A Neo-Formalist Approach to Mathematical Truth

1998 ◽  
pp. 41-47
Author(s):  
Alan Weir
Keyword(s):  
Ontological Commitment ◽  
Abstract Objects ◽  
Mathematical Truth ◽  
Truth Conditions ◽  
The Status ◽  
Formal Calculus

I outline a variant on the formalist approach to mathematics which rejects textbook formalism's highly counterintuitive denial that mathematical theorems express truths while still avoiding ontological commitment to a realm of abstract objects. The key idea is to distinguish the sense of a sentence from its explanatory truth conditions. I then look at various problems with the neo-formalist approach, in particular at the status of the notion of proof in a formal calculus and at problems which Gödelian results seem to pose for the tight link assumed between truth and proof.

Imperative logic

2018 ◽  
Author(s):  
Mitchell Green
Keyword(s):  
Moral Reasoning ◽  
Practical Reasoning ◽  
Literal Meaning ◽  
Empty Stomach ◽  
Truth Conditions ◽  
Proper Understanding ◽  
Imperative Logic ◽  
The Status ◽  
Logic Of Imperatives ◽  
Shed Light

Imperatives lie at the heart of both practical and moral reasoning, yet they have been overshadowed by propositions and relegated by many philosophers to the status of exclamations. One reason for this is that a sentence’s having literal meaning seems to require its having truth-conditions and ‘Keep your promises!’ appears to lack such conditions, just as ‘Ouch!’ does. One reductionist attempt to develop a logic of imperatives translates them into declaratives and construes inferential relations among the former in terms of inferential relations among the latter. Since no such reduction seems fully to capture the meaning of imperatives, others have expanded our notion of inference to include not just truth – but also satisfaction – preservation, according to which an imperative is satisfied just in case what it enjoins is brought about. A logic capturing what is distinctive about imperatives may shed light on the question whether an ‘ought’ is derivable from an ‘is’; and may elucidate the claim that morality is, or comprises, a system of hypothetical imperatives. Furthermore, instructions, which are often formulated as imperatives (‘Take two tablets on an empty stomach!’), are crucial to the construction of plans of action. A proper understanding of imperatives and their inferential properties may thus also illuminate practical reasoning.

The Metaphysics of Meaning: Propositions and Possible Worlds

2010 ◽  
Author(s):  
Scott Soames
Keyword(s):  
Possible Worlds ◽  
Abstract Objects ◽  
Possible World ◽  
Truth Conditions ◽  
Theories Of Truth ◽  
Cognitive States

This chapter examines two crucial aspects of the metaphysics of meaning—propositions and possible world-states. It reviews why propositions—needed as meanings of sentences and objects of the attitudes—can neither be extracted from theories of truth conditions, nor defined in terms of possible world-states, It then explains why they also cannot be the mysterious, inherently representational, abstract objects they have traditionally been taken to be. Instead of explaining the representationality of sentences and cognitive states in terms of their relations to the supposedly prior and independent representationality of propositions, we must explain the representationality of propositions in terms of the representationality of the cognitive states with which they are connected. A new account of is presented along these lines.

BENACERRAF’S DILEMMA AND INFORMAL MATHEMATICS

2009 ◽  
Vol 2 (4) ◽  
pp. 769-785 ◽  
Author(s):  
GREGORY LAVERS
Keyword(s):  
Abstract Objects ◽  
Mathematical Truth ◽  
Trial And Error ◽  
Informal Mathematics ◽  
Benacerraf's Dilemma ◽  
Informal Rigour

This paper puts forward and defends an account of mathematical truth, and in particular an account of the truth of mathematical axioms. The proposal attempts to be completely nonrevisionist. In this connection, it seeks to satisfy simultaneously both horns of Benacerraf’s dilemma. The account builds upon Georg Kreisel’s work on informal rigour. Kreisel defends the view that axioms are arrived at by a rigorous examination of our informal notions, as opposed to being stipulated or arrived at by trial and error. This view is then supplemented by a Fregean account of the objectivity and our knowledge of abstract objects. It is then argued that the resulting view faces no insurmountable metaphysical or epistemic obstacles.

Propositions, attitudinal objects, and the distinction between actions and products

2013 ◽  
Vol 43 (5-6) ◽  
pp. 679-701 ◽  
Author(s):  
Friederike Moltmann
Keyword(s):  
Propositional Attitudes ◽  
Abstract Objects ◽  
Truth Conditions ◽  
Work Of Art ◽  
Artistic Creation ◽  
Shared Objects ◽  
Cognitive Accessibility ◽  
The Law

Propositions as mind-independent abstract objects raise serious problems such as their cognitive accessibility and their ability to carry essential truth conditions, as a number of philosophers have recently pointed out. This paper argues that ‘attitudinal objects’ or kinds of them should replace propositions as truth bearers and as the (shared) objects of propositional attitudes. Attitudinal objects, entities like judgments, beliefs, and claims, are not states or actions, but rather their (spatio-temporally coincident) products, following the distinction between actions and products introduced by Twardowski (1912). The paper argues that the action–product distinction is not tied to particular terms in a particular language, but is to be understood as the more general distinction between an action and the (abstract or physically realized) artifact that it creates. It thus includes the distinction between the passing of a law and the law itself and an act of artistic creation and the created work of art.

