Notes prises par Jacques Lagrange lors du cours d'anthropologie donné par Michel Foucault en 1954-1955 à l'École Normale Supérieure.

While many philosophers of religion are familiar with the reconciliation of grace and freedom known as Molinism, fewer by far are familiar with that position initially developed by Molina’s erstwhile rival, Domingo Banez (i.e., Banezianism). My aim is to clarify a serious problem for the Banezian: how the Banezian can avoid the apparent conflict between a strong notion of freedom and apparently compatibilist conclusions. The most prominent attempt to defend Banezianism against compatibilism was (in)famously endorsed by Reginald Garrigou-Lagrange. (...) Even if it were true that freedom does not require alternative possibilities, Banezians have a grounding problem. (shrink)

My paper analyses the analogy between Computers and the Thomistic separate substances, and argues that Aquinas' account of angels as cognitively intuitive and non-discursive makes the analogical gap between these impossible to bridge. From there, I point the direction away from computers as the way for us to move up the order of cognitive excellence. Instead, the gifts of the Holy Spirit are the way to go, since by them we participate in this intuitivity. I then lay out the ascetical (...) presuppositions for the successful participation of this gifts, in particular the necessity for the passive purgations, according to the division of the ascetical life into three stages by Garrigou-Lagrange OP. (shrink)

I remain unsatisfied by a lack of philosophical clarity among Báñezian authors on the nature of freedom. In a recent paper, I therefore posed a problem for Báñezianism that resembles what is called the “grounding problem” for Molinism: where do the truths about alternative possibilities come from? And I illustrated the problem in the context of the account of grace given by one famous defender of the view, Fr. Réginald Garrigou-Lagrange, whose work in turn was recently promoted by (...) Taylor Patrick O’Neill. O’Neill responds in this present journal issue to that paper, arguing that I failed to be attentive to distinctions drawn by Garrigou-Lagrange which would resolve the dilemma. However, I fear that O’Neill’s presentation of both the dilemma and his own resolution is conceptually confused. I will try to illustrate that O’Neill’s own responses are at many points either a concession to a certain fork of the dilemma that I posed in the original paper, or change the subject and fail to address my concerns except superficially. However, since at many points O’Neill implies that I reject Báñezianism, despite my explicit qualification in the paper that I do not, I will end by proposing what would constitute a sufficient response to the dilemma. (shrink)

Operations research science is defined as the science that is concerned with applying scientific methods to complex problems in managing and directing large systems of people, including resources and tools in various fields, private and governmental work, peace and war, politics, administration, economics, planning and implementation in various domains. It uses scientific methods that take the language of mathematics as a basis for it and uses computer, without which it would not have been possible to achieve numerical solutions to the (...) raised problems, those that need correct solutions, when the solutions abound and the options are multiple, so we need a decision based on correct scientific foundations and takes into account all the circumstances and changes that you can encounter the decision-maker during the course of work, and nothing is left to chance or luck, but rather everything that enters into the account and plays its role in decision-making, and we get that when we use the concepts of neutrosophic science to reformulate what the science of operations research presented in terms of methods and methods to solve many practical problems, so we will present in this research a study aimed at shedding light on the most important methods used to solve nonlinear models, which is the Lagrangian multiplier method for nonlinear models constrained by equality and then reformulated using the concepts of neutrosophic science. (shrink)

A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The approach developed has a pronounced applied character and is based on the principle “The part is less than the whole” introduced by the ancient Greeks. This principle is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The point of view on infinities and infinitesimals (and in general, on Mathematics) presented in this paper (...) uses strongly physical ideas emphasizing interrelations that hold between a mathematical object under observation and the tools used for this observation. It is shown how a new numeral system allowing one to express different infinite and infinitesimal quantities in a unique framework can be used for theoretical and computational purposes. Numerous examples dealing with infinite sets, divergent series, limits, and probability theory are given. (shrink)

The concept of quantum information is introduced as both normed superposition of two orthogonal sub-spaces of the separable complex Hilbert space and in-variance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen. The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function (...) describing a state of a quantum system) as its value as the bound variable. A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is re-presentable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above. Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and in-variance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper. Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model. (shrink)

