The unreasonable effectiveness of mathematics

The presence of mathematics

Mathematics - both the best and the worst

Mathematics has a universal presence which manifests itself in extremely diverse ways. It would be more accurate to state that, while mathematics is universally used in social and technical practices, industry and everyday life, mathematics as a science seems to be increasingly distant from us in our culture. Mathematics appears to be covered up by the practices of its usage and buried in the implicit reality of that usage. It should be added that people's general apprehension of mathematics tends to provoke a certain form of rejection of or even hostility towards the abstract and obscure formalism mathematics represents. We would like to develop the idea that it is necessary to remedy this situation by starting to give definitions of mathematics and re-establishing some truths about the subject's true nature.

Definitions

I would like to start with a review of the characteristics that are traditionally attributed to mathematics. The first is mathematics' claimed and true universality in relation to individual subjects (it is recognised by any individual subject in possession of its statements), historical periods, cultures and civilisations. Mathematics probably possesses the only set of statements with such a set of positive characteristics. The first form of universality has given rise to many studies of cultural particularities - as with numbering systems for example - but among these, moving beyond ethnological studies and the great theses of comparatism between cultures, we should, for example, focus on the idea that numbers do not have their own intrinsic cultural determination even though it is only possible to understand their usage through reference to culturally determined institutions. Mathematics simultaneously represents both differences and universality. Regarding the characterisation by universality with respect to time, the philosopher Husserl substituted the notion of omnitemporality for that of timelessness.

Refining these criteria

Omnitemporality must be correlated with historicity. We may modify the designation of a concept or statement but only in the sense of passing from the implicit to the explicit as is the case when a context is included therein. Like Euclid, we shall state that the sum of the angles of a triangle is equal to two right angles and thus that the sum of the angles of every Euclidean triangle is equal to two right angles. As Mr. Caveing still says, the two statements say the same modulo a formalisation. This is not a question of truth and falsity succeeding each another as history progresses. There is no error in Euclid's treatise. As Husserl explains, the validation of a result in all cases is equivalent to its integration into the global unity of mathematics. Results are uprooted from all facticity of the "hic et nunc" and become perennial for all time to come. The question of precautions in the use of omnitemporality remains, particularly regressive omnitemporality. $\pi$ was not transcendent for Archimedes but nor was it non-transcendent.

Omnisubjectivity also needs to be clarified. A mathematical subject is defined as a normed subject that communicates with all others through rules. The universality of mathematics in relation to subjects is a form of omnisubjectivity. A minimal form of substitutability of one subject for any other does exist which is why any subject can access mathematics in law but not in fact. In this domain, every consciousness is in a perfect and transparent reciprocity with every other to the extent we could refer to logical intersubjectivity whose determinations are excluded from the facticity of the psychological "hic et nunc". Reactivation is an important term in this context as a mathematician subject operates by reactivating the meaning of his or her operations each time. Let us insist once more on two apparently contradictory movements that have an effect on this experiment. Universal mathematician subjects keep the inscription of the law of the operation in mind when they operate but access their own subjective singularity in the theoretical domain they find themselves in. In this way, mathematician subjects are individualised in their relation to idealities which explains the extreme diversity of mathematical 'temperaments', variations in the degrees of inventiveness, or even geniuses. This also explains the force of conviction and the importance of training for anyone practicing mathematics. An individual subject is caught up in the systematised set of statements in which s/he is inserted.

Taking this direction further leads us to note two classically noted and debatable modes for transmitting rules. The first is the unidirectional Babylonian model from master to apprentice with the rule simply exemplified and transmitted without demonstration. The other is the liberal Greek model. The communication space is a universe of argumentation. Anyone who teaches rules is obliged to justify them to an interlocutor who has an equal right to judge and criticise. This is also why mathematics establishes a form of democracy.

A characterisation of mathematical objectivity

The mathematician Alain Connes has put forward three criteria for objectivity:

• it needs to be possible to exhaustively classify the objects defined by an axiomatic that is thought to testify to objective constraints through which the universes of possibilities would be required;
• the overall inter-theoretical coherence and harmony of mathematical theories all of which testify to the unity of mathematics which cannot be reduced to a single calculation;
• the fact that worthwhile mathematical theories have an infinite information content. He adds that no other symbolic system, be it natural language or games etc., meets these criteria.

