Repetition, Seriality, Temporality and Reality in Mathematics

It took a long time for mathematicians to realize that symmetry is temporal: something is symmetric because one can act on it repetitively without disturbing it. Musicians were much faster is inventing symmetry, reverse symmetry, dilation, i.e. group theory, just because time is intrinsic to music. So symmetry has to do with temporality; it conforms perfectly to the triad of temporality, repetition and seriality: being the series of an infinite number of repetitions, which determines whether something is truly symmetric. But temporality in mathematics questions the notion of intrinsic realism: what is real in mathematics? And this refers immediately to Platonism, as I shall develop below.

Symmetry is often thought as identity in mathematics, or more precisely as conservation of identity (two parts of a symmetric object are equal). But this conservation can be seen with respect to very different forms of temporality: exchanges of paradigms, equivalence between objects of different types, symmetry between an equation and its solutions, etc. In other words symmetry in intrinsic mathematics calls necessarily for a precise notion of realism in mathematics.

Repetition, seriality and temporality are three concepts to be considered carefully when looking at how they operate in the field of mathematics. To be more precise, they act, at first glance, more naturally in a field, a space tied to mathematics. This field surrounds them, mostly inspires them, but does not boil down to them.

Repetition refers to the confrontation between identity and non-identity: to repeat means that we repeat the same, the identical. Any change breaks repetition. Seriality questions the possibility of decomposition of sequences, i.e. a succession of, say, objects in their full diversity into identical ones: decomposition of diversity into partial identity. Temporality addresses the phenomenon of successive actions parametrized by a ‘time’ belonging to an ordered set. The vocabulary used in this description of the three concepts is totally absent from mathematics. It rather belongs, among others and to restrict the view to academic directions, to the natural sciences or the philosophy of them. In other words, it belongs to real situations. In mathematics, there is no intrinsic time, i.e. an ordered set of parameters indexing mathematical objects; the question of identity (together with the strongly related concepts of repetition and seriality) is supposed to have been defined and settled once and for all in the mathematical concept of definition. From that naïve perspective, too, the three concepts of Repetition–Seriality–Temporality do not seem to belong to the core of mathematics. But they do in fact belong to mathematics, and in a fundamental way, to some space of upstream and downstream mathematics. Downstream is their domain of applications, such as physics, chemistry, biology, economy – the list can be long. In short, we are talking of real situations. The field upstream refers to the famous concept of Platonism in mathematics. Very broadly speaking, if one took a Platonistic point of view, the mathematical results would be located somewhere where the mathematician would catch them during his process of research. And it would then be in this area, this ‘somewhere’ upstream from mathematics, that Repetition–Seriality–Temporality in mathematics would be incarnated. That is, upstream mathematics would be a kind of a closet where real objects are placed and ordered by identity or chronology.

In conclusion, one can say, less naively than before, that the mathematical pertinence of Repetition–Seriality–Temporality is definitively tied to the general pertinence of realism in mathematics.

2.1. Dynamical Systems

Our first example deals with a quite new subject of mathematics: flows with low regularity properties.

The idea that the dynamical movement of rigid objects in our physical space should result from solving differential equations is the great revolutionary discovery of Newton. Let us try and put this in a nutshell as follows. Suppose our rigid body is ideally reduced to a point, a point like the one you get by posing the extremity of your pen on a sheet of paper. If you now draw a curve on the paper, you just draw the successive positions of your ideal rigid body. Successive refers to time and positions refers to space: at each moment the body occupies one point, and when times evolves it follows a trajectory consisting of the different, successive, points of the drawn curve. Exactly in the way you ask the internet the route from one town to another: the answer is a curve on a map.

