The crisis arose at the beginning of the 20th century with the discovery of several paradoxes or counter-intuitive results.1
When the foundational crisis arose, philosophical questions about mathematical truth was involved, presenting two main options for trying to avoid paradoxes:
- intuitionalism/constructivism: restrict logical rules for remaining closer to intuition
- formalism: considers a theorem is true if it can be deduced from axioms by applying inference rules, and no ‘trueness’ of the axioms is needed for the validity of a theorem
Formalism
The mathematician David Hilbert was a known formalist, proposing the famous Hilbert’s Program as a proposed solution to the crisis.
Axioms
Formal mathematics are built up from axioms. An axiom is a premise or statement so evident it can mostly be accepted without proof. Most of the time, this is due to the face that the axiom is so fundamental, it is difficult to break the statement down to prove it.
Different branches or schools of mathematics might have similar but different sets of axioms.
An example of an axiom is the Axiom of empty set, which proposes the existence of an empty set {}.
Theorem
A theorem is a proposition or a statement. A theorem is said to be proved if it can be derived from axioms.
Processes of proofs in algorithms SD include: induction, contradiction
Not in SD
Intuitionalism/constructivism
Not relevant to the study design.
Russel’s Paradox
(not in study design) a paradox that contributed to the crisis: X is a set of every single set that exists which don’t contain themselves. If X does not contain X, it is therefore a set that does not contain itself, and must be a part of X. But X does not contain X.
Russel himself uses the Barber Analogy:
In a small village, there is a barber who shaves all the men who do not shave themselves, and only those men. Who shaves the barber?
- If the barber does not shave himself, then he is a man who do not shave themselves, which states he must shave himself
- If the barber shaves himself, then he is shaving a man who does not shave himself.