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+From an AI based on Numbers to another based
+on Geometric Forms
+
+Grothendieck’s ideas on the notion of space were
+fundamentally based on his work in algebraic geometry,
+which led him to the notion of topos, and finally to the
+formulation of a geometry of forms. The form has fascinated
+Grothendieck, as a passage from Récoltes et Semailles testifies.
+“If there is one thing in mathematics that has always
+fascinated me more than any other, it is neither the number
+nor the size, nor magnitude, but always form. And among the
+thousand and one faces that form chooses to reveal itself to
+us, the one that has fascinated me more than any other and
+continues to do so is the structure hidden in mathematical
+things. The structure of a thing is not something we can invent.
+All we can do is patiently and humbly bring it up to date —get
+
+Belabes A. The Grothendieck’s Toposes as the Future Mathematics of AI. Philos Int J 2024, 7(3):
+000334.
+
+Copyright© Belabes A.
+
+6
+
+to know it, discover it” (p. 43) [12].
+
+In mathematical language, it appears that AI has
+developed more around numbers than geometric forms [32].
+In other words, deep learning techniques are based on a
+numerical approach to the subtle worlds that surround us,
+whereas the human mind tends to focus more on geometric
+forms, and then to distinguish geometric forms according to
+certain correspondences. This is where the notion of topos
+comes in by considering the correspondences. This is one of
+the reasons why the notion of topos is attracting the interest
+of AI engineers, who use machine learning techniques
+to develop systems that can help organizations increase
+efficiency, i.e. to minimize costs and turnaround times, and
+maximize volumes and profits.
+
+The notion of topos offers the opportunity to develop
+models for many aspects of life, closer to the way humans
+perceive the worlds around them, starting from the limits
+of the conceptualization of AI assimilating human mind
+to a digital system limited to calculation operations, to
+the detriment of the approach of the human mind being
+more interested in forms, then which distinguishes forms
+according to certain correspondences. It is preferable to use
+the notion of correspondence instead of that of invariants,
+which is based on the idea that in mathematics there can be
+different theories, possibly belonging to different branches of
+the speciality, which describe the same invariants in different
+languages. In this respect, the notion of topos is presented
+as a tool that offers the possibility of identifying invariants
+by eliminating aspects considered to be inessential while
+retaining what is essential [32].
+In the world of meanings where words arise from
+meaning through an underlying structure (p. 4) [33], the
+essential and insignificant dichotomy seems far from real
+life. Everything, no matter how small, is important if we take
+the time to look into it, as is the case in the animal world. In
+his novel Without the Orangutan, Eric Chevillard [34], which
+highlights a contradiction without imposing a conclusion,
+challenges the reader to the fact that humans did not suspect
+the importance of the orangutan in the general organization
+of the world nor that everything was held together thanks to
+him, to his discreet but decisive action. He was the subtle cog.
+It was enough for him to disappear for everything to collapse.
+How could the world live without him?
+
+The notion of correspondence refers to the idea that
+in exploring the relationship between things, the object of
+research is no longer the things in themselves, but the things
+left open to human interrogation, because every instrument
+carries within it the spirit in which it was created. “We only
+hear the questions to which we are able to find an answer”,
+
+Philosophy International Journal
+writes Nietzsche in Die fröhliche Wissenschaft (Gay Science)
+(p. 193) [35]. In this respect, the answers appear more
+like blockages to further exploration than as revealing
+elucidations. The primary virtue of the answers could even be
+to set prohibitions, and not to open the questioning. Deleuze
+said later: “language is a system of commands, not a means of
+information” (p. 60) [36]. Grothendieck is perceptive about
+the performative nature of language, as a passage from
+Récoltes et Semailles illustrates:
+“We are constantly having to invent the language capable
+of expressing in ever finer detail the intimate structure of the
+mathematical thing, and to ‘construct’ with the help of this
+language, as we go along and from scratch, the theories which
+are supposed to give an account of what has been apprehended
+and expressed” (p. 43) [12].
+
+By exploring the approaches specific to the knowledge
+of things, Nietzsche shows that the desire to penetrate the
+essence of things is always triggered by the observation of
+a correspondence of forms. Through such a perspective, the
+process of constructing knowledge appears as a matching
+of these forms, which presupposes an intervention of the
+mind, a work of sorting and selection, of transformation by
+deformation of what exists. This coherent deformation is
+achieved by eliminating the unusual, the irregular, the nonlinear, the rough, so that to know is essentially, in one way
+or the other, to reduce the unknown to the known because
+“what is known is recognized”, as Nietzsche said (p. 328) [35].
