The Forgotten Genius of Physics, a work on Marian Smoluchowski

Introduction

The forgotten genius of physics, a work on Marian Smoluchowski is the story of the greatest Polish physicist, all but forgotten by his compatriots.

It seems ironic that the name Smoluchowski is quite popular in my hometown of Gdańsk, but not due to his scientific achievements or mountaineering successes. Smoluchowski had no association with Gdańsk, he was never there and nothing connects him to the city, though quite an important street is named after him in the Aniołki district. Naming the street after him was decided by something of an accident; after the war most of the street names that had German connotations were changed. It is not an average street, however, differentiated by its location, leading to the Medical University with thousands of people walking down it every day. The polish physicist’s surname therefore remains in the common awareness of residents, although the majority of them think Smoluchowski was an eminent doctor working at the Gdańsk Medical Academy (the university had that status for decades after the war).

Two streets lead to the Medical University of Gdańsk, the first – for years the more important – led directly to the university’s former main gate and bears the name of Maria Skłodowska-Curie. A new, bigger hospital building is currently being constructed on Marian Smoluchowski street running parallel and it is slowly becoming the more important street in the opinion of Gdańsk residents. There is a high probability that the name Smoluchowski will gain in popularity.

In post-war Poland, Maria Skłodowska-Curie was the first lady of Polish science, her personality overshadowing the achievements of many other scientists, including Smoluchowski. This state of affairs was undoubtedly influenced by the objective greatness of our countrywoman, but that was not the only reason for her dominance. The dissemination of information at the time was monopolized and the authorities decided both which names of pre-war science could enter into social consciousness and which were consigned to be forgotten. Changes started in the 1970s when talk started of the Lvov-Warsaw School of Polish philosophy started by Kazimierz Twardowski and a little later of the Lvov school of mathematics – a group of academics active in the two decades between the wars focused around Stefan Banach (1892–1945) and Hugo Steinhaus (1887–1972). In the 1980s we became acquainted with the names of Witold Gombrowicz and Czesław Miłosz. It was only in free Poland, and then not immediately, that talk started of the brilliant logician Alfred Tarski (1901–1983) and the mathematicians Stanisław Ulam and Marek Kac. But the philosopher, logician and methodologist Henryk Mehlberg (1904–1979) remains almost unknown in the country to this day. The partitions of the 19th century and several decades of the communist system in the 20th century caused many outstanding Poles to be ousted from the national memory.

Beyond a short period in the 1950s, Smoluchowski disappeared from general consciousness. The people deciding on the culture of socialist Poland were not interested in paying homage to the brilliant physicist’s memory, perhaps because an attempt to make Smoluchowski a scientific icon of materialism failed. The book in the Reader’s hands is intended to at least in part fill in this national lack of memory. Smoluchowski’s achievements in the field of physics, described herein, have been supplemented with many biographical elements showing the extremely valuable personality of a man with broad interests. Research materials have been used presenting Smoluchowski as a pedagogue, scientist and father of a family, which were quite numerously published immediately after the physicist’s sudden death. Many parts have been devoted to Marian Smoluchowski’s mountaineering hobby, which he pursued with great love together with his brother Tadeusz, achieving many successes in this field and writing himself permanently into the history of European mountaineering.

Of interest to a Reader curious about the physicist’s non- scientific passions may be the Vernissage included at the end of the book, in which more than 30 of Marian Smoluchowski’s watercolours are presented. The graphic side is also an important aspect of the study – the book contains several dozen photos illustrating the life of Smoluchowski, his family and people connected with him. These will certainly help to bring the Reader closer to the scientist and the times in which he lived.

The monograph’s author wanted to reach a wider readership, especially the young. They should learn of the Polish scientist, whose famous achievements – beyond physics – also gained recognition in the realms of the philosophy of science and the philosophy of nature. This work is also intended to show the unusually colourful personality of Marian Smoluchowski, a man of the renaissance with many interests to which he devoted his life.