En su libro Mind and nature Gregory Bateson elabora un argumento que él llama ‘convencional’ para concluir que “nunca podremos ser capaces de reclamar conocimiento definitivo en asunto alguno [final knowledge of anything whatsoever]”. La conclusión es correcta, pero el argumento es engañoso y desorienta más de lo que ayuda. Además, Bateson, disimuladamente, introduce algunas afirmaciones falsas en medio de premisas verdaderas. Revisarlo nos permitirá aclarar algunos malentendidos comunes en filosofía de la ciencia, además de ser la excusa perfecta para (...) esbozar una introducción a la interpolación polinomial de Lagrange y una pequeña reflexión sobre los límites del conocimiento. (shrink)

This paper champions the view (REG) that the concept of a normative reason for an agent S to perform an action A is that of an explanation why it would be good (in some way, to some degree) for S to do A. REG has numerous virtues, but faces some significant challenges which prompt many philosophers to be skeptical that it can correctly account for all our reasons. I demonstrate how five different puzzles about normative reasons can be solved by (...) attention to the concept of goodness, and in particular observing the ways in which it—and consequently, talk about reasons—is sensitive to context (ends and information). Rather than asking simply whether or not certain facts are reasons for S to do A, we need to explore the contexts in which it is and is not correct to describe a certain fact as “a reason” for S to do A. These five puzzles concern: (1) reasons for attitudes of the “right kind”, (2) evidence as reasons, (3) normative facts as reasons, (4) subjective reasons, and (5) attitudes as reasons. (shrink)

Diagrams by Connor Camburn -/- The relationship between axiomatization, mechanization, creative individuation, and virtual/physical individuation presents a fascinating interplay of concepts that have significantly influenced various fields, including mathematics, physics, philosophy, and art. This essay explores these relationships by drawing insights from André Weil's "From Metaphysics to Mathematics," Gilles Châtelet's works, and Schmid's discussion on Gnostic Futurism. -/- Axiomatization: Weil and Grothendieck -/- Axiomatization, as discussed in André Weil's "From Metaphysics to Mathematics," represents the transformation of metaphysical concepts into formal (...) mathematical structures. Weil's document highlights the progression from vague metaphysical ideas to concrete mathematical theories. This process is evident in the works of mathematicians like Lagrange and Galois, where initial metaphysical notions eventually crystallized into the formal structures of modern algebra. Similarly, Alexander Grothendieck's work in algebraic geometry involved the creation of an abstract framework that could unify various mathematical concepts, demonstrating the power of axiomatization in providing a foundational structure for complex ideas. -/- Mechanization: Oppenheimer, Watson-Crick, and Von Neumann -/- Mechanization, in the context of scientific development, refers to the application of mechanical principles to solve problems and the automation of processes. Figures like Oppenheimer in physics, Watson and Crick in biology, and Von Neumann in computing and mathematics, represent the pinnacle of mechanization in their respective fields. Their work exemplifies how mechanization has enabled the simplification and automation of complex processes, leading to alienating discoveries such as the atomic bomb, the structure of DNA, and the development of modern computers. -/- Creative Individuation: Gnostic Futurism -/- Eric Schmid's discussion on Gnostic Futurism, as found in his document, offers a perspective on creative individuation. Gnostic Futurism, rooted in Kantian-Hegelian transcendental universalism, represents a departure from conventional realism, emphasizing a transcendental nature and integration of mythopoeia. This movement in art and philosophy underscores the role of creativity and individual expression in transcending traditional forms and norms. It embodies the idea of creative individuation, where individual creativity leads to the formation of new, often revolutionary, artistic and philosophical expressions. -/- Virtual/Physical Individuation: Châtelet's Perspective -/- Gilles Châtelet's works, particularly "Figuring Space: Philosophy, Mathematics, and Physics," delve into the concept of virtual/physical individuation. Châtelet discusses the configuration of space in mathematics and physics, emphasizing the role of diagrams in scientific discovery. This exploration of space, both virtual (as in mathematical abstractions) and physical (as in tangible representations), reflects the process of individuation in both realms. Châtelet's perspective highlights how mathematical and physical representations are not just tools for understanding the world but are also expressions of individual and collective intellectual endeavors. -/- Interconnections and Synthesis -/- The synthesis of these concepts reveals a complex tapestry of intellectual development across disciplines. Axiomatization provides a foundational structure that enables the mechanization of processes, as seen in the works of Oppenheimer, Watson-Crick, and Von Neumann. This mechanization, in turn, paves the way for creative individuation, where individuals can transcend established norms and frameworks, as exemplified in Gnostic Futurism. Finally, the concept of virtual/physical individuation, as explored by Châtelet, represents the culmination of these ideas, where the abstract and the tangible intersect, leading to new forms of understanding and expression. -/- In conclusion, the relationship between axiomatization, mechanization, creative individuation, and virtual/physical individuation is a testament to the dynamic and interconnected nature of intellectual progress. From the formal structures of mathematics to the revolutionary changes in art and philosophy, these concepts demonstrate how human thought and creativity continue to evolve, challenging existing paradigms and forging new paths of understanding and expression. (shrink)

