Royce's Model of the Absolute Steinhart, Eric
Transactions of the Charles S. Peirce Society,
06/2012, Letnik:
48, Številka:
3
Journal Article
Recenzirano
At the end of the 19th century, Royce uses the mathematical ideas of his day to describe the Absolute as a self-representative system. Working closely with Royce's texts, I will develop a model of ...the Absolute that is both more thoroughly formalized and that is stated in contemporary mathematical language. As I develop this more formal model, I will show how structures found within it are similar to structures widely discussed in current analytic metaphysics. The model contains structures found in the recent analytic metaphysics of modality; it contains Democritean worlds as defined by Quine; it contains Turing-computable sequences; and it contains networks of interacting software objects as defined by Dennett. Much of the content of recent analytic metaphysics is already implicit in Royce's study of the Absolute. Far from being an obsolete system of historical interest only, Royce's metaphysics is remarkably relevant today.
An axiomatic definition of fuzzy cardinalities for finite fuzzy sets, defined by means of a convex fuzzy set on the natural numbers, is presented in such a way that it includes the fuzzy ...cardinalities defined by authors like Zadeh, Ralescu, Dubois, Wygralak, and characterizes the cardinalities that fulfill the additivity property by means of the extended addition of fuzzy numbers. Such cardinalities result defined by two functions, one nondecreasing and the other nonincreasing in a similar way to the scalar cardinality (Fuzzy Sets and Systems 110 (2000) 175).
Schreier sets in Ramsey theory Farmaki, V.; Negrepontis, S.
Transactions of the American Mathematical Society,
02/2008, Letnik:
360, Številka:
2
Journal Article
Recenzirano
Odprti dostop
We show that Ramsey theory, a domain presently conceived to guarantee the existence of large homogeneous sets for partitions on k-tuples of words (for every natural number k) over a finite alphabet, ...can be extended to one for partitions on Schreier-type sets of words (of every countable ordinal). Indeed, we establish an extension of the partition theorem of Carlson about words and of the (more general) partition theorem of Furstenberg-Katznelson about combinatorial subspaces of the set of words (generated from k-tuples of words for any fixed natural number k) into a partition theorem about combinatorial subspaces (generated from Schreier-type sets of words of order any fixed countable ordinal). Furthermore, as a result we obtain a strengthening of Carlson's infinitary Nash-Williams type (and Ellentuck type) partition theorem about infinite sequences of variable words into a theorem, in which an infinite sequence of variable words and a binary partition of all the finite sequences of words, one of whose components is, in addition, a tree, are assumed, concluding that all the Schreier-type finite reductions of an infinite reduction of the given sequence have a behavior determined by the Cantor-Bendixson ordinal index of the tree-component of the partition, falling in the tree-component above that index and in its complement below it.
Calculating Pythagorean Triples Stocks, Collin RM; Lamb, Gerald; Colligan, J. Kevin ...
The Mathematics teacher,
09/2010, Letnik:
104, Številka:
2
Journal Article
Recenzirano
A Pythagorean triple is defined as a set of three positive integers that satisfy the Pythagorean theorem--a^ sup 2^+b^sup 2^=c^sup 2^--and can therefore represent the dimensions of a right triangle. ...Here, Stocks and Lamb investigate the possibility of finding a general representation for all Pythagorean triples using Mr. Lamb's triples.
Richard Heck and John Burgess have shown that Frege’s Basic Law V is consistent with predicative comprehension and that the resulting theory interprets Robinson Arithmetic. There are also many other ...ways to keep Frege from being contradictory. This paper shows that Basic Law V is also consistent with positive comprehension and that the resulting theory also interprets Robinson Arithmetic. In addition, the theory of positive Frege provides a new understanding of Dummett’s “indefinitely extensible concepts.”
Roy Sorensen's criticism of my use of margin for error principles to explain ignorance in borderline cases fails because it misidentifies the relevant margin for error principles. His alternative ...explanation in terms of truth-maker gaps is briefly criticized.
