Philosophy of Mathematics: A Preliminary Classification of Theories

This post revisits some of my earlier ones this year on epistemology and philosophy.

The essence of philosophy of mathematics is the study of the epistemology and ontology of mathematics: what are mathematical entities and how do know that mathematical statements are true?

Certain Austrian economists and libertarians influenced by Mises have an embarrassingly outdated and unconvincing Kantian view of mathematics and geometry, one which appears ignorant of the whole history of philosophy and epistemology as applied to mathematics over the past 130 years.

To see how rich and sophisticated modern philosophy of mathematics is we need only see a list of the various competing theories.

The first major division of theories in philosophy of mathematics is epistemological: how do we and can we know that mathematical statements are true, and how do we justify their truth?

The following is a first attempt at classifying epistemological theories in philosophy of mathematics:

I. Anti-Realism

(1) Logicism (Deductivism).

(1.1) Logicism (Frege, Russell);

(1.2) Conditionalism (if-thenism) (early Putnam, Hellman, Musgrave);

(1.3) Neo-logicism;

(2) Intuitionism (L.E.J. Brouwer, Arend Heyting, Michael Dummett);

(3) Formalism (Haskell Curry);

(4) Predicativism (Solomon Feferman).

II. Realism

(a) Platonic Realism

(1) Platonism

(2) Gödel’s Platonism

(b) Naturalist Realism/Empiricism (Anti-Platonist Realism)

(1) John Stuart Mill (mathematical physicalism), Philip Kitcher.

(2) Psychologism;

(3) Quine, Putnam, Penelope Maddy’s Realism in Mathematics

(4) Social constructivism or social realism;

(5) quasi-empiricism (Imre Lakatos, Karl Popper)

Another classification of theories in philosophy of mathematics is ontological.

That is, we have theories that explain what mathematical entities are and whether and in what sense they exist.

We could classify such ontological theories in philosophy of mathematics as follows:

I. Realism

(i) Platonism

(1) Platonism;

(2) full-blooded Platonism;

(3) Aristotelian realism;

(4) Structuralism (Resnik, Shapiro)

(ii) Realistic Anti-Platonism

(1) Psychologism (Edmund Husserl, L.E.J. Brouwer, Arend Heyting);

(2) Physicalism (David Armstrong, J. S, Mill);

(3) Conceptualism

II. Anti-Realism

(i) Anti-Realist Nominalism

(1) Paraphrase nominalism (or if-thenism, deductivism; Hilary Putnam, Geoffrey Hellman, Haskell Curry, Charles Chihara);

(2) Conventionalism

(3) Fictionalism (Hartry Field, Mark Balaguer, Gideon Rosen, Mary Leng, Stephen Yablo’s figuralism);

(4) neo-Meinongians (Richard Sylvan).

The major division in both types of theories is that between (1) realist and (2) anti-realist theories of mathematics.

Some realists think that mathematical entities have a real existence independently of the human mind, and human beings discover, rather than invent, mathematics.

Anti-realists, by contrast, think we invent mathematics or construct it in some sense. Mathematical entities are therefore not independent of conscious rational minds capable of understanding and perceiving these concepts.

Of course there is overlap in both classifications, and I may have made some errors and omissions here, and anyone who has specialist knowledge of philosophy of mathematics is welcome to comment on and criticise both lists.

Links
“Philosophy of Mathematics,” Stanford Encyclopedia of Philosophy, 2007 (rev. 2012)
http://plato.stanford.edu/entries/philosophy-mathematics/

“Philosophy of Mathematics,” Wikipedia
http://en.wikipedia.org/wiki/Philosophy_of_mathematics

BIBLIOGRAPHY
Brown, James Robert. 2012. Platonism, Naturalism, and Mathematical Knowledge. Routledge, New York and Abingdon.

Putnam, H. 1972. Philosophy of Logic. George Allen & Unwin, London.

Gray, Jeremy. 2008. Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton University Press, Princeton, N.J. and Woodstock.

Critics of the Classic Hayekian Business Cycle Theory (Updated)

This list is updated in light of Kuehn (2013), a great overview of the subject.

My list:

(1) Piero Sraffa:
He was really the first and possibly the most important in these articles:

Sraffa, P. 1932. “Dr. Hayek on Money and Capital,” Economic Journal 42: 42–53.

