If the old cliché about mathematicians were true, the lack of contact during the coronavirus pandemic would barely have hindered the work of Dr. Anton Freund. Don’t mathematicians usually just brood over tricky problems on their own? But it may be a surprise to learn that they also need close dialogue with their specialist colleagues. “It is unbelievably important”, emphasises Freund, a researcher in the Logic Group at the Technical University of Darmstadt. This is because he believes that mathematics is a creative process that can only succeed if mathematicians work together.
Freund says that he has really missed his discussions with colleagues over the last few months. That’s why he is particularly pleased about his successful application for funding from the Emmy Noether Programme at the German Research Foundation (DFG). It will allow the 31 year old mathematician to head his own junior research group. “The exchange of information and ideas in the group that I will head will be really amazing”, says Freund enthusiastically. Even if a conversation may not lead to a solution directly, “it is very often the case that people have new ideas after a discussion”. Freund is also looking forward to discussing his work with the students who attend his lectures, which he will hold from October 2021 onwards.
Logic as a fundamental thinking tool
His specialist field of logic was already a fundamental tool for Aristotle. The ancient Greek philosopher asked himself how new discoveries could be derived using logical reasoning. It was Aristotle who defined the concept of inference through syllogism. Two premises such as “all men are mortal” and “Socrates is a man” can be used to derive a third statement, in this case “Socrates is mortal”. These simple beginnings have been transformed into a highly complex logical structure over the centuries: mathematical logic.
The latter deals with mathematical proofs. Mathematicians use a few fundamental assumptions – known as axioms – to derive lots of new statements. One such axiom is for example: “Every natural number has a successor”. “This method has proven to be a very powerful tool”, explains Freund. Mathematicians explore further and further, like discoverers of a new continent, by constructing propositions out of axioms and finding new ways to combine them in order to build other propositions. In mathematical logic, these discoveries are analysed at a fundamental level. Logic thus creates a map of the continent of mathematics.
Anton Freund wants to further explore this mathematical landscape. When he describes the key questions behind his research, he starts to gesticulate and you can see the joyful anticipation of new, exciting insights on his face. “What mathematical axioms do I need to achieve which mathematical results?”, is one question that he asks. Or: “Can I differentiate between mathematical results that are achieved with fewer axioms and others that require more axioms?”
“This is how we discover new methods for answering our most fundamental questions”. “This is how we discover new methods for answering our most fundamental questions”.
Although mathematicians have already been exploring the landscape of their subject for many decades, it is a very diverse environment made up of rivers, mountains and forests. It is not as straightforward to forge into as many laypeople may imagine. The principle problem was identified by the Austrian-American mathematician Kurt Gödel 90 years ago. His incompleteness theorem says that mathematics as a whole can never be derived from one single set of axioms.
Therefore, there are multiple systems of axioms with which different areas of mathematics can be opened up. Described in pictures: Some summits can be reached with walking shoes, while others require climbing gear. Mathematical logicians like Freund investigate what is the minimum needed to achieve a certain goal. They try to make Gödel’s general finding more concrete. The fact that this is at all possible with the help of logic is what fascinates Freund the most about his specialist subject.
Freund is investigating, for example, a theorem that can only be proven using a complex system of axioms. This means that it doesn’t fit, so to speak, in its own natural environment in which things are easier to prove. “We do not yet know enough to be able to classify this graph minor theorem precisely”, explains Freund. It can only be derived in any case from extraordinarily strong axioms. But how strong do they actually have to be? And what precisely can be proven with the graph minor theorem?
“What we are doing is very abstract”, explains Freund. “However, it has links to practical applications”. The graph minor theorem helps computer scientists to, for example, decide whether certain complex algorithms will reach a result within an acceptable period of time or not. The findings of the researchers at TU Darmstadt may help to improve these methods.
“Darmstadt is one of the most renowned universities worldwide for proof theory”
Freund is thus following a long tradition of logic research at the Technical University of Darmstadt where there is a strong application-oriented focus, according to Ulrich Kohlenbach, who since 2004 has been the Professor of the Logic Group where Anton Freund works. “Darmstadt is one of the most renowned universities worldwide for proof theory”, says the mathematician. “We are known for applying mathematical logic to mathematical proofs and also to the area of computer science”. The roots of this work go back many decades. Pioneers in artificial intelligence – such as the computer scientist Wolfgang Bibel and the mathematician Rudolf Wille – worked in Darmstadt. According to Kohlenbach, the current Logic Group can be traced back, in particular, to Wille, who was a professor in Darmstadt from 1970, and to Klaus Keimel (who worked in Darmstadt from 1971).
There are also connections to the computer sciences in teaching, says Kohlenbach. “Logic-related lectures for the computer sciences at TU Darmstadt are traditionally held by mathematicians”. A third of the participants in the logic lectures are often computer science students. This may be the case in the lectures held by Anton Freund from this autumn onwards.
In addition, he will also supervise two doctoral students working on his research questions. “The opportunity to head a small team is something that is normally not possible at my career level”, says Freund excitedly. He hopes that the joint research will produce significant advances. “We want to combine two fields of research with one another”, explains the mathematician. “This is how we discover new methods for answering our most fundamental questions”, he says. Amongst other things, the research will also focus on the graph minor theorem mentioned above. Freund is looking forward to the group experience: “Although the doctoral students will be learning from me, they will also have the opportunity to contribute their own ideas”, says Freund. “The shared creative process will be extremely exciting”. Maybe the young researchers will even discover a whole new landscape on the continent of mathematics.