To read the real article selects: https://www.nature.com/articles/d41586-019-01171-0 Open access.
• The text in italics is copied from the article
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### Introduction

Mathematics is an immensely powerful tool for understanding the laws of the Universe.
Mathematics is an immensely powerful tool to describe the stable processes in the Universe.
That was demonstrated dramatically, for instance, by the 2012 discovery of the Higgs boson, predicted in the 1960s.
IMO one the first demonstrations was the prediction of the planet Neptune as a result of a series of irregularities detected in the path of the planet Uranus that could not be entirely explained by Newton's law of universal gravitation.
Yet an ongoing, often fervid debate over the direction of theoretical physics pivots on the relationship between physics and maths — specifically, whether maths has become too dominant.
In relation to the discovery of the planet Neptune the mathematics involved in Newton's law can never be called too dominant. In fact the mathematics is too simple because Newton's law can not be used to calculate the trajectory of the planet Mercury. There exists a small discrepency based on observations. Such a calculation involves all the planets around the Sun, because all the planets influence the trajectory of the planet Mercury vise versa.
Newton's Law is maybe too simple but it is successor General Relativity is maybe too complex to calculate the trajectory of the planet Mercury solely based on observations.
Next page 310.
The Universe isn't just speaking in numbers: it's singing and dancing.
The only thing that you can say is that the universe is evolving both slowly and hectic. Numbers per se, have nothing to do with this. That does not mean that in certain physical processes numbers are not important. When you study the periodic table the number of electrons, protons and neutrons is very important to clasify each element, but at the same time these numbers don't explain the physical differences between each element.
That constant speed of light led to Einstein's special theory of relativity in 1905.
This raises the question what types of problems tried Einstein to solve with the SR.
The first problem to solve is to establish by means of an experiment that the speed of light in two opposite directions is the same.
A next issue is that to measure the speed of light (observer independent) in two opposite directions. This is difficult. Part of the problem is that in order to measure the speed of light you need a clock and if this clock uses light signals to operate, the clock itself becomes part of the problem. As such to declare the speed of light a constant is an easy way out.

To declare the speed of light finite and constant also raises a whole different issue: How important is the speed of light, to be more specific electromagnetic radiation for the laws of nature?
How important is light physical in order to understand the movement of the stars in our Galaxy and the movement of the Galaxies in the Universe?
IMO zero. Light (photons) has nothing to do how objects move in space.

From this, in an amazing conceptual (and mathematically abetted) leap, Einstein conjured up general relativity in 1915, then the curvature of space-time, and eventually the gravitational waves discovered by LIGO 100 years later.
Space-time is mathematical a 'simple' concept, because it describes the universe in four space dimensions i.e. the 3 space dimensions x, y, and z and a space dimension t. This is a simple in an empty universe, but becomes very complicated when matter is introduced. The difficulties are extensively discussed in the book "The evolution of scientific thought from Einstein to Newton" by d'Abro. See for a review of that book: The evolution of Scientific Thought In real they require the calculation of the parameters gik In order to measure time clocks are used. A clock is also a physical process. The problem with clocks is that they do not always behave the same. When this is the case the concept of Space-time becomes very tricky.
Gravitational waves and LIGO are in some sense two 'complete' different areas of research. Gravitational waves are involved with all moving physical objects, specific with binary (large) objects. Ligo is used to detect gravitational waves. The detection as such is a huge accomplisment.
My guess is, that when gravitational waves are detected, always, when two large objects collide, a third smaller object is involved.
As Farmelo recounts, this is given interesting context by studies of Einstein’s notebooks, showing how he later overstated the role of mathematics, and underplayed that of physical insight, in his own breakthrough.
An onjective description specific about the issue of 'underplayed' sound to me important.

### A productive union

Farmelo’s argument is that mathematics and physics work effectively together, to the benefit of both.
Ofcourse if he can mention one case this statement is correct, but generally speaking this are two different scientific disciplines. Understanding physics requires detailed description of what is involved. Most of these descriptions are in text or in rather simple formula, specific in chemistry. Most of the mathematics used is rather simple except in the case when the physical processes are relatif stable, like in astronomy, respectivily celestial mechanics.
Newton's Law in that sense is too simple, because it cannot explain all astronomical phenomena. At the same time it is practical because it can easily describe the movements of the planets.
GR is much more complex and less practical. To describe the movements of the planets using curvature of Time-Space is very difficult i.e. the parameters gik.
Although the few mathematical physicists engaged in the field, notably Freeman Dyson, made important contributions, most physicists didn’t need to go beyond well-established mathematical techniques to progress.
That is what can be expected. The issue is really how does the evolution of the universe operates? It can not be very complex, however to predict exactly what is the situation to morrow can be very difficult, in casu impossible. A typical case is wheather prediction.
But others, led by luminaries such as Michael Atiyah, Edward Witten and pioneers of string theory including Michael Green and John Schwarz, were probing its mathematical boundaries.
String theory chalenges the limits of the physical boundaries. My understanding is that it is extremely difficult to test.
The standard model is a complex, subtle and immensely successful theoretical structure that leaves significant questions unanswered.
I think the major problem is the difference between experimental data and theoretical data. When this does not match or does not agree, the problem is in the theoretical model, which requires adaptations based on the underlying processes.
A typical case is dark matter, which is supposed to operate at astronomical scale and which we try to understand by performing laboratory experiments. The problem here is, if we can clearly demonstrate darkmatter in a laboratory, it does not mean that it operates the same inside a galaxy or the universe at large.
Farmelo makes a convincing case that, in attempting to answer those questions, mathematics has a crucial role.
Okay.
Yet whether theoretical physics has become too enamoured of beautiful mathematics will, I suspect, remain a topic of hot debate.
Mathematics is a tool. To understand the evolution of the world requires a model. The issue is the accuracy of this model but more important the actual parameters, which have to be calculated.
Newton's Law is simple. The most important parameter is the mass of each object investigated.
GR is much more complex. The most important parameter is a matrix of gik values for each object investigated
The need for evidence is even stronger if the argument is ‘it makes the maths look beautiful’.
To call an mathetical law beautifull is a misnomer.
The Universe might speak in numbers, but it uses empirical data to do so.
The Universe speaks in physics. In order to understand physics you need observations and experiments. An important are of mathematical reasearch, is sifting through the data (processing the data) to find the answers on your questions, on what you want.

