There it is: right in the dag gosh darn 1952 PREFACE.PREFACETwo successive eras of investigations of the foundations of mathematics in the nineteenth century, culminating in the theory of sets and the arithmetization of analysis, led around 1900 to a new crisis, and a new era dominated by the programs of Russell and Whitehead, Hilbert and of Brouwer.The appearance in 1931 of Godel’s two incompleteness theorems, in 1933 of Tarski’s work on the concept of truth in formalized languages, in 1934 of the Herbrand-Godel notion of ‘general recursive function’, and in 1936 of Church’s thesis concerning it, inaugurate a still newer era in which mathematical tools are being applied both to evaluating the earlier programs and in unforeseen directions.The aim of this book is to provide a connected introduction to the subjects of mathematical logic and recursive functions in particular, and to the newer foundational investigations in general.

Putting notes as I go as this is terse but dense information that can be useful later. .The ‘gist’ is: paradoxes arose immediately after Cantor and various ways attempted to scramble to fix them, one of which was axiomic set theory.

But the problems with set theory lay in the underlying logic.

“The first system of axiomatic set theory was Zermelo’s (1908). Refinements in the axiomatic treatment of sets are due to Fraenkel (1922, 1925), Skolem (1922-3, 1929), von Neumann (1925, 1928), Bernays (1937-48), and others. Analysis can be founded on the basis of axiomatic set theory, which perhaps is the simplest basis set up since the paradoxes for the deduction of existing mathematics. Some very interesting discoveries have been made in connection with axiomatic set theory, notably by Skolem (1922-3; cf. §75 below) and Godel (1938, 1939, 1940).

The broader problem of foundations. Assuming that the para¬ doxes are avoided in the axiomatization of set theory — and of this the only assurance we have is the negative one that so far none have been encountered — does it constitute a full solution of the problem posed by the paradoxes?

In the case of geometry, mathematicians have recognized since the discovery of a non-Euclidean geometry that more than one kind of space is possible. Axiom systems serve to single out one or another kind of space, or certain common features of several spaces, for the geometer to study. A contradiction arising in a formal axiomatic theory can mean simply that an unrealizable combination of features has been postulated.

But in the case of arithmetic and analysis, theories culminating in set theory, mathematicians prior to the current epoch of criticism general¬ ly supposed that they were dealing with systems of objects, set up genetically, by definitions purporting to establish their structure com¬ pletely. The theorems were thought of as expressing truths about these systems, rather than as propositions applying hypothetically to whatever systems of objects (if any) satisfy the axioms. But then how could con¬ tradictions have arisen in these subjects, unless there is some defect in the logic, some error in the methods of constructing and reasoning about mathematical objects, which we had hitherto trusted? “

I _love_ that this ‘jives’ with my understanding of the history – AND I didn’t know that Gauss said that.

The non-intuitionistic mathematics which culminated in the theories of Weierstrass, Dedekind and Cantor, and the intuitionistic mathematics of Brouwer, differ essentially in their view of the infinite. In the former, the infinite is treated as actual or completed or extended or existential. An infinite set is regarded as existing as a completed totality, prior to or in¬ dependently of any human process of generation or construction, and as though it could be spread out completely for our inspection. In the latter, the infinite is treated only as potential or becoming or constructive. The recognition of this distinction, in the case of infinite magnitudes, goes back to Gauss, who in 1831 wrote, “I protest … against the use of an infinite magnitude as something completed, which is never permissible in mathematics.”

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Ok, all finished with it. Rest of the book is proofs, which I don’t do. I believe it. I got the history, the names, the who-did-what-and-when and what is the significance-of and what was it called that they did and why i should care stuff. I’m satisfied.

Other books in the same Dewey Decimal category:

510.1

Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter

Introduction to Mathematical Philosophy by Bertrand Russell

The Principles of Mathematics by Bertrand Russell

Mind Tools: The Five Levels of Mathematical Reality by Rudolf Rucker

Descartes’s Secret Notebook: A True Tale of Mathematics, Mysticism, and the Quest to Understand the Universe by Amir D. Aczel

The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene

Remarks on the Foundations of Mathematics by Ludwig Wittgenstein

Descartes’ Dream: The World According to Mathematics by Philip J. Davis

How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics by Eugenia Cheng

Philosophy of Mathematics: Selected Readings by Paul Benacerraf

Mathematics: Is God Silent? by James Nickel

What Is Mathematics, Really? by Reuben Hersh

Mathematics: The Science of Patterns : The Search for Order in Life, Mind, and the Universe (Scientific American Library) by Keith Devlin

The Lifebox, the Seashell, and the Soul: What Gnarly Computation Taught Me About Ultimate Reality, the Meaning of Life, and How to Be Happy by Rudy Rucker

The Philosophy of Mathematics: An Introductory Essay by Stephan Körner

Introduction to Mathematical Thinking by Keith Devlin

Foundations and Fundamental Concepts of Mathematics by Howard Whitley Eves

Thinking about Mathematics: The Philosophy of Mathematics by Stewart Shapiro

and a bunch more: https://www.librarything.com/mds/510.1