A note on nominalism and recursive functions

1949 ◽  
Vol 14 (1) ◽  
pp. 27-31 ◽  
Author(s):  
R. M. Martin
Keyword(s):  
General Theory ◽  
Philosophy Of Science ◽  
Recursive Function ◽  
Homogeneous System ◽  
Formal Logic ◽  
Abstract Objects ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursive Functions ◽  
Definition Of

The purpose of this note is (i) to point out an important similarity between the nominalistic system discussed by Quine in his recent paper On universals and the system of logic (the system н) developed by the author in A homogeneous system for formal logic, (ii) to offer certain corrections to the latter, and (iii) to show that that system (н) is adequate for the general theory of ancestrale and for the definition of any general recursive function of natural numbers.Nominalism as a thesis in the philosophy of science, according to Quine, is the view that it is possible to construct a language adequate for the purposes of science, which in no wise admits classes, properties, relations, or other abstract objects as values for variables.

A note on the representation of general recursive functions and the μ quantifier

1959 ◽  
Vol 55 (2) ◽  
pp. 145-148
Author(s):  
Alan Rose
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursive Functions ◽  
Image Position ◽  
Primitive Recursive

It has been shown that every general recursive function is definable by application of the five schemata for primitive recursive functions together with the schemasubject to the condition that, for each n–tuple of natural numbers x1,…, xn there exists a natural number xn+1 such that

scholarly journals Only Two Letters: The Correspondence between Herbrand and Gödel

2005 ◽  
Vol 11 (2) ◽  
pp. 172-184 ◽  
Author(s):  
Wilfried Sieg
Keyword(s):  
Significant Role ◽  
Full Text ◽  
Recursive Function ◽  
Computability Theory ◽  
General Recursive Function ◽  
Kurt Gödel ◽  
Incompleteness Theorems ◽  
Potential Impact ◽  
Definition Of ◽  
Crucial Part

AbstractTwo young logicians, whose work had a dramatic impact on the direction of logic, exchanged two letters in early 1931. Jacques Herbrand initiated the correspondence on 7 April and Kurt Gödel responded on 25 July, just two days before Herbrand died in a mountaineering accident at La Bérarde (Isère). Herbrand's letter played a significant role in the development of computability theory. Gödel asserted in his 1934 Princeton Lectures and on later occasions that it suggested to him a crucial part of the definition of a general recursive function. Understanding this role in detail is of great interest as the notion is absolutely central. The full text of the letter had not been available until recently, and its content (as reported by Gödel) was not in accord with Herbrand's contemporaneous published work. Together, the letters reflect broader intellectual currents of the time: they are intimately linked to the discussion of the incompleteness theorems and their potential impact on Hilbert's Program.

Arithmetical representation of recursively enumerable sets

1956 ◽  
Vol 21 (2) ◽  
pp. 162-186 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursively Enumerable Set ◽  
Integer Coefficients ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Primitive Recursive

A set S of natural numbers is called recursively enumerable if there is a general recursive function F(x, y) such thatIn other words, S is the projection of a two-dimensional general recursive set. Actually, it is no restriction on S to assume that F(x, y) is primitive recursive. If S is not empty, it is the range of the primitive recursive functionwhere a is a fixed element of S. Using pairing functions, we see that any non-empty recursively enumerable set is also the range of a primitive recursive function of one variable.We use throughout the logical symbols ⋀ (and), ⋁ (or), → (if…then…), ↔ (if and only if), ∧ (for every), and ∨(there exists); negation does not occur explicitly. The variables range over the natural numbers, except as otherwise noted.Martin Davis has shown that every recursively enumerable set S of natural numbers can be represented in the formwhere P(y, b, w, x1 …, xλ) is a polynomial with integer coefficients. (Notice that this would not be correct if we replaced ≤ by <, since the right side of the equivalence would always be satisfied by b = 0.) Conversely, every set S represented by a formula of the above form is recursively enumerable. A basic unsolved problem is whether S can be defined using only existential quantifiers.

