Envisioning Mathematics – ESHS symposium’s call for contributions (deadline 13.12)

Dear colleagues,

I would like to invite you to participate in the proposal for the panel on Envisioning Mathematics for the European Society of History of Science conference in Bologna. If you would be interested in contributing an abstract about 200 words, please let me know by this Friday, December 13. 

What do we imagine, thinking of the foundations of mathematics? Could we not only think of, but also make conjectures of mathematical concepts or formulas? Is it possible to comprehend the ‘Absolute Infinite’ and could a world contain a symbol that helps people – philosophers or mathematicians – understand the absolute? For Georg Cantor an absolutely infinite sequence of numbers was the “appropriate symbol of the absolute”. As Cantor’s set theory has become the foundation of contemporary mathematics, it is essential to clarify the philosophical background that Cantor associated with it in order to elucidate his interpretation of the problem of infinite. On the one hand, in her article Cantor on Infinity in Nature, Number, and the Divine Mind, Anne Newstead (Newstead, 2009) delves into the symbol of a ladder formed by number classes. On the other hand, Galina Sinkevich maintains, that “functions and sets that were hard to imagine were introduced, and often the only thing that was postulated about them was either their sheer existence or some other characteristic quality” (Sinkevich, 2013).

The most abstract science, mathematics, is not always related to imagination as much as to the ability to build hypotheses. The art of mathematical conjecture could be grasped by the list of 23 mathematical problems, set out by Hilbert. Barry Mazur (Mazur, 1997) wrote, that the art of conjecturing has achieved a formidable, and quite formal, prominence in the mathematical landscape. Gödel’s incompleteness theorem — often seen as the greatest logical achievement since Aristotle — did not herald the end of mathematical logic. Instead it induced a blossoming that even led to the development of modern computers.

Returning to the mathematical imagination, let us remember the drawings and diagrams of erudites from different disciplines studying the basics of mathematics. Here the archival ‘scratchwork’ from the notebooks of Nikolai Bugaev (1837-1903), Ivan Lapshin (1870-1952), and other logicians could be presented. All polymaths, from differing disciplinary bases, these characters were investigating foundations of mathematics, with a particular interest in logic. A careful review of drawings and diagrammatic reasoning in this scratchwork elucidates various background assumptions and background intentions, which are noticeable in printed literature left by these scholars but never clearly explicated.

Alternative vision of the imagination in mathematics is the depiction of mathematical objects in the arts. The example here is an artist Alexander Pankin, who trained to be an architectect. The research experience pushed him to the next step - the realization of mathematical objects in the art space ("meta-abstraction"). Number series, irrational and transcendental numbers, scientific ideas became the impulse for the shaping of the artistic form.

For the symposium on Envisioning of Mathematics various topics related to imagination, hypothetical knowledge, conjectures, sketching, visualization will be very welcome. In addition, the reception of mathematics and mathematical concepts and formulas in the arts will be also interesting.

For expression of interest please contact: tvlevina@hse.ru

Tatiana Levina

School of Philosophy, Faculty of Humanities

Higher School of Economics

Moscow, Russia