# 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}
}

# On the Structure of Pleasure - Representing Quantities

On the Axiomatic Representational Theory Of Measurement

The Axiomatic Representational Theory of Measurement ('ARTM') is the default tradition of measurement theory in mathematical psychology. It arguably started in a seminal work by the late philosopher of science and polymath Patrick Suppes and the logician Dana Scott [1].

It is 'axiomatic' in that it uses rigorous axiomatic-deductive mathematical methods. It is 'representational' by employing set-theoretical structures to represent roughly both the features of the world world and the information expressed by these features [2].

This kind of theory is contrasted to what is called a classical or realist theory of measurement [3]. Theories of this sort carry the spirit of ancient Greek sage, philosopher and mathematician Pythagoras (depicted in the painting above, proselytizing about the "marvels" of vegetarianism); it is not merely the case that we can use numbers to represent the information about a given measurand - the values of measurands are numbers, and numbers exist right here, in the structure of the world.

I've mentioned that in the account of Tiresias, pleasure is a ratio-scalable feature of psychological states. What does that mean?

A bit on the ratio scale-type

The word 'scale' is ambiguous in contexts of measurement. First, it can mean the choice of what I prefer to call a representation standard of a given measurable feature. Some examples include m, the meter-standard of the feature length $[L]$ and Ω, the Ohm-standard of the feature electric resistance $[V]$$[I]$$^{-1}$. But for our purposes, a scale is the kind of mathematical structure under which the numerical values that we assign to a given feature are constrained. Scales in this second meaning come in different types and the modern terminology of scale types (of which we find the ratio scale) was baptized by psychophysicist Stanley Smith Stevens [4, 5].

Scales in this sense are a kind of mapping, an homomorphism from a structure representing the world to a structure representing the values of features. It's the transformation that allows the relational structure to be preserved, to remain invariant.

Take for example this excerpt from a book by contemporary moral philosopher Fred Feldman [6], a defender of the thesis that pleasure is a quantitative feature (just like Tiresias):

We assume that each episode of pleasure contains a certain ‘amount’ of pleasure, and that this amount is in principle subject to measurement. (We need not assume that these amounts can in practice be precisely determined either by introspection or by any existing technology.) The amount of pleasure in an episode depends upon intensity and duration, with longer-lasting and more intense pleasures being said to contain more total pleasure. For purposes of exposition I will imagine that there is a standard unit of measurement for these amounts. I call one unit of pleasure a “hedon”.

Imagine that Tiresias worked out his own theory of quantitative pleasure and proposed the representation standard eron for pleasure.

To say that pleasure is ratio-scalable is to say that we can switch from an hedon into an eron by a function of the form $y=ax$ where '$a$' is a positive real number. If one eron is 2.38 hedons, we can discover how many erons are in 10 hedons by dividing 10 by 2.38. Simple as that - the proportions are constrained and preserved.

There is another common type of scale that is intuitively considered to be "quantitative", the interval scale. For the interval scale, the admissable transformation is of the form $y=ax+b$, again with $a$' being a positive real number.

Extensive structures

Let's go a bit deeper on the proposal of the ARTM. First, we build a set-theoretical structure to represent the province of empirical reality that we're interested - in this case, it is the empirical feature 'pleasure'. Ex hypothesi, let's take Tiresias' word of it; a default contender to represent the quantity pleasure is the triple $\langle$$P$,$\succeq$,$\oplus$$\rangle$. I like to call this an example of a material relational structure, but you'll see it with other names throughout the literature - most likely empirical relational structure or system.

'$P$' is a nonempty set whose elements are interpreted as pleasure-events or pleasure-states - or 'P-events', for short.

'$\succeq$' represents an empirical relation that mimics the abstract binary relation of total order. This roughly means that there is an empirically verifiable procedure to rank two distinct P-events as being either just as pleasurable or one being more pleasurable than the other.

The last element of our triple, '$\oplus$', is representing an empirical operation that is traditionally called concatenation. This is also intuitively graspable; it means that there is an empirical procedure that allows us to "stack" or "pile up" our quantities.

The class of structure with this form most used in science are called continuous extensive structures [7]. They need to satisfy a number of axioms.

The pioneering analysis and proposal of a set of axioms describing quantities came from Hermann von Helmholtz in 1887 and Otto Hölder in 1901 [8]. Over the 20th and 21st century, the axiomatic system of Hölder was modified by many philosophers of science and measurement theorists. They include Ernest Nagel in 1931 [9], Patrick Suppes in 1951 [10] and Jean-Claude Falmagne in 1975 [11].

These various adjustments, both minor and major, were motivated by empiricist reasons in an attempt to ground the practice of measurement in a more empirically adequate setting. They involve, for instance, weakening the logical relation of identity '$=$' into a relation of equivalence and getting rid of background assumptions for non-standard numbers (infinitesimals).

