The Foundations of Arithmetic

2020 ◽  
Author(s):  
Gottlob Frege
Keyword(s):  
Foundations Of Arithmetic

C.S. Peirce: Belief, Truth, and Going from the Known to the Unknown

1998 ◽  
Vol 24 ◽  
pp. 9-29 ◽  
Author(s):  
David Wiggins
Keyword(s):  
High Probability ◽  
The Other ◽  
The Third ◽  
P 16 ◽  
Foundations Of Arithmetic

The third philosophical stratagem for cutting off inquiry consists in maintaining that this, that, or the other element of science is basic, ultimate, independent of aught else and utterly inexplicable- not so much from any defect in our knowing as because there is nothing beneath it to know. The only type of reasoning by which such a conclusion could be reached is retroduction.Now nothing justifies a retroductive inference except its affording an explanation of the facts. It is, however, no explanation at all of a fact to pronounce it inexplicable.That, therefore, is a conclusion which no reasoning can ever justify or excuse. (Peirce, Collected Papers 1.139)Abduction consists in studying facts and devising a theory to explain them. Its only justification is that, if we are ever to understand things at all, it must be in that way. (Peirce, Collected Papers, 5.145)[Scientific procedure] will at times find a high probability established by a single confirmatory instance, while at others it will dismiss a thousand as almost worthless. (Frege, Foundations of Arithmetic (1884), p. 16)

Arithmetic from Kant to Frege: Numbers, Pure Units, and the Limits of Conceptual Representation

2008 ◽  
Vol 63 ◽  
pp. 135-164 ◽  
Author(s):  
Daniel Sutherland
Keyword(s):  
Conceptual Representation ◽  
Space And Time ◽  
Foundations Of Arithmetic

There is evidence in Kant of the idea that concepts of particular numbers, such as the number 5, are derived from the representation of units, and in particular pure units, that is, units that are qualitatively indistinguishable. Frege, in contrast, rejects any attempt to derive concepts of number from the representation of units. In the Foundations of Arithmetic, he softens up his reader for his groundbreaking and unintuitive analysis of number by attacking alternative views, and he devotes the majority of this attack to the units view, with particular attention to pure units. Since Frege, the units view has been all but abandoned. Nevertheless, the idea that concepts of number are derived from the representation of units has a long history, beginning with the ancient Greeks, and was prevalent among Frege's contemporaries. I am not interested in resurrecting the units view or in righting wrongs in Frege's criticisms of his contemporaries. Rather, I am interested in the program of deriving concepts of number from pure units and its history from Kant to Frege. An examination of that history helps us understand the units view in a way that Frege's criticisms do not, and in the process uncovers important features of both Kant's and Frege's views. I will argue that, although they had deep differences, Kant and Frege share assumptions about what such a view would require and about the limits of conceptual representation. I will also argue that they would have rejected the accounts given by some of Frege's contemporaries for the same reasons. Despite these agreements, however, there is evidence that Kant thinks that space and time play a role in overcoming the limitations of conceptual representation, while Frege argues that they do not.

The anatytic conception of truth and the foundations of arithmetic

2000 ◽  
Vol 65 (1) ◽  
pp. 33-102 ◽  
Author(s):  
Peter Apostoli
Keyword(s):  
Philosophy Of Mathematics ◽  
Logical Truth ◽  
Numerical Identity ◽  
Hume’S Principle ◽  
Contextual Definition ◽  
Definition Of ◽  
Axiomatic Set Theory ◽  
In Virtue Of ◽  
Second Order Logic ◽  
Foundations Of Arithmetic

Until very recently, it was thought that there couldn't be any current interest in logicism as a philosophy of mathematics. Indeed, there is an old argument one often finds that logicism is a simple nonstarter just in virtue of the fact that if it were a logical truth that there are infinitely many natural numbers, then this would be in conflict with the existence of finite models. It is certainly true that from the perspective of model theory, arithmetic cannot be a part of logic. However, it is equally true that model theory's reliance on a background of axiomatic set theory renders it unable to match Frege's Theorem, the derivation within second order logic of the infinity of the number series from the contextual “definition” of the cardinality operator. Called “Hume's Principle” by Boolos, the contextual definition of the cardinality operator is presented in Section 63 of Grundlagen, as the statement that, for any concepts F and G,the number of F s = the number of G sif, and only if,F is equinumerous with G.The philosophical interest in Frege's Theorem derives from the thesis, defended for example by Crispin Wright, that Hume's principle expresses our pre-analytic conception of assertions of numerical identity. However, Boolos cites the very fact that Hume's principle has only infinite models as grounds for denying that it is logically true: For Boolos, Hume's principle is simply a disguised axiom of infinity.

Challenges to Predicative Foundations of Arithmetic

2000 ◽  
pp. 317-338 ◽  
Author(s):  
Solomon Feferman ◽  
Geoffrey Hellman
Keyword(s):  
Foundations Of Arithmetic

How far does Wittgenstein go with the Context Principle?

2014 ◽  
Vol 5 (1) ◽  
Author(s):  
Ana Falcato
Keyword(s):  
Internal Dynamics ◽  
Ways Of Thinking ◽  
Foundations Of Arithmetic

AbstractIn this paper I examine in which way the thought of late Wittgenstein can be said to have accommodated the so-called Fregean Context Principle. I suggest that such an evolution in late Wittgenstein’s philosophy - which could be conceived as an internal dynamics within Wittgenstein’s “two ways of thinking”, since he himself stated Frege’s Principle at Tractatus Logico-Philosophicus 3.3 - eventually lead to a deviation from Frege’s methodological statement in the Introduction to The Foundations of Arithmetic. This analysis is not merely historical-conceptual, since the “Context Principle” is claimed by some philosophers of language to lie in the background of contextualist proposals in philosophical semantics. I argue that this cannot be accurate, because the kind of context which late Wittgenstein thought of as indispensable in framing meaningful utterances is altogether different from the context of the “Context Principle”.

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