Metaphysics and Mathematics

(Pr. Jacques Vauthier, Sorbonne Paris 6)

 

“Ex umbris et imaginibus in veritatem”

— Card. J. H. Newman, Epitaph

 

 

In memoriam R.P. Stanley Jaki

“Mathematics has come to be identical with philosophy for modern thinkers, though they say that it should be studied for the sake of other things”[1]. This sentence taken from Aristotle’s Metaphysics would have suited perfectly the vision Father Jaki had of contemporary philosophy of Sciences. If philosophers were unable to grasp the impact of physics as it was developed in the 20th century, they were unable to see what they owe to mathematicians who slipped relentlessly from their science to philosophy. Harnack, a German physicist, once said in Berlin “People complain that our generation has no philosophers. Quite unjustly, it is merely that today Philosophers sit in another department. Their names are Planck and Einstein”[2]. This is to be related to Heisenberg’s advice “One can’t get anywhere in Philosophy nowadays without knowing something about modern Physics”.

Furthermore, in a world where the masters of uncertainty diffuse a prevalent desease, there is no doubt that mathematics appear as the rock of stability, despites Gödel’s theorem which stops any will of encompassing the whole reality. Let us recall that one of the jewel in Father Jaki’s opus magnum is the limit he shew about the impact in modern Physics by Gödel’s theorem.

Jean-François Revel is asking the following question in his Histoire de la philosophie occidentale: de Thales à Kant:

“Why metaphysics in 20 centuries of existence was unable to produce any consensus among poeple and obtain any indisputable proved result? It was but able to multiply different schools which excomunicated each other. On the contrary, we face a Science, namely Physics, which in less than a century established truths nobody could contest and a method everybody admits”.

But the core of Physics is Mathematics.

For instance, it was thanks to the theory of tensors of Levi-Civita that Einstein was able to elaborate his General Relativity. Infinite-dimensional Hilbert Spaces are crucial in elaborating Quantum Mechanics. There lies a kind of “miracle” as pointed out by our philosopher Simone Weil: “From a general point of view, Mathematics in their wider sense, i.e. encompassing under this name any study which is rigorous, theoretical and pure of necessary connections, will constitute both an unique knowledge of the material Universe where we live and the true glint of divine truths. No miracle, no prophecies can be compared to the marvel of this concordance”. Furthermore, she added “this very Mathematics is above all a kind of mystical poem composed by God Himself”[3].

One cannot but think of the letter to Solovine written by Einstein where he expressed his astonishment in seeing this concordance, speaking explicitly of a “miracle”. Indeed, “Everything was made with measure, numbers and weight” says the Book of Wisdom[4], a sentence which has been a light for Father Jaki in exploring sciences.

But why did Mathematics become the core of contemporary metaphysics? There is a road coming from antiquity to now, from Sextus Empiricus to our masters of uncertainty even if Sextus Empiricus wrote “If lines and surfaces are not objects, all Geometrical principles are ravish, all the foundation of geometry is a fake”. Descartes is a key player in this game, creating the doubt of the doubt! He designed a Geometry independant of senses, named after him “Cartesian Geometry”, for it was only with numbers and algebra that geometrical theorems where to be proved. Thankfully, Blaise Pascal objected in one of his “Pensées” “We have an incapacity of proof, insurmountable by all dogmatism. We have an idea of truth, invincible to all scepticism.»[5] in echo to “I am the light of the World”, this very Light which enlightens everyone[6]. Remember that Hegel pretended that Descartes’philosophy leads to materialism. Leibniz denounced the discontinuity installed by Descartes through Algebra and re-inforced continuity through his differential calculus. Furthermore, as Brunschwig pointed out “Causality, physically speaking, is nothing for Descartes, but a mathematical reason”. Death of metaphysics will be pronounced later by Kant after having “heard” of Newton and his “hard book”[7], namely the Principia, the impact of which gave rise to the so called “century of Lights”, preliminary to the French Revolution.

So we are left with a mathematical pace.

Most of the outstanding philosophers of the 20th century were former mathematicians. To begin with Husserl, whose mathematical thesis[8] was made under the supervision of Weierstrass, a German master in the field. After his thesis, he obtained his Habilitation under no less than Cantor, the father of set theory. It is therefore no surprise to understand that Phenomenology is nothing but a theory of relationships because existence in Mathematics is but a strong relation between mathematical objects. No wonder that the principal aim of Phenomenology is to describe phenomenology as shown by Edith Stein!

