Are Logic and Mathematics Identical?: An old thesis of Russell's is reexamined in the light of subsequent developments in mathematical logic

Abstract It is shown by using a rather elementary argument in Mathematical Logic that if indeed, quantum theory does violate the famous Bell Inequalities, then quantum theory must inevitably also violate all valid mathematical statements, and in particular, such basic algebraic relations like 0 = 0, 1 = 1, 2 = 2, 3 = 3, … and so on … An interest in that result is due to the following three alternatives which it imposes upon both Physics and Mathematics: Quantum Theory is inconsistent. Quantum Theory together with Mathematics are inconsistent. Mathematics is inconsistent. In this regard one should recall that, up until now, it is not known whether Mathematics is indeed consistent.

Download Full-text

Formenanalyse bei Marx und ihr Verhältnis zur Systemwissenschaft / Analysis of forms by Marx and its relation to systems research

Zeitschrift für Soziologie ◽

10.1515/zfsoz-1975-0305 ◽

1975 ◽

Vol 4 (3) ◽

Author(s):

Werner Loh

Keyword(s):

Mathematical Logic ◽

Empirical Analysis ◽

The Other ◽

Systems Research ◽

Other Hand ◽

Empirical Logic ◽

Formal Concepts ◽

And Mathematics ◽

Different Levels

AbstractMARX’s analysis of forms and modern systems research have in common the problem of form. MARX analyzed forms by functionally relating elements to each other on different levels. Contrary to modern systems theories and Marxism-Leninism elements are for MARX forms themselves and not non-formal elementary qualities. The analysis of forms, therefore, is able to characterize its objects only relationally-functionally. On the other hand modern systems theories integrate concepts like ‚action‘ or ‚goal‘ in an elementaristic manner. The analysis of forms must be controlled by systematic concretization and totalization adequate to the problem. The formal concepts of systems research are often interpreted as logical-mathematical. Logic and mathematics are usually understood as non-empirical. Empirical analysis of forms is in need of an empirical logic and mathematics.

Download Full-text

Does mathematics look certain in the front, but fallible in the back?

Social Studies of Science ◽

10.1177/0306312711424789 ◽

2011 ◽

Vol 41 (6) ◽

pp. 839-866 ◽

Cited By ~ 16

Author(s):

Christian Greiffenhagen ◽

Wes Sharrock

Keyword(s):

Mathematical Logic ◽

Doctoral Students ◽

Research Articles ◽

And Mathematics ◽

Writing Science ◽

The Way

In this paper we re-examine the implications of the differences between ‘doing’ and ‘writing’ science and mathematics, questioning whether the way that science and mathematics are presented in textbooks or research articles creates a misleading picture of these differences. We focus our discussion on mathematics, in particular on Reuben Hersh’s formulation of the contrast in terms of Goffman’s dramaturgical frontstage–backstage analogy and his claim that various myths about mathematics only fit with how mathematics is presented in the ‘front’, but not with how it is practised in the ‘back’. By investigating examples of both the ‘front’ (graduate lectures in mathematical logic) and the ‘back’ (meetings between supervisor and doctoral students) we examine, first, whether the ‘front’ of mathematics presents a misleading picture of mathematics, and, second, whether the ‘front’ and ‘back’ of mathematics are so discrepant that mathematics really does look certain in the ‘front’, but fallible in the ‘back’.

Download Full-text

Simplified foundations for mathematical logic

Journal of Symbolic Logic ◽

10.2307/2266898 ◽

1955 ◽

Vol 20 (2) ◽

pp. 123-139 ◽

Cited By ~ 1

Author(s):

Robert L. Stanley

Keyword(s):

Mathematical Logic ◽

Set Theory ◽

Natural Deduction ◽

Small Body ◽

The Other ◽

Consistency Proof ◽

Basic Notion ◽

Relative Consistency ◽

And Mathematics

A system SF, closely related to NF, is outlined here. SF has several novel points of simplicity and interest, (a) It uses only one basic notion, from which all the other concepts of logic and mathematics may be built definitionally. Three-notion systems are common, but Quine's two-notion IA has for some time represented the extreme in conceptual economy, (b) The theorems of SF are generated under just three rules of analysis, which unify into a single postulational principle, (c) SF is built solely in terms of what is commonly, known as the “natural deduction” method, under which each theorem is attacked primarily as it stands, by means of a very small body of rules, rather than less directly, through a very large, potentially infinite backlog of theorems. Although natural deduction is by no means new as a method, its exclusive applications have previously been relatively limited, not even reaching principles of identity, much less set theory, relations, or mathematics proper, (d) SF is at least as strong as NF, yielding all of its theorems, which are expressed here in forms analogous to those of the metatheorems in ML. If NF is consistent, so is SF. The main points in the relative consistency proof are set forth below in section seven.

Download Full-text

XXIII. On Mathematical Logic and Mathematics

Classical Mathematical Logic ◽

10.1515/9781400841554.461 ◽

2011 ◽

pp. 461-464

Keyword(s):

Mathematical Logic ◽

And Mathematics

Download Full-text

19th Century Logic Between Philosophy and Mathematics

Bulletin of Symbolic Logic ◽

10.2307/421117 ◽

1999 ◽

Vol 5 (4) ◽

pp. 433-450 ◽

Cited By ~ 14

Author(s):

Volker Peckhaus

Keyword(s):

Mathematical Logic ◽

19Th Century ◽

Early 20Th Century ◽

Symbolic Logic ◽

Modern Logic ◽

History Of ◽

And Mathematics ◽

Correct Reasoning ◽

The 19Th Century ◽

The Relationship

AbstractThe history of modern logic is usually written as the history of mathematical or, more general, symbolic logic. As such it was created by mathematicians. Not regarding its anticipations in Scholastic logic and in the rationalistic era, its continuous development began with George Boole's The Mathematical Analysis of Logic of 1847, and it became a mathematical subdiscipline in the early 20th century. This style of presentation cuts off one eminent line of development, the philosophical development of logic, although logic is evidently one of the basic disciplines of philosophy. One needs only to recall some of the standard 19th century definitions of logic as, e.g., the art and science of reasoning (Whateley) or as giving the normative rules of correct reasoning (Herbart).In the paper the relationship between the philosophical and the mathematical development of logic will be discussed. Answers to the following questions will be provided:1. What were the reasons for the philosophers' lack of interest in formal logic?2. What were the reasons for the mathematicians' interest in logic?3. What did “logic reform” mean in the 19th century? Were the systems of mathematical logic initially regarded as contributions to a reform of logic?4. Was mathematical logic regarded as art, as science or as both?

Download Full-text