MENTAL vs. The Generalist Mathematics of Grothendieck
MENTAL vs. Grothendieck's Generalist Mathematics
MENTAL vs. THE GENERALIST MATHEMATICS OF GROTHENDIECK
"If there is one thing in mathematics that fascinates me, more than anything else, it is not number or magnitude but form. And among the thousand and one faces that form chooses to reveal itself to us is the structure hidden in mathematical objects."(Alexander Grothendieck)
The Mathematics of Alexander Grothendieck
Alexander Grothendieck −mathematical genius of privileged mind and great capacity for work, visionary, idealist, libertarian, nonconformist, pacifist, ecologist, seeker of truth and mystic− revolutionized mathematics with innovative, unifying, synthesizing, extraordinarily general and abstract ideas.
This profound approach allowed him to establish connections between algebra, geometry, topology and number theory. It can be said that contemporary mathematics emerges fundamentally from Grothendieck's work.
Grothendieck completely and systematically reconstructed algebraic geometry, facilitating the solution of complex problems in number theory, among them Weil's conjectures, Mordell's conjecture, and Fermat's last theorem.
His written work is immense. The works "Elements of Algebraic Geometry" (EGA), written in collaboration with Jean Dieudonné, and "Seminar on Algebraic Geometry" (SGA) total about 10,000 pages. The seminar was given at the IHES (Institut des Hautes Études Scientifiques), which became the world center of algebraic geometry.
Very many of Grothendieck's original concepts have become part of the common heritage of mathematics, even retaining the names Grothendieck gave them.
His general strategy
Grothendieck's philosophy was based on several general principles, more or less explicit:
Naivete and innocence.
For Grothendieck, naivete or innocence was to look at things with one's own eyes rather than through the glasses of some human group vested with authority.
"In our knowledge of the things of the universe (whether mathematical or not), the renewing power that is in us is nothing but innocence [...]. It alone unites the humility and audacity that make us penetrate into the heart of things, and that allow us to let things penetrate us and permeate us" (Grothendieck in "Harvests and Sowings", 1985).
Simplicity and naturalness.
His concepts are simple, natural, almost self-evident. Grothendieck tried to contemplate everything as simply as possible, searching for its fundamental essence. His concern was to define "neatly natural" concepts. One of his passions was to "name" concepts as a means of apprehending them.
Of Grothendieck's two key concepts, Topos and Scheme, he said "The idea of Topos had everything one might expect to puzzle, mainly because of its naturalness and simplicity." "The very concept of Schema is one of childlike simplicity, it is so simple, so humble, that no one before me dared to take it seriously."
Depth.
Grothendieck would immerse himself in a problem by searching for its essence until he succeeded in dissolving it or getting the solution to emerge naturally. He identified this approach as feminine or yin. Her attitude was intuitive, passive, waiting, global observation, deep, gestalt, maturing. The opposite attitude is yang, masculine, active, superficial and rational, consisting of trying to solve a problem in a forced way.
Generalization.
Grothendieck always sought the most general possible solutions to problems in order to establish connections between different fields of mathematics. Faced with a specific problem, Grothendieck tended to perceive it as a particular case of a more general problem and approached it with as few restrictions or hypotheses as possible. Normally, in conventional mathematics, the approach is usually the opposite: one tries to prove something by adding to the initial situation additional hypotheses in an attempt to reduce the problem.
According to Freeman Dyson, there are two types of mathematicians: frogs and birds. Frogs study the details of the terrain. Birds contemplate the landscape from above. Grothendieck was a bird who wanted to contemplate the whole mathematical landscape. According to Grothendieck, a problem is not truly solved until it is viewed from a correct overall perspective, from which the problem can be solved effortlessly and in which it fits naturally into a larger framework.
Grothendieck's concepts were of a vertical type, where the generic or global connected with the specific or particular, in a continuous process of ascent and descent. Grothendieck had a unifying and synthetic vision of mathematics. He wanted to catch the One in the variety, the essential unity underlying diversity. He compared this unifying vision of mathematics to that of Newton and Einstein in physics, and to that of Darwin and Pasteur in biology.
Dialectics.
