The Crises of the Mathematical Sciences: ex falso sequitur quodlibet


Realism as Literalism

 

“With Democritus’s atomic theory emerges the possibility of a gap between representations and represented—“ appearance” makes its first appearance. Is the table a solid mass made of wood or an aggregate of discrete entities moving in the void? Atomism poses the question of which representation is real. The problem of realism in philosophy is a product of the atomistic worldview.” – Catherine Malabou

 

Looked at through a different diffraction in Malabou’s own sense, though, Democritus’ representations as literal (and hence arising from a relation to the written word or record) arises from the notion of record keeping itself, as such it is itself founded on currency as representation of value, and the real as property. Democritus’ atoms are a “currency of reality” in this sense.

 

What does it mean, though, that prior to Democritus’ theory the gap between that represented and representations wasn’t an issue? To understand this we have to understand the notion of the mathematical itself. From the modern perspective, Democritus’ theory isn’t perceived as particularly mathematical, yet it could only arise based on the mathematical.

 

Mathematics, until the later 19th century, was axiomatic. In this it corresponded to the original meaning of mathesis, as what we always already know. However the axioms of mathematics were already re-presentations of what we always already know rather than what we always already know as such, and so were already oversimplified. The work of Cantor and later mathematicians refounded mathematics, as measure, absolutely on the absolute notions of set theory. In doing so they were entirely correct, since as the measure of relation itself mathematics could not be correct if it had a priori relations, i.e. axiomatic ones. The absolute is always the self-referential insofar as the self is taken as self-identical.

 

However mathematics is simultaneously the ontological language of modern science. The recasting of mathematics though as purely epistemological, i.e. having no a priori relation to anything but itself, removes its apparent ontological foundation, which was properly always onto-theological. This ungrounds the sciences insofar as they are mathematical. To the degree that various sciences are more or less purely mathematical, the ensuing crisis became apparent more or less quickly. Physics, among the natural sciences, was the first affected, and to this point the only one to have changed any of its basic assumptions in accord with the reality of its mathematical ungrounding. For this reason the notion of appearance and reality is radically subverted in quantum mechanics and quantum field theory, the areas of physics that have recast their assumptions in the most fundamental sense.

 

To go back to Democritus and his atomic theory, when compared to modern quantum theory we have to say that modern quantum theory is not at all atomic. As the base of any possible homogeneity of reality (without which no physics of any kind is even possible) quanta, itself pure measure, has replaced the atom. Quanta is arbitrary, but not in the same sense as a mathematical set, in that the arbitrariness is not that of human though but of reality itself. This homogeneity of reality is itself, as a result, non-absolute and thus subject to complementarity – the absoluteness of an observation is only relative to the apparatus involved. But the most basic apparatus, since all measures from other apparatus must be interpreted through it, is the self itself.

 

In the setting of axioms (the original mysticism of number) what is always already known, the mathematical in the Greek sense, is only known intuitively, that is by the self and not by the hypokeimenon or measuring-facility of the self, which we know as the “I-Subject”. In founding mathematics proper as measure, this originary mathesis had to be re-presented to the hypokeimenon for judgment. The self itself, which having experienced presentation is the only entity capable of such re-presentation, always arbitrarily simplifies this “subjective” re-presentation; the origin of representationalism and thus the disparity between appearance and reality is in itself an oversimplification. Simultaneously the axiomatic nature of mathematics at the time produced a false appearance of ontological sufficiency. This onto-theological understanding of the results of re-presenting created an inconsistent and somewhat ironically, inexact mathematics.

 

As western thought and western science developed through the Greeks, to the thinkers and scientists of the Renaissance, this unnoticed inconsistency of both was compounded with the notion of a gap between representation and what it represents, along with a forgetting that what it re-presents must have initially been presentation itself, based on the recast understanding of the hypokeimenon as the “I-Subject”.   This recasting was based on an interpretation of quasi-magical perspectival tricks accomplished by Renaissance painters in Italy. The basic sense of the gap is the hole that was placed at the point of single point perspective paintings in the famous perspectival experiment. The perspective produced “indicates” something that cannot be contained, that is always subtracted from any representation, yet is simultaneously the proper topic, or subject of the painting, the “I-Subject”. That it is a gap is due to the difference, unnoticed by Descartes, between the self that represents and the I-Subject for which this representation is accomplished. In accordance with the perspectival trick at the root of the latter, the I-Subject is not in the exact place of the self, it is posited as an infinite perspectival focus behind the self. Thus when we represent to the measuring or judging facility, we experience it as a “step back”, as though it were somehow spatial. Yet factically we do not need to change location in order to accomplish this “stepping back”.

