In the philosophy lab

Enlarge

Professor Hannes Leitgeb is Chair Professor of Logic and Philosophy of Language, founder and co-director of the Munich Center for Mathematical Philosophy (MCMP), and winner of the Leibniz Prize. In the following interview, he explains how logic and mathematics are brought to bear on philosophical questions concerning belief, rationality, and paradoxes:

Mathematics and philosophy – for some people, they seem worlds apart. What connects these subjects in mathematical philosophy?

Hannes Leitgeb: In mathematical philosophy, mathematical and logical methods are applied to classical philosophical questions: What is truth? How can we justify our beliefs? Which sorts of things exist? And how should we act? When you formulate such questions precisely, certain structures are revealed – and these can be described mathematically. It’s not about reducing philosophy to mathematics, mind you, rather mathematics is a tool for philosophizing.

Mathematical philosophy is guided by the standards of empirical science – transparency, comprehensibility, and logical rigor. It defines concepts, formulates theses explicitly, and examines whether arguments bear up logically. This creates a network of relationships between abstract principles and concrete examples. When we do this, a philosophical theory is not just an inspiring idea, but a coherent system that makes apparent what follows from what – and where it might fail.

What are the benefits of formally investigating philosophical questions instead of using more traditional literary models?

It brings one thing above all: clarity. Formal methods force researchers to define concepts like truth, knowledge, and rationality such that their use becomes transparent. This does away with many ambiguities, which can easily arise in purely linguistic discussions. And the formalism makes it easier for us to systematically investigate language and rational thought and action.

In addition, mathematical models open up a sort of laboratory of philosophy. Just as physics idealizes a pendulum to understand its motion, philosophical problems can also be modeled in simplified terms. This allows us to vary assumptions, test theories, and compare alternatives. Moreover, formal models allow us to strictly evaluate arguments – which can lead to completely new insights.

Could you give a classic example?

In around 1900, the German philosopher and mathematician Gottlob Frege was trying to demonstrate that all of arithmetic could be derived from logical principles alone. Then the British logician and philosopher Bertrand Russell discovered a contradiction in Frege’s theory – and was able to logically prove it. While the letter from Cambridge was a shock for Frege, it was a boon for philosophy: Parts of modern logic developed out of Russell’s paradox, according to which not every property defines a set. And both mathematics and large swaths of theoretical philosophy today are founded on modern logic.

Scaffolding of reasons

Which philosophical questions do you research – and how can they be mathematically deciphered?

Aside from philosophical logic, the main focuses of my work are on epistemology, the philosophy of mathematics, and the philosophy of language. As part of my epistemological investigations, I ask what it means to rationally believe something. We all have opinions about the world – but with different levels of conviction: That 2 + 2 equals 4 we take as proven and therefore certain; that the sun will be shining in Munich tomorrow we see as probable – depending on the weather forecast.
These gradations of rational belief can be described mathematically. If you are 70 percent certain that the sun will shine tomorrow, then you must be 30 percent certain that it will not shine. If there is new information – such as an updated weather forecast – these degrees of belief should be adjusted according to the mathematical rules of conditional probability.

This foundation gives rise to a mathematical theory of rational yet uncertain inference – Bayesian epistemology – which allows us to precisely investigate classical epistemological questions about knowledge, learning, and the reasons for beliefs. At the same time, it builds bridges to statistics, psychology, economics, neuroscience, and computer science. After all, these disciplines are also concerned with how to draw rational inferences from uncertain data.

What’s your research into the philosophy of mathematics about?

Here I’m addressing the foundations of a subject that is otherwise considered the epitome of certainty. But what exactly are mathematical objects – numbers, sets, functions, geometric figures? Or more generally: What are mathematical structures? What makes a mathematical statement true – whether it’s a proposition about the solvability of an equation or one like “There are infinitely many prime numbers”? And, can the entirety of mathematics really be founded on pure logic – as Frege and Russell hoped?

Although such questions might sound abstract, they determine how mathematics understands itself. They relate to our understanding of mathematical proofs, the explanatory power of whole theories, and the question of whether the broad edifice of mathematics can be reduced to a few principles. The philosophy of mathematics reveals how closely logical precision, mathematical creativity, and philosophical analysis are interwoven – and why the foundations of a subject that seems so firmly established have to be continuously reassessed.

On sentences and their truth value

And what are you researching in the philosophy of language?

For example, the semantics and logic of if-then statements. Conditional statements like “If it rains tomorrow, I’ll take the umbrella” sound straightforward at first, but they’re highly complex from a theoretical perspective: Under what conditions are they true? Do they possess truth conditions at all, or do they rather express conditional probabilities?

