110 research outputs found
On the Failure of Fixed-Point Theorems for Chain-complete Lattices in the Effective Topos
In the effective topos there exists a chain-complete distributive lattice
with a monotone and progressive endomap which does not have a fixed point.
Consequently, the Bourbaki-Witt theorem and Tarski's fixed-point theorem for
chain-complete lattices do not have constructive (topos-valid) proofs
On the Bourbaki-Witt Principle in Toposes
The Bourbaki-Witt principle states that any progressive map on a
chain-complete poset has a fixed point above every point. It is provable
classically, but not intuitionistically.
We study this and related principles in an intuitionistic setting. Among
other things, we show that Bourbaki-Witt fails exactly when the trichotomous
ordinals form a set, but does not imply that fixed points can always be found
by transfinite iteration. Meanwhile, on the side of models, we see that the
principle fails in realisability toposes, and does not hold in the free topos,
but does hold in all cocomplete toposes
On the Bourbaki-Witt Principle in Toposes
Abstract The Bourbaki-Witt principle states that any progressive map on a chain-complete poset has a fixed point above every point. It is provable classically, but not intuitionistically. We study this and related principles in an intuitionistic setting. Among other things, we show that Bourbaki-Witt fails exactly when the trichotomous ordinals form a set, but does not imply that fixed points can always be found by transfinite iteration. Meanwhile, on the side of models, we see that the principle fails in realisability toposes, and does not hold in the free topos, but does hold in all cocomplete toposes
FICS 2010
International audienceInformal proceedings of the 7th workshop on Fixed Points in Computer Science (FICS 2010), held in Brno, 21-22 August 201
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Metascientific aspects of topoi of spaces
This thesis presents a study of the importance of topoi for Science. It is argued that whenever the concept of space enters the practice of Science then formal (mathematical) theories should be interpreted in a topos of spaces. It is claimed that these topoi encode knowledge of space arising directly out of the needs of Science, in that the constitutive questions of the Sciences can be traced back to their leading knowledge interests and these determine the role of mathematics as a methodical device. In the Natural Sciences the constitutive questions involve the study of non-intentional objects, in terms of a causal nexus to be explained geometrically, and this facilitates the introduction of geometric objects as the methodical device for posing questions to Nature. Although the study of intentional subjects in the Human Sciences requires ordinary language, not mathematics, to pose questions to each other, secondary methodological objectifications permit a conception of geometric objects analogous to that of the Natural Sciences. Lawvere*s axioms for the gros and petit topoi illustrate attempts to formalise the idea of topoi of spaces, as a rational reconstruction of categories in which geometric objects satisfying the formal theories of Science can be found. The catalysing function of this knowledge of topoi of spaces can lead to a diagnosis of mathematical difficulties caused by a failure to align mathematical conceptions with these topoi. This is illustrated through Varela's use of self-reference in Biology and Atkin's use of algebraic topology in Social Studies
Topos and Stacks of Deep Neural Networks
Every known artificial deep neural network (DNN) corresponds to an object in
a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of
morphisms in this topos. Invariance structures in the layers (like CNNs or
LSTMs) correspond to Giraud's stacks. This invariance is supposed to be
responsible of the generalization property, that is extrapolation from learning
data under constraints. The fibers represent pre-semantic categories (Culioli,
Thom), over which artificial languages are defined, with internal logics,
intuitionist, classical or linear (Girard). Semantic functioning of a network
is its ability to express theories in such a language for answering questions
in output about input data. Quantities and spaces of semantic information are
defined by analogy with the homological interpretation of Shannon's entropy
(P.Baudot and D.B. 2015). They generalize the measures found by Carnap and
Bar-Hillel (1952). Amazingly, the above semantical structures are classified by
geometric fibrant objects in a closed model category of Quillen, then they give
rise to homotopical invariants of DNNs and of their semantic functioning.
Intentional type theories (Martin-Loef) organize these objects and fibrations
between them. Information contents and exchanges are analyzed by Grothendieck's
derivators
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
Foundations of Quantum Theory: From Classical Concepts to Operator Algebras
Quantum physics; Mathematical physics; Matrix theory; Algebr
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