Shaping Logic: Formulas, Algebras, and Spaces

Hello Marco! This year you won two AILA prizes: the Gentilini Prize and the Lettieri Prize. Congratulations! The Gentilini Prize recognises your research career as a whole, while the Lettieri Prize recognises a specific paper that we have already discussed in part on MaddMaths! What do you work on, and how would you explain your research in accessible terms?

Thank you! It was a great honour to receive these prizes. I am also very glad that AILA awards them, because they provide important support for young researchers in logic.

In my work, I study logic using tools from algebra and geometry. Logical formulas have an algebraic structure, much as polynomials form a ring. For example, the formulas of classical propositional logic form a Boolean algebra, and different logics give rise to different algebraic structures. This makes algebraic tools available and provides a first layer of simplification.

A further simplification comes from algebra’s connection with topology, the study of spaces. By way of analogy, consider the polynomial x² + y² − 1. It is a symbolic expression that can be manipulated algebraically, but we can also associate it with the shape formed by its zeros—that is, the solutions of x² + y² − 1 = 0: a circle.

In logic, the connection between algebra and topology appears as a correspondence between algebras of formulas and spaces of models. My research develops and uses connections between algebraic and topological structures for different logics. The advantage is that, by crossing this bridge, I can use spatial intuitions to solve algebraic problems: questions about formulas and proofs become questions about the properties of certain spaces.

Thank you for that overview. We have seen just how broad logic is. What led you to choose this field, and how did your passion for it begin?

I have enjoyed puzzles since childhood, and over the years the ones I liked acquired an increasingly logical flavour. At university, logic intrigued me because it seemed to require ideas unlike those found elsewhere in mathematics. For example, how can one show that the continuum hypothesis can be neither proved nor disproved?

I was also struck by the discovery that what I thought I knew about sets was not true. I had assumed that any collection of things formed a set, but Russell showed that this approach leads to a contradiction: Russell’s famous paradox. This left me with many questions.

More recently, logic has also interested me for aesthetic reasons. I like proofs that handle every case uniformly, without artificial distinctions. I have noticed that many of the proofs I find most elegant are precisely those that work in intuitionistic logic, also known as constructive logic.

For example, if two sets defined in very different ways are shown to be in bijection, I wonder: is this merely a coincidence, or is there a genuinely natural correspondence between them? An intuitionistic proof tells us not only that the bijection exists, but also that it is natural in some sense, because it can be made explicit.

Algebraic logic often studies structures that “capture” properties of formal systems. Has an algebraic structure ever surprised you—has the formalism revealed something unexpected about the problem you started from?

A simple example arose from a question I considered while preparing a lecture: if I have a theory about certain objects, and then add a second theory about completely independent objects, can I obtain new information about the first ones?

At first glance, one would say no. If the two theories speak about separate worlds, why should the second tell me anything new about the first?

In fact, the answer is no—with one exceptional case. I realised this by translating the problem first into algebra and then, through the connection between algebra and topology, into a simple question about spaces. The question becomes: given two sets X and Y, is the projection X × Y → X surjective? (To be precise, we should assume that X and Y are Stone spaces, but this does not affect the answer.) The projection is always surjective with one exception: when Y is empty and X is non-empty.

Translating this back into algebra and then into logic, the only situation in which adding a completely unrelated theory changes what can be deduced in the original language is when the new theory is inconsistent—that is, it contains a contradiction, so its space of models is empty—while the original theory is consistent, so its space of models is non-empty.

In that case, although the new theory concerns objects wholly unrelated to the first ones, it still affects the original theory. This is not because it adds any genuinely interesting information, but because, in classical logic, a contradictory theory makes the system collapse: from a contradiction, anything can be deduced.

What fascinates me is that the formalism does more than restate the problem: sometimes it puts the problem into a form in which the answer becomes much easier to see.

Let us close with a broader reflection. It is increasingly difficult for a young researcher to secure a permanent position in Italy, and your path is a concrete example. How has this constant mobility affected your career and your life? What advice would you give to someone embarking on an academic career?

I have indeed moved several times. After my PhD in Milan, I was in Salerno and then in England; now, at the age of 33, I am in Belgium. I have a three-year contract and do not yet know where I will be in the future.

This constant mobility has had several kinds of effect. Scientifically and culturally, moving has been enriching because it has allowed me to encounter very different academic environments and people. On a personal level, however, every move involves separation and makes long-term planning difficult: when you expect to leave again soon, it is hard to put down deep roots. Uncertainty about the future is also a constant source of stress. It fuels a continuing pressure to be more productive, which has repeatedly undermined my work–life balance.

The difficulty of securing a permanent position is a major problem for young researchers, and unfortunately an academic career is not always compatible with the needs of one’s personal life. I have known highly talented young researchers who had to leave academia to find the stability they needed.

To anyone who decides to pursue an academic career, I would say: prepare for some sacrifices, but also approach it with enthusiasm, because it brings great joy and fulfilment: discovering the beauty of mathematics and enjoying the freedom of research; meeting brilliant new people; collaborating; writing a paper; creating something of your own, as an artist creates a work of art.

I recommend working with other colleagues: it is more enjoyable in a group, and the result is better. It is also important to attend conferences, meet other researchers, and present your results, even when they are still preliminary. Finally, I would encourage people not to be afraid of moving to another city or country to pursue their passion. For me, it has been worth it.