# Imagination and Mathematics: A Tale of Knowledge – 4/4

Isaac Newton formulated a theory for all of the universe known at his time, yet that was only part of his scientific work. Before he even spent one thought on mechanics, Newton laid the foundation of what is the most important observational tool of exploring the much greater known universe of today.

There is no hyperbole in that paragraph.

This is the concluding article in our series on Imagination and Mathematics. You can find part 3 here.

Image 1: Newton’s drawing of how to experiment with light rays. Source: The Royal Society (http://trailblazing.royalsociety.org/photos/1672SA1.jpg)

Young Newton was rather observant of his surroundings. To take an example, he wondered about where the colours of the rainbow would come from. In order to find out, he constructed an experiment, which showed what happened (image 1). When a ray of light from the Sun is isolated through a hole in the wall, then this ray can be led into a prism where the colours are separated and the ray as a whole is taken off its original path as well.

The Men of the Church did not protest. They probably should have, since this experiment showed that even the heavenly light itself was not perfect! But then, of course, a rainbow is undoubtedly beautiful, and the knowledge of how it comes to be does not need to subtract anything from the experience. In fact, one may argue that knowledge only adds to the fascination.

At any rate, being able to split light into its colours made it possible to examine the nature of its source – and the nature of what lay in between the source and the observer. The concept one needs to understand this is that of the wavelength and some quantum physics. The shorter the wavelength, the more energy the light has. Violet light, for example, has more energy than red light, since its wavelength is smaller.

Image 2: Blue Sky Spectrum. Source: Eric Bajart via Wikimedia Commons (http://upload.wikimedia.org/wikipedia/commons/b/b5/Spectrum_of_blue_sky.svg).

Image 2 shows the intensity of the sun light which is spread in the atmosphere (blue sky spectrum) compared with the wavelengths of the respective colours. The first thing we should note is that we cannot actually see all the colours of the spectrum in the picture, but only those at wavelengths from roughly 400 – 700 nanometers. Secondly, one could imagine that this would have to be a nice and smooth line, but it is not. It looks rather more like something is missing at some places. These bite-outs we call absorption lines, which, in this case, are due to the Earth’s atmosphere between us and the Sun.

Why? An atom or a molecule has different energy levels, which restricts the atom to only absorb or send out light of a particular wavelength. The molecules and atoms in the air all have such energy levels and absorb light of different wavelengths to different degrees. Turning the argument around, this means that light of a particular wavelength can be correlated to a particular chemical element, which sends out the light. The light’s intensity then gives us information on how much of this element is present in the light source. And by building instruments, which can see wavelengths our eyes cannot, the opportunities and possibilities for observation became enormous.

One can, for example, find out that there is a lot of hydrogen in the universe and why that is the case. One can find out about the structure of the universe around us and about how the Sun belongs to a galaxy of around 400 billion stars, such that, on the whole, it is not really all that special in the cosmical perspective.

Image 3: Redshift of spectral lines in the optical spectrum of object BAS11 (right) correlated to the spectrum of the Sun (left). Source: Georg Wiora via Wikimedia Commons (http://upload.wikimedia.org/wikipedia/commons/1/14/Redshift.png).

Another thing entirely is that the universe seems to be moving away from us. How do we know this? See image 3. In this example we see the absorption lines of the galaxy BAS11 to the right and the solar spectrum to the left. The lines which are connected by the arrows should be at the same colours, but they are not. Without going into detail, this means that BAS11 is moving away from us (Find information on the redshift here).

Truth be told, it may seem to the observer that the universe itself is expanding. This means, that everything was very close to each other once. Very close. We think that at one point there was nothing, which exploded.

That is a rather big conclusion, and perhaps it is not self-evident anymore where this conclusion is drawn from. What we have to talk about here are mathematical models, i.e. solutions to sets of equations, in particular to the Einstein field equations. The variables in these equations can be the time or the average distances of objects in the universe or a host of other things. And the solutions to the equations give us information about how these variables develop with respect to each other. There is a problem, however: Einstein’s system of equations is so complex that the solutions give information only under a certain set of conditions. At the time, the universe began, for example, our solutions become “infinite”. They do not work.

To repeat: We have models, which describe the beginning and progression of the universe to whatever fate it may have, yet we have no way of interpreting these equations as a whole. Still, we think that “in the beginning there was nothing, which exploded.” We expressed this rather directly. We have even given it a, by now, famous name: The Big Bang.

The Big Bang Theory is not the only example of this either. In the same system of equations by Einstein, one can also find Black Holes. These have inspired human imagination about as far as anything. Just watch this video or click through to this document and read, as far as you might, what the well-known American science communicator Sean Carroll has to say about the mathematics behind Einstein’s equations.

Did you understand everything? I did not either; but that is perhaps the point of it all.

“Big Bang” and “Black Holes” are inventions, virtually empty expressions, which are used to conjure a kind of description of the physical phenomena in question. That does not mean that the researchers are wrong or worse, that they can be charged with a scientist’s equivalent to heterodoxy! The scientists, which communicate this knowledge know what they are talking about. So, when the cameras are turned off, the words change. There are less “explosions out of nothing” and more “approximation limits”, “models” and “singularities”. Scientists do have a good understanding of what happens, what has happened, and what may happen in the universe. There are many mathematical approximation techniques, some of which bear strange names like “Perturbation Theory”, which make it possible to examine what happens when our models break down. The only things that are needed are a little experience in mathematics, the knowledge that we deal in approximations and not truths, and enough time to work intensively with the equations over an extended period of time.

Everybody makes experiences and learns from them. When we have gained enough experience, we begin to understand. This is just as true for my daughter learning about gravitation as it is for, say, the medieval church builder or Sean Carroll presenting the physics behind Black Holes. There is no principal difference for any of these three examples.

Learning about physics is like this, too. It is about finding mathematical equations and then using them often enough with ever changing the conditions applied to them. It is about learning what the equations can and, perhaps more importantly, what they cannot describe. Then understanding will, at the end, arise more or less by itself.

Remember how how this ale began and how it developed. The way the world’s thinkers approached the biggest problems never stopped changing. First, there was the mythical approach, which got softened and slightly changed by the advent of geometry and its reliance on the truth behind beauty all the way to mathematical equations. From what can be grasped to the ungraspable. It had to be like this, didn’t it? For the questions have not and will never end, while human imagination does have its limits.

Let us focus again on Black Holes towards the end. The whole point with those is that they draw to them all information around them. How would you know anything about them then? They should be invisible, simply invisible – unimaginable, if that is a word, yet people have predicted and seem to have discovered them.