At each point of the curve you have drawn, you can draw a straight line tangent to the original curve at the point you selected (the tangent to a curve at a chosen point is obtained by taking the straight line passing through two points near the point you chose and letting both points move towards the chosen point). This set of tangents reflects the dynamical process which produced the curve: when you draw in a row a curve, as you do spontaneously with a pencil of a sheet of paper, the curve you obtain is very nice, regular, smooth. On the contrary, if you stop your drawing because your phone rings or somebody touches your hand, the tangent will jump discontinuously and the curve will present at this point an angle, a singularity. The same is true for the online route finder: if you influence the dynamics of the process by asking, for example, not to take highways, or to avoid centres of towns, or to get there the cheapest way, you will find abrupt changes of directions in your route.

The fundamental problem of the theory of dynamical systems, one of the most productive fields in mathematics of the last century, consists of going the inverse way: instead of first drawing the curve and then tracing the tangents at each point, let us suppose that the dynamics is given first, that is, let us be given a straight line at each point of your sheet of paper. Can we start at any point we wish to pose our pen and draw a curve, the tangent at any point of it being the straight line which was given initially? And when this curve exists, another important question arises: is it unique or can we draw two curves having the same distributions of tangents? ‘Existence’, ‘uniqueness’, we are entering slowly the vocabulary of mathematics.

The intuition is that these two questions (existence, uniqueness) should have both a positive answer when the distribution of tangents is continuous: by this one means that, moving a little bit the point on the sheet, the associated tangent should change its direction just a little bit. The reader can have such an intuition by trying to draw a curve by following a distribution of tangents.

But this intuition is wrong.

It was proven by the end of the nineteenth century that the distribution of tangents must have a stronger property in order to allow the construction of a unique, nice curve. Without defining it for the moment, let us name the extra-property that the distribution of tangent must have the ‘Lipschitz continuity’. A natural question arises immediately: what happens when this Lipschitz continuity condition is not satisfied by the distribution of tangents?

It took more than one century to have an intimation regarding the answer to this question (DiPerna and Lions Reference DiPerna and Lions1989; Bouchut Reference Bouchut2001; Ambrosio Reference Ambrosio2004) and we will see that the answer necessitates that one destroy the underlying space. In order to understand the philosophical ideas behind this a priori negative phenomenon, we need to rephrase the preceding discussion in more mathematical language. But the non-mathematician reader should not be afraid of the mathematical ‘vocabulary’ used below, but should just concentrate on the (changes of) morphology of the syntax, almost at a graphical level, and try to adopt low-level thinking ‘à la Teissier’ (Reference Teissier, Grialou, Longo and Okada2005).

In mathematics (and in physics), a flow on a given space is defined, once again similar to a roadmap provided by an online route finder, by a curve (i.e. the route on a map) and a way of assigning to any value t the time a point X(t) in this curve (e.g. 0 hours: departure Paris, 2 hours: Tours, 3 hours: Poitiers, etc.). Such an assignment is called in mathematics a function t → X(t). When consulting an online route finder, you are asked certain constraints you wish your travel to satisfy. For example, you might decide not to pay any highway fees. The route finder will then decide to change brutally the direction of your trajectory when you are about to meet a highway toll booth: the tangent to your course will be modified and the trajectory will follow this new tangent indication. In other words, a certain trajectory is assigned to a certain dynamics (e.g. avoid fees). Such a dynamics is realized in mathematics by the fact of putting a vector field on your space, that is to say a way of associating with each point X a direction f(X), an (oriented) straight line, supposed to be the tangent of the trajectory at the point X. The problem is then to find a curve with the requirement that at each point the curve is a tangent to the direction assigned by the vector field at this point. It happens that solving this problem consists mathematically of solving a differential equation written as

(1) $${{\rm{d}}X\over {\rm{d}}t}(t) = f(X(t)),\quad X(0) = {X_0}$$

Here, X(t) is the point on the curve at time t and f(X(t)) is the direction (vector field) at the point X(t), as seen before. The new ingredients are X ₀, the point of departure, which can be chosen in principle anywhere in the space considered, and what is (a bit barbarically) denoted by ${{\rm{d}}X \over {\rm{d}}t}(t)$, the ‘velocity’ on the trajectory at time t (see the link with the definition of the tangent expressed earlier), a notion everybody knows intuitively. The equality between the ‘velocity’ ${{\rm{d}}X \over {\rm{d}}t}(t)$ and the direction f(X(t)) at each point X(t) is, in particular, the formalization of the statement ‘the curve is tangent to the vector field’.

