This chapter describes non‐standard philosophy, derived from non‐philosophy, and shows how this discipline and its use can be relevant to the discussion of innovation. The challenge of rebuilding innovation involves developing new processes and new ways of thinking. Multiple challenges, but also opportunities, arise when it comes to (re)thinking innovation with a non‐standard innovation attempt. The first concerns thinking innovation, without which the proposal is neither willing nor able to take root. Non‐standard innovation must be a quasi‐scientific (...) discipline. Because it is not a question of drawing up a new thought that is incapable of proving its interest or reduced to a proposal that does not have practical application. Non‐standard innovation cannot be entrusted solely to the field of organization management. It can only be understood as a form of overcoming, a radicalization or an accomplishment of innovation. (shrink)

Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for (...) anyone interested in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body. (shrink)

In this paper I am concerned with an analysis of negative existential sentences that contain proper names only by using negative or neutral free logic. I will compare different versions of neutral free logic with the standard system of negative free logic (Burge, Sainsbury) and aim to defend my version of neutral free logic that I have labeled non-standard neutral free logic.

We propose a set theory, called NRFST, in which the Cantorian axiom of infinity is negated, and a new notion of infinity is introduced via non standard methods, i. e. via adequate notions of standard and internal, two unary predicates added to the language of ZF. After some initial results on NRFST, we investigate its relative consistency with respect to ZF and Kawai's WNST.

The existence of non-standard numbers in first-order arithmetics is a semantic obstacle for modelling our arithmetical skills. This article argues that so far there is no adequate approach to overcome such a semantic obstacle, because we can also find out, and deal with, non-standard elements in Turing machines.

Much of contemporary philosophy of mind is marked by a dissatisfaction with the two main positions in the field, standard materialism and standard dualism, and hence with the search for alternatives. My concern in this paper is with two such alternatives. The first, which I will call non-standard materialism, is a position I have defended in a number of places, and which may take various forms. The second, panpsychism, has been defended and explored by a number of (...) recent writers. My main goals are: (a) to explain the differences between these positions; and (b) to suggest that non-standard materialism is more plausible than panpsychism. (shrink)

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...) teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

En associant philosophie et logique, cet ouvrage apporte un éclairage sur l'évolution de la logique de l'époque d'Aristote à nos jours. Il présente les différents formalismes des logiques non-standard en expliquant tout d'abord les besoins de ces logiques puis leurs descriptions syntaxiques et sémantiques. A l'aide d'exercices, il offre un panorama de ces logiques qui sont complexes à " utiliser " pour modéliser des connaissances permettant ainsi aux lecteurs d'appréhender ce domaine.

Within a weak subsystem of second‐order arithmetic WKL0, we develop basic part of non‐standard analysis up to the Peano existence theorem.

This research is motivated by the program of reverse mathematics and non-standard arguments in second-order arithmetic. Within a weak subsystem of second-order arithmetic ACA0, we investigate some aspects of non-standard analysis related to sequential compactness. Then, using arguments of non-standard analysis, we show the equivalence of the Riemann mapping theorem and ACA0 over WKL0. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

Non-standard models were introduced by Skolem, first for set theory, then for Peano arithmetic. In the former, Skolem found support for an anti-realist view of absolutely uncountable sets. But in the latter he saw evidence for the impossibility of capturing the intended interpretation by purely deductive methods. In the history of mathematics the concept of a nonstandard model is new. An analysis of some major innovations–the discovery of irrationals, the use of negative and complex numbers, the modern concept of (...) function, and non-Euclidean geometry–reveals them as essentially different from the introduction of non-standard models. Yet, non-Euclidean geometry, which is discussed at some length, is relevant to the present concern; for it raises the issue of intended interpretation. The standard model of natural numbers is the best candidate for an intended interpretation that cannot be captured by a deductive system. Next, I suggest, is the concept of a wellordered set, and then, perhaps, the concept of a constructible set. One may have doubts about a realistic conception of the standard natural numbers, but such doubts cannot gain support from non-standard models. Attempts to utilize non-standard models for an anti-realist position in mathematics, which appeal to meaning-as-use, or to arguments of the kind proposed by Putnam, fail through irrelevance, or lead to incoherence. Robinson’s skepticism, on the other hand, is a coherent position, though one that gives up on providing a detailed philosophical account. The last section enumerates various uses of non-standard models. (shrink)

