Use this tag for questions concerning history of mathematics, historical primacies of results, and evolution of terminology, symbols, and notations.

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-4
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0answers
42 views

Math: A discovery or a creation? [on hold]

I am just curious as to what math is at it's very basic state. Is math something that humans have invented? Or is it more of a discovery? Or possibly something completely different. If it is something ...
1
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0answers
82 views

Research Paper on Math [on hold]

I have to write a research paper for American history and it can be about anything so I want to write it about math. The thing is, in order to do so I have to be able to tie it back into American ...
2
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0answers
97 views

Is Euclidean geometry really a “dead” subject? If so, why? [on hold]

It seems that Euclidean geometry is a "dead" subject nowadays. In the time of the Greeks, mathematicians and geometers were one and the same. Today, very few professional mathematicians study ...
47
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6answers
5k views
+500

Mathematically, why was the Enigma machine so hard to crack?

Mathematically, why was the Enigma machine so hard to crack? In laymen terms, what was it exactly that made cracking the Enigma machine such a formidable task? Everything I have seen about the ...
1
vote
1answer
52 views

informal semantics regarding CH and AC

why is the assertion $\aleph_1=2^{\aleph_0}$ referred to as a hypothesis, whereas $$\forall \alpha( S_\alpha \ne \varnothing) \Rightarrow \prod_\alpha S_\alpha \ne \varnothing$$ is called an axiom? ...
0
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1answer
61 views

Which one of the following logical propositions is to be preferred?

I'm trying to update the symbolism of Giuseppe Peano's "Arithmetices Principia", to make the translation freely available. Might I ask you, which of the following might be a correct mathematical ...
5
votes
1answer
120 views

How fast was the Turing's machine for breaking the enigma code?

We know that, recently, personal computers make around $10^9$ calculations per second, and I'm just curious about how many calculations was able to compute the machine invented by Turing for breaking ...
3
votes
2answers
74 views

Why do we think of group compositions as multiplication?

This has bothered me for some time: The composition in a group is usually denoted $xy$ or $x\cdot y$. Powers (note the word) are denoted by $x^n$, inverses by $x^{-1}$, and the neutral element by $1$. ...
0
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2answers
72 views

What comes after seconds?

Angles can be measured in different ways. For example, one can measure angles in degrees/minutes/seconds. So $1^\circ$ is divded into $60$ min. and $1$ min is divided into $60$ sec. That way a tenth ...
2
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0answers
52 views

How much math was “Broken” by Russell's Paradox?

As you know, the phrase "the set of all sets that don't contain themselves" caused a paradox that "broke" (made trivial) Naive set theory. How much mathematics had to be redone because of this? Most ...
2
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0answers
41 views
+50

Why are logrithms of trigonometric functions useful

I have noticed that in many trigonometric tables the logarithm of the trigonometric values are given. Why this is given and not the actual values of the trigonometric functions? For example, instead ...
1
vote
2answers
35 views

What are these numbers in a logarithmic table?

Below is an image from a table of logarithms. As an example, one sees that $\log(661.3) = 2.82\color{red}{040}$. In this logarithmic table there are some numbers to the right. My question is: What is ...
87
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15answers
9k views

Has lack of mathematical rigour killed anybody before? [closed]

One of my friends was asking me about tertiary level mathematics as opposed to high school mathematics, and naturally the topic of rigour came up. To provide him with a brief glimpse as to the ...
1
vote
1answer
23 views

Derivation of the discriminant of a cubic polynomial by algebraic manipulation.

The problem was asked before: Using Vieta's theorem for cubic equations to derive the cubic discriminant . I tried to solve it by purely algebraic manipulation but was faced with an explosion of ...
1
vote
5answers
61 views

What does the solution of a PDE represent?

So I took a course in PDE's this semester and now the semester is over and I'm still having issue with what exactly we solved for. I mean it in this sense, in your usual first or second calculus ...
1
vote
1answer
83 views

What is the difference (or relationship) between geometric length and arithmetic numbers?

