- 1 The relative intransigence of quantification
- 2 The easy choice of ‘all mathematics’
- 3 The mathematical universe locked between a priori and a posteriori
- 4 Approximations hidden in the ‘=‘ sign
- 5 Engulfment in Mathematics
- 6 What can a genuine transcendental epistemology be?
- 7 The relational is self-defined, but the non-relational?
The relative intransigence of quantification
Let’s dive into the fundamental meaning of quantification. It implies an intransigent separation between individuations. Is the nature of this separation knowable? Yes, if we can reduce individuation to parties, whose relationships create discontinuity. The separation is therefore intransigent but not irreducible. Except by adopting foundationalism, that is, to postulate a quantified foundation inviolable to reality. But we will avoid postulates in this reflection. Especially since it leaves us perplexed by the nature of a separation that is no longer reducible.
In quantification, the nature of the separation is contradictory. It is intransigent but lets “pass” through the content of adjacent individuations. Mix of discontinuity and continuity. All this is very strange if we see individuations as substances. But physicists are not moved by effective distancing: they see quantification through a mathematical mask. Like any language, mathematics self-defines its rules and symbols. The separation of numbers cannot be analyzed. It is. Given like all other properties of language, implicit in the human mind. It is difficult in these conditions to look for its origin within the mind. What tool can it grab when all its tools come from the object of analysis itself?
The easy choice of ‘all mathematics’
Pragmatic, the physicist obliterates this insoluble problem and assimilates reality to its mathematical mask. Since the unconscious always finds a way out, the explicit symptom of this assimilation is the mathematical universe hypothesis (MUH). Its author, Max Tegmark, frankly assumes the simplification of an entirely mathematical reality. Between ‘nature is written in mathematical language’ and ‘everything is mathematical’, he chooses the latter. For him, physicists ‘discover’ the mathematical essence of the universe through its constituent equations. Also belonging to this universe, that is to say their minds being made of similar equations, they gradually put them in correspondence with those of matter.
A debate organized by the journal Foundations of Physics showed that Tegmark’s position is not widely held among physicists. But competing positions have not been stronger. The MUH is still in the running.
This position is universal structural realism. It clashes with Gödel’s incompleteness theorem. Incomplete does not mean false. Is it possible to philosophically invalidate MUH and in general the assimilation of physical microreality to math?
The mathematical universe locked between a priori and a posteriori
What is the nature of mathematics? Their structure was developed by Bourbaki. All symbols are integrated into it, from integers to topological spaces, with their operations. A set of remarkably self-defined language classes, which says nothing about what produced it. Again the mind is trapped in circularity, having no tool external to itself. Clues such as Gödel’s theorem persist to make us suspect the existence of an exterior. MUH is not decidable and therefore does not fall within the field of science.
The heaviest handicap of the MUH is elsewhere. Not only does it invent a foundation a priori, but it also decides the result a posteriori: entities that do not test themselves as mathematics, that use other languages and ways of perceiving. Assuming that the MUH is right, unfolding the fundamental equations would intuitively result in a great book of mathematics such as Bourbaki’s and not the sentient beings that we are. Without embarrassment, the MUH arbitrarily forges its airstrip as well as its runway. The reasoning is no longer circular but narrow-minded.
The MUH nevertheless remains attractive to many realists because of its simplicity, and with the reinforcement of the philosophy of science, which has also become very structuralist. It favors the analysis of reality as relationships in a context, as we saw in the previous article on Michel Bitbol and his ‘De l’intérieur du monde’ (not translated). However, Bitbol insists on the importance of getting rid of any a priori on the origin of things in relation, and everything a posteriori on the result. By ignoring this recommendation, Tegmark is symptomatic of a common attitude in the basic sciences: to make the supposed continuity of mathematics a parallel continuity of reality per se. Let’s see why this is philosophically untenable:
1) Any mathematically symbolized individuation is an approximation. The notion of ‘initial conditions’ consists in defining the individuations on which the mathematical symbols are affixed. The “elements” are axiomatic. Creating them in this way obscures the changes in their constitution. Intrinsic oscillations that fade in front of extrinsic stability; their properties are permanent. Yet no element has a fixed constitution. It varies on a different scale from extrinsic properties, without significant consequences on them. When science “fixes” the constitution of an element in the foundations of reality, for example the quanton as a quantification of a field, it does not have the means to verify its intrinsic stability. Need for a tool belonging to the level underlying the quanton, but it is energetically inaccessible. We know that this level exists: quantum vacuum and its virtual petulance. What is even more fundamental?
The quanton is therefore a physical approximation, to which is associated a mathematical term: the Hamiltonian. This approach works very well. Reality per se behaves faithfully to approximation; the model works. Two hypotheses: a) Reality is mathematical. But this forces us to confuse the physical reality of the quanton with the mathematical symbol. A priori untenable. The rigor of math cannot be satisfied with an approximate symbol. (b) Reality itself makes approximations. It defines elements in a context, thus creating levels of existence in relative independence. Such qualitative separations do not exist in mathematics, whose formalism can be applied indifferently to any level of existence. Something fundamental is lacking in math to claim the status of reality per se.