 

It is with this notion of the I-Subject that “rationality” reaches its full potential and is thus fully literal. Literal here though has to indicate that as measuring (ratio = measure) that what is measured, i.e. what is thought rationally is so thought as a literal referent, a referent at root to a written record, as the word literal itself indicates. It is this that makes rationalism and literalism, at root, based on an a priori written record keeping, and thus only sustainable within language and conscious use of language itself. The onto-theology at work thus fulfills itself in that the sense of being, of that which gives meaning, is purely located in the psyche, or soul, for the rest of reality, a meaningless mathematical projection, eventually understood as physical in the restrictive sense, takes place of the sense of that which we are always part of.

 

It is at this point and as the “age of reason” that the gap becomes fully realized and thus a problematic of its own, as dealt with variously by the philosophers of that time, most fully by Kant. Phenomena are firmly located in the conscious psyche as rational representations, while the directly unknowable noumena, the real of the real, are posited as an external reality that we can only know via their phenomenal representations.

 

Even the apparent reversals of medieval thinking in the modern are only that, and in fact are in general the most extreme consequence of prior assumptions rather than a reversal. The physical universe as meaningless replaces for instance, the “medieval world” as, although “fallen”, still meaningful, since all meaning has been located in the soul as rational consciousness. The very literalism of modern science is thus, not as people popularly imagine it, a direct knowledge, but precisely knowledge rooted only in language as the medium in which rational consciousness represents such reality.   Even in the Renaissance language was only seen as one figuring among others of world as inherently meaningful, whereas in the modern period it is the only locale of meaning, the rest of reality consigned to meaninglessness.

 

This reduction of non-human reality is, as reductionism, and invalid reduction. While reduction has been seen since Aristotle as a valid method of science, it was subject to restrictions centred on its being a valid reduction, and on not failing to take into account, after the reduction, what the reduction as such does not address. As an example, it is a valid reduction to understand certain metabolic processes of human cells via the similar processes of bacteria, but it is invalid to propose a reductionism, as one famous biologist maintains, that “all life is bacteria”. Numerous metabolic functions of human cells cannot, in fact, be understood on the basis of bacteria because they precisely fail to exhibit the behaviour our notion of such processes is based on. Another rather more famous example is that it is a valid reduction to understand certain elements of the genetic process via the self-replication of molecules that have that ability, primarily RNA and DNA, it is an invalid reductionism, though, to view human beings (or any organism) as merely an inessential by-product of such replication. The latter ignores the fundamental workings of the genetic system itself, which is by no means restricted to the genome. The word dialectic is often overused, but in the sense of understanding genetics it is appropriate in that the phenotype is determined not by the genotype, but by the interpretation of such, itself determined in dialogue between different types of cells and the identical genome, the body as a whole in the manner that cells differentiate based on bodily location, and the environs of that body.

 

At its most extreme in such reductionism, though, you find the inherent problem in the sciences as mathematical. What was taken as ontological is purely epistemological. In not approaching ontology science is predicated on nothing other than assumption. Science doesn’t “have” the truth because it doesn’t approach the realm in which truth can in any way be decided. It’s correctness is only valid within a restricted economy of thinking, restricted precisely by the unquestioned assumptions that only appear to ground it.

 

The proper sense in which science is literal is that it is predicated on a metaphorical application of the idea of written records, bookkeeping, onto non-human nature. At a more basic level it is predicated, as Democritus first shows, on an interminable exchangeability of one thing for another. Democritus’ atoms are the currency of that exchangeability. This metaphorical notion is in itself the foundation of the literal, the rational and the historical itself. Reason itself, far from being irreducible, is not even justifiable except in very restricted circumstances – those of accounting-for a given bookkeeping.

 

This literalism., along with the rationalism and historicism that accompanies it, is not only misleading, but invalid in the ways it is applied as modern science. Hence the crisis of the sciences based on a mathematical model that implies and, in its final form, requires it.

 

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