Other characteristics

Mathematics is the product of operative thinking which is only contemplative in its most powerful results such as someone contemplating a distance travelled or a landscape cleared. However it is also an activity which is why exercises are part of mathematical thinking and are necessary to enable concepts to be mastered and their substance grasped. Mathematics is an activity and it advances through acts. As Paul Valéry says "Mathematics is the science of acts without things - and through this, of things one can define by acts". I shall add another quotation by Valery (itself quoted by the philosopher Jean Largeault in Intuition et intuionisme, "Mathesis" Paris Vrin 1993: "When content is created by the operations themselves, that is to say, when noted operations are designated and isolated and combinations are formed from them, then we are in mathematics".

The question of the application of mathematics to physics

There has always been a link between mathematics and physics which is undoubtedly paradoxical and subtle. This is because what characterises Platonism is the thesis by which mathematics inhabits a world that is not the world of the senses however it is interpreted. The world of the senses is cut off from the intelligible world and a sense-based science is hardly conceivable, a fortiori a science of the latter. Major theoretical and philosophical transformations were required for the situation to change. I shall return briefly to this idea.

Mathematics in nature

The Renaissance is commonly considered to be the era in which Platonism came back down to earth thus making a science of nature possible, as Galileo opined in the well-known and often repeated phrase – "nature is written in mathematical language".

The idea that mathematics came back down to earth in the form of mathematical physics is a constantly repeated theme. This historical schema is certainly partly erroneous but one advantage is that it enables us to set out one aspect of the problem namely that mathematics is seen as being present in or adapted to nature even in these early developments of thought.

At the same time, mathematics develops endogenously outside of physics for its own benefit and most often in its most remarkable developments. The problems it poses and the theories it elaborates remain purely mathematical even when it is part of the most important episodes in the history of physics such as Einstein's Relativity or quantum mechanics.

I will take current scientific facts as my starting point. Immense progress has been made thanks to careful observations, extraordinary experiments, ever deeper and more ingenious reasoning in physics, mathematical tools ranging from the most complicated to the most common and highly impressive feats. Many magnificent achievements like special and general relativity, quantum mechanics at the origin of quantum field theory and the standard theory of particle physics and cosmology have required and involved impressive and highly developed mathematical theories. All of these theories are underpinned at various levels by uneven developments in mathematics.

A remark about aesthetics

One aspect that is fundamental - a term that juxtaposes with the term in question - to mathematical efficiency is linked to aesthetic considerations which actually play an essential role in the selection of mathematical theories. Mathematical beauty and coherence are intimately linked. Penrose stresses that the criterion of coherence is quite objective in nature as indeed have other mathematicians. Mathematical coherence can often only be discerned after years of work and often in unexpected ways. Those who approach these theories from the outside may be surprised and find it difficult to understand why a given property is accorded such value and why certain elements of theory should be easier to grasp than others.

I would like to insist on one point. The more complex and refined mathematics becomes, the more it deals with seemingly more complicated objects and the more effective its role in physics becomes (when such a role exists). At the end of the text, I will focus on complex numbers, complex functions and the functions of several complex variables from this standpoint.

Complex relations with physics

A considerable number of mathematical concepts have a physical meaning and there is no shortage of examples in geometry either. Let us start well before that. People have always searched for formal structures that can accurately convey and explain the behaviour of the physical world.

However therein lies a cause for astonishment. Mathematics develop far from what they were designed for in translating physical behaviour, such as calculating the length of the diagonal of a square and more generally the problems of generalising powerful notions or even theories. Let us return to real numbers. Nothing indicates that a notion of distance up to infinitely large scales exists and it is even less likely that such a notion could be applied to the infinitely small. And yet, as Penrose puts it, our physical theories are driven by the elegance and coherence of the real number system and indeed all rely on this ancient notion of real numbers.

Distances in cosmological theory range up to $10^{26}$ metres and the theory of particle physics reaches accuracies of the order of $10^{-17}$ metres. The scale at which some changes are seriously considered is $10^{-35}$ metres. This actually legitimises this unreasonable effectiveness of mathematics.

Numbers

As we know, the system of real numbers applies to areas and volumes while some physical quantities require descriptions involving real numbers with the most remarkable being time. This quantity associated with space forms the volume of space-time. In the theory of relativity this has 4 dimensions and the interval of real numbers describing time (of the order of $10^{43}$) should be included thus giving a total interval of the order of $10^{172}$.

I shall now return to real numbers. We should recall at this point that numbers (rational integers, real, complex) play a certain role in physics according to their specificity. In the framework of physical theory - from Archimedes, through Galileo and Newton, to Maxwell, Einstein, Schrödinger and Dirac - real numbers have provided a framework for the development of the conventional formulation of differential calculus.

