History and philosophy of mathematics, biographies of mathematicians, mathematics education, recreational mathematics, communication of mathematics.

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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 4/...
186
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36answers
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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 ...
118
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94answers
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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 ...
37
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37answers
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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 ...
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21answers
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Thinking and Explaining

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
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59answers
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Pseudonyms of famous mathematicians

Many mathematicians know that Lewis Carroll was quite a good mathematician, who wrote about logic (paradoxes) and determinants. He found an expansion formula, which bears his real name (Charles ...
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69answers
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Mathematical “urban legends”

When I was a young and impressionable graduate student at Princeton, we scared each other with the story of a Final Public Oral, where Jack Milnor was dragged in against his will to sit on a committee,...
91
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19answers
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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 ...
94
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6answers
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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 (...
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29answers
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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 ...
103
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6answers
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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 ...
261
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72answers
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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 ...
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136answers
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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 ...
155
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64answers
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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 ...
111
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27answers
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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 ...
71
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32answers
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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 ...
70
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26answers
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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. ...
42
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15answers
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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 ...
62
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15answers
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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 ...
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10answers
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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 ...
15
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3answers
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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. ...
39
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3answers
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What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?

This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields ...
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4answers
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Earliest diagonal proof of the uncountability of the reals.

I cited the diagonal proof of the uncountability of the reals as an example of a `common false belief' in mathematics, not because there is anything wrong with the proof but because it is commonly ...
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1answer
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Countable connected Hausdorff space

Let me start by reminding two constructions of topological spaces with such exotic combination of properties: 1) The elements are non-zero integers; base of topology are (infinite) arithmetic ...
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8answers
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Theorems first published in textbooks?

According to Wikipedia, the Bohr-Mollerup Theorem (discussed previously on MO here) was first published in a textbook. It says the authors did that instead of writing a paper because they didn't think ...
18
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6answers
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Uppercase Point Labels in High-School Diagrams: from Euclid?

I wonder if the convention of labeling points in geometric diagrams with uppercase symbols ultimately derives from Greek mathematics, which was originally written in "majuscule" (uppercase) Greek ...
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5answers
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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? ...
3
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2answers
804 views

Why unknowns are usually denoted by “X” ? [closed]

Why unknowns are usually denoted by "X" ? More precisely: is this answer really a serious answer or might be a 1 April joke ? Let me sketch it. But please watch it, it is really fun and cool and &...
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1answer
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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 ...
0
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1answer
474 views

The $\zeta-$word [closed]

I was wondering about classical notations in number theory. I will not ask here about special functions in general but about the more ubiquitous number theory functions. That which made me wonder ...
78
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16answers
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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, ...
67
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25answers
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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 ...
67
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7answers
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Is the boundary $\partial S$ analogous to a derivative?

Without prethought, I mentioned in class once that the reason the symbol $\partial$ is used to represent the boundary operator in topology is that its behavior is akin to a derivative. But after ...
87
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30answers
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Examples of theorems misapplied to non-mathematical contexts

For something I'm writing -- I'm interested in examples of bad arguments which involve the application of mathematical theorems in non-mathematical contexts. E.G. folks who make theological arguments ...
68
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6answers
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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 ...
55
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13answers
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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 ...
79
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23answers
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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 ...
58
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6answers
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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 ...
59
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22answers
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Examples of major theorems with very hard proofs that have NOT dramatically improved over time

This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time. I am looking for a list of Major theorems in mathematics whose proofs ...
53
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36answers
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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 $1+\frac{1}{...
35
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8answers
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Motivation for and history of pseudo-differential operators

Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...
45
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31answers
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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 ...
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19answers
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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 ...
67
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10answers
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Have you solved problems in your sleep?

I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending ...
55
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27answers
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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?
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9answers
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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 ...
67
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18answers
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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 ...
62
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6answers
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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, ...
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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 ...
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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 ...