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

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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 ...
13
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4answers
624 views

Why the terminology “monoid”?

As I am not a native English speaker, I sometimes am bothered a little with the word "monoid", which is by definition a semigroup with identity. But why this terminology? I searched some ...
13
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2answers
1k views

Etymology of $\arccos$, $\arcsin$ & $\arctan$?

Does anyone know the origin of the words $\arccos$, $\arcsin$ & $\arctan$? That is to say, why are they named like this? What connects "arc" with inverse? Can't seem to find out via Google. ...
13
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2answers
834 views

Why is analysis called “analysis”?

Just as the topic says, how did the name "analysis" come to denote the specific mathematical branch dealing with limits and stuff? The term "analysis" seems very generic compared to the words for the ...
13
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2answers
1k views

Who is a Math Historian?

In the context of classes, it is very often that discussion on the history of mathematics arises, whether it'd be on who should a lemma be attributed to or a certain event that occurred during the ...
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2answers
1k views

Why are Darboux integrals called Riemann integrals?

As far as I have seen, the majority of modern introductory real analysis texts introduce Darboux integrals, not Riemann integrals. Indeed, many do not even mention Riemann integrals as they are ...
13
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3answers
773 views

Journals of math history?

In a related question to this one, in what journals do math historians publish their article in? Brian M. Scott provided a link to Judy Grabiner's, who is a math historian, home page and it seems that ...
13
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4answers
803 views

Documentary of mathematics. [duplicate]

Possible Duplicate: List of Interesting Math Videos/ Documentaries I just watched a documentary of Fermat's last theorem. It is so good. I can feel how mathematician think and get excited. ...
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1answer
1k views

Why is logistic equation called “logistic”?

The logistic function solves the logistic ODE which is the continuous version of the logistic map. However, I was not able to find why any of these things are called "logistic".
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1answer
320 views

l'Hopital's questionable premise?

Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ ...
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8answers
7k views

Why are derivatives specified as d/dx?

Is the purpose of the derivative notation d/dx strictly for symbolic manipulation purposes? I remember being confused when I first saw the notation for derivatives - it looks vaguely like there's ...
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11answers
3k views

Good examples for mathemathical problems/statements that are easely solvable/provable in one theory and hard to solve/prove in another

Let $P$ be a mathematical statement or a mathematical problem. I am looking for a couple of nice examples for $P$ that satisfy the following criteria: Given two (or more) mathematical points of view ...
12
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5answers
516 views

How did Euler realize $x^4-4x^3+2x^2+4x+4=(x^2-(2+\alpha)x+1+\sqrt{7}+\alpha)(x^2-(2-\alpha)x+1+\sqrt{7}-\alpha)$?

How did Euler find this factorization? $$\small x^4 − 4x^3 + 2x^2 + 4x + 4=(x^2-(2+\alpha)x+1+\sqrt{7}+\alpha)(x^2-(2-\alpha)x+1+\sqrt{7}-\alpha)$$ where $\alpha = \sqrt{4+2\sqrt{7}}$ I know that ...
12
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7answers
964 views

What's the hard part of zero?

A lot of textbooks said it was hard for human to accept zero when it was first introduced. How could it be? It seems to me as natural as positive integer which represent there is no elements at all.
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2answers
905 views

Who decides after whom a theorem or conjecture is named?

Who decides after whom a theorem is named? When someone discovers and proves a theorem, it is almost always named after that person. But how about when person A conjectures a theorem, and B proves ...
12
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2answers
722 views

What is the mathematical intuition behind àl-jàbrà?

The term algebra comes from the arabic term àl-jàbrà that means "to force", "to restore". Over centuries mathematicians, in east and west, celebrate by this term mathematical disciplines. What is ...
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2answers
2k views

Motivation for Napier's Logarithms

In the wikipedia article on logarithms, I am clueless about the approach and motivation for the following computations done by Napier (and the mysterious appearance of Euler's number) in this section. ...
12
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4answers
461 views

Elementary problems that would've been hard for past mathematicians, but are easy to solve today? [closed]

I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous ...
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3answers
3k views

Motivation for Tom Lehrer's song “Lobachevsky”?

I am trying to understand the motivation for the jingle about plagiarism written by Tom Lehrer. A YouTube version can be found here http://www.youtube.com/watch?v=IL4vWJbwmqM . Where does history ...
12
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9answers
1k 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 ...
12
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2answers
977 views

Fermat's Last Theorem: implications (there is no new proof)

I am not experienced in Number Theory but what I know is that some results of this filed are applicable in other areas, e.g. algebra. For sure FLT made (and makes) people be interested in Number ...
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2answers
724 views

Where, specifically, did Principia Mathematica fail?

I'm very fascinated by the book Principia Mathematica. From what I've learned so far, Principia Mathematica set out to be, essentially, the bible of mathematics and logic, from which all mathematical ...
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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 ...
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3answers
404 views

What have been some of the most revolutionary philosophical shifts in perspective in mathematics?

Often times, great revolutions in mathematics come from shifts in philosophical perspective. The shift from extrinsic to intrinsic geometry yields manifolds (and much else). The shift in focus from ...
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2answers
326 views

Before Abel's proof, what did they used for trying to find the general solution for quintics?