Grthendieck's search was also of a horizontal type, based on dialectics or polarities, seeking complementary points of view. This dialectic was essentially based on the union between algebra and geometry, that is, the union between the discrete and the continuous.
Relations.
Grothendieck's method consisted in, before confronting a theory, unraveling the "yoga" (in his own words) of that theory, i.e., the relations or connections between the elements of the theory, and what the tool to be used should be (usually it was the theory of categories). The word "yoga" means "union".
In mathematics, it is relationships that are important. Mathematical objects can be defined or described by their relations to other mathematical objects. For example, a geometric object X can be conceived in terms of all morphisms that have object X as their destination. Therefore, X can be replaced by the functor that associates to each object Y the object X.
His mathematical strategy
Grothendieck's mathematical strategy was based on:
Primary concepts.
Grothendieck sought the primary concepts (or protoconcepts) from which all mathematical structures arise, attempting to connect the one (the form) and the variety (the structures).
Invariants.
Grothendieck also looked for invariants. In mathematics, an invariant is something that does not change when a set of transformations is applied to it.
The best known topological invariant is the homotopy. In algebraic topology, homotopy is the transformation of one topological space into another that makes them equivalent. For example, a sphere, a cylinder and a cube are topologically equivalent.
The invariants of shapes are cohomologies. Cohomology is a method of assigning invariants to a topological space.
Category theory.
Grothendieck considered category theory to be the most suitable general framework for all mathematical theories.
Category theory has very simple principles, but its developments are too abstract and complex. A category consists of objects and morphisms between them. It is a set with structure. A set is a particular case of a category when there are only objects and no morphisms.
Morphisms constitute the important thing about a category, i.e. its structure. Even objects can be considered as identity morphisms (which make each object correspond to the same object). There are higher order categories, i.e. categories whose objects are categories. Morphisms between categories are called "functors".
The revolution of algebraic geometry
Algebraic geometry has its origins in the concept of algebraic variety. An algebraic variety generalizes the notion of curve (1-variety), surface (2-variety), n-variety in general. An n-variety (or variety of dimension n) needs n parameters (or local coordinates) to define a point on it. Algebraic varieties are defined by a set of polynomial equations. Jean-Pierre Serre extended this concept to the notion of algebraic space.
In the 1960s and 1970s, Grothendieck reshaped functional analysis and algebraic geometry. He revolutionized functional analysis by generalizing the theory of Schwartz distributions. In mathematical analysis, a distribution is a generalized function, which generalizes the notion of function and that of measure. Schwartz's article (Fields Medal 1950) is considered a classic. Grothendieck did his doctoral thesis with Schwartz.
Subsequently Grthendieck left functional analysis to devote himself to geometry. This process he described as follows: "It was as if I had escaped from the arid, harsh steppes and suddenly found myself transported to a kind of 'promised land' of superabundant wealth that multiplied to infinity wherever I put my hand, whether to search or to gather." Grothendieck's ideas transformed algebraic geometry and made it one of the most abstract mathematical fields in mathematics.
In 1949, André Weil proposed four mathematical conjectures. These conjectures were very precise, but Weil could only prove them in particular cases, not in general. One of Weil's conjectures proposed that a type of algebra invented for the study of continuous functions could be used to find the number of solutions of a diophantine equation (an equation whose solutions are integers).
Weil's conjectures were for Grothendieck the main source of reflection between 1958 and 1969. Grothendieck proved the second conjecture. The fourth conjecture (the most difficult) was proved by his disciple Pierre Deligne (winner of the Fields medal in 1978 and the Abel prize in 2013).
To prove these conjectures, a new mathematical theory had to be created that required generalizing the concept of geometric space, a theory that could unify the discrete and the continuous, where the powerful homology and cohomology techniques of topology were also valid in the field of integers.
In mathematics there are many types of geometric spaces: Euclidean, projective, affine, topological, etc.; even infinite-dimensional spaces, such as those used in quantum physics. All these conceptions of space have in common that they are based on "points" and subject to constraints of some kind.
A new concept of space
In 1958, generalizing Serre's ideas of algebraic space, Grothendieck proposed a new concept of space: a space without points, based exclusively on algebraic expressions and their relations. This space was the generalization of the notion of algebraic variety. Grothendieck's generalist strategy was to "algebrize everything" based on the concepts of ring and ideal:
Ring.
