Abstract
I have argued that the unreasonable effectiveness of mathematics in fundamental physics leads to a Pythagorean view of the world in which the underlying principles are structural, mathematical principles. This naturally raises one of the key questions which Wigner addressed in his investigation of the relationship between physics and mathematics: ‘What is mathematics?’ That is the question which I address in the next two chapters. I am particularly interested in exploring the implications of a Pythagorean view for one’s philosophy of mathematics. What should Pythagoreans believe about the nature of mathematics? What are the potential problems for this view? It seems clear that, for Pythagoreans, mathematics is necessarily true, eternal and unchanging. Pythagoreans should be realists about mathematics. They should regard the pursuit of mathematics as discovering facts about the world. They should look to mathematics as the ultimate arbiter of questions about existence and truth. It is the last point that I am mainly concerned with in this chapter: can mathematics serve as the ultimate arbiter of questions about existence and truth? More particularly, I investigate whether there is one background theory of mathematics in which any mathematical problem can be interpreted and (potentially) decided. If there is—as universalists believe—then the various disciplines of mathematics are part of one, true mathematical universe. If not—as held by pluralists—then there are alternative mathematical universes and mathematical statements may be true in some universes and false in others, not necessarily true. That would be a problem for Pythagoreanism.
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Notes
- 1.
One interesting exception is Harvey Friedman’s efforts to ‘reverse engineer’ textbook mathematics in order to reveal exactly what set-theoretical machinery is used (Friedman and Simpson 2000).
- 2.
Following Hilbert’s famous exclamation: “No one shall expel us from the paradise which Cantor has created for us” (Boyer 1991: 570).
- 3.
Let T1 and T2 be recursively enumerable axiom systems. We say that T1 is interpretable in T2 (T1 ≤ T2) when, roughly speaking, there is a translation τ from the language of T1 to the language of T2 such that, for each sentence φ of the language of T1, if T1 proves φ then T2 proves τ(φ). T1 and T2 are said to be mutually interpretable when both T1 ≤ T2 and T2 ≤ T1 (Koellner 2010: 5).
- 4.
The Axiom of Infinity asserts that there are infinite sets, that is, completed infinite collections.
- 5.
Cantor used the term inconsistent manifold.
- 6.
Axiom systems other than ZFC may omit the Axiom of Foundation, and may even include its negation, allowing for non-well-founded sets. Such sets may be “pathological” in some sense and not suitable for some mathematical applications.
- 7.
- 8.
The statement that ZF has any models at all is equivalent to the statement that ZF is consistent, which is an assumption that cannot be proved within ZF. This follows from Gödel’s Second Incompleteness Theorem together with his completeness theorem for first-order logic.
- 9.
A set X is logically definable from parameters in Lα if there exist elements a1,…,an of Lα and a logical formula Ф(x1,…,xn) in the formal language for set theory such that X = {a Є Lα | (Lα, Є) ╞ Ф(a,a1,…,an)]} where the order (Lα, Є) is the set Lα with the membership relation Є restricted to it and ╞ is the symbol for consequence (Woodin 2010a: 3).
- 10.
For example, it is not necessary for the ground model to be a countable, standard, transitive model and not all forcing extensions are cardinal preserving.
- 11.
The Lowenheim–Skolem theorem says that if there exists an infinite model of PA, then there exist models of all cardinality, from countable upwards.
- 12.
Note that, according to Gödel’s Second Incompleteness Theorem, such models must exist if PA is consistent because it is consistent with its own inconsistency statement.
- 13.
Externalists think that all that it is needed to justify a belief is for the belief to be acquired by a reliable method (in this case, the construction of consistent mathematical theories); we don’t need to know that the method is reliable or be able to explain its reliability (Balaguer 1995: 309).
- 14.
Magidor (2011: 2) also advocates this view, although he is prepared to contemplate a small number of alternative universes.
- 15.
- 16.
Finitism is often associated with the German mathematician Kronecker, who is attributed with the statement: “God made the integers, all the rest is the work of man” (Boyer 1991: 570).
- 17.
- 18.
See Kanamori (1994: 17) for a definition of a Mahlo cardinal (N.B. the definition of large cardinals is very technical).
- 19.
By Gödel’s Second Incompleteness Theorem, large cardinal axioms at the level of a strongly inaccessible or higher cardinal cannot be shown to be relatively consistent with ZFC because, if x is a strongly inaccessible or higher cardinal, then it can be shown that Vx is a model of ZFC and, thus, that ZFC is consistent (Maddy 1997: 74).
- 20.
For technical reasons, Gödel thought that the resolution of the continuum hypothesis would require new axioms of an extrinsic kind (i.e. based on hitherto unknown principles). In fact, it is now known that the continuum hypothesis is independent of all remotely plausible large cardinal axioms that have been considered to date.
- 21.
Here, as always, one must be careful to keep separate the concepts of proper class and set, for example, V has the property of containing all sets, but no set (and, therefore, no initial segment) can have that property.
- 22.
See Koellner (2009a: 212–213) for a definition of these large cardinals.
- 23.
See the discussion of Gödel’s constructible universe L in Sect. 4.2.4 for a definition of definable sets. By starting with the real numbers R and iterating the definable power set operation along the ordinals, we get the hierarchy L(R) of definable sets of reals.
- 24.
See Woodin (2010a: 4) for details.
- 25.
See Steel (2007) for a definition of Woodin cardinals.
- 26.
See Kanamori (1994: 298) for a definition of a supercompact cardinal.
- 27.
For more information on Ω-logic and its relation to full first- and second-order logic, see Koellner (2010).
- 28.
The Strong Ω Conjecture is the Ω Conjecture plus a conjecture about an extended form of the Axiom of Determinacy. See Koellner (2011c: 16–17).
- 29.
Note that the continuum hypothesis is just a particular case; there are other suitable sentences which could be forced to differ between Ω-complete theories of Vα.
- 30.
For example, it would be correct about singular cardinals and compute their successors correctly (Woodin 2011c: 465).
- 31.
According to predicativists, the natural numbers form a definite totality but the collection of all sets of natural numbers (i.e. the power set) does not. For more on predicativity, see Feferman (2005).
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McDonnell, J. (2017). One True Mathematics. In: The Pythagorean World. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-40976-4_4
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