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Why is Mathematics so Effective in Science?
"Mathematics is the language of the natural sciences."
This is how Galileo Galilei one of the co-founders of modern science once put it. And the more you study science, the more you realize that this statement is indeed true.
I myself was surprised at the beginning of my studies at how much one could deduce about the natural sciences with the help of mathematics.
It required only a few axioms, in this case those of Newton, and
and a law of gravitation, and you could deduce a lot of other facts. Eventually, you could even deduce all of Newtonian mechanics including the law of conservation of energy.
Later in experimental experiments one sees that these are not only beautiful mathematical constructs, but, that they can be found exactly like this in nature.
This fascinating effect of mathematics, obviously to be perfectly suitable for the description of natural science, can be found again and again.
I would like to mention here only a few examples:
From the observations of Tycho Brahe, a Danish astronomer, Johannes Kepler succeeded in deriving exact mathematical formulas, which then could be formed by Isaac Newton to a precise mathematical law of gravity.
Newton himself attributed this order in his General Scholium to the existence of a God.
James Clerk Maxwell generalized the findings of electricity and magnetism to the Maxwell equations named after him.
From these equations a mathematical solution can be found for charge- and current-free fields, which propagate as waves. These electromagnetic waves were unknown until then and could be confirmed by experiments of Heinrich Hertz. Later it was noticed that light itself is composed of such electromagnetic waves in a certain frequency range.
While physics was formulated mathematically from the beginning, this was initially only partly the case in other natural sciences.
However, it was possible to describe the statements of chemistry more and more by insights of physics.
In particular, the advent of quantum mechanics made this possible. But also in the field of biology mathematics has shown itself to be helpful. This can be seen in the DNA sequences which follow a mathematical structure.
Eugene Wigner was an important mathematician from the last century. Also he saw himself confronted with this, as he called it, unreasonable effectiveness of mathematics in the natural sciences and published an essay on it.
There he describes this among other things using an anecdote, in which old schoolmates meet again after years.
While one of them has become a statistician, the other has chosen a non-academic career.
The statistician proudly shows him the formula for the Gaussian distribution. The worker asks what the Greek letter Pi means. This is the ratio of the circumference of a circle to its diameter.
The worker reacts confused, because he cannot understand what this Gaussian distribution has to do with this number derived from a circle.
And in fact it is not obvious. Only if one computes a Gauss integral to normalize this distribution it becomes obvious.
Nevertheless, it is fascinating how often this quantity Pi appears in mathematics in the seemingly most improbable places.
And the question that naturally arises is: Why?
How can this unreasonable effectiveness of mathematics in the natural sciences be explained?
At first one may think, that man wants to describe the world by science and on the way there mathematics gets developed. Thus it is a simple consequence of the human researcher spirit and not further remarkable. I also thought this for a long time, but this falls short.
Because this would not explain why mathematics is so precise in describing nature. Oftentimes even after the experiments have been refined and are more precise than when the theories were established, the results only further confirm the mathematical predictions. If mathematics were simply a developed tool to describe the existing results, this effect should not occur as often as it does.
Newton's law of gravity, for example, has withstood many more precise measurements than those of Tycho Brahe and only the perihelion of Mercury led to deviations. This was almost 400 years after Newton had established this theory and could be explained by the general theory of relativity.
The question remains therefore: What is the reason for this?
After Eugene Wigner, another mathematician and co-founder of computer science also dealt with this question: R.W. Hamming.
In his essay he finds four partial explanations for it:
We see what we are looking for: Many scientific statements are already a consequence of the used mathematics. Galileo came up with his law of inertia by reducing the possibility that differently heavy bodies move differently fast without a force acting on them to absurdity. In the same way in quantum mechanics Fourier integrals are used from which the uncertainty relation follows as a mathematical consequence.
Thus everything can be deduced apart. This view was already held by Arthur Eddington. However this is only partly true. At least one can assume that there is a limit, as Eddington himself shows using an example: In a lake, the fishermen all fish with nets that have holes the size of a square centimeter.
From their catch they conclude that there are no fish smaller than one square centimeter. Since their nets are holes of this size, it is not possible to determine whether this is really true or whether it is due to the fact that smaller fish literally pass through the net of the fishermen.
Hence, there may well be a limit to deductibility.
We choose our mathematics: At first, nature was described in terms of simple values, called scalars.
With forces and velocities, as well as many other quantities, it became clear that here not only one value is sufficient for the description, because these quantities always have a direction. Therefore vectors were introduced. Still later this was not sufficient for the description of nature anymore and to describe e.g. rotations so-called tensors were introduced. Therefore, one has chosen specific mathematics based on the scientific knowledge, which therefore works so well. Nevertheless, as described before, it cannot explain why this just works so precisely.
Science does not answer all questions: This means that science is limited in what it is able to describe.
I definitely believe that is the case. This would reduce the extent of this effect, but it is still there.
Evolution: Over time, science has continued to evolve. This can possibly also be explained by evolutionary effects. The enormously short time scales on which scientific progress has been achieved can be explained only partly by such effects, as Hamming himself notes.
Hamming himself calls all these points only partial explanations, which are not able to provide a complete explanation.
Thus, his conclusion is that he does not have a complete answer either, but that it is important to pursue the question.
How can this be explained then? Does the Christian faith perhaps provide answers to this?
Indeed it does:
So God created mankind in his own image, in the image of God he created them; male and female he created them.
- Genesis 1:27
According to Galileo, mathematics is the language of science. As creator of the world, which is described by the science, God must know this language. As it says in Genesis 1, 27 God has created us in his image. Thus, it is reasonable to assume that man as a being created in the image of God is also proficient in this language of science.
This would explain why man is able to describe the world so well with mathematics: Because it is the language of nature given to his mind by God. But why? For what purpose?
The Bible gives us answers to this too, right one verse further on:
God blessed them and said to them, ‘Be fruitful and increase in number; fill the earth and subdue it. Rule over the fish in the sea and the birds in the sky and over every living creature that moves on the ground.’
- Genesis 1:28
God instructs man to subdue the earth (nature). And if one looks at what is actually achieved with science, it is exactly that. Because on the one hand the natural science or an understanding of it, allows to make use of the forces of nature. For example, an understanding of electromagnetism makes it possible to build devices that enable something like the internet to create an information exchange that allows you to read this text anywhere in the world. Another example is cars, where the knowledge of the laws of thermodynamics are used to build machines that allow us to move around.
On the other hand, an understanding of science also allows us to overcome these same forces of nature. For example, the moon landing made it possible to overcome the gravitational force of the earth and send a man to another celestial body.
Through an understanding of the natural sciences humans succeed to subdue the earth, as God has instructed them in the Bible.
This fascination about this high position of man is also found in the Psalms:
What is mankind that you are mindful of them, human beings that you care for them? You have made them a little lower than the angels and crowned them with glory and honour. You made them rulers over the works of your hands; you put everything under their feet.
- Psalm 8:4-6
Thus, God has indeed called us to rule over this earth and one of the means by which we can do this is through mathematics, which allows us to describe nature and thus to subdue it.
Therefore, this unreasonable effectiveness of mathematics in the natural sciences does not only point to an order indicating a creator as Newton already noted. It points beyond that also to the fact, that this creator has indeed created us in his image, since we are able to understand this creation to a certain extent and are able to make use of this knowledge.
C.S. Lewis once said:
"Men became scientific because they expected Law in Nature, and they expected Law in Nature because they believed in a Legislator."
I think one can extend this with these findings such that the lawfulness of nature does point to a God, but our ability to grasp it points to a special position of man in creation, as the Bible also clarifies.