(Rodrigo Peñaloza, January 1st, 2017)

One of the most beautiful aspects of Mathematics is its relation to the human soul and mind. I once wrote about how Russel’s paradox of Set Theory is intimately related to the Greek thought about being, genus and species, that is, the idea that being is not a summum genus, it cannot be reached by taking genera over genera in an increasing order of continence. Being is not a genus because every specific difference applied to a genus in order to move from that genus to one of its species is already part of being. Aristotle’s concept of specific difference reappeared in modern Mathematics by the hands of Russel and Whitehead under the role of a property defining a class of sets (click here to read it).

It is unfortunate, then, that Mathematics is seen with so much rejection by many. If you love Humanities and do not like Math, the reason most likely lies on the way your teachers taught Math in the school. I would also say Humanities. Then you go to Humanities and from that moment on, Math becomes something off your horizon of thoughts. You just cannot understand that when you talk about necessity and possibility in the Philosophy of Language, you are also talking about mathematical concepts in Modal Logics. When you think of justice in the society you live in, you may not realize that your conceptions of justice have mathematical counterparts. The same happens the other way round. If you love Math and do not like Humanities, you just have no patience to relate it, for instance, with the idea of justice according to the Greek thinkers. Maybe your Humanities teacher did not teach you how to connect them. The same goes to your Math teacher.

As I said, I blame teachers for that. There is a kind of universal agreement about this. However, my position is slightly different from the common agreement, as you have likely perceived. I not only blame Math teachers, I also blame Humanities teachers for not knowing how to connect Humanities to mathematical thought. Both are blamed for not understanding the common philosophical background in either side.

The purpose of this post is to give a little insight about how to build a bridge between such apparently different worlds. The topic I chose is the mathematical mean or average. This is a topic usually taught to kids around the seventh grade, in which they learn three kinds of means: arithmetic, harmonic and geometric. They are not taught to see them in any unified way, neither to extrapolate their meanings to other spheres of human life. Here I want to show that, depending on the way you teach it, you can let the door ajar for the student to think about different concepts of justice and then about Moral Philosophy, Politics, and the debate about economic equality and even Music and visual art. In the end I also say something about Physics and Humanities, so in a sense, when I wrote the term Mathematics in the title of this text, I actually meant Science as opposed to Humanities. What I want to do, if I succeed, is to show that this opposition should be replaced by the word colaboration: they (Science and Humanities) are the two legs of the same walker and that which makes them coordinate their movement in an elegant and efficient way is Classical Education.

Since Mathematics and Humanities have to walk pari passu, it is important that, at such grade, the student be already exposed to basic Trivium (Grammar, Logic, and Rhetoric), specially the concepts of substance and accident and the 10 categories Aristotle derived from them, and to basic Quadrivium (Arithmetics, Geometry, Music, and Astronomy). The advantages of classical education here become very clear. It is, again, very unfortunate that standard modern Paedagogy does not have any clue about the necessity of teaching Latin, Trivium, Quadrivium, Humanities and Mathematics in such a coordinated manner. Let us proceed though.

Take two numbers, say 1 and 9. Take from Aristotle the categories of quantity (ποσóν) and place (ποῦ). Number is to be understood as the amount of place. We assume for granted Aristotle’s concept of justice according to which justice is a medium-term between what is less and what is more:

Aequale autem est realiter medium inter maius et minus, ut dicitur in X Metaphysica. Unde iustitia habet medium rei. — Thomas Aquinae, Summa Theologiae, IIa IIae, quaest. 58, art. 10 (“But equality really is a medium-term between the greater and the smaller, as is told in book X of [Aristotle’s] Metaphysics, whence justice comprises a real medium-term”).

Then the number that equilibrates 1 and 9 is the one which divides the spatial quantity between them in a quantitatively equal manner. The spatial quantity between 1 and 9 is the distance 9–1=8. However, we do not need this piece of information for the subsequent reasoning. Look at the figure below:

The spatial quantity between 1 and 9 is divided equally into two smaller spatial quantities. The smaller spatial quantity to the left will be given to the smaller number, 1, whereas the smaller spatial quantity to the left will be taken from the bigger number, 9.

