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In Recursive predicates and quantifiers (Trans. Amer. Math. Soc. 53 (1943), 41-73) Kleene gives a description of general recursive functions acording to Herbrand and Gödel, as understood before the pa …

According to Wikipedia, in 5th century BCE, Bryson of Heraclea spoke of a special case of the intermediate value theorem. If we're very generous, that would be an early occurrence of a constructively …

If I understand the proof correctly, it uses the following fact: Statement: A monotone sequence $b : \mathbb{N} \to \{0,1\}$ stabilizes, i.e., there is $d \in \{0, 1\}$ and $n$ such that $b_m = d$ …

What did the logicians of the 20th century think? Perhaps this was best described by Michael Beeson in his book ("Foundations of constructive mathematics: metamathematical studies", 1985, Springer): …

According to the BHK interpretation of intuitionistic logic we have that: A proof of $\exists x \in A . \phi(x)$ consists of a pair $(a, p)$ where $a \in A$ and $p$ is a proof of $\phi(a)$. A proof …

You are probably referring to Brouwer's considerations of the Creative Subject, which can be formulated mathematically as Kripke's schema. It implies that all subsets of $\mathbb{N}$ are countable, fo …

You seem to think that synthetic differential geometry only handles squarenil infinitesimals, but this is not so. The generalized Kock-Lawvere axiom allows us to work with infinitesimals of any order, …

I do not know whether the early calculus was consistent, but it surely can be made as consistent as modern mathematics, with practically no modifications of the basic setup. This goes under the name S …

Brouwer's intuitionistic mathematics was heavily criticized by his contemporaries, most notably Hilbert. For almost a century it was casually ridiculed by mathematicians who had no clue whatsoever abo …

It is easier to list the differences than similarities between the two kinds of logic. Mathematical logic is the branch of mathematics that studies mathematical activity. It has all the usual propert …

Please print this and hang it on your office door for your colleagues to see. The inference rule for $\forall x \in A . \phi(x)$ is as follows (in natural deduction style): $$\frac{\begin{matrix}[x\i …