Philosophy of mathematics

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.


  1. Algebra
    1. Introduction
    2. History
    3. Linear algebra
    4. Finite groups
    5. Lie groups
    6. More
  2. Analysis
    1. Introduction
    2. History
    3. Development of calculus
    4. Differential geometry
  3. Number theory
    1. Introduction
    2. Set theory
    3. Transfinite numbers
  4. Logic
    1. Introduction
    2. History
    3. Propositional logic
    4. First-order logic
    5. Second-order logic
    6. Modal logic
    7. Alternative logics
    8. Proof theory
  5. Model theory
    1. Introduction
    2. History
    3. Incompleteness
    4. Complexity theory
  6. Category theory
    1. Introduction
    2. History
    3. Homotopy type theory
    4. Cobordism hypothesis
    5. Topos theory
  7. Unification programs
    1. Introduction
    2. Erlangen program
    3. Langlands program
    4. Univalent foundations
  8. Platonism
    1. Introduction
    2. Pythagoreanism
    3. Plato
    4. Contemporary platonism
    5. Visual proofs
  9. Nominalism
    1. Introduction
    2. History
    3. Science Without Numbers
  10. Logicism
    1. Introduction
    2. Principia Mathematica
    3. ZFC
    4. Neologicism
  11. Formalism
    1. Introduction
    2. Hilbert’s program
  12. Intuitionism
    1. Introduction
    2. Linear logic
    3. Criticism
  13. Fictionalism
  14. Structuralism
  15. Naturalism
    1. Unreasonable effectiveness
    2. Quine-Putnam indispensability thesis
    3. Mathematical naturalism
  16. My thoughts
  17. Annotated bibliography
    1. Wigner, E.P. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
    2. Field, H. (1980). Science Without Numbers.
    3. Snapper, E. (1979). The Three Crises in Mathematics: Logicism, Intuitionism, Formalism.
    4. More articles to do
  18. Links and encyclopedia articles
    1. SEP
    2. IEP
    3. Wikipedia
    4. Others
    5. Videos
  19. References





Linear algebra

Figure 1: Matrix World: Classifying all matrices (source: The Art of Linear Algebra).

See also:

Finite groups

Lie groups


See also:





Development of calculus

Figure 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.

Matrix calculus

Differential geometry

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

See also:

Number theory


Set theory

Naive Set Theory

Zermelo-Fraenkel set theory

Other approaches

Transfinite numbers


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.14






Propositional logic

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.”

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 1921.25

Material implication:

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

Modus ponens:

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

Modus tollens:

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

Peirce’s law:

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

First-order logic


See also:

Second-order logic

Incompleteness of second-order logic

See also:


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

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

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

De Morgan duality:

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

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


Alternative logics


See also:

Proof theory

See also:

Model theory



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).53

Weiss & D’Mello:

Model Theory is the part of mathematics which shows how to apply logic to the study of structures in pure mathematics.54



See also:



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).63



[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.68


Relationship to mind:

See also:

Complexity theory

See also:

Category theory


See also:



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.78

Homotopy type theory

Cobordism hypothesis

See also:

Topos theory

Unification programs


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.81

Erlangen program

Langlands program

Univalent foundations

See also:



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.82


Figure 3: Bronnikov, F. (1869). Pythagoreans celebrate sunrise. (Wikimedia, 2007).


Figure 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.84

Contemporary platonism

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.86

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.87

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).






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


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.91

Johwn Stuart Mill:

Nominalists, who, repudiating Universal Substances, held that there is nothing general except names.92

See also:

Science Without Numbers

Field: Abstract platonic entities, were they to exist, could not interact causally with the world.96

significant in context but naming nothing.100



Principia Mathematica



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.107




We must know. We shall know.


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.108

Hilbert’s program

Hilbert wanted math to be

See also:



See also:

Linear logic



Taking the Principle of the Excluded Middle from the mathematician … is the same as … prohibiting the boxer the use of his fists.112




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).115

Quine-Putnam indispensability thesis


[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.122

Mathematical naturalism

My thoughts

Annotated bibliography

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

  • Wigner (1960)

My thoughts

  • TODO.

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

  • Field (2016)

My thoughts

  • TODO.

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

  • Snapper (1979)

My thoughts

  • TODO.

  • TODO.







