application of the conscious, mathematical mode by a clinician of the following generation, John Haygarth.
At the outset of this report I quoted John Gregory b. He had also had a mathematical education and had taught mathematics : In his view, rather than getting stuck in endless argumentations any. He repeated this argument three more times in his book pp 15, , And describing the psychological hindrances to doing so he concluded:.
It is, indeed, difficult and painful for men to give up favourite opinions, and to sink from a state of security and confidence into one of suspence [sic! They were re-edited in and in in Philadelphia and translated into French, German, Italian and Spanish. In , John Haygarth b. It occurred to me that it might be computed arithmetically by the doctrine of chances, according to the data, if one, if two, or if three persons were exposed, for the first time, to the variolous infection, what degree of probability there was that one or more of them would catch the distemper.
At my request a mathematical friend made the following computation, on each suppos[it]ion Haygarth , pp These efforts carried with them the notion of probability of success of a therapy rather than certainty — and a host of new problems. For instance, could averages derived from documenting outcomes in groups be applied to an individual? And how could comparable data be assembled? Indeed, exact day-to-day record keeping in tabular form was repeatedly propagated and practised in Britain by many doctors throughout the 18 th century.
Quantification of data derived from them was a new research tool. This 18 th century mental bent was shared by many clinical investigators. They participated in the endeavours and communicated results in an informal network between various cities.
It cumulated in a textbook by William Black b. Clinical arithmetic continued in the early 19 th century. In , Sir Gilbert Blane b. At the age of seventy he published an account of it in his Elements of medical logick in terms of a typical British compromise between two epistemic camps, the rationalists and the empiricists.
It was a plea for rational empiricism as we would define it:. This is a compromise. And further: It is only by a sort of arithmetical computation, founded upon large averages, that truth can be ascertained; and hence the danger of founding a general practice on the experience of a single case, or a few cases [be avoided] Blane , pp , This was unconscious, informal probabilistic thinking at its best as it would henceforth be characteristic of British epistemological literature.
And a decade later, Tweedy J Todd b. Alcock repeated 18 th century positions when writing in that tables. The advantages thus obtained, by enabling the student to generalize the facts, to compare the result of various modes of treatment, […] are too obvious to be dwelt upon Alcock , p Alcock mentioned probability en passant in a footnote p Neither dealt explicitly with quantification.
Yet their endeavour implied unconscious probabilistic reasoning…. In this updated book in English on medical statistics, the young London physician b. He summarized the status quo as follows:. Statistics has become the key to several sciences. So, clearly, there was methodological awareness in Britain tied to probabilistic thinking throughout all these decades, mostly in an unconscious mode.
Louis seen through British eyes As we know, Louis contributed notably to two fields — to anatomo-clinical research, and to the notion of numerical evaluation of therapies. He was influential in Britain in both areas — although British writers did not refrain from pointing to earlier British authors. The late Mr. Another example was Thomas Hodgkin.
Quantification in anatomo-clinical research When, in the s, Thomas Hodgkin b. He deplored the conjectural state of medicine. This could only be overcome by strict adherence to precise descriptions of many clinical cases followed by equally precise inspections at autopsy among those ending fatally; to grouping and presenting them in tabular form for comparison and statistical analysis. The unquestionable talent and powerful sarcasms of [this] author…under which many other systems and authors have seemed to give way, the physiological views upon which he so arrogantly plumes himself, and the authority of his name… make unitedly but a miserable figure when confronted with the counted facts of his accurate and statistical opponent Hodgkin , p Remarkably, Hodgkin revealed his conscious mode of probabilistic thinking when concluding that, with respect to the past, the numerical system was,.
The method adopted by M. Louis may be easily stated. If the [natural] mortality in pneumonia were known to be 25 in , and its mean duration twenty-one days, it would only be necessary to subject a considerable number of similar cases to a particular treatment, to count the deaths or recoveries, and to take the mean duration, in order to state in precise terms the modifying power of the treatment.
The same process would serve to compare or contrast the effects of two systems of treatment …From data thus furnished the results could be calculated, thrown into tables, and readily compared Lancet . Essay review p Indeed, McGrigor b. A few weeks later another review of an earlier work by Louis, was also published in The Lancet.
It had meanwhile been translated into English by Charles Cowan. The editor was. Charles Cowan b. He had spent nearly four years in the hospitals of Paris; and he was personally acquainted with Louis, having followed his clinical rounds and assisted in his post-mortem room. First, he drew attention to the principles of observation according to that famous 17 th century Englishman, Thomas Sydenham b.
Cowan then went on once again to emphasize the need for large numbers of cases, to be presented in tables. One had to overcome preconceived ideas and selection bias to obtain the necessary fair comparisons. Put in one sentence:. It is not our intention, in advocating the numerical method, to conceal for a moment its difficulties; these are great and numerous, but at the same time they can never form any solid argument against its utility, though they will necessarily curtail the number of its disciples pp What sober, yet sensible and farsighted words!
Probabilistic reasoning in 19 th century British clinics. William A. Guy b. He had become interested in statistics, and he wrote for one of the first issues of the Journal of the Statistical Society of London. The idea of hypothesis-testing with the help of quantification was an immense epistemological step. Unconsciously it implied probabilistic reasoning.
Numerical probability became the natural substitute when numerical certainty was not available Guy , pp , 34, Now, this was obviously the case in medicine, with its variable quantities and events. And so, medicine was also amenable to perfection through adoption of the numerical method as had been the case in astronomy and chemistry. It was the only reference to a statistical work he gave.
For medicine this was obvious in vital statistics and nosography description of diseases. And then he continued:. This statement was comparative, quantifying and consciously, yet informally, probabilistic. But as a current example of formal probabilistic thinking, Guy referred to the trustworthy estimate of the benefit of vaccination against smallpox, and of the extent of protection it gave p This was the example in which a calculus of probabilities had been used 60 years earlier to show the utility of inoculation of smallpox.
Numbers added precision to words of doubtful meaning. Yet Guy admitted that the application of this method to individual cases was limited. This was conscious probabilistic reasoning. Although Guy could have taken these arguments from 18 th century Scottish doctors, or literally — yet improbably — from John Gregory see p 1above , his qualification of the numerical method was new.
In his words it was. These calculations supply but one element for the solution of the problem, for they apply only to cases of average severity p In other words, they applied to the average of a group. And it was not obstinately doctrinaire, but pragmatically realistic. His was the method of averages p For the errors were precisely those to which the results of common observation, expressed in common language, were equally exposed p Indeed, at the outset of his reflections, he had enumerated the necessity, the difficulties and pitfalls of precise observation, of grouping of comparable facts in tables, of large numbers etc.
As the French clinicians had done, Guy distinguished medicine considered as a science from medicine considered as an art. The text provides a kind of theoretical standard of probabilistic reasoning after British and Foreign Medical Review ; Agnew But then the review developed two novelties: the clarification of definitions, and the inclusion of the calculus of probabilities:.
The numerical method is sometimes erroneously regarded as a mere substitution of figures for words. Against this mistake Gavarret strongly protests, and with good reason, though th[is] mere substitution is [already] a great improvement in our scientific methods, seeing that figures admit of strict comparison which words do not p This terminological clarification was deemed necessary to specify that the use of numbers as a research tool differed from the historical meaning of statistics as the science of the state p The calculus of probabilities supplied a method to determine the limits of error of our observations, a method, for instance, to specify the limits of confidence of a difference between two treatments p And he continued:.
If even Louis […] lies open to censure, what shall we say of the majority of his followers, and in what terms shall we speak of those who still persist […] in drawing important conclusions from one or two scattered and not comparable facts p Her language must be the language of figures; her test, the calculus of probabilities; her example, the most perfect and exact among the sciences of observation and experimentation [i.
This was certainly a truly remarkable passage, an example of the proverbial British pragmatism re-uniting the perennial quest for dogmatic certainty with the proposed yet reasonably unattainable model of probability under review to form a practical modus vivendi.
If only these words had been taken to heart! Yet the application of formal probabilistic procedures was hampered, within medicine, by ignorance, many inherent difficulties, for example in collecting comparable cases , confidence in presupposed certainty of technological innovations, and socio-culturally, by hierarchical authoritarian structures within medicine.
Despite the methodological insights into the possible sources of errors described, many but not all doctors continued to compare small groups, often incomparable, to select cases, to fail to take account of the natural course of diseases, to fall into the post hoc-ergo-propter-hoc fallacy, for instance in the appreciation of homeopathy , even to cheat.
Interestingly enough, as I have shown above , Hodgkin saw no necessity to change anything in his text, when re-producing his lecture twenty years later Hodgkin , rather he had a new motive. Yes, there was one. These describe a controlled trial to test the claim that belladonna reduced the likelihood of developing scarlet fever.
Two boys in each group developed scarlet fever. Balfour properly cautioned against inferring on the basis of such small numbers that belladonna had no prophylactic effect Balfour , whereas for Guy this evidence was sufficient! This Edinburgh and later Cambridge MD lectured to students for twenty years, till , on pathology and therapeutics at St.
Thus, he was a man of the establishment. His attitude towards the methodological aspect of medical research may be derived from his Lumleian Lectures for which bore the title Medical errors…Fallacies…of the inductive method of reasoning to the science of medicine Barclay They were all about the ways to find laws of cause-effect by experimentation in the Baconian tradition.
