Mathematics is the purest subject

The spirit in matter

For some, mathematics is the noblest and purest form of thinking. Others simply regard it as a breadless art. Even the well-known English number theorist Godfrey Harold Hardy was proud that he had never done anything useful in his field. "None of my discoveries - for better or for worse - had the slightest significance for the world, and nothing should change that in the future," he wrote in 1940. A mistake. A few years later, based on number theory, sophisticated techniques for encrypting secret messages were developed. That even called the security authorities on the scene in the USA. For example, the FBI tried to prevent certain works on number theory that were directly related to cryptography from being accessible to everyone in public publications.

This example is not an isolated case. Many things in modern life are based on the preparatory work of mathematicians, who are often themselves surprised by the practical use of their formulas and theorems. Let's just take non-Euclidean geometry, a consistent mathematical theory developed in the 19th century by Carl Friedrich Gauß, among others, in which the Euclidean axiom of parallels does not apply. Instead, two or more straight lines parallel to g can run through a point P outside a straight line g. Or none at all. For decades, the work on non-Euclidean geometry went unnoticed - until Albert Einstein used them to describe the curvature of cosmic space-time within the framework of the general theory of relativity.

Or let's think of Gottfried Wilhelm Leibniz, who created the binary number code in 1703, without which there would be no computer today. When trying to define the square root of a negative number, complex numbers were invented in the 16th century, which even mathematicians initially considered a gimmick to calculate with. The astonishment was therefore great when it was found that such "gimmicks" lead to practically useful results. Today, complex numbers are part of the tool of every engineer who, for example, wants to calculate an electric motor or represent the dynamics of flowing liquids in a mathematically elegant way.

"The book of nature is written in the language of mathematics," Galileo Galilei once asserted, although he and other naturalists of course did not understand why. And so one question is asked again and again to this day: How is it possible that mathematical formulas that arose from the human mind fit so well into reality? To answer that God created the world according to a mathematical plan is just as of little help here as the reference to the biological adaptation of the brain. After all, what survival advantage should abstract mathematics have given our early ancestors? If, on the other hand, one adheres to evolutionary epistemology, and there is much to be said for doing so, then the ability to mathematically grasp reality is only a by-product of human brain and mental development, which was only useful to its creators at an advanced stage of culture.

As is well known, the German philosopher Immanuel Kant took a different view. He believed that humans read the mathematical order into nature and not from it. A cursory look at such a point of view suggests that there is no clear correlation between mathematics and reality. This means that there are often different mathematical representations for the same physical phenomena. For example, quantum mechanics in the Heisenberg or Schrödinger picture can be described mathematically without changing anything in the physical content of the theory.

While Galileo still hoped to be able to completely replace physics with mathematics, modern science teaches that mathematically formulated theories are only models that do not depict the entire complexity of reality, but a reality under ideal conditions. Strictly speaking, the famous Galilean Laws of Fall only apply if one disregards air resistance. Basically, when experimenting, humans adapt nature to their minds, explains the Hamburg mathematician Claus Peter Ortlieb with a view to Kant, and concludes: “If the ideal conditions assumed in the model cannot be established, or only inadequately, the› laws of nature ‹to be observed ultimately remain mathematical fictions. «According to Ortlieb, it starts with the hypothesis that reality follows mathematical laws. And only if this is the case, the search for a mathematical structure that corresponds to the empirical data begins in science. Often this endeavor is crowned with success, but for Ortlieb this does not mean that this has to be the case always and everywhere. From this point of view, mathematics would be only one, albeit the most effective, method of describing reality, but it also contains areas in which mathematics alone cannot get you very far.

Nevertheless, anyone who assumes that the world can largely be represented mathematically will hardly be able to deny that objectively it also has a kind of basic mathematical structure that is considered to be one of the greatest and most difficult challenges of science to uncover.

The writer Hans Magnus Enzensberger also compares the work of a mathematician with that of an artist: both are not primarily concerned with practical usefulness. Rather, they are eager to leave their contemporaries and posterity behind elegant and aesthetically pleasing intellectual creations. Why these retrospectively match reality so well in the case of mathematics will probably remain a mystery forever. But that doesn't change the fact that mathematics is one of the most impressive human cultural achievements - and an engine of what is commonly called progress.

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