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# A God-like view over maths

‘Logicians in my field consider themselves above mathematicians,’ jokes Bahareh Afshari. She is a logician, who started her MacGillavry Fellowship at the Institute for Logic, Language and Computation in the beginning of 2019. ‘Not to sound arrogant, but logicians have a God-like view over what is going on in maths. They analyse how mathematicians work, how they reflect on things, and establish new theorems.’

She clarifies the field of logic with a comparison: ‘When you consider mathematics as consisting of different languages, the language of algebra, the language of analysis and the language of number theory, then logic would be the science of languages.

So a logician could say: “I know an infinite number of languages because I can abstractly look at them and say something amazing or useful about it.” But you might get a different answer if you ask a mathematician.’

## The science behind proofs

Afshari mainly works in the mathematical field called Proof Theory. ‘Which means I study the correctness of arguments in different mathematical languages. I check what kind of arguments are acceptable and which are faulty. Every step should click together to a conclusion.’

In 2017 she published a breakthrough paper on “cyclic proofs”, which answered a question that had been open for over two decades. ‘Cyclic proofs are arguments that go round and round,’ Afshari explains.

‘It is a phenomenon that often occurs in mathematics when you want to prove something. You first reduce the problem into something else, and then to something else again, and after a while you come back to the same problem you started with. You can then ask yourself: “Did I make any progress in this argument?” Surprisingly, sometimes you do. We call these cyclic proofs.’

Afshari’s work is theoretical and has no immediate applications. But her work could be used to analyse the underlying mathematical structures of reactive computer systems. Reactive computer systems are systems that engage with the environment and it is critical to ensure their correctness for safety reasons.

Afshari: ‘Take for example the computer systems that are used for managing railways. Obviously there should be only one train on a track at a time. In this case logic expresses the safety and efficiency properties: is the system operating correctly? Are there any faults? With the help of logic one can ensure that no two trains are allowed on the same track at the same time and a train waiting at a signal will eventually be permitted to pass.’

### The pattern of mathematics

The process of problem solving is what’s driving Afshari in her work. ‘I constantly think about my work. On the train, while cooking, walking or stroking my cat on the sofa. It is hard to put the problems down. It feels so good when you solve a little bit of it.’

‘I had a hard time memorising things at school,’ Afshari tells about her youth in Iran. ‘But mathematics had a pattern, and that I could remember.’ So it was a simple decision for her to study maths. And as soon as she followed some logic classes, she fell in love with it. ‘It enables me to have an abstract view of things. I really liked that it wasn’t specialised and it gave me a perspective on many areas at the same time. I like that level of abstraction. It makes me feel in control.’

### Making proofs simpler

She moved to the UK to pursue a PhD in Leeds, a postdoc in Edinburgh and another one in Oxford. After that Afshari worked at the mathematics department in Vienna, where she worked on making proof arguments simpler and shorter.

Afshari: ‘Proof compressions, as we call it, is a key step in the automatic generation of proofs. The latter has wide applications ranging from Philosophy –understanding different reasonings – to Artificial Intelligence – generating proofs not by humans but by programmes.’

### Is it a fair position?

When she got the MacGillavry Fellowship at ILLC, Afshari was thrilled. ‘And I am still thrilled. I cannot ask for more. It is a heaven for logicians,’ she says. ‘The institute is absolutely unique, it is the only institute in the world that contains all branches of logic: philosophy, computer science and mathematics.

It is very inspiring because there are so many researchers you can talk to. The level is very high. I love the students, they are so clever and they approach you very easily, which brings the classroom to life. And I also love my colleagues, who are very collaborative and inspiring.’

Afshari had mixed feelings about the MacGillavry Fellowship itself, being for women only, but now is happy that the programme exists. ‘I must say I didn’t like the fellowship in the beginning. I asked myself: did I get this job because I am this good or because I am female? I also doubted whether it was fair to the men who could not apply. But when I talked with other MacGillavry fellows, I realised that women lose many opportunities due to bias. So if this position helps us to overcome these biases at a certain point in our careers, I feel okay about it.’

### Female presence in logic

She does not feel like talking too much about what it is like to be a woman in her field of research. ‘There are very few women in my field. If you look at the ILLC list of staff, there’s only a handful of us, most work in philosophy rather than maths or computation. I would like to have more female colleagues. I’m used to working with men, but I would like the experience of collaborating with women.’

Afshari does feel that she is in the position to raise her voice when it comes to female presence in logic. Therefore, she is actively trying to motivate female students, put more women on stage during conferences and makes committee members aware of their biases while hiring staff. ‘I mentor a number of female students. These students are so good, with some encouragement I can see them flourish into very capable researchers.’

### A fashionable topic

Afshari now shares her time between Sweden and Amsterdam. She is also a researcher at the University of Gothenburg, where she works on logics of time and space. ‘I am assessing the limits of these logics in terms of their expressive capabilities so the work has a more philosophical and mathematical flavour.’

In Amsterdam she is busy with setting up her own research group in computational logic. ‘I want to stay at the frontier of this cyclic proof research. I am hiring PhD students and organising a workshop on cyclic proofs which will be held in Amsterdam next year. It is a fashionable topic, and it’s nice to be at the heart of it.’

### CV

 2004-2008: PhD in Mathematics, University of Leeds, UK 2009-2016: Postdoctoral Researcher: Laboratory for Foundations of Computer Science, University of Edinburgh, UK Department of Computer Science, University of Oxford, UK Institute for Discrete Mathematics and Geometry, TU Wien, Austria 2017-present: Associate Professor, University of Gothenburg, Sweden 2019-present: MacGillavry fellow, ILLC, University of Amsterdam, the Netherlands 2021: Member of The Royal Swedish Academy of Sciences