Sequential time, mathematics and narrative

First Act of Intuitionism. Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognizing that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained in memory. If the twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities. And it is this common substratum, this empty form, which is the basic intuition of mathematics. [Brouwer. 2011, 44]

If we understand Brouwer’s first act of intuitionism devoid of its mystical overtones, propositions such as a basic mind’s intuition of the passage of time, or the passing of twoities into the empty form of the common substratum of all twoities, his distinction of a life moving into two separate things, gives us all the psychological foundation that mathematics needs to start its conceptual edifice. The proposal is rather simple, as it could be expected from such a foundational question: cognitive acts are based on a intuitional pre-logical distinction, a condition of possibility for thinking, the intuition of sequentiality However, if we do not ground the act of distinction on biological basis, we run the risk of ascribing to the twoity or foundation of sequentiality thus defined -as Brouwer himself does- to a transcendental content. His understanding of language uniquely on human terms justifies his separation between language and mathematics, but it does not make any psycho-biological sense. The act of distinction which characterizes the intuition of time in Brouwer’s sense precedes the use of our human languages but not of languages in general, i.e., of communication among living creatures. A twoity, therefore, precedes also the linguistic construction that we call number -even the concept of numerosity- and denotes an action of separation, giving the condition of possibility for any verbal communication and narrative.

Contemporary neuroscience has confirmed the sequential character of the linking of two mental operations. Despite previous suppositions about the essentially parallel processing mode of the human brain, Sackur and Dehaene [2009] have shown that the linking of two conscious processes of the brain have an almost-sequential character, and can be accurately described by the model of a Von-Neuman like machine (or a Turing machine as well). Curiously, according to the experience gathered by their tests with basic arithmetic operations, there is a parallel functioning of the brain for computation at an unconscious level, but such workings would produce stochastic nodes of data that would be further processed and composed in a sequential manner. Expressed in musical terms: the unconscious processes would give the vertical or harmonic, the multidimensional aspects of the procedure, a complex way for the checking out of information and data of a multiprocessor architecture in which the same information is processed through different channels at the same time. On the other hand, conscious processes are given by the horizontal or contrapuntal passing of one chord (object) to the next according to a set of rules different to the vertical ones. Brouwer’s twoity would refer to this concatenation, the syntactic monoid structure of language, but says nothing about the vertical multiplicity of the objects of the monoid, the semantic memory checking in which the past is recombined to produce a synthetic image for the conscious syntactic process. The interpretation given by Sackur and Dehaene suggests a continuous semi-stochastic filtering out of the plurality of the unconscious processing of information into a single line of consciousness and determination whose performance is conditioned by the new data entering the system, i.e., by the action of the human being in a cultural and natural milieu. The experiments performed have little semantic content, and it would be interesting to carry out further experimentation with narrative verbal processing in order to ascertain the mechanisms of the stochastic vertical processing of communication. However, if there is a general constraint of sequentiality in the conscious chaining of mental operations, as Sackur and Dehaene’s experiments seem to prove, narrative constructions (and mathematical ones) would be conditioned by such biological constraints, reflecting general epistemological strategies for survival and adaptation rooted in the unconscious addition and complexification of numerous simple neural processes, but not by Brouwer’s substratum of twoities, or by any other transcendental condition of possibility for human thinking.

References

Brouwer, L.E.J.  Brouwer’s Cambridge Lectures on Intuitionism. Cambridge University Press. Cambridge. 2011.

Sackur, Jérôme and Dehaene, Stanislas. The cognitive architecture for chaining of two mental operations. Cognition 111 (2009) 187–211.