## Group 4: Mathematics, Numbers and Value

Wednesday 9th of December 2020

15:30-17:00 GMT

Adrian Liu, Stanford University

‘Incomparability in Comparison and in Choice’

*Abstract:*

We regularly face choices wherein our options resist comparison. In November, I will have to vote for a cash-bail or an algorithm-bail system in California, where neither choice is clearly better, but neither do they seem equally choice worthy. Rather, they resist comparison. Nonetheless, faced with the question what to do, I must choose some course of action. If we must compare options to determine how to proceed, then rational choice is threatened if options cannot be compared. But the literature on such incomparability has largely assumed that the possibility or impossibility of comparing certain items is unrelated to whether or not the comparison occurs in the context of some practical choice.

I think this assumption is mistaken. My paper motivates the significance, for studying incomparability, of a distinction between choice situations, wherein there is a set of options for choice and I am to choose one of the options, and comparison situations, wherein there is a set of items (not necessarily choices) and I am to compare them with respect to some value (but not necessarily to choose one). To explain the putative incomparability of options for choice, I argue, we must pay closer attention to the differences between choice situations and comparison situations.

The paper is in two parts. In the first part, I argue that choice situations and comparison situations are structurally different in terms of what sort of moves are rationally justified within each situation. In non-choice comparison situations, we have agency over the parameters: the items to be compared, the values by which we compare them, and the sort of conclusion we must reach. But in practical choice situations we have no such agency. The necessity of choice places rational limitations on how we can narrow the scope of our options and the values we use to compare them. And the ultimate conclusion we must reach is always the same: we must make a choice.

These are significant structural differences between choice situations and comparison situations, and in the second part I explain why the differences are relevant for theories of incomparability. First, I argue that the distinction creates new explanatory burdens for both of the main types of theories of incomparability: hard- failure theories and vague-failure theories. Hard-failure theories posit a determinate failure of standard comparative relations. But their plausibility depends chiefly on thought experiments of comparison situations, and the theory is less appealing when we move to choice situations. Vague-failure theories claim that when items are incomparable it is simply vague whether each of the standard comparative relations applies. But certain features of comparison situations make vague-failure less compelling, meaning that the vague failure theorist must explain why features of choice in particular can preserve vague-failure for choice situations.

Second, I suggest that my distinction can deflate a tension between two basic intuitions about comparison in choice situations: first, that choices must always be made on the grounds of comparisons, and second, that comparisons of values and bearers of values is often inappropriate or distasteful.

Noëlle Rohde, University of Oxford

‘Value by the Numbers: Quantificational Discrimination and the School Mark’

*Abstract:*

One of the key forms in which value is fixed and communicated is the number. From university rankings to salaries, from 10-point attractiveness scales to the Chinese Social Credit score - numbers seem to offer a clear, simple and transculturally understandable means of expressing value.

However, one of the crucial drawbacks of quantifying (human) value has to date hardly been discussed: discrimination. Scholarship on discrimination has been traditionally skewed in two respects. Firstly, philosophers have focused on what I call “classical discrimination,” that is unjust, disadvantageous treatment based on non-quantified features such as gender or ethnicity. Secondly, discrimination has always been described as an act which requires at least two individuals - the “discriminator” and the “discriminee,” if you will. In this paper, I aim to broaden the scope of the debate by redressing these two shortcomings.

Firstly, I introduce what I call “quantificational discrimination”. I define the phenomenon as any act, practice or policy that imposes a relative disadvantage on persons based on their membership in a numerically defined group. I show that quantificational discrimination is harder to detect and harder to call out as wrong than its classical counterpart. What is more, it is often mistaken for legitimate selection and it affects not only marginalised groups but has a much broader social impact. Secondly, I advance the argument that it is possible to discriminate against oneself, that is to engage in what I call reflexive discrimination.

The liaison of both insights yields the concept of reflexive quantificational discrimination. Prompted by the (implicit or explicit) value judgements encapsulated in numbers, individuals treat themselves unjustly. In particular, they unduly withhold internal goods such as self-trust, self-esteem or credit from themselves.

In order to illustrate the social significance of reflexive quantificational discrimination, I draw on empirical examples from my philosophical-anthropological doctoral research. During a year of ethnographic fieldwork in Germany, I am investigating a site in which numbers and value are inextricably intertwined: the school. Using students’ lived experiences with being graded as a window onto social quantification at large, I offer insights into how reflexive quantificational discrimination plays out in practice and I argue for its theoretical recognition.

Deniz Sarikaya, University of Hamburg

‘How mathematization might render narratives of mathematics better (and why this matters!)’

*Abstract:*

There are different narratives on mathematics as part of our world, some of which are more appropriate than others. Such narratives might be of the form ‘Mathematics is useful’, ‘Mathematics is beautiful’, or ‘Mathematicians aim at theorem-credit’. These narratives play a crucial role in mathematics education and in society as they are influencing people's willingness to engage with the subject or the way they interpret mathematical results in relation to real-world questions; the latter yielding important normative considerations.

In this talk, we want to analyze different narratives of mathematics and suggest that mathematizing as a virtuous practice in its own right is a better narrative of mathematics than, for example, extrinsic narratives which focus on the results of mathematical activity and the application of mathematics in non-mathematical contexts. By ‘better’ we mean that the mathematizing-narrative describes mathematical practice more adequately and that it allows for a shift in mathematics education that yields beneficial outcomes for our society. We argue that the fundamental activity of doing mathematics, or, more precisely, of introducing, using, varying, applying ... mathematical symbolism is a virtuous practice — what we call mathematizing, drawing on Freudenthal’s research in mathematics education.

includes individual choices on the component factors of the model. We argue that mathematizing, parallel to virtues such as art appreciation or art production, is beneficial for personal flourishing as it opens up a new aspect of reality — or at least a new perspective on it — that is not available without mathematizing. This virtue narrative focusing on mathematizing is better than other competing narratives that are currently more present in society. The latter often hide the arbitrary component factors of mathematical models which depart from the real-world context for reasons of reducing complexity or favoring simplicity of the mathematical tools for example. A mathematical model is often perceived as an objective and true representation of a societal context. If it, however, becomes clear that any normative conclusion, which is (partly) grounded in such a model, is directly connected to the choices made by building the model, then we can reduce the risk associated with the authority of formal tools in public debates.

Mathematizing means modelling a context in mathematical terms, which

We start with a short exposition of Freudenthal, analyse the ethical consequences as to be found in Ernest’s Philosophy of Mathematics Education, we than look at four narratives in particular, which we might title by the simplified slogans:

1. Mathematics is useful

2. Mathematics is beautiful

3. Mathematicians aim at deep understanding

4. Mathematicians aim at theorem-credit