Note on a conjecture of Skolem

1946 ◽  
Vol 11 (3) ◽  
pp. 73-74 ◽  
Author(s):  
Emil L. Post
Keyword(s):  
Recursive Function ◽  
Recursive Relation ◽  
Natural Numbers ◽  
Excellent Review ◽  
Recursive Functions ◽  
Diagonal Argument ◽  
Primitive Recursive ◽  
Recursive Relations

In his excellent review of four notes of Skolem on recursive functions of natural numbers Bernays states: “The question whether every relation y = f(x1,…, xn) with a recursive function ƒ is primitive recursive remains undecided.” Actually, the question is easily answered in the negative by a form of the familiar diagonal argument.We start with the ternary recursive relation R, referred to in the review, such that R(x, y, 0), R(x, y, 1), … is an enumeration of all binary primitive recursive relations.

Recursive Colorings of Highly Recursive Graphs

1981 ◽  
Vol 33 (6) ◽  
pp. 1279-1290 ◽  
Author(s):  
Henry A. Kierstead
Keyword(s):  
Characteristic Function ◽  
Computer Program ◽  
Finite Number ◽  
Natural Numbers ◽  
Recursive Functions ◽  
Explicit Constructions ◽  
Recursive Structures ◽  
Definition Of ◽  
Recursive Relations

One of the attractions of finite combinatorics is its explicit constructions. This paper is part of a program to enlarge the domain of finite combinatorics to certain infinite structures while preserving the explicit constructions of the smaller domain. The larger domain to be considered consists of the recursive structures. While recursive structures may be infinite they are still amenable to explicit constructions. In this paper we shall concentrate on recursive colorings of highly recursive graphs.A function f: Nk → N, where N is the set of natural numbers, is recursive if and only if there exists an algorithm (i.e., a finite computer program) which upon input of a sequence of natural numbers , after a finite number of steps, outputs . A subset of Nk is recursive provided that its characteristic function is recursive. For a more thorough definition of recursive functions and recursive relations see [10].

Dualism QM and GR

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Noncommutative Geometry ◽  
Quantum Tunneling ◽  
Closed Surface ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Surface Integral ◽  
Definition Of

Quantum tunneling of noncommutative geometry gives the definition of time in the form of holography, that is, in the form of a closed surface integral. Ultimately, the holography of time shows the dualism between quantum mechanics and the general theory of relativity.

Transcendence of cardinals

1965 ◽  
Vol 30 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
Recursive Functions ◽  
Power Set ◽  
Recursive Predicate

In this paper, by a function of ordinals we understand a function which is defined for all ordinals and each of whose value is an ordinal. In [7] (also cf. [8] or [9]) we defined recursive functions and predicates of ordinals, following Kleene's definition on natural numbers. A predicate will be called arithmetical, if it is obtained from a recursive predicate by prefixing a sequence of alternating quantifiers. A function will be called arithmetical, if its representing predicate is arithmetical.The cardinals are identified with those ordinals a which have larger power than all smaller ordinals than a. For any given ordinal a, we denote by the cardinal of a and by 2a the cardinal which is of the same power as the power set of a. Let χ be the function such that χ(a) is the least cardinal which is greater than a.Now there are functions of ordinals such that they are easily defined in set theory, but it seems impossible to define them as arithmetical ones; χ is such a function. If we define χ in making use of only the language on ordinals, it seems necessary to use the notion of all the functions from ordinals, e.g., as in [6].

Two Refutations of the Vorhanden Reading of Heidegger’s Philosophy of Science in Being and Time

2021 ◽  
Vol 55 ◽  
pp. 174-187
Author(s):  
Paul Goldberg ◽  
Keyword(s):  
Philosophy Of Science ◽  
Natural Science ◽  
Being And Time ◽  
Robert Brandom ◽  
Definition Of ◽  
Hubert Dreyfus

The dominant interpretation of Heidegger’s philosophy of science in Being and Time is that he defines science, or natural science, in terms of presence-at-hand (Vorhandenheit). I argue that this interpretation is false. I call this dominant view about Heidegger’s definition of science the vorhanden claim; interpreters who argue in favor of this claim I call vorhanden readers. In the essay, I reconstruct and then refute two major arguments for the vorhanden claim: respectively, I call them equipmental breakdown (Section 1) and theoretical assertion (Section 2). The equipmental breakdown argument, stemming mainly from Hubert Dreyfus, advances a vorhanden reading on the basis of three other interpretive claims: I call them, respectively, the primacy of practice claim, the decontextualization claim, and the breakdown claim. While I remain agnostic on the first claim, the argument fails because of decisive textual counterevidence to the latter two claims. Meanwhile, the theoretical assertion argument, which I reconstruct mainly from Robert Brandom, premises its vorhanden claim on the basis of some remarks in Being and Time indicating that theoretical assertions, as such, refer to present-at-hand things. Since science is taken to be a paradigmatic case of an activity that makes theoretical assertions, the vorhanden claim is supposed to follow. I refute this argument on the grounds that it equivocates on Heidegger’s concept of “theoretical assertion” and cannot account for his insistence that science does not principally consist in the production of such assertions. I conclude that, with the failure of these two arguments, the case for the vorhanden claim is severely weakened.