Then arrived Whitehead for whom Philosophy is nothing but a series of footnotes to the work of Plato. Concrescence is his key word where a Deus ex machina creates objects from an emporium of Plato’s ideas. Remember that Plato’s god is a geometer... Deleuze used to say “tell me what you think of Plato and I will tell you who you are”. Unfortunately for Whitehead, his work with Russell consisting in founding Mathematics on a firm basis was destroyed by Gödel: demonstrability will never exhaust veracity...Still, for most of the mathematicians, Mathematics is a terra incognita existing for ever and which is to be discovered.

The architecture of knowledge uses one’s senses in capturing the different objects. The synthesis of it is given in this figure :

Where, through one’s senses, objects are detected by intelligence and described by one’s language using common senses which lead to science in measuring the “accidents” of the being. From this, all mathematics come in a realistic way.

Today metaphysics are no more than a theory of knowledge.

A new position in the prestigious French College de France — the motto of which is Docet omnia — was created a few months ago. The title of this chair is “Metaphysics and theory of knowledge” which shows the shift from the basic interrogation about “to be” to the explanation of “how it is”. For Ms. Tiercelin, nominated on this chair, our conception of matter should contain an intrinsic probabilist data. Indeed, scientific knowledge is obtained through experimentations and procedures. So, she said, the central role played in phenomena by variations, errors in procedures, made almost impossible any certitude on experimental data and probabilistic analysis became compulsory and is a natural methodology. “Probability without necessity is a nonsense” insisted Simone Weil[9]. But for Tiercelin, her approach receives acknowledgment from the so called Heisenberg principle. She should have read the incisive comment of Father Jaki on this principle — ∆E. ∆T ≥ h, where E stands for energy and T for time — : “an interaction which cannot measured exactly cannot be ontological and produce particles out of nothing”. But for Heisenberg, this principle proved “the invalidity of the principle of causality” as causality became probabilistic. Tiercelin comments that “the main teaching of contemporary science is to reveal an almost impossible access to certain knowledge”. Science is now an infinite quest and metaphysics a science of observation...

What would be the fate for reality in front of this wave of probabilities? She defines it as an opinion of Habermas-type along with a general consensus. Language is now the core of reality in its expression. But which language? Mathematical language of course! Wittgenstein is not far, for whom ethics was the fight against the limit of language... Unfortunately, science in the 20th century shifted from differential calculus with its efficiency in Newton’s Physics to deep algebra where particles appear as elements of an abstract structure. The amusing feature is that if you apply Ockham’s razor to all this, it will remain... nothing. But Algebra is everywhere in today’s Physics: matrix theory, tensors, Hilbert Spaces even if for Carnap “mathematical theorems are only conventions and rules of language which do not tell anything”. Unfortunately for him and the Vienna School, isomorphism like the one between square integrable functions and square sommability for numbers expresses exactly duality between waves and particles... Algebra appears as a tectonic dictionary. Remember that for G.K. Chesterton, nothing is more metaphysics than a dictionary!

Undoubtedly, the desappearance of causality, drowned into propension, is the main concern. Both finality and causality have disappeared even if Schopenhauer argued that causality has to be before intuition and experience, contradicting Hume. Tiercelin insists on this, saying “propensions as well as Newtonian attactive forces are invisible but as these, they can act, they are effective, they are real. World is no longer a causal machine, one has to consider it as an universe full of propensions, a process of enlargement, of possibilities waiting for actualisation”. Incredibly, she is mixing a physical concept “attractive forces” with a philosophical one “propension”... So stochastics is now an a-priori which kills any will to explain phenomena. Only descriptions remain. Kojève, one of our master in Hegelianism, pointed out that, thanks to contemporary Physics, which show that everything is stochastic, there is no place for a divine intelligence who would know and control everything ; “new Physics imply an incoercive atheism” he said. On the contrary, Simone Weil insists that Probability in this context is nothing but a résumé of statistics which are only justified by a practical use. She complained “Never more than in our days, one has produced interpretations or hypothesis, which were never allowed to be produced with such a liberty”[10]. For her, the link between the mysterious wisdom and the quest of the scientist is expressed by the presence of Mathematics in the description of Nature. She pointed out that Arithmos and Logos are synonymous in Greek, for instance in Plato’s Philebos.

