MATH-lec 15

Question 1

Intuitionism’

Answer 1

in the philosophy of mathematics is the view that a mathematical
statement is true iff it is (in principle) provable by us and false iff it is in principle provably
unproveable, where it is not assumed that every well formed mathematical statement will be
either proveable or proveably unproveable.

Note 1 This is verificationism applied to the special case of mathematical statements.
Note 2 Because it is not assumed that we will be able, for every well-formed mathematical
statement p, either to prove p or prove that p cannot be proved, the intuitionist is committed to
denying that every well-formed mathematical statement is either true or false.

This is denial of
the Principle of Bivalence for the mathematical case. But with the Principle of Bivalence gone,
intuitionists must give up various laws and proof strategies of classical logic. In particular, they
must give up the Law of Excluded Middle and proof by reductio

Question 2

Classical ‘¬’ introduction (CNI)

Answer 2

A proof by reductio assumes the negation of what is to be proved; derives an absurdity;
moves to the negation of the negation of the target statement (by CNI)

Question 3

CNE

Answer 3

then moves to the
target statement itself (by CNE).
The intuitionist keeps CNI (to show that p is absurd is to show that p is unproveable). But
with bivalence gone, CNE goes too (without bivalence, ¬¬p is consistent with both ‘p is
true’ and ‘p is neither true nor false’).

Question 4

The mathematician’s argument

Answer 4

The mathematician’s argument for intuitionism begins with a claim about our understanding of
mathematical terms. The basic idea is that our understanding of mathematical terms is grounded
in understanding of the nature of progression. For the case of natural numbers, the suggestion is
that the basic component of mathematical understanding is our intuitive grasp of the practice of
adding one – that is, our intuitive grasp of the ‘successor’ relation.
Given this account of our understanding of natural number arithmetic, we can construct an
argument for intuitionism (in the natural number case) as follows:

1 Construction of mathematical proofs must be constrained by our understanding of
mathematical terms.
2 Our understanding of mathematical terms traces to our intuitive grasp of the basic step of
adding one.
So

3 A mathematical proof is legitimate only if it can be understood as constructed from a series of
such basic steps (or a series of steps which make sense relative to such basic steps).
In addition
4 We have no grasp of mathematical truth aside from what is established by mathematical proof.
So

5 Our grasp of mathematical truth (for the natural number case) just is grasp of the notion of
proveability relative to the standards of legitimacy at 3