MATH-lec 12

Question 1

semantic theory’

Answer 1

for a language is a theory laying down how the truth and falsity of sentences in the
language depend on contributions made by their parts.

In general, a semantic theory will presuppose a syntax which specifies the kinds of expressions in the language, and lays
down which combinations of expressions count as well formed formulae (sentences)

Question 2

The semantic theory stipulates

Answer 2

a) the kind of contribution to determining the truth or falsity of a sentence that is made by each unstructured expression (the
expression’s semantic value), and
b) how the semantic value of a syntactically complex expression depends on the semantic values of its simpler constituents
and the way these constituents are combined

The semantic value of a sentence is a truth value (‘TRUE’ or ‘FALSE’)

Question 3

Platonism’

Answer 3

Realism’ in the philosophy of mathematics is the view that mathematics has a mind-independent
subject matter, and that a mathematical statement is true iff it characterizes this subject matter
correctly. (A mathematical statement is true iff it corresponds to a mathematical fact.)

‘Platonism’ is a form of realism according to which the subject matter of mathematics is a realm
of mind-independent and non-physical mathematical objects

Question 4

Mathematical anti-realism (1) – Formalism

Answer 4

Anti-realism’ in the philosophy of mathematics is the denial of realism. An anti-realist says that
mathematics does not have a mind-independent subject matter/ that correctness in mathematical
proofs is not a matter of generating results that correspond with mathematical reality.

‘Formalism’ is an anti-realist view according to which mathematical statements are not true or
false at all. According to a formalist, the activity of engaging in mathematical proof is an
exercise in manipulating symbols according to rules. A proof is ‘correct’ iff it is constructed by
steps that are in good order relative to the rules, where the rules themselves are in good order as long as they do not generate inconsistencies. The question of whether a statement arrived at by
such a series of steps is ‘true’ simply does not arise.

One of the main objections to formalism is that it seems to be unable to account for the
usefulness of mathematics.

Question 5

Intuitionism

Answer 5

‘Intuitionism’ is an anti-realist view according to which mathematical statements are assessable
as true or false, but mathematical truth is not regarded as correspondence with mind-independent
mathematical reality. According to an intuitionist, a mathematical statement is true iff it is
proven [strong form of intuitionism] or (in principle) provable by us [weaker form], and false iff
its negation is proven/provable.
Because a well-formed statement may (for all we know) be neither provable nor disprovable,
intuitionists reject the claim that every mathematical statement is either true or false, and all
forms of reasoning that rest on this claim. In particular intuitionists repudiate reductio arguments
as a means of proof in mathematics