A ring is an algebraic system (A, +, *) consisting of a set A (non-empty) and two inner operations, usually called "sum" (+) and "product" (*), such that (A, +) is a commutative group with neutral element (0), and the product is associative and distributive with respect to the sum. The inverse operation of addition is subtraction (−). The product has no inverse. If the product is commutative, the ring is said to be commutative. If the product has neutral element (1), the ring is said to be unitary. An example of a commutative and unitary ring is the set of integers with the operations of addition, subtraction and product. The set of all polynomials, which is closed with respect to the arithmetic operations of addition and product, is a commutative ring.
An element x of a ring is nilpotent if there exists an integer n such that xn = 0.
A body is a ring in which we can divide (except by zero).
Ideal.
If A is a commutative ring, we call "ideal of A" any additive subgroup I of A such that for every element x of A and for every element y of I, the element x*y = y*x belongs to I. For example, for all integer k the set of elements k*Z of the integer multiples of k is an ideal (Z is the set of integers).
Grothendieck's idea consisted in associating to any kind of commutative ring a superstructure which, together with a number of axiomatic properties would make it a "space". These point-free spaces have cohomologies, that is, they are algebraic structures that allow us to classify topological spaces.
According to Grothendieck, this new geometry was the synthesis of two worlds: the arithmetic (discrete) world and the world of continuous magnitudes (geometric space). Grothendieck proposed the name "arithmetic geometry".
The main concepts
Grothendieck was mainly based on the concepts of Scheme, Sheaf, Topos and Motive.
Schemes.
Schemes have their origin in the rings. In the 1960s, Oscar Zariski, studying algebraic varieties, discovered that every variety gives rise to a ring.
To every variety V can be associated a ring of polynomials and an equivalence relation between the elements of the ring that give rise to ideals. For example, if we have the variety V defined by the equation x2+y2− 1 = 0, we can divide the ring of all polynomials into equivalence classes based on the property that it yields the same value for every point of V. For example, the polynomials 2x2+2y2+5 and 3x< sup>2+3y22+4 = 0 produce the same value (7) for every point of V. The difference between them is a multiple of x2+y2− 1 = 0, i.e., they belong to the same ideal of the multiples of x2+y2−1 = 0.
But there were rings that had no variety associated with them. It was precisely this asymmetry that motivated Grothendieck to redefine geometry in terms of arbitrary commutative rings. The resulting structures are known as "Schemes". A Scheme gives rise to a geometric object determined by the structure of a commutative ring.
The notion of Scheme is a generalization of the notion of "algebraic variety". This generalization was for Grothendieck "the heart of the new geometry". Varieties and Schemes are the (abstract) descendants of elementary geometric figures such as the straight line or the circle.
Schemes were a generalization that introduced, among others, two important features: 1) commutative algebra (as part of algebraic geometry); 2) the introduction of nilpotent elements in rings, which play an important role in the study of infinitesimal properties of algebraic varieties.
Around 1960, Grothendieck (in collaboration with Jean Dieudonné) began publishing the monumental treatise EGA (Elements of Algebraic Geometry) in which he proposes to lay the foundations of algebraic geometry within the framework of Scheme theory.
The concept of Scheme has been the key to the profound renewal of algebraic geometry. Schemes are today the main objects of algebraic geometry.
Today, most of the relevant works in algebraic geometry employ, more or less explicitly, the language of Schemes. And much of the recent progress in number theory would have been impossible without the geometric intuition provided by Scheme theory.
Sheaves.
Historically, the notion of Sheaves was introduced in mathematics in the 1940s by Jean Leray to deal with some fundamental problems in function theory. It was then extended to algebraic geometry and is today widely used in algebraic geometry and algebraic analysis. Leray conceived of Sheaves while doing research while he was a prisoner of the Germans during World War II. The same concept was introduced by Kiyoshi Oka when studying analytic functions of complex variables. The Sheaf concept quickly became one of the main tools of algebraic geometry. To study a space was to study the Sheafs of that space.