As we mentioned earlier, solving this problem in a unique way and for any initial point X ₀ necessitates that the way f(X) depends on X is regular enough, not too hectic (Lipschitz continuity): otherwise the curve might not exist or the problem might have several solutions.

When the Lipschitz condition is not satisfied and replaced by a weaker hypothesis, ‘BV’ regularity,Footnote ³ it happens that equation (1) is still solvable in a unique way, but only for almost all initial points X ₀, not all initial points X ₀. When we release the Lipschitz condition, though equation (1) is still very cute, very smooth so that we naively expect as before the solution to be smooth too, the actual solution is often quite hectic: it still exists but only ‘almost everywhere’, not ‘everywhere’. What is meant by ‘exists almost everywhere’? It means that if you select by chance an initial point, then almost surely, with probability one, as one says in mathematics, everything will go smoothly, as if the equation were more regular. But nothing prevents you from picking as an initial point, ‘by chance’, one of these rare points where, for example, two different trajectories (or even worse, none) can be born at the same time.

The flow can then be defined to meet the requirements of the original task, although not on the whole space but rather on an ‘almost everywhere’ defined space. There is no trouble with that a priori: we know such spaces, real numbers deprived of the rational numbers is such an example. But in our case, the set of remaining points of our space is not known. Or equivalently the bad points are not known (contrary to the rationals imbedded in the reals which are perfectly identifiable). But there is more: the bad points of this ‘almost everywhere space’ can change if you compose flows, that is, if you stop and restart again. A good point can happen to become bad, and a bad one good. In fact nothing is known about that, except the fact that almost all points are again good when you restart.

By identifying a ‘space’ by the entity consisting of all the trajectories which are the solutions of a vector fieldʼs equation, and not by an ‘object’ made of points given (platonically?) a priori, you see that the two definitions merge when the vector field is Lipschitz. But when the vector filed has only BV regularity, there is no ‘object’ counterpart to the ‘entity-like’ notion of space.

We have the two following diagrams.

When the vector field f is Lipschitz,

When the vector field f has only BV regularity

In the case of Lipschitz vector fields, there was a symmetry/identity between the two spaces where the equation and its solution were sitting: the two spaces were the same. The willingness to preserve this symmetry in the non-Lipschitz situation leads to a new type of space, a new paradigm, a new form of reality.

2.2. Quantum Mathematics

The, nowadays, quite popular notions of quantum groups and quantum (= noncommutative) geometries have no underlying group or geometry-type objects. Nevertheless, they fulfil completely, according to us, the paradigm of realism in the sense that their structures provide an arsenal of study methods, comparable to the ones available in the ‘classical’ situation.

How are they constructed?

A rigid body, a human body, a landscape, is a geometrical object, a manifold, perfectly understood if one knows enough drawings of it (it can be reconstructed from them). There is no need to explain this fact experienced by all of us. What is a drawing? It is a set of points on a sheet of paper. What does the action of drawing consist of? It consists of assigning to each point of the (surface of the) body a point on the sheet, that is, precisely, a function from the body to the sheet. A single drawing does not capture the body entirely but, and Picasso understood this very well, several drawings do. This beautiful fact turns into a theorem in mathematics: the set of all the functions on a manifold (e.g. the surface of a rigid body) determine the manifold itself.Footnote ⁴ Looking closely at this statement, we see that stating (and using) it consists of building a higher level, as in the preceding section. This theorem refers to a dynamical action (‘drawing, looking at’) and to a formalism: it belongs definitively to the side of ‘entities’.