Yes, large case collections, tabulated in statistical form, were better than to trust in memory, and there might even be a way to induce a causal relationship p Barclay shrewdly analysed the therapeutic inquiries then issued recently by the British Medical Association on therapies of pneumonia with and without bleeding , non-syphilitic psoriasis, tapeworm, and scarlatina. He quoted huge accumulated Parisian and London statistics not deemed necessary , Dr.
He did this from the standpoint of principles of logic without entering in the ways of thinking behind them. No kind of probability was mentioned, let alone the calculus. Rather he repeated a typically paternalistic hackneyed saying:. But how? Here he quietened down. The numerical method has not yet been applied to any great extent in therapeutical [sic! The difficulties attending its employment are so great, and the method itself so open to fallacy, that the results are not likely to be very available for scientific purposes pp , This attitude, which he repeated, less outspokenly, 17 years later in his Harveian Oration to the Royal College of Physicians Barclay , was not constructive, yet it may very well have been representative of the views of a great majority of physicians:.
Under these circumstances, for the few who thought about methodological issues, a numerical method was, for the time being, the solution of the complicated problems of day-to-day clinical practice. Insofar as Guy, for instance, had found some well-designed and cautiously interpreted trials, as judged by contemporary insights , he confirmed a genuine British tradition of enlightened pragmatism Pickstone , whereas the French mathematical tradition, sophisticated formally by Gavarret, was only theoretically valuable.
The general situation in practical therapeutics was one of laisser-faire. But German authors would not leave things there. They developed his mathematical concepts and applied them in practice. Introducing German dramatis personae Theory. Books by Lind, Gregory, Haygarth and Black quoted above had all been translated into German by the end of the 18 th century — but not into French with the exception of Lind.
I have been unable so far to find any of their methodological probabilistic passages referred to in the wider contemporary and early 19 th century German literature. German medicine was trapped for a time by the speculative philosophical systems of romantic medicine Wiesing That explains why we find references to these issues by German doctors from this latter period onwards, for instance by Jacob Henle.
Jacob Henle b. I seem to hear Gilbert Blane — his work had been edited in German in — when I read. But not only to fill the deficiencies of both parts should empiric and rational medicine be linked to each other, but to foster one another where both can be applied simultaneously Transl. Two years later, he was the first German I have been able to trace so far who referred to it. In admirably worded sentences he summarized the contemporary epistemological basis of therapeutics and, in a farsighted way, looked ahead.
Of course, he also came to speak of Louis, whose. Henle then gave precise methodological guidance: The number of cures obtained after a particular therapy had always to be given in relation to the untreated or otherwise treated patients, i. They are still huge problems today. Henle did not follow this track further. After all, he was a professor of anatomy and not a clinician.
But younger German clinicians who might have read this early book of his during their studies , took up Gavarret. Carl Theodor August Wunderlich b. According to him, the numerical method was practised sloppily. Its usefulness was anyhow very restricted:. But above all, he considered this method to be inhumane, when implying human experimentation. As an experience in support he mentioned, in remarkably sarcastic words, an experiment for which a French physician divided patients with typhus into three groups bloodletting, laxatives or nothing.
I could not help the impression that we live in times more barbaric than when criminals sentenced to death were used for [testing] operations or physiological experiments. He realized the classic confusion of a method as such with its incorrect execution, both scientifically and ethically. He now also recognized that therapeutics was in a crisis. This also made clear that statistics had so far not achieved much. Wilhelm Griesinger b.
He too had been in Paris twice , , and in Vienna as had Wunderlich. Subsequently, Griesinger became Privatdozent, and extra-ordinary professor there, before leaving for Kiel in Very early on, he grappled with the methodological issues that had been discussed in France. Griesinger was all in favour of numerical and statistical methods. Quoting Guy Griesinger , p 39 , he noted:. Provided the method was used correctly! Lamentably, doctors were still unfamiliar with the notions of precise observation, note-taking, comparability of cases, comparison without selection of cases etc.
To reform all this one needed to bring together rational theory and empiric facts, both based on accurate observations, not on the philosophical speculations of German romantic medicine. This was the rational empiricism propagated in Britain since the 18 th century arithmetic observationists.
Mathematisation would be the next step, as in every true science. This needed time and confidence — and a method for a posteriori calculus of probabilities. In the whole world one would not find an institution allowing for the assembly of similar cases, at least, per group, that is for two groups, to be compared! Adding up cases from the literature did not work because of their heterogeneity. In that sense, smaller groups could also be valuable.
And there was the problem of the relevance of mean values the individual cases. Much remained to be done p 22! Friedrich Oesterlen b. Three years later he left for a full professorship at Dorpat, whence he had returned to Heidelberg as a Privatdozent in With the hope of an academic re-start at home he started to publish extensively, for instance, Medicinische Logik , which was published in English as Medical Logic, by the Sydenham Society Oesterlen had also read Gavarret.
He now aimed to set the issue of statistics in a theoretical context by applying in medicine pVI the teaching of J. But valid scientific results consisted in the discovery of causation, not just in the discovery of statistical correlations. This held as long as one kept to the rules of extremely precise observation, compared comparables, considered the natural course of diseases, collected large numbers to establish high grades of probability.
He was very cautious about generalizations Oesterlen , pp and hasty conclusions, as had been the case with Louis. These could lead to nonsense, and the general error of internal medicine was the post-hoc-propter-hoc fallacy. Comparison was needed pp This work would prove to be a major contribution to the methodological discussion in Germany Rothschuh However, neither Oesterlen nor Wunderlich mentioned the calculus of probabilities in this context, in contrast to their friend Griesinger. It was beyond their horizon at this time.
Oesterlen did not succeed academically in Germany. After much publishing on various issues he retired to private practice, eventually in Switzerland. The field was still in its infancy Schweig , pp , Schweig started his article by clarifying definitions: medical statistics were for him a special method for drawing conclusions Schlussziehung p He had clearly read Jacob Bernouilli, Poisson, and Gavarret p , and he wrote at length on the establishment of arithmetic averages means of groups of cases.
Such averages were only of any significance if compared with other averages. These had to be as small as possible. But the exact determination of the sufficient number of cases or groups to achieve this by the method of least-squares was too complicated.
Thus finally, he set up the following rules:. These rules were certainly clarifying, but they were not acknowledged by the medical world. Schweig was not quoted by a group of contemporaneous, yet somewhat younger mathematician-physicist-physiologist-physicians who advanced the methodology by developing tests of significance for assessing the meaning of differences between groups. All these theoretical contributions of the s and early s reflected probabilistic thinking in the unconscious Wunderlich and conscious, pre-mathematical modes Henle, Griesinger, Oesterlen, Schweig.
In the next two decades two generations of younger men acted in compliance with the formal, mathematical mode. Gustav Radicke b. He was only a professor extraordinarius of physics, that is without any strong institutional ties. It had a word title which, when abridged, reads Die Bedeutung und Werth arithmetischer Mittel …und Regeln zur exacten Beurtheilung… On the value of arithmetical means… and rules for the exact assessment….
This method was designed not only for physiological experiments, but also for enquiries dealing with purportedly effective therapeutic measures. Instead, he proposed comparing the differences between the means including their standard errors. This would show the degree of confidence that could be attributed to such a difference. When Radicke applied his test to some physiological and therapeutic examples it suggested that no effects had been produced Coleman This was deemed impossible.
A storm in a teacup ensued over the next few years. In the end, determinism prevailed; Radicke and his test disappeared from the German literature Coleman Adolf Fick b. He was ordinary professor of physiology in Zurich when he published Anwendung der Wahrscheinlichkeitsrechnung auf medicinische Statistik On the application of the calculus of probabilities to medical statistics as an Anhang appendix to the second edition of this textbook of Medicinische Physik Medical physics, Fick For, above all, Fick noted, it was thanks to Gavarret that.
And, after quoting from puzzling Beneke at length, he added: Yes, even quite frequently, weighty voices have risen against them in principle Transl. Fick took the application of numbers for granted. That was quite something. The next problem to solve was the elaboration of.
A certain measure is naturally to be observed. Since probability is more or less to replace certainty one must not be satisfied with too scanty a probability, e. Then Fick said that one should neither go too far in the opposite direction. So, this value was still near unity, the symbol of certainty. Fick now developed a formula and calculated a logarithmic table which permitted determination of the limits within which a probability was included.
It functioned for a rather large number of cases, at least not fewer than a hundred pp That was a methodological advance, yet Fick did not contribute to solving the practical difficulty of the computation of hundreds of comparable cases. Consequently, the large number of patients required according to Gavarret and Fick continued to be criticized. Several ways to solve the problem were suggested. Wunderlich had, irrelevantly, proposed concentrating on the effects of a given remedy rather than on a disease because of the diagnostic uncertainties Wunderlich , p So, questions remained open.
But new inputs were soon to be propounded by three physicians of an even younger generation born in the s and then elaborated by an older, remarkably versatile colleague. The crude death rates were This yielded a probability of above the Therefore, [he said] it is also permitted to choose this stricter form of calculation, although the absolute numbers are not very large.
This science, albeit hardly existing today… will in the future solve problems of which we now have not the slightest idea Transl. Willers Jessen b. He used this term meaning mathematical probability, for he had obviously read Poisson and Gavarret in German translations.
Accordingly, he had devised tables for the application of the calculus of probabilities, Fick had simplified them, and Jessen now provided one even more easily to use p He concluded, with foresight:. This was also the case of Julius Hirschberg. Julius Hirschberg b. In , when still a Privatdozent in Berlin , he wrote a book with the enticing title Die mathematischen Grundlagen der medizinischen Statistik elementar dargestellt The mathematical bases of medical statistics elementarily presented, Hirschberg This allowed the comparison of much smaller groups.
