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

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5answers
554 views

who invented division and why we do division in those steps told?

i know how to divide but i dont quit understand why we use those steps told in schools. like for example ...
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6answers
394 views

Evaluating the reception of (epsilon, delta) definitions

There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. ...
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0answers
43 views

The “enabler” of Maxwell's equations

Is it possible to point to a specific development in mathematics that allowed Maxwell's equations to happen? Similarly to Newton's laws of physics that depended on the invention of calculus? And ...
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0answers
215 views

How did Cohen invent forcing?

A couple of popular maths book, I forget which stated that Cohen invented Forcing. Now, generally I've noticed that there is a history which allows one in hindsight to show that how certain ...
3
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1answer
79 views

Source request of axiom of Archimedes

I'm a little confused with axiom of Archimedes has a proof since it is an axiom. So I'm guessing there's a historical reason that this property of ordered field was given such a name. Is there any ...
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2answers
468 views

What fields (and operators acting on those fields) might form the basis of alien mathematics?

Addition and multiplication, according to most histories, arose in human civilizations out of a need to count a finite number of objects, and then later on especially, to measure land. What might be ...
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9answers
933 views

Why did we define the concept of continuity originally, and why it is defined the way it is?

The concept of continuity is a very important idea in topology. Though I am using it all the time, but indeed I don't know what is the original purpose for us to define this concept. And I also don't ...
3
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1answer
85 views

Why might one be inclined to think that polynomials of the form $\cos(n\arccos{x})$ would minimize error in Lagrange interpolation?

I was first introduced to Chebyshev polynomials (of the first kind) in the form $T_n(x)=\cos\left(n \operatorname{arccos}(x)\right)$. The usual recurrence relation was then derived from using trig ...
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1answer
606 views

sin(x) infinite product formula: how did Euler prove it?

I know that $\sin(x)$ can be expressed as an infinite product, and I've seen proofs of it (e.g. Infinite product of sine function). I found How was Euler able to create an infinite product for sinc by ...
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2answers
166 views

Which small area of mathematics had fully developed already and thus no more research in this area?

Which small area of mathematics had fully developed already and thus no more research in this area? For example, no more PHD research in Euclidean Geometry anymore.
2
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3answers
143 views

Mathematician's names in structures.

I would like to know how it is that mathematical objects come to receive the name of a mathematician. Do these mostly happen through the author's proposal, or is it a process that takes more time? ...
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2answers
214 views

Popular Topics in mathematical analysis(Functional analysis)

I am writing a text(as a duty by my mentor) dealing with the recently popular topics(including open problems) in mathematical analysis. At first part, I briefly introduced the mathematical ...
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0answers
159 views

Accuracy of maths papers wrote in the 1950's

I am writing a code that will calculate the solid angle subtended by an off axis disk. I am using the table of answers in this paper on page 257 to check if my code is giving correct answers. The ...
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2answers
136 views

Understanding Euler's paper on curvature

We've recently discussed, in the course on differential geometry which I am taking, Euler's theorem regarding curvature of sections of surfaces in $\mathbf{R}^3$. Being curious, and knowing that ...
11
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1answer
176 views

Why is $e$ the Identity?

Some authors use $e$ to be the identity element of a group instead of $1$. What is the origin of this notation? Was this before or after we used $e$ to represent the base of the natural logarithm? ...
2
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1answer
68 views

Multiply (as a Babylonian): 141 times 17 1/5

How do we multiply 141 times 17 1/5 as a Babylonian? I wasn't sure the space between 17 and 1/5, now I see that 17 1/5 is 17.2 in our notation. Is there a formula that I can solve this? Any hint, ...
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0answers
130 views

Why is the Mazur swindle named so?

Often results or techniques in mathematics are called 'theorems'. Sometimes they are called 'tricks'. In no other context have I seen a result called a 'swindle'. Is there a historical reason for this ...
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3answers
176 views

When and where the concept of valid logic formula was defined?

I was stimulated by a recent question about Gödel Completeness Theorem. All my citations are from Jean van Heijenoort (editor) From Frege to Gödel A Source Book in Mathematical Logic (1967). Gödel's ...
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1answer
112 views

Why can't ✳1.1 be expressed symbollically in Whitehead and Russell's PM?

✳1.1. Anything implied by a true elementary proposition is true. Pp. In the follow passage, it says, "we cannot express the principle symbolically, partly because any symbolism in which p is ...
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0answers
46 views

pierre simon laplace and his knowledge of the (Laplacian) matrices

so as we all know, there is a graph matrix called the Laplacian that is used in some eigenvalue/eigenvector/graph theory/spectral theory problems. i'm wondering if the name of this matrix is ...
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0answers
33 views

Is there a source linking Robinson's work in wing theory with his theory of infinitesimals?

Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book Robinson, A.; Laurmann, J. A. Wing ...
2
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0answers
66 views

The first proof for Poincare lemma in history

How can I get a reference about the first proof of Poincare lemma in history? I already know some methods of proof, but I do want to know the original approach. Thanks for your help!
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3answers
63 views

Should the notion of continuity, usually ascribed to Cauchy, be ascribed to Leibniz?

In his text, Deleuze and the History of Mathematics, Simon Duffy writes: Leibniz also thought the following to be a requirement to continuity: "When the difference between two instances in a ...
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1answer
115 views

History of the method of adding and subtracting the same number

We know that when a number is added and subtracted, the effect is null (same number of course). I want to know the first occurrence of this documented method. Is it from Euclid's elements in Book ...
3
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2answers
55 views

Fundamental roles that astronomy played in the development of mathematics

I'm currently a third year undergrad maths student, and am particularly interested in how astronomy changed maths as we know it today? Are there any particular sources that could be recommended? Or ...
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0answers
17 views

original reference for Gauss' integration formula

The Gauss $n$-point quadrature formula provides an approximation for an integral in terms of a weighted sum of $n$ function values and this approximation is exact for all polynomials of degree at most ...
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2answers
1k views

Are infinitesimals dangerous?

Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions: (1) In what sense are ...
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1answer
124 views

How to get a top-notch Math education (high school level) online?

For the past years, it is becoming more and more accessible to get college level content from many different sources, and, if one is willing can get very far with his math education (not only by ...
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1answer
671 views

Why are there so few Euclidean geometry problems that remain unsolved?

Stillwell mentions in his book Mathematics and its History that: Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers... What is it ...
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4answers
6k views

Why are so many of the oldest unsolved problems in mathematics about number theory?

Stillwell mentions in his book, Mathematics and its History that: Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers... Have ...
3
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1answer
140 views

Recommendations for books that provide a good survey of the history of mathematics, and are meant to be read by mathematicians/mathematics students?

Preferably: The book should not be fixated upon the "standard/popular" accounts of the Greeks (which usually begin with Pythagoras, move on to Euclid and Aristotle, and end with Hypatia). The book ...
3
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1answer
140 views

Important results of calculus before Newton and Leibniz?

We have all come to know that calculus was invented by Newton and Leibniz, right? But many calculus results were already proven by the time. I have read that Fermat already found how to calculate ...
57
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1answer
3k views

Theorem that von Neumann proved in five minutes.

In "How To Solve It", George Pólya writes: "There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it ...
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3answers
340 views

What does “meaning” mean in Whitehead and Russell's PM?

In Principia Mathematica's Introduction, there is a definition for "incomplete" symbol: By an "incomplete" symbol we mean a symbol which is not supposed to have any meaning in isolation, but is ...
3
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1answer
68 views

the difference between the proof in time of Euclid and the proof in nineteenth and twentieth centuries

What is the difference between a proof at the time of Euclid and proof in the nineteenth and twentieth centuries? Thanks for your help
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2answers
149 views

Identity of a Mathematician Mentioned in Euler

I and several others are in the process of translating one of Euler's papers from Latin to English, in particular the one that the Euler Archive lists as E36. In it Euler proves the Chinese Remainder ...
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0answers
32 views

Terminology Regarding Basic Properties of Functions

Is there a cultural difference between saying that a function is 1-to-1 or injective, onto or surjective and a 1-to-1 correspondence or bijective?
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2answers
148 views

What is the necessary condition for the process of “proceeding to the limit” in Whitehead and Russell's PM?

I read this from Introduction of the 1st edition of Principia Mathematica by Whitehead and Russell: Since the orders of functions are only defined step by step, there can be no process of ...
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0answers
51 views

Original proof of the Invariance of Domain Theorem (in English)?

Does anyone know where I can find a translation of the original proof of the Invariance of Domain Theorem in English? Wikipedia cites the original proof to be in: Beweis der Invarianz des ...
3
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0answers
114 views

History behind the choice of letters $h$ and $k$ for the vertex of a parabola?

After failing to find a historical explanation for usage of letters $h$ and $k$ for the vertex of a parabola in most relatively recent textbooks in anglosphere, I turn to math.SE. Is there any ...
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1answer
43 views

Math Mindeset: Historical Learning vs Generality of Concepts

I started math four months ago with modules like measure theory and topology. It was unavoidable to notice how many concepts are more general than what I thought before. For example the ...
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1answer
221 views

Why is it called 'discrete' mathematics?

I understand why you would refer to mathematics which concerns itself with all of the numbers on the number line as 'continuous' but why would you refer to countable or finite mathematics as ...
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4answers
236 views

What is the difference between asserting “$\phi(a)$” and asserting “$\phi(a)$ is true” in Whitehead and Russell's PM?

The first edition of Principia Mathematica clearly distinguishes "Socrates is a man" and "'Socrates is a man' is true." Judging from the context, the distinction is neither a primitive idea nor a ...
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0answers
783 views

Is there any English version of Récoltes et Semailles?

I felt like my question isn't appropriate for MO, so I though maybe I should post it here. I want to read Alexander Grothendieck's "Récoltes et Semailles", but I don't know any French. I can easily ...
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1answer
104 views

How can I find who discovered this integral?

I need to find the first paper/author to document this integral $$\int\log^nx\;\mathrm dx=(-1)^n\;\Gamma(n+1,-\log x)\quad n\in\Bbb N_0$$ To prevent this in the future, is there a service in which I ...
5
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2answers
281 views

Origin of the modern definition of the tensor product

Due to whom is the modern (i.e. via its universal property) definition of the tensor product, and in which article was it communicated?
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1answer
57 views

Historical question about irrationals.

Which beliefs of the Pythagoreans were invalidated by the discovery of irrationals?
3
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1answer
96 views

Is the construction of $\mathbb{R}$ by Cauchy sequences due to Cauchy? For that matter, are Cauchy sequences due to Cauchy?

A little bit of cursory searching around on Wikipedia reveals only that Cauchy sequences are named after Cauchy—but I already knew that.
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0answers
116 views

History of incenter and Euler line

It is easy to see that if a triangle is isosceles, then its incenter lies on its Euler line. Who first proved the converse of this result and what technique was used? (See the post "The incenter and ...
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1answer
188 views

What was the planned topic of Gödel's second paper on incompleteness?

Gödel's incompleteness theorems first appeared together in a paper titled (translated to English) "On formally undecidable propositions of Principia Mathematica and related systems I," with the Roman ...