But identity (again identity) in mathematics allows the semantic shift:

This is a dynamical point of view, an operating one. And this algebra of functions inherits from numbers a nice property: one can multiply functions like numbers. The order in which you perform this operation is insignificant: one calls such an algebra a commutative algebra since we can commute the different functions/drawings that we multiply without changing the result. (It seems to us that this multiplication is truly incarnate in the cubistʼs portraits. On a single drawing one put/multiplies several different views of the body to be drawn. And, obviously, the order between the different takes is insignificant.)

And, somehow even more importantly, the converse is also true: give me any algebra with this property, that is, any commutative algebra, then I can construct the underlying manifold. This is the Gelfand Theorem (Gelfand Reference Gelfand1941). We can start building our diagram: at the lower level, the ground floor, we put what we would like to call, by convention, objects such as a manifold, considered as a set of points. At the upper level we put an alternative ‘entity’, the algebra of functions on the corresponding object.

We will now remove one property of the algebra, the commutativity. And we will think about this displacement from commutative to noncommutative as a left to right movement. By doing that we leave a commutative algebra, isomorphic to a lower level object, the manifold, and we get a new algebra, noncommutative, with a priori no corresponding object under it, at the lower level. Nevertheless, this right-upper level exists and inherits from the left-upper one all the properties you need, in principle, to generalize the construction made at the lower level. The question is: can we move by staying on the ground floor and construct an object that would be the classical object corresponding to the new upper level? Can we move left/right by remaining at the lower level?

Let us draw a picture.

The answer is: no!

There is no manifold whose space of functions is noncommutative. But then, what becomes of the arrow at the ground floor? The diagrammatic answer is the following:

The object ‘manifold’ joins the upper level and becomes the entity ‘noncommutative algebra’, and by another semantic shift one can say:

This presentation might let us think that the noncommutative manifold arrives in the field of mathematics just by a fantasy of generalization that mathematicians are known to be fond of. This is not true, and the most interesting situations where the lower floor disappears are the ones in which this destruction is performed at (by) the lower floor itself. The simplest case comes from the concept of quotient space, which is defined as the set of families of elements of a space, a family being the subset of elements sharing a certain property. For example, take a sheet of paper and fill it completely by drawing straight lines on it. The quotient appears as the set of such lines. The sheet itself is a sweet set of points, each line is itself a collection of points. What about the quotient? Well, it is a fact in mathematics (only in mathematics?) that the set of nice objects included in a nice object might be not nice at all.

This can be metaphorically visualized as follows: when you talk about a packet of spaghetti, you mention in fact two different things. One is the set of the 250 grams of flour its volume contains, the other is the set of 50 spaghetti it contains. The second is the quotient of the first when you regroup the grams of flour sharing the property of belonging to the same spaghetti. When the spaghetti are very well stored in the packet, you can count the spaghetti and easily determine one from the other. But drop the packet of spaghetti on the floor and try to count them without putting them back into the packet.

The reader interested in going a bit further and treating her/himself a very simple example, though totally meaningful, can try the following experiment.

But let us suppose now that one does exactly the same construction but without taking care of how we choose the point where you are going to cross the right side. In other words, let us choose this point randomly. ‘In general’, the drawn straight lines will not be closed any more: you will not go back to the starting point but you will, in general, miss it narrowly after many rounds (try this!), the line will intersect the left side at an infinite number of points. But there is more: if you continue going around the torus on this line, the set of these intersection points will accumulate everywhere and become dense on the left side. If you now choose another starting point and draw a straight line parallel to the latter, the set of intersections with the left side will look exactly the same as the one for the first straight line, and you will not be able to distinguish which intersection point belongs to which straight line, although each point of the left side belongs to one and only one of the two straight lines.