The multi-talented Carl von Liebermeister considered this a great step forward. Since his youth, Carl Liebermeister b. Obviously the two colleagues met. This had to be explained. But he also had in mind to find a mathematical solution to the meaning of a statistical difference between two therapies. For this he developed a test of significance for such a difference He lectured on the issue, and sent a manuscript to two professors of mathematics, former colleagues from the University of Basel, for critical examination.
They approved. The ensuing publication bore a similar, yet more specific, title than Fick had chosen, namely Ueber Wahrscheinlichkeitsrechnung in Anwendung auf therapeutische Statistik. On the calculus of probabilities applied to therapeutic statistics Liebermeister He included examples of both situations Ineichen , Senata et al Two comments from outside and inside Germany These contributions had not passed unnoticed.
A conscientious young medical historian doctor sensed that issues of evaluation and probability were in the air: Julius Petersen b. In 29 pages he dwelled on Poisson, Gavarret, Louis, Wunderlich, and others, and on the numerical method. He quoted, rightly and at length , Gavarret saying that there was much loose verbiage about probability whereas only the calculus of probabilities could really help to estimate the worth of mean values averages.
Albeit still far from being perfect, this method was important for future developments p Petersen qualified the confusion between Benecke and Vierordt: the former explained the effect of a cure, the latter thought that it was being demonstrated statistically. Again, this was the old debate between rationalism and empiricism. A young German insider engaged in an overview of the methods available in clinical research, particularly in therapeutics.
Friedrich Martius b. Later he became professor of internal medicine at Rostock and, typically, did not publish any longer on the subject. As Oesterlen and Schweig had done some thirty years previously, Martius begun by clarifying the confused terminology.
He analysed the French and German works Radicke was not mentioned! Numerical induction he understood as being based on statistics and probability calculus. Indeed, they complemented each other. The calculus of probabilities needs for its application materials collected according the strict rules of statistics and the latter, without the calculus, would not […] always find its critical utilization and the elaboration of which it is capable Transl.
Gavarret was following Laplace, who had declared that all knowledge was based upon probability Martius , p This was obviously not true. One had just to think of anatomy. This showed the arbitrariness and unreliability of such ratios and of the whole process: which of these haphazardly proposed probabilities excluded the hazard?
And this for any […] observational material, be it ever so small […provided] the comparability of the cases, this eternal crux of all statistical data collections can be demonstrated Transl. This did not mean that he proposed neglecting statistics. On the contrary, through mass observation and reliable assessment of treatment successes, the probability of obtaining important indications for practical action increased Martius It was not, as Gavarret had deemed, the ripest fruit of modern thought, or.
Rather it is and remains a makeshift, albeit a very important one […], that is undoubtedly worth an even deeper foundation and more extensive application Transl. These were clever insights, and such efforts would effectively be made in the 20 th century. But before, new difficulties, and consequently, new desiderata were recognized by two practitioners from Breslau now Wrocklaw, Poland , Alfred Ephraim and Ottomar Rosenbach.
When reading these two testimonies one realises that formal, mathematical probabilistic reasoning had clearly made an impact on its authors. But on the practical side the consequences were limited, whilst on the theoretical new requirements for scientific evaluation were identified — for the new century. So, Alfred Ephraim b. The methodology of clinical evaluation was eclipsed by new technical methods of examination. Ephraim noted that a recent discussion between two eminent German physicians made clear that the numerical method continued to have both detractors and supporters.
He claimed that the reversal of previously statistically founded claims did not help to convince the medical world of the value of such work Ephraim , pp That is why Ephraim answered the two eternal questions of the statistical endeavour — i what was to be counted? But while these theoretical difficulties could be dealt with, one should not overlook the practical ones; and here he enumerated three new criteria for solid comparisons:.
The untreated cases were necessary for comparison, but difficult to find. If lack of treatment seemed inhuman, it could be justified because most treatments had actually not been demonstrated to be useful. Ephraim concluded that those who deemed these requirements insurmountable must be aware that they are renouncing trustworthy therapeutic knowledge. Yet, reliable identification of less dramatic treatment effects could only be assured by the results of statistical research.
However, he did not mention the calculation of probabilities as a complementary method of evaluation, thus once more ignoring terminological precision. Ottomar Rosenbach b. By he had resigned his position as chief of the medical department and his associate-professorship and retired to private practice in Berlin, but continued publishing. It is probable that he knew young Ephraim since they had lived in Breslau at the same time.
Like Ephraim, Martius and many others before them, Rosenbach criticized the misuse of therapeutic statistics:. And he further emphasized, the arbitrarily defined, often inadequate duration of trials, the failure to use modern diagnostic criteria for example, the use of clinical signs and bacteriology , and the differences among cases an issue that had already been proclaimed innumerable times. In short, these statistics served only to reinforce preconceived opinions and frequently, when there was no comparison group, to fall into the trap of the post hoc-ergo-propter hoc fallacy pp And maybe they did not like him: had he not, while still an aspiring Privatdozent , written quite aggressively in a book on the Foundations, Duties and Limits of Therapeutics Grundlagen, Aufgaben und Grenzen der Therapie, Rosenbach :.
Statistics — what would they not have sanctioned in the hands of able arrangers. And later: The history of medicine furnishes enough examples of friends and foes fighting with equal obstinacy and equal certainty for a dogma established on the basis of such contradictory [statistical] results Transl. Returning to the serum-therapy of diphtheria, he repeated the need for the experimentum crucis the decisive experiment , namely,.
And of course, this holds not only for the treatment of diphtheria Transl. If one did not want to, or could not perform this process of evaluation, which Rosenbach felt was easy to carry out, one deprived oneself straightaway of the possibility of doing scientific research Rosenbach ; Chalmers et al. Germany by According to Petersen and many of his contemporaries and later historians , the overall situation in the 19 th century may have been similar in practice to that in France and Britain: if considered at all, the actuarial method of counting and statistical analysis prevailed.
In the end, selection bias, making results positive by changing criteria, and quantification of preconceived ideas were decried as misuses Rosenbach The notion of probability was understood by many clinicians, and a few of them actually struggled to apply formal mathematical probability and its consequences throughout the whole second half of the 19 th century.
The influence of Poisson and particularly of his medical pupil, Gavarret, was pivotal. Arguably the first outflow of probabilistic thinking in medicine was evaluation of the inoculation of smallpox in 18 th century England. Numerical comparisons of death rates of inoculated and uninoculated groups were made by mathematically- inclined clinicians such as James Jurin. From then, probability became a problem of numbers, of quantification Daston , p 4. This implied probabilism in that proportions of average mortalities of groups were calculated and compared.
The probabilistic reasoning behind these approaches was unconscious Mode I in Table 1. Quantification remained informal: simple counting, summation and calculating averages, rates, proportions and frequencies. In other words, it was pre-mathematical in the strict sense of the word. This mode of practice was widespread. It prevailed with different intensities throughout the two centuries covered in my research. Another mode spoke of probability explicitly, but in practice still used informal, pre-mathematical quantification, i.
I found scattered examples of this authored by clinicians from onwards, and increasingly after It was in the s, and in the issue of inoculation of smallpox in particular, that reasoning became mathematically probabilistic, and therefore conscious Mode III in Table 1. This mode of thinking was present among subsequent generations of French mathematicians, until , when it became practical again with the young French mathematician-clinician Jules Gavarret.
Mode III was had definitely been launched, but on a small scale. A list of those who unconsciously propagated probability by fostering informal, pre-mathematical numerical evaluation of therapy Mode I in Table 1 cou ld easily be compiled.
It became common practice from the second half of the 19 th century onwards. In the same way I identified authors who consciously propagated pre-mathematical probabilistic aspects Mode II , or even evolved formal, mathematical probability Mode III in clinical medicine. After extensive research, this list seems to me fairly exhaustive. It suggests the relative rarity of Mode III probabilistic thinking in clinical medicine.
In other words, they were looking for laws of nature. Such quantifications fitted neatly into the contemporary statistical movements that became so active in Europe and North America Porter , p In a more democratic society, trustable action would be called for, and, for many, numbers seemed trustworthy. The fight against superstition, fixed ideas, prejudices, and newly the church, also played a role. Besides this general societal trend there were certainly individual psychological stimuli.
I can only speculate about these. Rather, let me enquire about the intellectual motives behind the phenomenon I have observed. In clinical medicine, thinking became probabilistic when new interventions and therapies were invented. Enlightened doctors wanted to compare them with older ones to find which one was to be preferred: was inoculation valuable in preventing smallpox compared to leaving the disease to take its course?
In other words what were the risks of medical innovation? Another motive came from young men bluntly recognising, time and again, that therapeutics was chaotic. The truth lay in observed facts, and in many of them, assembled in groups for calculation. Fostering this philosophy motivated some clinicians. In turn it meant probabilistic thinking would be involved, as we now know, in counting, comparing and, in the end, mathematical analysis.
Objectivation and scientification brought new problems, however. It had been identified and addressed in the s in the Paris inoculation debate, later, for instance, in the Paris disputes in the s, and then by Henle , Griesinger , and Trousseau And the conundrum remains and seems likely to remain a bone of contention.
