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Results tagged with ho.history-overview
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user 1176
History and philosophy of mathematics, biographies of mathematicians, mathematics education, recreational mathematics, communication of mathematics.
38
votes
Logic in mathematics and philosophy
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 …
- 48.8k
29
votes
Bishop quote stating that axiom of choice is constructively valid
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 …
- 48.8k
25
votes
"Let $x \in A$", beginning a proof of "$\forall x \in A$ ...", if A were empty
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 …
- 48.8k
23
votes
Mathematicians whose works were criticized by contemporaries but became widely accepted later
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 …
18
votes
What's the earliest result (outside of logic) that cannot be proven constructively?
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 …
- 48.8k
13
votes
Was the early calculus inconsistent?
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 …
- 48.8k
11
votes
Accepted
What did the Intuitionists want to do with applied mathematics?
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):
…
- 48.8k
10
votes
Salvaging Leibnizian formalism?
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, …
- 48.8k
9
votes
Accepted
Did Brouwer evade uncountability?
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 …
- 48.8k
8
votes
Hilbert's and Gödel's expanded definition of "Recursive Function"
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 …
- 48.8k
3
votes
Accepted
Nonconstructive reasoning in Skolem's proof of the Löwenheim-Skolem Theorem
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 $b_m = d$ …
- 48.8k