In mathematical language this means that, in general (namely, for almost all values of the slope of the straight line), the set of straight lines, still ideally well defined as a set, is totally unreachable by any approximation, this last property reflects the ‘looks the same’ expressed before. Defining the set of drawn straight lines only as ‘a set’ is tautologically possible. But this view is unsatisfactory, as the simple drawing shows, since one cannot differentiate any line from the other by its trace on the left side. In fact, if one looks at the (algebra of) continuous functions on this set, one can prove the following theorem:

The set of drawings of our set, the algebra of functions on it, is reduced to trivial ones, the ones making no differences between the points: i.e. a fully black drawing, with not even a texture ‘à la Soulages’. Nothing. No classical structure, nothing. But if we lift the whole construction we made on the lower level to the upper one, we find that there is a possibility of describing and ‘understanding’ this space by identifying it with a noncommutative algebra (Connes Reference Connes1994). One can calculate, manipulate this set-entity, construct a topology on it, although there is: no underlying object to this new entity.

The symmetry/equivalence between manifolds and algebras is lost, but, by disappearing, it creates the incredibly rich mathematical structure of quantum mechanics.

We insisted a bit heavily on the preceding construction not because we wanted to torture the reader, but rather because we believe that it gives a quite faithful image of the mathematician at work: tedious drawings, computations, failures, repetitions, etc. – using and pushing to its very extremities a formalism leading to a new paradigm. In a nutshell: the mathematical formalism is a formalism in action (Benoist and Paul Reference Benoist, Paul, Benoist and Paul2013a; Benoist and Paul Reference Benoist and Paul2013b).

In order to conclude this section, we would like to point outFootnote ⁶ a clear difference between the object–entity dialectic present in this article and the dynamics of ‘generalization–extension’ so familiar in mathematics, where, though widely generalized, the underlying object never really disappears. An example of this in group theory is already hinted at in Footnote 1: abstract groups are transformations of … nothing. Yet, it happens that a very efficient way of studying groups is to let them ‘act’ on different types of objects in the framework of representations theory. Let us give another example: Analysis situs by Poincaré (Reference Poincaré1885) views objects as new entities but without removing the lower level. Perhaps the upper level becomes more important, the concepts of fundamental group and simplicial homology becoming even necessary to an understanding of the underlying level, but the underlying object never disappears.

2.3. Partial Differential Equations

Our last example will be a bit more technical and might be skipped by the uninterested reader without affecting the comprehension of the core of this article.

Solving the Navier–Stokes equations, fundamental equations of hydrodynamics, has up to now been limited to defining ‘weak’ solutions. In this section we would like to show how this concept of weak solutions, which we are going to explain later on, conforms perfectly to the notion of entity as it was defined before: it is an entity considered as the solution (because the important fact here is that there is only one), as a substitute to a ‘true’ solution; an object which is, for the Navier–Stokes problem, still unknown and might never exist.

Let us be a little bit precise, without too much technicality (once again the reader should not be afraid by the presence of equations whose precise meaning is irrelevant for the purpose developed here). A partial differential equation (PDE) consists of a function u (the unknown), for example a function x → u(x) as defined in Section 2.1 which sends (real) numbers x to (real) numbers u(x), an operator P : u → P(u), a ‘function’ with sends the ‘function x → u(x)’ to another function P(u) : x → P(u)(x), and an ‘equality to zero’:

(2) $$P(u) = 0$$

What is meant by this is that P(u) is a function. And one looks at a function u such that P(u) is the null function, i.e. the function identically equals to zero: we want to find the u such that P(u)(x) = 0 for all numbers x.

We see here a question arising concerning identity: ‘identically equal to 0’ means what we just wrote. But checking something ‘for all numbers’ is a long task, a very long task. And doing such a task patiently is boring and the risk of missing ‘some x’ is high. An alternative way consists in taking averages of p(u) with a given probability distribution. What does this mean? Just that we will add up the numbers u(x) defined out of ‘almost’ all numbers x by weighting them according to the importance we want to lend to each x – just as insurance or political survey companies do. The reader might argue that we still miss some values of x through the concept of ‘almost’, intrinsic to the concept of average. The answer is that if we take all the averages with all probability distributions then we determine the value of P(u)(x) for all x. Actually, the reason is very simple: take a probability asserting the maximal value to a number x ₀ and the value zero to all others. Obviously, the corresponding average will be equal to P(u)(x ₀). What else?

