Matryoshkas and semantics

Representations are everywhere around us. We use words to describe concepts, perceptions, or emotions, and letters and drawings to capture abstract concepts and the relations between them. However, words and their combinations are usually not representations themselves, but merely references to mental items that, at least in a number of frameworks, can be adequately --and productively-- represented. Elsewhere, I have proposed a methodology to approach human language in terms of categories and relations, which might lead to a holistic formalization of semantics.

How comprehensive that approach may ultimately be, is still to be elucidated. For now, I would like to refer to one particular case where the lack of a representation implies a non-concept, i.e. a concept syntactically formulated, but actually devoid of any possible meaning.

To that end, we will start by defining a matryoshka as any spatial object having a specific matryoshka shape S and zero thickness. This means that all possible matryoshkas will have exactly the same shape, i.e. S, and will differ only in their volume. Because each matryoshka has thus been unequivocally defined, we may now define the mathematical set of all matryoshkas. Note that it will not be possible to identify the biggest matryoshka, because, by definition, whatever the matryoshka we may select, there will always exist a bigger one. Of course, the converse is also true, and therefore it will not be possible to identify the smallest matryoshka in that set.

Now, let's associate to each matryoshka M a different list Lm, selected from the set of all lists. Once we have done that, we associate to each matryoshka M the list Um of all the lists associated to the matryoshkas nested within M. Whatever M we choose, the list Um will always be smaller than the list of all lists, because there will always be a matryoshka M' larger than M and, therefore, a list Lm' different from Lm.

This means that, to consistently denote the list of all lists, we should use the symbol associated to the matryoshka of all matryoshkas. In fact, we have been using the concept of a matryoshka as a representation of the concept of a list. Is it an acceptable representation? As far as we can tell, there is a one-to-one correspondence between the set of lists and the set of matryoshkas, and the relation between lists has a clear correlate in the relation between matryoshkas. So, what is that thing sometimes referred to as “the list of all lists”? It is easy to see that that question is a non-question. “The matryoshka of all matryoshkas” is just a string of words without a referent, and we cannot possibly identify a concept for which we do not have a representation. When I say identify, I mean locate in the data aggregate of our mental concepts.

Now, you can repeat the matryoshka-list association process for any set that may be claimed to be a member of itself. The notion of a set that is a member of itself lacks a representation, and therefore is a non-concept. This is a first step to delve into Russell's paradox, which postulates the existence of a 'set of all sets that are not a member of themselves'. A non-subject cannot possibly be the subject of a paradox, e.g., I could state any number of (meaningless) theorems about 'the point where a circle ends'.

Even if we are usually not aware of it, semantics is strongly linked to spatial mental models, onto which it gets represented, then processed. That is why I propose that all semantic tokens are, at a deep (subconscious) level, a representation, and therefore susceptible to get algebraically formalized, as I have expounded in my papers. If that is true, any research on semantics, whether distributional or not, should be focussed on structures, rather than metrics. That is the reason distributional semantics models cannot be expected to fully capture the semantics of human language. At least, insofar as their output is presented in terms of spatial fields, rather than articulated data.

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El mercado de futuros

Mi amigo Jesús me envió ayer un enlace en el que alguien se despachaba a gusto contra los especuladores, la bolsa, el mercado, etc. etc. A partir de una noticia que el autor del artículo no había entendido (y no se había molestado en entender), el artículo se limitaba a ensartar los tópicos habituales sobre el 'malvado' capitalismo y sus secuaces, tal como los describe machaconamente la propaganda izquierdista. Como Jesús es una de las pocas personas que conozco que escucha sin prejuzgar, se me ocurrió escribir un texto para aclararle las ideas. Sí, esas mismas ideas que miles de obtusos propagandistas repiten día a día por todas partes como loritos amaestrados. Pero al final, después de reflexionar un poco, he decidido publicarlas aquí. Quién sabe. Tal vez a alguien le sirvan de ayuda para salir del oscurantismo imperante. Allá voy.