Speaking of which, now we need some numbers to build our numerical relational structure. This is seemingly more straightforward. For quantities, a common candidate is the triple $\langle$$Re_+$,$\geq$,$+$$\rangle$. Here, '$Re_+$' is the set of positive reals, '$\geq$' is the standard relation of total order and '$+$' is the standard arithmetic operation of addition. A large share of the ARTM is focused on proving theorems concerning homomorphisms from material relational structures to numerical relational structures.

I hold that there are a series of problems with this particular choice of numerical structure but I'll utter them elsewhere.

Having set this amount of framework, I still have some things to say about quantities, the qualitative-quantitative distinction and I also have bad news for the defenders of the hypothesis that pleasure is quantitative. It will be discussed in a future entry.

[1] D. Scott and P. Suppes, "Foundational aspects of theories of measurement," Journal of symbolic logic, vol. 23, iss. 2, pp. 113-128, 1958.
[Bibtex]
@article{Suppes1958a,
author = {Scott, Dana and Suppes, Patrick},
citeulike-article-id = {13531611},
journal = {Journal of Symbolic Logic},
number = {2},
pages = {113--128},
posted-at = {2015-03-02 22:50:45},
priority = {2},
title = {Foundational Aspects of Theories of Measurement},
volume = {23},
year = {1958}
}
[2] A. Frigerio, A. Giordani, and L. Mari, "Outline of a general model of measurement," Synthese, vol. 175, pp. 123-149, 2009.
[Bibtex]
@article{Mari2009a,
author = {Frigerio, Aldo and Giordani, Alessandro and Mari, Luca},
citeulike-article-id = {12534798},
journal = {Synthese},
pages = {123--149},
posted-at = {2013-07-30 07:13:11},
priority = {2},
title = {Outline of a general model of measurement},
volume = {175},
year = {2009}
}
[3] J. Michell, "The logic of measurement: a realist overview," Measurement, vol. 38, pp. 285-294, 2005.
[Bibtex]
@article{Michell2005a,
author = {Michell, Joel},
citeulike-article-id = {13531643},
journal = {Measurement},
pages = {285--294},
posted-at = {2015-03-02 22:50:46},
priority = {2},
title = {The logic of measurement: A realist overview},
volume = {38},
year = {2005}
}
[4] S. S. Stevens, "On the theory of scales of measurement," Science, vol. 103, iss. 2684, pp. 677-680, 1946.
[Bibtex]
@article{Stevens1946a,
author = {Stevens, S. S.},
citeulike-article-id = {13112840},
journal = {Science},
number = {2684},
pages = {677--680},
posted-at = {2014-03-21 00:50:29},
priority = {2},
title = {On the Theory of Scales of Measurement},
volume = {103},
year = {1946}
}
[5] S. S. Stevens, "Mathematics, measurement and psychophysics," in Handbook of experimental psychology, Wiley, 1951, pp. 1-49.
[Bibtex]
@incollection{Stevens1951a,
author = {Stevens, S. S.},
booktitle = {Handbook of experimental psychology},
citeulike-article-id = {13112820},
pages = {1--49},
posted-at = {2014-03-21 00:50:01},
priority = {2},
publisher = {Wiley},
title = {Mathematics, measurement and psychophysics},
year = {1951}
}
[6] F. Feldman, Pleasure and the good life: concerning the nature, varieties and plausibility of hedonism, Clarendon Press, 2004.
[Bibtex]
@book{Feldman2004a,
abstract = {Fred Feldman's fascinating new book sets out to defend hedonism as a theory about the Good Life. He tries to show that, when carefully and charitably interpreted, certain forms of hedonism yield plausible evaluations of human lives. Feldman begins by explaining the question about the Good Life. As he understands it, the question is not about the morally good life or about the beneficial life. Rather, the question concerns the general features of the life that is good in itself for the one who lives it. Hedonism says (roughly) that the Good Life is the pleasant life. After showing that received formulations of hedonism are often confused or incoherent, Feldman presents a simple, clear, coherent form of sensory hedonism that provides a starting point for discussion. He then presents a catalogue of classic objections to hedonism, coming from sources as diverse as Plato, Aristotle, Brentano, Ross, Moore, Rawls, Kagan, Nozick, Brandt, and