# 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.

TODO

## Analysis

TODO

### History

• René Descartes (1596-1650)
• Geometry and coordinates
• Isaac Newton (1642-1726/7)
• Gottfried Wilhelm Leibniz (1646-1716)
• Jacob Bernoulli (1655-1705)
• Leonhard Euler (1707-1783)
• 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

### Development of calculus Figure 1: 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
• Wikipedia: 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?

## Number theory

### Set theory

• Membership: Axiom of extensionality
• Axiom of unrestricted comprehension and Naive Set Theory
• Axiom schema of specification AKA Axiom of restricted comprehension
• von Neumann’s set theoretical definition of numbers
• Zermelo-Fraenkel set theory + Axiom of choice = ZFC

### Transfinite numbers

• Ordinal (index) vs cardinal (size) numbers
• Transfinite numbers:
• $$\omega$$: the lowest transfinite ordinal number; the order type of the natural numbers.
• $$\aleph_0$$: the first transfinite cardinal number; the cardinality of the natural numbers.
• Transfinite numbers were anticipated by Robert Grosseteste (ca. 1168-1253).
• 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 the natural numbers: $$\aleph_0 \equiv |\mathbb{N}|$$
• The cardinality of the reals: $$C \equiv |\mathbb{R}| = |P(\mathbb{N})| = 2^{|\mathbb{N}|} = 2^{\aleph_0} > \aleph_0$$
• Continuum hypothesis (CH)
• CH: There is no set $$S$$ such that $$\aleph_0 < |S| < 2^{\aleph_0}$$
• Under the CH, $$C = \aleph_1 = 2^{\aleph_0}$$.
• Axiom of choice
• Paul Cohen showed the CH is undecidable in ZFC (1963).

## Logic

### Introduction

Pedagogy:

• Hunter, G. (1971). Metalogic: An Introduction to the Metatheory of Standard First-Order Logic.9
• Teller, P. (1989). A Modern Formal Logic Primer.10
• Bonevac, D. (2003). Deduction: Introductory to Symbolic Logic.11
• MacFarlane, J. (2021). Philosophical Logic: A Contemporary Introduction.12
• Smith, P. (2020). An Introduction to Formal Logic.13
• Smith, P. (2022). Beginning Mathematical Logic A Study Guide.14
• logicmatters.net

More:

### 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 $$A$$.

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 notationally 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 1921.18

• Emil Post and his anticipation of Gödel and Turing19

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:

$((P \rightarrow Q) \rightarrow P) \rightarrow P$

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

### First-order logic

• First-order logic
• AKA predicate logic
• Adds (non-logical) predicates and quantification over elements
• Domain of discourse
• C.S Peirce was first to distinguish between propositional logic, first-order logic, and second-order logic in 1885.21
• Consistency, completeness, expressivity
• Gödel’s completeness theorem
• Establishes a correspondence between semantic truth and syntactic provability 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.
• Awodey, S. & Forssell, H. (2013). First-order logical duality.22

### Second-order logic

• Second-order logic
• Second-order arithmetic
• 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.
• 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.23
• Higher-order logics, type theory
• Russell’s theory of types
• Alonzo Church’s lambda calculus
• Michael Dummett
• Setwart Shapiro
• Foundations without Foundationalism: A Case for Second-Order Logic (1991)24
• Gillian Russell
• “The justification of the basic laws of logic”25
• Eliezer Yudkowsky
• Second-Order Logic: The Controversy
• “Second-order logic is sound, in the sense that anything syntactically provable from a set of premises, is true in any model obeying those premises. But second-order logic isn’t complete; there are semantic consequences you can’t derive. If you take second-order logic at face value, there’s no effectively computable way of deriving all the consequences of what you say you ‘believe’… which is a major reason some mathematicians are suspicious of second-order logic. What does it mean to believe something whose consequences you can’t derive?”
• Jerzak, E. (2009). Second-order logic, or: How I learned to stop worrying and love the incompleteness theorems.26
• Bueno, O. (2010). A defense of second-order logic.27