Taking the average of a function f with a probability distribution φ is written in mathematics the following wayFootnote ⁷

$$\int {\phi (x)f(x){\rm{d}}s} $$

And what we just wrote can be formalized as

(3) $$[P(u)(x) = 0] \Leftrightarrow \left[ {\int {\phi (x)P(u)(x){\rm{d}}s = 0\ {\rm{for}}\ all\ {\rm{functions}}\ \phi } } \right]$$

It is a matter of fact (a very unpleasant fact) that in many cases the mathematician finds out that a prototype model of solution u she/heʼs working on is such that, for some few points y, P(u)(y) is infinite and therefore cannot be properly defined.Footnote ⁸ Of course, in this case, u cannot pretend to be a solution of equation (2) as we just defined, but these ‘bad’ points y are sometimes so few that, in the interest of proceeding, we would like to be just able to ignore them for the moment. In order to do that we first remark that, usually in these situations, these points are so few and so isolated that they disappear when taking an average of the left hand side of equation (2). But one does not want to take averages with all probabilities since, because of equation (3), this would be equivalent to equation (2). What to do?

It happens that solving the right-hand side of equation (3) by restricting it to some special functions, called, meaningfully, ‘test functions’, is easier than solving the left-hand side, namely equation (2).

We will say that u is a weak solution of equation (2) if

(4) $$\int \phi (x)P(u)(x){\rm{d}}s = 0\ {\rm{for}}\ {\rm{all}}\ {\curr ‘}test\ {\rm{functions}}{\curr ’}\ \phi $$

Why do we say that such a function u satisfying equation (4) is a weak solution of equation (2)? Well because if u was a nice solution of equation (2), then equation (4) would be true for every function, not only a test function and therefore equation (4) would be equivalent to equation (2), as every function whose integration against every function is equal to 0 is itself equal to 0.

We propose to define as ‘objects’ the contents of equation (2) and the bracket in the right place in equation (3), and as ‘entities’ the brackets in the left place in equation (3) and the contents of equation (4). We have

$$\underbrace {[P(u) = 0]}_{{\rm{object}}} \Leftrightarrow \underbrace {\left[\int \phi (x)P(u)(x){\rm{d}}s = 0\ {\rm{for}}\ all\ {\rm{functions}}\ \phi \right]}_{{\rm{entity}}}$$

and, since test functions are functions, after all,

$$\underbrace {[P(u) = 0]}_{{\rm{object}}} \Rightarrow \underbrace {\left[\int \phi (x)P(u)(x){\rm{d}}s = 0\ {\rm{for}}\ {\rm{all}}\ {\curr ‘}test\ {\rm{functions{\curr ’}}}\ \phi \right]}_{{\rm{entity}}}$$

But as we mentioned before

$$\underbrace {\left[\int \phi (x)P(u)(x){\rm{d}}s = 0\ {\rm{for}}\ {\rm{all}}\ {\curr ‘}test\ {\rm{functions{\curr ’}}}\ \phi \right]}_{{\rm{entity}}} {{\curr/\hskip-9.5pt\Rightarrow}\underbrace {[P(u) = 0]}_{{\rm{object}}}$$

Under this last entity there is no clear existence of a true solution, nothing which plays the role of the complex plane for the second degree equation. Equation (4) is definitively different from equation (2), so:

We get the two following diagrams.

Here also we see that it is the need of symmetry, taking the form of an equivalence between entity and object in the good case (column on the left), which leads to an identity: there is only the upper level in the column on the right.

What Happened?

We claim that one attends here to a sort of mathematization of non-Platonism in the following sense: not only are the mathematical entities somewhere and waiting to be discovered (as being upstairs, over a non-existing ground floor), but they are not even expressible by the standard essentialist language of mathematics available at the time of their creation. And more than that, they influence language right up to its paradoxical extremities: a noncommutative manifold is stricto senso a non-sense as there is nothing in the definition of a manifold to be multiplied, commutatively or not (check the definition on Wikipedia).