The forgotten genius of physics, a work on Marian Smoluchowski is intended as a popular science book. Readers interested in deeper and wider analyses of physics, and especially of Smoluchowski’s philosophy, are invited by the author to read Between physics and philosophy. The philosophy of nature and physics in the writings of Marian Smoluchowski, published in 2020 by Towarzystwo Autorów i Wydawców Prac Naukowych “Universitas”. This publication uses fragments of scientific deliberations from that work.

An accident in the common understanding is a random, unplanned event. For the last few years of his life, Smoluchowski studied the essence and role of chance, a fundamental factor in accidents. He wondered what chance was, could it be researched scientifically? He reduced the essence of chance to an incomplete knowledge of the laws functioning in nature or to an ignorance of all causes at play. The line of research the Pole set out had an influence on the emergence of a serious branch of mathematics – statistical physics, which deals with probability in research in physics.

Smoluchowski thought that by treating chance as a negation of regularity certain contradictions appear that are a source of dilemmas and, moreover, such an understanding of chance cannot be reconciled with the generally prevailing determinism. From a deterministic position, cause and effect are perceived as a constant necessity. We can talk about a lack of necessity – Smoluchowski argued – in a relative sense, in so far as necessity is not externally recognisable. Despite the evident causal relationship between cause and effect, the nature of this relationship is unknowable since the phenomenon itself is too complex, hence the impression of a seeming break with regularity². Determinism demands cause and effect to be treated as events related to one another through internal relationships of necessity and so a visible lack of necessity is an apparent phenomenon resulting from the fact that part of the cause is unknowable³. Chance is then defined as a hidden causal relationship existing between cause and effect.

In overly complex phenomena, chance manifests as an apparent break with regularity. Understood in this way, chance was scientifically unacceptable because the probability of an event occurring can depend exclusively on the conditions acting upon it, rather than on the extent of our knowledge.⁴ Smoluchowski therefore postulated removing the subject from the concept of chance, which would result in its objectification. Strict natural science is interested not in subjective statements and presumptions but in objective or mathematical probability, i.e. the relative frequency of occurrence of the random events in question. In this narrower sense, the concept of probability becomes accessible in strictly mathematical terms⁵

A similar problem was presented by the Polish physicist in his work, Notes on the Concept of Chance in Physical Phenomena. Smoluchowski posits the thesis that laws of nature are a subject of physics and the assumption that causality and determinism should represent an antithesis to randomness is wrong. It should be considered how chance can arise in phenomena proceeding in accordance with the immutable laws of nature. How can physics describe it with the aid of deterministic laws and is there a place for it in nature governed by those laws? If a truly unpredictable chance event, negating causal regularity, plays a certain role in physical phenomena, how can the correct course of these phenomena be predicted?⁶

Smoluchowski recognised that finding answers to such questions depends on a correct understanding of chance, which in turn can become a factor in applying it to research on probability theory. Assuming that determinism is the fundamental concept for conducting deliberations in this field, he sought to define chance by moving away from its everyday understanding and giving it the status of a scientific notion and then performing further analyses – this time in the context of generally accepted determinism. In the article On the Concept of Chance and the Origin of the Laws of Probability in Physics, Smoluchowski conducts an analysis of the relation of chance to stable laws of physics. To the question as to which events fall within the scope of the application of probability theory, he answers that they are usually said to be events whose occurrence depends on chance.⁷

Science studies objective regularities and scientific laws must assume the existence of a causal relationship, which does not mean that the concept of cause can be replaced by the concept of a scientific law. Scientific laws not only clarify when, where and how something happens but they also try to answer the question why as a tool to explain events occurring in nature. Probabilistic statistical forecasts can however be an additional tool for explaining events.

Smoluchowski initially specified two conditions a phenomenon must meet in order to be called accidental. They are “small cause – great effect”⁸ and “different causes – same effects”⁹. The first condition states that a small change in the input parameters causes serious changes to the effect. A small change is understood in a relative sense – it is small compared to the possible range of changes. The second condition describes that fact that in the range of initial states there exist many configurations that lead to the same final state.

