Which came first Math or Reality
Interview with Claus Peter Ortlieb
brand eins: Mr. Ortlieb, why is it that most people have difficulties with mathematics or are even afraid of them?
Claus Peter Ortlieb: I have often asked myself why that is. Of course, there is no simple, universal answer, and certainly not from me, because as a mathematician I never had these problems. But it's true, other people's difficulties with math can be determined. And even if I don't know a direct answer, I can at least say: The aversion to mathematics shows that the mathematical view of the world is not generally human.
You have to explain that.
Modern mathematics certainly did not emerge from a growing human need that could be justified beyond history. People got along very well without modern math. Ultimately, it is a modern invention. Most people have always grasped the world sensually and directly, empathically and emotionally. The explanation for the cause of an effect of processes is mostly derived from one's own, self-experienced experiences, which are considered plausible and understandable. The mathematical view of the world, on the other hand, is not based on direct experience, it absorbs the processes in an abstract way and tries to subordinate them to mathematical laws. Most people are reluctant to do this, they simply have a different, directly empirical approach to the things that surround them. Which, historically, is more likely to correspond to the human nature.
Human nature as a reason for being bad at math. You give the students a nice new argument.
I'm talking about a mathematical view of the world and not just about mathematics, which has been around for about as long as human language. They are not the same things. In addition, of course, cultural reasons also play a role. For example, it is a very German phenomenon to flirt with the fact that one was bad at maths, which is very en vogue in this country. In France, for example, it is completely different.
Why is that?
Nobody there could afford that. There, mathematics is more part of the national culture and has a higher reputation than in this country. The Germans have always defined themselves more in terms of the humanities.
How has mathematics changed our view of the world?
It is of tremendous importance. Today this is no longer clear because we have internalized the modern description of the world and take it for granted. But the mathematical view only emerged in the 17th century, when Galileo put forward the thesis that the entire world functions according to mathematical laws and can be described, explained and calculated using the language of mathematics. During this time, mathematicians made and thought for the first time what we do and think in mathematics and natural sciences to this day. In the pre-modern era, society could not do without God and the explanations of religion. In their place came the mathematical and scientific consideration of the world, which God made a private matter. Meanwhile, the mathematical and scientific interpretation of things is unrivaled. Which is not justified.
Because it is, of course, a mistake to believe that the whole world can be grasped in this way.
Is that what you say as a mathematician?
I say that as a mathematician.
How did you get there?
The claim that the modern sciences, in contrast to the dark ages, face facts, is nonsense. In truth, they are based on assumptions that are often completely fictional and precisely not based on experience.
Do you have an example?
Newton's first law on the principle of inertia says that a body moving in a force-free space maintains its speed and does not change it. The assumption of a force-free space is pure fiction. There is no such space. In this respect, Newton describes something that cannot exist in this way. The interesting thing is that it is one of the laws on which Newtonian mechanics is based.
What does that say about the mechanics?
That it is purely mathematical. The science historian Alexandre Koyré put it this way: Galilean science tries to explain the real by the impossible. Newton's axiom describes a fictional, real, non-producible, yet mathematical situation. An ideal mathematical state that does not actually exist. Building on this, mathematics is developed that describes reality. And if I'm lucky, you can prove it in experiments, but the original axiom can't. Only the conclusions one draws from it. This example explains the mathematical view of the world well. You describe an ideal, mathematical situation and say how the world would behave in this situation, although this cannot exist.
Yes, at least if you pretend that the whole world follows the derivations of a purely mathematical model and that the world can be forced under its own rules. One has to realize that mathematics can capture the world with limits. The assumption that it works solely according to mathematical laws leads to the fact that one only looks for these laws. Of course, I will also find it in the natural sciences, but I have to be aware that I am looking at the world through glasses that block out large parts from the outset. I must not forget that this is not the way the world is, but rather that I alone act that way. This method has a long tradition that goes back to the Enlightenment. Immanuel Kant, for example, described this approach in the natural sciences very correctly and claimed that it was the only way in which knowledge could be obtained.
Do you share his view?
