Is math or physics the ultimate truth

The role of mathematics in a science

Views of a physicist

“I maintain, however, that in any particular theory of nature there can only be found as much real science as there is mathematics in it”, wrote Immanuel Kant in his work in 1786 “Metaphysical Foundations of Natural Science, A VIII ”. At that time he was under the influence of Newtonian mechanics, which had shown unprecedented success in the explanation and prediction of all possible movements in the sky and on earth and thus had become the standard for a science of nature. And this "theory of nature" arose from the symbiosis of experiment and mathematics that Galileo conceived two generations before Newton. I consider this statement by Kant to be problematic and misleading, and I want to give my reasons for it here. At the same time, however, I also want to sing the high song of the mathematization of the sciences, only that I do not see this as a guarantee and the real cause of a scientific nature.

Science versus customer

Physics was already “born through mathematics” - precisely because of this it became something different from natural philosophy. Galileo investigated free fall, experimented with a gutter and measured the distances that a ball travels in the gutter at certain time intervals. He summarized his result in a mathematical equation: The distance increases quadratically with time. And he realized that this result is not due to the coincidence of a specific test arrangement, but rather represents a general law of nature. This relationship was now a "public" fact; it could be verified or used by anyone. Johannes Kepler also recognized such facts. From the very precise observations of Tycho Brahe, he distilled insights into the course of the planets around the sun, including that their orbits are ellipses, with the sun in one of its focal points.

So Galileo and Kepler formulated the laws of nature, and in fact in mathematical form, so that they can be checked at any time "tip on button". If you consider what prior knowledge, what preparatory work and what intellectual achievement is behind it, then we would definitely like to declare this as science. One can, however, ask whether a collection of facts, even if they come in mathematical form and can be checked quantitatively, so that “actually” can already constitute a science. In German we know the beautiful word “customer”. There is material science, geography and even natural history. These are the areas in which one has news of such regularities. And, depending on the area, these are formulated more or less mathematically.

A customer becomes a science if one can explain the accumulation of facts from a more abstract point of view, ideally inferred from more general principles in strict mathematical terms. The so-called Newtonian axioms, i.e. basic assumptions, lead to an equation for the case of the planetary orbits, from which all Kepler's laws can be strictly derived and of course also the relationship between distance and time in free fall, discovered by Galileo. As I have pointed out many times in my blog posts, a physical theory consists of a set of relationships between measurable quantities and a basic assumption from which all these relationships follow. The proprium of a science, the “essential”, is this hierarchical interweaving, the logical order of its statements - and not the mathematics.

So when you work scientifically, you move within this web of statements. If one is on the trail of a new relationship between measurable quantities, one has to discuss their possible relationship to this network, analyze how it fits with it and how it will enrich or modify it.

In addition to the consideration and elaboration of the networking of statements and facts, it is the formation of terms that is part of the “actual” scientific work. In physics, one knows the troubles one has had to establish concepts such as energy, entropy or the state of a quantum.

Advantages of mathematics as the language of a science

In all of this intellectual, scientific work, the language in which it is done naturally plays a major role. In physics, this language is mathematics. The subject area of ​​physics allows this. It was Galileo's achievement to recognize this.

In this language one has to take the relationships and relations between measurable quantities absolutely “seriously”. They are available in quantitative form and can be verified in this form. The building of a theory, the network of statements thus contains highly stable elements and their connection is just as reliable, since they can also be formulated as mathematical derivations. The whole structure of such a theory is thus extremely stable; Newtonian mechanics is the best example of this. It is not for nothing that it has been considered the ideal of a science for 200 years and has shaped the thinking habits of all those interested in science.

Language also guides our thinking, and so mathematics has also influenced the formation of concepts and even the formation of creative hypotheses in physics, as I further explained in the article: “The language of physics”.

Anyone who works in such a quantitatively arguing science has a vivid impression of the difference between qualitative and quantitative statements. In many cases one can blame several influences for one effect, but has no knowledge of how strong they are in each case. This cannot always be said from the outset from the known laws of nature. This is a very common situation where a lot of time is wasted on discussions where different opinions about the strength of each influence clash. If you manage to formulate a mathematical model based on these natural laws, which takes into account all known influences and passes all tests, then you have also determined the strength of each influence. Sometimes you wonder how wrong you were with your original ideas about the respective strength.

About the effect of the mathematization of physics on outsiders

If you want to study physics, you first have to go through a hard mathematical training course in the first semesters. Opinions are already divided, because higher mathematics is and remains for many a book with seven seals; I think there is a definite talent for mathematics as there is for music. Without mastering the mathematical language, however, one cannot gain an understanding of physical theories. You can tell more or less aptly about the equations, but that is no substitute for dealing with them personally. Anyone who has ever carried out the calculations of the planetary orbits themselves, who has once deduced from the Maxwell equations that there must be electromagnetic waves, who has calculated the energy levels of an electron in the hydrogen atom from the Schrödinger equation, is extremely impressed how powerful and reliable this method of knowledge is.