• C.I. Lewis (1883-1964)
• Founded modern modal logic.
• Criticism of material implication. Introduced strict implication.28
• Strict implication is not truth-functional. It requires asking about the truth-values that propositions take in worlds other than the actual world.
• Saul Kripke (1940-2022)
• Kripke, S.A. (1959). A completeness theorem in modal logic.29
• Revives Leibniz’s idea that necessity is truth in all possible worlds.
• David Lewis (1941-2001)
• On the Plurality of Worlds (1986)30

$$\Box$$ 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

• Intuitionistic Logic
• Law of excluded middle (LEM): Tertium non datur.
$$\vdash A \lor \lnot A$$
• Intuitionistic logic rejects LEM.
• 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.
• Priest, G. (1998). What is so bad about contradictions?31
• Dialetheism rejects LNC.
• Martínez-Ordaz, M. del R. (2021). The ignorance behind inconsistency toleration.32

Criticism:

• Quine, W. V. (1986). Philosophy of logic. Harvard University Press.33

## Model theory

### Introduction

• Wikipedia: Model theory
• 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.

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).35

### Incompleteness

• Kurt Gödel (1906-1978)
• Goldstein38
• 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.39

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).40

From Wikipedia:

• Gödel showed for that any formal system $$S$$ powerful enough to represent arithmetic, there is a theorem $$G$$ which is true but the system is unable to prove. $$G$$ could be added as an additional axiom to the system in place of a proof. However this would create a new system $$S'$$ with its own unprovable true theorem $$G'$$, and so on.
• 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 $$T_1$$ satisfying the technical conditions outlined above cannot prove the consistency of any theory $$T_2$$ that proves the consistency of $$T_1$$. This is because such a theory $$T_1$$ can prove that if $$T_2$$ proves the consistency of $$T_1$$, then $$T_1$$ is in fact consistent. For the claim that $$T_1$$ is consistent has form “for all numbers $$n$$, $$n$$ has the decidable property of not being a code for a proof of contradiction in $$T_1$$.” If $$T_1$$ were in fact inconsistent, then $$T_2$$ would prove for some $$n$$ that $$n$$ is the code of a contradiction in $$T_1$$. But if $$T_2$$ also proved that $$T_1$$ is consistent (that is, that there is no such $$n$$), then it would itself be inconsistent. This reasoning can be formalized in $$T_1$$ to show that if $$T_2$$ is consistent, then $$T_1$$ is consistent. Since, by second incompleteness theorem, $$T_1$$ does not prove its consistency, it cannot prove the consistency of $$T_2$$ 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.

Related:

• Tarski’s undefinability theorem on the formal undefinability of truth
• Church’s proof that Hilbert’s Entscheidungsproblem is unsolvable
• 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.41

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.42

More:

Relationship to mind:

## Category theory

### 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.49

## Unification programs

### 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.52

## Platonism

### 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.53

• TODO

### Pythagoreanism

• Monism: everything is math.
• Tegmark, M. (2014). Our Mathematical Universe.54 Figure 2: Bronnikov, F. (1869). Pythagoreans celebrate sunrise. (Wikimedia, 2007).

### 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 3: 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.55

### Contemporary platonism

• Quine’s “reluctant platonism”
• Indispensability argument
• Quine, W.V.O. (1948). On what there is.56
• distinction between meaning and naming

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.57

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.58

### 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 $$\sum_{k=1}^{\infty} \frac{1}{4^k} = \frac{1}{3}$$.

## Nominalism

• Antirealist

### History

Antisthenes:

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

• Plato: “Third Man Argument” in Parmenides
• Medieval nominalism
• Modern nominalism

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.61

Johwn Stuart Mill:

Nominalists, who, repudiating Universal Substances, held that there is nothing general except names.62

• Positivism, Carnap, ESO
• Goodman & Quine63
• Azzouni, J. (2015). Nominalism, the nonexistence of mathematical objects.64

### Science Without Numbers

Field: Abstract platonic entities, were they to exist, could not interact causally with the world.65

• Field: Science Without Numbers66
• John Burgess67
• Bueno68

significant in context but naming nothing.69

## Logicism

### 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)
• Bertrand Russell (1872-1970)
• Alfred North Whitehead (1861-1947)
• Ernst Zermelo (1871-1953)
• Sets are not simply the extensions of arbitrary conditions. Some conditions have no corresponding set.
• Realist?
• Relationship with positivism?
• Russell, B. (1905). On denoting.70
• Russsell expresses support for the platonic “universals”71
• Snapper, E. (1979). The three crises in mathematics: Logicism, intuitionism and formalism.72
• Shapiro, S. (2000). Thinking about Mathematics: The Philosophy of Mathematics.73
• Hume’s principle
• Frege’s theorem

### ZFC

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

### Neologicism

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.75

## Formalism

### Introduction

• Hilbert
• Antirealist?