That's nonsense. Kant ignores the fact that before his time people also came to knowledge in other ways, and of course also that there are areas that cannot be recognized with this method. The ancient Greeks, for example, used mathematics, but not in the way modern science does. They practiced mathematics as a kind of philosophy of pure form, and at the same time they knew that the world is more complex than this pure form. In modern scientific historiography there is a tendency to glorify the new disciplines over those of the times before and to embellish the way of gaining knowledge with nice stories. Galileo, for example, did not climb the Leaning Tower of Pisa in front of the assembled crew in order to drop two bullets from above. We have known that for sure since the 1930s. Galileo wrote a lot and was a gifted self-promoter in his writings. He never mentioned the scene with the tower itself; if it had happened he would certainly have written it down. Rather, he has described his attempts on an incline as a substitute for free fall, on which he let the balls roll down. But even that is controversial.
So the world remains unpredictable?
Of course it stays that way. That does not mean that the mathematical view of the world is nonsense per se, on the contrary: It is a real success story, and we owe a lot of knowledge, our entire scientific and technical development and the way we do to it live today. But the success story is also the problem. Because it not only gives rise to the illusion of believing that everything can be grasped and deciphered in this way, but through this illusion one is also forced to press the world into this form. And that's dangerous.
Because it leads to decisions being made that affect people's lives. The mathematical method has long been adopted by scientists from almost all disciplines and is used in all possible areas where it actually has no place. Parts of the social sciences, for example, see themselves as a kind of social physics and believe that the coexistence of people in a society functions according to certain mathematical laws that need to be recognized. Unfortunately, the prerequisites of the mathematical method are clearly not met there. In the social sciences, you cannot do experiments that connect mathematics and reality in the first place. In some social sciences this may not be so serious, provided that people's behavior is only described in statistical terms and admitted that the individual can deviate from it. The own methodical limitation is recognized and admitted. The prevailing economics, for example, does not do that. She's abusing math.
What do you mean by that?
Unlike in other social sciences, the limited expressiveness is not conceded, it is no longer even recognized. The prevailing economics is actually a mere mathematical discipline, it creates mathematical models that could never be reproduced in real life and that are nevertheless used to make calculations on their basis and to reduce complex economic processes to a few numbers. There, too, an attempt is made to describe the real with the impossible. In principle it is the same process, only one cannot combine the deductions from the assumptions of the mathematical model, which is nothing more than a fictitious ideal state, with reality in an experiment, as the natural sciences can. For this reason alone, it is legitimate to doubt whether mathematics is even allowed to be used in economics. In addition, economic processes are ultimately made by people and never follow natural laws. People always have freedom of choice. In the natural sciences, it is possible to start from regularities and to describe processes in a clearly determinable way if I know their conditions. As soon as people come into play, things are different, especially when their behavior is considered in a complex social space. History is made. It is not a natural process that just happens. The neoclassical theory ignores this and comes to absurd results.
This happens, for example, because the mathematical method, which is doubtful per se, is also incorrectly applied within economics. The neoclassical theory of the market, for example, is mistakenly transferred from the goods market to other markets. The goods market is described in terms of supply and demand. Supply is a monotonically increasing function of price and demand is a monotonically decreasing function of price. These two lines cross somewhere, and that's where the equilibrium arises to which the market adapts. For the sake of clarity, this is described with the famous picture of a marketplace where suppliers and buyers meet and negotiate prices. This is how all economics textbooks begin, the intersecting lines of the functions of supply and demand form the so-called Marshall Cross, which every economics student knows. And then the mistake: this model is being applied to all sorts of situations like hell, such as the job market.
Why is that not legitimate?
Because the basic assumptions of the goods market model simply do not apply to him. In the low-wage sector, the assumption of a monotonically growing supply function is incorrect. If I lower wages, someone who wants to make a living will have to work harder to get the same amount. However, the model assumption assumes that someone would then work less because the use of their labor is not attractive to them. This completely ignores reality, but is simply claimed and used as an argument against collective wages or minimum wages. If these were set too high, the equilibrium could not be achieved and unemployment would result. That is the prevailing view of neoclassical economics. One can read books by Harvard professors who, in relation to the labor market, argue in this way, although they have shown a hundred pages beforehand in another model that the assumptions are by no means fulfilled.
Don't these authors want to admit it?
In any case, it's hard to imagine someone doing this by mistake, especially not when it comes to a book that is now in its third edition. The neoclassical theory of economics is based on a kind of theory of harmony in the markets. If you leave the markets to their own devices, everything will turn out for the best. To prove this opinion, pseudo-arguments are used that make use of mathematics and misuse it to convey ideology. Mathematics is very well suited for this because it has the success story of the exact natural sciences on its side and is the measure of all things in terms of accuracy. What has been calculated mathematically exactly cannot be wrong. That is why many people trust the information that comes in the form of numbers. At first sight, numbers seem easy to understand, especially in economic contexts.
