Philosophy of Mathematics Study Guide

Analytic

A truth p is ________ iff there is a proof of p from general logical laws and definitions.

Synthetic

A truth p is ________ iff it is not analytic.

Logicism

Arithmetic is analytic (Frege's thesis)

Hume's Principle

The number of Fs = the number of Gs iff the Fs are equinumerous with the Gs. This is abbreviated as N(F) = N(G) iff F @ G.

Truth Value Realism

Every well-formed mathematical statement has a unique and objective truth value which is independent of whether it can be known by us or proved from our current mathematical theories.

Abstractness

Mathematical objects are abstract.

Mathematical Truth

Most sentences accepted as mathematical theorems are true.

Classical Semantics

The singular terms of the language of mathematics are supposed to refer to mathematical objects, and its first order quantifiers are supposed to range over such objects.

Object Realism

There are mathematical objects.

Platonism

Object Realism + Abstractness + Reality.

Reality

Mathematical objects are at least as real as ordinary physical objects.

Deductivism

Pure mathematics is the investigation of the deductive consequences of arbitrarily chosen sets of axioms, where the axioms are uninterpreted.

Semantics

The study of the meanings of linguistic expressions: it concerns notions such as truth, reference, linguistic meaning, and synonymy.

Syntax

The study of linguistic expressions that abstracts from their meanings: it considers expressions inasmuch as they are strings of symbols formed according to rules.

Formalism

The view that mathematics has no need for semantic notions, or at least none that cannot be reduced to syntactic ones.

Term Formalism

The singular terms of mathematics refer to themselves (or to other terms).

Game Formalism

Mathematics is like a game in that its symbols have no meanings, and its sentences are neither true nor false.

Empiricism

The view that all substantive knowledge is based on sense experience; this view holds that all mathematical knowledge either is not substantive or is based on sense experience.

Rationalism

The opposite of empiricism; this viewpoint holds that mathematical knowledge is substantive and is not based on sense experience.

Nominalism

The denial of object realism. According to this view, there are no mathematical objects.

Fictionalism

The denial of mathematical truth and object realism. According to this view, mathematical sentences purport to make true claims about mathematical objects. But there are no mathematical objects and no mathematical sentence is true.

Structuralism

The view that mathematics is the study of abstract structures.

Noneliminative Structuralism

This view holds that there are really abstract structures, and mathematical objects are positions in these abstract structures.

Quine's Indispensability Argument

This argument concludes that there are mathematical objects, and assumes as premises that we have reason to believe that natural science is true, that natural science makes indispensable use of mathematical sentences, and that mathematical sentences commit us to the existence of mathematical objects.

Benacerraf's Dilemma

A philosophy of mathematics can give a plausible semantics for mathematics, or a plausible epistemology, but not both.

Maddy's Objective Mathematics

Maddy thinks that it is incontestable that mathematics is objective. She argues that two kinds of facts are objective: facts about what follows from what, and facts about mathematical depth.

Tropospheric Complacency

Our tendency to overestimate the determinateness of our concepts. (Maddy borrows from Mark Wilson).

___________________ about our concepts of truth and existence leads philosophers to think that realism about mathematics must be correct or incorrect. For Maddy, there is no substantive fact that could decide whether mathematical sentences really are true or whether mathematical objects really do exist.

Coexact Properties

Properties of a figure that are NOT sensitive to slight changes in the drawn diagram

Exact Properties

Properties of a figure that ARE sensitive to slight changes in the drawn diagram

Diagram-based Inferences

Allow us to draw conclusions about co-exact properties

Text-based Inferences

Allow us to draw conclusions about exact properties

3 features of the customary concept of a proof

Convincing, formalizable, and surveyable.

Tymoczko

We should revise our customary concept of proofs, since some computer proofs are not surveyable, thus similar to experiments, therefore empirical and not apriori

Peano Axioms

The standard axioms for arithmetic

Russell's Paradox

A paradox in set theory concerning the set of all sets that do not contain themselves.

Parallel postulate

An axiom included in Euclidean geometry; that, for any line and point not on that line, there is one and only one parallel line through that point.

Non-Euclidean Geometry

Includes a denial of the parallel postulate.

Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC)

The axioms for set theory that are now standard

Crisis in Intuition

Arose from results such as the existence of continuous but nowhere differentiable curves.

Gaps in Euclid

It is often claimed that there are gaps in Euclid, because he does not give explicit axioms for continuity or betweenness.

Four Color Theorem

Any map can be colored by only four colors; the proof is a proof by cases and requires computer assistance to prove.

What did Cantor do?

Proved that the rational numbers are countable, but the real numbers are uncountable; there are just as many rational numbers as natural numbers, but more real numbers than natural numbers.