We give an elementary introduction to non-standard analysis and its applications to the theory of stochastic processes. This is based on a joint book with J. E. Fenstad, R. Høegh-Krohn and T. Lindstrøm. In particular we give a discussion of an hyperfinite theory of Dirichlet forms with applications to the study of the Hamiltonian for a quantum mechanical particle in the potential created by a polymer. We also discuss new results on the existence of attractive polymer measures in dimension (...) d ≤ 5, with applications to the (φ 1 2 φ 2 2 )d-mode1 of interacting quantum fields. (shrink)

I review some theoretical ideas in cosmology different from the standard “Big Bang”: the quasi-steady state model, the plasma cosmology model, non-cosmological redshifts, alternatives to non-baryonic dark matter and/or dark energy, and others. Cosmologists do not usually work within the framework of alternative cosmologies because they feel that these are not at present as competitive as the standard model. Certainly, they are not so developed, and they are not so developed because cosmologists do not work on them. It (...) is a vicious circle. The fact that most cosmologists do not pay them any attention and only dedicate their research time to the standard model is to a great extent due to a sociological phenomenon . We might well wonder whether cosmology, our knowledge of the Universe as a whole, is a science like other fields of physics or a predominant ideology. (shrink)

Gideon Rosen has recently sketched an argument which aims to establish that the notion of metaphysical modality is systematically ambiguous. His argument contains a crucial sub-argument which has been used to argue for Metaphysical Contingentism, the view that some claims of fundamental metaphysics are metaphysically contingent rather than necessary. In this paper, Rosen’s argument is explicated in detail and it is argued that the most straight-forward reconstruction fails to support its intended conclusion. Two possible ways to save the argument are (...) rebutted and it is furthermore argued that the crucial sub-argument only supports a rather particular variant of Metaphysical Contingentism. (shrink)

Forthcoming in S. Artemov and R. Parikh, Proceedings of the ESSLLI 2006 Workshop on Rationality and Knowledge.

In this paper, we introduce the systems ns-ACA₀ and ns-WKL₀ of non-standard second-order arithmetic in which we can formalize non-standard arguments in ACA₀ and WKL₀, respectively. Then, we give direct transformations from non-standard proofs in ns-ACA₀ or ns-WKL₀ into proofs in ACA₀ or WKL₀.

Summary The historical development of the non-standard analysis is sketched. With the help of this mathematical branch infinite and infinitesimal quantities are placed in an extension of the real numbers and so find their justification. In this way an old mathematical and philosophical problem is solved in the 20th century, but not in such a manner, mathematicians with classical „standard methods thought of.

In this paper we study a new operator of strong modality ⊞, related to the non-contingency operator ∆. We then provide soundness and completeness theorems for the minimal logic of the ⊞-operator.

In his doctor's thesis [1], Henkin has shown that if a formal logic is consistent, and sufficiently complex, then it must admit a non-standard model. In particular, he showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of ≦.For (...) the purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either: The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model. That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation ≦. That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation ≦. (shrink)

For cognitivist accounts of aesthetic appreciation, appreciation requires an agent (1) to perceptually respond to the relevant aesthetic features of an object o on good evidential grounds, (2) to have an autonomous grasp of the reasons that make the claim about the aesthetic features of o true by pointing out the connection between non-aesthetic features and the aesthetic features of o, (3) to be able to provide an explanation of why those features contribute to the overall aesthetic value of o. (...) In this framework, aesthetic emotions have traditionally been confined to the level of aesthetic perception (1) and dismissed from the process of reason-giving (2, 3). I argue that this dismissal is due, firstly, to a questionable perceptual reading of the connection between emotional experience and value, and, secondly, to a narrow focus on the basic emotions. My argument will reveal that the non-standard or ‘intellectual’ emotions, the emotions which are in fact most important to appreciation, can play a significant epistemic role in our appreciative practices. They can do this because they (a) help us to deliberately focus our attention and (b) place the appreciator in a state of second-order awareness of their mental states. I conclude the paper by showing how these two epistemic tools (a, b) can help the appreciator to meet the explanatory/justificatory conditions (2) and (3). (shrink)

