6  Philosophy of mathematics

Published

October 10, 2025

These next several outlines deal with philosophy of certain specialized topics, starting with this one on the philosophy of mathematics. Here we dig into issues of what is abstraction.

First we survey the following branches of mathematics: algebra, analysis, numbers theory, logic, model theory, and category theory. Then we discuss unification programs across branches, followed by sections on positions in philosophy of mathematics, in particular dealing with the realism/antirealism or platonism/nominalism debate.

6.1 Algebra

6.1.1 Introduction

TODO

Prove all things; hold fast that which is good.

– 1 Thessalonians, 5:21–28, KJV

6.1.2 History

  • roots from Babylonians
  • Al-Karaji (c. 953-1029)
  • Gerolamo Cardano (1501-1576)
  • Carl Friedrich Gauss (1777-1855)
    • published a proof of the fundamental theorem of algebra (1797)
    • The theorem states that the field of complex numbers is algebraically closed.
  • Évariste Galois (1811-1832)
  • Arthur Cayley (1821-1895)
  • Leopold Kronecker (1823-1891)
  • arithmetic vs abstract algebra

6.1.3 Linear algebra

  • Linear equations and transformation
  • Vector spaces and dual spaces
    • For an infinite dimensional vector space, V, while elements of V must have “finite support” (only finitely many nonzero coordinates in any basis representation), elements of V* can have “infinite support.”
  • Axler, S. (2024). Linear Algebra Done Right.
  • Spectral theorem: \(A\) is normal, \(A A^\dagger = A^\dagger A \iff A = U D U^\dagger\) is an eigendecomposition of \(A\), where the column vectors of \(U\) are the eigenvectors of \(A\) with corresponding eigenvalues on the diagonal of \(D\).
  • Principal Component Analysis (PCA)
  • Rayleigh quotient
Figure 6.1: Matrix World: Classifying all matrices (source: The Art of Linear Algebra).

See also:

6.1.4 Lie groups

1 Ramond (1976).

2 Hall (2000).

3 Zee (2016).

4 Siegel (2014).

Cartan’s classification:

  • \(\mathrm{A}_{n} = \mathrm{SU}(n+1)\)
  • \(\mathrm{B}_{n} = \mathrm{SO}(2n+1)\)
  • \(\mathrm{C}_{n} = \mathrm{Sp}(n)\)
  • \(\mathrm{D}_{n} = \mathrm{SO}(2n)\)
  • Exceptional groups: \(\mathrm{E}_6\), \(\mathrm{E}_7\), \(\mathrm{E}_8\), \(\mathrm{F}_4\), \(\mathrm{G}_2\)

6.1.5 Finite groups

5 Carter (2009).

Summarizing the classification of finite simple groups:

  • Cyclic groups of prime order, \(\mathrm{Z}_n\)
  • Alternating groups, \(\mathrm{Alt}_n\) for \(n \geq 5\)
  • Lie-type groups:
    • Classical/Chevalley groups: \(\mathrm{A}_n\), \(\mathrm{B}_n\), \(\mathrm{C}_n\), \(\mathrm{D}_n\)
    • Exceptional groups: \(\mathrm{E}_6\), \(\mathrm{E}_7\), \(\mathrm{E}_8\), \(\mathrm{F}_4\), \(\mathrm{G}_2\)
    • Twisted groups: \({}^{2}\mathrm{A}_{n}\), \({}^{2}\mathrm{D}_{n}\), \({}^{3}\mathrm{D}_{4}\), \({}^{2}\mathrm{E}_{6}\), plus the Suzuki and Ree groups \({}^{2}\mathrm{B}_{2}\), \({}^{2}\mathrm{F}_{4}\), \({}^{2}\mathrm{G}_{2}\)
  • Tits group: \({}^{2}\mathrm{F}_{4}(2)^\prime\)
  • 26 Sporadic groups
    • 20 groups in the Happy Family
    • 6 Pariahs

6.1.6 Abstract algebra

  • Groups, rings, fields
  • A semigroup is a set with a closed, associative, binary operation.
  • A monoid is a semigroup with an identity element.
  • A group is a monoid with inverse elements.
  • An abelian group is a group where the binary operation is commutative.
  • A ring is an abelian group (under addition) with a second closed, associative, binary operation (multiplication) that distributes over the first.
  • A division ring is a ring where every non-zero element has a multiplicative inverse.
  • A field is a commutative division ring (where multiplication commutes).
  • Vector space
  • Modules are to rings as vector spaces are to fields.

\[ \begin{align} \mathrm{sets} &\supset \mathrm{semigroups} \supset \mathrm{monoids} \supset \mathrm{groups} \supset \mathrm{abelian\ groups} \\ &\supset \mathrm{rings} \supset \mathrm{division\ rings} \supset \mathrm{fields} \end{align} \]