The modern definition of Sheaf is due to Élie Cartan (in 1952). Later, in 1953, the postwar French school of algebraic geometry led by Jean-Pierre Serre introduced Sheaves into algebraic geometry. The idea of Sheaves on topological spaces first appeared in Hermann Weyl's (1913) work on Riemann surfaces (A Riemann surface is a complex variety of a complex variable).
The idea behind the notion of Sheaf is simple and arose from the study of functions defined on the same sets: all functions are manifestations of a function of higher order.
A simple example of Sheaf is the set of all lines tangent to all points on a given circle of radius R. Another example is the set of all tangent planes to a surface in 3D space. In general, we can assign to each point of a variety a certain algebraic structure. This was precisely Grothendieck's idea: to generalize the concept of Sheaf as the set of algebraic objects associated to each of the points of a variety.
Sheaves can be considered generic sets or structures, although Lawvere considered Sheaves to be variable structures. A Sheafis a structure extended over a base space, which is the variety.
Sheaves are mathematical entities that unite algebra and geometry: the real and the imaginary, the interior and the exterior, the essence and the existence, the analytic and the synthetic, the local and the global, the discrete and the continuous.
Topos theory.
The term "Topos" is singular (the plural is "Topoi" or "Toposes"). A Topos is a category. The objects of the category are Sheaves and the morphisms are relations between Sheaves. The notion of Topos is a redefinition or generalization of the notion of traditional topological space. For Grothendieck, the concept of Topos is the maximum generalization of the concept of space. It is a more general concept than that of Scheme. A Topos is the envelope or abode of a Scheme. A Topos is associated to every Scheme.
Grothendieck, by generalizing the notion of sheaf, showed that it was possible to contemplate the category of classical sets from the category of all Sheaves (the category of Topos). In 1960, Grothendieck introduced the category of "generalized Sheaves", the category called "Grothendieck's Topos". The essential properties of Grothendieck's Topos gave rise to William Lawvere and Myles Tierney's general notion of "elementary Topos", which is more general and abstract than Grothendieck's Topos and is what today is simply called "Topos".
For Grothendieck, a space X is described by the Topos T(X) of Sheaves over X. The toposes contain all possible structures of .
In a topological space what really counts are not its "points" (or the subsets of points) and the proximity relations between them, but the Sheaves on that space and the category they form.
The concept of Topos unites traditional (continuous) topological spaces with the (discrete) "spaces" or "varieties" of algebraic geometry.
The theory of Topos opened the doors to a foundation of mathematics on concepts different from the conventional one (the one based on the theory of sets). Moreover, Topos theory has many applications in quantum physics, artificial intelligence, computer science, etc.
Motives.
Grothendieck introduced the notion of Motive (in French "Motif", pattern) in a letter to Serre in 1964. He claimed that, among the objects he had been privileged to discover, they were the most charged with mystery and were perhaps the most powerful tool for discovery.
The Motives were introduced by Grothendieck when he believed that it was possible to define a universal cohomology theory, i.e., one that would contemplate the essence of all possible cohomology theories there might be about the category of algebraic varieties. Grothendieck conjectured that the Motives provided such a universal theory in a series of problems he called "standard conjectures," which remain unproven. This is an attempt to find a universal way to linearly combine simple geometric varieties to create progressively more complex varieties.
Motives are intermediate elements between algebraic varieties and their linear invariants (cohomology).
Each algebraic variety X has an associated Motive [X]. The idea is that a Motive has the same structure as any cohomology. If you get the Motive of a variety, you have all the information about all cohomology theories.
In algebraic geometry, a Motive denotes an essential part of an algebraic variety. A Motive is a structure associated with the shape as invariant. A Motive can have different manifestations (structures). The relation is "Form → Motive → Structure".
According to Grothendieck, the Motives are "the heart of the heart of the new geometry", the most profound theory and his greatest contribution to mathematics. Grothendieck spoke of "motive yoga" because of its importance and depth. Grothendieck's dream was that the theory of Motives would unify all of mathematics, especially the unification of Galois theory and topology.
Grothendieck never published anything on Motives, but he mentioned them frequently in his letters to Serre. There is currently only one book on Motives, which is by Yves André [2004].
It is recognized that the theory of Motives is ambiguous. There are even different versions of it. Many mathematicians have tried to pin it down to no avail. At present we have only small, somewhat fuzzy fragments of this theory.
