Challenging the Infinite: About the Conference
The conference aims to foster greater debate and dialogue about the costs and benefits of potentialism in mathematics. In recent years there has been a resurgence of interest in potential infinity in a variety of forms, most notably (though certainly not exclusively) in Modal Potentialism.
The exploration of potentialism emerges from different motivations and philosophical projects. For some it offers a more rigorous account of the iterative conception of sets; for others it is part of a broader pluralism about mathematics; whilst for others still who have lingering scepticism about the philosophical plausibility of actual
infinity, potentialism offers a less problematic way to interpret much of the work traditionally done in actualist set theory and mathematics.
For Aristotle's mathematical entities like numbers, lines, and fractions were inherently potential. Though this view was certainly challenged, it remained the dominant view until Cantor whose work in set theory brought actual-infinity into the core of mathematics. The sheer utility and unifying breadth of set theory meant that even sceptics like Hilbert were drawn to ‘Cantor’s paradise’.
Yet potentialism in various forms persisted. For some such as Kroenicker, Brouwer, Hilbert and Dummett, the notion of having a completed infinite set was philosophically problematic. More recently the iterative conception of the set appears to offer a compelling motivation for the ZFC axioms by appealing to a picture of the universe that is inherently potentialistic.
But potentialism is not without its own challenges and problems. One major question for the potentialist is whether the potentialist is tacitly employing an actually infinite domain. If not, then can they account for claims such as those about ‘all numbers’ or ‘all sets’? Other questions arise as to whether the potentialist might fall victim to the success of their own translation schemes between potentialist and actualist mathematics, blurring any actual differences between the two, and if so whether they have simply added an unnatural and bloated syntax without sufficient benefit.
Whilst the technical work on potentialism has grown in scope and sophistication, there has been to date only limited conversation about the fundamental metaphysical and epistemological questions that underpin such programmes. It is our view that philosophical ideas are sharpened when in dialogue with alternative perspectives; when ideas are challenged they become sharpened and refined.
This conference brings together philosophers who have worked at the heart of potentialist programmes with those who have begun to challenge such work. Apart from the talks, there will be plenty of breaks to allow for the real dialogue to take place. We hope that attendees will not merely be spectators but will be able to take an active role in furthering the discussion and debate thus helping to enrich the philosophy of mathematics at its core.