Whenever I read about the history of algebra, I end up with the same conclusion: They solved the general cubic, then the general quartic and then spent lots of years trying to solve the general ...
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2answers
490 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 ...
12
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4answers
740 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 ...
12
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1answer
898 views

What is the name of the $\in$ symbol and where does it come from?

It looks like a lower-case epsilon, but the Wikipedia page on epsilon states that they are not the same. Does this symbol have a typographic identification outside of mathematics? Where did the ...
12
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1answer
2k views

Proof of Euler's Theorem without abstract algebra?

Every proof I've seen of Euler's Theorem (that $\gcd(a,m) = 1 \implies a^{\phi(m)} \equiv 1 \pmod m$) involves the fact that the units of $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. ...
12
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1answer
218 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? ...
12
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2answers
399 views

What's the history of the result that $p_{n+1} < p_n^2$, and how difficult is the proof?

In Edsger Dijkstra's monograph "Notes on Structured Programming", he describes a simple imperative program for generating an array of the first $n$ primes. For each prime $p_n$, it finds the next ...
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3answers
448 views

Is there any difference between a math invention and a math discovery? [closed]

From wikipekia: The calculus controversy was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates – ...
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2answers
813 views

What do Greek Mathematicians use when they use our equivalent Greek letters in formulas and equations?

Like for example, it's common to use the Greek letter $\theta$ to represent an angle right? So what would a Greek person doing math use to represent an angle? Would they also use $\theta$? Or is there ...
12
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1answer
597 views

What is “inner” about the inner product?

The inner product I am asking about is the one that generalizes the dot product for an arbitrary inner product space. Why is it called an "inner" product? Is there an outer product? Who named it ...
12
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3answers
310 views

Why “integralis” over “summatorius”?

It is written that Johann Bernoulli suggested to Leibniz that he (Leibniz) change the name of his calculus from "calculus summatorius" to "calculus integralis", but I cannot find their correspondence ...
12
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1answer
516 views

Sperner's theorem on antichains - where does it come from?

Sperner proved in 1927 (the paper was published in 1928) his theorem stating that the maximal size of an antichain of subsets of $[n]$ is $\binom{n}{n/2}$. In the introduction to his paper, he ...
12
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1answer
488 views

Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets): a) compositions with parts from the set {1,2} (e.g., 2+2 = ...
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2answers
340 views

What meaning did Riemann assign to $dx$?

Detlef Laugwitz wrote a monumental biography of Riemann. The book was translated into English by Shenitzer. Laugwitz discusses Riemann's fundamental essay Uber die Hypothesen, welche der Geometrie ...
12
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1answer
554 views

history of the contraction mapping technique

If $|f(x)-f(y)| \leq k|x-y|$ for all $x,y$ then $f$ is Lipschitz with constant $k$, if $k<1$ then $f$ is called a contraction mapping. The beautiful result that a fixed point is associated to a ...
12
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1answer
167 views

Original Formulation of Hilbert's 14th Problem

I have a problem seeing how the original formulation of Hilbert's 14th Problem is "the same" as the one found on wikipedia. Hopefully someone in here can help me with that. Let me quote Hilbert first: ...
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2answers
367 views

Hao Wang's $\mathfrak S$ system/$\Sigma$ system: a “transfinite type” theory that avoids the Goedel's theorems.

Long ago, while I was reading a book ($*$) about the various way to build set theories (Zermelo-Freankel, Von Neumann–Bernays–Gödel, and type theories), I read about a variant of type theory with ...
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1answer
190 views

Who is responsible for the analytical/topological proof of FTA?

The fundamental theorem of algebra asserts: Theorem Let $P$ be a polynomial of degree $\geq 1$ in $\Bbb C$. Then there exists a $z_1\in\Bbb C$ such that $P(z_1)=0$. The proof sketch goes as ...
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2answers
241 views

Lie and Weierstrass' visualization of complex functions

I am reading Whittaker and Watson's A Course of Modern Analysis. In the third chapter where they discuss different ways to visualize functions that map the complex plane to the complex plane, they ...
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0answers
513 views

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor ...
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2answers
3k views

Which symbol should be used for an empty set?

Currently, a discussion started on the German Wikipedia article for Empty Set (the German discussion), whether $\emptyset$ or $\varnothing$ should be used or is more common as a symbol for an empty ...
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3answers
2k views

Provenance of Hilbert quote on table, chair, beer mug

All over the web one can find statements to the effect that: "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs" There are many ...
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5answers
1k views

What is the meaning of set-theoretic notation {}=0 and {{}}=1?

I'm told by very intelligent set-theorists that 0={} and 1={{}}. First and foremost I'm not saying that this is false, I'm just a pretty dumb and stupid fellow who can't handle this concept in his ...
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2answers
2k views

Why the name 'FACTORIAL'?

Factorial is defined as $n! = n(n-1)(n-2)\cdots 1$ But why mathematicians named this thing as FACTORIAL? Has it got something to do with factors?
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3answers
893 views

History of notation: “!”

Does anyone know where the factorial "!" symbol came from? I can't decide if it is my favorite or least favorite notation in mathematics...
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2answers
2k 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 ...