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4 - Artificial Economics versus Mathematics?
from Part I - Artificial Economics and Mainstream Economics
Ruben Mercado
Book:

Artificial Economics

Published online:

28 October 2021

Print publication:

04 November 2021, pp 60-70
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Summary

Chapter 4 focuses in the methodological and instrumental contrasts between artificial economics and mainstream economics. It discusses the mathematical methods of mainstream economics (centered around the use of optimization methods and systems of equations representations) versus the computational methods of artificial economics (characterized by the use of algorithms, software, and computer hardware). Presents basic notions on algorithms, recursion, and Turing machines. And discusses the methodological and instrumental differences between artificial economics and mainstream economics as derived from differences between classical mathematics and constructive mathematics.

Chapter 10 - Logics and worlds
from Part II - History and Authors
- By Ermanno Bencivenga
Edited by Penelope Rush, University of Tasmania
Book:

The Metaphysics of Logic

Published online:

05 October 2014

Print publication:

16 October 2014, pp 178-188
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Summary

Wittgenstein's thought on mathematics had undergone a major, if often undetected, change. The idea that adopting an algorithm like "plus" determines in some physical, mental, or metaphysical way one's response to infinitely many exercises is nothing but covert Platonism, in many ways worse than the Platonism of objects. Wittgenstein agrees entirely with the Intuitionist critique of the law of excluded middle. For the Goldbach conjecture to be true in the sense of classical mathematics, we have to say that the operations of arithmetic determine in advance that every even number, no matter how large, can be partitioned into two primes. The law of excluded middle cannot be regarded as a hardened regularity in cases in which it is applied it to a putative infinite totality. But precisely because of this, there is no direct comparison possible between empirical observations and mathematical theorems in this type of proof.

8 - Kripke on the Incoherency of Adopting a Logic
- By Alan Berger
Edited by Alan Berger, Brandeis University, Massachusetts
Book:

Saul Kripke

Published online:

05 June 2012

Print publication:

06 June 2011, pp 177-208
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This chapter discusses Saul Kripke's general objections to the notion of adopting a logic. The main issue Kripke chose to talk about first is whether logic is a set of statements, and not whether logic is revisable. Kripke also criticized the view that the notion of "adopting a logic" is a coherent one. The whole point of introducing quantum logic is to put quantum mechanics on a sound foundation, making it paradox-free. Hilary Putnam refers to a formal system that he calls "quantum logic", which he says can "be read off from Hilbert space". Unlike quantum logic, intuitionist logic is commonly given as the most standard example of a change in logic. Intuitionists have supposedly adopted a logic different from "the received one" and have based a whole different system of mathematics upon it, as well as having rejected classical mathematics.