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4
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69 views

Unclear on why Meissel's approach to counting primes works

I am reading through the Wikipedia article on prime counting. The following is presented to describe Meissel's approach: Let $p_1, p_2, \dots, p_n$ be the first $n$ primes. Let $\Phi(m,n)$ be the ...
1
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0answers
43 views

Deriving the wave equation in 3 dimensions and the history of it

I'm trying to find how the wave equation was derived in 3 dimensions. Surprisingly, there isn't much information available on this apart from wikipedia of all places https://en.wikipedia.org/wiki/...
8
votes
0answers
136 views

Who is “R. Drabek”?

The book "Algebra für Einsteiger" bei Bewersdorff (I think the English edition is called "Galois Theory for Beginners") starts with a nice quotation: Math is like love; a simple idea, but it can ...
6
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0answers
63 views

Proofs of Simplicity of $A_n$

There are different proofs of simplicity of the group $A_n$, and one can get at least two proofs by choosing randomly 10 books of the subject, so I will not go into what are these proofs? Rather, I ...
2
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0answers
54 views

Separability and second countability is the same thing to Halmos

I was browsing through Paul Halmos' classic book on measure theory, when I came by the following definition of separability on page $3$ in the chapter on prerequisites: Today a separable space is ...
3
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0answers
35 views

Characterizations of Nilpotent Groups

There are several characterizations of finite nilpotent groups (they are, as in wiki): $G$ is (finite) nilpotent group. Normalizer of every proper subgroup is bigger than the subgroup. Every ...
4
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0answers
30 views

History of graph minor concept

I can't find the correct reference to the first introduction of graph minor. There are plenty of strong results on minors (Kuratowski theorem, well-quasi-ordering by Robertson and Seymour, ...) and ...
14
votes
8answers
558 views

Serendipitous mathematical discoveries in recent times

As of today, most important results in mathematics are conjectured long before they are proven. Are there any examples of (important) mathematical discoveries that were proven by chance rather than ...
106
votes
19answers
4k views

Past open problems with sudden and easy-to-understand solutions

What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved ...
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3answers
625 views

Why is the word associative used to represent the concept of the associative property?

For the commutative property ... According to wikipedia: The word "commutative" is a combination of the French word commuter meaning "to substitute or switch" and the suffix -ative meaning "...
0
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0answers
24 views

What is the origin of the distinction between assignable and inassignable number?

Leibniz described his infinitesimals as being inassignable numbers in a number of texts, e.g., in his Cum Produisset that was analyzed in detail by H. Bos in a seminal text dating from the 1970s. The ...
17
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1answer
306 views

Errors in math research papers [duplicate]

Have there been cases of errors in math papers, that were undetected for so long, that they caused subsequent errors in research, citing those papers. ie: errors getting propagated along. My ...
4
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0answers
36 views

Reference on the history of ergodic theory

I'm looking for some good books on the history of ergodic theory. I'm a Ph.D student in the field, and I am taking Steven Weinberg's advice to learn about the history of my field: http://math....
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0answers
34 views

Why do we define a limit/continuous function/vector space etc. the way we do?

I am looking for any material dealing with the evolution of what now are standard mathematical definitions. One example what I am looking for: Let $(a_n)_{n\in\mathbb N}$ be a sequence with $a_n\in\...
0
votes
0answers
69 views

Proportion and disproportion in the Pythagorean theorem.

Is there any accepted explanation about why the square areas of the Pythagorean theorem are proportionated if the referential lengths of the legs and the hypothenuse are disproportionated? I think ...
0
votes
0answers
65 views

On the History of the Concept of Module

I am interested in knowing a little bit more about the history of the concept of module. As far as I know, there are two primary meanings of the word in mathematics, namely, modules as derived from ...
5
votes
2answers
93 views

The name $\mathcal{C}^\omega$

Let A be a set (of real numbers); define $\mathcal{C}^\omega (A)$ as the set of all real-valued functions that are defined, bounded, and analytic on A. My question is simply this: how did $\mathcal{C}...
2
votes
0answers
15 views

Local Solid Angle Units

This is a cultural question: Are there any, even moderately or historically used, units that measure solid angles which are not steradians? Basically, is there a unit x such that x:sr::grad:rad?
2
votes
1answer
82 views

Where is the definition of the derivative formula derived from?

