**134**

votes

**36**answers

36k views

### Widely accepted mathematical results that were later shown wrong?

I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ...

**21**

votes

**2**answers

2k views

### Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated ...

**37**

votes

**37**answers

13k views

### Major mathematical advances past age fifty [closed]

From A Mathematician’s Apology, G. H. Hardy, 1940:
"I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever ...

**79**

votes

**19**answers

12k views

### Do you read the masters?

I often hear the advice, "Read the masters" (i.e., read old, classic texts by great mathematicians). But frankly, I have hardly ever followed it. What I am wondering is, is this a principle that ...

**78**

votes

**6**answers

8k views

### what mistakes did the Italian algebraic geometers actually make?

It's "well-known" that the 19th century Italian school of algebraic geometry made great progress but also started to flounder due to lack of rigour, possibly in part due to the fact that foundations ...

**56**

votes

**29**answers

6k views

### The half-life of a theorem, or Arnold's principle at work

Suppose you prove a theorem, and then sleep well at night knowing that future generations will remember your name in conjunction with the great advance in human wisdom. In fact, sadly, it seems that ...

**208**

votes

**72**answers

84k views

### Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...

**123**

votes

**132**answers

28k views

### Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)
I'd love to learn about ...

**142**

votes

**64**answers

24k views

### Proofs that require fundamentally new ways of thinking [closed]

I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is ...

**98**

votes

**97**answers

57k views

### Famous mathematical quotes [closed]

Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?
Standard community wiki rules apply: one ...

**55**

votes

**32**answers

35k views

### Why do we teach calculus students the derivative as a limit?

I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher ...

**55**

votes

**14**answers

6k views

### Mathematical research published in the form of poems

The article
Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen,
Math. Z. 127 (1972), no. 1, 10-16
is written in the form of a lengthy poem, in a style similar to that
of the ...

**30**

votes

**15**answers

5k views

### Abstract Thought vs Calculation

Jeremy Avigad and Erich Reck in their remarkable historical paper "Clarifying the nature of the infinite: the development of metamathematics and proof theory" claim that one of the factors of becoming ...

**49**

votes

**36**answers

11k views

### What are some correct results discovered with incorrect (or no) proofs?

Many famous results were discovered through non-rigorous proofs, with
correct proofs being found only later and with greater difficulty. One that is well
known is Euler's 1737 proof that
...

**33**

votes

**10**answers

3k views

### Is there a mathematical axiomatization of time (other than, perhaps, entropy)?

Since Euclid's axiomatization of space, we have developed a sophisticated mathematical model of space. Given a category of structures (measures), local space is modeled the spectrum of measurements ...

**85**

votes

**5**answers

6k views

### Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference:
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
where the ...

**18**

votes

**18**answers

6k 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 ...

**14**

votes

**5**answers

3k views

### When did the career of 1 as a prime number begin and when did it end? [closed]

The old Greek did not consider 1 a number, so it was not a prime. The theorem of unique prime factorization excludes 1 to be a prime number. But in between probably at Euler's and Goldbach's times? ...

**8**

votes

**1**answer

2k views

### What is the source of this E̶r̶d̶ő̶s̶ quote?

Namely, the following one
"All problems appeared once in the [American Mathematical] Monthly."
I remember reading it several years ago... When I first posed the question, I believed that I had ...

**74**

votes

**16**answers

17k views

### What if Current Foundations of Mathematics are Inconsistent? [closed]

The title of the question is also the title of a talk by Vladimir Voevodsky, available here.
Had this kind of opinion been expressed before?
EDIT. Thanks to all answerers, commentators, voters, ...

**90**

votes

**26**answers

12k views

### Extremely messy proofs

Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what ...

**63**

votes

**24**answers

7k views

### Modern Mathematical Achievements Accessible to Undergraduates

While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even ...

**65**

votes

**6**answers

8k views

### How to find ICM talks?

I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in ...

**81**

votes

**90**answers

11k views

### What would you want to see at the Museum of Mathematics?

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...

**54**

votes

**13**answers

10k views

### Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic.
The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...

**54**

votes

**6**answers

16k views

### What are Jacob Lurie's key insights?

This question is inspired by this Tim Gowers blogpost.
I have some familiarity with the work of Jacob Lurie, which contains ideas which are simply astounding; but what I don't understand is which key ...

**53**

votes

**26**answers

8k views

### What are some famous rejections of correct mathematics?

Dick Lipton has a blog post that motivated this question. He recalled the Stark-Heegner
Theorem: There are only a finite
number of imaginary quadratic fields
that have unique factorization. ...

**39**

votes

**19**answers

6k views

### Mathematicians whose works were criticized by contemporaries but became widely accepted later

Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's ...

