International Journal of Algebra, Vol. 14, 2020, no. 2, 43 - 137 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2020.91249 Algebraic Logic and Topoi; a Philosophical Holistic Approach Tarek Sayed Ahmed Department of Mathematics, Faculty of Science Cairo University, Giza, Egypt This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 2020 Hikari Ltd. Abstract We take a magical tour in algebraic logic and its most novel applica- tions. In algebraic logic we start from classical results on neat embed- dings due to Andreka,´ Henkin, N´emeti, Monk and Tarski, all the way to recent results in algebraic logic using so{called rainbow construc- tions. Highlighting the connections with graph theory, model theory, finite combinatorics, and in the last decade with the theory of general relativity and hypercomputation, this article aspires to present topics of broad interest in a way that is hopefully accessible to a large audience. Other topics deallt with include the interaction of algebraic and modal logic, the so{called (central still active) finitizability problem, G¨odels's incompleteness Theorem in guarded fragments, counting the number of subvarieties of RCA! which is reminiscent of Shelah's stability theory and the interaction of algebraic logic and descriptive set theory as means to approach Vaught's conjecture in model theory. The interconections between algebraic geometry and cylindric algebra theory is surveyed and elaborated upon as a Sheaf theoretc duality. This article is not purely expository; far from it. It contains new results and new approaches to old paradigms. Furthermore, various scattered results in the literature are presented from a holistic perspective highlighting similarities be- tween seemingly remote areas in the literature. For example topoi and category theory are approached as means to unify apparently scattered results in the literature. Mathematics Subject Classification: 05C30, 03B45, 03C0f, 83C05, 81805 44 Tarek Sayed Ahmed 1 Introduction 1.1 Holism vs atomism Holism as a philosophical concept is diametrically opposed to atomism. Where the atomist believes that any whole can be broken down or reduced into sep- arate parts and to the interaction between such local smaller parts, the holist maintains that the whole is primary and more often than not reveals more than the sum of its parts. The atomist divides things up in order to know them more accurately; the holist looks at things or systems in aggregate and argues that we can know more about them and better understand their nature and their purpose if they are viewed in this global, so to speak, manner. The early Greek atomists Leucippus and Democritus (fifth century B.C.) were an- cestors of new physics. There view is a forerunner to the intricate philosophy underlying quantum physics. Everything in the universe consists of indivisible, atoms of various kinds. Such atoms are indistructible; we cannot go further by splitting them into smaller 'subatoms'. The visible change on the macro level is nothing more than merely a re-arrangement of these atoms when viewed on the microlevel. This view was a reaction to the earlier holistic philoso- phy of the Greek philosopher Parmenides, who argued that at some primary level the world is a changeless static unity. Like Plato, change for Parmenides was disturbing and thus illusionary. In his theory of Forms and Ideals, Plato postulates that things around us are nothing but a copy of their perfect im- ages.These images, he claims, are invariant, stable - transcending the world of senses - they lie beyond space and time and can be only visualized through the mind. Plato living in a period of great social disturbance was trying to arrest all forms of change by discovering unchanging rules in a `world of ideals' to a world of flux based on the concept of relativism. He further suggests that his world of ideals is the real world and that what we percieve in our everyday life is actually an illusion, an imperfect copy of this perfect reality. Plato was try- ing to make a projection of the world of senses (which, being only appearance, is deceptive) onto the world of Forms (which he claimed to be the world of things as they are). On the other hand, Paramenidus said \All is One. Nor is it divisible, wherefore it is wholly continuous. It is complete on every side like the mass of a rounded sphere." In the seventeenth century, at the same time that classical physics ushered a renewed boost to atomism and reductionism, Spinoza developed a holistic philosophy, oriented more to ethics and morality, reminiscent of Parmenides. According to Spinoza, all the differences and ap- parent divisions we see in the world are really only aspects of an underlying single substance, which he called God or nature. All things existing are there- fore only ripples on a universal ponde. This ardent emphasis on an underlying unity is reflected in the mystical thinking of most major spiritual traditions. Hegel, too, had