I know what the definition of the derivative is , however, I am curious where this comes from mathematically.
6
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2answers
63 views

What “linguistic and philosophical problems” might be inherent in trigonometry?

In "A Mathematician’s Lament", Paul Lockhart derides the "status quo" of math education, claiming that "mathematics is an art form done by human beings for pleasure" but instead is taught "devoid of ...
0
votes
1answer
21 views

Did the symbol for expectation value originate in bra-ket notation?

I was just wondering, whether the common $\langle x \rangle$ symbol for the expectation value of a variable originates in the bra-ket notation of quantum physics? I would think that a fundamental ...
1
vote
1answer
20 views

On non-modular lattices and orto-modularity

I would like to have a definition for non-modular lattices which clearly sets them appart from their modular counterparts, thereby focusing on their main distinctive feature. Besides, I would be very ...
1
vote
1answer
158 views

Why did Euclid call 6 a perfect number?

The old Greek did not consider $1$ a number. Nevertheless Euclid called $6 = 1+2+3$ a perfect number. How could he use $1$ which was not a number?
5
votes
1answer
101 views

Connection of Fourier's work with Fredholm's

Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with ...
0
votes
1answer
49 views

Why do we use a single 'dash' for difference : $-$ and a double 'dash' for sum: $+$? [closed]

Why do we use a single 'dash' for difference : $-$ and a double 'dash' for sum: $+$? Just a shower thought: Who came up with this notation? It kind of makes it look like the difference is simpler ...
0
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0answers
22 views

On Symmetric Bilinear Forms by Milnor and Husemoller

Note: I am not sure the question is worth to be asked. But I have always been curious about this... The usual practice in mathematics is to put the names of authors in alphabetic order. However, ...
2
votes
1answer
80 views

History of Riesz spectral theory on compact maps and the Fredholm alternative's place in it.

Im reading Lax book in functional analysis. He proves the Fredholm alternative for compact operators. I.e For compact maps $C$ and for $T=I-C$ we have ; i)$u \in R_{T}$ iff $(u,\ell)=0$ for all $\...
12
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5answers
1k views

Mathematicians who overcame academic failure to achieve success [closed]

Does anyone have any story of mathematicians who overcame "academic failure" or setbacks to achieve success later as a result of their perseverance? This is a soft question, that hopefully can inspire ...
0
votes
0answers
28 views

Power Set, Theory of Linear Algebra, and Axiomatic Systems

So for homework, my History of Math professor gave us these three questions: Explain why the power set of a set S (collection of all subsets) has the same cardinality as the set [all functions from ...
3
votes
2answers
238 views

Why we still diagonalize compact operators even tho we lack invertibility.

We know that any compact symmetric operator on a Hilbert space, has a orthogonal base of eigenvectors. But we also know that $0$ is in the spectrum if $X$ is infinite dimensional, which makes the ...
3
votes
0answers
66 views

The motivation for quivers? [duplicate]

I would like to know about the reasons (I mean, methodological reasons, not just a penchant for innovation in terminology) for Pierre Gabriel to make use of quivers. Is it fair to say he wanted to ...
2
votes
2answers
89 views

Proving $n\sin(\frac{\pi}{n})<\pi<n\tan(\frac{\pi}{n})$ ; obtaining results from it.

I was reading The Simpsons and the Mathematical Secrets when I encountered the story of $\pi$. It mentions how Archimedes devised a method to place a lower and upper bound on $\pi$ by bounding a ...
2
votes
1answer
42 views

Solving cubics with complex numbers, before complex numbers.

An aside in another website reads: Complex numbers were used to solve cubic polynomials, before complex numbers were invented. I tried Googling this technique but didn't get anywhere. What is ...
2
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0answers
25 views

What is a point to give the Abel's Test for product series convergence a place in introductory textbooks?