En la bolsa no cotizan materias primas, porque las materias primas no son empresas. En la bolsa cotizan sólo empresas que deciden dividir su capital en pequeñas fracciones para que los pequeños inversores puedan participar como copropietarios de la empresa, con sus beneficios y sus riesgos. Un inversor comprará acciones de una empresa porque la ha investigado a fondo, y confía en que esa empresa creará riqueza y puestos de trabajo. Un especulador, en cambio, compra sólo porque cree que el precio va a subir y espera vender con beneficio.

Parece lo mismo, pero no es lo mismo. Los inversores de bolsa esperan ganar dinero a largo plazo. A corto plazo las fluctuaciones son enormes, y a menudo es más fácil perder dinero que ganarlo. Ahí entran en juego los especuladores, que prevén que el precio va a subir o bajar a causa de factores externos: los tipos de interés, la moda de un producto, los datos de empleo o de PIB, las epidemias, las vacunas... Todo eso afecta a los precios a corto plazo, pero especulando es más fácil perder dinero que ganarlo. Con la burbuja inmobiliaria de los años 2000, muchos ganaron mucho dinero (si supieron vender a tiempo, que no es nada fácil), pero muchos más perdieron muchísimo cuando explotó la burbuja. Con las vacunas, muchos han ganado dinero comprando acciones de Pfizer, pero son muchos más los que han perdido comprando acciones, por ejemplo, de Sanofi.

En bolsa se negocian también los llamados 'contratos derivados', que están asociados al precio de las acciones. Los orígenes de esos contratos fueron los 'contratos futuros', que inventaron los agricultores para asegurarse un ingreso fijo a cambio de su cosecha. Meses antes de la recolección, el comprador se compromete a pagar al agricultor un precio fijo por la cosecha, que es menos de lo que el agricultor podría ganar, pero más de lo que podría perder si hay pedrisco, langosta, sequía, etc. El comprador, por su parte, espera ganar dinero si todo va bien, pero también se arriesga a perderlo todo si las cosas van muy mal.

En cualquier caso, siempre hay un riesgo, y los especuladores a menudo pierden hasta la camisa. Cuanto más dinero quieres ganar, más tienes que arriesgar, así que los especuladores son más parecidos a los clientes de un casino que a unos buitres despiadados.

A veces, el agua escasea. Como no es posible saber cuándo lloverá --y, por lo tanto, si su precio bajará o subirá--, puede haber consumidores o empresas que prefieran contratar un precio fijo por hectolitro para el año que viene. Al igual que los agricultores, se aseguran un precio fijo, que no dependerá de las variaciones del consumo ni del tiempo meteorológico, y evitan así correr riesgos. Quienes se comprometan a pagar ese precio se arriesgarán a que el año próximo, en lugar de haber sequía, haya lluvias torrenciales y el precio del agua baje. En ese caso, perderán dinero. Pero los contratos futuros los firman libremente ambas partes, y el hecho de que haya un mercado de contratos es una garantía de que habrá muchos compradores que competirán entre sí y, por lo tanto, los precios no serán abusivos. La libertad de mercado es la mejor garantía contra los abusos.

Con el tiempo, los 'contratos futuros' evolucionaron. En lugar de asociarlos a productos básicos, como las naranjas, el petróleo o el agua, ahora es posible asociarlos también al precio de una acción determinada. Las ganancias pueden ser espectaculares, pero el riesgo también es enorme, y son muchos los que se arruinan comprándolos. Personalmente, yo me sentiría más seguro jugando a la ruleta.

Resumen: 

ganar mucho dinero = correr mucho riesgo

libertad de mercado = garantía contra los abusos

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Continuity, acuity

What is continuity? Or, better still, how do we conceive continuity? For example, if an arrow moves along a trajectory, we conceive its movement as continuous, that is, we would not allow the arrow to ever 'skip' any empty stretch of space. And that should be so from the very beginning. For an arrow to even start moving, it would have to pass from its initial position to a contiguous one smoothly, and then onwards. Unfortunately, this mental model raises a number of issues.

For one, if we were able to identify a contiguous position, then we would also be able to identify the location where both positions meet. Would that location be an intermediate position? Hardly so. Any intermediate position would be contiguous to the previous one and to the next one, and therefore the initial contiguity would be lost.