Sraffa, P. 1932. “A Rejoinder,” Economic Journal 42 (June): 249–251.

See also:

Kurz, H. D. 2000. “Hayek-Keynes-Sraffa Controversy Reconsidered,” in H. D. Kurz (ed.), Critical Essays on Piero Sraffa’s Legacy in Economics. Cambridge University Press, Cambridge. 257-302.

Lawlor, M. S. and Horn, B. 1992. “Notes on the Hayek-Sraffa exchange,” Review of Political Economy 4: 317–340.

Lachmann, L. M. 1986. “Austrian Economics under Fire: The Hayek-Sraffa Duel in Retrospect,” in W. Grassl and B. Smith (eds.), Austrian Economics: Historical and Philosophical Background, Croom Helm, London. 225–242.

(2) Karl Gunnar Myrdal:

Myrdal, G. 1933. “Der Gleichgewichtsbegriff als Instrument der geld-theoretischen Analyse,” in F. A. Hayek (ed.), Beitrage zur Geldtheorie, Vienna. 361–487.

(3) Paul Rosenstein-Rodan:
However, I don’t think his criticisms were ever published (I could be wrong), but were certainly conveyed to Ludwig Lachmann in conversations in the 1930s as reported in this Austrian Economics Newsletter (AEN) (an interview with Lachmann):

“AEN: You have talked a number of times about the importance of expectations in business cycle theory. What first drew your interest to expectations as far as the business cycle question was concerned.

Lachmann: Talking to Paul Rosenstein-Rodan, who was then a lecturer at University College, London – not technically in the London School of Economics – but he gave a course on the history of economic thought to which all of us who were research students then went.
It was Rosenstein-Rodan who in discussing Austrian trade cycle theory with me said, ‘Ah yes, but whatever happens in the business cycle is in the first place determined by expectations.’ And then he told me of the work that had been done in Sweden.”
Ludwig Lachmann, “An Interview with Ludwig Lachmann,” The Austrian Economics Newsletter, Volume 1, Number 3 (Fall 1978), Mises.org.
http://mises.org/journals/aen/lachmann.asp

Rosenstein-Rodan was possibly referring to Gunnar Myrdal’s work listed in (2) above.

(4) John Maynard Keynes:
Keynes’s criticisms were in a response to Hayek’s review of his earlier work, and mostly in his private correspondence with Hayek:

Keynes, J. M. 1931. “The Pure Theory of Money. A Reply to Dr. Hayek,” Economica 34 (November): 387–397.

Ingrao, B. 2005. “When the Abyss Yawns and After. The Correspondence between Keynes and Hayek,” in M. C. Marcuzzo and A. Roselli (eds.), Economists in Cambridge. A Study Through their Correspondence, 1907–1946. Routledge, London. 236-256.

Ingrao, B. and F. Ranchetti. 2005. “Hayek and Cambridge: Dialogue and Contention,” in M. C. Marcuzzo and A. Roselli (eds.), Economists in Cambridge. A Study Through their Correspondence, 1907–1946. Routledge, London. 392-413.

Moggridge, D. (ed.). 1973. The Collected Writings of John Maynard Keynes (vol. 13). Macmillan for the Royal Economic Society, London.

It was in Keynes (1931) that he delivered his rather harsh verdict on Prices and Production:

“The book, as it stands, seems to me to be one of the most frightful muddles I have ever read, with scarcely a sound proposition in it beginning with page 45 … It is an extraordinary example of how, starting with a mistake, a remorseless logician can end up in Bedlam.” (Keynes 1931: 394).

(5) Ludwig M. Lachmann:

Lachmann, L. M. 1943. “The Role of Expectations in Economics as a Social Science,” Economica n.s. 10.37: 12–23.

(6) Frank H. Knight:

Knight, Frank H. 1935. “Professor Hayek and the Theory of Investment,” Economic Journal 45.177 (March): 77-94.

Here is one of Hayek’s responses to Knight:

Hayek, F. A. von. 1936. “The Mythology of Capital,” Quarterly Journal of Economics 50.2: 199-228.