Conceptualizing Computer Simulations in Philosophy of Science

2021 ◽  
Vol 58 (2) ◽  
pp. 151-169
Author(s):  
Timur V. Khamdamov ◽  
Mikhail Yu. Voloshin ◽  
Keyword(s):  
Philosophy Of Science ◽  
Computer Simulations ◽  
Russian Philosophy ◽  
Main Role ◽  
Controversial Issues ◽  
Taxonomic Problem ◽  
Basic Characteristics ◽  
The One ◽  
Definition Of ◽  
The Impact

In the modern Russian philosophy, discussions about the phenomenon of computer simulations in the scientific research practice of conducting experiments are just beginning to pass the stage of initiation in small interdisciplinary groups studying this new direction for the philosophy of science. At the same time, in Western philosophy by the current moment there have been formed entire directions for the study of computer simulations. Different groups of researchers in different ways form ideas about the basic characteristics of simulations: from skeptical views on their nature, which are of no philosophical interest, to extremely revolutionary attitudes that assign simulations to the main role in the next expected turn of philosophy, comparable in its power to the linguistic turn in early XX century. One of the main controversial issues in Western philosophical thought was the search for relevant criteria and signs of simulations that could create a solid basis for formulating a rigorous definition of this phenomenon. Thus, through the definition, researchers first of all try, on the one hand, to solve the taxonomic problem of the correlation and interconnection of simulations with other types of experiment: natural, laboratory, mental, mathematical. On the other hand, to reveal for philosophy ontological and epistemological foundations of simulations, which carry the potential of new philosophical knowledge. This article is devoted to a brief review of the existing concepts of representatives of Western schools of thought on the phenomenon of computer simulations in the context of the philosophy of science. The structure of the review is built on three basic conceptual directions: 1) definition of the term "computer simulation"; 2) computer simulations as an experiment; 3) the epistemic value of simulations. Such a review can become the subject of discussion for Russian researchers interested in the impact of computer simulations on science and philosophy.

scholarly journals UKRAINIAN REALIA OF THE XIX CENTURY AND ITS TRANSLATION (BASED ON THE POEM “KATERINA” BY TARAS SHEVCHENKO)

2019 ◽  
Vol 1 (16) ◽  
pp. 124-130
Author(s):  
E.I. Panchenko
Keyword(s):  
General Theory ◽  
Cultural Differences ◽  
Real World ◽  
Species Replacement ◽  
Different Types ◽  
General Meaning ◽  
Definition Of ◽  
Xix Century ◽  
Entire Array

The article is written in line with current research, since the problem of studying Ukrainian realities is of unquestionable interest for several reasons. First, understanding the realities will promote bettermutual understanding of different peoples; and secondly, the definition of optimal means of translating the realities is a definite contribution to the general theory of translation. Different types of real-world classifications are proposed, the difficulties associated with the adequate transfer into the translated text of an entire array of cultural information encoded in the realities contained in the origina text are investigated. Basing on the analysis of numerous translations of literary works, Ukrainian researchers (R. Zorivchak, V. Koptilov, O. Kundzich, O. Cherednichenko, etc.) show ways to overcome linguistic obstacles caused by cultural differences. But, as far as we know, the problem of the translation of Ukrainian realities in the works of T. Shevchenko is not yet exhaustively highlighted. The purpose of this article is to analyze the peculiarities of the use of realities in the work of Taras Shevchenko "Katerina" and their translation into English. We have given an ideographic classification of lexical units - Ukrainian realities in fiction and analyzed such means of their translation as calque, renomination, transcription with explanation, the introduction of neologism, the principle of generic-species replacement, which allows  conveying (approximately) the content of the realities by a broader, general meaning, that is, the reception of generalization. The results of our analysis allow us to make an ideographic classification of Ukrainian realities that are used in fiction, as well as to summarize the prevalence of their means of translation. Prospects for further research are seen in the analysis of certain translation failures in the translation of realities and to offer the best options for their translation.

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