Chance plays an important role in physical phenomena and a scientific approach to it required the use of probability calculus methods. Although at the time they had not been sufficiently developed, they had to be used as researching chance through traditional means was ineffective as it went beyond the field of calculus methods.¹⁰ The use of probability calculus persuaded the researcher that to illustrate some phenomena in physics, the language of probability theory can be successfully used as a tool to describe events, which had earlier been questioned.

In the modern period, the application of elements of probability calculus has been quite popular, especially for the requirements of common gambling. Many enlightened minds, such as the mathematician Giovanni Francesco Peverone (1509–1559) or Galileo Galilei (1564–1642) himself considered gambling’s mathematical dilemmas¹¹. Girolamo Cardano (1501–1576) was the first mathematician to note that scientific rules exist governing probability and the simple throw of the dice can be viewed in terms of mathematical knowledge. He proved that the possibility of getting the elusive “double six” beyond mere luck or chance could be calculated mathematically. He included his thoughts in the book Liber de Ludo Aleae, probably written in 1563. However, in popular opinion, Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665) are considered to be the creators of modern probability theory. In the history of science there are few stories of important discoveries as spectacular as that of the role of Pascal and Fermat, and they are the most memorable. Pascal’s interest in stochastic processes was prompted by a chance encounter with the gambler Antoine Gombaud, known as “Chevalier de Méré” (1607–1684). He sought solutions to several dilemmas encountered in card and dice games which were de facto mathematical problems.

The classic problem of a game of chance, resolved by Pascal for Chevalier de Méré and entering the history of probability calculus, was a situation in which during a game of dice, betting on the appearance of a six once within four throws of the die, the gambler who bet against the occurrence – that a six would not appear – won more often than lost. Wanting to make the game more complicated so that players would not be aware of the system he was using, de Méré added another die and bet on a double six on the roll of two dice. He assumed that the extra die represented only six times more possibilities, so he would have to roll the two dice six more times. This reasoning proved false as after adding another die, he lost more often than he won. In his reasoning, he had used the arithmetic rule of three, which enables the calculation of a fourth value on the basis of the remaining three known values. Pascal found the solution to this problem. He asked Fermat, living in Toulouse, what he thought of his conception. He concluded his positive response with the famous statement: “I plainly see that the truth is the same at Toulouse and at Paris.”¹²

Another important figure that made a huge contribution to the development of probability calculus was Pierre Simon de Laplace (1749–1827). For the needs of his deliberations, he created a figure of omniscient intellect, which later became known in science as Laplace’s demon. It was a mind that at a given moment knew all the forces of nature and the current position of all the bodies making up the Universe and which would be sufficiently powerful to analyse the data and describe the movements of all the bodies in the Cosmos – from the heaviest to the lightest atoms. To such an intellect, nothing would be uncertain and it would see both the past and the future. According to this concept, “The curve described by a simple molecule of air or vapor is regulated in a manner just as certain as the planetary orbits”¹³.

According to Fernando Corbalána, Laplace was not talking about some omniscient mind. The author of The Taming of Chance believes that the French mathematician meant the demon metaphorically. It was intended to be a scientific method of calculating the probability of an event’s occurrence with the aid of probability calculus, enabling the prediction of nature’s behaviours and the learning of its laws. The essence of Laplace’s idea is a situation in which the demon knows the future.¹⁴ In probability calculus, this is an extreme chance, a state in which we foresee the occurrence of a given event with 100-percent certainty. In such a situation, the essence of the problem is missed. In writing about the demon, then, was Laplace thinking about calculus? It is possible, although according to Smoluchowski, “chance of that type is removed from all a priori calculation and as such can never be the basis for applying probability calculus. Because as long as we do not know the causes with sufficient precision (…), nothing at all can be foreseen regarding the outcome. But when we know, the result can be predicted with certainty so that there remains no place for probability”¹⁵