Many of the founding ideas of physics are underpinned by differential concepts such as velocity, momentum, energy. The system of real numbers permeates physical theories in a fundamental manner and makes it possible to describe various different quantities. I shall discuss the case of velocity.

Before this, we shall consider the reasonable effectiveness of Euclidean geometry as applied to physical reality and more particularly the role it plays in rational mechanics. Euclidean geometry is characterised by the group of displacements which leave the figures of this geometry invariant. This is indeed the geometry of solids which cannot be deformed by local motion as was noted by the great physicist, mathematician and philosopher Helmholtz. We base the demonstration of triangles being equal on this invariance of measurements when we bring two triangles into coincidence. The figure studied by geometricians is invariant with respect to displacement and this is the way we can understand the coupling with rational mechanics. This consists of adding local motion to the objects of a theory which this motion does not subject to any changes. Their structure "predestines" them to be insensitive to the motion added to them.

It is important to stress that these are elaborate idealities that no perception can provide, like uniform rectilinear motion, accelerated motion or instantaneous velocity. We are dealing with a form of dual entanglement. Firstly, movement is geometrically determined by fixing the trajectory as a curve as defined in Euclidean geometry. Secondly, in contrast, there is the mathematical system which allows measurement like the ratio between a well-known geometric quantity, the length of the path travelled, and a "quantity" which does not appear in geometry but which is geometrically defined by the model of a straight line.

It is necessary to understand that there is a kind of affinity between the concepts of rational mechanics and those of Euclidean geometry. A deeper analysis of this affinity is required because geometry acts on mechanics but this affinity shows that the "real" to which mathematics is supposed to be applicable has already been constituted a priori by the determinations Euclidean geometry imposes. It should be noted that both processes play a role. Sometimes the physical phenomena concerned undergo a form of conceptual preparation in such as way that mathematical concepts can be applied to them in their original form. However, on other occasions it seems the mathematical concepts used need to be modified or even created to construct a system capable of dealing with the physical data involved. Finally the two processes can be combined.

All historians of this period concur that the privileged secular coupling of Euclidean geometry and Newtonian mechanics is linked to the fact explained by Caveing that the general assumption that Euclidean space is the locus of physical phenomena ensuring that mathematical concepts have a hold on such phenomena from the start. Mathematics plays a constitutive role in almost all the main concepts of classical physics.

This is not the end of the analysis however. Why is Newtonian mechanics constituted to this extent by the effects of geometry in the sense that it is a process of constitution? The philosopher Immanuel Kant provided elements for an answer that remain to be developed in his theory of schematism. This describes the process by which quantity develops through the schema it possesses, namely space, thanks to the imagination that produces schemas. The same applies to the schema of quality which adds determinations to a physical object which is in turn provided with differential elements and so on. The great philosopher Kant demonstrates in great depth how the question is analysed. The mystery of the effectiveness of mathematics for physics lies in the depths of the human soul, to use a Kantian expression, and we take this line of analysis further herein. Let us add a basic question which I will also only indicate. Why was elementary geometry developed in a Euclidean form? It must be stressed that geometry cannot be derived from pure perceptual data unless a radical empiricist point of view is adopted and this is a philosophy whose programme has never been fulfilled. Why should we choose an Euclidean structure rather than another? As Elie Cartan pointed out, we know that Euclidean space is the only space that permits a triple orthogonal system of totally geodesic surfaces.

This presents as a torsion-free space with constant curvature (zero). It is the most degenerate of spaces with the largest group of automorphisms. Another analysis of each of these properties is required with greater study of the history of geometry to understand the reasons for the relevance of Euclidean geometry.

At this stage of the analysis we can state that geometry has physical "forms" and physics has geometric forms which I have only sketched out so far. However the earlier analysis of Euclidean geometry provides the first premises of this idea.

Let us return to certain of Mr Caveing's expressions. Mathematics does not denote, it determines. Euclidean geometry is the true operator of "pre-established harmony" between physical phenomenality and mathematics. There actually is an affinity between Euclidean geometry and classical (Newtonian) mechanics and - taking the idea further - between various disciplines of classical physics.

Let us refer back to the example of the mathematical theory of electrodynamic phenomena deduced from the experiment alone. The law of the action of an electric current on another is considered independently of any hypotheses about the nature of electricity. The aim is to find the law at the level of the infinitesimal element of current taken over an arbitrarily small length of a wire conductor. Diverse geometrical situations involving the relative position and shape of the circuits are examined.

