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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