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

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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 ...
1
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3answers
144 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 ...
2
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1answer
104 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
41 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 ...
1
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0answers
31 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
58 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!
0
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3answers
62 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 ...
1
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1answer
100 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 ...
14
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2answers
747 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 ...
0
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1answer
98 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 ...
15
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1answer
626 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 ...
25
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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
111 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
105 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 ...
51
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0answers
2k 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 ...
4
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3answers
314 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
66 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
13
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2answers
142 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 ...
3
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0answers
30 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?
1
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2answers
141 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 ...
4
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0answers
43 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
94 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 ...
0
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1answer
37 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 ...
1
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1answer
120 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 ...
4
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4answers
217 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 ...
2
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0answers
268 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 ...
1
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1answer
98 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
votes
2answers
258 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?
1
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1answer
57 views

Historical question about irrationals.

Which beliefs of the Pythagoreans were invalidated by the discovery of irrationals?
3
votes
1answer
82 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.
3
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0answers
73 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
261 views

Cantor and a paradox of naive set theory [closed]

He was the creator of set theory. Did he recognize a paradox of the naive set theory? In other words, did he recognize that the naive set theory leads to a contradiction?
9
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1answer
181 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 ...
2
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1answer
73 views

Measure and Probability

Can someone tell me that how did the idea to relate measure and probability come?(What's the conceptual history of measure and probability?)
5
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5answers
575 views

Why are second-order 'things' studied so much in mathematics?

In many areas of math, I've been surprised at how much research, past and present, focuses on second order 'things'. Examples: Number theory: quadratic reciprocity, quadratic number fields Analysis: ...
2
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2answers
300 views

Fundamental theorem of linear algebra

When I studied linear algebra we (our books, our professors) used to call Fundamental theorem of linear algebra the theorem that says: Fundamental theorem of linear algebra: A linear ...
11
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4answers
172 views

The origin of the function $f(x)$ notation

What are the historical origins of the $f(x)$ notation used for functions? That is when did people start to use this notation instead of just thinking in terms of two different variables one being ...
6
votes
1answer
174 views

How does Schröder explain the apparent oddity of ❋5.11.12.13.14 in 1st ed of Whitehead and Russell's PM?

The footnote refers to Schröder's work. I'd appreciate if someone can explain Schroder's insights and spare me some hard reading.
10
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1answer
100 views

History of the point at infinity?

I'm curious to learn more about the history of the introduction of the concept of the point at infinity into mathematics. The sum of my knowledge of the historical aspect is from this paragraph (which ...
4
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2answers
195 views

How to prove ❋4.86 in 1st ed of Whitehead and Russell's PM?

This one has a great degree of self-evidence. Paradoxically, I find it difficult to deduce it from primitive propositions. The book only hinted ❋4.21 and ❋4.22.
3
votes
1answer
113 views

unique factorisation fails for cyclotomic integers $p>23$

Background: I have stopped doing algebra a long time ago and I am not that interested in the nitty-gritty details of proofs, but I am interested in maths history. ...
13
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2answers
268 views

Motivation for introducing algebraic topology?

What kind of topological questions does algebraic topology answer where point set topology is not enough? Phrased differently: Where is the line (or maybe intersection) between point set topology ...
9
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1answer
119 views

The average of the roots of a polynomial equals the average of the roots of its derivative

Background: It's straightforward to check that the average (i.e. the mean) of the roots of a nonlinear polynomial equals the average of the roots of its derivative: if $$f(x) = x^n + a_{n-1} x^{n-1} ...
5
votes
3answers
88 views

Swapping Theorems with definitions

My question is motivated from the following passage of Gian-Carlo Rota's Indiscrete Thoughts, 'Suppose you are given two formal presentations of the same mathematical theory. The definitions of the ...
10
votes
5answers
521 views

What are reasons why some symbols in mathematical logic are not standardized?

Why is so hard to find a standardisation regarding symbolism and/or terminology in Mathematical Logic ? We see again and again students asking if e.g. $\rightarrow$ and $\implies$ means the same ...
5
votes
1answer
97 views

the word “derivative”

When did the word "derivative" come into use in calculus, and why? As in Can the word "derive" be used to mean "take the derivative of"? the word "derivative" in normal English ...
3
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0answers
68 views

Is there a link between level of abstraction and use of numbers?

One of my friend who stopped studying maths in high school told me once You study maths, can you help me fill my tax forms? In her mind, advancing in maths studies implied manipulating an ...
5
votes
1answer
110 views

Does Whitehead and Russells' PM distinguish Proof from Demonstration?

I'm currently at Chapter 4, vol. 1 and 1st ed. I have to ask this question because the most important thing about this book is in its minute details. Thanks. Take *3.3 for example. Acording to this ...