Ontology, Analyticity, and Meaning

2014 ◽  
Author(s):  
Scott Soames
Keyword(s):  
Ontological Commitment ◽  
Theoretical Framework ◽  
Abstract Objects

This chapter examines the dispute between Quine and Carnap about how to understand ontological commitment and what ontology to adopt. The central dispute is over Carnap’s acceptance of abstract objects, including numbers, properties, and propositions, which Quine characterizes in “On What There Is” (1948) as a form of Platonism. Carnap vigorously disagrees, responding in “Empiricism, Semantics, and Ontology” (1950, 1956). For him, commitments to these things are unproblematic consequences of accepting an optimal theoretical framework for doing science. Philosophers haven’t seen this because, he believes, they have approached ontology in an unscientific way.

Non-Factualism about Abstract Objects

2021 ◽  
pp. 123-160
Author(s):  
Mark Balaguer
Keyword(s):  
Abstract Objects ◽  
Abstract Object ◽  
Truth Conditions ◽  
Truth Value ◽  
Argument Proceeds

Chapter 5 provides an argument for a non-factualist view of the abstract-object question; in other words, it argues that there’s no fact of the matter whether there are any such things as abstract objects like numbers and sets and propositions (where an abstract object is a non-physical, non-mental, unextended, acausal, non-spatiotemporal object). Roughly speaking, the argument proceeds by showing that the sentence ‘There are abstract objects’ is catastrophically unclear and indeterminate—i.e., that it’s so unclear that it doesn’t have any truth conditions and, hence, doesn’t have a truth value. In addition, the chapter also argues against necessitarian versions of platonism and anti-platonism.

Kitcher’s Circumlocutionary Structuralism

1991 ◽  
Vol 21 (1) ◽  
pp. 81-89
Author(s):  
Michael Hand
Keyword(s):  
Subject Matter ◽  
A Priori ◽  
Physical Reality ◽  
Physical World ◽  
Structural Features ◽  
Mathematical Truth ◽  
Truth Conditions ◽  
Philip Kitcher ◽  
Physical Knowledge ◽  
The World

Philip Kitcher has proposed an account of mathematical truth which he hopes avoids platonistic commitment to abstract mathematical objects. His idea is that the truth-conditions of mathematical statements consist in certain general structural features of physical reality. He codifies these structural features by reference to various operations which are performable on objects: the world is structured in such a way that these operations are possible. Which operations are performable cannot be known a priori; rather, we hypothesize, conjecture, idealize, and eventually wind up with theories which are true of the world (taking into account our idealizations), just as we do in the sciences. Kitcher argues that mathematical and physical knowledge are continuous, in that they concern the same subject matter (the physical world) and are subject to the same epistemological and methodological constraints.

The study of interfacial reactions of nickel thin films on GaAs

1983 ◽  
Vol 41 ◽  
pp. 156-157
Author(s):  
L.J. Chen ◽  
Y.F. Hsieh
Keyword(s):  
Thin Films ◽  
Integrated Circuits ◽  
Interfacial Reactions ◽  
Metal Contacts ◽  
Nickel Thin Films ◽  
Thin Foils ◽  
The Status ◽  
Si Doped ◽  
Electron Microscopes

One measure of the maturity of a device technology is the ease and reliability of applying contact metallurgy. Compared to metal contact of silicon, the status of GaAs metallization is still at its primitive stage. With the advent of GaAs MESFET and integrated circuits, very stringent requirements were placed on their metal contacts. During the past few years, extensive researches have been conducted in the area of Au-Ge-Ni in order to lower contact resistances and improve uniformity. In this paper, we report the results of TEM study of interfacial reactions between Ni and GaAs as part of the attempt to understand the role of nickel in Au-Ge-Ni contact of GaAs.N-type, Si-doped, (001) oriented GaAs wafers, 15 mil in thickness, were grown by gradient-freeze method. Nickel thin films, 300Å in thickness, were e-gun deposited on GaAs wafers. The samples were then annealed in dry N2 in a 3-zone diffusion furnace at temperatures 200°C - 600°C for 5-180 minutes. Thin foils for TEM examinations were prepared by chemical polishing from the GaA.s side. TEM investigations were performed with JE0L- 100B and JE0L-200CX electron microscopes.

Fate of sperm membrane in fertilized ova

1993 ◽  
Vol 51 ◽  
pp. 40-41
Author(s):  
Frank J. Longo
Keyword(s):  
Electron Microscopy ◽  
Electrical Activity ◽  
Sea Urchins ◽  
Morphological Changes ◽  
Integrated System ◽  
Electrophysiological Recording ◽  
Experimental Conditions ◽  
The Status ◽  
Sperm Membrane ◽  
Temporal And Spatial

Measurement of the egg's electrical activity, the fertilization potential or the activation current (in voltage clamped eggs), provides a means of detecting the earliest perceivable response of the egg to the fertilizing sperm. By using the electrical physiological record as a “real time” indicator of the instant of electrical continuity between the gametes, eggs can be inseminated with sperm at lower, more physiological densities, thereby assuring that only one sperm interacts with the egg. Integrating techniques of intracellular electrophysiological recording, video-imaging, and electron microscopy, we are able to identify the fertilizing sperm precisely and correlate the status of gamete organelles with the first indication (fertilization potential/activation current) of the egg's response to the attached sperm. Hence, this integrated system provides improved temporal and spatial resolution of morphological changes at the site of gamete interaction, under a variety of experimental conditions. Using these integrated techniques, we have investigated when sperm-egg plasma membrane fusion occurs in sea urchins with respect to the onset of the egg's change in electrical activity.

Revisiting the status of dental ethics instruction

2000 ◽  
Vol 64 (11) ◽  
pp. 772-774 ◽  
Author(s):  
JG Odom ◽  
PL Beemsterboer ◽  
TD Pate ◽  
NK Haden
Keyword(s):  
Ethics Instruction ◽  
The Status
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