On the other hand, numerical work was also decried on moral grounds as inhumane because it stubbornly adhered to a research protocol instead of a true treatment plan Wunderlich The issue seems eternal to me. Another controversial issue concerned the essential interpretation of data. Since it could imply value judgements, inferences and generalizations could be considered correct or injudicious. And there was the phenomenon of apparently contradictory results of, say, two or more successive clinical trials.
Another line of argument concerned the question of the risks of medical innovations. While traditional treatments such as bleeding and purging were just there, unquestioned from time immemorial, innovations met not only with approval or repudiation, but also with scepticism and uncertainty. A specific strategy for dealing with uncertainty was the new notion of risk. It was based on the calculation of probabilities. It produced a new kind of knowledge in order to reduce uncertainty possibly to certainty… , namely numerical data.
And there was a fact that we know with hindsight: the human brain does not recognise probability; prima vista it is neither perceptible, discernible, nor evident; it must either be believed or calculated; and calculations are barriers. On top of these intellectual and psychological difficulties there were continuing practical obstacles: the elaboration of statistics was a cumbersome and time-consuming enterprise.
In fact, the prerequisites for meaningful statistical comparisons increased over time. There was the number of cases theoretically deemed necessary, their comparability, and difficulties of concurrent comparisons. However, those who were convinced of the need to use probabalistic thinking underestimated these practical difficulties and thereby marginalised themselves, whilst deeming clinicians to be mathematically incapable Martius.
It is important to realise that all these problems and impediments were Janus-headed: They were challenges on the one side and reasons for criticism on the other. In the 19 th century, however, the preoccupations developed in a new direction. They could not be bothered with complicated epistemic issues.
Yet they complained that therapeutic chaos could no longer be ignored; it should be vanquished by exactly the methods maligned by those who simply muddled through. But who were they, these propagators, doubters, critics and opponents of methods involving probabilistic thinking?
To answer this question, it is helpful to consider their social status within their respective communities. Who was concerned about probabilism? But as mathematicians they were only marginally interested in practical real-world issues, and published their results in learned books, journals and societies.
Daniel Bernoulli was in fact a first medically qualified adopter. He was followed two generations later by Condorcet, Pinel, and Poisson in words and formulae ; and finally, after a further generation, by the pivotal young physician-mathematician Jules Gavarret in practice. Jurin, the secretary, and Scheuchzer, a fellow of the Royal Society, were both part of the British scientific establishment.
Both held Cambridge MDs, Jurin after having also read mathematics. They were learned physicians. British arithmetic observationists such as Lind, Gregory, Haygarth, Black, Millar a particularly militant author , McGrigor and many others emulated them. They acted upon Modes I and II probabilistic thinking. They were practical clinicians. It fits also for Blane and Balfour, for after active service, they rose to high posts in naval and army medical administration, respectively. Yet they both abandoned medical practice early in their lives and became distinguished London figures, FRCPs Croonian, Gulstonian and Lumleian Lecturer, and Harveian Orator in the case of Guy and FRSs vice-president , acknowledged for their commissioned public work and active in public health statistics.
He taught pathology at two Parisian Hospitals; yet in a way he was a loner, who occupied no important posts. His efforts had waned by the end of the s, after his son had died. In the second half of the 19 th century, after Gavarret, formal probabilistic evaluation became further fostered and sophisticated mathematically up to tests of statistical significance; authors were chiefly German.
Yet Schweig , a physician turned civil servant, and Radicke , a physicist, were hardly mentioned in the medical literature. Fick was already a professor of physiology and may have had some temporary impact. Interestingly, six of these nine clinicians were of Jewish origin.
This religious peculiarity was a feature of 19 th century German probabilists, just as non-conformism was a feature among18 th century British arithmetic observationists. Whether anti-semitism was a hampering reason is an open question that could only be addressed using detailed individual biographical studies Hammerstein ; Weber I reckon that the temporally limited marginal social positions of most clinicians when publishing on epistemological questions, thereby thinking probabilistically and actively trying to foster it, was fairly typical of the 18 th century British and the 19 th century French, British, and German authors I have studied.
Antagonists on the contrary were well established members of the academic community, interested in maintaining the status quo and their personal prestige. The question now arises whether there were nevertheless national differences in the emergence, reception and dissemination of probabilistic thinking.
Does the evidence suggest different national models of emergence of a science of therapeutic evaluation, and were there differences in the communication of ideas? In France the issues of evaluation and of risks were first treated theoretically by scientists interested in probability, who saw this notion as applicable to the real world of clinical medicine.
This developed in Paris, first among colleagues, over four generations in a master-to-pupil-chain. Probability stayed mathematical even when Gavarret finally meant to apply it formally in — in a practically unusable mode. This state-of-the-art remained for the rest of the century. The French ignored the pragmatic mode of British arithmetical observationism, as well as the later German thinking.
Had they been interested in these matters they would have taken notice of the prior and concurrent British pre-mathematical probabilistic evaluations and quantified nosography, of which I have not so far found any translations into French see Table 2. When some realized by the mids that their type of informal probabilistic thinking was also being practised in Paris, it was favourably but also critically reviewed.
This pragmatism can be seen as a genuine British tradition Pickstone When German authors became involved in the s, they could — and actually did — draw from French and British sources. They tackled some pending problems in applying formal probabilistic techniques to clinical practice. Over time, a solution was presented, deemed too complicated, a new answer was proposed, and so on, in a dynamic, indefatigable manner.
Thus far, I do not know for sure whether any of those German solutions were taken into account in Britain or France, but I doubt it. Authors with an international outlook were rare among those I have studied. As Table 2 suggests, if they did not know the languages, they were unable to keep abreast of the reported developments in probabilistic thinking.
Maybe also, they simply did not care about this specific topic. When surveying historical evidence covering years one expects today to identify what has changed. And indeed, I have mentioned many changes between and On the other hand, there were also remarkably constant features.
During the two centuries, and in all three of the countries studied, there was increasing awareness among clinicians of a need to publish dependable information about medical achievements. The simple fact that evolving probabilistic modes of thinking and clinical action had encroached on minds over years makes a difference, particularly in the long run. The meaning of probability had changed. A new kind of knowledge was being generated, and this new situation created new problems.
As there were more and more innovations, the epistemic issues seem likely to have concerned more people. However, the typology of those tackling these issues remained the same. The ways the questions were considered or not, why and by whom, suggest ten enduring features.
First, it reveals that most French, British and German clinicians throughout the two centuries considered were not aware of the underlying probabilistic nature of their thinking and action when counting and analysing their cases using Mode I quantification. Second, it is clear that relatively few of them consciously mentioned probability, according to Mode II, let alone to apply the formal mathematical techniques of Mode III to estimate the value of a medical measure.
Third, it turns out that, with few exceptions, the probabilistically orientated clinical authors whom I have studied were young when publishing references to this topic. Fifth, most of the clinicians were still in marginal social positions when publishing on mathematically orientated probabilistic evaluation. For ever one who later achieved a recognized academic position, another dropped the manifested interest — for various reasons.
Some may have resigned themselves to the conservative influence of the established clinical community. Sixth, others remained marginal men — again for various reasons. At any rate, neither the field of epistemology nor the clinicians who tackled it ever received academic recognition for this endeavour.
Seventh, by ignoring or underestimating the practical difficulties in propagating their ideas among clinicians, authors marginalized themselves and the calculus of probabilities intellectually over the years covered in this study.
Questions and answers remain. Ninth, the long-term perspective makes clear that the arguments advanced for or against probabilistic solutions persisted over time, and they were used in whatever way supported the analyses and perspectives, in other words the interpretations of the disputatious parties. In fact, interpreting the results of calculations is an essential step in the circle of probabilistic reasoning [ see graph ]. Tenth, the uphill battle fought by the supporters of probabilistic thinking engaged in research suggests that only very rarely before was clinical practice guided by conscious probabilistic reasoning pursued through informal quantification, let alone through formal methods.
The above ten persistent features may help to explain why, on the whole, clinical medicine had never really adopted even informal probabilistic thinking by the end of the 19 th century, let alone the rigorous conditions of statistical testing as required by the calculus. Certainly, the active, published opposition, and, above all, the passive daily resistance or simple lack of interest in teaching and practice, were major contributory reasons.
In other words, it is the insight that medicine is neither theoretically dogmatic-rational, nor empirically knowledgeable. It is both — a rational-empiric unity. Perhaps by accident, but typical anyway, it was a British clinician who arrived at such a pragmatic conclusion. The 18 th century William Cullen b. My study has now extended the timespan over another years converging to a century prior to our present days. This could be learned by long-term experience.
It was also understood to be the application of results provided by Science — a quantifying science producing and evaluating average data that were useful for everyone, albeit requiring specialist knowledge. In that restricted sense clinical practice was probabilistic. Both aspects represented dogmas and had flaws: authoritatively proclaimed certainty was more easily believed by practitioners and patients than calculated probable estimates; but the scientifically minded considered such certainty a phantom.
The second half of the 19 th century, had witnessed a rapid rise in operative surgery thanks to pathological anatomy, anaesthesia and anti-sepsis and asepsis. New therapeutic possibilities had been introduced on a much larger scale than during the 18 th century.
This increasingly opened up important therapeutic options. There were also advances in non-operative medical disciplines. The era was optimistic. This conclusion was characteristic motive for scientifying evaluation procedures which had emerged since. Two Germans, Ephraim and Rosenbach, stated this again towards the end of the 19 th century. They formulated a host of indispensable conditions, old and new, to improve the relevance of quantitative evaluations of therapies: comparison to an untreated or differently treated group of patients in the same conditions; using uniformity of diagnostic methods; documented adherence to treatment; trials extended over sufficient time; the concepts of placebo and blinding.
