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Why are $\pi$ and $e$ simply referred to as “pi” and “e”?

I'm aware of the names "Archimedes' constant" and "Euler's number" for $\pi$ and $e$ respectively, but these don't seem to be used very commonly. Even in school I remember $\pi$ and $e$ being almost ...
6
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1answer
379 views

Generalizing complex numbers: Is there a mathematical system isomorphic to 3 dimensional space?

As I understand it, complex numbers: $ax+i$ are isomorphic to two-dimensional space. Quaternions consist of $4$ dimensions. Is that right? Wikipedia says "quaternions form a four-dimensional ...
0
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0answers
17 views

Prime Counting: In truncation rule #2 mentioned in an AMS.org article, I'm unclear how “special leaves” work?

I'm reading through an AMS.org article on prime counting. The article covers the history of prime counting and focuses on improvements to the Meissel-Lehmer method. It is the improvement on page ...
0
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0answers
36 views

Did Ackermann produce a finitary consistency proof of second-order $PRA$?

In Wilhelm Ackermann's Doctoral Thesis (it is claimed, by Richard Zach, for one, in his paper "The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program", arXiv: math/...
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4answers
276 views

What are the disadvantages of non-standard analysis?

Most students are first taught standard analysis, and the might learn about NSA later on. However, what has kept NSA from becoming the standard? At least from what I've been told, it is much more ...
0
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1answer
63 views

What is the origin of the name Hermitian and Unitary matrix?

A matrix $H$ is Hermitian if $H ^\dagger = H$. A matrix $U$ is Unitary if $U^\dagger=U^{-1}$. My question is: Why do we name matrices of such properties Hermitian and Unitary? These names are non-...
4
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2answers
60 views

Where do hash functions come from?

I have some basic understanding of how hash functions work, however, I have no idea of how mathematicians created them. Were them a byproduct of a non cryptografics related research or were them a ...
1
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10answers
1k views

What's an example of an infinitesimal?

If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give them? What I am asking for are specific techniques for explaining infinitesimals to students, ...
2
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0answers
33 views

Trying to understand how Lehmer's method represents a simplification of Meissel's method for counting primes

My question stems from a wikipedia article on prime counting. The details on Meissel's method can be found in the wikipedia article. As I understand, Meissel proposed two formulas which I asked ...
4
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2answers
139 views

Peano Arithmetic before Gödel

If I understand correctly, Gödel was the one to discover how to encode finite sequences of integers in Peano Arithmetic, with his Chinese Remainder Theorem trick (his "beta function"). As a ...
16
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3answers
703 views

What did Whitehead and Russell's “Principia Mathematica” achieve?

In philosophical contexts, the Principia Mathematica is sometimes held in high regard as a demonstration of a logical system. But what did Whitehead and Russell's Principia Mathematica achieve for ...
3
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1answer
67 views

Why Frobenis concerned the groups which today called “Frobenius Group”?

From their work, it seems that the Ancient mathematicians were investigating a mathematical object not as a fun, but to solve some problem occurred in earlier work of someone. Lagrange, Galois, Abel ...
4
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0answers
70 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
45 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
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0answers
140 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
56 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
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8answers
572 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
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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 ...
16
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3answers
629 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 ...
16
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1answer
307 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
37 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....
0
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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
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0answers
70 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
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0answers
66 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
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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
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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
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1answer
84 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
64 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
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1answer
22 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 ...
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1answer
22 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
160 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
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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 ...
-1
votes
1answer
52 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
24 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
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0answers
29 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
240 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
67 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
92 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
43 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
26 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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2answers
49 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 ...
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0answers
58 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
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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
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0answers
76 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 ...