How effective is mathematics in the description of the world?




Mathematics known as the “queen of the sciences.” The language of the universe. Scientists and engineers often talk about the elegance of mathematics in describing the physical reality, citing the examples of π, E = mc 2, and even abstract number for the account of real objects. However, although these examples show how useful mathematics us, does it mean that the physical world follows the rules of mathematics, as if it was his mother tongue? And that math exists by itself, just waiting until we make one of its separate discoveries? This point of view regarding the nature of the relationship between mathematics and the physical world is called “Platonism,” but not all of it is shared.

Derek Abbott, Professor of Electrical Engineering and Electronics at the University of Adelaide in Australia, has written an interesting work, which argues that the mathematical Platonism – this is a distorted view of reality. In return to this, he says quite the opposite point of view in favor of a non-Platonist: mathematics – a product of the human imagination, which we customize the description of reality.

This argument is not new. According to Abbott (according to his own experience), although 80% of mathematicians adhere to the Platonic views, engineers are mostly non-Platonists. Physics, according to the professor, are the “latent non-Platonists,” that is, they are often the Platonists in public. But if pressed to the wall of physics, he admits that he does not believe in Platonism.

So if mathematicians, engineers and physicists can work in peace no matter what point of view is supported in this philosophical question is why the true nature of mathematics in relation to the physical world so important?

Abbott says the reason is that when you realize that math is just a mental construct – just an approximation of reality, which has its own weaknesses and faults and breaks at some point, because the ideal mathematical form does not exist in the physical universe, – you can see how math is inefficient.

Grain thought Abbott (and most controversial moment) is: math is not entirely suitable in describing reality, and certainly not a “miracle”, admired by some scientists. Einstein, the mathematical non-Platonist, was one of the scientists who marveled at the power of mathematics. He asked: “How can it be that mathematics, being after all a product of human thought which is independent of experience, so well describes the real objects? ‘.

In 1959, physicist and mathematician Eugene Wigner has described it as being “unreasonable effectiveness of mathematics.” Abbott’s work is called “reasonable ineffectiveness of mathematics.” Both views are based on a non-platonic idea that mathematics – the invention of humankind. But if Wigner’s and Einstein’s math may be called optimists, who acknowledged that the mathematical way of describing reality effective Abbott pessimistic notes that these mathematical models are almost always break down after a while.

As a general looks “effective mathematics”? Abbott explains that effective mathematics provides a compact and idealized representation of the physical world inevitably noisy.

“Analytical mathematical expressions – is one way to make a compact description of our observations,” – said the scientist. – “As a people, we are looking for similar” compression “, which provides us with math, because we lack the power of the mind. Mathematics is effective when it provides a simple compact expression, we can regularly apply in many situations. It is inefficient as this can not provide an elegant compact. It is compact making it useful and practical, if only we can achieve this compactness without the need to seriously sacrifice accuracy. ”

Abbott argues that there are more situations in which mathematics is not effective (not compact) than effective (compact). Mathematics only creates the illusion of efficiency, when we focus on the success stories. But our success stories are just a small fraction of the possible questions that we are looking for answers in this universe. ”

Some of Abbott’s arguments are based on the ideas of mathematics, Richard W. Hamming, who in 1980 identified four reasons why math is not as effective as it seems. But if the Hamming resigned to the idea that mathematics is unreasonably effective, Abbott shows that the causes of the Hamming actually confirm the non-Platonism, greatly reducing the level of effectiveness of mathematics.

Here are a few reasons for Abbott, according to which mathematics reasonably ineffective, most of them are on the point of view of non-Platonist: mathematics – it’s only a human invention, the fruit of the collective mind of sentient beings from the planet Earth.

1. Math is successful because we choose the task to which we can apply a mathematical approach. Most likely, there were millions of unfortunate mathematical models, but no one paid any attention to them. “Genius” – writes Abbott, – “it’s just someone who has a great idea, as well as common sense to keep silent about the thousands of other crazy ideas.”

2. Our application of mathematics varies at different scales. For example, in 1970, when the length of the transistor of the order of micrometers, the engineers were able to describe the behavior of transistors, using the elegant equation. Today submicron transistors include complex effects that were neglected before, so engineers have turned to computer building a miniature model of the transistor. Effective formula should be described transistors at all levels, but such a compact formula does not exist.

3. Despite the fact that our model can be used in any time frame, we are likely to create descriptions that are based on the length of human life. For example, we see the sun a source of energy for our planet, but if a person’s life is equal to the life of the universe, the sun will likely be short-lived fluctuations that quickly would sleep the planet and “bahnul” into a red giant. From this standpoint, earth can not obtain useful energy from the sun.

4. Even the accounts have limits. When counting bananas, for example, at some point, their number would be so large that the gravitational attraction of bananas suck them into the black hole. At some point, we can no longer rely on the account in the figures.

5. What to do with the concept of integers? Where one ends and the other begins a banana? Although we think we know visually, we do not have a formal mathematical definition. To bring this to the logical extreme, if people were not solid, and gaseous and lived in the clouds, counting discrete objects would not be so easy. Thus, the axioms based on the notion of simple accounts are not native to our universe, but merely human constructs. And there is no guarantee that the mathematical description that we will create will be universal.

Abbott to these items and more evident: mathematics – not a miraculous discovery, which is true with strange regularity. Mathematics is a human invention, useful, and limited working as it should. Well, mathematics – not only a human invention, in which philosophy can not do without.

See more: 50 of the most well-known popularizers of science of all time.
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