The Pythagorean theorem describes the properties of a right triangle.
Foto: iStock/Dmitro2009
The supposedly brittle and sober mathematics provides real benefits. Yes, one can definitely say: without this science the modern world would not function. Data compression, encryption, imaging processes in medicine and technology, satellite-based navigation, climate forecasts are just a few examples of important applications of higher mathematics.
Although mathematics is a model of precision, there is disagreement about its philosophical interpretation. Three major directions emerge; we call them empiricism, rationalism and platonism in relation to mathematics and briefly introduce them here.
Gained from experience?
Empiricism counts mathematics as an empirical science. The difference to physics, chemistry and biology, for example, is only of degree; Mathematics pushes abstraction a little further. Empiricist epistemology says this: Experiences (observations, experiments, practice) are the raw material for an abstraction process. At the end there is a natural law, another general law or a mathematical theorem. Something is discovered that is in principle already contained in reality, but is not obvious. Implicit connections are presented explicitly.
Friedrich Engels was convinced of such empiricism. In his polemical work “Anti-Dühring” (F. Engels: Mr. Eugen Dühring’s revolution in science. Dietz-Verlag, Berlin 1952, p. 36) he wrote in 1878: “The concepts of number and figure have come from nowhere other than from the real world (…). Pure mathematics has as its subject the spatial forms and quantitative relationships of the real world, i.e. a very real material. The fact that this material appears in a highly abstract form can only superficially conceal its origin from the outside world. The disadvantage here is the commitment to specific content, because these can change. In fact, new areas arose later that are now part of the core of mathematics and whose subjects are far removed from “numbers and figures” – we name set theory, topology, category theory.
Creation of the human mind?
Rationalism considers mathematics to be an intellectual exercise that does not rely on references to external reality. She imposes strict internal rules on herself and sticks to them; that is what makes the difference to other forms of thinking. Mathematical theories are therefore creations of the human mind with strict rules. A theorem is not discovered, but – in a context – invented. Mathematics has a lot in common with strategy games such as chess and Go. Hermann Hesse probably had mathematics in mind when writing his poem “The Glass Bead Game” in his well-known novel of the same name. (Perhaps he was thinking of a future variant.) The analogy between mathematical operations and gaming operations extends to psychosocial and aesthetic aspects. Mathematicians are drawn to mental challenges, and they strive to carry them out as elegantly as possible. The analogy ends in the fact that in addition to the mental work, a game also has a material side – a game board, game pieces or the like.
The above provision is expressly open in terms of content. A variant of rationalism, called structuralism, goes further and assigns an object to mathematics: it is the science of conceivable structures. “Possible to think” means located in thinking and free of contradictions. “Structure” is understood abstractly; all realizations or accompanying circumstances are disregarded. “Structure” can be defined internally mathematically; Elements, relations and operations are ingredients of the definition. As structuralism, rationalism has influenced the didactics of mathematics. The major textbook by the Bourbaki author collective is an implementation of structuralism.
“Insofar as the propositions of mathematics relate to reality, they are not certain, and insofar as they are certain, they do not relate to reality.”
Albert Einstein physicist
Albert Einstein argued strongly for rationalism. In his ceremonial lecture and article “Geometry and Experience” (A. Einstein: Geometry and Experience. Meeting Reports Preuss. AdW. February 1921) he says: “Mathematics enjoys a special reputation above all other sciences for one reason: its propositions are absolute certain and indisputable (…). Insofar as the propositions of mathematics relate to reality, they are not certain, and insofar as they are certain, they do not relate to reality.”
The usefulness of mathematics
A speech given by Eugene Wigner in 1959 is entitled “The unreasonable effectiveness of mathematics”; the paper on this, published in 1960, is often cited. The famous physicist Wigner draws attention to the application problem; he doesn’t solve it. “We don’t understand the usefulness of mathematics and we don’t deserve it,” he writes.
A moderate rationalism admits that mathematical thinking can certainly be stimulated from outside – by reality or by another area of thought. But after each such initiative, the window closes and the suggestion is then processed mathematically. After all, this kind of opening is one of the explanations for the indisputable applicability of mathematics. Because what started from real phenomena probably finds its way back there. Another explanation of potential benefit is a selection effect: mathematics produces many possible structures in a playful way. Unplanned, some of them turn out to be hits, that is, practically applicable; These naturally receive increased attention. Finally, there is an ontological argument (ontology = theory of being): the human brain is ultimately from this world. Although thinking is free in principle, it remains relatively down-to-earth and does not stray too far from this world of ours. We believe that the three reasons taken together solve the application problem.
A fictional conversation between philosophers
Fictional sophisticated conversations used to be a popular form of philosophizing. Here we take a well-known mathematical theorem and let an empiricist (E) and a rationalist (R) comment on it. A Platonist (P) should have the last word.
The Pythagorean theorem states: Let a, b be the lengths of the sides of a right-angled triangle and let c be the length of the hypotenuse. The following applies: a² + b² = c².
E: Experiences with real triangles led to the theorem. Conversely, it has proven itself again and again in practice with very satisfactory accuracy.
R: No real triangle has an interior angle of exactly 90 degrees. In general, absolutely exact measurements of angles and lengths are an illusion, simply because of the atomic structure of matter. The “Pythagoras” applies to ideal right-angled triangles. Such a thing is a thing of thought, and the sentence is the result of mental effort.
E: The concept of the ideal triangle was inspired by real triangles. Small deviations of the real triangles from the ideal play practically no role.
R: Yes, admittedly. But the stimulus only sets the thinking in motion and is then no longer needed.
P: The sequence of cognition is just the opposite. The ideal triangle came first. The idea structure of geometry has always been there.Rainer Schimming
A gift from heaven?
We summarize: According to empiricism, a mathematical idea is wrested from reality with a certain amount of effort. According to rationalism, on the other hand, the idea is generated informally by the human mind. There is a third possibility – the idea has long existed in its pure form “somewhere out there” and is brought “down here” by the recognizing subject. This view goes back to Plato. Platonism considers mathematics to be a participation in the realm of absolute ideas. Such an afterlife exists independently of this world, that is, the visible world, and is prior to the latter. The Absolute Ideas or Eternal Truths are unchangeable and perfect. The visible things and events are imperfect imitations of the ideal archetypes. Mathematics now opens up privileged access to Plato’s realm of ideas. In fact, the researching mathematician often has the feeling that he is discovering a truth that is “out there” waiting to be discovered. If successful, the researcher believes he or she is participating in something greater. Of course, the status of being of the absolute ideas outside of humans is unclear. By the way, Platonism is not just a historical phenomenon. There are still Platonists today. The versatile thinker Roger Penrose is the best known.
Prof. Dr. Rainer Schimming taught mathematics at the University of Greifswald and conducted research in the fields of mathematical physics, differential geometry and mathematical biology. Since his retirement in 2010 he has turned to philosophy.
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