To these hypotheses of mathematical predeterminations a force is added which follows the line joining the interacting electric current elements and that of the form of the law $A/r^{n }$ in which $r$ is the distance between the elements, $A$ is a function of the size of the elementary currents and their relative orientation and $n$ is an integer. Ampère was convinced that he was only expressing observational and experimental data in algebraic and analytical language but in fact he was putting forward very strong hypotheses that determined reality. There is no direct expression of reality. Instead there is a prior determination of reality.

To be or not to be a mathematician

Calculating and demonstrating

What kind of discourse is at work in mathematics? What can be found in the treatises? Generally speaking, statements and demonstrations are found and this has essentially characterised mathematics since Euclid. However this discourse is essentially linked to calculations. Mathematics is calculation as much as it involves demonstration, elementary "arithmetic" calculation, algebraic calculation, differential and integral calculation, vector calculation, calculus of variations, tensor calculus, probability calculation, numerical analysis and so forth. In the history of mathematics, the Greeks restricted calculus to a subsidiary discipline they called logistics which required algorithms to be used as long as they involved specified numbers, the aim being to obtain a numerical result. Calculation as it was understood by the founders of Analysis included demonstrations that should (and have to) validate the calculation procedures that are used. Calculation and demonstration are intertwined in both ancient and modern mathematics and this form of entanglement was always part of Euclid's geometry. Here, even if calculation is confined to logic, it reappears, applied to symbols in demonstrations whose discursivity only consists of the regulated succession of calculation steps, according to M. Caveing's analyses and expressions. This remains the case in even the slightest demonstrations and in elementary geometry. We may observe that Euclid's axioms which are recognised as authentic exclusively involve properties of equality (or inequality) and that the equality of geometrical elements is specified therein. As further analysis demonstrates, this is linked to the logical structure of proof or the way the propositions are linked together which enables progress from a hypothesis to a conclusion. Also, the transformations of equalities obey logical rules which guarantee the preservation of the validity of the equality. We should add at this point that calculation is based on a structure that organises a set. Calculation elements belong to the body of real numbers which are used to create formulas of different kinds. A distinction must be made between primitive formulas and other deduced formulas. The aim of these observations is to provide brief indications of how closely demonstration and calculation are intertwined and to point out that calculation is not just calculation and that demonstration does not just involve creating a sequence.

It is important to note the rise of geometry, as the great mathematician M. Gromov put it, and this is indeed one of today's major characteristics. We can distinguish discrete geometry, differential geometry, complex geometry, algebraic geometry and new syntheses of disciplines, differential topology, so-called "tropical" geometry and diffeology. New fields have opened up and intercommunicate which in turn leads to the birth of new disciplines.

Relations

Mathematics is above all the study of relations as Poincaré pointed out - "Mathematicians do not study objects but relations between objects…Content to them is irrelevant: they are only interested in form " (Henri Poincaré, La science et l'hypothèse, Paris Flammarion 1902). It is probable that Pascal was the first of the Modern Mathematicians who clearly considered mathematical thinking to mean thinking about relations as indeed M. Caveing and others believe. Pascal considered mathematics to involve work on relationships, correspondences, transfers of properties and variations of points of view as the arithmetical triangle or conics show. A mathematical object is an identifiable thematic moment within the synthetic unity of a system of relations.

This idea should be taken. "Whatever the phase of historical development, what has secretly and silently guided mathematical research is the presence of relational patterns with anticipatory value which ultimately opens up the explicitation or invention of relations in the domain or subject field under investigation" (again quoting Caveing). This has become particularly clear in the case of category theory.

As you can see, mathematical or physical explanations can be provided but are these sufficient? I would like to develop the analysis of the properties of complex numbers and complex geometry as such properties can explain the concepts in physics that use them. However I can take the analysis of the question of their "efficiency" further.

Almost all the theories of classical mathematical physics can be subjected to this form of analysis. This time I shall consider complex geometry with functions of one or even several complex variables. Let us reiterate the structure of complex numbers.

I shall now describe the mechanism of manipulation of the complex numbers even the history of their appearance is not respected. A quantity should be introduced, $i$, the square root of $1$ and the number le $i$ combined with real numbers can be written as $a +ib$.

Also, we can add up these various combinations to add, multiply and divide complex numbers.

