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HLF Blogs: Is mathematics idealistic or realistic?

In September, Katie and Paul spent a week blogging from the Heidelberg Laureate Forum – a week-long maths conference where current young researchers in maths and computer science can meet and hear talks by top-level prize-winning researchers. For more information about the HLF, visit the Heidelberg Laureate Forum website.

Stephen Smale

5th Heidelberg Laureate Forum 2017, Heidelberg, Germany, Picture/Credit: Christian Flemming/HLF

The closing talk of the HLF’s main lecture programme (before the young researchers and laureates head off to participate in scientific interaction with SAP representatives to discuss maths and computer science in industry) was given by Fields Medalist Steve Smale.

Speaking without slides, Smale shared with us some of his recent work in the crossover between mathematics and biology, but the central theme of his talk was one which goes to the heart of what mathematics is. Mathematics is embedded in science, and is used to describe and understand many aspects of scientific discovery.

The question was of whether mathematics is realistic or idealistic – do the mathematical models we use to solve problems and understand the universe give a realistic picture of how the world works, or is it all fantasy and we’re ignoring the fine details in order to get a model that works nicely? It’s a constant struggle, and Smale illustrated this with several historical examples.

Heroes of maths and science

Alan Turing’s theory of computation was an inspiring vision of how we can understand and use computers, and has influenced the whole field since. But was it realistic? For some applications, Turing’s approach is the correct model, but for others it fails. Modern study of NP-completeness in computability assumes an infinite amount of input, but obviously this doesn’t model the real world – it’s an idealisation.

Moving on to another giant of maths history, Smale turned to Isaac Newton. Newton’s work on physics, differential equations and mechanics was all an idealisation – to the extent that his calculations didn’t even include friction, which was added to the theory 100 years later.

John Von Neumann, who created the early models of quantum mechanics, introduced the concept of a Hilbert space – again, an idealisation. And even in other fields – Watson and Crick discovered the structure of DNA, but didn’t include the protein core of chromatin, later discovered to be a fundamental part of the more detailed structure.

This idealism is possible because this major work was often done without Newton, Turing or Von Neumann doing lab experiments – they used experimental data from other people’s work, but data which had already been ‘digested’ by the rest of the scientific community. Newton built on the work of Galileo, Copernicus and Kepler before him.

Getting to the heart of the matter

Smale’s current research is on the human body, and in particular the heart. How does it manage to beat in such a coordinated way, with all the myocytes, or muscle cells, acting together to create a regular heartbeat?

The synchronisation was compared to what happens in a crowd applauding for a long time, when the clapping falls into synchronisation almost accidentally. It’s a dynamical system, and can be understood through maths, much like the beating heart.

Smale is working towards a deeper understanding of the heart through mathematics, and in this case standing on the shoulders of Alan Turing. Turing’s final paper on morphogenesis – the biological processes that lead to stripes on a zebra, or the arrangements of seeds in plants – included some differential equations. These were again an idealisation of the reality, as they were based on what happens as the numbers of cells increases to infinity (obviously not the real situation – some biologists disliked Turing’s work for this reason, as they mainly worked with small quantities of cells).

The cells making up the heart are of the same cell type – they all behave in the same way. Smale is looking at how this happens, and outlined how if you consider the set of genes as a graph, with individual genes as the nodes, and directed edges indicating how each transcription factor controls the adjacent gene, elevating protein production, you can build a system of ordinary differential equations to describe how it works.

This system of equations can be seen to reach an equilibrium – a global basin of attraction where the levels of each gene work in exactly the right way to determine how the cell behaves. These basins define the cell types – so a liver cell, or a heart cell, knows exactly what quantities of each protein to produce based on these equations, which are hard-wired into the DNA. You can even consider stable periodic behaviour in the system, to understand the heartbeat’s regular cycle.

Of course, this is again all idealisation – but the workings of the heart are something that have been understood from the biological angle for some time, and now mathematics is providing new ways to model and understand it – which will hopefully lead to powerful medical advances we can implement in reality.

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