The reality (or lack thereof) of numbers is the kind of problem some philosophers consider overwhelmingly important, but it's of no consequence to just about everyone else. It does not make a wink of difference to your life whether the figures in your bank account or the digits on your clock are, in a philosophical sense, really real, so long as they work as expected. The mathematician Paolo Zellini's book, now translated by Simon Carnell and Erica Segre from the 2016 Italian original, does not exactly elevate the number-reality problem to a matter of concern to non-philosophers, and certainly does not explain the problem in a way that will make it tractable to them. But Zellini does offer a creative shift in perspective that challenges certain philosophers and philosophy-minded mathematicians to see the problem differently.

Where one might expect numbers to get their reality from the things they enumerate -- canonically, two apples come before the number two -- Zellini argues that this gets the story backward. Rather, the most philosophically significant examples of enumeration from ancient to modern times used numbers to give reality to the things they enumerated. He reaches this conclusion by setting to one side the bulk of historical enumeration and focusing on philosophical texts about divine and natural existence. Sure enough, in these texts numbers appear to be the source of reality, often by way of a divine agency or inspiration: hence the titular 'mathematics of the Gods'.

The book's second intervention, about the 'algorithms of men', connects 19th- and early 20th-century debates about how to define what numbers really are to subsequent developments in the theory of computing and computability. Zellini links the book's two themes by identifying a trans-historical through-line of interest in how numbers scale and grow through sacred and secular calculations. Such transformations structure questions about what exactly remains stable or immutable, as a basis of understanding what is real.

The reality of numbers, Zellini argues, comes not from their abstract definition but from what he calls their calculability. Seen this way, not all numbers are alike, nor are they equally real. Principles of algorithmic efficiency place hard limits on what can be calculated and with what precision.