**37**

votes

**32**answers

6k views

### Trichotomies in mathematics

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...

**62**

votes

**6**answers

8k views

### Why didn't Vladimir Arnold get the Fields Medal in 1974?

As you all probably know, Vladimir I. Arnold passed away yesterday. In the obituaries, I found the following statement (AFP)
In 1974 the Soviet Union opposed Arnold's award of the Fields Medal, ...

**47**

votes

**24**answers

7k views

### Writing papers in pre-LaTeX era?

I wonder how people wrote papers in the pre-LaTeX era? I mean, when typewriters and simple computers were (60th-70th?). Did they indeed put formulas by hand in the already printed articles?

**30**

votes

**8**answers

3k views

### What do named “tricks” share?

There are a number of theorems or lemmas or mathematical ideas that come to be known as eponymous
tricks, a term which in this context is in no sense derogatory.
Here is a list of 10 such tricks (the ...

**40**

votes

**4**answers

3k views

### Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative ...

**25**

votes

**2**answers

1k views

### The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.

I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...

**25**

votes

**9**answers

22k views

### What is the shortest Ph.D. thesis? [closed]

The question is self-explanatory, but I want to make some remarks in order to prevent the responses from going off into undesirable directions.
It seems that every few years I hear someone ask this ...

**17**

votes

**10**answers

3k views

### New proofs to major theorems leading to new insights and results?

I am wondering, historically, when has a new proof of an old theorem been particularly fruitful. A few examples I have in mind (all number theoretic) are:
First example is classical... which is ...

**26**

votes

**7**answers

983 views

### Concise model of modern fiat money and its non-conservation

A confession: I have never really understood the basic model of fiat money and central banking, by which a central bank controls the money supply. By the standards of someone trained in mathematics, ...

**25**

votes

**10**answers

4k views

### real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?

**37**

votes

**8**answers

3k views

### What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?

I think we all occasionally come across terminology that we'd like to see supplanted (e.g. by something more systematic). What I'd like to know is, under what circumstances is it reasonable to believe ...

**22**

votes

**5**answers

4k views

### Origins of names of algebraic structures

Consider the names of basic algebraic structures: 'group', 'ring', 'space', 'field', 'Körper', even the name 'structure' itself - all of them time-honoured terms, deeply rooted in our history and ...

**18**

votes

**9**answers

3k views

### Was the early calculus inconsistent?

This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY.
George Berkeley wrote in 1734 with reference to the early calculus that ...

**10**

votes

**4**answers

2k views

### History of the high-dimensional volume paradox

Inscribe an $n$-ball in an $n$-dimensional hypercube of side equal to 1, and let $n \rightarrow \infty$. The hypercube will always have volume 1, while it is a fun folk fact (FFF) that the volume of ...

**33**

votes

**9**answers

3k views

### Who first proved that the value of C/d is independent of the choice of circle?

I have an elementary question about the history of $\pi$. I thought the answer would be easy to find. But, to the contrary, after quite a bit of searching and after consulting math historians, I have ...

**31**

votes

**1**answer

3k views

### A topologist is not a mathematician - a small question

Years ago I read about a topologist who was to enter the states as an immigrant and was asked a question about his profession. He indicated he was a topologist, but as this was not included on the ...

**18**

votes

**19**answers

5k views

### What are some mathematical concepts that were (pretty much) created from scratch and do not owe a debt to previous work?

Almost any mathematical concept has antecedents; it builds on, or is related to, previously known concepts. But are there concepts that owe little or nothing to previous work?
The only example I know ...

**13**

votes

**3**answers

3k views

### Why is a ring called a “ring”?

Why is a ring called "ring" (or Zahlring in German)? There seems to (naive) me nothing more ring-like to a ring than there is to a group or a field. I am particularly interested to learn why the ...

**11**

votes

**3**answers

588 views

### Riemann's formula for the metric in a normal neighborhood

I would love to understand the famous formula $g_{ij}(x) = \delta_{ij} + \frac{1}{3}R_{kijl}x^kx^l +O(||x||^3)$, which is valid in Riemannian normal coordinates and possibly more general situations.
...

**29**

votes

**1**answer

737 views

### Producing finite objects by forcing!

It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations,
where we can use forcing to prove the existence of finite objects with some ...

**25**

votes

**2**answers

803 views

### Klein's Protocols: A window into our mathematical past

Klein's Protocols in over 8,000 pages recording seminars organized from 1872 to 1913 by Felix Klein and given by Klein, his colleagues, students and other invited speakers, including luminaries such ...

**18**

votes

**14**answers

2k views

### Insightful books about elementary mathematics

What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful ...