- Paperback: 144 pages
- Publisher: Northwestern University Press; 2nd Revised edition (December 1, 1980)
- Language: English
- ISBN-10: 0810106051
- ISBN-13: 978-0810106055
- Product Dimensions: 5.5 x 0.2 x 8.5 inches
- Shipping Weight: 6.4 ounces (View shipping rates and policies)
- Average Customer Review: 14 customer reviews
- Amazon Best Sellers Rank: #670,953 in Books (See Top 100 in Books)
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The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number 2nd Revised Edition
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Text: English (translation)
Original Language: German
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Some will say that mathematical philosophy, or the closely related philosophy of mathematics, only began in the 1980's in earnest. But reading the "big 5" shows threads going back to antiquity. The field is far from settled, and the two aspects--the philosophy of math itself, and the closely related field of applying math and logic TO other branches of philosophy, has enough active journalized information in the mid 2014+ years to fill 50 volumes. Since thousands have been written in mainline philosophy, and even the philosophy of science as well as logic, this is not without surprise and mystery.
The good news is that an invested, energetic reader can pick up this handful of keys and be in the top percent of folks on the planet with a good foundation! This is hardly true of any other field. I'd start with Shapiro's Oxford Encylopedia, study Benacerraf and Putnam's classic collection of essays, then finish with Shapiro's deep and difficult "Thinking about" and of course Russell and Frege for historic and specialized puzzle pieces.
One "sleeper" I'd like to recommend that is not usually included in comparisons of books in this field is Steinhart: More Precisely: The Math You Need to Do Philosophy.
Eric helps with both math within philosophy (the basics) and tangentially helps with the math used as examples within the philosophy OF math. Beyond the issues of categorization, discovery, math as model vs. underpinning reality ala the Matrix, there of course is the whole field of logic, induction, deduction, etc. which has thousands of volumes. The six mentioned here cover logic, but are much more specific in the broader subject area of mathematics, which now includes dynamical systems and differential equations undreamt of in the past, and bringing many new mental tools to bear, from intuition to analytic, qualitative, numeric, perturbative and of course stochastic. Here are the other links to those mentioned in this survey:
The Oxford Handbook of Philosophy of Mathematics and Logic (Oxford Handbooks)
Thinking about Mathematics: The Philosophy of Mathematics
Philosophy of Mathematics: Selected Readings
The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number
Introduction to Mathematical Philosophy
Frege regards the number zero as belonging to to the "natural" or "counting" numbers, whereas we subscribe to the view that zero is not a counting number at all (the first of the counting numbers being 1) and is only properly used when we regard a number as a "relative-magnitude," zero being the relative-magnitude of two equal counting numbers.
This work of Frege's has considerable historical interest as a forerunner of the work of Whitehead and Russell. The translation is excellent and the printing leaves nothing to be desired.
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Note that he is very consistently hard on Mill.Read more