One of the hypotheses of the Abel's test for product series convergence is stronger than the corresponding Dirichlet's test; that is, the former imposes the convergence of one of the series and the ...
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votes
2answers
48 views

Value of a number

How does one define a value of a number? What is the value of the number 4? Asked differently, how does one show that a certain number is greater than another number? After this, one might ask how do ...
0
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0answers
53 views

Was Ramsey mistaken in thinking that the same proposition can be both elementary and non-elementary in form?

According to Ramsey's Foundations of Mathematics, chapter III, suppose $'a', 'b', ..., 'z'$ were all the individuals, then $\phi{a}.\phi(b)...\phi(z)$ expresses the same proposition as $(x)\phi(x)$ ...
3
votes
1answer
64 views

Why is the notation $\frac{\partial f}{\partial x}$ used? [duplicate]

I am wondering why the notation $\frac{df}{dx}$ isn't used for partial derivatives, because it seems to me like someone could tell that it was a partial derivative if they knew that $f$ was a function ...
3
votes
0answers
75 views

Motivation of Vieta's transformation

The depressed cubic equation $y^3 +py + q = 0$ can be solved with Vieta's transformation (or Vieta's substitution) $y = z - \frac{p}{3 \cdot z}.$ This reduces the cubic equation to a quadratic ...
5
votes
1answer
95 views

Is the natural exponential function defined as being its own derivative?

Is $e^x$ actually defined as being the function $f$ for which $\dfrac{d}{dx}f=f$? By which I mean not "does the identity hold", of course I know it does and that this definition is sufficient for $e$,...
6
votes
1answer
92 views

History: Probability Theory

Of course they're both major oversimplifications, but which of (1) and (2) is closer to the truth? Lebesgue invents measure theory and then Kolmogorov notices that measure theory can be used to ...
21
votes
1answer
1k views

What did Hilbert actually want for his second problem?

When I read about the historical background of Gödel's incompleteness theorems, it is often mentioned that he was essentially responding to Hilbert, who was trying to prove the consistency of (...
7
votes
2answers
140 views

Where did the angle convention originate?

Where did the angle convention (in mathematics) come from? One would imagine that a clockwise direction would be more 'natural' (given sundials & the like, also a magnetic compass dial). Also, ...
2
votes
2answers
66 views

Who invented the notation $Df$ for the derivative?

We are often taught that $f'$ came from Newton and $\frac{df}{dx}$ came from Leibniz, but who introduced $Df$? Are there other notations for this simple idea by famous mathematicians?
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3answers
65 views

The meaning of notation with two letters inside of parentheses

What does the notation in the red box mean?
3
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0answers
59 views

Mathematical texts: white background or tan [closed]

Why is it that while the vast majority of mathematical textbooks are printed on white paper, a select few are printed on that tan, sand-colored paper that sometimes shows up? I find the latter very ...
5
votes
1answer
344 views

What exactly did Hermann Weyl mean?

"The introduction of numbers as coordinates is an act of violence." - Hermann Weyl. A lot of people like this quote, apparently. They also seem to associate it to the manifold context in the obvious ...
1
vote
0answers
27 views

On a curve every point of which is a point of ramification

The title of my post is the same as the title of a known article written by Sierpinski where he introduced its famous triangle. In the book Handbook of the history of general topology by Lowen said ...
2
votes
1answer
114 views

Euler's derivation of e?

Does anyone know where I can read Euler's original derivation of the infinite series used to define $e$? I mean the series as defined in the wikipedia page about $e$.
6
votes
2answers
504 views

How to convert Roman numerals with dashes?

What does the dash mean over the symbols here? How to convert these Roman numerals to numbers? Textual equivalent of the image: $$\overline{\text{M}}\,\overline{\text{L}}\,\overline{\text{V}}\...
1
vote
3answers
148 views

Why is variable called “variable” in mathematics if in fact it's immutable?

I've never thought of this issue until recently when I've been using Haskell to build a substantial project. In Haskell (and functional programming languages in general), most so-called "variables" ...