Complex numbers have existed for four centuries and enabled mathematical thinking to become much more in-depth than with real numbers alone. Penrose considers complex numbers to contain something that can only be described as magic. During the last three hundred years, complex numbers have shown themselves to be a system of concepts that play a role in the latest physical theories and, through them, the laws for the behaviour of the universe at the smallest scales that are fundamentally regulated by the complex number system.

Complex numbers

I shall now discuss complex numbers. First $-1$ was said to possess a square root. It seemed impossible to find a number whose square would be negative. And yet we recognised, following on from the Pythagoreans, that it was necessary to generalise our system of numbers from rational numbers to a larger system namely the system of numbers which were to be called real. Raphael Bombelli introduced the transition to these complex numbers in Algèbre and they first appeared in Jerome Cardan's Ars Magna. An enormous number of calculations are made possible by complex numbers.

There is a mathematical magic and a physical magic for complex numbers. Let us focus on the mathematical form of magic – the aim was to give existence to the square root of minus $1$ (to find solutions of polynomial equations for calculation purposes) by requiring that the ordinary laws of calculation be consistently satisfied. I would now like to focus and insist on some of the effects of this magic. Expressions that seem strange will appear for those reading which will actually enable certain forms of understanding to eventually emerge. I will only mention a few of these.

You will see that this form of understanding derives from the fact that known operations carried out with the numbers we know are also performed with this new form. For example $-2$ has a root: $i \sqrt{2}$ or $-i\sqrt{2}$ in which the small number $i$ makes an effect. There is nothing magic about this. But

$\sqrt{{ 1\over 2} ( \sqrt{ + a + \sqrt{ a^{2} + b^{2})}} } + i \sqrt {{1 \over 2} ( \sqrt{-a + \sqrt {a^{2} + b^{2}) } }}$

squared gives $a+ib$ (as well as its opposite). Even if we had only assigned a square root to one number ($-1$), we still find all numbers in the system automatically have a square root. Now that's magic!

A few calculations of solutions

We can ask ourselves about cube roots, fifth roots, roots, $999^{es}$, the n-th roots and the i-th roots. And we find that there is always a solution to a problem involving any choice of complex numbers. This is what corresponds to the fundamental theorem of algebra. Any equation with the form

$$a_{0} +a_{1}z + a_{2} z^{2} + \cdots a_{n} z^{n} = 0$$

must have complex solutions. Cardan found an expression for a cubic equation. There were many rivalries and dramas scattering the historical path leading to the solution from Tartaglia to Scipio del Ferro. Here it is in modern terms:

$$x = (p+w) ^{1/3} + (q-w) ^{1/3} )$$

$$w = (q^{3} -p ^{3})^{1/3} $$

The formula contains three real solutions and the square root of the negative number. This only makes sense if complex numbers are introduced. The real solutions are given by this formula, the expression giving the solution $x$ is a sum of two complex numbers whose parts in $i$ cancel each other out. The mysterious thing about all of this is that we have had to take a trip to this strange complex world even though the problem has nothing to do with complex numbers at first sight. And the formula brings us back to real numbers which we had to move away from.

We can continue to discuss these reconstruction elements once the complex numbers are in place and admire the wonders they reveal to us.

As you will imagine, many miracles may occur with operations related to series. Complex numbers enable us to understand the behaviour of whole series. As they shed light on the real numbers (and just because of their complexity), does this mean that they allow us to see the physical characteristics of reality? But does this role inherited by complex numbers explain their involvement in physics?

The forms of understanding that complex numbers introduce into mathematics and beyond the subject also open up new determinations of physical concepts from the elementary standpoint (I will give examples of this) and the more elaborate and explicitly algebraic standpoint. This is of immense benefit for physical theories.

I will not develop the remark which follows. The process known as complexification plunges a space or situation expressed using real numbers into a space expressed using complex numbers and introduces new possibilities for determining physical concepts into the new space being considered. Let us just refer to the case of the Minkowski space from Einstein's theory of relativity - the space-time framework in which we understand physical phenomena. If we make it more complex, we are then dealing with complex light cones. I cannot provide an in-depth explanation of why the geometric cone brings us into physics. Let us just note that photons travel along a trajectory in space-time. Trajectories passing through a given point in space-time form a cone of light.

I shall now explain why one of the deepest mathematical structures takes us very far into physics, namely the fibred structure. This notion has an attraction for physics. We need new spaces for physics to understand, and design and determine interactions between particles. These spaces are such that it is necessary to refer to a form of spatial dimensions in addition to the ordinary dimensions of space and time. These are called "internal dimensions".

