Nietzsche’s Philosophy of Mathematics

Nietzsche has a surprisingly significant and strikingly positive assessment of mathematics. I discuss Nietzsche's theory of the origin of mathematical practice in the division of the continuum of force, his theory of numbers, his conception of the finite and the infinite, and the relations between Nietzschean mathematics and formalism and intuitionism. I talk about the relations between math, illusion, life, and the will to truth. I distinguish life and world affirming mathematical practice from its ascetic perversion. For Nietzsche, math is an artistic and moral activity that has an essential role to play in the joyful wisdom

Infinity

This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks

`Iconoclastic', Categorical Quantum Gravity

This is a two-part, `2-in-1' paper. In Part I, the introductory talk at `Glafka--2004: Iconoclastic Approaches to Quantum Gravity' international theoretical physics conference is presented in paper form (without references). In Part II, the more technical talk, originally titled ``Abstract Differential Geometric Excursion to Classical and Quantum Gravity'', is presented in paper form (with citations). The two parts are closely entwined, as Part I makes general motivating remarks for Part II.Comment: 34 pages, in paper form 2 talks given at ``Glafka--2004: Iconoclastic Approaches to Quantum Gravity'' international theoretical physics conference, Athens, Greece (summer 2004

Introduction to the Ontology of Knowledge iss. 20211125

We can only know what determines us as being and by the fact that it determines us as being. Our knowledge is therefore logically limited to what determines us as being. Since representation is defined as the act that makes knowledge dicible, our representation is logically limited to what dynamically determines us as being. Our representation is included in our becoming. Nothing that we represent, no infinite, can exceed the mere necessity of our becoming. The world, my physical being and my consciousness are subsumed by the necessity of my becoming. We know nothing but “we become”  To the question "Is there anything else to know?" we can give no logical answer Summary: Reality is pure logical interdependence, immanent, formless, unspeakable. Logos is a principle of order in this interdependence. Individuation is the necessary asymptote of any instance of the Logos. Each knowing subject is Individuation, a mode of order among infinites of infinites of possible modes of order. Everything that appears to the subject as Existing participates in his Individuation. This convergence into Individuation defines a perspective that gives meaning. The subject is representation. It is in this representation that exist the subject, objects and laws of the world. Without subject there are no objects, no laws, no framework. The representation is not isomorphism but morphogenesis. The physical world and the Spirit have the same logical nature: they are categories of representation. The representation is animated because meaning is an Act. Representation is limited by a horizon of meaning. Below this horizon the subject represents the universe and itself. Beyond this horizon there is no prevailing space, time or form. The predicate expresses, below the horizon of meaning, a necessity whose source is beyond this horizon, unfathomable. The OK is neither materialism nor idealism and frees itself from any psychological preconceptions. The OK does not propose an "other reality" than that described by common sense or science, but another mode of representation. The OK is compatible with the current state of science, while offering new interpretive avenues. The OK differs from ontic structural realism (OSR) in various ways: Just like being, the relationship is representation, The knowing subject is present in any representation, the real is non-founded

Newton vs. Leibniz: Intransparency vs. Inconsistency

We investigate the structure common to causal theories that attempt to explain a (part of) the world. Causality implies conservation of identity, itself a far from simple notion. It imposes strong demands on the universalizing power of the theories concerned. These demands are often met by the introduction of a metalevel which encompasses the notions of 'system' and 'lawful behaviour'. In classical mechanics, the division between universal and particular leaves its traces in the separate treatment of cinematics and dynamics. This analysis is applied to the mechanical theories of Newton and Leibniz, with some surprising results

Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity

Previous work on applications of Abstract Differential Geometry (ADG) to discrete Lorentzian quantum gravity is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantum causal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure. This topos is seen to be a finitary instance of both an elementary and a Grothendieck topos, generalizing in a differential geometric setting, as befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies. The paper closes with a thorough discussion of four future routes we could take in order to further develop our topos-theoretic perspective on ADG-gravity along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references polished) Submitted to the International Journal of Theoretical Physic

Linear superposition as a core theorem of quantum empiricism

Clarifying the nature of the quantum state $|\Psi\rangle$ is at the root of the problems with insight into (counterintuitive) quantum postulates. We provide a direct-and math-axiom free-empirical derivation of this object as an element of a vector space. Establishing the linearity of this structure-quantum superposition-is based on a set-theoretic creation of ensemble formations and invokes the following three principia: $(\textsf{I})$ quantum statics, $(\textsf{II})$ doctrine of a number in the physical theory, and $(\textsf{III})$ mathematization of matching the two observations with each other; quantum invariance. All of the constructs rest upon a formalization of the minimal experimental entity: observed micro-event, detector click. This is sufficient for producing the $\mathbb C$-numbers, axioms of linear vector space (superposition principle), statistical mixtures of states, eigenstates and their spectra, and non-commutativity of observables. No use is required of the concept of time. As a result, the foundations of theory are liberated to a significant extent from the issues associated with physical interpretations, philosophical exegeses, and mathematical reconstruction of the entire quantum edifice.Comment: No figures. 64 pages; 68 pages(+4), overall substantial improvements; 70 pages(+2), further improvement

Quantum Field Theory

I discuss the general principles underlying quantum field theory, and attempt to identify its most profound consequences. The deepest of these consequences result from the infinite number of degrees of freedom invoked to implement locality. I mention a few of its most striking successes, both achieved and prospective. Possible limitations of quantum field theory are viewed in the light of its history.Comment: LaTeX, 12 pages, 3 figures. Will appear in Centenary issue of Rev. of Mod. Phys., March 1999. Incorporated minor corrections suggested by edito

Zeno's Paradoxes. A Cardinal Problem 1. On Zenonian Plurality

In this paper the claim that Zeno's paradoxes have been solved is contested. Although no one has ever touched Zeno without refuting him (Whitehead), it will be our aim to show that, whatever it was that was refuted, it was certainly not Zeno. The paper is organised in two parts. In the first part we will demonstrate that upon direct analysis of the Greek sources, an underlying structure common to both the Paradoxes of Plurality and the Paradoxes of Motion can be exposed. This structure bears on a correct - Zenonian - interpretation of the concept of division through and through. The key feature, generally overlooked but essential to a correct understanding of all his arguments, is that they do not presuppose time. Division takes place simultaneously. This holds true for both PP and PM. In the second part a mathematical representation will be set up that catches this common structure, hence the essence of all Zeno's arguments, however without refuting them. Its central tenet is an aequivalence proof for Zeno's procedure and Cantor's Continuum Hypothesis. Some number theoretic and geometric implications will be shortly discussed. Furthermore, it will be shown how the Received View on the motion-arguments can easely be derived by the introduction of time as a (non-Zenonian) premiss, thus causing their collapse into arguments which can be approached and refuted by Aristotle's limit-like concept of the potentially infinite, which remained - though in different disguises - at the core of the refutational strategies that have been in use up to the present. Finally, an interesting link to Newtonian mechanics via Cremona geometry can be established.Comment: 41 pages, 7 figure