In this case, justice requires that the medium term M be such that: the quantity M-1 between 1 and the medium-term M be equal to the quantity 9-M between the medium-term M and 9. Therefore M-1=9-M. If we solve this equation for M, we get:

This is the arithmetical mean between the numbers 1 and 9. In general, the arithmetic mean between two numbers A and B is:

According to Thomas Aquinae:

[S]icut Philosophus dicit in V Ethica [Nicomachea], omne superfluum in his quae ad iustitiam pertinent lucrum, extenso nomine, vocatur; sicut et omne quod minus est vocatur damnum. — Thomas Aquinae, Summa Theologiae, IIa IIae, quaest. 58, art. 11 (“As the Philosopher [Aristotle] says in book V of Ethics to Nicomachus, everything that is superfluous in these things pertinent to justice, are called, by extension, profit, as well as everything that is less is called damage”).

Justice corrects excess and privation. Likewise, the arithmetic mean corrects excessiveness of 9 over 1 by means of redistributing according to quantitative equality this excessive quantity between them.

When we compare countries’ income distributions by means of the Gini coefficient, we are taking a measure of distance between actual income distribution and equal (uniform) income distribution, hence comparing these distances accross countries. It is a political goal to reduce Gini coefficient to its lowest levels. This is why policy-makers implement redistribution mechanisms. However, the standard of comparison is uniform distribution of income (equal per capita income). If economic inequality is considered to be unjust, then the use of the Gini coefficient pressuposes that the just distribution is given by equal income. It is therefore a concept of justice based on quantity. It is a very primitive concept indeed.

The Math teacher at this moment makes the student understand how important it is to evaluate which instance of injustice in life can be analysed in terms of quantitative justice. He will see this analysis as theoretically equivalent to the mathematical concept of arithmetic mean. Such equivalence will hold as well when the student, should this be the case, becomes a Mathematician or a Statistician and learns measure theory. From arithmetic mean to weighted mean, hence Lebesgue integral and concepts such as Radon-Nikodym derivatives, it is just a simple step further.

Let us take now the categories of relation (πρòς τι) and place (ποῦ). Remember that number is the amount of place, so no wonder this category shows up again. We now have to specify what kind of relation we have in mind with respect to the numbers 1 and 9. If we only consider the category of quantity, we will only be able to say that 9 is a bigger quantity than 1. In order to talk about relation, we have to be able to say, for example, how much bigger, in terms of proportion, is 9 with respect to 1. Equality of proportions is what the Greeks thought about harmony.

In order to introduce the idea of harmonic justice, one in terms of a particular kind of relation, harmony, let us say that the medium-term between 1 and 9 is that number H defined in such a way that: the relative increase from 1 to the medium-term H must be equal to the relative decrease from 9 to the medium-term H.

The medium-term is the number H. Its relation to 1 is given by the fact that there is a rate of increase h such that H=1x(1+h), that is, H equals 1 times (1+h). On the other hand, its relation to 9 is given by the fact that the same rate h is now the rate of decrease from 9 to H, that is, H equals 9 times (1-h).

In order to find the just number H that harmonically equilibrates the difference between 1 and 9, we first have to find out the common rate h. This is given by applying this concept of harmonic justice to 1 and 9. In other words, since, by definition, H=1x(1+h) and H=9x(1-h), we have to solve the equation 1(1+h)=9(1-h) for h. Once h is found, we just substitute it into the expression H=1+h. It must be clear that if we rather substitute it into the expression H=9(1-h), then we will get to the same answer. The solution for h is:

Therefore, once we substitute this rate h into, say, H=1+h, we then find out that the medium-term H is equal to:

This is the harmonic mean between 1 and 9. Notice that H=1.8. The change from 1 to H=1.8 corresponds to an increase of 80%. This is value h=0.8. The change from 9 to H=1.8 corresponds to a decrease of 80%. The rate of increase on 1 and decrease from 9 is the same. This rate had to be calculated in a such a way to make both the amount of increase over 1 and the amount of decrease from 9 meet at the same place between 1 and 9: the harmonic mean. In general, the harmonic mean between two numbers A and B is:

The harmonic mean corrects excessiveness of 9 over 1 by means of redistributing according to relative equality the excessive quantity between them. Relativeness in this case is given by proportion, hence harmony.

