# Can numbers be racist?

It appears to be the case - at least according to historian Sarah E. Bond, an expert in Classical Studies. In her essay Why We Need to Start Seeing the Classical World in Color, Professor Bond claims that the cephalic index is racist.

But what is the cephalic index? Every metazoan from a lineage that endured cephalization (that is, every animal with a head) potentially exhibits a cephalic index. The cephalic index is defined as the ratio of the width of a head times a hundred divided by the length of the same head.

Heads evolved at least four times in our planet (taken from Berkeley's excellent Understanding Evolution)

From the point of view of orthodox dimensional analysis, the cephalic index $[CI]$ is a dimensionless quantity expressed through the formula $[CI]=\frac{a L}{L}$ where:

• The coefficient $a$ is a real number conventionally fixed as $100$
• $L$ is the base or fundamental quantity "length"
• As it is commonly taught, the operation of division is said to "cancel out" the dimension "length" ($L$) of the numerator with another of its instance in the denominator and thus we say that the cephalic index is "dimensionless".

The numerical value of dimensionless quantities remains invariant under different representation standards (for instance, using inches instead of centimeters to calculate a certain cephalic index). Dimensionless quantities are typically simply identified with real numbers. These numbers are deemed to be "pure", that is, unitless, devoid of dimensional content. So claiming that the cephalic index is racist is claiming that pure numbers can be racist.

But I digress; if we were to only acknowledge established orthodoxy, the world would become boring really fast. The fact of the matter is that there appears to be something really troubling going on conceptually with the readied identification of dimensionless quantities with real numbers (see for instance Chapter 10 of [1] and Chapter 5 of [2]). Cephalic indexes in some ways are like angles; both are features defined as the ratio of two lengths. But despite being prototypically "dimensionless", angles are similar to quantities such as times, lengths, masses and resistances in that they can be extensively measured. And we also liberally assign units such as radians and degrees to angles. No wonder Krantz et al. aptly named angles "the bastard quantity of dimensional analysis". This leads to quirky characterizations; for instance, the late dimensional analysis theorist Ain Sonin [3] took angles to be dimensionless quantities that were also derived from the base quantity of length. Some modern treatments of dimensional analysis frown upon tradition and openly include the dimension of angle $\alpha$ side by side with other base quantities [4].

Similar vexations happen with a large number of quantities sporting obvious empirical significance which are expressed logarithmically and thus, by default, dimensionless. This standard narrative of dimensionless quantities and pure numbers is troublesome when one investigates the formal semantics of these features under measurement theory.

But back to our original issue; even if the cephalic index is not a "pure" number, this hardly improves the case that it is "racist". Everything with a head exhibits a cephalic index and the cephalic index has reliably tracked real patterns in the behavior and cognition of metazoans hundreds of millions of years before the first human being displayed enmity towards conspecifics.

[1] D. Luce, D. Krantz, P. Suppes, and A. Tversky, Foundations of measurement, vol. i: additive and polynomial representations, New York Academic Press, 1971.
[Bibtex]
@book{Suppes1971a,
author = {Luce, Duncan and Krantz, David and Suppes, Patrick and Tversky, Amos},
citeulike-article-id = {13531614},
posted-at = {2015-03-02 22:50:45},
priority = {2},
publisher = {New York Academic Press},
title = {Foundations of Measurement, Vol. I: Additive and polynomial representations},
year = {1971}
}
[2] L. Narens, Theories of meaningfulness, Lawrence Erlbaum Associates, 2002.
[Bibtex]
@book{Narens2002a,
author = {Narens, Louis},
citeulike-article-id = {13531638},
posted-at = {2015-03-02 22:50:46},
priority = {2},
publisher = {Lawrence Erlbaum Associates},
series = {Scientific Psychology Series},
title = {Theories of Meaningfulness},
year = {2002}
}
[3] A. A. Sonin, "The physical basis of dimensional analysis," Massachusetts Institute of Technology 2001.
[Bibtex]
@techreport{Sonin2001a,
author = {Sonin, Ain A.},
citeulike-article-id = {13531648},
institution = {Massachusetts Institute of Technology},
posted-at = {2015-03-02 22:50:46},
priority = {2},
title = {The Physical Basis of Dimensional Analysis},
year = {2001}
}
[4] J. C. Gibbings, Dimensional analysis, Springer Science & Business Media, 2011.
[Bibtex]
@book{Gibbings2011a,
author = {Gibbings, John C.},
citeulike-article-id = {14387318},
posted-at = {2017-07-03 19:35:45},
priority = {2},
publisher = {Springer Science \& Business Media},
title = {Dimensional analysis},
year = {2011}
}