6.1.7 More

6 Baez (2002).

7 Westbury (2010).

8 McCrimmon (2000).

See also:

6.2 Analysis

6.2.1 Introduction

TODO

6.2.2 History

  • René Descartes (1596-1650)
    • Geometry and coordinates, modern notation, geometric problems forumalted with algebra
    • La Géométrie (1637)
  • Isaac Newton (1642-1726/7)
  • Gottfried Wilhelm Leibniz (1646-1716)
  • Jacob Bernoulli (1655-1705)
  • Leonhard Euler (1707-1783)
    • Mechanica (1736)
  • Pierre-Simon Laplace (1749-1827)
  • Carl Friedrich Gauss (1777-1855)
  • Augustin-Louis Cauchy (1789-1857)
  • Karl Weierstrass (1815-1897)
  • George Stokes (1819-1903)
  • Differential forms
  • Geometry and the Erlangen program

6.2.3 Development of calculus

Figure 6.2: Leibniz’s notation of integration: \(\int\) (summa) and differentiation: \(d\) (differentia) summarized in the margin of his notes in 1675. Note that \(\Pi\) was Leibniz’s notation for equality. Photo by S. Wolfram (2013) of notes in the Leibniz-Archiv in Hanover, Germany.
  • James Gregory (1638-1675)
  • Isaac Barrow (1630-1677)
    • Fundamental theorem of calculus in Proposition 11, Lecture 10, of his Lectiones Geometricae, published in 1670.
  • Isaac Newton (1642-1727)
    • Newton claimed to have the ideas of calculus in the mid 1660s.
    • In 1669, Newton wrote an article on infinite series with ideas leading to calculus: “De analysi per aequationes numero terminorum infinitas”, which wasn’t published until 1711, 42 years later.
    • Philosophiæ Naturalis Principia Mathematica (1687)
  • Gottfried Wilhelm Leibniz (1646-1716)
    • Leibniz first used \(dx\) in publication in
      Leibniz, G.W. (1684). Nova methodus pro maximis et minimis. Acta Eruditorum.
    • Leibniz first used his intergral sign, \(\int\), in publication in
      Leibniz, G.W. (1686). De geometria recondita et analysi indivisibilium atque infinitorum. Acta Eruditorum.
    • Wolfram, S. (2013). Dropping in on Gottfried Leibniz.
  • Aldrich, John. (?). Earliest uses of symbols of calculus.
  • Leibniz-Newton calculus controversy
    • In 1849, C. I. Gerhardt, while going through Leibniz’s manuscripts, found extracts from Newton’s “De analysi per aequationes numero terminorum infinitas”.
    • Starbird, M. (2016). Who invented calculus?

6.2.3.1 Matrix calculus

9 Dwyer (1967).

6.2.4 Differential geometry

10 Burke (1985).

11 Frankel (1997).

12 Frè (2013), ch. 2.

13 Varadarajan (2003).

14 Tao (2007).

15 Sussman & Wisdom (2013).

16 Tu (2017).

17 Bronstein, Bruna, Cohen, & Velickovic (2021), p. 56–60.

18 Connes (1985).

\[ a \times b = \star(a \wedge b) \]

See also:

6.3 Number theory

6.3.1 Introduction

6.3.2 Set theory

  • Membership: Axiom of extensionality
  • von Neumann’s set theoretical definition of numbers

6.3.2.1 Naive Set Theory

6.3.2.2 Zermelo-Fraenkel set theory

6.3.2.3 Other approaches

6.3.3 Transfinite numbers

  • Transfinite numbers were anticipated by Robert Grosseteste (ca. 1168-1253).
  • Ordinal (index) vs cardinal (size) numbers
  • Developed by Georg Cantor (1845-1918) in 1895
  • Cantor’s theorem
    • Let \(|A| \equiv \mathrm{card}(A)\) denote the cardinality (i.e size) of a set, \(A\).
    • A power set, \(P(A)\), of a set, \(A\), is the set of all subsets of \(A\).
    • The cardinality of a power set is \(|P(A)| = 2^{|A|}\)
    • The cardinality of a power set is strictly larger than the set: \(|A| < |P(A)|\)
  • Cantor-Bernstein-Schröeder theorem
  • Transfinite numbers:
    • \(\omega\): the smallest transfinite ordinal number; the order type of the natural numbers.
    • \(\aleph_0\): the first transfinite cardinal number; the cardinality of the natural numbers, \(\aleph_0 \equiv |\mathbb{N}|\)
    • There is a one-to-one correspondence between ordinal and cardinal numbers. 19 \(\omega = \aleph_0\)
    • The \(\aleph_n\) hierarchy of cardinals is defined by transfinite recursion:
      • \(\aleph_0\) is the smallest infinite cardinal.
      • \(\aleph_{n+1}\) is the successor cardinal to \(\aleph_{n}\)
      • \(\aleph_{\lambda} = \mathrm{sup}_{n<\lambda} \aleph_n\) for limit ordinals \(\lambda\)
    • \(\aleph_0\) is the smallest infinite cardinal; it is countable.
    • \(\aleph_1\) is the first uncountable cardinal.
    • \(\aleph_2\) is the second uncountable cardinal.
  • Cardinality of the continuum
    • The cardinality of the reals: \(C \equiv |\mathbb{R}|\)
    • The cardinality of the reals is the power set of the natural numbers: \(C \equiv |\mathbb{R}| = |P(\mathbb{N})| = 2^{|\mathbb{N}|} = 2^{\aleph_0} > \aleph_0\)
    • Examples of sets with cardinality = \(C\)
      • real numbers, \(\mathbb{R}\)
      • closed or open intervals on \(\mathbb{R}\)
      • Euclidean space, \(\mathbb{R}^n\)
      • complex numbers, \(\mathbb{C}\)
      • set of all continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\)
      • power set of the natural numbers, \(P(\mathbb{N})\)
    • Examples of sets with cardinality > \(C\)
      • set of all functions from \(\mathbb{R}\) to \(\mathbb{R}\)
      • power set of the real numbers, \(P(\mathbb{R})\)
  • Continuum Hypothesis (CH)
    • CH: There is no set \(S\) such that \(\aleph_0 < |S| < 2^{\aleph_0}\)
    • \(C = 2^{\aleph_0}\), and under CH, \(C = 2^{\aleph_0} = \aleph_1\).
    • Relationship with the axiom of choice
    • Paul Cohen showed the CH is undecidable in ZFC (1963).
  • Generalized Continuum Hypothesis (GCH)
    • \(\aleph_{n+1} = 2^{\aleph_n}\)
  • Surreal numbers
  • Pedagogy

19 Trioni, S. (2020). Cantor’s attic - Omega.

20 Bajnok (2013).

Hilbert:

There is, however, a completely satisfactory way of avoiding the paradoxes without betraying our science. The desires and attitudes which help us find this way and show us what direction to take are these:

  1. Wherever there is any hope of salvage, we will carefully investigate fruitful definitions and deductive methods. We will nurse them, strengthen them, and make them useful. No one shall drive us out of the paradise which Cantor has created for us.
  2. We must establish throughout mathematics the same certitude for our deductions as exists in ordinary elementary number theory, which no one doubts and where contradictions and paradoxes arise only through our own carelessness. 21

21 Hilbert (1926), p. 191.

6.4 Logic

6.4.1 Introduction

Pedagogy:

  • Hunter, G. (1971). Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. 22
  • Monk, J.D. (1976). Mathematical Logic. 23
  • Sullivan, B.W. (2013). Everything You Always Wanted To Know About Mathematics. 24
  • Smith, P. (2020). An Introduction to Formal Logic. 25
  • Smith, P. (2022). Beginning Mathematical Logic A Study Guide. 26

22 Hunter (1971).

23 Monk (1976).

24 Sullivan (2013).

25 Smith (2020).

26 Smith (2022).

More:

  • Carnap, R. (1958). Introduction to Symbolic Logic and its Applications. 27
  • Teller, P. (1989). A Modern Formal Logic Primer. 28
  • Bonevac, D. (2003). Deduction: Introductory to Symbolic Logic. 29
  • MacFarlane, J. (2021). Philosophical Logic: A Contemporary Introduction. 30
  • The Open Logic Text 31
  • logicinaction.org
  • logicmatters.net

27 Carnap (1958).

28 Teller (1989).

29 Bonevac (2003).

30 MacFarlane (2021).

31 Open Logic Project (2020).

6.4.2 History

32 Sheffer (1913).

6.4.3 Propositional logic

  • Propositional logic
    • AKA propositional calculus and zeroth-order logic
  • Validity and soundness:
    • An argument is valid iff for any assignment of the truth values in the argument where all of the premises are true, the conclusion is always true.
    • An argument is sound iff it is valid and all of its premises are true.
  • Sequent calculus

Syntactic consequence:

\[ A \vdash B \]

means that \(B\) is logically derivable/provable from \(A\); \(B\) is a theorem of the premises, \(A\).

Semantic consequence:

\[ A \models B \]

means that in all possible valuations in which \(A\) is true, \(B\) is also true. One says that \(A\) “entails” \(B\), or \(A\) “models” \(B\).