The outsider cannot empathize with the experience and the conviction that goes with it, even the way to this experience is closed to him. He often feels a sense of discomfort: There is an important development with consequences that affect his view of the world, but which he cannot control. Although he cannot judge the fruits of other sciences either, these do not intervene so strongly in his ideas of the possibilities of our knowledge and of the “ultimate things” of this world.

The experience naturally also shapes the worldview and the general view of science in general. This is how physicists, yes natural scientists in general, mostly become naturalists: Everything in nature “works with the right things”, i.e. everything in nature will proceed according to certain laws which, if not already known, can be formulated at some point. On the other hand: Anyone who has found treasures deep in the ground is in danger of despising earthworms and forgetting that the soil is not always made in such a way that deep digging is possible. The physicist sometimes appears arrogant and his expectations of the method, no matter how impressive the successes to date, are often set too high. Just think of the time when classical mechanics celebrated its greatest triumphs and it was believed that everything would soon be understood in terms of classical mechanics. They were right that we are learning to understand nature better and better with the method initiated by Galileo - namely, formulating hypotheses in the language of mathematics and testing them through experiments. But they were wrong about the speed with which this would happen and had no idea what prejudices would have to be thrown overboard in the future and what new terms and ideas would result.

Problems and limits

Those who study physics first learn to use the language of mathematics to solve simple physics problems. The so-called two-body problem - the situation where a single planet orbits the sun - is a prominent example of this. So one disregards the other planets, one can estimate that the influence of this on the orbit to be calculated would be only very weak. But if you also want to take into account the influences of the other planets, you are dealing with a multi-body problem, and mathematically that is much more difficult. In general, one makes the experience in mechanics, in which the movement of material bodies always plays a role, that with regard to the number of particles involved, only the extreme cases are somewhat mathematically under control, i.e. problems with very few and those with so many Particles, so that the individual trajectories of the particles no longer matter. What one wants to know about the system of so many particles can then be derived from a statistical distribution of the positions and speeds of the particles. In statistical mechanics one learns how one arrives at plausible assumptions about such distributions, how useful and measurable quantities for the description of the systems result from them and how one finally arrives at verifiable relationships between these quantities. For example, for a gas that consists of myriads of atoms, the terms pressure and temperature can be introduced and relationships between them can be derived.

With the help of today's computers and sophisticated numerical procedures, however, it is now also possible to deal with multi-body problems and to simulate even more complex situations.

The problem of mathematization in other sciences

The fact that the mathematical language has contributed so decisively to the success of physics has of course not gone unnoticed by scientists in other fields. After physics, the process of mathematization first took hold of the natural sciences chemistry and biology. In biology, new fields of work such as biosignal analysis and bioinformatics are experiencing great growth and are attracting mathematicians, physicists and computer scientists. But also in the subjects of economics, psychology and linguistics, mathematical methods have long been established wherever quantitative matters are concerned. However, the more complex the subject area of ​​research, the more confusing the advantages of mathematical models become. Here some go beyond Kant and tacitly assume that their subject, even if it is no longer a “science of nature”, only becomes an actual science when “mathematics is to be found” in it. Of course, this must lead to a dispute within the subject [1].

Indeed, the mathematical language belongs to a quantitative science, but a lot can be told in any language [2]. But it also essentially includes the formation of terms and the formation of more general principles or principles, as well as a clear distinction between facts and their connection through superordinate hypotheses. The story you tell has to make sense. I realize that this is much more difficult in sciences where human behavior plays a role than in physics. In addition, ethical aspects will also have to play a role here.

But there is no getting around mathematization if a science can be about something quantitative. There is always resistance to this, something that is also known from the history of physics. Maxwell's theory was only slowly able to gain a foothold on the European continent because most physicists could not deal with its mathematical terms. And the relationship between experimental physicists and representatives of “mathematical physics” is not always free of tension even today.

Incidentally, it's like everyday life: a good rhetorician can impress. If he tells a meaningful story that also tells you something new, it's twice as nice. Otherwise one feels somehow betrayed.

 

[1] see e.g. S. Thieme: Mathematisierung, http://wissenschaftlichefreiheit.de/?p=203

[2] M. Binswanger reports on a particularly adventurous story in mathematical language in http://www.oekonomenstimme.org/artikel/2012/01/wie-die-uni-oekonomen-versagen–die-theorie-der-prostitution- as-memorial /

Josef Honerkamp was professor for theoretical physics for more than 30 years, first at the University of Bonn, then for many years at the University of Freiburg. He has worked in the fields of quantum field theory, statistical mechanics and stochastic dynamic systems and is the author of several textbooks and non-fiction books. After his retirement in 2006, he would like to devote himself even more to interdisciplinary discussions. He is particularly interested in the respective self-image of a science, its methods as well as its basic starting points and questions and can report on the views a physicist comes to in view of the development of his subject. Overall, he sees himself today as a physicist and "really free writer".