+
+The notion of topos could prove fruitful as long as the
+one who uses it realizes that the will to know could become a
+veil to knowledge [37]. This invites to take steps to minimise
+as far as possible the gap between what is targeted and
+what is actually achieved. The notion of conceptual strata
+[38] will undoubtedly constitute an important step forward,
+in this awareness to the extent that the exploration of
+correspondences is carried out through a layered approach.
+What seems insignificant through one layer could prove to be
+significant through another layer. This allows those who use
+the notion of topos to move from ‘Invariant’ to ‘invariant for’
+and from ‘Insignificant’ to ‘insignificant for’.
+
+If the notion of invariants suggests that it is possible
+through the notion of topos to link parts supposed to
+exist separately and independently, the epistemology of
+quantum physics shows that the parts supposed to function
+independently are simply particular forms and fortuitous
+within a set that is capable of expansion [39]. The notion of
+conceptual strata is part of an approach in which the world
+is perceived as a complex web of facts, in which relationships
+of various kinds are superimposed, thereby determining the
+fabric of the set [40].
+
+Belabes A. The Grothendieck’s Toposes as the Future Mathematics of AI. Philos Int J 2024, 7(3):
+000334.
+
+Copyright© Belabes A.
+
+Philosophy International Journal
+
+7
+
+But as someone digs into anything, sooner or later he
+will discover that the layers are not infinite and that they will
+be confronted with the abyss of nothingness that Stéphane
+Malarmé referred to in a letter dated 28 April 1866 to Henri
+Cazalis (p. 696) [41]. However, nothingness by definition
+does not exist, since it is precisely the opposite of what exists.
+In his poem Peri Physeos (On Nature), Parmenides writes in
+fragment 6: “Being is, non-being is not”.
+With the theory of loop quantum gravitation [42],
+it is possible to illustrate what happens at the bottom of
+conceptual strata at a moment when time seems to stop and
+space is torn apart. From their depths, time and space are
+reversed to the point where new conceptual strata are born.
+So what we cannot see or imagine, we can no longer ignore.
+
+Conclusion
+
+In view of the data collected from press articles, TV
+reports, interviews, lectures, and papers, the notion of topos
+is likely to become the future mathematics of AI, with a
+greater emphasis on geometric forms than the mainstream
+approach in the business world, which favors numbers
+through statistical procedures to make predictions and more
+informed decisions. Numbers have no meaning in themselves.
+As a result, when things are reduced to collections of
+numbers, their meanings disappear.
+
+AI engineering requires models capable of formalizing
+the basic variables that shape its practice, as well as the basic
+operations that impact both the structure and functioning of
+the human brain. The notion of topos could help to satisfy
+these two main aspirations of AI engineering geometric
+forms by connecting things through a common essence after
+removing the inessential aspects.
+My study is part of an investigation into the nature of
+intelligence and supposedly intelligent machines, as well as
+the interpretation of Grothendieck’s toposes as the future
+mathematics of AI in terms of impact and consistency. In
+this regard, it would be useful to enrich the approach which
+assimilates the topos to a sort of bridge which removes all the
+inessential aspects of the things to be linked to retain only a
+common essence.
+
+One of the most promising avenues in the archeology of
+discourse on the topos is reading through the senses to which
+Heraclitus referred based on the extremely lucid observation
+that at the very heart of language there was a very deep gap
+which separates the man between what he said and what he
+meant. This lucidity is found in Grothendieck who invites in
+his autobiography Récoltes et Semailles to constantly invent
+the language capable of expressing more and more finely
+what has been apprehended and seen.
+
+From this art of reading well stems the notion of
+conceptual strata, which shows that the things to be
+connected have a stratified configuration that invites us to
+discover the complexity of the relationship between what
+happens deep inside ourselves and what we designate
+by words. Rather than calling something ‘insignificant’, a
+consistent relativism of ‘insignificant for’ would be more
+appropriate, depending on the conceptual stratum under
+consideration. As Gorthendieck writes:
+“What makes for the quality of the researcher’s
+inventiveness and imagination is the quality of his attention,
+listening to the voice of things. For the things of the Universe
+never tire of speaking for themselves and revealing themselves
+to those who care to listen” (p. 43-44) [12].
+Unfortunately, we often mishear what we think we hear
+well, just as we often misread what we think we read well.
+
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+Belabes A. The Grothendieck’s Toposes as the Future Mathematics of AI. Philos Int J 2024, 7(3):
+000334.
+
+Copyright© Belabes A.