Philosophy Notes Philosophy of Mathematics Notes
Notes made for the Philosophy of Mathematics paper at the University of Oxford.
Each set of notes brings together in detail all the major areas needed to write a first-class essay on that topic. Key arguments and positions from both primary and secondary sources are summarised clearly: perfect as a basis for an exam essay or as a primer on the subject. Also includes a summary of key positions, including strengths and weaknesses, within the philosophy of mathematics.
The notes draw on key re...
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Philosophies of Mathematics
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).
Meaningful | Declarative | True | Surface form is logical form | |
---|---|---|---|---|
Formalist | X | In the system | ||
Intuitionist | Intuitionist part only | Intuitionist part only | ||
Platonist | ||||
Logicist | ||||
Nominalist | X | |||
Eliminative Structuralist | X implicitly conditional on axioms | |||
Fictionalist |
Physical | Mental | Neither physical nor mental | Non-existent | |
---|---|---|---|---|
Formalist | Y | |||
Intuitionist | Y | |||
Platonist | Y | |||
Logicist | Y in the Fregean sense | |||
Nominalist | Y | |||
Eliminative Structuralist | Y if structures taken as concrete | Y if structures taken as abstract | Y nominalist approach | |
Fictionalist | 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.
Statement: let X be a non-empty set of disjoint non-empty sets. Then there exists a set B such that for all AϵB we have that |AB|=1
Equivalent statements:
Well-ordering: every set X can be well-ordered;
Cardinal comparability: if X and Y are sets then |X| |Y| or |Y| |X|;
Löwenheim-Skolem-Tarski Theorem: a first-order sentence having a model of infinite cardinality κ also has a model of any infinite cardinality μ such that μ κ (proven equivalent by Tarski);
Every CRI has a maximal ideal (proven equivalent by Hodges);
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 ω-sequence) 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 | Non-algebraic | |
---|---|---|
Group theory | Arithmetic | |
Topology | Mathematical analysis | |
Graph theory |
Instrumentalism is the view that the use or existence of something is justified by its ability to fulfil the role required of it.
Predicativism embodied the attempt to reform mathematics along purely predicative lines. A definition is predicative if it only picks out entities that exist independently from the defined collection. In other words, predicative definitions do not violate the vicious circle principle, which rules out impredicative definitions, ones that define a collection S by implicit reference to S itself.
Proponents: Hermann Weyl. According to Leon Horsten’s SEP entry ‘Philosophy of Mathematics’, Weyl took the natural numbers as unproblematically given, and showed that often in analysis we can bypass impredicative definitions in favour of predicative ones. (Though Weyl later converted to Brouwer’s intuitionism).
Platonism (in mathematics) is the view that the subject matter of mathematics consists of abstract entities, usually taken to be somewhat (though not necessarily precisely) akin to Platonic Forms. These abstract entities are usually taken to be (1) unchanging; (2) not contingent of any features of the universe, so independent of human activity.
Constructivism (in mathematics) is the view that mathematics is an essentially constructive activity. Accordingly, a constructivist will only admit constructive proofs: proofs which exhibit the required object in question, as opposed to a pure existence proof.
Intuitionism is the view that mathematics is a kind of mental construction; precisely, the kind of mental construction capable of being created by an ‘ideal’ mathematician, not contingent on practical considerations. This ideal mathematician is, nonetheless, finite: the intuitionist rejects the completed infinite. All the ideal mathematician can create is the arbitrarily large finite.
Proponents: Dummett, Brouwer
Problems:
fleshing out the capabilities of an ‘ideal’ mathematician’
destructive to mathematics
wrong epistemology?
Nominalism (in mathematics) is the view that denies the existence of abstract objects to which singular terms can refer, but maintains that...
Buy the full version of these notes or essay plans and more in our Philosophy of Mathematics Notes.
Notes made for the Philosophy of Mathematics paper at the University of Oxford.
Each set of notes brings together in detail all the major areas needed to write a first-class essay on that topic. Key arguments and positions from both primary and secondary sources are summarised clearly: perfect as a basis for an exam essay or as a primer on the subject. Also includes a summary of key positions, including strengths and weaknesses, within the philosophy of mathematics.
The notes draw on key re...
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