Gottlob Frege and Bertrand Russel
André Deledicq and Mickaël Launay tell in Dictionnaire amoureux des mathématiques how Frege saw his beautiful set theory dynamited by a seemingly innocuous firecracker, posed by Russel.
The year is 1902. Logicians like Frege try to make sets the fundamental core on which to build all mathematics. Arithmetic operations thus become special cases of manipulations on sets. One of Frege’s postulates is that regardless of the type of mathematical object, it is always possible to consider the set of objects with the same property. The postulate applies to the sets themselves, always likely to form another.
A logical paradox
This is where Russel places his firecracker. Consider the set of all sets. By being one itself, it is therefore inside itself! Why not? Let’s then take this bizarre property to find the set of all the sets that don’t have it (those that don’t contain themselves). Does this set contain itself?
There is no right answer. This is a logical paradox. Russel illustrated it with a more concrete story: In a village the barber shaves all the men who do not shave themselves. Does the barber shave himself? No logical answer. If he shaves he must stop since he shaves only those who do not shave themselves. If he stops he joins the group of those who do not shave themselves and should therefore shave. Such a barber cannot exist. As well as the set of all sets that do not contain themselves. Frege’s postulate does not hold.
Complexity would have saved Frege
There was, however, a way to save set theory, which Russel proposed: to define these recalcitrant supersets differently, to call them ‘classes’, to get them out of the properties applicable to sets.
Sadly enough, Deledicq and Launay report that Frege didn’t publish much after this story. He isolated himself and died demoralized in 1925. This is very sad because his theory is not a failure. It was only inscribed in a horizontal look, without vertical thought. Frege did not go the wrong way, as the authors say. It lacked the complex framework in which to locate it. And Russel has no more formalized this framework than Frege. He simply showed that logic could not remain in the previous one.
Tantalizing equation of the Whole
Mathematicians and physicists alike are fascinated by the idea that there can be a fundamental principle, possibly materialized by an equation, from which all the objects of their discipline would be born. “Theory of Everything” among physicists, set theory at Frege. In the induction/deduction couple, which makes it possible to move from the general to the particular and vice versa, it is the universal aspect of the general that fascinates. An ultimate equation is the equivalent of the divine in the scientist. The world, in its incommensurability, is finally surrendering to our minds. From insignificant fetu, it becomes personalization of the creative principle.
Frege believed he had attained this enlightenment. Hope brutally discredited. Russell’s letter, in its delicate and polite guise, “I agree with you on almost everything that is essential […] There is only one place where I have encountered a difficulty,” is a stab from which Frege’s soul will not rise. This is far from a simple intellectual debate. The mathematician is human and therefore mystical. Mathematical research is theological in many ways, with an enormous stimulus: the seeker writes the gospel as much as he consults it.
Who owns the math?
If I insist on this aspect of mathematics, it is because it greatly influences the question of whether it is a language or reality per se, translated into signs. Is our mind only interpreting reality per se in its own way, owner, without being able to go further? Or does it access the fundamental essence of reality, in accordance with mathematical ideals?
I deliberately accentuate a difference between the 2 positions, which is not so clear-cut. It is, after all, a spirit-real relationship, in which no participant can fully encompass the other. Each communicates, that is, accesses the part of the other that it can represent. This similar part is codified by the language of communication. The previous question becomes: Who developed this language? The spirit or the real? Difficult to answer, in this matter of chicken and egg. Except to seek a divine ancestry, the spirit thinks it comes from reality. So mathematics is, originally, the property of reality. But the mind has built a relative independence, in cooperation with others. It lives in a representation of reality that moves away from it in many aspects.
Touching the real
As much as artistic languages seem dedicated to distancing it further from the real, mathematics seems to bring it closer. The mathematician experiences himself as an asteroid that orbited far into the darkness, crossing ghosts of reality, and approaching its solar essence when it follows the flows of equations. Mathematics, because it is discovered, seems foreign to his mind. So if it is necessary to designate an original owner, it is the real.
This process of discovery is very different from the spoken languages, which are learned, and sometimes have no logic. Their “reasons” are lost in human history. Strictly human. The separation of oral and mathematical language thus appears as an iteration of the mind/matter dualism. Everyone has their own way of expressing themselves. But is this dualism still tenable? Contemporary science is erasing the gap. Mathematicians are generally materialistic, reluctant to dub a biblical origin in mind. Isn’t the separation of spoken and mathematical languages, then, the survival of an obsolete dualism?
Human vs. Supernatural source
Difference is a feeling. We easily trace the birth of language in our minds. Parents, educators, have transmitted it to us by learning. There is nothing universal about it. Another people speaks of it as an entirely foreign one, which describes reality so precisely. Human creation.
Mathematics presents itself to us as if by magic. Regardless of the people, any researcher who analyzes reality forms the same language, with an equivalence between the signs, identical theorems. The one attributed to Pythagoras was known to Mesopotamian scribes several centuries before his birth. The mathematician has the impression of updating a structure of reality that has always been there, which any other mind will grasp in the same way if it seeks to understand it. Real creation.
Final conscious access, not full
But do we access our entire mind? Not. Consciousness is only a tiny part of it. Sophisticated attentional layer that back-controls thoughts, most of them popping up out of nowhere. Even when their filiation is clear to us, we would be hard pressed to predict where they will go. We are a mind, and simultaneously if a part emancipates itself to observe it it knows only a tiny fraction.
