Bibliography - Metamathematics

The text above includes direct links to specific documents and references. Here we’ll give a slightly more general bibliography, though most of it should be considered background, since the approach taken here represents a significant departure from traditional directions, and builds more or less directly on extremely low-level concepts.

The earliest known large-scale axiomatic presentation of mathematics was:

Euclid (300 BC), Στοιχεῖα (in Ancient Greek) [Elements].

Empirical metamathematics from this was given in:

S. Wolfram (2020), “The Empirical Metamathematics of Euclid and Beyond”.
arXiv: 2107.07337.

The concept that there is underlying reality in mathematics was discussed in:

Plato (375 BC), πολιτεία (in Ancient Greek) [The Republic].

Plato (360 BC), Τίμαιος (in Ancient Greek) [Timaeus].

Modern explorations of these ideas include:

M. Balaguer (1998), Platonism and Anti-Platonism in Mathematics, Oxford University Press.

J. Gray (2008), Plato’s Ghost: The Modernist Transformation of Mathematics, Princeton University Press.

R. Tieszen (2011), After Gödel: Platonism and Rationalism in Mathematics and Logic, Oxford University Press.

The contemporary axiomatic formulation of mathematics was developed in:

F. L. G. Frege (1879), Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (in German), Verlag von Louis Neber. (Translated in J. v. Heijenoort (1967), as “Begriffsschrift: A Formal Language, Modeled upon That of Arithmetic, for Pure Thought” in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, 1–82.)

R. Dedekind (1888), Was sind und was sollen die Zahlen? (in German), F. Vieweg und Sohn. (Translated in H. Pogorzelski, et al. (1995), as What Are Numbers and What Should They Be?, Research Institute for Mathematics.)

G. Peano (1889), Arithmetices principia, nova methodo exposita (in Italian), Fratres Bocca. (Translated by H. C. Kennedy (1973), as “The Principles of Arithmetic, Presented by a New Method”, in Selected Works of Giuseppe Peano, University of Toronto Press, 101–134.)

D. Hilbert (1903), Grundlagen der geometrie (in German), B. G. Teubner. (Translated by E. G. Townsend (1902), as The Foundations of Geometry, Open Court.)

E. Zermelo (1908), “Untersuchungen über die Grundlagen der Mengenlehre I” (in German), Mathematische Annalen 65: 261–281. doi:10.1007/BF01449999. (Translated by J. v. Heijenoort (1967), as “Investigations in the Foundations of Set Theory” in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, 199–215.)

A. N. Whitehead and B. A. W. Russell (1910–1913), Principia Mathematica, Volumes I–III, Cambridge University Press.

A standard collection of source documents is:

J. v. Heijenoort (1967), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press.

Among general commentaries on axiomatic formalism are:

N. Bourbaki (1950), “The Architecture of Mathematics”, The American Mathematical Monthly 57: 221–232. doi: 10.2307/2305937.

S. Feferman, et al. (2000), “Does Mathematics Need New Axioms?”, The Bulletin of Symbolic Logic 6, 401–446. doi: 10.2307/420965.

W. Sieg (2013), Hilbert’s Programs and Beyond, Oxford University Press.

A. Weir (2019), “Formalism in the Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy. plato.stanford.edu/archives/spr2022/entries/formalism-mathematics.

Expositions of metamathematics and mathematical logic ideas appear for example in:

D. Hilbert and W. Ackermann (1928), Grundzüge der theoretischen Logik (in German), Springer. (Translated by L. M. Hammond, et al. (2000), as Principles of Mathematical Logic, AMS Chelsea.)

W. V. O. Quine (1940), Mathematical Logic, W. W. Norton & Company.

A. Church (1956), Introduction to Mathematical Logic, Princeton University Press.

H. Wang (1962), A Survey of Mathematical Logic, Science Press North-Holland.

H. B. Curry (1963), Foundations of Mathematical Logic, Dover.

H. Rasiowa (1963), The Mathematics of Metamathematics, Panstwowe Wydawnictwo Naukowe.

E . Mendelson (1964), Intro to Mathematical Logic, Van Nostrand Reinhold.

G. Kreisel and J. L. Krivine (1967), Elements of Mathematical Logic: Model Theory,
North-Holland.

S. C. Kleene (1971), Introduction to Metamathematics, Wolters-Noordhoff.

H. B. Enderton (1972), A Mathematical Introduction to Logic, Harcourt/Academic Press.

A. Yasuhara (1975), “Recursive Function Theory and Logic”, Journal of Symbolic Logic 40: 619–620. doi: 10.2307/2271829.

J. Barwise (1982), Handbook of Mathematical Logic, Elsevier.

G. E. Sacks (2003), Mathematical Logic in the 20th Century, World Scientific.

Works on the metamodeling of mathematics and on universal algebra include:

A. N. Whitehead (1898), A Treatise on Universal Algebra with Applications, Cambridge University Press.

A. Robinson (1963), Introduction to Model Theory and to the Metamathematics of Algebra, North-Holland.

N. G. de Bruijn (1970), “The Mathematical Language AUTOMATH, Its Usage, and Some of Its Extensions”, in Symposium on Automatic Demonstration, M. Laudet, et al. (eds.), Springer.

A. I. Mal’Cev (1971), The Metamathematics of Algebraic Systems, Collected Papers: 1936–1967, North-Holland.

A. S. Toelstra (1973), Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Springer.

S. Burris and H. P. Sankappanavar (1981), A Course in Universal Algebra, Springer.

Low-level symbolic representations of mathematics were developed in:

M. Schönfinkel (1924), “Über die Bausteine der mathematischen Logik” (in German), Mathematische Annalen 92, 305–316. doi: 10.1007/BF01448013. (Translated by S. Bauer-Mengelberg (1967), as “On the Building Blocks of Mathematical Logic”, in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, J. v. Heijenoort, Harvard University Press, 357–366.)

E. Post (1936), “Finite Combinatory Processes—Formulation 1”, The Journal of Symbolic Logic 1, 103–105. doi: 10.2307/2269031.

36 Bibliography