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.76

### Hilbert’s program

Hilbert wanted math to be

• Consistent?
• Decidable
• Complete

## Intuitionism

### Introduction

• Antirealist
• Leopold Kronecker’s finitism was a forerunner of intuitionism in foundations of mathematics.
• Kronecker: “God made the integers, all else is the work of man.”
• L.E.J. Brouwer (1881-1966)
• Constructive mathematics

### Criticism

Hilbert:

Taking the Principle of the Excluded Middle from the mathematician … is the same as … prohibiting the boxer the use of his fists.78

• Tait, W.W. (1983). Against intuitionism: Constructive mathematics is part of classical mathematics.79

## Structuralism

• Realist
• Awodey, S. (2014). Structuralism, invariance, and univalence.80

## Naturalism

### 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).81

### Quine-Putnam indispensability thesis

• Quine
• Putnam86

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.87

## My thoughts

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

## Annotated bibliography

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

• Wigner (1960)

• TODO.

• Field (2016)

• TODO.

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

• Snapper (1979)

#### My thoughts

• TODO.

• TODO.

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Awodey, S. (2014). Structuralism, invariance, and univalence. Philosophia Mathematica, 22, 1–11.
Awodey, S. & Forssell, H. (2013). First-order logical duality. Annals of Pure and Applied Logic, 164, 319–348.
Azzouni, J. (2015). Nominalism, the nonexistence of mathematical objects. In E. Davis & P. Davis (Eds.), Mathematics, Substance and Surmise (pp. 133–145). Springer.
Baez, J. C. (2002). The octonions. Bulletin of the American Mathematical Society, 39, 145–205. https://arxiv.org/abs/math/0105155
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Bayes, T. (1736). An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of The Analyst. London. (published anonymously).
Bonevac, D. (2003). Deduction: Introductory to Symbolic Logic (2nd ed.). Blackwell.
Bradley, T. D. (2018). What is applied category theory? https://arxiv.org/abs/1809.05923
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Brouwer, L. E. J. (1908). Unreliability of the logical principles. https://arxiv.org/abs/1511.01113
Bueno, O. (2010). A defense of second-order logic. Axiomathes, 20, 365–383. https://web.as.miami.edu/personal/obueno/Site/Online_Papers_files/SecnOrd_FINAL.pdf
———. (2013). Nominalism in the philosophy of mathematics. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/nominalism-mathematics/
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Button, T. & Walsh, S. (2018). Philosophy and Model Theory. Oxford University Press.
Carnap, R. (1958). Introduction to Symbolic Logic and its Applications. New York: Dover Publications.
Carter, N. (2009). Visual Group Theory. Mathematical Association of America.
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Drake, S. (1957). Discoveries and Opinions of Galileo. New York: Doubleday and Co.
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Enderton, H. B. (2009). Second-order and higher-order logic. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/archives/sum2019/entries/logic-higher-order/
Ewald, W. (2018). The emergence of first-order logic. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/logic-firstorder-emergence/
Field, H. (1989). Realism, Mathematics, and Modality. Oxford: Blackwell.
———. (2016). Science Without Numbers (2nd ed.). Oxford University Press. (Originally published in 1980 by Princeton University Press).
Fong, B. & Spivak, D. I. (2018). Seven sketches in compositionality: An invitation to applied category theory. https://arxiv.org/abs/1803.05316
Franzén, T. (2005). Gödel’s Theorem: An incomplete guide to its use and abuse. A K Peters.
Frè, P. G. (2013). Gravity, a Geometrical Course, Volume 1: Development of the Theory and Basic Physical Applications. Springer.
Goldstein, R. (2005). Incompleteness: The Proof and Paradox of Kurt Gödel. Norton.
Goodman, N. & Quine, W. V. O. (1947). Steps toward a constructive nominalism. The Journal of Symbolic Logic, 12, 105–122.
Hall, B. C. (2000). An Elementary Introduction to Groups and Representations. https://arxiv.org/abs/math-ph/0005032
Hamming, R. W. (1980). The unreasonable effectiveness of mathematics. The American Mathematical Monthly, 87, 81–90.
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).
Hunter, G. (1971). Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California Press.
Jerzak, E. (2009). Second-order logic, or: How I learned to stop worrying and love the incompleteness theorems. http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Jerzak.pdf
Kripke, S. A. (1959). A completeness theorem in modal logic. The Journal of Symbolic Logic, 24, 1–14. http://naturalthinker.net/trl/texts/Kripke,Saul/Kripke%20S.%20-%20A%20Completeness%20Theorem%20in%20Modal%20Logic.pdf
Lewis, C. I. (1917). The issues concerning material implication. The Journal of Philosophy, Psychology and Scientific Methods, 14, 350–356. https://www.jstor.org/stable/2940255
Lewis, D. (1986). On the Plurality of Worlds. Oxford: Blackwell.
Linsky, B. & Zalta, E. N. (2006). What is Neologicism? The Bulletin of Symbolic Logic, 12, 60–99. http://mally.stanford.edu/Papers/neologicism2.pdf
MacFarlane, J. (2021). Philosophical Logic: A contemporary introduction. Routledge.
Martínez-Ordaz, M. R. (2021). The ignorance behind inconsistency toleration. Synthese, 198, 8665–8686. http://philsci-archive.pitt.edu/19672/
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1. Carter (2009).↩︎