See also:

Boettke, P. and K. Vaughn. 2002. “Knight and the Austrians on Capital and the Problem of Socialism,” History of Political Economy 34: 155–176.

Cohen, A. J. 2003. “Hayek/Knight Capital Controversy: The Irrelevance of Roundaboutness, or Purging Processes in Time?,” History of Political Economy 35.3: 469-490.

(7) Nicholas Kaldor:

Kaldor, N. 1939. “Capital Intensity and the Trade Cycle,” Economica n.s. 6.21: 40–66.

Kaldor, N. 1940. “The Trade Cycle and Capital Intensity: A Reply,” Economica n.s. 7.25: 16–22.

Kaldor, N. 1942. “Professor Hayek and the Concertina-Effect,” Economica n.s. 9.36: 359–382.

(8) George L. S. Shackle:

Shackle, George L. S. 1981. “F. A. Hayek, 1899– ,” in D. P. O’Brien and J. R. Presley (eds.), Pioneers of Modern Economics in Britain. Macmillan, London. 234–261, at p. 240.

(9) Gottfried von Haberler:

Haberler, G. 1986. “Reflections on Hayek’s Business Cycle Theory,” Cato Journal 6: 421–435.

Reprinted in Haberler, G. 1991. “Reflections on Hayek’s Business Cycle Theory,” in John Cunningham Wood and Ronald N. Woods (eds.), Friedrich A. Hayek: Critical Assessments (vol. 4). Routledge, London. 249–262.

(10) Gordon Tullock:

Tullock, G. 1988. “Why the Austrians are Wrong About Depressions,” The Review of Austrian Economics 2: 73–78.

(11) David Ramsay Steele
His views (quite insightful) are described in this comment:

http://consultingbyrpm.com/blog/2012/05/federal-government-outlays-and-receipts-as-of-nominal-gdp.html#comment-39021

(12) Allin Cottrel:

Cottrell, A. 1993. “Hayek’s Early Cycle Theory Re-examined,” Cambridge Journal of Economics 18: 197–212.

(13) Ulrich Witt:

Witt, U. 1997. “The Hayekian Puzzle: Spontaneous Order and the Business Cycle,” Scottish Journal of Political Economy 44: 44–58.

(14) Tyler Cowen:

Cowen, Tyler. 1997. Risk and Business Cycles: New and Old Austrian Perspectives. Routledge, London.

(15) Robert P. Murphy:
Although I suppose Murphy sees himself as a defender of the ABCT, nevertheless his work here is nothing but a critique of the classic Hayekian theory (where Hayek uses the Wicksellian natural rate of interest):

Murphy, Robert P. 2003. Unanticipated Intertemporal Change in Theories of Interest, PhD dissert., Department of Economics, New York University.

Murphy, Robert P. “Multiple Interest Rates and Austrian Business Cycle Theory.”
http://consultingbyrpm.com/uploads/Multiple%20Interest%20Rates%20and%20ABCT.pdf

(16) Robert L. Vienneau:

Vienneau, R. L. 2006. “Some Fallacies of Austrian Economics,” September
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=921183

Vienneau, R. L. 2010. “Some Capital-Theoretic Fallacies in Garrison’s Exposition of Austrian Business Cycle Theory,” September 4
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1671886

(17) Daniel Kuehn:

Kuehn, Daniel. 2013. “Hayek’s Business-Cycle Theory: Half Right,” Critical Review 25.3–4: 497–529.

Another very good work on this issue is Constatinos Repapis, “Hayek’s Business Cycle Theory during the 1930s: A Critical Account of its Development,” History of Political Economy 43 (2011): 699–742.

In addition, here are some criticisms of Mises’s version of the ABCT:

Vasséi, Arash Molavi. 2010. “Ludwig von Mises's Business Cycle Theory: Static Tools for Dynamic Analysis,” in Harald Hagemann, Tamotsu Nishizawa, Yukihiro Ikeda (eds.). Austrian Economics in Transition: From Carl Menger to Friedrich Hayek. Palgrave Macmillan, Basingstoke. 196–217.

Vasséi, Arash Molavi. 2010. “The Foundation of Ludwig von Mises’s Business Cycle Theory: Real Analysis as a Chain of Tautologies,” SSRN paper, June 5, 2012.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2077604