Some saw these as possibilities to potentially improve the approach to impartial statistics aiming at objective probabilities: a science for the future. And it came. With hindsight, the 18 th and 19 th centuries were the long dawn of this science of probabilistic testing.
By , the time seemed ripe for it, and it started soon after in Britain with statisticians — Karl Pearson b. Gosset b. Fisher b. They had all forgotten their 19 th century forerunners. During the second half of the 20 th century, evaluation science developed into a recognized discipline, widely accepted by doctors, whether researchers or practitioners, and young idealists from a variety of backgrounds, for example, those who initiated the Cochrane Collaboration.
What about the answer? Science never comes to an end — except when it is mathematised. Latest Financial Press Releases and Reports. Annual General Meeting of Shareholders. Share Information. Specialty Products. Catalogs, Flyers and Price Lists. Open Access. Open Access for Authors. Open Access and Research Funding. Open Access for Librarians. Open Access for Academic Societies. About us. Stay updated. Corporate Social Responsiblity.
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Review a Brill Book. Making Sense of Illustrated Handwritten Archives. The Enlightenment was a watershed event of reform and renewal that transformed society. Motion is a major metaphor for mental operations. Like the English radical freethinker Anthony Collins seventy years before him, Christoph Martin Wieland firmly believed that great benefits would accrue to society through the freedom to philosophize on any matter.
His approach is the middle way between the extremes of too much and too little freedom of speech. Reference Works. Primary source collections. Open Access Content. Contact us. Sales contacts. Publishing contacts. Social Media Overview. Terms and Conditions. We are used to g being a symmetric tensor field, i. In the following this need not hold, so that the decomposition obtains :. An asymmetric metric was considered in one of the first attempts at unifying gravitation and electromagnetism after the advent of general relativity.
We also note that. In a manifold with Lorentzian metric, a non-trivial real conformal structure always exists; from the equation. In view of the physical interpretation of the light cone as the locus of light signals, a causal structure is provided by the equivalence class of metrics [ 67 ]. A special case of a space with a Lorentzian metric is Minkowski space, whose metrical components, in Cartesian coordinates, are given by.
A geometrical characterization of Minkowski space as an uncurved, flat space is given below. Let be the Lie derivative with respect to the tangent vector X ; then holds for the Lorentz group of generators X p. The metric tensor g may also be defined indirectly through D vector fields forming an orthonormal D -leg -bein. From the geometrical point of view, this can always be done cf.
By introducing 1-forms , Equation 11 may be brought into the form. The fibre at each point of the manifold contains, in the case of an orthonormal D -bein tetrad , all D -beins tetrads related to each other by transformations of the group O D , or the Lorentz group, and so on.
In Finsler geometry , the line element depends not only on the coordinates x i of a point on the manifold, but also on the infinitesimal elements of direction between neighbouring points dx i :. The second structure to be introduced is a linear connection L with D 3 components L ij k ; it is a geometrical object but not a tensor field and its components change inhomogeneously under local coordinate transformations. The connection is a device introduced for establishing a comparison of vectors in different points of the manifold.
For each vector field and each tangent vector it provides another unique vector field. The expressions and are abbreviated by and X i ,k , respectively, while for a scalar f covariant and partial derivative coincide:. We have adopted the notational convention used by Schouten [ , , ].
Eisenhart and others [ , ] change the order of indices of the components of the connection:. As long as the connection is symmetric, this does not make any difference as. For both kinds of derivatives we have:. Both derivatives are used in versions of unified field theory by Einstein and others. A manifold provided with only a linear connection L is called affine space. From the point of view of group theory, the affine group linear inhomogeneous coordinate transformations plays a special role: With regard to it the connection transforms as a tensor cf.
Section 2. For a vector density cf. A curve is called an autoparallel if its tangent vector is parallely transported along it at each point :. The equivalence class of autoparallels defined by Equation 18 defines a projective structure on M D [ , ]. From the connection L ij k further connections may be constructed by adding an arbitrary tensor field T to its symmetrised part :.
By special choice of T we can regain all connections used in work on unified field theories. We will encounter examples in later sections. The antisymmetric part of the connection, i. The trace of the torsion tensor is called torsion vector ; it connects to the two traces of the affine connection as. Various subcases of affine spaces will occur, dependent on whether the connection is asymmetric or symmetric , i.
In physical applications, a metric always seems to be needed; hence in affine geometry it must be derived solely by help of the connection or, rather, by tensorial objects constructed from it. This is in stark contrast to Riemannian geometry where, vice versa, the connection is derived from the metric. Such tensorial objects are the two affine curvature tensors defined by. In a geometry with symmetric affine connection both tensors coincide because of.
In particular, in Riemannian geometry , both affine curvature tensors reduce to the one and only Riemann curvature tensor. The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity :. The curvature tensor 22 satisfies two algebraic identities:. From both affine curvature tensors we may form two different tensorial traces each. In the first case , and. V kl is called homothetic curvature , while K jk is the first of the two affine generalisations from and of the Ricci tensor in Riemannian geometry.
We get. While V kl is antisymmetric, K jk has both tensorial symmetric and antisymmetric parts:. We use the notation in order to exclude the index k from the symmetrisation bracket. In order to shorten the presentation of affine geometry, we refrain from listing the corresponding set of equations for the other affine curvature tensor cf.
For a symmetric affine connection, the preceding results reduce considerably due to. From Equations 29 , 30 , 32 we obtain the identities:. For the antisymmetric part of the Ricci tensor holds. This equation will be important for the physical interpretation of affine geometry.
It may be desirable to derive the metric from a Lagrangian; then the simplest scalar density that could be used as such is given by det K ij. As a final result in this section, we give the curvature tensor calculated from the connection cf. Equation 20 , expressed by the curvature tensor of and by the tensor :. A manifold carrying both structural elements, i. If the first fundamental form is taken to be asymmetric , i.
Equation It depends on the physical interpretation, i. Thus, in metric-affine and in mixed geometry, two different connections arise in a natural way. With the help of the symmetric affine connection, we may define the tensor of non-metricity by.
The inner product of two tangent vectors A i , B k is not conserved under parallel transport of the vectors along X l if the non-metricity tensor does not vanish:. Thomas introduced a combination of the terms appearing in and to define a covariant derivative for the metric [ ], p. We will have to deal with Equation 47 in Section 6. Connections that are not metric-compatible have been used in unified field theory right from the beginning.
In case of such a relationship, the geometry is called semi-metrical [ , ]. We may also abbreviate the last term in the identity 42 by introducing. Then, from Equation 39 , the curvature tensor of a torsionless affine space is given by. Riemann-Cartan geometry is the subcase of a metric-affine geometry in which the metric-compatible connection contains torsion, i.
A linear connection whose antisymmetric part has the form. Riemannian geometry is the further subcase with vanishing torsion of a metric-affine geometry with metric-compatible connection. In this case, the connection is derived from the metric: , where is the usual Christoffel symbol The covariant derivative of A with respect to the Levi-Civita connection is abbreviated by A ; k.
The Riemann curvature tensor is denoted. An especially simple case of a Riemanian space is Minkowski space, the curvature of which vanishes:. However, in metric affine and in mixed geometry geodesic and autoparallel curves will have to be distinguished. As a consequence, the Riemann curvature tensor is also changed; if, however, can be reached by a conformal transformation, then the corresponding spacetime is called conformally flat.
Even before Weyl, the question had been asked and answered as to what extent the conformal and the projective structures were determining the geometry: According to Kretschmann and then to Weyl they fix the metric up to a constant factor [ ]; see also [ ], Appendix 1; for a modern approach, cf.
The geometry needed for the pre- and non-relativistic approaches to unified field theory will have to be dealt with separately. There, the metric tensor of space is Euclidean and not of full rank; time is described by help of a linear form Newton-Cartan geometry, cf. In the following we shall deal only with relativistic unified field theories.
The connection to the inhomogeneous coordinates x i is given by homogeneous functions of degree zero, e. Thus, the themselves form the components of a tangent vector. Furthermore, the quadratic form is adopted with being a homogeneous function of degree A tensor field cf. If we define , with , then transforms like a tangent vector under point transformations of the x i , and as a covariant vector under homogeneous transformations of the.
The may be used to relate covariant vectors a i and by. Thus, the metric tensor in the space of homogeneous coordinates and the metric tensor of M D are related by with. The inverse relationship is given by with. The covariant derivative for tensor fields in the space of homogeneous coordinates is defined as before cf. The covariant derivative of the quantity interconnecting both spaces is given by.
Cartan introduces one-forms by. The reciprocal basis in tangent space is given by. The metric is then given by. We have. The link to the components of the affine connection is given by. In Equation 65 the curvatureform appears, which is given by. Up to here, no definitions of a tensor and a tensor field were given: A tensor T p M D of type r , s at a point p on the manifold M D is a multi-linear function on the Cartesian product of r cotangent- and s tangent spaces in p.
A tensor field is the assignment of a tensor to each point of M D. Usually, this definition is stated as a linear, homogeneous transformation law for the tensor components in local coordinates:. Strictly speaking, tensors are representations of the abstract group at a point on the manifold. Note that. The dual to a 2-form skew-symmetric tensor then is defined by. In connection with conformal transformations , the concept of the gauge-weight of a tensor is introduced. A tensor is said to be of gauge weight q if it transforms by Equation 56 as.