In Abbott's Understanding Analysis there was a phrase like, "Ancient Greeks did not understand the difference (or relationship) between geometric length and arithmetic numbers." What is this ...
2
votes
1answer
17 views

Noether comment to Dedekind and Weber's work

I am trying to consult Emmy Noether's “Erläuterungen zur vorstehenden Abhandlung”, some sort of epilogue or comment to Richard Dedekind and Heinrich Weber's “Theorie der algebraischen Funktionen einer ...
0
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3answers
55 views

History and early development of Mathematics

Please provide references (books, articles, websites) that describe the conceptual development of calculus, complex numbers, group theory and matrix. I am curious about how the Mathematicians ...
0
votes
1answer
30 views

On Dedekind's prime ideals

Prime ideals were an essential tool for Dedekind to save or restore unique factorization. Is it fair to say that the shift from Kummer's ideal numbers to Dedekind's ideals (with prime ideals, and so ...
5
votes
1answer
218 views

What does “hom” stand for in hom-sets and hom-functors?

With given category $\mathcal{C}$ and its objects $A$ and $B$, a hom-set $\hom_\mathcal{C}(A, B)$ is the collection of all morphisms from $A$ to $B$. There is also a related notion of hom-functor ...
3
votes
1answer
64 views

A question regarding Kummer [closed]

As you know, Ernst Kummer noticed that examples such as $$6 = 2\cdot 3 \text{ or } 6 = 3 \cdot 2 \text{ and, crucially } 6 = (1 + \sqrt{-5}) (1 -\sqrt{-5}) $$ proved the failure of unique ...
3
votes
0answers
56 views

Why didn't Bernoulli and Euler use an integral comparison to estimate the solution to the Basel problem?

I was reading the history of the Basel problem in William Duhnam's book, Euler - The Master of Us All. The book tells how Jakob Bernouili did some clever manipulation to show that the sum of $1/n^2 ...
2
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1answer
39 views

Original paper by Gauss on gaussian integers

Could anyone provide me with the title and date of Gauss's paper where he first introduces gaussian integers and proves their unique factorization? If you could also provide me with his exact proof ...
0
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0answers
22 views

Maltsev on Algebraic Systems

As far as I know, it was A.I Maltsev who fist coined the term "Algebraic systems" in a paper from 1953. Then Birkoff, MacLane and others extended its usage and appplications. My question is a simple ...
1
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0answers
26 views

Notations for interior product

There are two symbols in the Unicode "Supplementary Mathematical Operators" range whose names intrigue me 2A3C: INTERIOR PRODUCT: ⨼ (like $\lnot$ upside down) 2A3D: RIGHTHAND INTERIOR PRODUCT: ⨽ ...
1
vote
2answers
67 views

Why is the letter “F” used for the curvature 2-form?

Given a differentiable manifold $X$, a vector bundle $E\to X$ and a connection $A$ on $E$. The curvature $2$-form of the connection is a $2$-form with values on the endomorphisms of $E$ defined as ...
7
votes
1answer
79 views

(Co)homology theory and electrical circuit

I have read that one of the origins of the theory of (co)homology is the study of electrical circuits by Poincare. I'd like to know more about that. Could someone sugest any reference on this subject? ...
4
votes
0answers
74 views

In the mean value theorem, we are guaranteed $c$ such that $f'(c) = (f(b)-f(a))/(b-a)$. Does $c$ have a name?

The Mean Value Theorem says approximately that for differentiable $f$, there is a $c \in (a,b)$ such that $$ f'(c) = \frac{f(b)-f(a)}{b - a}. $$ I presume that the number $f'(c)$ is the mean value. My ...
0
votes
1answer
45 views

On Gaussian Primes

Some primes in the ring of integers (17, for example) cease to behave as such in the ring of gaussian integers, while others (7, for instance) keep being prime there as well. The former are of the ...
1
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0answers
51 views

When did Liouville come up with the first transcendental numbers?

There are some conflicting sources regarding this. It is a matter of fact that Liouville defined what it was for a number to be approximated to degree $n$ by rational numbers. He then effectively ...
-1
votes
0answers
67 views

diophantine-equations

Why there are no solutions in positive coprime integers for the following diophantine equation $$2x^3 + y^2 = z^k$$ where, (x,y,z) are (pairwise) positive coprime integers, and k is positive integer ...
3
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1answer
45 views

who coined the prime ideals?

I know that Ernst Kummer first made used of "ideal complex numbers", and, hinging on that, Dedekind later introduced his "ideals" in Vorlesungen über Zahlentheorie. But, who coined the term "prime ...
2
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2answers
70 views

Calculus without functions (or, how did Newton differentiate?)

