Philosophy of mathematics

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Issues and positions

Logic

• Ancient logic and Aristotle
• Modus ponens, modus tollens, Affirming the consequent, Proof by contrapositive
• Frege, Gottlob (1848-1925)
• first-order and second-order logic

Algebra

• roots from Babylonians
• Kronecker, Leopold (1823-1891)

Analysis

• Newton, Isaac (1642-1726/7)
• Leibniz, Gottfried Wilhelm (1646-1716)
• Euler, Leonhard (1707-1783)
• Gauss, Carl Friedrich (1777-1855)
• Cauchy, Augustin-Louis (1789-1857)
• Weierstrass, Karl (1815-1897)
• Stokes, George (1819-1903)
• Struggles with the continuum1.

Platonism

• Realist
• The Academy (387 BCE - 529 CE): “Let no one ignorant of geometry enter”
• Mathematics is descriptive of a real but trans-empirical realm.
• A very platonist math documentary

Logicism

• Formalism
• Antirealist, Positivism
• Wigner2
• Review article on the philosophy of math by Snapper3

Hilbert:

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

Incompleteness

Gödel, Kurt (1906-1978)

From Wikipedia:

• The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an “effective procedure” (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers.
• The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency. Sufficiently strong proof theories cannot prove their own consistency (provided that they are in fact consistent).
• Gödel’s second incompleteness theorem also implies that a theory T1 satisfying the technical conditions outlined above cannot prove the consistency of any theory T2 that proves the consistency of T1. This is because such a theory T1 can prove that if T2 proves the consistency of T1, then T1 is in fact consistent. For the claim that T1 is consistent has form “for all numbers n, n has the decidable property of not being a code for a proof of contradiction in T1”. If T1 were in fact inconsistent, then T2 would prove for some n that n is the code of a contradiction in T1. But if T2 also proved that T1 is consistent (that is, that there is no such n), then it would itself be inconsistent. This reasoning can be formalized in T1 to show that if T2 is consistent, then T1 is consistent. Since, by second incompleteness theorem, T1 does not prove its consistency, it cannot prove the consistency of T2 either.
• The corollary also indicates the epistemological relevance of the second incompleteness theorem. It would actually provide no interesting information if a theory T proved its consistency. This is because inconsistent theories prove everything, including their consistency. Thus a consistency proof of T in T would give us no clue as to whether T really is consistent; no doubts about the consistency of T would be resolved by such a consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a theory T in some theory T’ that is in some sense less doubtful than T itself, for example weaker than T. For many naturally occurring theories T and T’, such as T = Zermelo-Fraenkel set theory and T’ = primitive recursive arithmetic, the consistency of T’ is provable in T, and thus T’ can’t prove the consistency of T by the above corollary of the second incompleteness theorem.
• The second incompleteness theorem does not rule out consistency proofs altogether, only consistency proofs that could be formalized in the theory that is proved consistent. For example, Gerhard Gentzen proved the consistency of Peano arithmetic (PA) in a different theory that includes an axiom asserting that the ordinal called $$\varepsilon_0$$ is wellfounded; see Gentzen’s consistency proof. Gentzen’s theorem spurred the development of ordinal analysis in proof theory.

More:

• von Neumann
• Gödel was a platonist, a (religious) realist.

Intuitionism

• Antirealist
• Kronecker’s finitism was a forerunner of intuitionism in foundations of mathematics.
• L.E.J. Brouwer (1881-1966)

Hilbert:

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.

• Antirealist

Nominalism

• Antirealist
• Science Without Numbers5
• Bueno6

• Realist

My thoughts

• What would happen if you asked an Alien to solve a Rubik’s cube?

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.

• Carnap (1950)

1. The problem of abstract entities

• Empiricists tend to prefer to restrict themselves to nominalistic language – without containing references to abstract entities.

My thoughts

• TODO.
• The map is not the territory (Jorge Luis Borges).

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

• Wigner (1960)

• TODO.

• Field (1980)

• TODO.

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

• Snapper (1979)

• TODO.

• TODO.

References

Baez, J. C. (2016). Struggles with the continuum. https://arxiv.org/abs/1609.01421

Bueno, O. (2013). Nominalism in the philosophy of mathematics. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/nominalism-mathematics/

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. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

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).