Note that \(\vdash\), \(\models\), and \(\equiv\) are all metalogical symbols, not part of the rules of logic; they are shorthands.

\(\vdash\) means “from which, it is derivable that”.
\(\models\) means “entails that, i.e. in every case this is true, that is true (regardless of provability)”.
\(\equiv\) means “is identical to”.

TODO: discuss the differences between logical equivalence and material equivalence.

Lecture notes on soundness and completeness:

A formal system is sound if everything that is provable is in fact true, i.e. if \(A_1, A_2, \ldots A_n \vdash B\), then \(A_1, A_2, \ldots A_n \models B\).

A formal system is complete if everything that is true has a proof, i.e. if \(A_1, A_2, \ldots A_n \models B\), then \(A_1, A_2, \ldots A_n \vdash B\).

Propositional logic was proven to be sound (\(\vdash\) implies \(\models\)) and complete (\(\models\) implies \(\vdash\)) by Emil Post in 1921. 33

33 Post (1921).

  • Emil Post and his anticipation of Gödel and Turing 34

34 Stillwell (2004).

Material implication:

\[ P \rightarrow Q \equiv \lnot P \lor Q \]

35 von Fintel (2011).

Modus ponens:

\[ P \rightarrow Q, P \vdash Q \]

Modus tollens:

\[ P \rightarrow Q, \lnot Q \vdash \lnot P \]

Peirce’s law:

\[ ((P \rightarrow Q) \rightarrow P) \rightarrow P \]

  • TODO: More basic examples in propositional logic
  • TODO: Use-mention distinction: P vs ‘P’ vs Quine quotes
  • Cut-elimination theorem

6.4.4 First-order logic

  • First-order logic
    • AKA predicate logic
    • Domain of discourse: \(\{x\}\)
    • Adds (non-logical) predicates, \(Fx\), and quantification over elements, \(\exists x\ Fx\).
  • C.S Peirce was first to distinguish between propositional logic, first-order logic, and second-order logic in 1885. 36
  • Consistency, completeness, expressivity
  • Gödel’s completeness theorem
    • Establishes a correspondence between semantic truth and syntactic provability in first-order logic.
    • Gödel, K. (1929). Über die Vollständigkeit des Logikkalküls. 37
      • His doctoral dissertation, University Of Vienna.
      • The first proof of the completeness theorem.
    • Henkin, L. (1996). The discovery of my completeness proofs. 38
    • Awodey, S. & Forssell, H. (2013). First-order logical duality. 39
    • Lindström’s theorem
  • Presburger arithmetic
    • First-order theory of the natural numbers with addition, but without multiplication.
    • Presburger arithmetic is consistent, complete, and decidable.

36 Ewald (2018).

37 Gödel (1929).

38 Henkin (1996).

39 Awodey & Forssell (2013).

6.4.4.1 Limitations

  • Löwenheim-Skolem theorem
    • The Löwenheim-Skolem theorem implies that infinite structures cannot be categorically axiomatized in first-order logic.
    • No first-order theory has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line.
  • In second-order logic, it is possible to define the addition and multiplication operations from the successor operation, but this cannot be done in the more restrictive setting of first-order logic.
  • Tennenbaum’s theorem
  • Quantifiers other than \(\forall\) and \(\exists\) are only definable within second-order logic or higher-order logics.

40 Bès (2002).

41 Bès & Choffrut (2022).

See also:

6.4.5 Second-order logic

  • Second-order logic
  • Second-order and higher-order logic, SEP
  • Includes relation variables in addition to object variables and allows quantification over both.
    • Extends first-order logic to allow predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted.
    • \(\exists P\ P(x)\)
  • Addition and multiplication are definabile in second-order logic.
  • The power set can be written in terms of second-order logic.
    • This second-order expressibility of the power-set operation permits the simulation of higher-order logic within second order. 42
  • \(\mathbb{N} \models \mathrm{PA}\)
    • The natural numbers model PA, but not uniquely in FOL.
    • In SOL, PA is categorical, meaning it has only one model up to isomorphism.
  • Henkin semantics
    • SOL with Henkin semantics is complete. 43
  • Higher-order logics, type theory
    • Russell’s theory of types 44
    • Alonzo Church’s lambda calculus 45
    • All higher-order logics have the same expressive power as second-order logic.
    • Modern type theory and its relationship with category theory

42 Enderton (2009).

43 Henkin (1950).

44 B. Russell (1908).

45 Church (1940).

6.4.5.1 Incompleteness of second-order logic

46 Kleene (1943).

47 Rossberg (2004).

See also:

6.4.5.2 Discussion

48 Kleene (1952).

49 Shapiro (1991).

50 G. Russell (2015).

51 Boolos (1984).

52 Jerzak (2009).

53 Bueno (2010).

54 Sider (2022).

6.4.6 Modal logic

  • C.I. Lewis (1883-1964)
    • Founded modern modal logic.
    • Criticism of material implication. Introduced strict implication. 55
    • Strict implication is not truth-functional. It requires asking about the truth-values that propositions take in worlds other than the actual world.
  • Rudolf Carnap (1891-1970)
    • Carnap, R. (1947). Meaning and Necessity. 56
  • Saul Kripke (1940-2022)
    • Kripke, S.A. (1959). A completeness theorem in modal logic. 57
    • Revives Leibniz’s idea that necessity is truth in all possible worlds.
  • David Lewis (1941-2001)
    • On the Plurality of Worlds (1986) 58

55 C. I. Lewis (1917).

56 Carnap (1947).

57 Kripke (1959).

58 D. Lewis (1986).

\(\Box\) means “necessarily”. \(\Diamond\) means “possibly”.

If necessarily \(P\), then necessarily necessarily \(P\):

\[ \Box P \rightarrow \Box \Box P \]

De Morgan duality:

\[ \Diamond P = \lnot \Box \lnot P \]

\[ \Box P = \lnot \Diamond \lnot P \]

More:

6.4.7 Alternative logics

  • Liar paradox
  • Intuitionistic Logic
    • Law of excluded middle (LEM), Tertium non datur:
      \(\vdash A \lor \lnot A\)
    • Law of double negation (LDN):
      \(\lnot \lnot A \leftrightarrow A\)
    • Intuitionistic logic rejects LEM and LDN.
    • Constructive mathematics
  • Paraconsistent logic
    • Principle of explosion
    • Ex contradictione quodlibet (ECQ): from a contradiction anything follows.
      \(A, \lnot A \models B\)
    • Law of No Contradiction (LNC).
      \(\models \lnot ( A \land \lnot A )\)
    • Paraconsistent logics reject ECQ, and may or may not invalidate LNC.
    • Dialetheism rejects LNC.
    • Priest, G. (1998). What is so bad about contradictions? 60
    • Martínez-Ordaz, M. del R. (2021). The ignorance behind inconsistency toleration. 61

59 Carroll (1895).

60 Priest (1998).

61 Martínez-Ordaz (2021).

Criticism:

  • Quine, W.V.O. (1986). Philosophy of Logic. 62

62 Quine (1986).

See also:

6.4.8 Proof theory

63 Viteri & DeDeo (2022).

See also:

6.5 Model theory

6.5.1 Introduction

  • Wikipedia: Model theory
  • Hodges: 64
    • model theory = universal algebra + logic
    • model theory = algebraic geometry - fields
  • SEP: Model theory
    • Model theory is the study of the interpretation of any language, formal or natural, by means of set-theoretic structures, with Alfred Tarski’s truth definition as a paradigm.

64 Hodges (1997), p. vii.

Hunter:

Model theory is the theory of interpretations of formal languages (a model of a formula of a language is an interpretation of the language for which the formula comes out true). 65

65 Hunter (1971), p. 6.

Weiss & D’Mello:

Model Theory is the part of mathematics which shows how to apply logic to the study of structures in pure mathematics. 66

66 Weiss & D’Mello (2015), p. 1.

Pedagogy:

67 Weiss & D’Mello (2015).

68 Button & Walsh (2018).

More:

69 Makowsky (1995).

See also:

6.5.2 History

  • William Rowan Hamilton (1805-1865)
  • Alfred North Whitehead (1861-1947)
    • Whitehead, A.N. (1898). A Treatise on Universal Algebra.
  • Löwenheim-Skolem theorem (1915, 1920)
  • Rudolf Carnap (1891-1970)
  • Alfred Tarski (1901-1983)
  • Kurt Gödel (1906-1978)
  • Leon Henkin (1921-2006)
  • Jaakko Hintikka (1929-2015)
  • Wilfrid Hodges (b. 1941)

70 Hodges (1985).

6.5.3 Incompleteness

  • Kurt Gödel (1906-1978)
    • Carnap inspired Gödel to study logic. 71
  • Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. 72
  • Gödel, K. (1951). Some basic theorems on the foundations of mathematics and their implications. 73
  • Gödel presented his incompleteness theorems at the Second Conference on the Epistemology of the Exact Sciences in Königsberg September 5-7, 1930.
    • von Neumann recognized the importance first.
    • Carnap had warning of Gödel’s results the month before. 74
  • Goldstein 75