Where does the illumination of a theorem come from? A posteriori its mechanics are clear, a priori we know nothing about how this mechanism engages. There are cogs in our unconscious that are mathematical enough to make them. Consciously we contemplate the final fabric, represent the weaving technique, and return orders to improve it. But the machine is not in the conscious field.
The real as an unconscious machine
This machine out of reach of consciousness is really what the mathematician calls “real” and not reality per se. The two are of course in close relationship. No mind without matter. But the mind must remain warned that its own independence, which allows it to represent, automatically creates a distance from reality.
It finds proof of this easily when a mathematical demonstration is false. The language stands, respecting its internal coherence, but the application to reality does not hold. Language is not reality. Mathematics creates a mental universe that frequently intersects with reality per se. Other parties do not. The mental universe is less vast than the real, in volume of information, but more varied. Because much higher located in the complex dimension.
Platonism is a religion
Mathematicians hide another dualism, also very old: the world of Plato’s ideals. As reality does not seem to host all possible mathematics, it is necessary to house them in a larger virtual world, superimposed on reality. How do these two worlds communicate? Deep mystery that puts Platonism in the rank of religions.
To escape this criticism, some mathematicians have broadened the field of reality to subordinate it to mathematics. If it is the essence of reality, then all its forms must really exist. Any new mathematical structure discovered necessarily corresponds to a universe, even if we have no way of observing it. The intrinsic coherence of language is transposed to reality, by extraversion. The mathematician’s mental universe becomes entirely realistic, in a computational version of solipsism.
Escaping the religious through complexity
Solipsism is not derogable in the absolute. I have dealt with this subject elsewhere. Group solipsism is even less easy to denigrate, hidden behind a false collectivism: “Look, we’ve all come to the same conclusion, so it’s a universal principle.” I therefore prefer to offer you an alternative solution, less axiomatic, using the complex dimension.
This approach does not aim to get rid of the separation of mind and matter but on the contrary to multiply it, between successive planes of reality, to such an extent that it ends up resembling a continuity. Monism born of a frantic division, family of systems in relative independence.
Let’s start from mathematics as a language, the only aspect of which we are certain, since manipulated directly by our mind. Is it ownership of reality? Let us leave this issue open for now.
Like any language, mathematics can be applied to a multitude of different objects. Multi-realizability. For example, I take the language of parallelepipeds to store slabs in a lean-to or books in a cardboard box. The same equations allow me to know the volume of the container and the contents, to verify that they are compatible. I applied a unique language to very different objects.
The unconscious knows math
Additional clue: even ignorant in mathematics, even without taking out ruler and calculator, I unconsciously use mathematics to arrange objects. It seems that learning only formalizes a language that I already possess, which is used by the inaccessible layers of my reason.
Multirealizability also for abstractions. Our mind is constantly digging into its existing models for new representations. And between abstract and concrete objects? Similar equations apply to flows of biologic exchange and finance.
Each system creates its own language
The complex dimension attributes relative independence to the systems. A system is self-determined by the relationships of its elements. Each system creates its autonomous language, spoken collectively by its elements. Just as the strength of English is to be known to a majority of humans on the planet, the strength of mathematics is to be spoken in a common way by physical systems.
Does the rapprochement seem specious to you? Between a language learned and another expressed ontologically? Yet we can say that the physical elements “learn” their language from their own ascendants, from what gave birth to them. Adopting this view is easy by abandoning the religion of the ‘fundamental laws’, very mysterious in their own origin.
Community is not structure
Mathematics that applies with equal happiness to different systems reflects a close community between different parts of the complex dimension. This translation is not the structure. Not until we find an equation that turns each model into its successor. The principle certainly exists since reality is one. But is this principle an equation? Not. An equation will always be a mental representation, partial property of the human and not the principle in itself. This point is justified more fully in Surimposium.
The 1st advantage of mathematics is its incomparable intrinsic coherence. The second is its flexibility, through the multitude of its sublanguages. Let’s push this flexibility to the point that each system owns its relationships, it also has partial ownership of its maths. To make it a universal mathematics is to make reality an indivisible Great Whole, which it clearly is not.
Universal or human mathematics?
Indeed, universal math is its appropriation by the human mind, not its delegation to reality. It facilitates the work of the mind but do not reflects the versatility of reality, which builds its fantasies with a dynamism that no mathematics can encompass. Let us then relativize our fragments of language to what the fragments of reality seem to speak. If we can connect these fragments, so much the better. But without the theory of connection, it is impossible to make it a continuity.
Each system speaks its own iteration of math, to which we apply our standardized version. It works very well at the bottom of complexity, less so when rising. But it is still an approximation. By adopting this idea, the system must also be given back the relative ownership of its framework. There is no longer a universal framework. Only the one self-created by a system in its level of reality.
A framework for each system
The universal spatio-temporal framework is obsolete. It is generalist only for physical elements fully described, collectively, by this framework. Because they own it. Other more complex systems include the spatio-temporal framework, but surimpose on it other layers of meaning that personalize their ‘framework-system’. Example: the diffusion of memes, these mental objects shared by human minds, does not obey the spatio-temporal framework of physical particles, although it is always present, by the location of brains.
All this is a fundamental change of view of mathematics. Purely philosophical? Mathematicians often pout at the lack of practical outlets. Here is one, in the form of a theorem: When they come across a logical paradox, they are faced with a crossing of mathematical complexity.
The framework has changed. The previous one has not disappeared. It must be surimposed on another one. ‘Classes’ over the ‘sets’…