MENTAL vs. Grothendieck's Mathematics
Unification.
Grothendieck delved deep into mathematics, searching for primary, general or universal concepts to bridge the various mathematical fields, but he failed to find the primary archetypes, the "boundary concepts" that unite the deep with the superficial in order to unify mathematics.
Grothendieck focused mainly on algebraic geometry and category theory. He claimed that all mathematics should be founded on the theory of Topos.
MENTAL is based on primitives of supreme power of abstraction and generality. All mathematical structures are manifestations of the primal archetypes. MENTAL is the universal language and the foundation of mathematics and the formal sciences in general.
Philosophy.
Grothendieck did not contemplate philosophical topics. He focused exclusively on "pure" mathematics. MENTAL emerges from philosophical categories.
Relationships.
Grothendieck emphasized the theme of relationships, "yoga". In MENTAL, semantics lies in relationships and the true "yoga" resides in the primary archetypes, the center from which the unity of all things is contemplated and from which all mathematical expressions emerge. Moreover, the relationships envisioned by Grothendieck were static. In MENTAL they are dynamic.
Space-time.
For Grothendieck, space is based exclusively on algebraic expressions and their relations.
In MENTAL abstract space arises naturally from the relations between expressions. Abstract time arises from the evaluation of expressions. Abstract space is not a primary concept, but a derived one. In this space there are no points, only related dynamic expressions. It is not an absolute space.
The true space, the deep, generic and universal space, is the abstract space where expressions "live", where they interact dynamically. In MENTAL the distinction between algebra and geometry is truly diluted. The different types of spaces are particular cases of this universal space.
Category theory.
Grothendieck contributed decisively to category theory, the most abstract and generic theory of mathematics, mainly with his theory of Topos.
The main problem with categories is that they have no definite semantics, since a morphism is an ambiguous concept that can be interpreted in many ways (function, transformation, connection, implication, etc.) [see Comparisons - MENTAL vs. Category Theory].
Given the close relationship between abstraction and simplicity, Einstein's statement "Everything should be made as simple as possible, but not simpler" we can turn it into "Everything should be made as abstract as possible, but not more abstract". Category theory is "too abstract". Paradoxically the too abstract and the too simple leads to the extremely complex because the most important thing is lost (or weakened), the basis of everything: semantics.
The true categories are the primary archetypes of MENTAL, the dimensions of reality, which establish perfectly defined relationships.
Union of opposites.
Grothendieck tried to unite several opposites, based on the union between algebra and geometry: qualitative-quantitative, continuous-discrete, etc. MENTAL is the integral union of opposites.
Naivety and innocence.
Grothendieck used as his philosophy the principle of innocence or naivety. MENTAL is an essentially naive language.
Simplicity.
Grothendieck sought simplicity, but (paradoxically) he relied on category theory and his theory of Topos, a theory that is complex, difficult to understand. MENTAL is the supreme simplicity. The key to generality and universality lies in simplicity.
Consciousness.
When Grothendieck switched from the study of functional analysis to geometry, what he did was to basically switch modes of consciousness: from the analytic to the synthetic mode, from the local to the global. But Grothendieck realized that it was necessary to unite algebra and geometry, to unite the two modes of consciousness.
MENTAL is the language of consciousness for two reasons: 1) because it is based on primary archetypes, the archetypes of consciousness; 2) because it is the integral union of opposites. MENTAL is consciousness, freedom, simplicity and creativity, the true "promised land" of which Grothendieck spoke.
Topos.
Grothendieck pretended to found mathematics by means of the abstruse theory of Topos. But the foundation must necessarily be simple. Moreover, the concept of Topos is not primary, for it combines different concepts. Lawvere's elementary Topos theory is even more abstract, and also has the same problems. The Topos theory imposes a fixed model, restricting the freedom of creation of mathematical entities.
MENTAL is the foundation of mathematics. All mathematics (and not only mathematics) is built from primary archetypes, which are degrees of freedom.
Sheaves.
Sheaves unite algebra and geometry. But the concept of Sheaf is retrenched, it is not general enough, because there are no higher order Sheaves (Sheaves of Sheaves). In MENTAL, a Sheaf is a parameterized generic expression representing several mathematical objects that have a common pattern, and higher order parameterized generic expressions can be defined to represent higher order Sheaves.