This notion of harmonic justice is the same found in Music. It was Pithagoras who said, for instance, that if the length of two strings are in relation of 2:3 to each other, the difference in pitch was a fifth. Musical harmonics is related to proportions.

When the student reaches the Quadrivium in his education, and starts to learn Geometry and Music, we have to understand that these liberal arts are the means to understand Art in general, that is, Aesthetics. They are not relevant per se. They are the tools with which the student, as a free thinker and citizen, can think about the world he lives in. This is why classical education is liberating and modern education is encapsulating.

Jumping to the issue of justice in more day-to-day matters, consider Leibniz’s definition of equality in the realm of Logic: “two terms are equal when one can be substituted by the other within the same context without any change in the value of the context”. If we conveniently define what we mean by context, we can get a pretty good concept of legal justice, which was exactly what Leibniz did: “There is legal equality between two individuals if one individual, being defendant in a legal process with respect to a crime committed under some circumstance, can be substituted by the other, also a defendant who committed the same crime under the same circumstance, without any change in the legal process”. It is now evident that this concept of legal equality is defined in terms of the category of relation: it is proportional justice. Redistribution of punishment and reward is relative to the circumstance. Suppose for instance, that Chaim kills his brother Abel. According to quantitative justice, Chaim should receive the same damage he imposed on Abel. In other words, he should be killed. Nothing else than Lex Talionis, “eye for an eye” or retributive justice. According to relative justice, Chaim should be judged according to circumstance. If he killed on self-defence, his punishment is alleviated to a lesser one. This is harmonic justice. In spite of the obvious differences of realms, the philosophical substract is the same. (To see more about equality, click here).

Finally, let us take now the categories of quality (ποιóν) and place (ποῦ). The particular quality we will take is the form of rectangles (quadrilateral rectangular figures). Any two numbers A and B allow us to conceive of geometric figures called rectangles. For example, from the numbers 1 and 9 we can think of the rectangle with sides 1 and 9.

To the Greek mind, asymmetries are not beautiful nor just. Among rectangles we have to find that particular form which equilibrates the ugly rectangle above to its beautiful representative figure. In the realm of rectangles, the most beautiful figure is the square, since all its sides are equal. In order to move from the rectangle above to the square, we have to take away from the bigger number, 9, and somehow give more to the smallest number, 1, until the point where both find justice in this redistribution. Since we fixed the category of quality, and particularly, the form of rectangles, we have to find the square that equilibrates 1 and 9, and to do that, we have to realize something that is equal to both, a common geometric denominator. This will be the area of the rectangle. The area of the (ugly and unjust) rectangle is 1x9=9. The area of the (beautiful and just) square of side G is GxG=G². Notice that G must be a number between 1 and 9, once we realize it should be a reduction from 9 and an augment to 1. It is therefore a candidate to medium-term between 1 and 9. If we want it to be the medium-term prescribed by geometric justice, the only solution is that it be the side G of the square with the same rectangular area 1x9. In other words, G²=1x9, hence:

This is the geometric mean between 1 and 9: the square root of the product 1 times 9. It substitutes the two numbers (1 and 9) that form the ugly and unjust rectangular figure by a single number G=3 that gives form to the beautiful and just rectangular figure that equilibrates those numbers 1 and 9: the square of side G=3. In general, the geometric mean between two numbers A and B is:

The geometric mean equilibrates 1 and 9 by taking away from the bigger number, 9, and giving to the smaller, 1, in such a way that so redistributing it makes them reach perfect beauty according to form. It is therefore very platonic in nature.