However, the demon’s powerful mind, which could determine all the consequences of the laws of nature, can be reduced to a calculus of probability developed to know these laws of nature as well as possible.¹⁶ However, even assuming the 100- percent efficiency of Laplace’s demon, that is assuming a situation in which we managed to know in physics with absolute accuracy the momentary state of the system of all atoms and calculate its changes, then we will find that the calculus of probability maintains its value¹⁷

According to Smoluchowski, even absolute knowledge of the initial state of all gas particles would not cancel out the usefulness of probability calculus to describe that system and therefore the utility of the concept of probability itself. The claim can be made, therefore, that for Maxwell’s demon¹⁸ the concept of probability calculus would be useful as a convenient tool to describe complex systems¹⁹. Despite the acceptance of the existence of a demon whose predictive abilities surpass those of a person, it must be assumed that he would not be able to determine all intermediate states without the help of the calculus of probability. Today, in the age of quantum physics, we know that Smoluchowski’s final suggestion proved correct. It is not possible to predict the movement of particles with mathematical certainty; it will always be hampered by the uncertainty of velocity (or position). Stephen Hawking notes that even predicting a specific set of positions and velocities ceases to be possible once the existence of black holes is taken into account.²⁰

Chance suited to the calculation of probability, so-called normalised chance, differs significantly from chance in the wider sense in that the effect shows a certain regularity through frequent repetition of the phenomenon regardless of the type of cause.²¹ Such normalised chance enables a predicted event to be included in an empirically verifiable mathematical formula.

In the search for solutions, Smoluchowski gives an example illustrating a situation in which the values enabling the application of probability calculus are known.

A shooter fires a fixed shotgun at a spinning circular disc divided into sectors and painted alternately black and white. Whether he hits a black or white sector depends on the moment he pulls the trigger. The disc can be spun so fast that certainty of firing a shot on target can be eliminated. At whatever moment the shooter pulls the trigger, the time that has elapsed from taking the decision to fire will vary within certain limits such that the probability of the shot occurring at time t is expressed by the function φ(t) (perceptibly different from zero in the range of t to t +​ τ), however for this function it must be assumed that there are no unique features. If there are many revolutions of the disc in the time range τ, the influence of the individual form that the function φ(t) could have disappears and the probability of hitting a white or black sector depends on the relative size of the fields²² Smoluchowski disagreed with Meinong’s thesis that “between a given cause and effect there exists a causal relationship but it is unknowable to us as the phenomenon is too complex. We are therefore dealing with an apparent break with regularity and chance is defined as ‘a partial cause unknown to us’”²³

He proved that ignorance of the partial cause is no obstacle to calculating probability. It is possible to calculate the effect of a “partial cause unknown to us” by invoking the law of large numbers, a rule that cannot be proven but which has shown itself to be empirically irrefutable. (…) The law of large numbers causes the irregularities wrought in the world by chance events to disappear in the overall result… our mind probably cannot reconcile itself with that in order to accept a similar principle simply because here and there its accuracy has been confirmed²⁴.

Probability is a mathematical concept, hence with every probabilistic theory comes the issue of its physical and operational interpretation. A frequency interpretation of probability is generally accepted. In line with this, in a set of randomly selected identical systems to which a given theory is applied, certain dynamically possible behaviours should occur with relative frequencies proportional to theoretical probabilities and this consistency is greater the more individual instances we take into consideration²⁵ We build probabilistic consistency on the basis of the law of large numbers, enabling physical and operational interpretation through the greatest possible use of individual instances.

1 Z. Klemensieicz, Marian Smoluchowski, “Mountaineer. A unit of the Tourist Section of the Polish Tatra Society,” Kraków, 1915–1921, p. 4 (photocopy without number).

2 M. Smoluchoski, On the Concept of Chance and the Origin of the Laws of Probability in Physics, “Wiadomości Matematyczne” (Mathematical News), 1923, volume 27, book 2, p. 29.

3 Ibidem.

4 Ibidem.

5 Ibidem.

6 Ibidem.