How great is the power of numbers?
They are extremely powerful, modern people believe in numbers and can be manipulated very easily using numbers. Numbers embody simple objectivity, they easily take on a life of their own and thus quickly become a fetish. For example, it is basically insane to reduce Germany's economic output to a single number, the gross domestic product, which then has to grow every year for the world to be in order. The number as such is comparatively meaningless compared to what is behind that number in human action.
What would the alternative be?
There are none, at least I don't see any in our modern, complex world. You have to make information manageable in order to be able to act. However, it is important not to make the numbers a fetish. Especially when their derivation is doubtful and leads to results that result in constraints.
What are you thinking about?
In everyday life, for example, I think of the awarding of grades that determine résumés. I am a university professor myself and I have to sum up the work of students. Ultimately, that is not possible. A complex intellectual achievement can only be described and evaluated in a complex manner. I also write a multi-page report for every thesis, but at the end I have to put a number below it. But grades are never objective; ultimately, an evaluation scheme levels off. In the mathematics department, for example, we have a fairly high grade point average for diploma theses, which is roughly a grade of "two". In the case of lawyers, on the other hand, a grade of "three" is a good result in the exam. That already shows how little objective and comparable numbers are.
And outside of everyday life?
Numbers are always questionable when they lead to normalization, although no one can understand how the numbers came about. The derivation of the numbers is always cut off at some point because it cannot be conveyed, nobody understands that. The numbers take on a life of their own and then stand alone without being questioned. Think of the two degree goal that is always mentioned in connection with climate warming. The assertion that we can barely afford a temperature rise of two degrees, anything above that leads to disaster. That is a figured number. Why two degrees? Why not a degree? Why not three? Nobody really knows. It is pretended that these are objective findings that have been calculated in computer models. But it is wrong and unscientific to say that there is an absolute numerical limit below which everything is fine, while disaster breaks out if it is exceeded. Of course, one degree is better than two degrees and two degrees is better than three degrees. This is also the case with other legal limit values. How many pollutants can a food contain? How high can the concentration of fine dust in the air be? These numbers are simply put, and the experts know that. But this is practically not communicated to the public.
Do you doubt global warming?
No, absolutely not. The people who predict climate change do so to the best of their knowledge. You work with high quality methods and with reasonably hardened, scientific data. Those who doubt it have no reason in comparison. However, the numbers that are mentioned are nothing more than the abbreviated representation of a scientific hypothesis. And every hypothesis should be viewed with skepticism, that is science. Ultimately, we have the same situation in climate sciences as in economics or social sciences: You can't do experiments. The numbers are not based on experience, they are pure forecast and the result of mathematical models. In truth, nobody really knows what temperature we will have in the year 2100.
So does mathematics and its models always reach a limit when it comes to questions that exceed a certain complexity?
She does. Any sufficiently powerful formal system is contradictory or incomplete, including mathematics.We owe this insight to the mathematician Kurt Gödel, who showed in the 1930s that the idea that had prevailed until then was wrong, that everything in mathematics can be formalized in the sense that one only has to use one algorithm to find all true sentences to get out. In any sufficiently complex formal theory there are propositions which one can neither prove nor disprove within the theory. This is an important point: if there are things within mathematics that cannot be clarified, then of course this is all the more true when mathematics is applied to informal systems such as climate, economy or societies. There are many math problems, conjectures, and questions that cannot be said to be true, false, or ever provable.
The Goldbach Hypothesis, for example, says that every even number that is greater than two can be represented as the sum of two prime numbers. Eight is three plus five, ten is seven plus three, and so on. No one has ever discovered an even number that is not. On the other hand, no one has ever been able to prove that it really always is.
And that can't be proven?
So far, nobody has made it. That doesn't sound like a big deal now, but around 1900 mathematics got into a real fundamental crisis because of such unsolvable questions. Best known in this context is probably Bertrand Russell, who showed up in 1903 with the antinomies named after him, and suddenly there were paradoxes in mathematics that have been popularized and are quite entertaining. The math found that less entertaining, the basis fluctuated tremendously. Do you know the barber's paradox?
It goes like this: "A barber can be defined as someone who shaves people who don't shave themselves. And the question is, does a barber shave himself?" Just think about it.
On occasion, gladly. You said the base was swaying. How did mathematics deal with its fundamental crisis?Very human. At some point she decided not to bother about it any further. ---
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