The aim of this paper is to extend the classical recurrence theorem of A.Y. Khintchine, as well as certain multiple recurrence results of H. Furstenberg concerning weakly mixing and almost periodic ...measure preserving transformations, to the framework of C*-algebras 𝔄 and positive linear maps Φ : 𝔄 → 𝔄 preserving a state φ on 𝔄. For the proof of the multiple weak mixing results we use a slight extension of a convergence result of Furstenberg in Hilbert spaces, which is derived from a non-commutative generalization of Van der Corput's "Fundamental Inequality" in Theory of uniform distribution modulo 1, proved in Appendix A.
This paper is motivated by the question whether there exists a logic capturing polynomial time computation over unordered structures. We consider several algorithmic problems near the border of the ...known, logically defined complexity classes contained in polynomial time. We show that fixpoint logic plus counting is stronger than might be expected, in that it can express the existence of a complete matching in a bipartite graph. We revisit the known examples that separate polynomial time from fixpoint plus counting. We show that the examples in a paper of Cai, Fürer, and Immerman, when suitably padded, are in choiceless polynomial time yet not in fixpoint plus counting. Without padding, they remain in polynomial time but appear not to be in choiceless polynomial time plus counting. Similar results hold for the multipede examples of Gurevich and Shelah, except that their final version of multipedes is, in a sense, already suitably padded. Finally, we describe another possible candidate, involving determinants, for the task of separating polynomial time from choiceless polynomial time plus counting.
Quasi-MV algebras are generalisations of MV algebras arising in quantum computational logic. Although a reasonably complete description of the lattice of subvarieties of quasi-MV algebras has already ...been provided, the problem of extending this description to the setting of quasivarieties has so far remained open. Given its apparent logical repercussions, we tackle the issue in the present paper. We especially focus on quasivarieties whose generators either are subalgebras of the standard square quasi-MV algebra S, or can be obtained therefrom through the addition of some fixpoints for the inverse.
For a positive integer m, set \zeta_{m}=\exp(2\pi i/m) and let {\bf Z}_{m}^{*} denote the group of reduced residues modulo m. Fix a congruence group H of conductor m and of order f. Choose integers ...t_{1},\dots,t_{e} to represent the e=\phi(m)/f cosets of H in {\bf Z}_{m}^{*}. The Gauss periods \displaylines{ \theta_{j} =\sum_{x \in H} \zeta_{m}^{t_{j}x} \;\;; (1 \leq j \leq e) } corresponding to H are conjugate and distinct over {\bf Q} with minimal polynomial \displaylines{ g(x) = x^{e} + c_{1}x^{e-1} + \cdots + c_{e-1} x + c_{e}. } To determine the coefficients of the period polynomial g(x) (or equivalently, its reciprocal polynomial G(X)=X^{e}g(X^{-1})) is a classical problem dating back to Gauss. Previous work of the author, and Gupta and Zagier, primarily treated the case m=p, an odd prime, with f >1 fixed. In this setting, it is known for certain integral power series A(X) and B(X), that for any positive integer N \displaylines{ G(X) \equiv A(X)·B(X)^{\frac{p-1}{f}} \;\;\;({\rm mod}\;X^{N}) } holds in {\bf Z}X for all primes p \equiv 1({\rm mod}; f) except those in an effectively determinable finite set. Here we describe an analogous result for the case m=p^{\alpha}, a prime power (\alpha > 1). The methods extend for odd prime powers p^{\alpha} to give a similar result for certain twisted Gauss periods of the form \displaylines{ \psi_{j} = i^{*} \sqrt{p} \sum_{x \in H} (\frac{t_{j}x}{p}) \zeta_{p^{\alpha}}^{t_{j}x} \;\;(1 \leq j \leq e),} where (\frac{ }{p}) denotes the usual Legendre symbol and i^{*}= i^{\frac{(p-1)^{2}}{4}}.
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