71 Goldfarb (2005).

72 Gödel (1931).

73 Gödel (1995).

74 Edmonds (2020), p. 96 and TODO: Carnap’s diary.

75 Goldstein (2005).

Fom the SEP:

The first incompleteness theorem states that in any consistent formal system \(F\) within which a certain amount of arithmetic can be carried out, there are statements of the language of \(F\) which can neither be proved nor disproved in \(F\). According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent). 76

76 Raatikainen (2020).

Related:

  • Tarski’s undefinability theorem on the formal undefinability of truth
    • Tarski, A. (1936). The concept of truth in formalized languages. 77
    • Tarski, A. (1969). Truth and proof 78
  • Church’s proof that Hilbert’s Entscheidungsproblem is unsolvable
    • Church, A. (1936). A note on the Entscheidungsproblem. 79
  • Turing’s theorem that there is no algorithm to solve the halting problem
    • Turing, A.M. (1937). On computable numbers, with an application to the Entscheidungsproblem. 80
  • Rice’s theorem that all non-trivial semantic properties of programs are undecidable

77 Tarski (1983).

78 Tarski (1969).

79 Church (1936).

80 Turing (1937).

Chaitin:

[M]y information-theoretic approach to incompleteness makes incompleteness appear pervasive and natural. This is because algorithmic information theory sometimes enables one to measure the information content of a set of axioms and of a theorem and to deduce that the theorem cannot be obtained from the axioms because it contains too much information.

This suggests to me that sometimes to prove more one must assume more, in other words, that sometimes one must put more in to get more out. 81

81 Atiyah, M. et al. (1994), p. 182.

More:

82 Franzén (2005).

83 Lloyd (1993).

84 Cubitt, Perez-Garcia, & Wolf (2015).

Relationship to mind:

See also:

6.5.4 Complexity theory

85 Ord (2024).

86 Aaronson (2011).

See also:

6.6 Category theory

6.6.1 Introduction

87 Eilenberg & MacLane (1945).

88 Rodin (2012).

89 Spivak (2013).

90 Fong & Spivak (2018).

91 Bradley (2018).

92 Lawvere (1963).

See also:

6.6.2 History

Grothendieck:

If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither “number” nor “size”, but always form. And among the thousand-and-one faces whereby form chooses to reveal itself to us, the one that fascinates me more than any other and continues to fascinate me, is the structure hidden in mathematical things. 93

93 TODO

6.6.3 Homotopy type theory

See also:

6.6.4 Cobordism hypothesis

94 Baez & Dolan (1995).

95 Baez & Stay (2009).

See also:

6.6.5 Topos theory

6.6.6 Applications

96 Bradley, Terilla, & Vlassopoulos (2021).

See also:

6.7 Unification programs

6.7.1 Introduction

Yanofsky, N.S. (2016). Why mathematics works so well:

When you have two distinct fields of mathematics and they are shown to be intimately related in a way that results of one field can be used to get results of another field, you have a type of unification. An example of such unification is monstrous moonshine. This subject describes the shocking connection between the monster group and modular functions. Another example is the Langlands program which connects Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups. The Erlangen program can also be seen as a way of unifying different types of geometries using group theory. In all these examples of unifications, there is a function (or an isomorphism) from the domain of discourse of one field to the domain of discourse of another field. The symmetries of one field (the true mathematical statements) will than map to symmetries of the other field. Category theory is an entire branch of mathematics that was created to describe such unifications. The founders of category theory invented a language that was based on algebraic topology, which is a branch of mathematics that unifies algebra and topology. Category theory is now used in many areas to show that seemingly different parts of mathematics (and physics and theoretical computer science) are closely related. 97

97 Yanofsky (2016) In Aguirre, Foster, & Merali (2016), p. 153.

6.7.2 Erlangen program

  • Erlangen program
  • A program proposed by Felix Klein in 1872 to classify geometries based on their symmetry groups.

6.7.3 Langlands program

98 Frenkel (2005).

6.7.4 Univalent foundations

See also:

6.8 Platonism

6.8.1 Introduction

What has been is what will be,
and what has been done is what will be done;
there is nothing new under the sun.
Is there a thing of which it is said,
“See, this is new”?
It has already been,
in the ages before us. 99

99 Ecclesiastes 1:9–10 (NRSV).

6.8.2 Pythagoreanism

  • Radical realism
  • Monism: everything is math.
  • Tegmark
Figure 6.3: Bronnikov, F. (1869). Pythagoreans celebrate sunrise. (Wikimedia, 2007).