This conundrum was already pointed to, although in different terms, by Zeno of Elea in his renowned paradoxes, particularly the one about the 'impossibility' for an arrow to ever move. But also the one about Achilles and a tortoise, whereby the speedy Achilles could never get to reach a tortoise sluggishly pacing ahead of him. 

As centuries went by, the development of mathematics --specifically, mathematical analysis-- brought about the notion of real number, which formally solved the problem of the discontinuity of rational numbers, i.e. any numbers which could be expressed as the quotient of two integers. The problem with real numbers, however, is that, by introducing infinity to define continuity, they exclude the notion of contiguity. Two real numbers can be selected as close to each other as we choose, but could never be contiguous to each other. There will always be an infinity of real numbers between them.

The mathematical definition of continuity has proved to be a most useful and indispensable tool in engineering, architecture, physics... and mathematics, among others, so it would be ludicrous to even think of dispensing with it. The problem seems to reside not in the formal rules of mathematics themselves, but in the realm of mental concepts and, hence, human language. We would very much like to think that, as Achilles sprints, he does not get lost in an dizzying maze of infinite infinities before being able to reach a crawling tortoise ahead of him. 

A solution to this quandary could perhaps be arrived at by formalizing the concept of acuity. Depending on the means we use to assign a number to a physical magnitude, the precision of such assignment will be variable, but unavoidable. Insofar as we could never exactly pinpoint a location in space, or determine a mass, a pressure value, or an atom's velocity, some degree of fuzzines is to be inexorably counted on, no matter what.

We could specify that degree of fuzziness, e.g., by expressing the value of a magnitude not as one single number, but two, the second one being the threshold of our accuracy. As an example, Achilles' position in his pursuit of the tortoise would be expressed not as 0.35, but (0.35, 3), 3 being the number of decimal places we are confident to have measured with precision. We could refer to it as the 'acuity' of the measurement.

According to this convention, then, one specific measurement could be expressed as any number of different pairs (n, q), where n would be a number, and q its acuity. Thus, if Achilles' position were determined as (0.35, 3), then we would know for sure that he is not at 0.36, but the positions 0.350001 and 0.349998 would be, as far as our determination is concerned, equally acceptable.

The above is not a quantitative definition. Any pair (n, q) could be used to describe Achilles' location at any instant (t, u), and what mathematicians call the equivalence class of all such pairs could be used as a definition of the concept of 'location'.

How would this new concept get over Zeno's paradoxes? Insofar as locations have been defined as fuzzy concepts, both Achilles and the tortoise would also occupy a fuzzy territory, and there would always be at least a pair (n, q) that would describe both Achilles' and the tortoise's location. By increasing the acuity, their positions could be discriminated, e.g. as resp. (n, q), (n', q),  in such a way that, when Achilles's next step is greater than n' - n, we could always find an acuity that discriminates him as ahead of the tortoise.

Being a qualitative definition, the above does not seem to help much to the advancement of science--that is, until we enter the quantum realm. At a quantum level, we will find that there is a limit to the discriminating potential of a measurement, essentially determined by the Planck constant. Furthermore, the acuity of time measurements could also be expected to reach some limit, not because of the inherent nature of time, but rather because of the inherent nature of our own relation to reality, including our instruments' readings.

But the main lesson to be learnt from the above is semantic. Numbers are not really useful at describing mental processes. Instead, qualitative concepts such as 'range' and 'contiguity' are. Computers may provide a fashionable model to represent mental processes, but the human mind, I am afraid, does not generally compute. It categorizes, compares and connects. A hardware configuration that duly reflects those processes might therefore provide the key to understanding the human mind.

Fortunately, hardly anybody will ever read this post, much less take it seriously. For finding out how the mind works would --eventually will, but I will not be to blame-- open a door to the hell of total control of the population, which to this date is already worrysome enough. So, let this post remain insignificant, drifting forever in the vast ocean of the Internet. This is what I have found. It may be stupid, or useful, who knows. But I have to content myself with just having written it and getting it published. It is intimately frustrating, but morally reassuring.

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