Motives.
Motives are conjectures. The primary archetypes of MENTAL constitute a universal thesis. Are the Motives the "causes" or primary concepts that Grothendieck sought as the foundation of the entire mathematical universe? There are several clues that reinforce this hypothesis:
Archetypes are "forms without content" (according to Jung's conception) and Motives, according to Grothendieck, are intermediaries between (geometrical) forms and (algebraic) structures.
The French term "Motif" means "pattern".
According to Claire Voisin (former director of Jussien (at the University of Paris VI) the theory of Motives is the "Holy Grail" of mathematics.
According to Grothendieck, the concept of Motive is "at the heart of the heart" of mathematics, the common source of all mathematics.
Algebrization.
Grothendieck "algebrized" everything. In MENTAL all are algebraic expressions, but he goes beyond abstract algebra, for he contemplates imaginary algebra based on imaginary expressions, which are of the form (x = y), where x and y can be any expressions. And it contemplates all kinds of relations between expressions.
Language.
Grothendieck did not go so far as to define a new language for mathematics. MENTAL is a formal universal operational and descriptive language applicable to the mathematical world.
Practice.
Grothendieck built a whole theoretical edifice; he was not too interested in the practical aspect. MENTAL unites theory and practice, which are aspects of the same thing.
Logic.
According to category theory, logic has a categorial character (precisely because of the ambiguous character of morphism which can be interpreted as implication). Grothendieck never explicitly dealt with logical issues. In MENTAL, logic is one of the dimensions of reality based on the primitive "Condition", supported by the rest of the primitives.
Algebraic geometry vs. geometric algebra.
Algebraic geometry combines abstract algebra with analytic geometry. It can be thought of as the analysis of systems of equations and their solutions, along with their geometric interpretation.
In contrast, geometric algebra (or Clifford algebra) is an algebra based on the geometric product, a generalization of the scalar product and the vector product operating on multivectors. A multivector is the n-dimensional generalization of the vector concept.
MENTAL, as a universal formal language, contemplates algebraic geometry and geometric algebra, and both fields can be combined to create new mathematical entities.
An example of Sheaf
A simple example of Sheaf is that of tangents to a circle of radius r. We assume that the center of the circle is the origin of coordinates. The equation of the circle is x2 + y2 = r2. A unit vector between the center (0, 0) and of angle φ is u = (cos φ, sin φ ), which cuts the circle at the point (x, y) = (r•cos φ, r•sin φ). A tangent vector at the point (x, y) is v = (−sin φ, cos φ). The equation of the tangent at point (x, y) −points (x', y')− is.
x' = r•cos φ - s•sin φ
y' = r•sin φ + s•cos φ
Therefore, the sheaf constituted by all the tangents of a circle of radius r is defined by three parameters: r, φ and s:
Grothendieck was born on March 28, 1928, in Berlin (Free State of Prussia), the only son of a Jewish activist (Alexandre Shapiro) and a journalist (Hanka Grothendieck). His parents participated in the Spanish Civil War.
Between 1934 and 1939, Grothendieck lived in Hamburg with a foster family while his parents were in France. In 1939, he was reunited with his mother in France.
His studies in mathematics begin at the University of Montpellier (between 1945 and 1948). After a short period in Paris, in 1950 he went to Nancy to do his doctorate with Laurent Schwarz in functional analysis. It was then that he began to make his mark. He was given 14 possible questions to work on. He solved them all. The problem he chose for his thesis defense in 1953, he approached with a novel and general approach, applicable to broad fields of mathematics.
Upon completion of his thesis he changed domains and switched to geometry. In 1956, on his return to Paris, he proposed an entirely new approach to algebraic geometry. At some point he was part of the group of mathematicians gathered under the name of Nicolas Bourbaki.
His first permanent job was at IHES, a private research institute founded in 1958 in Paris. There he initiated, with the help of the best of the international community, the "Seminars in Algebraic Geometry" (SGA), of which 7 volumes were published; and the writing of his "Elements of Algebraic Geometry" (EGA), of which he published 4 of the 12 projected books. These writings were a revolution in geometry, mainly because of their deepening of the basic concept of space.