6.8.3 Plato

  • Realist
  • Theory of the forms
  • Dualism: the world and forms?
  • The Academy (387 BCE - 529 CE): “Let no one ignorant of geometry enter”
  • Mathematics is descriptive of a real but trans-empirical realm.
  • Plato perhaps doubts Platonism in Parmenides
  • Aristotle on universals
Figure 6.4: Detail of School of Athens by Raphael (1511), showing Plato and Aristotle (Wikimedia, 2013).

I think you know that the students of geometry, calculation, and the like hypothesize the odd and the even, the various figures, the three kinds of angles, and other things akin to these in each of their investigations, as if they knew them. They make their hypotheses and don’t think it necessary to give any account of them, either to themselves or to others, as if they were clear to everyone. And going from these first principles through the remaining steps, they arrive in full agreement.

I certainly know as much.

Then you also know that, although they use visible figures and make claims about them, their thought isn’t directed to them but to those other things that they are like. They make claims for the sake of the square itself and the diagonal itself, not the diagonal they draw, and similarly with the others. These figures that they make and draw, of which shadows and reflections in water are images, they now in turn use as images, in seeking to see those others themselves that one cannot see except by means of thought. 100

100 Plato, Republic VI 510c, Cooper & Hutchinson (1997), p. 1131.

6.8.4 Contemporary platonism

  • Quine’s “reluctant platonism”
    • Indispensability argument
    • Quine, W.V.O. (1948). On what there is. 101
    • distinction between meaning and naming

101 Quine (1948).

Whatever we say with the help of names can be said in a language which shuns names altogether. To be assumed as an entity is, purely and simply, to be reckoned as the value of a variable. In terms of the categories of traditional grammar, this amounts roughly to saying that to be is to be in the range of reference of a pronoun. Pronouns are the basic media of reference; nouns might better have been named propronouns. The variables of quantification, ‘something’, ‘nothing’, ‘everything’, range over our whole ontology, whatever it may be; and we are convicted of a particular ontological presupposition if, and only if, the alleged presuppositum has to be reckoned among the entities over which our variables range in order to render one of our affirmations true. 102

102 Quine (1948), p. 7.

a theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true. 103

103 Quine (1948), p. 9.

104 Zalta (1983).

105 Linsky & Zalta (1995).

106 Zalta (2007).

107 Zalta (2011).

108 Zalta (2025).

109 Tegmark (2008).

110 Tegmark (2014).

111 De Cruz (2016).

112 Zalta (1983), p. 12.

Zalta on ordinary objects exemplifing properties and abstract objects encoding properties: 112

\[ F^{n} x_{1} \ldots x_{n} = x_{1} \ldots x_{n}\ \mathrm{exemplify\ property}\ F^{n} \]

\[ x F^{1} = \mathrm{individual}\ x\ \mathrm{encodes\ property}\ F^{1} \]

If possibly an abstract object encodes a property, then it does so necessarily: 113

113 Leitgeb, Nodelman, & Zalta (2025), p. 129.

\[ \Diamond x F \rightarrow \Box x F \]

Ordinary objects possibly exist concretely; abstract objects do not possibly exist concretely: 114

114 Zalta (2025), §15.2, p. 866.

\[ O! x \equiv \Diamond E! x \]

\[ A! x \equiv \neg \Diamond E! x \]

See also:

6.8.5 Visual proofs

Composite numbers can be arranged into rectangles but prime numbers cannot (source: Wikimedia).

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

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

Visualization of the binomial theorem (source: Wikimedia).

Criticism:

6.9 Nominalism

6.9.1 Introduction

  • Antirealist

6.9.2 History

Antisthenes:

A horse I can see, but horsehood I cannot see.

Bayes:

It is not the business of a mathematician to show that a straight line or circle can be drawn, but he tells you what he means by these; and if you understand him, you may proceed further with him; and it would not be to the purpose to object that there is no such thing in nature as a true straight line or perfect circle, for this is none of his concern: he is not inquiring how things are in matter of fact, but supposing things to be in a certain way, what are the consequences to be deduced from them. 115

115 Bayes (1736), p. 9–10.

Johwn Stuart Mill:

Nominalists, who, repudiating Universal Substances, held that there is nothing general except names. 116

116 Mill (1877).

  • Positivism, Carnap, ESO
  • Goodman & Quine 117
  • Henkin, L. (1953). Some notes on nominalism. 118
  • Azzouni, J. (2015). Nominalism, the nonexistence of mathematical objects. 119

117 Goodman & Quine (1947).

118 Henkin (1953).

119 Azzouni (2015).

See also:

6.9.3 Science Without Numbers

Field: Abstract platonic entities, were they to exist, could not interact causally with the world. 120