Objects that transform as in Equation 67 but with respect to a subgroup, e. All the subgroups mentioned are Lie -groups, i. Spinors are representations of the Lorentz group only; as such they are related strictly to the tangent space of the space-time manifold. To see how spinor representations can be obtained, we must use the homomorphism of the group SL 2,C and the proper orthochronous Lorentz group, a subgroup of the full Lorentz group. By picking the special Hermitian matrix.
The link between the representation of a Lorentz transformation L ik in space-time and the unimodular matrix A mapping spin space cf. The spinor is called elementary if it transforms under a Lorentz-transformation as. Covariant and covariant dotted 2-spinors correspondingly transform with the inverse matrices,. Higher-order spinors with dotted and undotted indices transform correspondingly. Next to a spinor, bispinors of the form , etc.
Often the quantity is introduced. The reciprocal matrix is defined by. The simplest spinorial equation is the Weyl equation:. The Dirac equation is in 4-spinor formalism [ 53 , 54 ]:. The group of coordinate transformations acts on the Latin indices.
For the example of the Dirac spinor, the adjoint representation of the Lorentz group must be used. In Section 2. With its help we may formulate the concept of isometries of a manifold, i. If a group G r is prescribed, e. A Riemannian space is called locally stationary if it admits a timelike Killing vector; it is called locally static if this Killing vector is hypersurface orthogonal. In purely affine spaces, another type of symmetry may be defined: ; they are called affine motions [ ].
Within a particular geometry, usually various options for the dynamics of the fields field equations, in particular as following from a Lagrangian exist as well as different possibilities for the identification of physical observables with the mathematical objects of the formalism. Thus, in general relativity, the field equations are derived from the Lagrangian. This Lagrangian leads to the well-known field equations of general relativity,.
In empty space, i. If only an electromagnetic field derived from the 4-vector potential A k is present in the energy-momentum tensor, then the Einstein. Maxwell equations follow:. The components of the metrical tensor are identified with gravitational potentials. The equations of motion of material particles should follow, in principle, from Equation 92 through the relation.
For point particles, due to the singularities appearing, in general this is a tricky task, up to now solved only approximately. However, the world lines for point particles falling freely in the gravitational field are, by definition, the geodesics of the Riemannian metric. This definition is consistent with the rigourous derivation of the geodesic equation for non-interacting dust particles in a fluid matter description. It is also consistent with all observations. For most of the unified field theories to be discussed in the following, such identifications were made on internal, structural reasons, as no link-up to empirical data was possible.
Due to the inherent wealth of constructive possibilities, unified field theory never would have come off the ground proper as a physical theory even if all the necessary formal requirements could have been satisfied. The latter choice obtains likewise in a purely affine theory in which the metric is a derived secondary concept.
In this case, among the many possible choices for the metric, one may take it proportional to the variational derivative of the Lagrangian with respect to the symmetric part of the Ricci tensor. This does neither guarantee the proper signature of the metric nor its full rank. Several identifications for the electromagnetic 4-potential and the electric current vector density have also been suggested cf. Complex fields may also be introduced on a real manifold. Such fields have also been used for the construction of unified field theories, although mostly after the period dealt with here cf.
Part II, in preparation. In particular, manifolds with a complex fundamental form were studied, e. Also, geometries based on Hermitian forms were studied [ ]. In later periods, hypercomplex numbers, quaternions, and octonions also were used as basic number fields for gravitational or unified theories cf. Part II, forthcoming. In place of the real numbers, by which the concept of manifold has been defined so far, we could take other number fields and thus arrive, e.
In this part of the article we do not need to take into account this generalisation. In most of the cases, the additional dimensions were taken to be spacelike; nevertheless, manifolds with more than one direction of time also have been studied. In his letter to Einstein of 11 November , he writes [ ], Doc. Perhaps, you are luckier in the search. I am totally convinced that in the end all field quantities will look alike in essence.
But it is easier to suspect something than to discover it. Various reasons instilled in me strong reservations: […] your other remarks are interesting in themselves and new to me. Ishiwara, and G. The result is contained in Hilbert , p. The hints dropped by you on your postcards bring me to expect the greatest. According to him, the deviation from the Minkowski metric is due to the electromagnetic field tensor:.
He claims to obtain the same value for the perihelion shift of Mercury as Einstein [ ], p. The meeting was amicable. In this context, we must also keep in mind that the generalisation of the metric tensor toward asymmetry or complex values was more or less synchronous with the development of Finsler geometry [ ]. Although Finsler himself did not apply his geometry to physics it soon became used in attempts at the unification of gravitation and electromagnetism [ ].
The idea that they keep together the dispersing electrical charges lies close at hand. Thus, the idea of a program for building the extended constituents of matter from the fields the source of which they are, was very much alive around Naturforscherversammlung, 19—25 September [ ] showed that not everybody was a believer in it. He claimed that in bodies smaller than those carrying the elementary charge electrons , an electric field could not be measured.
I wish to see this reason in the fact that it is altogether not permitted to describe the electromagnetic field in the interior of an electron as a continuous space function. The electrical field is defined as the force on a charged test particle, and if no smaller test particles exist than the electron vice versa the nucleus , the concept of electrical field at a certain point in the interior of the electron — with which all continuum theories are working — seems to be an empty fiction, because there are no arbitrarily small measures.
Einstein whether he approves of the opinion that a solution of the problem of matter may be expected only from a modification of our perception of space perhaps also of time and of electricity in the sense of atomism, or whether he thinks that the mentioned reservations are unconvincing and is of the opinion that the fundaments of continuum theory must be upheld. If, in a certain stage of scientific investigation, it is seen that a concept can no longer be linked with a certain event, there is a choice to let the concept go, or to keep it; in the latter case, we are forced to replace the system of relations among concepts and events by a more complicated one.
The same alternative obtains with respect to the concepts of timeand space-distances. In my opinion, an answer can be given only under the aspect of feasibility; the outcome appears dubious to me. But a more precise reasoning shows that in this way no reasonable world function is obtained. It is to be noted that Weyl, at the end of , already had given up on a possible field theory of matter:. To me, field physics no longer appears as the key to reality; in contrary, the field, the ether, for me simply is the totally powerless transmitter of causations, yet matter is a reality beyond the field and causes its states.
Klein on 28 December , see [ ], p. Yet it retains part of its meaning also with regard to questions concerning the constitution of elementary particles. Because one may try to ascribe to these field concepts […] a physical meaning even if a description of the electrical elementary particles which constitute matter is to be made.
Only success can decide whether such a procedure finds its justification […]. During the twenties Einstein changed his mind and looked for solutions of his field equations which were everywhere regular to represent matter particles:. Let us move into the field chosen by him without too much surprise to see him apparently follow a road opposed to the one successfully walked by the contemporary physicists. After , Einstein first was busy with extracting mathematical and physical consequences from general relativity Hamiltonian, exact solutions, the energy conservation law, cosmology, gravitational waves.
Thus, while lengths of vectors at different points can be compared without a connection, directions cannot. This seemed too special an assumption to Weyl for a genuine infinitesimal geometry:. A metrical relationship from point to point will only then be infused into [the manifold] if a principle for carrying the unit of length from one point to its infinitesimal neighbours is given. In contrast to this, Riemann made the much stronger assumption that line elements may be compared not only at the same place but also at two arbitrary places at a finite distance.
At a point, Equation 98 induces a local recalibration of lengths l while preserving angles, i. If, as Weyl does, the connection is assumed to be symmetric i. With regard to the gauge transformations 98 , remains invariant. From the 1-form dQ , by exterior derivation a gauge-invariant 2-form with follows. Let us now look at what happens to parallel transport of a length, e. If X is taken to be tangent to C , i. The same holds for the angle between two tangent vectors in a point cf.
Yet, also today, the circumstances are such that our trees do not grow into the sky. Due to the additional group of gauge transformations, it is useful to introduce the new concept of gauge-weight within tensor calculus as in Section 2. Weyl did calculate the curvature tensor formed from his connection but did not get the correct result ; it is given by Schouten [ ], p. His Lagrangian is given by , where the invariants are defined by. Weyl had arranged that the page proofs be sent to Einstein.
In communicating this on 1 March , he also stated that. In the most general case, the equations will be of 4th order, though. He then asked whether Einstein would be willing to communicate a paper on this new unified theory to the Berlin Academy [ ], Volume 8B , Document , pp. Einstein was impressed: In April , he wrote four letters and two postcards to Weyl on his new unified field theory — with a tone varying between praise and criticism. His first response of 6 April on a postcard was enthusiastic:.
It is a stroke of genious of first rank. Nevertheless, up to now I was not able to do away with my objection concerning the scale. However, as long as measurements are made with infinitesimally small rigid rulers and clocks, there is no indeterminacy in the metric as Weyl would have it : Proper time can be measured.
As a consequence follows: If in nature length and time would depend on the pre-history of the measuring instrument, then no uniquely defined frequencies of the spectral lines of a chemical element could exist, i. He concluded with the words. Only for a vanishing electromagnetic field does this objection not hold. Only in a static gravitational field, and in the absence of electromagnetic fields, does this hold:.
Einstein saw the problem, then unsolved within his general relativity, that Weyl alluded to, i. Presumably, such a theory would have to include microphysics. But I find: If the ds , as measured by a clock or a ruler , is something independent of pre-history, construction and the material, then this invariant as such must also play a fundamental role in theory.