The door is now open to Greek Philosophy and the Greek culture, the very foundation of western Civilization.

The reader should not believe I have used purposedly the example of mathematical means only to defend a general procedure towards Education. Nothing far from the truth. Imagine, for instance, that our kid advances his studies and is now in high school. He begins to study Newtonian Physics. Before facing the mathematical formulae of Newtonian Mechanics, he is intellectually prepared to understand why the concept of space is absolute to Newton. This is because space was interpreted as an attribute of God, the sensorium Dei. All the attributes the Scholastic philosophers assigned to the Supreme Being were transposed one by one to the notion of Space. Indeed, the attributes of Space were:

Unum, simplex, immobile, aeternum, completum, independens, a se existens, per se subsistens, incorruptibile, necessarium, immensum, increatum, incircumscriptum, incomprehensibile, omnipr[a]esens, incorporeum, omnia permeans et complectans, ens per essentiam, ens actu, purus actus. — Henry More, Enchiridion Metaphysicum (1671) [One, simple, immovable, eternal, complete, independent, existing by itself, subsistent through itself, incorruptible, necessary, measureless, uncreated, unbounded, incomprehensible, omnipresent, incorporeal, all-pervading and all-embracing, being in essence, being in act, pure act]. (See also Milic Capek, The Philosophical Impact of Contemporary Physics, pp. 9–10).

This is fundamental to understand how later thinkers came across with the Law of Conservation of Motion, which states that total quantity of motion remains constant in a isolated system.

Space is absolute, it took from God, by the hands of natural philosophers, all the divine attributes. Thus, according to Newton, a body moving in absolute space is in absolute motion. A body at rest in absolute space is in absolute rest. Motion occurs in space, but cannot affect space. Motion occurs in time, but cannot affect time. As Leibniz pointed out in a letter to Thomasius:

Cum enim corpus nihil aliud sit, quam materia et figura, et vero nec ex materia nec figura intelligi possit causa motus, necesse est causam motus esse extra corpus. — [“Since body is nothing but matter and form, and since the cause of motion cannot be understood as derived from matter nor form, it is necessary that the cause of motion be outside body”]. (See Capek, p. 70).

This implies that movement in space and time cannot alter space and time. Motion and matter are mutally uncovertible. Threfore, only motion can explain motion. If this is true, the total quantity of motion must remain constant. It was this kind of philosophical reasoning that led Physicists to postulate the Law of Conservation of Motion. They got to it because they went into intellectual reasoning in the first place. They were humanists, they knew Aristotle and Philosophy. With this in mind, the student gets a cristal clear vision of Einstein’s intellectual revolution in comparison to what humankind has believed for milennia since it started to think scientifically and philosophically about the world. If we could pick two individuals on whom the whole history of Physics leans on, those will be Aristotle and Einstein. Everybody else are followers in some sense.

The lesson from this last remark on Physics is that Greek thought is still important even to modern science. The philosophical background that led to scientific statements such as the Law of Conservation of Motion, for example, has much in common to scholastic Theology. If the student is studying medieval history of the Western Civilization and is then confronted, in his classes of Physics, with Newtonian Physics, he will see the connection through the lenses of the concepts of matter and space. He will realize that medieval thought was not really put aside entirely in the end of the 15th century by Francis Bacon’s Organon, which established the basis for empirical science. It took rather a longer time until Einstein really overcame the Aristotelian paradigm of space and time. If the student knew these things, he would see with much criticism and even skepticism the ridiculous simplifications of modern standard History textbooks.

There is a philosophical background in the mathematical formulation of Physics that connects Mathematics and Humanities. Neither Math teachers understand Humanities, neither Humanities teachers understand Math in a way sufficient enough for them to converse clearly with each other and to coordinate their teachings. With respect to these common backgrounds, when we separate Mathematics from Humanities and Humanities from Mathematics, we prevent students from really understanding Science and Humanities in essence. The common background is Classical Education. It is Latin and Greek, it is Trivium and Quadrivium.