others. One of Feldman's central themes is that there is an important distinction between the forms of hedonism that emphasize sensory pleasure and those that emphasize attitudinal pleasure. Feldman formulates several kinds of hedonism based on the idea that attitudinal pleasure is the Good. He claims that attitudinal forms of hedonism - which have often been ignored in the literature -- are worthy of more careful attention. Another main theme of the book is the plasticity of hedonism. Hedonism comes in many forms. Attitudinal hedonism is especially receptive to variations and modifications. Feldman illustrates this plasticity by formulating several variants of attitudinal hedonism and showing how they evade some of the objections. He also shows how it is possible to develop forms of hedonism that are equivalent to the allegedly anti-hedonistic theory of G. E. Moore and the Aristotelian theory according to which the Good Life is the life of virtue, or flourishing. He also formulates hedonisms relevantly like the ones defended by Aristippus and Mill. Feldman argues that a carefully developed form of attitudinal hedonism is not refuted by objections concerning 'the shape of a life'. He also defends the claim that all of the alleged forms of hedonism discussed in the book genuinely deserve to be called 'hedonism'. Finally, after dealing with the last of the objections, he gives a sketch of his hedonistic vision of the Good Life},
author = {Feldman, Fred},
citeulike-article-id = {13562104},
number = {1},
posted-at = {2015-03-25 19:43:16},
priority = {2},
publisher = {Clarendon Press},
title = {Pleasure and the Good Life: Concerning the Nature, Varieties and Plausibility of Hedonism},
year = {2004}
}
[7] 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}
}
[8] O. Hölder, "Die axiome der quantität und die lehre vom mass," Berichte uber die verhandlungen der koeniglich sachsischen gesellschaft der wissenschaften zu leipzig, mathematisch-physikaliche klasse, vol. 53, pp. 1-46, 1901.
[Bibtex]
@article{Holder1901a,
author = {H\"{o}lder, Otto},
citeulike-article-id = {13531626},
journal = {Berichte uber die Verhandlungen der Koeniglich Sachsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physikaliche Klasse},
pages = {1--46},
posted-at = {2015-03-02 22:50:46},
priority = {2},
title = {Die Axiome der Quantit\"{a}t und die Lehre vom Mass},
volume = {53},
year = {1901}
}
[9] E. Nagel and C. G. Hempel, "Measurement," Erkenntnis, vol. 2, iss. 1, pp. 313-335, 1931.
[Bibtex]
@article{Nagel1931a,
author = {Nagel, Ernest and Hempel, C. G.},
citeulike-article-id = {13559115},
journal = {Erkenntnis},
number = {1},
pages = {313--335},
posted-at = {2015-03-23 22:50:07},
priority = {2},
publisher = {Springer},
title = {Measurement},
volume = {2},
year = {1931}
}
[10] P. Suppes, "A set of independent axioms for extensive quantities," Portugaliae mathematica, vol. 10, iss. 4, pp. 163-172, 1951.
[Bibtex]
@article{Suppes1951a,
author = {Suppes, Patrick},
citeulike-article-id = {13559116},
journal = {Portugaliae Mathematica},
number = {4},
pages = {163--172},
posted-at = {2015-03-23 22:50:07},
priority = {2},
title = {A Set of Independent Axioms for Extensive Quantities},
volume = {10},
year = {1951}
}
[11] J. Falmagne, "A set of independent axioms for positive holder systems," Philosophy of science, vol. 42, iss. 2, pp. 137-151, 1975.
[Bibtex]
@article{Falmagne1975a,
abstract = {Current axiomatizations for extensive measurement postulate the existence of infinitely small objects. This assumption is neither necessary nor reasonable. This paper develops this theme and presents a more acceptable axiom system. A representation theorem is stated and proved in detail. This work improves some previous results of the author},
author = {Falmagne, Jean-Claude},
citeulike-article-id = {13559114},
journal = {Philosophy of Science},
number = {2},
pages = {137--151},
posted-at = {2015-03-23 22:50:07},
priority = {2},
publisher = {University of Chicago Press},
title = {A Set of Independent Axioms for Positive Holder Systems},
volume = {42},
year = {1975}
}
|

