61,637 research outputs found
Classical Mathematics for a Constructive World
Interactive theorem provers based on dependent type theory have the
flexibility to support both constructive and classical reasoning. Constructive
reasoning is supported natively by dependent type theory and classical
reasoning is typically supported by adding additional non-constructive axioms.
However, there is another perspective that views constructive logic as an
extension of classical logic. This paper will illustrate how classical
reasoning can be supported in a practical manner inside dependent type theory
without additional axioms. We will see several examples of how classical
results can be applied to constructive mathematics. Finally, we will see how to
extend this perspective from logic to mathematics by representing classical
function spaces using a weak value monad.Comment: v2: Final copy for publicatio
Agent-Based Modeling: The Right Mathematics for the Social Sciences?
This study provides a basic introduction to agent-based modeling (ABM) as a powerful blend of classical and constructive mathematics, with a primary focus on its applicability for social science research.ďż˝ The typical goals of ABM social science researchers are discussed along with the culture-dish nature of their computer experiments. The applicability of ABM for science more generally is also considered, with special attention to physics. Finally, two distinct types of ABM applications are summarized in order to illustrate concretely the duality of ABM: Real-world systems can not only be simulated with verisimilitude using ABM; they can also be efficiently and robustly designed and constructed on the basis of ABM principles. ďż˝
Quantum Immortality and Non-Classical Logic
The Everett Box is a device in which an observer and a lethal quantum
apparatus are isolated from the rest of the universe. On a regular basis,
successive trials occur, in each of which an automatic measurement of a quantum
superposition inside the apparatus either causes instant death or does nothing
to the observer. From the observer's perspective, the chances of surviving
trials monotonically decreases with increasing . As a result, if the
observer is still alive for sufficiently large she must reject any
interpretation of quantum mechanics which is not the many-worlds interpretation
(MWI), since surviving trials becomes vanishingly unlikely in a single
world, whereas a version of the observer will necessarily survive in the
branching MWI universe. Here we ask whether this conclusion still holds if
rather than a classical understanding of limits built on classical logic we
instead require our physics to satisfy a computability requirement by
investigating the Everett Box in a model of a computational universe running on
a variety of constructive logic, Recursive Constructive Mathematics. We show
that although the standard Everett argument rejecting non-MWI interpretations
is no longer valid, we can show that Everett's conclusion still holds within a
computable universe. Thus we argue that Everett's argument is strengthened and
any counter-argument must be strengthened, since it holds not only in classical
logic (with embedded notions of continuity and infinity) but also in a
computable logic.Comment: 12 page
An Objection to Naturalism and Atheism from Logic
I proffer a success argument for classical logical consequence. I articulate in what sense that notion of consequence should be regarded as the privileged notion for metaphysical inquiry aimed at uncovering the fundamental nature of the world. Classical logic breeds necessitism. I use necessitism to produce problems for both ontological naturalism and atheism
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
On algorithm and robustness in a non-standard sense
In this paper, we investigate the invariance properties, i.e. robust- ness, of phenomena related to the notions of algorithm, finite procedure and explicit construction. First of all, we provide two examples of objects for which small changes completely change their (non)computational behavior. We then isolate robust phenomena in two disciplines related to computability
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