2. Hall (2000).↩︎

3. Baez (2002).↩︎

4. Frè (2013), ch. 2.↩︎

6. Tao (2007).↩︎

7. Sussman & Wisdom (2013).↩︎

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

9. Hunter (1971).↩︎

10. Teller (1989).↩︎

11. Bonevac (2003).↩︎

12. MacFarlane (2021).↩︎

13. Smith (2020).↩︎

14. Smith (2022).↩︎

15. Carnap (1958).↩︎

16. Open Logic Project (2020).↩︎

17. Sheffer (1913).↩︎

18. Post (1921).↩︎

19. Stillwell (2004).↩︎

20. von Fintel (2011).↩︎

21. Ewald (2018).↩︎

22. Awodey & Forssell (2013).↩︎

23. Enderton (2009).↩︎

24. Shapiro (1991).↩︎

25. G. Russell (2015).↩︎

26. Jerzak (2009).↩︎

27. Bueno (2010).↩︎

28. C. I. Lewis (1917).↩︎

29. Kripke (1959).↩︎

30. D. Lewis (1986).↩︎

31. Priest (1998).↩︎

32. Martínez-Ordaz (2021).↩︎

33. Quine (1986).↩︎

34. Viteri & DeDeo (2022).↩︎

35. Hunter (1971), p. 6.↩︎

36. Button & Walsh (2018).↩︎

37. Weiss & D’Mello (2015).↩︎

38. Goldstein (2005).↩︎

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

40. Raatikainen (2020).↩︎

41. Turing (1937).↩︎

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

43. Franzén (2005).↩︎

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

45. Aaronson (2011).↩︎

46. Spivak (2013).↩︎

47. Fong & Spivak (2018).↩︎

49. TODO↩︎

50. Baez & Dolan (1995).↩︎

51. Baez & Stay (2009).↩︎

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

53. Ecclesiastes 1:9–10 (NRSV).↩︎

54. Tegmark (2014).↩︎

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

56. Quine (1948).↩︎

57. Quine (1948), p. 7.↩︎

58. Quine (1948), p. 9.↩︎

59. Tegmark (2008).↩︎

60. Chen (2018).↩︎

61. Bayes (1736), p. 9–10.↩︎

62. Mill (1877).↩︎

63. Goodman & Quine (1947).↩︎

64. Azzouni (2015).↩︎

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

66. Field (2016).↩︎

67. Burgess (1983).↩︎

68. Bueno (2013).↩︎

69. Goodman & Quine (1947), p. 105.↩︎

70. B. Russell (1905).↩︎

71. B. Russell (1912), p. 91–118.↩︎

72. Snapper (1979).↩︎

73. Shapiro (2000).↩︎

74. Linsky & Zalta (2006).↩︎

75. Linsky & Zalta (2006), p. 43.↩︎

76. 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.

↩︎
77. Brouwer (1908).↩︎

78. Reid (1996), p. 149.↩︎

79. Tait (1983).↩︎

80. Awodey (2014).↩︎

81. Drake (1957), p.237-8.↩︎

82. Wigner (1960).↩︎

83. Putnam (1975b).↩︎

84. Hamming (1980).↩︎

85. Roberts (2021).↩︎

86. Putnam (1975a), p. TODO.↩︎

87. Putnam (1971), p. 57.↩︎