# On the Structure of Pleasure - A Tale of Measurement Theory and Classical Mythology

Tiresias, mythical mathematical psychologist

Tiresias is a legendary blind prophet in Greek myth. His story has many versions. In this tract, I'll draw primarily from the account of ancient mythographer Pseudo-Apollodorus.

As the narrative goes, while traveling in Mount Cyllene, Tiresias beheld a pair copulating snakes and struck the female with his walking stick. For this display of animal cruelty, he was cursed to live as a woman. With the body and mind of a woman, Tiresias experienced sex; according to versions of the legend, he did so quite intensely, becoming a prostitute of great renown.

Seven years had passed after this incident and female Tiresias had the opportunity to re-encounter an event similar to the one that caused his transformation. This time, he left the reptiles undisturbed and in that, he was allowed to regain his former manhood.

Having lived as both a man and a woman, Tiresias was privileged with a unique perspective on the varieties of sexual experience across the sexes. For that, he was consulted by the Olympians themselves. Zeus and Hera willed to settle once and for all the question of who enjoys greater pleasure in carnal love - man or woman. Tiresias provided an outstanding answer for the Gods:

Of ten parts [of sexual pleasure] a man enjoys one only; but a woman enjoys the full ten parts in her heart.

This passage is remarkable in theoretical depth. The divine couple requested from the mortal soothsayer an ordinal evaluation and received more than what they had bargained. In his statement, Tiresias is claiming that events of (sexual) pleasure not only can be ranked, but that they have magnitudes whose proportions can be meaningfully compared and expressed.

In measurement-theoretic terms, this is the claim that pleasure is a ratio-scalable property or attribute. This is what one usually has in mind when someone says that something is 'quantitative'.

It's a bold ontological claim. The story of Tiresias however is also daring epistemologically; it stipulates that information about the value of the magnitude of pleasure instantiated in a given psychological state is obtainable through introspection with a certain precision. It is also supposedly reliably registrable in long-term memory; the mind is used as a truthful scale of itself.

From ancient myth to modern science

It took over two millenia for the ideal of Tiresias to come into fruition. The earliest robust attempts happened in the German tradition of psychophysics, by physicists Gustav Fechner and Hermann von Helmholtz. Somewhat unexpectedly, this tradition arguably culminated with mathematician Otto Hölder [1].

The quest of psychophysics has been to establish an interface between subjective mind and objective world by uncovering functional relations between 3rd-person fact and 1st-person unknown. For example, the investigation of how psychological brightness co-varies with physical luminance. For these tasks, psychophysicists have developed several methods for subjective magnitude estimation of sensations from given controlled stimuli as input to verbal and motor reports as output.

Measurement theorist Louis Narens has defended that formally, these forms of measurement can have a foundation as strong as other prototypical practices of experimental science [2].

A lot of progress has been made on the investigation of the structure of sensations. Affective states such as pleasure and pain, however, still remain very murky.

Is pleasure quantifiable?

On closer inspection, Tiresias' proposition that pleasure is quantitative does not seem to be particularly bold after all. It has received widespread uncritical support for quite some time - from the dawn of utilitarianism in modern moral philosophy to the use of visual analog scales for pain assessments in contemporary clinical medicine and cardinal utility in economics. They have become ingrained in our common sense. Queries like "from a scale to 0 to 10, how do you feel?" have become staples of modern managerial society.

But is it true? Unfortunately for Tiresias, the hypothesis that pleasure and other similar affective states such as pain have a quantitative structure has not been empirically established with satisfaction. Not even the other rather intuitive hypothesis that pleasure and pain have a structure that allows representation as opposite intervals of a same axis.

To stipulate that a given feature has a quantitative structure is not a matter of pragmatic choice but of experimental fact. This claim has been vigorously defended by measurement theorist Joel Michell [3]. Michell condemns gratuitous cursory admissions that a given psychological feature is quantitative without previous evidential justification. Conceptual residue from operationalist philosophy in experimental psychology is at fault.

The fact of the matter is that the stakes for an empirical feature to be ratio-scalable are very high. This will be exposed in greater detail in another entry.

[1] J. Michell and C. Ernst, "The axioms of quantity and the theory of measurement," Journal of mathematical psychology, vol. 41, iss. 4, pp. 345-356, 1997.
[Bibtex]
@article{Michell1997c,
author = {Michell, Joel and Ernst, Catherine},
citeulike-article-id = {13559112},
journal = {Journal of mathematical psychology},
number = {4},
pages = {345--356},
posted-at = {2015-03-23 22:50:07},
priority = {2},
publisher = {Elsevier},
title = {The axioms of quantity and the theory of measurement},
volume = {41},
year = {1997}
}
[2] L. Narens, "The irony of measurement by subjective estimations," Journal of mathematical psychology, vol. 46, iss. 6, pp. 769-788, 2002.
[Bibtex]
@article{Narens2002b,
author = {Narens, Louis},
citeulike-article-id = {13559113},
journal = {Journal of Mathematical Psychology},
number = {6},
pages = {769--788},
posted-at = {2015-03-23 22:50:07},
priority = {2},
publisher = {Elsevier},
title = {The irony of measurement by subjective estimations},
volume = {46},
year = {2002}
}
[3] J. Michell, "Quantitative science and the definition of measurement in psychology," British journal of psychology, vol. 88, pp. 355-383, 1997.
[Bibtex]
@article{Michell1997a,
author = {Michell, Joel},
citeulike-article-id = {13531650},
journal = {British Journal of Psychology},
pages = {355--383},
posted-at = {2015-03-02 22:50:46},
priority = {2},
title = {Quantitative science and the definition of measurement in psychology},
volume = {88},
year = {1997}
}
|