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

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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 ...
-2
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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
129 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
571 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
279 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
166 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
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1answer
171 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
96 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
186 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
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1answer
110 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
257 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
117 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
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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
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5answers
508 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
96 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
109 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 ...
3
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0answers
123 views

What makes Beal's conjecture “beautiful” enough to make people offer a million dollar prize? [closed]

What makes Beal's conjecture "beautiful" enough to make people offer a million dollar prize? Is it just a challenge or does it have real applications?
7
votes
3answers
566 views

Why are compact sets called “compact” in topology?

Given a topological space $X$ and a subset of it $S$, $S$ is compact iff for every open cover of $S$, there is a finite subcover of $S$. Just curiosity: I've done some search in Internet why compact ...
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0answers
45 views

Mathematical journals (maybe in the past) with regular competitions?

I just finished reading Yōko Ogawa's "The Housekeeper and the Professor". One of the main characters - "the professor" - is a retired mathematician who regularly takes part in contents published in a ...
4
votes
1answer
107 views

What is Euler's proof of his formula ${e^{ix}=\cos(x)+i\sin(x)}$

I've read several proofs of the Euler's formula $$e^{ix}=\cos(x)+i\sin(x)$$ but I want to know how Euler's himself prove it at the first time, how did he think about it?
2
votes
1answer
99 views

Who first proved that the second-order theory of real numbers is categorical?

The second-order theory of real numbers is obtained by taking the axioms of ordered fields and adding a (Dedekind) completeness axiom, which states that every set which has an upper bound has a least ...
3
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2answers
219 views

History of Calculus

Newton/ Leibniz invented calculus on approximately 1680's. Cauchy/Weierstrauss defined the $\epsilon - \delta$ definition of a limit in approximately 1820's. So how did they define derivatives? I ...
5
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1answer
77 views

What is $M_x$ in Frege's Basic Law IIb?

Gottlob Frege's magnum opus, "The Basic Laws of Arithmetic" (Die Grundgesetze der Arithmetic in German) constitutes one of most impressive and meticulous attempts at developing a rigorous foundation ...
13
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2answers
568 views

A problem V.I. Arnold solved as a primary school student

According to a 1995 interview that Vladimir I. Arnold gave to the Notices of the AMS, his primary school teacher I.V. Morozkin gave in 1949 (when Vladimir I. Arnold was 12 years old) to a Soviet ...
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1answer
491 views

The most active fields of mathematics?

Which fields of mathematics are the most active at this time -- by number of papers published, grant money, people working in them or by any other measure? Any trends in this regard?
5
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1answer
162 views

Could Euclid have bisected a line segment without his method of superposition?

In Book I Proposition 10 of the Elements, Euclid performs the bisection (i.e. finding a midpoint) of a line segment. In the course of doing so, he uses Book I Proposition 4, the Side-Angle-Side ...
3
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2answers
99 views

Notation and the name choice for meet and join (in order theory)

I have two simple questions: From where do the names meet and join come from? I don't see any intuition between those names in context of order theory. From where does the notation come? I have to ...
0
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2answers
456 views

Mathematics and slavery [closed]

I think that ancient Greek mathematics is a miracle. Think about Euclid. Developing mathematical arguments from a small set of axioms is incredibly modern. And their influence on modern mathematics is ...
3
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1answer
74 views

Cauchy's theorem or Frobenius' lemma

A textbook exercises asks: The goal of this exercise is to prove Frobenius's lemma, which asserts that if the order of the group $G$ is divisible by the prime $p$, then $G$ contains an element of ...
20
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3answers
750 views

Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis?

I've seen a lot of definitions of things like boundary points, accumulation points, continuity, etc, and axioms of the real numbers. But I have a hard time accepting these as "true" definitions or ...
0
votes
1answer
49 views

Who first described commutative algebraic theories explicitly?

Lately, I've been thinking that the concept of a commutative algebraic theory is really, really important. So I'm curious; who had the honor of first discovering this concept? In particular, I'd like ...
2
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0answers
86 views

What were some Mexican contributions to high school level algebra and statistics?