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———. (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).
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  1. Carter (2009).↩︎

  2. Hall (2000).↩︎

  3. Zee (2016).↩︎

  4. Baez (2002).↩︎

  5. Westbury (2010).↩︎

  6. Frè (2013), ch. 2.↩︎

  7. Varadarajan (2003).↩︎

  8. Tao (2007).↩︎

  9. Sussman & Wisdom (2013).↩︎

  10. Tu (2017).↩︎

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

  12. Connes (1985).↩︎

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

  14. Hilbert (1926), p. 191.↩︎

  15. Hunter (1971).↩︎

  16. Monk (1976).↩︎

  17. Smith (2020).↩︎

  18. Smith (2022).↩︎

  19. Carnap (1958).↩︎

  20. Teller (1989).↩︎

  21. Bonevac (2003).↩︎

  22. MacFarlane (2021).↩︎

  23. Open Logic Project (2020).↩︎

  24. Sheffer (1913).↩︎

  25. Post (1921).↩︎

  26. Stillwell (2004).↩︎

  27. von Fintel (2011).↩︎

  28. Ewald (2018).↩︎

  29. Gödel (1929).↩︎

  30. Henkin (1996).↩︎

  31. Awodey & Forssell (2013).↩︎

  32. Bès (2002).↩︎

  33. Bès & Choffrut (2022).↩︎

  34. Enderton (2009).↩︎

  35. Kleene (1943).↩︎

  36. Rossberg (2004).↩︎

  37. Kleene (1952).↩︎

  38. Shapiro (1991).↩︎

  39. G. Russell (2015).↩︎

  40. Boolos (1984).↩︎

  41. Jerzak (2009).↩︎

  42. Bueno (2010).↩︎

  43. Sider (2022).↩︎

  44. C. I. Lewis (1917).↩︎

  45. Carnap (1947).↩︎

  46. Kripke (1959).↩︎

  47. D. Lewis (1986).↩︎

  48. Carroll (1895).↩︎

  49. Priest (1998).↩︎

  50. Martínez-Ordaz (2021).↩︎

  51. Quine (1986).↩︎

  52. Viteri & DeDeo (2022).↩︎

  53. Hunter (1971), p. 6.↩︎

  54. Weiss & D’Mello (2015), p. 1.↩︎

  55. Weiss & D’Mello (2015).↩︎

  56. Button & Walsh (2018).↩︎

  57. Makowsky (1995).↩︎

  58. Hodges (1985).↩︎

  59. Gödel (1931).↩︎

  60. Gödel (1995).↩︎

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

  62. Goldstein (2005).↩︎

  63. Raatikainen (2020).↩︎

  64. Tarski (1983).↩︎

  65. Tarski (1969).↩︎

  66. Church (1936).↩︎

  67. Turing (1937).↩︎

  68. Atiyah, M. et al. (1994), p. 182.↩︎

  69. Franzén (2005).↩︎

  70. Lloyd (1993).↩︎

  71. Cubitt, Perez-Garcia, & Wolf (2015).↩︎

  72. Aaronson (2011).↩︎

  73. Eilenberg & MacLane (1945).↩︎

  74. Rodin (2012).↩︎

  75. Spivak (2013).↩︎

  76. Fong & Spivak (2018).↩︎

  77. Bradley (2018).↩︎

  78. TODO↩︎

  79. Baez & Dolan (1995).↩︎

  80. Baez & Stay (2009).↩︎

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

  82. Ecclesiastes 1:9–10 (NRSV).↩︎

  83. Tegmark (2014).↩︎

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

  85. Quine (1948).↩︎

  86. Quine (1948), p. 7.↩︎

  87. Quine (1948), p. 9.↩︎

  88. Tegmark (2008).↩︎

  89. De Cruz (2016).↩︎

  90. Chen (2018).↩︎

  91. Bayes (1736), p. 9–10.↩︎

  92. Mill (1877).↩︎

  93. Goodman & Quine (1947).↩︎

  94. Henkin (1953).↩︎

  95. Azzouni (2015).↩︎

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

  97. Field (2016).↩︎

  98. Burgess (1983).↩︎

  99. Bueno (2013).↩︎

  100. Goodman & Quine (1947), p. 105.↩︎

  101. B. Russell (1905).↩︎

  102. B. Russell (1912), p. 91–118.↩︎

  103. Carnap (1983).↩︎

  104. Snapper (1979).↩︎

  105. Shapiro (2000).↩︎

  106. Linsky & Zalta (2006).↩︎

  107. Linsky & Zalta (2006), p. 43.↩︎

  108. 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.

  109. Hilbert (1926).↩︎

  110. Brouwer (1908).↩︎

  111. Girard (1987).↩︎

  112. Reid (1996), p. 149.↩︎

  113. Tait (1983).↩︎

  114. Awodey (2014).↩︎

  115. Drake (1957), p.237-8.↩︎

  116. Wigner (1960).↩︎

  117. Putnam (1975b).↩︎

  118. Hamming (1980).↩︎

  119. Zee (2016), p. 564.↩︎

  120. Roberts (2021).↩︎

  121. Putnam (1975a), p. TODO.↩︎

  122. Putnam (1971), p. 57.↩︎