Yet, if the manner in which nature really behaves would be otherwise, then spectral lines and well-defined chemical elements would not exist. Another famous theoretician who could not side with Weyl was H. However, Weyl still believed in the physical value of his theory. There exists an intensive correspondence between Einstein and Weyl, now completely available in volume 8 of the Collected Papers of Einstein [ ].
We subsume some of the relevant discussions. Weyl did not give in:. Einstein then suggested the affine group as the more general setting for a generalisation of Riemannian geometry [ ], Vol. In particular, it is unimportant which value of the integral is assigned to their world line.
Otherwise, sodium atoms and electrons of all sizes would exist. But if the relative size of rigid bodies does not depend on past history, then a measurable distance between two neighbouring world-points exists. As far as I can see, there is not a single physical reason for it being valid for the gravitational field.
The gravitational field equations will be of fourth order, against which speaks all experience until now […]. The quadratic form Rg ik dx i dx k is an absolute invariant, i. If this expression would be taken as the measurable distance in place of ds , then.
A very small change of the measuring path would strongly influence the integral of the square root of this quantity. Einstein added:. The last remarks are interesting for the way in which Einstein imagined a successful unified field theory. In the same way in which Mie glued to his consequential electrodynamics a gravitation which was not organically linked to it, Einstein glued to his consequential gravitation an electrodynamics i.
You establish a real unity. Understandably, no comments about the physics are given [ ], pp. Of course, as he noted, no progress had been made with regard to the explanation of the constituents of matter; on the one hand because the differential equations were too complicated to be solved, on the other because the observed mass difference between the elementary particles with positive and negative electrical charge remained unexplained.
In his general remarks about this problem at the very end of his article, Pauli points to a link of the asymmetry with time-reflection symmetry see [ ], pp. Now as before I believe that one must look for such an overdetermination by differential equations that the solutions no longer have the character of a continuum.
But how? The idea of gauging lengths independently at different events was the central theme. This must also give rise to an identity; and it is found that the new identity expresses the law of conservation of electric charge. Section 4. As he had abandoned the idea of describing matter as a classical field theory since , the linking of the electromagnetic field via the gauge idea could only be done through the matter variables.
In October , in the preface for the first American printing of the English translation of the fourth edition of his book Space, Time, Matter from , Weyl clearly expressed that he had given up only the particular idea of a link between the electromagnetic field and the local calibration of length:.
This attempt has failed. Weyl himself continued to develop the dynamics of his theory. As an equivalent Lagrangian Weyl gave, up to a divergence. Due to his constraint, Weyl had navigated around another problem, i. In the paper in , he changed his Lagrangian slightly into. The changes, which Weyl had introduced in the 4th edition of his book [ ], and which, according to him, were of fundamental importance for the understanding of relativity theory, were discussed by him in a further paper [ ].
His colleague in Vienna, Wirtinger , had helped him in this. If J has gauge-weight -1, then Jg ik is such a metric. In order to reduce the new theory to general relativity, in addition only the differential equation. More important, however, for later work was the gauge invariant tensor calculus by a fellow of St. Newman [ ]. In this calculus, tensor equations preserve their form both under a change of coordinates and a change of gauge.
Newman applied his scheme to a variational principle with Lagrangian K 2 and concluded:. We shall discuss these topics in Part II of this article. What is now called Kaluza-Klein theory in the physics community is a mixture of quite different contributions by both scientists. But you understand that, in view of the existing factual concerns, I cannot take sides as planned originally.
Kaluza did not normalize the Killing vector to a constant, i. Of the 15 components of , five had to get a new physical interpretation, i. The component g 55 turned out to be a scalar gravitational potential which, in the static case, satisfies the equation.
The Lorentz force appears augmented by an additional term containing g 55 of the order which thus may be neglected. For him, any theory claiming universal validity was endangered by quantum theory, anyway. The remaining covariance group G 5 is given by. The objects transforming properly under are: the scalar , the vector-potential , and the projected metric.
Klein identified the group; however, he did not comment on the fact that now further invariants are available for a Lagrangian, but started right away from the Ricci scalar of M 5 [ ]. The group G 5 is isomorphic to the group H 5 of transformations for five homogeneous coordinates with homogeneous functions of degree 1. I value your approach more than the one followed by H. If you wish, I will present your paper to the Academy after all. The negative result of his own paper, i. His motivation went beyond the unification of gravitation and electromagnetism:.
Clearly, the non-Maxwellian binding forces which hold together an electron. In the first, shorter, part of two, Eddington describes affine geometry; in the second he relates mathematical objects to physical variables. He starts by calculating both the curvature and Ricci tensors from the symmetric connection according to Equation By this, Eddington claims to guarantee charge conservation:.
However, for a tensor density, due to Equation 16 we obtain. Who shall say what is the ordinary gauge inside the electron? Only connections leading to a Lorentz metric can be used if a physical interpretation is wanted. Thus, in general, g kl is not metric-compatible; in order to make it such, we are led to the differential equations for , an equation not considered by Eddington.
This is due to the expression for the inverse of the metric, a function cubic in R kl. Note also that Eddington does not explicitly say how to obtain the contravariant form of the electromagnetic field F ij from F ij ; we must assume that he thought of raising indices with the complicated inverse metric tensor.
Now, Eddington was able to identify the energy-momentum tensor T ik of the electromagnetic field by decomposing the Ricci tensor K ij formed from Equation 51 into a metric part R ik and the rest. His aim was reached in the sense that all three quantities were fixed entirely by the connection; they could no longer be given from the outside. If so, then it must be a purely phenomenological one without any recourse to the nature of the charged elementary particles cf.
At first, Einstein seems to have been reserved cf. To Bohr, Einstein wrote from Singapore on 11 January Eddington has come closer to the truth than Weyl. Like Eddington, Einstein used a symmetric connection and wrote down the equation. By this, the metric was defined as the symmetric part of the Ricci tensor.
Due to. Let us note, however, that while transforms inhomogeneously, its transformation law. For a Lagrangian, Einstein used ; he claims that for vanishing electromagnetic field the vacuum field equations of general relativity, with the cosmological term included, hold. If , then the electric current density j l is defined by.
The field equations are obtained from the Lagrangian by variation with regard to the connection and are Einstein worked in space-time. From Equation the connection can be obtained. This equation is an identity if a solution of the field equations is inserted. From Equation ,. In order that this makes sense, the identifications in Equation are always to be made after the variation of the Lagrangian is performed.
For non-vanishing electromagnetic field, due to Equation the Equation now becomes. Einstein concluded:. Except for singular positions, the current density is practically vanishing. Up to the same order,. In general however,. Also, the geometrical theory presented here is energetically closed, i. His final conclusion was:. Until the end of May , two further publications followed in which Einstein elaborated on the theory.
In the second paper, he exchanged the Lagrangian for a new one, i. The resulting equations for the gravitational and electromagnetic fields are the symmetric and skew-symmetric part, respectively, of. Although the theory offered, for every solution with positive charge, also a solution with negative charge, the masses in the two cases were the same. However, the only known particle with positive charge at the time what is now called the proton had a mass greatly different from the particle with negative charge, the electron.
Einstein noted:. The logic of the subsequent derivations in his paper is quite involved. The first step consisted in the definition of tensor densities. By using both Equation and Equation , Einstein obtained the Einstein. After a field rescaling, he then took a third expression to become his Lagrangian. Nobody can determine empirically an affine connection for vectors at neighbouring points if he has not obtained the line element before.
He criticised a theory that keeps only the connection as a fundamental building block for its lack of a guarantee that it would also house the conformal structure light cone structure. This is needed for special relativity to be incorporated in some sense, and thus must be an independent fundamental input [ ].
From a recent conversation with Einstein I learn that he is of much the same opinion. His outlook on the state of the theory now was rather bleak:. To me, the quantum-problem seems to require something like a special scalar, for the introduction of which I have found a plausible way. But I fail to succeed in giving my pet idea a tangible form: to understand the quantum-structure through an overdetermination by differential equations. The initial state of an electron moving around a hydrogen nucleus cannot be chosen freely; its choice must correspond to the quantum conditions.
In general: not only the evolution in time but also the initial state obey laws. He then ventured the hope that a system of overdetermined differential equations is able to determine. One of the crucial tests for an acceptable unified field theory for him now was:. In such a way, the un-ambiguity of the initial conditions ought to be understood without leaving field theory. In the introduction to his book, Struik distinguished three directions in the development of the theory of linear connections [ ]:.
The generalisation of parallel transport in the sense of Levi-Civita and Weyl. Schouten is the leading figure in this approach [ ]. Thomas [ ], and T. Thomas [ , ]. Here, only symmetric connections can appear.
The idea of mapping a manifold at one point to a manifold at a neighbouring point is central affine, conformal, projective mappings. In his assessment, Eisenhart [ ] adds to this all the geometries whose metric is.
Developments of this theory have been made by Finsler, Berwald, Synge, and J. In this geometry the paths are the shortest lines, and in that sense are a generalisation of geodesics. Affine properties of these spaces are obtained from a natural generalisation of the definition of Levi-Civita for Riemannian spaces. In fact, already in May Jan Arnoldus Schouten in Delft had submitted two papers classifying all possible connections [ , ]. In the first he wrote:. Weyl, Raum-Zeit-Materie , 2.