In ${\mathbf{H}}$ , a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of ${\mathbf{H}}$ , and we prove an analogy of Hájek’s theorem with a very simple procedure.

Few views are as widely held as the Standard View of Identity. Here I am concerned with minority views that depart from the standard account. First, I attempt to illuminate such views and the debates concerning them by identifying the principles of identity at issue, articulating some of the assumptions underlying the debates, and presenting some of the evidence used against the Standard View of Identity. Second, I enter two of these debates myself. I first defend two (...) Non-Standard Views of Identity from the charge that they violate a principle of identity, namely the Transitivity of Identity. I then present an overlooked consequence of another Non-Standard View of Identity that challenges the view on one of its own methodological principles. Third, I draw on recent work in ontological and parthood pluralism to show how one might be led to think that there is more than one way of being identical. That is, I show how one might be an identity pluralist. (shrink)

L'auteur consacre sa réflexion à l'amplification et à l'achèvement de la non-philosophie, en combinant science et philosophie qui sont considérées comme des variables définissant un espace ondulatoire et particulaire de l'opération de penser.

In this paper, we show within RCA 0 that weak Konig's lemma is necessary and sufficient to prove that any (separable) compact group has a Haar measure. Within WKL 0 , a Haar measure is constructed by a non-standard method based on a fact that every countable non-standard model of WKL 0 has a proper initial part isomorphic to itself [10].

In his Mathesis rationis Leibniz discounted out of hand four categorical propositions that would have considerably broadened the resultant syllogistic logic. He did this despite the facts both that he had devised a suitable manner for expressing the latent quantification over terms, and that he had reasoned adequately to determine which of the syllogisms in the resulting broadened logic were valid. Leibniz's reasons for discounting these non-standard propositions are shown to be inadequate, and the resultant syllogistic logic is outlined.

"Stains" can serve as a metaphor for the role allotted to meaninglessness not only by partisans of the deterritorializing force of "brute matter", but also by diagnosers of symbolic incompleteness. For both, the blindspot that will lead to the disturbance of a given regime of meaning must be determined through a smear or glitch which that regime cannot sublate: the mark of a Real stripped of systematising mediation. However, we argue that it is all too easy to allow the stringency (...) of this Real to be undermined by the inflation in its name of merely contingent empirical instances. Such blockages to theoretical and artistic practice can be removed with the aid of the articulation of incompleteness and inconsistency implied by François Laruelle's conception of the Real as non-consistent but hypercomplete "radical immanence". À rebours of Laruelle himself, different types of meaninglessness can then be distinguished, de-metaphorized, and conceptualized as "noise". (shrink)

The kilogram prototype is an object manufactured in late nineteenth century to serve as the standard unit of mass worldwide. Since then, the kilogram definition has been determined by this unique object, making it a peculiar case in metrological studies. However, after realizing that the prototype’s mass was unexpectedly changing, scientists are seeking a new way to define the kilogram in pure mathematical terms.