What logical inferences do they permit?

To find out, researchers in logic and the philosophy of language analyze the linguistic units that make up statements – linking words like “if-then,” “and,” “or,” “not,” “there is,” etc. – which form the logical scaffolding of our thought. For simple statements like “Snow is white,” we can define truth without any difficulty. At the other end of the scale, we have extreme cases like the sentence “This sentence is not true,” the so-called liar paradox, where language seems to encounter its own logical limits. And yet we can also explain the truth or falsehood of such sentences with logical rigor. Examples like these show that formal methods can be applied not only to numerical quantities, but also to everyday things like the structure of ordinary sentences.

So mathematical philosophy doesn’t just work with numbers?

No. You see, modern mathematics is primarily a structural theory. Wherever structures can be exactly defined, we can employ logical and mathematical methods. This can be the probability with which we rationally believe something, but also the logical structure of arguments, relations between concepts, or models for truth and meaning.

At the MCMP, therefore, we don’t just “juggle numbers.” We work with logic, set theory, graph theory, probability theory, decision theory, game theory, computer simulations – and often very specifically with linguistic structures. We’re trying to make the aforementioned “scaffolding of thought” visible – and want to understand which types of inference are valid within it – regardless of whether these inferences are expressed with the help of numbers, concepts, or sentences.

Does this run the risk of overly simplifying or specifying complex philosophical relationships?

These risks are always present, as any method can be applied inappropriately or ambiguously. As such, my first question when students present me with a formal treatment of a philosophical problem is: Does this formalization pay off philosophically? If not, I urge them to drop it.

Philosophical concepts and questions should never be hastily formalized without good reason!
Conversely, there are now philosophical fields the key findings of which simply couldn’t be justified without formalization. So it’s important to be circumspect: to mathematize where it creates insight – and hold back where it would only generate spurious precision.

Making AI systems verifiable

What are you working on at the moment?

One of the main focuses of my present research is on the logic of reasons, particularly reasons for beliefs and actions. A practical application of this research in philosophical logic could consist in reappraising how logic relates to neural networks – that is, to those types of computer programs which, rather like our brains, learn from varieties of examples and are able to recognize patterns. Locally, such networks can be described like physical systems. But I’m interested in a question that already fascinated me while writing my dissertation around the year 2000: Can the workings of such networks also be described ‘symbolically’ – which is to say, following certain rules to get from premises to a conclusion as we do in logic? Back then, I was able to demonstrate that the answer was “yes” in principle – although my results were mere proofs of existence, which could not easily be translated into practical applications. Moreover, there was not that much interest in neural networks in those days.

Today, however, these networks are actually making loan decisions and carrying out risk analyses – sensitive decisions and forecasts, the justification or lack of justification of which we must be able to understand. Accordingly, I’m working on a theory of ‘reasons’ that can be applied to human and machine inference, and which could explain the rationale for a network arriving at a certain result. My MCMP colleague Levin Hornischer and I have just written an article on this subject.

What role could logic play in AI in the future?

Although I don’t believe in a return to good old symbolic AI, I see very good grounds for supposing that there will be something of a resurgence of logical methods in computer science – this time around, as a tool for reconstructing and critically evaluating the methods by which neural networks draw conclusions. I’m therefore very interested in whether logic and today’s AI could be brought back closer together.

If we cleverly combine philosophical theories of reasons, logical methods, and machine learning, we could create AI systems that are not just powerfully effective but which would also be understandable and rationally verifiable. The fact that the MCMP is working at precisely this interface makes me optimistic – for philosophy, to be sure, but also for our understanding of machines which are increasingly becoming part of our everyday lives.

Profile:

Prof. Hannes Leitgeb is Chair of Logic and Philosophy of Language and co-director of the Munich Center for Mathematical Philosophy at LMU. Leitgeb studied mathematics, computer science, and philosophy at the University of Salzburg, where he obtained a doctorate in mathematics in 1998 and a second PhD in philosophy in 2001. Subsequently, he worked as a non-tenured assistant professor (Universitätsassistent) at the University of Salzburg, before spending a year as an Erwin Schrödinger Fellow at Stanford University in 2004. From 2005, he worked in the Departments of Philosophy and Mathematics at the University of Bristol, first as a reader and then from 2007 as a professor. In 2010, Leitgeb was awarded an Alexander von Humboldt Professorship and came to LMU as Chair of Logic and Philosophy of Language. In 2025, he was awarded a Gottfried Wilhelm Leibniz Prize by the German Research Foundation (DFG) and the Frege Prize by the German Society for Analytic Philosophy (GAP).