I was recently reading about how functions did not really exist at the time of Newton and Leibniz; They thought in terms of geometry. That makes me curious. I can understand that derivation would be ...
0
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0answers
39 views

DeMoivre's approximation to the ratio of $\binom{n}{n/2}$ to $2^n$

I'm reading Stigler's History of Statistics and am trying to understand as many of the derivations as I can. Stigler begins his discussion of DeMoivre's contributions by stating the result that the ...
10
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2answers
1k views

Who discovered the first explicit formula for the n-th prime?

I just found out on Wolfram that there is a formula for the n-th prime in terms of elementary functions. I wonder who found it and if he was rewarded for this. The formula (here) is: Also shown at ...
2
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4answers
53 views

motivating diagonalization of a matrix [duplicate]

I have to teach about diagonalization of a matrix to a first year undergrad student and I was wondering what would be a good way to motivate this concept. I would appreciate any suggestions. Thanks!
0
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0answers
51 views

Historical calculations of $tan^{-1}x $ and $e^x$

SineBhaskara_I One reads that $tan^{-1}(x) $ series expansion existed in early (Indian) history. But like the Sine trigonometric function, did any similar approximation exist as well? The query ...
0
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0answers
19 views

Who is the inventor of slovin's formula?

And how can I use it in the population contain 10000 people with confidence interval 95%? Also, why there is only a few information about the inventor in the web?
0
votes
1answer
18 views

Table of Contents from André Weil's Edition of Kummer's papers

I would be very grateful if someone could provide me with the table of contents of Volume 1 (pertaining Number Theory) of Andre Weil's edition of Ernst Kummer's papers, published by Springer Verlag in ...
0
votes
3answers
36 views

Cross products and determinants in $\mathbb{R}^3$

I know that the absolute value of determinant of three vectors in $\mathbb{R}^3$ is the volume of the parallelepiped determined by the three vectors. The volume can be computed by basic calculation ...
2
votes
1answer
43 views

estimation of a unit circle - how to show a relationship

It has been eons since I've done any trigonometry, but I just can't prove how this following relationship holds for $n = 4, 8, 16, 32, \dots$ The relation is: $$ 2 \biggl( \! \frac{A_{2n}}{n} \! ...
3
votes
3answers
147 views

Why did it take mathematicians so long to discover non-Euclidean geometry?

Why did it take mathematicians so long to realise that Euclid's fifth postulate is independent of the other 4? Why didn't people like Lagrange notice that a sphere is a model for a non-Euclidean ...
0
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0answers
28 views

Who discovered the Inverse Function Theorem?

I was wondering who discovered this theorem, I can't find this information in Wiki or with a simple google research and all my books do not report the author.
0
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0answers
18 views

How Leibniz invented the Binary System?

Do you know which reasoning and observations made Leibniz invent the Binary system ? Some say that he was inspired by Chinese mathematicians do we have any record of how he came with this idea ?
10
votes
2answers
144 views

When, how & who first gave this calculation of $\pi$

I came across this interesting method to calculate $\pi$. Why is it true and who first presented it? To calculate $\pi$, multiply by $4$, the product of fractions formed by using, as the numerators. ...
0
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2answers
41 views

Functions applied from the right

In some of the older books by Nathan Jacobson (like Lie Algebras and Lectures in Abstract Algebra), a convention is used that is quite uncommon at least today: Functions are applied from the right. ...
2
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1answer
44 views

History Behind Integral Error Between $\pi$ and $22/7$

Looking at an expression for $\pi$ $$\pi = \frac{22}7-\int_0^1 \frac{x^4(1-x)^4}{1+x^2} \ dx$$ it seems to me that the integral expression is the error between the approximation $\frac{22}7$ and ...
4
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1answer
73 views

Maths Discoveries thanks to Computer Science

Which discoveries have been made in mathematics thanks to computer science ? For example fractals have been discovered thanks to computers (correct me if im wrong) do you know any similar discoveries ...
7
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1answer
94 views

What came first, the $\forall$ or the $\exists$? [closed]

I imagine that these symbols originated in one of the following ways: "I will declare a symbol for "for all." I will just use the letter "A" flipped upside-down. Yes, let $\forall$ represent "for ...
3
votes
1answer
71 views

How was the functorial factorization axiom “frequently misstated”?

In modern times, a model category is frequently required to have functorial factorizations of morphisms into (fibration/acyclic cofibration) and (acyclic fibration/cofibration). But the precise ...