mystical visions of the unity of all things, on which he based Algebraic logic and topoi; a philosophical holistic approach 45 his own holistic philosophy of nature and the state. In his holistic philosophy of society he argued using dialectical logic or simply dialectics; progress occurs as a result of the unity of thesis and antithesis, the outcome is their synthesis. Progress is achieved due to, not inspite of, contradictions. The main consensus among dialecticians is that dialectics do not violate the law of contradiction of formal logic, although attempts have been made to create a `paraconsistent logic' which is 'inconsistency tolerant'. Quoting Hegel: `When the difference of reality is taken into account, it develops from difference into opposition, and from this into contradiction, so that in the end the sum total of all realities simply becomes absolute contradiction within itself.' Hegel's state is a quasi- mystical collective, an `invisible and higher reality,' from which participating individuals derive their authentic identity, and to which they owe their loyalty and obedience. Karl Marx `turned Hegel on his head' by turning the idealis- tic dialectic into a materialistic one, in proposing that material circumstances shape ideas, instead of the other way around. In this, Marx was following the lead of Feuerbach. All modern collectivist political thinkers - including, of course, Marx - stress some higher collective reality, the unity, the whole, the group, though nearly always at the cost of minimizing the importance of difference, the part, the individual. Against individualism, all emphasize that the social forces can be viewed as one social entity that somehow possess a holistic character and have a will of their own, not only over and above the characters, but in a way transcending the wills of individual members. Where atomism was apparently legitimized by the stupendous sucesses of quantum physics, holism found no such solace in the basic sciences, like mathematics and physics. It remained a change of emphasis rather than a new philosoph- ical position. There were attempts to found holsim in other basic sciences. For example the idea of organism in biology, the emergence of biological form and the cooperative relation between biological and ecological systems; but these, too, were ultimately reducible to simpler parts, their properties, and the relation between them. Even systems theory, although it highlights the complexity of aggregates, it does so in terms of causal feedback loops between various constituent parts. It is only with quantum theory and the holistic approach to general relativiity, based on a (logistic positivist) axiomatic ap- proach, allowing the maneavering of attack of what once belonged to the 'why' (rather than the 'how') question(s), a new form of deep holism is emerging. In this paper, our prime concern is to give a comprehensive survey of this new holism in dealing with mathematics and physics. The exhilarating work of Andr´eka and N´emetiin axiomatizing new physics, is a new venture into the realm of the axiomatic method applied to relativity theory and quantum mechanics. It raises deep questions of the form, do we need new axioms? Is such an axiom better than another? is this or that axiom a redundency? The central problem in the philosophy of natural science is when and why the sorts of facts scientists cite as evidence really are evidence, a fortoiri the same is 46 Tarek Sayed Ahmed true in the case of mathematics and physics. Historically, philosophers have given considerable attention to the question of when and why various forms of logical inference are truth-preserving and what is the real guide of our search of new axioms is. The word `intution ' is vague enough as it stands, and does nothing but renaming the real problem. For some reason, there has been little attention to the understanding and classification of the sorts of facts mathe- matical scientists choose, let alone to the philosophical question of when and why those facts constitute evidence. The question of how the unproven can be justified is especially pressing in current set theory, where the search goes on for new axioms, that certainly goes far beyond self-evident facts, to determine the size of the continuum one way or another. (This will be further elaborated upon in a while) This is a pressing problem till this moment of time; it is is also perhaps one of the deepest, if not the deepest that contemporary math- ematics presents to the contemporary philosopher of mathematics. Progress towards understanding the process of mathematical hypothesis formation and confirmation contribute to our philosophical understanding of the nature of mathematics. It might even be of solace to those mathematicians actively en- gaged in the axiom search. According to the Andr´eka and N´emetiunfolding project for the last decade, the same can be said out the mind boggling new advances in contemporary physics.