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Philosophies of Mathematics Basic things to remember????Metaphysics: what the subject is about Epistemology: how we come into cognitive contact with the subject matter Distinguish between the representatives (e.g. numerals 1, one, un) and what they represent (the number one) Distinguish how we learn the concept (for example by playing with blocks) from what the concept is about (number): our thought transcends the basic triggers we use to learn a concept. Adjectival number usage: 'there are three X' (roughly like 'there are green X') Nominal number usage: 'the number of X is three' (roughly like 'the colour of X is green') Abstraction: generalises out specific concrete content Generalisation: increases the scope or range of the theorem in question, for example by applying to any dimension rather than just the one-dimensional case. Abstract: not spatial, mental or causal, and its intrinsic properties do not change over time. Anything non-abstract is called concrete. (Distinguish this from the mathematical usage in which, for example, a ring is abstract and the integers a concrete instantiation).

Treatment of Mathematical Statements Meaningfu l Formalist

X Intuitionist

Intuitionist part only

Declarative

True

Surface form is logical form

In the system Intuitionist part only

Platonist Logicist Nominalist Eliminative Structuralist Fictionalist

X X implicitly conditional on axioms

Treatment of Mathematical Objects Physical Formalist Intuitionist Platonist Logicist

Mental

Neither physical nor mental

Non-existent Y

Y Y Y in the Fregean sense

Nominalist Eliminative Structuralist

Y Y if structures taken as concrete

Y if structures taken as abstract

Fictionalist

Y nominalist approach Y strictly speaking

Conventionalism (in mathematics) is the view that mathematical truths are true only by convention - because we agree, implicitly or otherwise, to take them as true - and the laws and procedures of mathematics are merely conventions.

The Axiom of Choice Statement: let X be a non-empty set of disjoint non-empty sets. Then there exists a set B such that for all A [?] B we have that | A [?] B|=1 Equivalent statements:

1. Well-ordering: every set X can be well-ordered;

2. Cardinal comparability: if X and Y are sets then |X| [?] |Y| or |Y| [?] |X|;

3. Lowenheim-Skolem-Tarski Theorem: a first-order sentence having a model of infinite cardinality k also has a model of any infinite cardinality m such that m [?] k (proven equivalent by Tarski);

4. Every CRI has a maximal ideal (proven equivalent by Hodges);

5. Tychenov's Theorem: the product of compact topological spaces is compact (proven equivalent by Kelley).

Set-theoretic reductionism is the view that all of mathematics can be reduced to set theory. According to this view, mathematics is exhausted by what is expressible in set theory, such as ZFC: mathematics is just 'set theory in disguise'.

Structuralism (in mathematics) is the view that mathematics is the study, not of objects in particular, but of certain kinds of structure. Leon Horsten calls a particular instantiation of a kind of structure (for example an osequence) a system. He also calls attention to Shapiro's distinction between algebraic mathematical theories and non-algebraic mathematical theories. A non-algebraic mathematical theory has an intended model; as Horsten puts it, such theories "appear at first sight to be about a unique model (p.26)." By contrast, algebraic mathematical theories do not even appear to be about a unique model. Algebraic Group theory Topology Graph theory

Non-algebraic Arithmetic Mathematical analysis

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