But how could mathematics be non-Platonist, even sometimes anti-Platonist, in itself without being Platonic when viewed by the mathematicians? More precisely, if mathematics did the job itself – that is, the job of inventing a new realism over the traditional ones – it could just mean that mathematics itself is somewhere, i.e. that it happens to be outside the thinking of mathematicians inside of which it could only be a construction.

The answer to this ‘paradox’ lies, according to us, in the finality of these entities without objects, which is to compensate for the lack of a classical, standard definition of underlying objects or to compensate for their lack of ability in solving an equation. It is when they face a dead end that mathematics take off to the upper level: this realism is not a new one, it is just the right one.

The realism of an equation lies in its solutions. In the case of a nice, gentle partial differential equation (PDE) the solutions are nice functions on a nice space, and it is to this space, this nice one, that they, the solutions, provide the status of realism. But for the Navier–Stokes equations, the solutions exist only in a weak sense (up to nowadays), i.e. without a clear spatial counterpart. But if we set aside this lack of a ‘clear spatial counterpart’, the situation is the same, one continues to do mathematics, to compute, to estimate: realism = solutions. In fact:

The realism belongs to the movement, identified with a nice space in the good situations, to an ‘almost everywhere’, a noncommutative one in the bad ones: realism = movement.Footnote ¹⁰ In the three situations examined in Section 2, realism lies on the upper floor, isomorphic to the ‘natural’ object for the good situations, isomorphic to itself, and only to itself, in the ‘bad’ ones. This is the meaning of the north-east oriented arrows in the diagrams.

The essentialist diagrams do not commute, the existentialist ones do.

Identity: Last Call for Immediate Boarding to Temporality!

Let us return to the theme evoked in the Prelude: Identity–Repetition–Seriality or, more generally and synthetically, identity versus temporality. Indeed, discussing identity together with repetition–seriality seems to us like realizing an attempt to look at identity versus temporality: repetition–seriality refers to identity in a temporality of repetition.

Naively, identity belongs to the ground floor and temporality (namely action, process, dynamics) is located on the upper level. We tried to show in Section 3 that temporality possibly creates a kind of non-existing objects that we named entities. Weak solutions of PDEs, almost everywhere defined as flows, and noncommutative manifolds are examples of such entities without clear underlying identified objects. The word identified clearly refers to the concept of identity, a concept that disappeared from the lower level in such situations.

But one of the goals of mathematics, under the angle we chose to look at in this article, is precisely to give an identity to these entities. Weak solutions, which appeared first as worse options (or the worst solution), or as the lesser evil, are nowadays perfectly identifiable: they acquired their own identity by themselves.

The temporality of … nothing, happens to be only a transient phenomenon, which, after taking off, creates its own identity. And in the three examples we treated, it was the willing of preserving a type of symmetry/identity/equivalence which was the strong engine responsible for this precious temporality which creates new paradigms, new ‘levels’ in mathematical reality.

Mathematics versus Philosophy: The Other Way

We believe that this discussion offers the opportunity of considering an example of interaction between mathematics and philosophy which goes ‘the other way’. Indeed, the traditional discussion concerning Platonism involves mathematics as outside philosophy, watched by it, and looks at what they do when creativity is in action: do they invent or do they discover?

It seems to us that in the situations described in this article, which belong to recent mathematics, everything works the other way. It is inside mathematics that the question of discovering (something already existing, i.e. at the lower level) or creating (something new, i.e. something which belongs only to the upper level) is considered. A noncommutative manifold does not exist somewhere else than in the framework of ‘its’ noncommutative algebra of functions, that is, functions … on the (non-existing) manifold itself.

By bringing mathematical Platonism inside their own interior, the mathematics itself may well have closed the debate on its own reality.