A good experiment generates new data to test a hypothesis on the cause of a pattern seen in existing data. These new data can then be subjected to more statistical analysis. If the experiment provides support for a hypothesis, we end up at a point where we have seen some pattern in data, posed some hypotheses as to what generated that pattern, and generated data from experiments that test which of the hypotheses is plausible. If a hypothesis is not supported, this is useful information, and could lead to us devising a new experiment or set of experiments.
In some cases, experiments are just not possible. For example, we are unable to re-create conditions seen at the birth of the universe as it is beyond our technical expertise. A consequence of this is that we must rely on observations made by telescopes of distant objects in the night sky. When experiments are impossible, the next best thing is to build a mathematical model and see whether it generates predictions that match observations. Scientists have built models of the development of the universe, and they examine the outputs of each model to see which look most similar to what we observe. The models that most closely match reality are most likely to capture processes that generated the patterns we see.
Mathematics is an exact and unforgiving language. It is abstract and describes how different things are related to one another, and how they change over time, or across space, or even in abstract dimensions that we cannot really envisage. A bit like many human endeavours such as sports, music, art or learning to speak other languages, there is a huge amount of variation in mathematical ability between people. To some people it is all obvious, while for others it will always be gobbledegook, however much effort they invest in trying to master it. Not all science requires the use of mathematics. Many superb experimentalists, fieldworkers and lab-based researchers are brilliant scientists yet poor or mediocre mathematicians. I am reasonable, but not exceptional at mathematics, but I am hopeless at fieldwork. I do tend to work with collaborators who are experts in fields where I lack skills, and one of my skills is being able to bring people with complementary skill sets together to address interesting questions.
There are mathematical models not only of the development of the universe but also of the behaviour of solar systems and galaxies, of chemical reactions, of the diversification of life and of the functioning of the human brain. I cannot think of an aspect of science that has not been modelled using equations. Much like there are different genres of music and art, there are different genres of mathematical models. Some models are kept deliberately simple to examine how one process might create a particular pattern, while others are highly complicated, providing real-time, accurate descriptions of the system under study. Each genre of model is useful in some way, and the trick is knowing what sort of model to use. Albert Einstein, one of the greatest scientists ever, stated that ‘everything should be made as simple as possible, but not simpler’, and this has sometimes been interpreted to mean that models should be kept super-simple, but that is not an appropriate interpretation. Sometimes models need to be complex to achieve their aims. For example, imagine you developed a complicated mathematical model that perfectly predicted the stock market. You would be extremely happy. If each attempt to simplify the model resulted in its predictive accuracy disappearing, the model no longer does what you want it to do, and you conclude the full complexity of your model is necessary to allow early retirement. You probably wouldn’t care if you didn’t understand why your model worked, you’d just be happy that it did. In this case, your model’s complexity is at the right level because it does what it was designed to do: make you rich.
Models are also an extremely important tool in generating new hypotheses. Once a model is constructed, it can be used to make predictions, and these predictions can sometimes be tested with new observations or experiments. The Standard Model of particle physics is a complicated equation that describes the workings of three of the four fundamental forces of the universe: the weak and strong nuclear forces and electromagnetism. Analysis of the model led to prediction of a particle called the Higgs boson. The LHC, the world’s largest machine, was constructed to search for the boson, and it found it. A mathematical model predicted a new fundamental particle and, when scientists looked, it was where the model said it would be. That is an impressive use of a mathematical model to pose a hypothesis that experiment proved was supported.
The LHC is spectacular. It is an example of the remarkable technological progress that scientists and engineers have made over the last couple of centuries. We can now make new types of measurements that were unthinkable even just a decade ago, and we can also now measure things much more precisely and with much less error than ever before. We have also learned to manipulate nature in new ways, altering some aspect of it and seeing what the outcome is.
Genetics is one area where our ability to understand, and alter, nature has increased dramatically. When I was at university in the late 1980s, the genetics we did almost entirely revolved around fruit flies, Drosophila melanogaster. We would take flies with different characteristics, breed them and then score the traits in offspring. The characteristics would be things like eye colour, or the number of bristles on their legs. It was dull work, but we were able to infer inheritance patterns of the traits and work out how many genes were involved. Although I always enjoyed statistics and evolution, I found these practical sessions uninspiring. But I did get taught something, and I now work on a project where we have been able to take things much further than studying flies that look a little different from one another. It is also a project where a mathematical model provided new understanding.
One of my study systems is the population of grey wolves living in Yellowstone National Park. It is a wonderful place to work, and the reintroduction and re-establishment of wolves in and around the park is one of the great conservation success stories, thanks largely to the tireless work of my friend and collaborator Doug Smith. Not everyone likes wolves in their backyards, and despite the improvements to the landscape due to wolves suppressing elk numbers, there are various groups that do not see it this way. For example, hunters now have to get out of their cars to shoot an elk and for that reason they dislike wolves.
Although the common name for the species Canis lupis is the grey wolf, wolves can have either a black or grey coat colour. The genetic variant that causes the black coat colour evolved in domestic dogs and was passed on to wolves when they mated with dogs shortly after people arrived in North America. Black wolf numbers then started to increase. Black wolves are not found everywhere in North America but tend to increase in number from the far north-east near Nova Scotia, where they do not occur, to the south-west edge of their range towards Mexico, where they are common. Elsewhere in the world, black wolves are absent or extremely unusual.
If we had conducted the fly experiments I did at school and university with wolves, we would have learned that coat colour is caused by genetic variants at one gene. If a grey wolf mates with another grey wolf the pups that are produced are always grey. If a black wolf mates with a grey wolf, or if a black wolf mates with another black wolf, the young can be either black or grey. The frequency of the black and grey pups reveals that there are two variants, or alleles, at the gene; let’s call them Grey and Black, or G and B. Each wolf has two alleles. If they are both G, the wolf has a grey coat, and if they are both B, the wolf is black. If one is G and the other B, then the wolf is also black. The B allele is said to be dominant over the G one. I would have really enjoyed doing a wolf breeding experiment at school rather than working with fruit flies, but health and safety regulations forbade it.