120 Field (1989), p. 68. TODO: Get exact quote.

121 Field (2016).

122 Burgess (1983).

123 Bueno (2013).

124 Chen (2018).

significant in context but naming nothing. 125

125 Goodman & Quine (1947), p. 105.

6.10 Logicism

6.10.1 Introduction

  • A program to derive all or part of mathematics from logic.
  • Richard Dedekind (1831-1916)
  • Gottlob Frege (1848-1925)
    • All mathematical structures can be constructed from sets and natural numbers.
    • Natural numbers can be constructed from sets.
    • The properties of sets can be reduced to logic. Sets are the extensions of arbitrary conditions.
  • Giuseppe Peano (1858-1932)
  • Ernst Zermelo (1871-1953)
    • Sets are not simply the extensions of arbitrary conditions. Some conditions have no corresponding set.
  • Alfred North Whitehead (1861-1947)
  • Bertrand Russell (1872-1970)
  • Realist
  • Relationship with positivism?
  • Russell, B. (1905). On denoting. 126
  • Russell expresses support for the platonic “universals” 127
  • Carnap’s logicism
  • Snapper, E. (1979). The three crises in mathematics: Logicism, intuitionism and formalism. 131
  • Shapiro, S. (2000). Thinking about Mathematics: The Philosophy of Mathematics. 132
  • Frege’s theorem

126 B. Russell (1905).

127 B. Russell (1912), p. 91–118.

128 Carnap (1983).

129 Marschall (2021).

130 Marschall (2022).

131 Snapper (1979).

132 Shapiro (2000).

6.10.2 Principia Mathematica

6.10.3 ZFC

  • ZFC
  • ZFC actually does circumvent Russell’s paradox by restricting the comprehension axiom to already existing sets by the use of subset axioms.

6.10.4 Neologicism

133 Zalta (2000).

134 Linsky & Zalta (2006).

135 Leitgeb et al. (2025).

Zalta:

Our thesis is that mathematical objects just are (reducible to) the abstract objects systematized by a certain axiomatic, mathematics-free metaphysical theory. This thesis appears to be a version of mathematical platonism, for if correct, it would make a certain simple and intuitive philosophical position about mathematics much more rigorous, namely, that mathematics describes a realm of abstract objects. 136

136 Zalta (2000), p. 219.

Linsky & Zalta:

Our knowledge of mathematics is to be explained in terms of the faculty we use to understand language, since that is the only faculty we need to understand object abstraction. 137

137 Linsky & Zalta (2006), p. 43.

See also:

6.11 Formalism

6.11.1 Introduction

  • Hilbert
  • Antirealist?
  • Michael Hallett

Hilbert:

We must know. We shall know.

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. 138

138 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.

6.11.2 Hilbert’s program

  • Hilbert’s program
  • Influenced by Principia Mathematica
  • Hilbert, D. (1926). On the infinite. 139
  • Gödel derailed the program
  • TODO: What do we expect from proof theory now?

139 Hilbert (1926).

Hilbert wanted math to be

  • Consistent?
  • Decidable
  • Complete

See also:

6.12 Intuitionism

6.12.1 Introduction

140 Brouwer (1908).

141 Gisin (2020).

See also:

6.12.2 Linear logic

142 Girard (1987).

6.12.3 Criticism

Hilbert:

Taking the Principle of the Excluded Middle from the mathematician … is the same as … prohibiting the boxer the use of his fists. 143

143 Reid (1996), p. 149.

  • Tait, W.W. (1983). Against intuitionism: Constructive mathematics is part of classical mathematics. 144

144 Tait (1983).

6.13 Fictionalism

6.14 Structuralism

145 Awodey (2014).

146 Nodelman & Zalta (2014).

6.15 Naturalism

6.15.1 Unreasonable effectiveness

Philosophy is written in that great book which ever lies before our eyes—I mean the Universe—but we cannot understand it if we do not learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures without whose help it is impossible to comprehend a single word of it, without which one wanders in vain through a dark labyrinth.

– Galileo Galilei. (1623). Il Saggiatore (The Assayer). 147

147 Drake (1957), p.237-8.

148 Wigner (1960).

149 Putnam (1975b).

150 Hamming (1980).

151 Zee (2016), p. 564.

152 Roberts (2021).

6.15.2 Quine-Putnam indispensability thesis

  • Quine
  • Putnam 153

153 Putnam (1975a), p. TODO.

Putnam:

[Q]uantification over mathematical entities is indispensable for science…; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question. 154

154 Putnam (1971), p. 57.

6.15.3 Mathematical naturalism

6.16 My thoughts

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

6.17 Annotated bibliography

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

6.17.1.1 My thoughts

  • TODO.

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

6.17.2.1 My thoughts

  • TODO.

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

6.17.3.1 My thoughts

  • TODO.