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Results tagged with soft-question
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user 1946
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
263
votes
What are the most misleading alternate definitions in taught mathematics?
Many topics in linear algebra suffer from the issue in the
question. For example:
In linear algebra, one often sees the determinant of a
matrix defined by some ungodly formula, often even with
specia …
105
votes
Accepted
Have you solved problems in your sleep?
On several occasions it has happened that I have made a key insight while sleeping or drifting in and out of sleep.
For example, one of the critical ideas in my paper
Joel David Hamkins, Gap forcing, …
104
votes
Theorems with unexpected conclusions
My favorite example of this phenomenon is Goodstein's Theorem.
Take any positive number $a_2$, such as the number $73$, and write it in complete base $2$, which means write it as a sum of powers of $2 …
98
votes
Accepted
Adapting arguments and plagiarism
What you describe seems to me to be a normal mode of mathematical
progress, and I would urge you simply to carry on! Ride that train
as far as you can.
It often happens that someone's mathematical re …
94
votes
Mistakes in mathematics, false illusions about conjectures
Computer designers and programmers dreamed, from the earliest days of the computer, of a computer that could play chess and win. Even Alan Turing had that dream, and designed turochamp, the first ches …
86
votes
Has incorrect notation ever led to a mistaken proof?
Here is an example from set theory.
Set theorists commonly study not only the theory $\newcommand\ZFC{\text{ZFC}}\ZFC$ and its models, but also various fragments of this theory, such as the theory o …
82
votes
Value of "of course" in the mathematical literature
I don't agree that if something is obvious, then it is obvious that it is obvious. When an author declares in a mathematical exposition that a fact is obvious, or says "of course" or something with a …
80
votes
Which mathematical ideas have done most to change history?
Turing's work on computability, extending those of Goedel and the other early logicians, paved the way for the development of modern computers. Before Turing and Goedel, the concept of computability w …
74
votes
Accepted
What's wrong with the surreals?
At a recent conference in Paris on Philosophy and Model Theory (at which I also spoke), Philip Ehrlich gave a fascinating talk on the surreal numbers and new developments, showcasing it as unifying ma …
- 236k
73
votes
Jokes in the sense of Littlewood: examples?
The fundamental axioms of mathematics are inconsistent if
and only if we can prove that they are consistent.
(Because, you know, it follows from "logic." See Second
Incompleteness
theorem)
68
votes
Accepted
Capitalization of theorem names
In English, proper nouns are capitalized. The numbered instances you mention are all usages as proper nouns, but merely refering to a lemma or corollary not by its name is not using a proper noun, and …
67
votes
How has "what every mathematician should know" changed?
As mathematics grows and diversifies beyond belief, surely the collection of topics that every mathematician must know is shrinking fast. One can carry out serious mathematical research in one area wh …
61
votes
Naming in math: from red herrings to very long names
Let me mention as a counterpoint that there is less need for
new terminology than one might expect. Mathematical exposition
is often more successful and clearer without new terminology, and
one should …
60
votes
Accepted
How do you select an interesting and reasonable problem for a student?
Let me first answer a slightly different question, how to organize one's thoughts about such problems. I simply maintain a list of suitable projects, with ideas on how to approach them, and put them i …
48
votes
Awfully sophisticated proof for simple facts
There is no largest natural number. The reason is that
by Cantor's theorem, the power set of a finite set is a
strictly larger set, and one can prove inductively that
the power set of a finite set is …