Movement in one of these internal dimensions does not take us away from the point in which we find ourselves in space-time. We need to introduce the notion of the fibre bundle to visualise this idea geometrically. This notion proved to be very useful in pure mathematics long before physicists became aware that the physical notions they were using were understood in the language of fibre bundles.

The additional internal dimensions come from the arrangement of these additional spatial dimensions. Penrose suggests an analogy with a garden hose. The hose appears to be one-dimensional (length) when seen from a distance but closer examination reveals it actually to be two-dimensional. This image is the example of how physical space-time must be perceived in Kaluza-Klein's pentadimensional space-time. This idea is extremely attractive and has been taken up by advocates of modern theories like supergravity or string theory.

Other interactions than electromagnetism appear in these theories. Rather than considering them as emanating from Kaluza Klein's extended space-time, as some have attempted to do, Penrose suggests considering these internal dimensions as being at the origin of a fibre bundle in space-time. Thus they should not be considered as part of a higher dimensional space. This is where we need to concentrate to understand the way in which the physical concepts involved in this mathematical theory are made more in-depth.

The notion of the fibre bundle would require a long analysis but I shall simply state that it is now one of the most enlightening conceptual languages in theoretical physics. Also, certain fibre bundles, like the Hopf-Clifford fibres for example, have made it possible to formulate entire physical theories and even to conquer objective elements of physics.

A certain kind of magic rules over complex numbers but nature itself seems to take advantage of this magic right down to even the most fundamental levels of the universe. Was it the convenience of these numbers alone that led us to give them such importance in our physical theories? Why does the role of these numbers seem so universal, given that they underpin the fundamental principle of quantum superposition and, in a different form, the Schrödinger equation? At this point, let us once again refer to the condition of positivity of frequencies and the complex structure of infinite dimension. Penrose considers the importance of complex numbers (holomorphicity) for the foundations of physics to be perfectly natural. And here an opposite enigma appears – why do real numbers play such an important role in physics? The formalism of quantum mechanics based on the system of complex numbers is not a complex (holomorphic) theory. Many properties are linked to the condition that the result of measurements need to be real numbers. Also unitarity depends on the fact that probability is preserved or, in other words, that the modulus square rule (for measurements) is preserved.

I will not take this idea any further but I would like to insist on the fact that the twister theory - the great unifying theory (at least it has succeeded in parts of QM and GR) which is in full development - is based on identifying a mathematical structure (a complex projective line) with a ray of light.

The precision, subtlety and sophistication of mathematics operating in fundamental physics are the expression of a deep interweaving of physics and mathematics of a similar order to Kantian schematism. Eugen Wigner famously referred to the excessive efficiency of mathematics in the physical sciences. Conversely the great mathematician Andrew Gleason trivialised the existence of a harmony between mathematics and physics which he thought to reflect the fact that mathematics is the science of order. I have borrowed these references from Roger Penrose who thinks and shows that this explanation of order is insufficient.

The absolute

The absolute and God

In our dialogue, we mentioned the divine nature of mathematics and even its relationship with a form of absolute. This is an element that I must also mention herein. We initially experience this feeling of being overwhelmed when faced with the immensity of the mathematical field which seems to lead us into the unknown. Many analyses of such feelings can be found in passages from the history of philosophy. I shall only quote Kant's analysis of the sublime - there is indeed something sublime in mathematics. Secondly, important mathematicians have explicitly analysed absolutes. These analyses relate to concepts or systems of concepts whose power seems to complete a theory or even surpass mastery of any sort. Galois theory gave rise to these formulations which relate to the absolute – the ascent towards the absolute, an absolute Galois group referred to by the mathematician Grothendieck.

I shall end by reiterating the idea that mathematics always involves symbolism and that practicing mathematics involves working on or even experimenting with these symbols. However I would like to add that these symbols are nothing less than formal. Symbols are one of the forms of conceptual work and cannot act as a substitute for an idea or a remedy for our conceptual weakness that requires assistance. They can undoubtedly enable us to shorten or make easier a calculation but more often than not the symbol is the concept itself or actually one side of the concept, to put it more accurately. We could take the case of tensors for example but are many others exist.

I would like to finish by insisting that mathematics fully belong to our culture and are even an actual part of that culture's essential matter.

This text results from a dialogue with the philosopher Jean-Michel Salanskis, to whom the author owes a great deal. He also owes a great deal to the organisers of the France Culture radio programme who invited us.