Section, Leipzig 3. The most general connection is characterised by two fields of third degree, one tensor field of second degree, and a vector field […]. The fields referred to are the torsion tensor S ij k , the tensor of non-metricity Q ij k , the metric g ij , and the tensor C ij k which, in unified field theory, was rarely used. It arose because Schouten introduced different linear connections for tangent vectors and linear forms. He defined the covariant derivative of a 1-form not by the connection L ij k in Equation 13 , but by.
In fact. Furthermore, on p. For such an extension an invariant fixing of the connection is needed, because a physical phenomenon can correspond only to an invariant expression. According to Schouten. In the following pages will be shown that this difficulty disappears when the more general supposition is made that the original deplacement is not necessarily symmetrical.
He then restricted the generality of his approach; in modern parlance, he did allow for vector torsion only:. On the same topic, Schouten wrote a paper with Friedman in Leningrad [ ]. He relied on the curvature, torsion and homothetic curvature 2-forms [ 32 ], Section III; cf.
The spreading of knowledge about properties of differential geometric objects like connection and curvature took time, however, even in Leningrad. After an uninterrupted search during the past two years I now believe to have found the true solution. After some manipulations, the variation with regard to the metric and to the connection led to the following equations:. However, he cautioned:. The process of generalisation consists in abandoning assumptions of symmetry and in adopting a definition of covariant differentiation which is not the usual one, but which reduces to the usual one in case the connection is symmetric.
The two covariant derivatives introduced by J. Thomas are and. Thomas then could reformulate Equation in the form. Toward the end of the paper Einstein discussed time-reversal; according to him, by it the sign of the magnetic field is changed, while the sign of the electric field vector is left unchanged. As he wanted to obtain charge-symmetric solutions from his equations, Einstein now proposed to change the roles of the magnetic fields and the electric fields in the electromagnetic field tensor.
He went on to say:. This is surely a magnificent possibility which likely corresponds to reality. The question now is whether this field theory is consistent with the existence of quanta and atoms. In the macroscopic realm, I do not doubt its correctness.
Yet, in the end, also this novel approach did not convince Einstein. Soon after the publication discussed, he found his argument concerning charge symmetric solutions not to be helpful. The link between the occurrence of solutions with both signs of the charge with time-symmetry of the field equations induced him to doubt, if only for a moment, whether the endeavour of unifying electricity and gravitation made sense at all:. In this, electrodynamics is basically different from gravitation; therefore, the endeavour to melt electrodynamics with the law of gravitation into one unity, to me no longer seems to be justified.
First , the attempts of all of us were directed to arrive, along the path taken by Weyl and Eddington or a similar one, at a theory melting into a formal unity the gravitational and electromagnetic fields; but by lasting failure I now have laboured to convince myself that truth cannot be approached along this path. The new field equation was picked up by R.
In the same spirit as the one of his paper, Einstein said good bye to his theory in a letter to Besso on Christmas in words similar to those in his letter in June:. Anyway, I now am convinced that, unfortunately, nothing can be made with the complex of ideas by Weyl-Eddington. The equations. I take as the best we have nowadays. But it appears doubtful whether there is room in them for the quanta.
It does not allow for electrical masses free from singularities. Moreover, I cannot bring myself to gluing together two items as the l. Research on affine geometry as a frame for unified field theory was also carried on by mathematicians of the Princeton school. Thus J. We met J. During the period considered here, a few physicists followed the path of Eddington and Einstein.
He showed that, in first approximation, he got what is wanted, i. Three months later, Infeld published a note in Comptes Rendus of the Parisian Academy in which he now presented the exact connection as. Thus, he is back at vector torsion treated before by Schouten [ ]. The Japanese physicist Hattori embarked on a metric-affine geometry derived purely from an asymmetric metrical tensor.
He defined an affine connection. The electromagnetic field was not identified with f ik by Hattori, but with the skew-symmetric part of the generalised Ricci tensor formed from. By introducing the tensor , he could write the generalised Ricci tensor as. F ikl is formed from F ik as f ikl from f ik. The proper world geometry which ought to lead to a unified theory of gravitation and electricity can only be found by an investigation of its physical content.
Infeld could as well have applied this admonishment to his own unified field theory discussed above. Thus, Straneo suggested a unified field theory with only vector torsion as Schouten had done 8 years earlier [ , ] without referring to him. The field equations Straneo wrote down, i. Straneo wrote further papers on the subject [ , ].
By this, he claimed to have made superfluous the five-vectors of Einstein and Mayer [ ]. This must be read in the sense that he could obtain the Einstein. Mayer equations from his formalism without introducing a connecting quantity leading from the space of 5-vectors to space-time [ ].
Einstein, in his papers, did not comment on the missing metric compatibility in his theory and its physical meaning. In this work a generalisation of the equation for metric compatibility, i. The continuation of this research line will be presented in Part II of this article. Lorentz, 16 February On the next day 17 February , and ten days later Einstein was to give papers of his own in front of the Prussian Academy in which he pointed out the gauge-group, wrote down the geodesic equation, and derived exactly the Einstein.
Maxwell equations — not just in first order as Kaluza had done [ 81 , 82 ]. He came too late: Klein had already shown the same before [ ]. Einstein himself acknowledged indirectly that his two notes in the report of the Berlin Academy did not contain any new material. In his second communication, he added a postscript:. Mandel brings to my attention that the results reported by me here are not new. The entire content can be found in the paper by O. That Klein had published another important clarifying note in Nature , in which he closed the fifth dimension, seems to have escaped Einstein [ ].
Maxwell equations [ ], p. Mandel of Leningrad was not given credit by Einstein although he also had rediscovered by a different method some of O. From the geodesics in M 5 he derived the equations of motion of a charged point particle. One of the two additional terms appearing besides the Lorentz force could be removed by a weakness assumption; as to the second, Mandel opinioned.
Fock derived the general relativistic wave equation and the equations of motion of a charged point particle; the latter is identified with the null geodesics of M 5. A main motivation for Klein was to relate the fifth dimension with quantum physics. From a postulated five-dimensional wave equation. By this, the reduction of five-dimensional equations as e. Klein had only the lowest term in the series. Beyond incredibly complicated field equations nothing much had been gained [ ].
Even L. Klein and Einstein. He then discussed conformally invariant field equations, and tried to relate them to equations of wave mechanics [ ]. In , N. Presently, the different contributions of Kaluza and O. An early criticism of this unhistorical attitude has been voiced in [ ]. Indices are raised and lowered with the metrics of V 5 or V 4 , respectively. A consequence then is. Both covariant derivatives are abbreviated by the same symbol A ; k.
The covariant derivative of tensors with both indices referring to V 5 and those referring to V 4 , is formed correspondingly. The autoparallels of V 5 lead to the exact equations of motion of a charged particle, not the geodesics of V 4. From them follows. They also noted that a symmetric tensor F kl could have been interpreted as the second fundamental form, and the formalism would then be the same as local isometric embedding of V 4 into V 5.
It is related to the Riemannian curvature of V 4 by. From , by transvection with , the 5-curvature itself appears:. By contraction, and. Two new quantities are introduced:. It turns out that. Also, in a lecture given on 14 October in the Physics Institute of the University of Wien, he still was proud of the 5-vector approach. However, following an idea half of which came from myself and half from my collaborator, Prof.
Mayer, a startlingly simple construction became successful. In this way, we succeeded to recognise the gravitational and electromagnetic fields as a logical unity. Electrical and mass-density are non-existent; here, splendour ends; perhaps this already belongs to the quantum problem, which up to now is unattainable from the point of view of field [theory] in the same way as relativity is from the point of view of quantum mechanics.
The witty point is the introduction of 5-vectors in fourdimensional space, which are bound to space by a linear mechanism. Let a s be the 4-vector belonging to ; then such a relation obtains. In the theory equations are meaningful which hold independently of the special relationship generated by.
Infinitesimal transport of in fourdimensional space is defined, likewise the corresponding 5-curvature from which spring the field equations. In his report for the Macy-Foundation, which appeared in Science on the very same day in October , Einstein had to be more optimistic:.
It furnishes, however, clues to a natural development, from which we may anticipate further developments in this direction. In any event, the results thus far obtained represent a definite advance in knowledge of the structure of physical space.
However, according to Pauli, Veblen and Hoffmann had spoiled the advantage of projective theory:. The five-dimensional space is just a mathematical device to represent the events points of space-time by these curves. Thus, Veblen and Hoffmann also gained the Klein. Gordon equation in curved space, i. Nevertheless, Hoffman remained optimistic:. In particular, we do not demand a relationship between electrical charge and a fifth coordinate; our theory is strictly four-dimensional.
Of the three basic assumptions of the previous paper, the second had to be given up. The expression in the middle of Equation is replaced by. The field equations were set up according to the method of the first paper; now the 5-curvature scalar was. It also turned out that with , i. In the last paragraph, the compatibility of the equations was proven, and at the end Cartan was acknowledged:. At about the same time as Einstein and Mayer wrote their second note, van Dantzig continued his work on projective geometry [ , , ].
Together with him, Schouten wrote a series of papers on projective geometry as the basis of a unified field theory [ , , , ] , which, according to Pauli, combine. In this paper [ ], p. The formalism of Schouten and van Dantzig allows for taking the additional dimension to be timelike; in their physical applications the metric of spacetime is taken as a Lorentz metric; torsion is also included in their geometry.
Pauli, with his student J. The authors pointed out that. In a sequel to this publication, Pauli and Solomon corrected an error:.
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