We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a lattice. We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invariant of the group at hand. The analysis first goes through a local level, where a pertinent notion of pregeometry and (...) generic elements is each time introduced. (shrink)

In the context of life sciences, we are constantly confronted with information that possesses precise semantic values and appears essentially immersed in a specific evolutionary trend. In such a framework, Nature appears, in Monod’s words, as a tinkerer characterized by the presence of precise principles of self-organization. However, while Monod was obliged to incorporate his brilliant intuitions into the framework of first-order cybernetics and a theory of information with an exclusively syntactic character such as that defined by Shannon, research advances (...) in recent decades have led not only to the definition of a second-order cybernetics but also to an exploration of the boundaries of semantic information. As H. Atlan states, on a biological level "the function self-organizes together with its meaning". Hence the need to refer to a conceptual theory of complexity and to a theory of self-organization characterized in an intentional sense. There is also a need to introduce, at the genetic level, a distinction between coder and ruler as well as the opportunity to define a real software space for natural evolution. The recourse to non-standard model theory, the opening to a new general semantics, and the innovative definition of the relationship between coder and ruler can be considered, today, among the most powerful theoretical tools at our disposal in order to correctly define the contours of that new conceptual revolution increasingly referred to as metabiology. This book focuses on identifying and investigating the role played by these particular theoretical tools in the development of this new scientific paradigm. Nature "speaks" by means of mathematical forms: we can observe these forms, but they are, at the same time, inside us as they populate our organs of cognition. In this context, the volume highlights how metabiology appears primarily to refer to the growth itself of our instruments of participatory knowledge of the world. (shrink)

Internal sets and the Boolean algebras of the collection of the internal sets are of central importance in non-standard analysis. Boolean algebras are the algebraization of propositional logic while the logic applied in non-standard analysis (in non-standard stochastics) is the first order or the higher order logic (type theory). We present here a first order logic algebraization for the collection of internal sets rather than the Boolean one. Further, we define an unusual probability on this algebraization.

In this short paper, I outline a non-standard account of what it feels like to have an intuition. According to this account, intuitive contents are ontologically ambiguous. Because intuition alone is liable to persuade us of both motivated inferences and necessary truths, it is not a reliable source of evidence. However, we would not be able to grasp the concept of necessity without intuitions. Hence, I do not think it is any good to ignore or quarantine our intuitions when (...) forming judgments. (shrink)

In this paper the ultrafilter properties of canonical frames are used to produce a non-standard map between canonical frames of different cardinalities. While this map is not a p-morphism, it is presented as a step towards the full understanding of canonical structures.

In response to Bhupinder Singh Anand''s article CAN WE REALLY FALSIFY TRUTH BY DICTAT? in THE REASONER II, 1, January 2008,that denies the existence of nonstandard models of Peano Arithmetic, we prove from Compactness the existence of such models.

Twenty years after cultivating a new orientation for aesthetics via the concept of non-photography, François Laruelle returns, having further developed his notion of a non-standard aesthetics. Published for the first time in a bilingual edition, _Photo-Fiction, a Non-Standard Aesthetics_ expounds on Laruelle’s current explorations into a photographic thinking as an alternative to the worn-out notions of aesthetics based on an assumed domination of philosophy over art. He proposes a new philosophical photo-fictional apparatus, or philo-fiction, that strives for a (...) discursive mimesis of the photographic apparatus and the flash of the Real entailed in its process of image making. “A bit like if an artisan, to use a Socratic example, instead of making a camera based off of diagrams found in manuals, on the contrary had as his or her project the designing of a completely new apparatus of philo-fiction, thus capable of producing not simply photos, but photo-fictions.” One must enter into a space for seeing the vectorial and the imaginary number. Laruelle’s philo-fictions become not art installations, but “theoretical installations” calling for the consideration of the possibility of a non-standard aesthetics being of an equal or superior power to art and philosophy, an aesthetics in-the-last-instance that is itself an inventive and creative act of the most contemporary kind. (shrink)

Although there are a few books available that give brief surveys of a variety of nonstandard logics, there is a growing need for a critical presentation providing both a greater depth and breadth of insight into these logics. This book assembles a wider and deeper view of the many potentially applicable logics. Three appendixes provide short tutorials on classical logic and modal logics, and give a brief introduction to the existing literature on the logical aspects of probability theory. These tutorials (...) and the bibliography included in the appendixes provide useful reference material for the reader. (shrink)

It has been known for years that it is only the round-trip speed of light that has physical significance. We show that even using this freedom, the synchronization gap, which is useful for testing predictions of general relativity, still remains.