In 1966, the International Congress of Mathematicians meeting in Moscow decided to award him the Fields Medal (awarded every 4 years), the highest mathematical award, for his contributions to homological algebra and algebraic geometry. Grothendieck refused to attend the award ceremony and collect the prize because of the repressive policies of the Soviet regime.
In 1970, at the age of 42, at the height of his international fame and creative ability, Grothendieck left the IEHS because of his pacifist convictions, when he learned that 5% of the budget came from the French Ministry of Defense.
In 1970 he founded, together with two colleagues, the pacifist-ecologist organization "Survivre et Vivre" (Survive and Live), for the defense of the environment, and retired to a small village on the outskirts of Montpellier.
In 1972, he acquired French nationality (until then he was stateless) in order to obtain a teaching position at the University of Montpellier. He worked at this university until the day of his official retirement in 1988. During this period he continued his mathematical research but outside the official standards: without publishing anything and with few contacts with other colleagues. It seems that his exclusive dedication to mathematical research, and his feverish pace of work, caused him (in his own words) "a long period of spiritual stagnation".
Between 1983 and 1988 he wrote thousands of pages of non-mathematical meditations, which he distributed to his closest friends and colleagues. In "Récoltes et Semailles" (Harvests and Sowings), a work of more than a thousand pages, he combines personal and mathematical reflections. In "La Clef des Songes" (The Key to Dreams) he recounts his discovery of God.
In 1988, Sweden awarded him the Crafoord Prize of the Royal Swedish Academy of Sciences, shared with his disciple Pierre Deligne. The recognition was accompanied by a large sum of money, which he refused because "given the decline in scientific ethics, participating in the prize game means endorsing a spirit in the scientific community that seems to me unhealthy" and because "my pension is more than enough for my material needs and those who depend on me."
In 1990 he disappeared and cut off all contact with family and friends. He retired permanently to live in a small village in the French Pyrenees. His whereabouts, at his express wish, remained unknown to the mathematical community and the general public. There he continued to publish nothing and to socialize with his neighbors. In the last decade he decided to go a step further and restricted all contact with the outside world, living his last years as a hermit, dedicated to meditation and the search for truth, oblivious to the impact that his ideas continue to have today.
Grothendieck passed away on November 13, 2014, at the age of 86 at the Arège Conserans hospital in Saint-Girons.
Grothendieck gave orders to burn all his writings. He himself burned numerous documents. On the second floor of a building in the center of Montpellier are now five boxes containing 20,000 pages of notes written between 1970 and 1991. Despite the destruction order, the person in charge of the university's heritage managed to save them. In Paris there is a Grothendieck Circle, which collects, translates and publishes his writings.
Evaluation of his figure
Grothendieck is, for many, the greatest mathematician of the 20th century. His work in algebraic geometry opened new horizons, some of which remain to be explored. "Alexander Grothendieck's ideas, so to speak, have penetrated the unconscious of mathematicians," his most brilliant pupil, Pierre Deligne, went so far as to assert.
Grothendieck's mathematical stature is comparable to that of Gauss, Riemann or Galois. He has been called "the Einstein of mathematics" because of his generalist philosophy. He has also been called "the Freud of mathematics" for having delved into the depths of mathematics.
Grothendieck himself identified with Einstein by drawing two parallels:
By the transformation of our conception of space.
Einstein revolutionized the concept of space. Space is not a pre-existing entity (as Newton claimed), but space emerges from the relational properties between its elements.
Grothendieck revolutionized the most fundamental concept of geometry: space. Space is not constituted by its points, but by the relations between algebraic expressions. This conception of space is richer than traditional Euclidean space; it has more possibilities.
For its unifying vision.
Einstein unified space and time, as well as mass and energy.
Grothendieck revolutionized mathematics by profoundly connecting algebra and geometry.
Grothendieck felt twinned with Galois, for having both given the keys for algebra to follow new paths, for knowing how to see the general in the particular. In the case of Galois, the conditions for solving equations gave rise to group theory. In the case of Grothendieck, the structural generalization of the concept of space.
Gromthendieck, following the path of Descartes, Pascal or Leibniz, has contributed to introduce mathematics as a way to transcendence.
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