Creative duo: philosophy and mathematics

Hannes Leitgeb’s research brings together two disciplines which initially seem worlds apart: philosophy and mathematics. Having taken a doctorate in both, the Austrian academic then dedicated himself to philosophy without ever quite forsaking mathematics. Recognised internationally as an eminent researcher, he is now at LMU München establishing the world’s only research centre for mathematical philosophy, with the aim of formulating philosophical questions as precise mathematical problems and applying different disciplines to find the answers.

In modern science, mathematics comes into its own as a lingua franca: a language every discipline can and must use when theories get more complex. It is also the tool that sets the tone in Hannes Leitgeb’s new Munich Centre for Mathematical Philosophy – Language and Cognition. “Just like scientists, we use the methods of logic and mathematics and apply them to all the fields of philosophy. But we don’t reduce philosophy to mathematics on the way,” emphasises Leitgeb. Research still focuses on the traditional philosophical questions such as “What is truth?” and “How should man behave?”

“We mathematical philosophers address the classic questions rather like natural scientists describing a phenomenon,” explains the researcher. Mathematics helps philosophers to draw inferences from their premises. Leitgeb expects this approach to produce clarity: “Mathematics is great for elucidating structures.” Some of the classic questions can be made much more precise or actually answered with the help of mathematical proofs or models. Effectively, philosophical arguments are put under the microscope. In the best case, this reveals exactly what it is that a certain argument states. It might, for example, emerge that the argument is inconsistent, which could not be determined before. Such ‘stripping down’ to the basic formal structure opens up unexpected paths. “In this way, mathematics can lead to the formulation of new, more precise philosophical questions which take us further,” Leitgeb explains.

Much of what goes on in terms of methodology at the Munich Centre has been shaped by logical empiricism. In the orbit of Rudolf Carnap, one of Leitgeb’s role models, this revolutionised philosophy in the 1920s. Logic has developed enormously since then, and other mathematical theories have emerged, too. For example, probability calculus is used in epistemology; graph theory in metaphysics. Leitgeb thinks mathematics will become the most important tool in philosophy. Thus it is very important to him to introduce first-semester students to the fundamentals of logic himself.

A bridge to neuroscience

At the intersection between philosophy and neuroscience the philosopher/mathematician has also forged new paths – in Munich he wants to initiate interdisciplinary projects in this area. Leitgeb is particularly involved in investigating neural networks – both natural, like the brain, and artificial. Brain researchers try to determine empirically which of the brain’s physicochemical processes relate to which intellectual abilities, like recognising faces. “As a philosopher I ask myself why it is these particular relationships that exist and not others. And how we should formulate the scientific theory,” explains Leitgeb.

One of his results is that neural networks draw inferences of the “if A, then B” variety. The philosopher gives an example: “If Oswald did not kill President John F. Kennedy, then it was someone else.” This sentence is valid with a high degree of certainty, but not so the sentence: “If Oswald had not killed Kennedy someone else would have done it.” It is in the nature of if-when sentences that the factor experience determines whether they are accepted or not. The conditions, under which a neural network identifies a sentence like this as logical, are defined. From this Leitgeb has derived rules. “My results are of interest to computer scientists as well. They no longer have to think of an artificial neural network as an impenetrable box inexplicably producing certain outputs from inputs.”

Leitgeb plans to cooperate with brain researchers from the Munich Centre for Neurosciences – Brain and Mind on questions like ‘How do people learn?’ As a basis for neuroscientific experiments he wants to draw up theories which justify the process of learning from a philosophical/rational point of view. However, if the results of the neuroscientists’ experiments reveal that learning functions differently, the philosopher would have to re-examine his assumptions. “An exciting project for both sides which is bound to be productive,” says the researcher.

Mathematics and philosophy – this combination attracts a lot of outstanding young academics to the centre in Munich. “We have enormous drawing power for junior researchers from all over the world,” Leitgeb claims. Thanks to his own excellent connections and international reputation, the Humboldt Professor has been able to initiate numerous collaborations with important partners in the United States, England, the Netherlands and France. He himself is convinced: “We want to play in the premier league.”