I want to do a presentation on Mexican mathematicians' contribution to either high school level algebra or statistics. What kind of resources are out there? Does anyone know of a contribution that I ...
2
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1answer
25 views

What is a cofactor in this case?

Im looking at a homework problem I have and I am a bit confused. The first part of the question is to show that 8128 is a perfect number. This is simple enough: $1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + ...
3
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1answer
123 views

Why $\operatorname e$ was named e ? What is the history and reason behind it?

Why the constant that Euler discovered has symbol $\operatorname e$ and not other symbols? What is the history and reason behind it? Thanks.
2
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0answers
43 views

Order of operations - is there any standard?

Is there any standard regarding order of mathematical operations? I only found ISO 31-11 that deals with mathematical symbols, but not with the order of operations. If there's not, I'd be interested ...
2
votes
2answers
90 views

why function argument is on right side $f(x)$ rather than on left side as $xf$

Is there an advantage for writing function arguments on the right side as $f(x)$ rather than on the left side as $xf$? The latter looks more natural if we think about it in diagram as $domain ...
1
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1answer
77 views

From $\mathsf{O}$ to $\mathsf{I}$ via $\infty$

The following is not true mathematics, but a little imaginary story about mathematical symbols. I wonder if there is - in parts - a true (etymological) story behind it. Once there was a symbol ...
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2answers
148 views

Is there any surprising elementary probability problem that result in surprising solution like the Monty Hall problem?

For recreational purpose, i haven't seen a interesting elemetary probability question quite a while. Is there any surprising elementary probability problem that result in surprising solution like the ...
1
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1answer
73 views

Russell's definition of finite cardinals

whether the thought had been previously adumbrated, perhaps confusedly, i know not, but the name of Bertrand Russell has become associated with the assertion that: the number $2$ is the set of all ...
1
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1answer
112 views

Reform of math symbols for high school texts

I am looking for references to papers and resources related to reforming math symbols for introductory courses at middle or high school level. Pointers to other forums also welcome. Eidt: For ...
4
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0answers
69 views

The Leibniz rule in Euler's works

Does anyone know if the Leibniz rule (the method of differentiation under the integral sign), or a variation thereof, has ever appeared in any of Euler's papers? Any references would be appreciated. ...
0
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2answers
109 views

General questions about theorems and laws

I have doubts about the construction of mathematical elements. There are proofs, that are proven using other theorems (corollaries) and/or axioms or definitions, such as Fermat's Last Theorem, the ...
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1answer
63 views

Probability of World Series - Using Pascal and Fermat “Problem of Points”

This is a question I have for a history of math class, but I can't figure it out. I need to use the three method that Pascal and Fermat used on their problem of points, and it doesn't seem to work ...
13
votes
1answer
186 views

Why are proofs written in first person plural? Were they ever written differently?

It's probably a silly question but it interests me when was the convention of writing proofs in first person plural introduced? Is there any historical examples of a different POV for proof writing?
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0answers
84 views

Ludwig Sylow 12 december 1832 [closed]

Not a question, but just to commemorate that Ludwig Sylow was born today exactly 181 years ago, on 12-12-1832. Note that $1832=2^3.229$, and hence by Sylow's theorems there is no simple group of order ...
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1answer
28 views

Pascals first method

So pascals first method was to first solve a simple problem,this was before the pascal triangle. This is in relation to De Meres problem: Each player stakes $32$ pistoles. One player has 1 round ...
2
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1answer
100 views

Mathematician as a title [closed]

After some googling around, I can't seem to find a definite answer to this question. When can someone call themselves a mathematician? Is it after a b.sc. in mathematics? After graduate school? Or ...
3
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0answers
51 views

Question about the first step in Mann's original proof of the Schnirelmann-Landau Conjecture

I was reading Henry Mann's proof for the Schnirelmann-Landau Conjecture from 1942 which can be found in JSTOR here Today, the Schnirelmann-Landau Conjecture is known as Mann's Theorem: $$d(C) \ge ...