Experiments with remote perception and Random Event Generators (REG) performed over the last decades show small but significant anomalous effects. Since these effects seem to be independent of spatial and temporal distance, they appear to be in disagreement with the standard scientific worldview. A very simple explanation of quantum mechanics is pre- sented, rejecting all unjustified claims about the world. A view of mind in agreement with cognitive neuroscience is introduced. It is argued that mind and consciousness are emer- gent (...) properties of the brain and are understandable without any nonphysical assumptions. A plausible explanation of the results of anomalous experiments, based on the concept of synchronicity, introduced by C.G. Jung and advocated by W. Pauli, is offered. A proof is given that strong correlations should exist between any systems that once interacted. Synchronicity events between parts of the brain and physical objects may be sufficient to explain the results of anomalous experiments. Standard physics is sufficient to understand these phenomena. Motto. (shrink)

Without doubt, the principle of least action is a fundamental principle in classical mechanics. Contemporary physicists, however, consider the PLA as a purely mathematical principle – even an axiom which they cannot completely justify. Such an account stands in sharp contrast with the historical meaning of the PLA. When the principle was introduced in the 1740s, by Pierre-Louis Moreau de Maupertuis, its meaning was much more versatile. For Maupertuis the principle of least action signified that nature is thrifty or economical (...) in all its actions, i.e., that nature avoids to do anything unnecessary. Maupertuis understood the principle in teleological terms and even considered the principle as an expression of God’s wisdom. It has been correctly pointed out by historians that Maupertuis in his later years moved towards a more speculative and metaphysical approach, whereas his contemporary Euler and later Lagrange, wanted to avoid such theological and metaphysical implications and frame the PLA in purely mathematical terms. Such readings, however, have had the unintended side-effect that they lose out of sight the question how the mathematical and metaphysical aspect of the principle of least action fit together within Maupertuis’ own work. Investigating if and how the mathematical and metaphysical aspects of the PLA are compatible within Maupertuis’ thought will be the main goal of this paper. (shrink)

Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing machine – (...) Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem. (shrink)

The principle of least action, which has so successfully been applied to diverse fields of physics looks back at three centuries of philosophical and mathematical discussions and controversies. They could not explain why nature is applying the principle and why scalar energy quantities succeed in describing dynamic motion. When the least action integral is subdivided into infinitesimal small sections each one has to maintain the ability to minimise. This however has the mathematical consequence that the Lagrange function at a (...) given point of the trajectory, the dynamic, available energy generating motion, must itself have a fundamental property to minimize. Since a scalar quantity, a pure number, cannot do that, energy must fundamentally be dynamic and time oriented for a consistent understanding. It must have vectorial properties in aiming at a decrease of free energy per state (which would also allow derivation of the second law of thermodynamics). Present physics is ignoring that and applying variation calculus as a formal mathematical tool to impose a minimisation of scalar assumed energy quantities for obtaining dynamic motion. When, however, the dynamic property of energy is taken seriously it is fundamental and has also to be applied to quantum processes. A consequence is that particle and wave are not equivalent, but the wave (distributed energy) follows from the first (concentrated energy). Information, provided from the beginning, an information self-image of matter, is additionally needed to recreate the particle from the wave, shaping a “dynamic” particle-wave duality. It is shown that this new concept of a “dynamic” quantum state rationally explains quantization, the double slit experiment and quantum correlation, which has not been possible before. Some more general considerations on the link between quantum processes, gravitation and cosmological phenomena are also advanced. (shrink)