More fundamentally, intuitionism is best seen as a theory about mathematical assertion and denial.
The book's "lucidity," he wrote — the idea that a mathematical assertion could "be proved with such certainty that any doubt appeared to be out of the question" — provoked "wonder of a totally different nature".
With Husserl, Gödel saw the problem of mathematical evidence, i.e., the question: what sort of data should count as evidence for the truth of a mathematical assertion, as fundamental.
Variables are also used to express generality, as in the mathematical assertion that for any natural number x, there is a natural number y, such that y>x and y is prime.
Recently, it has been shown that within the framework of classical replicator dynamics, the MinxEnt principle is a rigorous mathematical assertion that precisely describes the replicator dynamics [ 3, 5].
Any topos may be conceived as possible "universe of discourse" in which mathematical assertions may be interpreted and mathematical constructions may be performed.
It is, instead, intuitionistic logic, that is, the logic derived from the constructive interpretation of mathematical assertions.
Mathematical assertions are rendered interpretable in a topos E by expression within E's internal language — a type-theoretic version of the usual language of set theory.
His explanation of the difference between non-normative and normative assertions (between, as he put it, 'the assertion that someone was obliged to do something and the assertion that he had an obligation to do it') was merely that the latter sort of assertion is used to display a distinctive sort of attitude.
Therefore a musical interval between two notes can be expressed as a mathematical ratio between the lengths of two strings.
