The question of how mathematics could lay the foundation for a machine that sustains such a wide variety of practices is really quite well understood from the point of view of the mathematical theory of computation. From a humanities standpoint however, despite the number of texts commenting on the genius of key figures such as Gödel, Turing, Shannon, and Church, there is still a certain awkwardness when it comes to situating the key steps in mathematical reasoning that lead up to the birth of the computer in the larger context of mathematics itself. One of the questions I find really quite interesting is the role of the formalist stance in mathematics.

In the philosophy of mathematics, there are many different positions. The realist stance for example holds that mathematical objects exist. For the platonist, they exist in some kind of extra spatio-temporal realm of ideas. For the physicalist, they are intrinsically connected to material existence, even if that relationship is not necessarily simple. Then there is formalism and this is where things get interesting. In a tale we can read in many social sciences and humanities books on the computer, there is the young Kurt Gödel that smashes the coherent world of the “establishment” mathematician David Hilbert, inventing the metamathematical tools that will later prove essential for the practical realization of computing machinery in the process. What is most often overlooked in that story is that Hilbert’s formalist position is already an extremely important step in the preparation for what is to come. For Hilbert, the question of the ontological status of mathematical objects is already a no-go – truth is no longer defined via any kind of correspondence to an external system but as a function of the internal coherence of the symbolic system. As Bettina Heintz says, Hilbert’s work rendered mathematical concepts “self-sufficient” (autark) by liberating them from any kind of external benchmark and opening a purely mechanical world where symbolic machinery can be built at will, like in a game.

If we want to think about computing today, I think we should remember this break from an ontological concept of truth to a purely formalistic one (even if that mean Gödel put a pretty big crack in it lateron). Because in a way, programming is like a “game” with formulas and if the algorithm works, that means it is “true”. In this sense, Google’s PageRank algorithm is true. But without the reference to an external system, this “truth” is purely mechanical, internal. In a similar way, an algorithm’s claim to objectivity, impartiality, or neutrality should be seen as internal only. The moment we apply mathematics to the description of some external mechanism (gravity, for example), there is a second truth criterion that intervenes, which refers to the establishment of correspondence between the formal system and the external reality. In the same way, if an algorithm is applied to, let’s say the filtering of information, the formal world of the game is mapped onto another world. There is an important difference however. When mathematics are applied to physical phenomena, the gesture is descriptive and epistemological (verb: is). When an algorithms is applied to tasks such as information filtering, the gesture is prescriptive and political (verb: ought).

The fact than an automatic procedure works makes it true in a formal sense. The moment we apply it to a certain task, other criteria intervene. Hilbert’s formalism pulled mathematics from the empirical world and if we bring the two together again by writing software, the criteria by which we judge the quality of that action should be seen as political because there are no mathematical criteria to judge the mapping of on world onto the other. No Hilbert to hold our hand…

Post filed under computing, epistemolgy, mathematics.

2 Comments

  1. First, Turing’s thesis that underlies Computer Science is based on the rejection of Hilbert’s principles. The thesis has proved that mathematics is not powerful enough to uncover truth as it was earlier thought to be. Second, the problem solving using computers has two sides. While the formal side is based on logic pure mathematics, the empirical side is subject to interpretation because it has to do with practical applications that needs to engage with human reality. The latter is really outside the realm of mathematical reasoning; otherwise, we would not have read Penrose’s Emperor’s New Mind or about Searle’s Chinese Room Experiment!

  2. bernhard says:

    Hello mk, thanks for your comment! I do believe however, as I wrote in the post, that the idea that Turing’s work on the Entscheidungsproblem is “based on the rejection of Hilbert’s principles” is flawed, because it only takes a very narrow historical window into account. Certainly, Turing put another nail into the coffin of an idea of mathematics where every problem is in principle solvable, but his work was done inside of Hilbert’s formalist paradigm. The idea that mathematics is nothing but symbol manipulation without any referent to an external world as judge of correctness was most forcefully defended by Hilbert – and the Grundlagen der Geometrie indeed became a model for 20th century axiomatic mathematics – and both Gödel’s and Turing’s work is, in that sense, a continuation rather than a break. Sure, Hilbert’s program would never be completely fulfilled, but the logical and metamathematical space opened by the German mathematician is precisely the one that would later give birth to the computer…

    For the second part of you argument, I completely agree, but the real problem is probably not the logical question but the sociological attraction of what Gallison calls mechanical objectivity…

    best,
    B.

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