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As a philosopher, Pieter Booij specializes in logic and the foundations of mathematics. In the next four short essays Pieter will interpret a selection of Wittgensteins aphorisms in the context of the Corona pandemic. Wittgensteins second masterpiece is the Philosophical Investigations, which consists of a collection of aphorisms. These aphorisms read like poetry to Pieter, and like poetry, they are not only open to interpretation, but it are intentionally meant to be interpreted.
Corona-statistics and reality
The more closely we examine actual language, the greater becomes the conflict between it and our requirement. (For the crystalline purity of logic was, of course, not something I had discovered; it was a requirement.) The conflict becomes intolerable; the requirement is now in danger of becoming vacuous. – We have got on to slippery ice where there is no friction, and so, in a certain sense, the conditions are ideal; but also, just because of that, we are unable to walk. We want to walk: so we need friction. Back to rough ground!’ 1
Since the beginning of the corona crisis, mathematics, especially statistical modeling, seems to have caught the public interest like never before. Talk of ‘flattening the curve’ probably made a lot of people reminisce what they did in high school during those lovely mathematics class hours. Perhaps it even made some people wish they had paid more attention to their teachers. 2 As a teacher of mathematics myself, I encourage any interest in mathematics and like the fact that more people are being remembered of mathematics and the value it has for society, but we need to be cautious. Mathematics can give us a false sense of security. The clarity of numbers and graphs can offer us a feeling of safety and hope. Currently, the corona-infection graphs show us that we are over the top of the bell-curve and on the downward slope towards an ever-declining daily infection-rate. We should, however, not blindly put our faith in those graphs that represent the amount of corona-infections. Neither in their power to be genuinely representative, nor in their predictive power. We have to understand the limits of those statistical models and the limits of mathematics as a whole.
A plethora of problems arises when relating mathematics to the natural world; imperfect models, observational noise, false negatives and positives, uncertainty of the initial condition, etc. All those problems allow for uncertainty and an exponential growth of that uncertainty over time, debilitating any predictive power. However, the most important problem is not one regarding prediction but one about representing reality. It is not the problem that when viewing reality from a mathematical perspective we are in danger of disregarding the complexity of the situation of corona in the natural world and the empirical reality it creates. To be considered is a fundamental problem, a problem impossible to solve or overcome, even in theory. It is the fact that any model, even if it is a perfect model, is merely a model.
Mathematical models are either an abstract interpretation of empirical reality, distilling certain structures out of it, or they are coherent systems with structures that may reflect those of the empirical reality they are meant to describe. Applicable to it, but independent from it. They are not the structures of empirical reality itself. Making the mistake that they are, creates the illusion that reality should follow those mathematical models and that we can therefore know the present and predict the future.
Confusion in this demarcation of mathematics as it relates to empirical reality, is the same kind of confusion that was being made in the first decades of twentieth century philosophy of language: confusion of how logic relates to language. The “crystalline purity of logic” represents Wittgenstein’s belief in the possibility of a normative reformation of language as a meaningful endeavor. This normative reformation of language was what he was trying to accomplish with his first masterpiece: the Tractatus Logico-Philosophicus, written during World War One. In this work, he develops a logical system which tries to mirror the structures of meaningful language and, parallel to it, the structures of empirical reality. Wittgenstein believed at that time that he was truly describing, analyzing the structures of meaningful language. However, what he was doing was creating a normative framework for language in order to be meaningful. His logical system was therefore a requirement, not a description of actual language. Actual language is much more complex, it does not follow one logical structure and sometimes the logic is very difficult to see or understand, if it is there at all. Think of poetry, prayers, emotive utterances, i.e. all instances of language that convey meaning in their own way. And even in those parts of language where the grammar’s logic is clear, we should not have the illusion that the logical structure trying to describe the grammatical structure are one and the same.
Statistical models that try to describe corona-infections create “conditions that are ideal”, in the sense that they create a clear picture and a path out of the current crisis. But we should be weary that those models don’t become vacuous so that we will be unable to walk on that path. If we have the illusion that they really are the descriptions of empirical reality and are not aware of the normative framework they place on that reality, we will create a dangerous dynamic. If we believe empirical reality corresponds to statistical models we could take measures and actions entirely based on them. Those actions will alter the empirical reality that corona creates itself and thereby change the correctness of the predictions those models offer for the worse. Human agency has the potential to be very disruptive for correctness of statistical models. If we all decide to stay at home tomorrow, we can. If we all decide to go to a party tomorrow, we can. We should keep our eyes open to the reality around us. Even if the numbers of the daily corona-infection are low, we should still wash our hands. Turn away when someone sneezes. Back to rough ground!
Text: Pieter Booij en Heleen Janssens.
Edit: Kees Muller.