Within this context, no wonder that Duns Scot is the reference and no longer Thomas Aquinas. For the former, one has the capacity to know the singularity of any being; on the contrary to the latter singularity is second, “esse” being first. We are far from the classical sentence “Primum in intellectu cadit ens”. No wonder to find at the end, Jean-Luc Marion with his thesis “God without to be” for whom metaphysics of being is to be replaced by a metaphysics of gift. The subtil nuance is between “the one” (singularity) and “being”. In the human soul, one distinghishes between reason and intelligence. Reason catches unity and intelligence catches the being as such without any comment on it. Individuality is not made out localisation. An electron is not there, it is a-priori everywhere together with a probability of presence according to the experiment performed. For Duns Scot “the first object of any perception is unity which exists in a singular object”. “there is” replaced by “this is”. No wonder that Duns Scot together with William of Ockham are now the references for a science which concentrates only on stochastic phenomena. Charles Pierce insisted that these two philosophers were the grestest metaphysicians ever... So, after Ockham, reason is absorbing intelligence and after Kant metaphysics is just knowledge theory.

So the question is now “is a true metaphysics still possible?”

Cardinal Newman, who Father Jaki appreciated so much, is certainly one of the references for a re-fundation. Newman was an excellent mathematician as his library in Birmingham proves. “Chance is not a cause”, he wrote to Pusey. This single sentence destroys Darwin’s theory in its essence. Let me recall, the last book by Father Jaki on The Impassable Divide in this context. “A real science culminates with a differential equation” used to say famous French Mathematician Henri Poincaré.

In his Grammar of Assent, Newman shows the difference between assent and inference. The former is immediate when the latter is an intellectual process starting from objective data. Doubt appears as an assent to an inconsistent proposal related to a thesis. Assent is unconditional whereas inference is conditional as it supposes assumptions. Assent is in relation with concrete proposals, inference with abstract ones. “Science starts with matter and ends with matter” so “ne sutor ultra crepidam”: do not overpass the domain where you are expert. For instance, the last book by Harari on Sapiens[11] is a caricature of extrapolation made from his domain to what is a human being. Of course, there is a blind evolutionary process devoid of any purpose leading to the birth of individuals. But we are no better than spiders. He concludes by demanding that Biology department mix with the Law department in Universities as we are governed by chemical processes.

Newman in his realistic approach underlines that: “What the physical creation presents to us in itself is a piece of machinery, and when men speak of a Divine Intelligence as its Author, this god of theirs is not the Living and True, unless the spring is the god of a watch, or steam the creator of the engine». One has to recall the definitive sentence of saint Thomas Quod mundum non semper fuisse, sola fide tenetur et demonstrative probari non potest, sicut et supra de mysterio Trinitatis dictum est. Which Physics could “teach Holy Trinity, truth, justice and mercy?» asked Newman. “Nobody would die for the sake of computations but for realities”. Nobody would trust abstract experts who jeopardize empathy in pretending that it is a strategy of Nature to optimize natural selection! Newman concludes saying “Would you ask chemists to be our cooks or geologists to be our masons?”.

In his Grammar of Assent, Newman shows that nowadays scientific proofs are taken for granted and deliver truths which are the pillows of a new gnose. He emphasizes the fact that the Church is the only place where reason is at work. Chesterton used to say exactly the same.

To conclude by an anecdote, last year in Paris, a seminar was organised on the well known sentence of Leibniz “why is there is something instead of nothing?”, sentence which is certainly one of the most truly metaphysics among other assertions. Even for Heidegger, it was so. The outcomes of the seminar were reduced — even if a number of philosophers came to deliver their point of view — to the following conclusion “this question is preposterous”. We are indeed in a “teratological metaphysics” to quote É. Gilson. The last book by Carlo Rovelli[12] epitomised what Gilson described as a “spinozism of the becoming”. For him, Nature laws determine our behaviour because they act directly on our brains. Any individual is a process issued from complexity (Cf. Von Neumann results on complexity!) and continuous flux of information is producing our thougths. Imagine what could be a “Contra Averroes” by Saint Thomas nowadays...

Metaphysics reshaped by phenomenology shows the underlying mathematics level. Heidegger wanted to reshape Universities according to three departments: Philosophy and Mathematics, Physics, Biological Sciences. Here we are where mathematical being is now reality. It is time to go back to Aristotle and the to on ê on which is the way to God.