2) The actual nature of separations changes for each mathematical quantification. Each system is defined by specific elements in relation, in a particular context. For example, a molecule is a set of atoms floating in a “vacuum” that is not one at the quantum level. The mathematical model uses quantifications. It assigns symbols to each atom and calculates the fate of the molecule. Can we say that the separations between these quanta of information are of the same nature as those between hadrons, quarks and virtual particles? Rigor answers no. Qualitatively no. Yet this is what the observer is forced to do when he declares reality per se mathematical. It considers all separations between numbers to be identical, regardless of the element to which the number is symbolically associated. The same goes for operative signs. The sign ‘=’ in particular has the same meaning whether it applies within the system defined at the beginning or moves to another. The mutation of the framework is ignored. One of the best examples of this mathematical prestidigitation is the so-called Boltzmann equation: S = kB logW. which relates a physical value (S, entropy) to a probabilistic value (W, the number of possible microstates). The terms of the equation each concern a level of reality irreducible to the other, but are nevertheless amalgamated by the sign ‘=’, which here does not have at all the same meaning as in the addition of identical properties.
Caricatural example, but all mathematics is concerned with this strange confluence of discontinuities between numbers in a single nature. If this position were correct, reality would be a large, single, open-ended mathematical system, which it obviously is not. The mathematical universe has trapped us in the mask of a reality per se stripped of its complex dimension.
Engulfment in Mathematics
The more complex the entities of a system, the more their separations hide a host of criteria. Using integers to individualize them becomes a challenge. Between a first human and a second, the separation is not that between a first atom and a second. The equations do not take this into account. The qualitative is absent. A container associated with each model is missing. Mathematics is only sufficient in itself by placing the observer inside the language. Engulfment that seized Tegmark: I am mathematics and I want to be reality. No more exterior.
Getting rid of the a priori and a posteriori is a valuable recommendation to build a fragment of objective reality / faithful to its being. It is in this objectivity that mathematics finds its strength. They find a custom language for these fragments because each is a single relational set. Exceptions exist: renormalization manages to be transcendental because reality sometimes opens the barriers between its levels, adjusts them to each other. But always it is mathematics that adjusts to reality, not the other way around. There is no one or more founding equations of reality. They are in fact more numerous than particles in the universe, separated by qualities whose nature is not descriptible by pure mathematics. We need a constituted being, an observer, not necessarily brain or even alive, who can enter into relationship with this quality. Observers certainly exist: we are. Their quality comes from their offbeat positioning, in the complex dimension, of what they observe. No need for a mathematical model. The transformation is purely conceptual: from content to container. Change of look.
What can a genuine transcendental epistemology be?
Philosophers and scientists agree to elect the relational system as a fundamental unit of knowledge. The system is self-defined and abstracted from a priori and a posteriori. Autonomous existence associated with a mathematical language; we will now avoid confusion between the two. A direct implication is astonishing: the mathematics of the system must also be considered autonomous, relative to its use in other contexts. Numbers, operators, separations between quantifications, have a specificity relative to the system. In this singular identity appears the qualitative aspect of the system.
But to whom or to what does it appear? Refocusing on the system the entry of knowledge then makes look at the adjacent rooms, rid of a priori and a posteriori. Did we really enter the first with a virgin mind? impossible. The mind is made of representations that seek each other. Ours cannot see in the system something that would be completely foreign to them. Familiarities are discovered with our mind, owners of the system, which must be coordinated with the adjacent rooms. Linking them firmly together will consolidate the fabric of knowledge. Self-generated fabric in a dimension greater than the system, that of complexity.
We are immersed in a paradox very specific to the complex dimension: one can only enter it through the small door of a system, before extending its global vision; but it takes a multiple vision to enter and discern something. Two contradictory views, the multiple of the constitution and the global of the observation, which have no mathematical translation but form the container.
The relational is self-defined, but the non-relational?
Mathematics has both an intrinsic property of multiple realization and is, applied, the property of the system to which it is addressed. Systems thus become constitutive units of reality. But this is not a usual mathematical set. The units are not juxtaposed; they are surimposed in the complex dimension. Surimposition is the notion that one level of information is added to another without making the precedents disappear. It imposes itself on them but its constitution remains dependent on it. The self-definition of a level is relational, of the same nature as that of systems but inverse: it is non-relational, separating elements that cannot interact together. Surimposition is the equivalent in the complex dimension of juxtaposition in the spatial dimension.
If mathematics is segmented between levels of reality, how can it be linked other than by current empiricism? Is there a single metaprincipe of complexity or several? Surimposium proposes a candidate: the conflict, modeled in the principle T<>D. To follow…