Philosophy of mathematics

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  1. Issues and positions
    1. Logic
    2. Algebra
    3. Analysis
    4. Language
    5. Pythagoreanism
    6. Platonism
    7. Logicism
    8. Incompleteness
    9. Intuitionism
    10. Fictionalism
    11. Nominalism
    12. Structuralism
    13. Naturalism
  2. My thoughts
  3. Annotated bibliography
    1. Carnap, R. (1950). Empiricism, Semantics, and Ontology.
    2. Wigner, E.P. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
    3. Field, H. (1980). Science Without Numbers.
    4. Snapper, E. (1979). The Three Crises in Mathematics: Logicism, Intuitionism, Formalism.
    5. More articles to do
  4. Links and encyclopedia articles
    1. SEP
    2. IEP
    3. Wikipedia
    4. Others
    5. Videos
  5. References

Issues and positions





A visual proof that \sum_{k=1}^{n} k = (n^2+n)/2.

A visual proof that \(\sum_{k=1}^{n} k = (n^2+n)/2\).

A visual proof that \sum_{k=1}^{\infty} \frac{1}{4^k} = \frac{1}{3}.

A visual proof that \(\sum_{k=1}^{\infty} \frac{1}{4^k} = \frac{1}{3}\).





Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.4


Gödel, Kurt (1906-1978)

From Wikipedia:




Taking the Principle of the Excluded Middle from the mathematician … is the same as … prohibiting the boxer the use of his fists.

Reid 1996, p. 150.





My thoughts

Jon Lawhead - One of the climate people just asked me “how do philosophers get paid?” He was blown away that most people don’t do grant writing, and that we tend to juggle our writing with teaching. He also got very excited about the unreasonable effectiveness of mathematics problem, and marveled that we get paid to think about things like that.

It is awesome. Some level of necessitarianism gots to be on the right track there right?

Jon Lawhead - I’m not convinced of that, Ryan Reece, though I’m far from an expert on this area. His intuitions leaned in the constructivist (David Hilbert-y) direction, as do mine. That is, that math is effective because we designed it to be that way, since it’s a general language in which to talk about patterns (which is, at bottom, what science is all about). Discovering more mathematical truths involves discovering more implications of the formal system we’ve designed, and/or extending that formal system deliberately.

Of course I’m not an expert here either, but lots of “…buts” come to mind when I’m told that math is just a language. (Forgive my brain dump.)

There’s a lot of hangups here because common language is imprecise about differentiating a mathematical concept from its notation (obviously constructed). I find it really hard not to be convinced that, for example, if we made contact or found evidence of intelligent life in another star system, and we were able to comb their mathematical journals, there would be a metaphysical fact of the matter to whether or not they had a theory of differential and integral calculus. There would similarly be a separate fact of the matter as to whether they knew the pythagorean theorem or whether they had discovered that there are finite number of simple Lie groups. Regardless of its notation or history of construction, there will be mathematical concepts represented that we could identify.

We can make the same argument with disconnected cultures here on earth, and identify that ancient Indians and Egyptians both knew about fractions, even if they didn’t have a concept of groups. Similarly, independent of the notational construction, we celebrate that both Newton and Leibniz developed fundamental concepts in calculus independently (even with the controversy about how much of each other’s documents they may have seen).

Surely this against-formalism type of argument has been made by people better versed in this than me. I’m reacting to reading Carnap’s ESO recently. Carnap agreed with you that math is a construction, but my list of “buts” to him would start with pointing out that in the Abstract vs Nominalistic divide, we should also be careful to further divide abstract concepts as to whether or not they are natural. For example, Vector spaces are different kinds of abstractions than unicorns. Why? Because they are natural. It’s those natural kinds that we can be confident we could identify across linguistic barriers.

I’m still trying to piece together who has threaded this argument together the best. But I think the right road to countering the positivists/nominalists rejection of the reality of all abstractions is by pointing out that some of our abstractions are natural kinds, which in some sense, cut nature at its joints and describe a real structure in nature. The digits of pi, for example, are discovered, not constructed.

I realize I’ve left things question begging as to what makes a natural kind, but at least theres a direction to march now, and I don’t think clarifying that definition is insurmountable. Probably many philosophers have already done it for me.

Annotated bibliography

Carnap, R. (1950). Empiricism, Semantics, and Ontology.

1. The problem of abstract entities

My thoughts

Wigner, E.P. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

My thoughts

Field, H. (1980). Science Without Numbers.

My thoughts

Snapper, E. (1979). The Three Crises in Mathematics: Logicism, Intuitionism, Formalism.

My thoughts







Baez, J. C. (2016). Struggles with the continuum.

Bueno, O. (2013). Nominalism in the philosophy of mathematics. Stanford Encyclopedia of Philosophy.

Carnap, R. (1950). Empiricism, semantics, and ontology. Revue Internationale de Philosophie, 4, 20–40.

Field, H. (1980). Science Without Numbers. Princeton University Press.

Hilbert, D. (1967). The foundations of mathematics. In J. van Heijenoort (Ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (pp. 464–479). Harvard University Press. From a lecture given by Hilbert in 1927.

Snapper, E. (1979). The three crises in mathematics: Logicism, intuitionism and formalism. Mathematics Magazine, 52, 207–216.

Wigner, E. P. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics, 13, 1–14. Richard courant lecture in mathematical sciences delivered at New York University, May 11, 1959.

  1. Baez (2016).

  2. Wigner (1960).

  3. Snapper (1979).

  4. Hilbert (1967), p. 479. Discussion by Michael J. O’Donnell:

    Well, the tone sounds like hubris again, but this is actually a relatively modest claim, and largely substantiated by events. Mathematics may arguably be understood as the science of forms, which have an objective quality independent of individual beliefs. What is missing here is the claim that a single formal system may embrace all of mathematics at once, and that we may prove that it contains no error. Notice that Hilbert objects to the “actual, contentual assumptions” of Russell and Whitehead. But, recall that Hilbert claims to treat numerical equations contentually, and even seems to regard that as a virtue. It is not the mere contentual quality of Russell’s and Whitehead’s assumptions that Hilbert objects to, but the fact that each particular assumptions is not verifiable by a single computation or finite observation, and furthermore that there is not even a proof that they are consistent with basic numerical equations. Of course, Hilbert’s proposed system never gets its consistency proof either. Oh well.

  5. Field (1980).

  6. Bueno (2013).