Related papers: Permanence as a Principle of Practice
The primary aim of Hilbert's proof theory was to establish the consistency of classical mathematics using finitary means only. Hilbert's strategy for doing this was to eliminate the infinite (in the form of unbounded quantifiers) from…
Tarski gave a general semantics for deductive reasoning: a formula a may be deduced from a set A of formulas iff a holds in all models in which each of the elements of A holds. A more liberal semantics has been considered: a formula a may…
We introduce a framework uniting algorithmic randomness with exchangeable credences to address foundational questions in philosophy of probability and philosophy of science. To demonstrate its power, we show how one might use the framework…
The aim of this article is twofold. First, we shall review and analyse the Neo-Kantian justification for the application of probabilistic concepts in physics that was defended by Hans Reichenbach early in his career, notably in his…
The author advocates two specific mathematical notations from his popular course and joint textbook, "Concrete Mathematics". The first of these, extending an idea of Iverson, is the notation "[P]" for the function which is 1 when the…
Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume's Principle is analytic then in the standard setting the answer appears to be yes. Hodes's work pointed to a way out by…
In his 1879 paper on the Begriffsschrift, Gottlob Frege introduced a notation to formalize mathematical arguments. In this note we explain Frege's notation by using the nowadays common notions from elementary propositional logic. We compare…
This article explores the following methodological principle for theory construction in physics: if an ontological theory predicts two scenarios that are ontologically distinct but empirically indiscernible, then this theory should be…
We give a concise introduction to the theory of continuants and show how Perron used them in his proof of Tietze theorem on the convergence of infinite semi-regular continued fractions, as well as for the study of the convergence of purely…
The social and economic importance of large bodies of programs and data that are potentially long-lived has attracted much attention in the commercial and research communities. Here we concentrate on a set of methodologies and technologies…
According to Karl Popper assumptions are statements used to construct theories. During the construction of a theory whether the assumptions are either true or false turn out to be irrelevant in view of the fact that, actually, they gain…
The notion of slow provability for Peano Arithmetic ($\mathsf{PA}$) was introduced by S.D. Friedman, M. Rathjen, and A. Weiermann. They studied the slow consistency statement $\mathrm{Con}_{\mathsf{s}}$ that asserts that a contradiction is…
Philosopher Benardete challenged both the conventional wisdom and the received mathematical treatment of zero, dot, nine recurring. An initially puzzling passage in Benardete on the intelligibility of the continuum reveals challenging…
In a comparison of the principles of special relativity and of quantum mechanics, the former theory is marked by its relative economy and apparent explanatory simplicity. A number of theorists have thus been led to search for a small number…
This is the first of five papers comprising The Semantic Arrow of Time. The argument begins with a claim: computing's arrow of time is semantic, not thermodynamic. The direction in which meaning is preserved or destroyed across transactions…
This article was motivated by the discovery of a potential new foundation for mainstream mathematics. The goals are to clarify the relationships between primitives, foundations, and deductive practice; to understand how to determine what…
Classical theory proves that every primitive recursive function is strongly representable in PA; that formal Peano Arithmetic, PA, and formal primitive recursive arithmetic, PRA, can both be interpreted in Zermelo-Fraenkel Set Theory, ZF;…
The strong continuity principle reads "every pointwise continuous function from a complete separable metric space to a metric space is uniformly continuous near each compact image." We show that this principle is equivalent to the fan…
Some recent philosophical debate about persistence has focussed on an argument against perdurantism that discusses rotating perfectly homogeneous discs (the `rotating discs argument'; RDA). The argument has been mostly discussed by…
It was shown by Visser that Peano Arithmetic has the property that any two bi-interpretable extensions of it (in the same language) are equivalent. Enayat proposed to refer to this property of a theory as tightness and to carry out a more…