### Reflection 1 The philosophy of science.

If you want to study the way the universe at all scales operates we must first layout a set of rules of how we perform science. This is very important. When you do that there is a mutual agreement, that we understand the same, what is the same or roughly the same.
A very important starting point is that the defintions of all the concepts used should be clear.
It is also important that the opposite of the concept also makes sense. As such it does not make sense to call paintings beautifull. A bended surface only makes sense in relation to a flat surface. The concept absolute and relatif only make sense if there are examples of each to explain the differences. This last is not as simple as it seems, because there is a tendency, specific when we study SR and GR, to declare all physical observations relatif. That is correct if we assume that all observations are observer dependent, but that does not make sense if you try to describe the physical state of the solar system or the universe. At least something should be absolute or both words should not be used.
In the mathematical context this discussion can be different.

An important concept in the philosophy of science is the concept: The laws of physics. The laws of physics are descriptions of subsets of physical phenomena, partly in writing, partly by means of mathematics i.e. in the form of mathematical equations. When different equations are used each describes often a smaller detail. It is important that the variables used should have a clear link of what physical is involved and what can be measured or calculated based on measurements.
A typical case is the variable mass used in Newton's Law, which in order to be calculated, requires both observations (in time) and Newton's law it self.

A different concept is the use of clocks in physics versus the concept of time. Time is not something that exists, it is something abstract, which raises the issue what means abstract. The problem is that any clock is also a physical process, which raises the issue if clocks are a proper tool as an indication of time, specific in relation of the evolution of the universe.

### Reflection 2 - The Universe Speaks in Numbers: How Modern Maths Reveals Nature's Deepest Secrets.

This reflection is about the book "The Universe Speaks in Numbers: How Modern Maths Reveals Nature's Deepest Secrets." by Graham Farmelo. Raised in a different the question is: How important is mathematics in order to understand the evolution of physical processes at all levels of details.
To answer that question a certain amount of reservedness or aloofness is required, because we maybe never able to find a complete and satifying answer. The problem with many physical processes is they evolve in some sense completely indepent, at first sight without any structure and there is no way how and why this is, assuming you have a clear definition of what the word why means. That is true, globally, for the behaviour of the stars in our Galaxy, for the planets in our solar system and for the whole evolution of human life on earth, including the behaviour of the clouds in the air.

At a more local level we humans can observe that the stars in a galaxy move around a common centre in a circle and generally speaking that there are 2 types of galaxies: either all stars move in a flat plane, or in all directions like in a cloud. How come?
When we study the periodic table, the number of electrons and protons of each element is explained by the position of the element in the table. The number of protons is also a function of the position but increases more sharply. When we study the three elementary particles, each contains three quarks, but they are of different types.
What is important neither of this explains the chemical or physical properties of each element.

Observing the planets of the solar system for a certain duration and by using Newton's Law (i.e. the concept the sum that all forces are zero) we can calculate the masses (a number) of all the planets. That is a number. However that is not enough to predict the positions of the planets at some future date, we also need the initial conditions at some date in the past. Using Newton's Law, the initial conditions and the masses of the planets we can predict the future (within a certain accuracy).
But does that mean that Newton's Law (or GR) governs the motions of the planets? No, it does not. The motions of the planets generally speaking are stable and this stability is inherent in the internal qualities (i.e. the mutual attractive force between objects) of the planets (or objects). Newton's Law (or GR) is a mathematical description of this behaviour.
What is even more remarkable, this quality (force) has nothing to do with light or photons, nor have light or photons anything to do with the behaviour or motions of the objects. Objects (stars) emit light, but this effect is so small on the mass of an object that it can be neglected.
An important physical phenomenon is the comparison between mass and size of an object. What this relation shows, is, that the heaviest objects also can be the smallest i.e. these objects are contracted. This has a huge impact on the internal physical state of the object and has nothing to do with mathematics.

In order to study the evolution of any physical process in time you need a clock i.e. something reliable that maintains a regular time count. The problem is that a clock is also a physical process which has its own limitations for time keeping.

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Created: 18 April 2019