Please read as well:

Economics and Classics
* A República Romana e os instrumentos de crédito
* On the notion of wrong decision in the Greek tragedy: the epistemic rôle of uncertainty, risk and ignorance
* O mercado de escravos em Roma e o problema da seleção adversa
* The Roman slave market and the problem of adverse selection
* O bolsa-família do Império Romano
* Entre a insciência e a literatura
* On the Greek origins of the idea of spontaneous order versus deliberate order
* Cícero sobre o Estado e os direitos de propriedade
* Tacendo cavillamur cives: ou Columella sobre a má gestão econômica

“Microeconomia em doses” series:
* Microeconomia em doses: curva de oferta
* Microeconomia em doses: custo de eficiência
* Microeconomia em doses: depreciação e obsolescência
* Microeconomia em doses: curto-prazo versus longo-prazo
* Microeconomia em doses: obsolescência programada
* Microeconomia em doses: valor de Shapley
* Microeconomia em doses: teorema de Allen-Alchian

Economics and doxa 
Competição perfeita: o que é isso?
* Externalidades e o teorema de Coase
* Custo de eficiência do imposto de renda
* There is no such thing as a free coffee 
* Microeconomia contra a Lei Pelé
* Como desenhar contratos sob seleção adversa
* Microcredit: a Bosnian tragedy
* FIES: quando vale a pena? (No blog do EconomistaX)
* Reinterpretando o índice de Gini
* Marshall e Walras sobre lucro zero
* O pobre gosta menos de educação do que o rico?
* The top 10 Economics books of the past century: what is the problem?
* Reflexões sobre o lucro segundo Schumpeter, Clark, Knight e Kirzner
* Microeconomia da variação da renda nacional
* Jevons on clearing houses
* O decálogo de Mankiw e a tétrada da Economia
* Teoria dos jogos versus estatística
* “Res perit domino” ou “codex civile est nobis dolori” ou ainda: do Direito ineficiente 
* O paradoxo do voto
* A mecânica democrática da rejeição ao governo do PT: majoritarimente eleito mas rejeitado pela maioria
* Gabriel Biel sobre o monopólio e o valor da moeda
* Francis Bacon sobre os juros

Philosophical thoughts
* The deteleologization of Joseph McMill
* O princípio platônico e o livre-arbítrio
* Russel’s paradox and Greek ontology
* Minha interpretação da carta de Paulo aos romanos 12:19
* On the rationality of the Golden Rule or: why Eve should not be blamed alone for the fall
* A fine example of true irony: Friedrich Hayek on constructivist rationalism
* On the concept of soul among the Greeks
* O paradoxo da nave de Teseu
* On Thoreau’s Essay on Walking
* Cícero, Sêneca e a amizade

Writings in Latin and Classical Greek
* Dilemma captivorum
* Ad bestias
* ῥώμα
* Insula Itamaracá
* De malo iudicio in tragoedia graeca: quae sint personae gerendae ab alea et ignorantia?
* De consolamento more catharorum
* De caelo sive certus et exquisitus modus intellegendi omnes res, non solum mundi sed etiam hominum
* Fabella sepulcralis
* De chao apud veteros graecos
* A symbolis aegiptiis usque ad veritatem
* De iactu nucleoli
* De aequalitate
* De monade
* Epistula
* Bellum iaponicum
* De trivio
* De paradoxo russeliano et ontologia graeca
* De censura in Re Publica Romana
* εἲς μούσαν τῆς χρωμογραφίας
* περì τῆς ἐκλείψεως
* De coloribus verborum
* Cur viris liberalibus sit incohaerens capitis praedicare poenam
* Quid accidit Aristoteli Americoque Castro, philologo hispanico, nuper in Paradiso
* Iustitia
* Utinam
* El Chapo
* De via caritatis

* Egregora
* Terra Brasilis
* A conundrum about Proto-Indo-European
* Mal de bem-querer
* Encômio para um congressista