All Questions
10 questions
152
votes
26
answers
39k
views
Has philosophy ever clarified mathematics?
I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...
104
votes
19
answers
14k
views
Can a mathematical definition be wrong?
This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...
68
votes
9
answers
12k
views
When have we lost a body of mathematics because errors were found?
The history of mathematics over the last 200 years has many occasions when the fundamental assumptions of an area have been shown to be flawed, or even wrong. Yet I cannot think of any examples where, ...
60
votes
15
answers
11k
views
Abstract thought vs calculation
Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was the use of more abstract notions ...
51
votes
30
answers
8k
views
Taking a theorem as a definition and proving the original definition as a theorem
Gian-Carlo Rota's famous 1991 essay, "The pernicious influence of mathematics upon philosophy" contains the following passage:
Perform the following thought experiment. Suppose that you are ...
47
votes
7
answers
8k
views
Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians
I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics....
39
votes
10
answers
4k
views
Believing the Conjectures
In Believing the axioms (I and II), Penelope Maddy proposes five "rules of thumb" that she then uses to justify large cardinal axioms in set theory. These extrinsic rules are modeled after the ...
32
votes
21
answers
16k
views
What are some applications of other fields to mathematics?
It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely:
What are some applications ...
15
votes
4
answers
3k
views
Statements which were given as axioms, which later turned out to be false.
I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness axiom....
8
votes
1
answer
388
views
Formalisation of intuitive concepts in the language leading to mathematical progress
In his work, Albert Lautman thinks the genesis of some mathematical works as a dialectic that takes place between opposite notions, like between global and local. He argues that while those notions, ...
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