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7
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2answers
236 views

Who conjectured that there are only finitely many biplanes, and why?

This question on MathOverflow motivates me to ask what the reasoning is behind the conjecture that there are only finitely many biplanes. More generally, it has been conjectured that for fixed $\...
35
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2answers
603 views

Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider $$G=\operatorname{Gal}(\mathbb{Q}(r_1,\...
38
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3answers
5k views

Why, historically, do we multiply matrices as we do?

Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very ...
6
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2answers
3k views

Which notable mathematicans have tried solving the Riemann hypothesis?

I have read that the Riemann hypothesis is the most important open question in mathematics and has been open since 1859. I am wondering which famous mathematicians have actually tried to solve it and ...
22
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2answers
2k views

Does there exist a copy of Euclid's Elements with modern notation and no figures?

I am working through Euclid's Elements for fun, but I find the propositions difficult to understand without referencing the provided figures. Unfortunately, the figures usually give away the proofs, ...
7
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1answer
1k views

Liouville's proof of the existence of transcendental numbers

The existence of transcendental numbers can be shown easily by considering the cardinality of the set of solutions to polynomials with integer cofficents and the cardinality of the real numbers. It ...
3
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0answers
170 views

Is there a way to determine how many solution does “ The hundred Fowls problems” have looking at the coefficients?

I was looking at the different versions of the hundred fowls problems. I came across the problem posed by Chinese mathematicians posed in 5th century,and then Indians in 9th, 10th centuries. Alcuin of ...
3
votes
4answers
446 views

Why $\sqrt{\frac {\sum(x-\mu)^2} {N}}$ instead of $\frac {\sum{\Bigl|x-\mu\Bigr|}} {N}$? [duplicate]

Possible Duplicate: Motivation behind standard deviation? In statistics very often you see something of the sort: $$ \textrm{quantity}=\sqrt{\frac {\sum(x-\mu)^2} {N}} $$ to measure things ...
4
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2answers
168 views

Plato's Disc of Gold

In the book Mathematical Cranks, Underwood Dudley describes the following problem on page 36: Dear Archimedes, Your problem is solved but:-- About twenty years ago he lived on Crete and was ...
0
votes
1answer
325 views

Square root principle value convention

Why is the principal square root of a complex number defined as $\sqrt z = \sqrt r e^{-i \varphi / 2}$ for $\varphi \in (-\pi, \pi]$ ? Wouldn't it be more natural to let $\varphi \in [0, 2\pi)$ as it ...
12
votes
1answer
199 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 ...
8
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2answers
299 views

What was Klein working on when he “replaces his Riemann surface by a metallic surface”?

I am reading The Value of Science by Poincare, and the following paragraph from Chapter I seems rather interesting: Look at Professor Klein: he is studying one of the most abstract questions of ...
63
votes
3answers
2k views

Paul Erdos's Two-Line Functional Analysis Proof

Legends hold that once upon a time, some mathematicians were rather pleased about a 30-ish page result in functional analysis. Paul Erdos, upon learning of the problem, spent ten or so minutes ...
5
votes
2answers
671 views

Who proved the Master Theorem?

In all of the classes I've had on algorithms, and the books I've seen that talk about the master theorem, none of them mention where it came from, which is pretty odd. Certainly, it didn't just kind ...
2
votes
0answers
91 views

History of imperative in hypotheses

In mathematical hypotheses it is traditional to use the imperative instead of a declarative sentence. What is the origin of this tradition? Does it go back to ancient Greek mathematics? Or maybe ...
6
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2answers
1k views

How many classification of mathematical topics exists?

I found only one Mathematics Subject Classification, are there more?
13
votes
4answers
878 views

How do mathematicians think about the existence of numbers?

Question: How do mathematicians think about the existence of numbers? And how did Newton, Euler, and other famous mathematicians thought about this concept? I know that existence of numbers is a big ...
13
votes
2answers
413 views

A quote from Arnold

Arnold said the following in a talk on teaching: Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as ...
3
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2answers
2k views

How did people calculate numerical values of transcendental and trigonometric functions?

I know that back in the Stone Age, people used tables on this thing called paper to look up values for functions like $\sin$ and $\ln$. But how did the guys who wrote the tables calculate those values?...
4
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3answers
1k views

Where did these symbols come from?

Where did these symbols come from? Like Pi, Fee and this weird E/sideways M and the triangle.
7
votes
1answer
4k views

Why are vector spaces sometimes called linear spaces?

I have never come across the term 'linear space' as a synonym for 'vector space' and it seems from the book I am using (Linear Algebra by Kostrikin and Manin) that the term linear space is more ...
19
votes
2answers
886 views

Who first called the Grothendieck's schéma scheme?

Grothendieck called "schemes" schémas in French. I find it strange we call them schemes. In fact, Grothendieck called them (pre-) schemas(this is an English word) in his talk(in English) at Proceeding ...
4
votes
1answer
98 views

Examples of concepts, definitions or areas of study that were later abandoned

I've been recently thinking about what I've learned in mathematics, and I realised that in contrast to physics (or the other sciences), I tend to take the concepts and definitions for granted in that ...
1
vote
2answers
188 views

Semantic parsing of a sentence from “The mathematical analysis of logic” By Goerge Boole, 1847

Having the pleasure of reading some original text, I was wondering if someone can translate two small statements on the second half of page 11 from http://archive.org/stream/mathematicalanal00booluoft#...
10
votes
1answer
186 views

Who first explicitly noted that second-order logic is unaxiomatizable?

As every student now knows, second-order logical consequence is unaxiomatizable. (At least when we read the second-order quantifiers in the natural way, as running over all possible properties on the ...
16
votes
7answers
2k views

Films about math: a question about math education and motivation for learning math [closed]

I'm interested in movies about or related with mathematics or physics, I mean not documentaries which I also consider movies, but artistic or mainstream films about math. Now I have the following in ...
9
votes
1answer
593 views

When was the term “mathematics” first used?

By the second century, in the Almagest, Ptolemy provides a modern conception of "mathematics" as a "science": 'Mathematics' ... is an attribute of all existing things, without exception, both ...
2
votes
1answer
214 views

Earliest proof of completeness for axiomatization of Boolean Algebra

Suppose we define Boolean algebra as the system of algebraic rules (logical equivalences) obeyed by AND, OR, NOT with AND, OR, NOT defined by the usual truth tables. We also have variables, which can ...
4
votes
1answer
21k views

Why do some people state that 'Zero is not a number'?

Every now and then I read about people who wonder whether zero is a number. It never occurred to me to question this, so I checked the Wikipedia page which, when talking about the Rules of Brahmagupta ...
0
votes
1answer
482 views

Early proofs of Leibniz's formula

Wikipedia attributes Leibniz's formula to Madhava of Sangamagrama, James Gregory and Gottfried Leibniz. But what were their proofs?
12
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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 ...
4
votes
1answer
346 views

Early history of lower bounds on the prime counting function

Let $\pi (x)$ be the number of prime numbers less than or equal to $x$. Euclid's proof of the infinitude of primes gives a horrible lower bound of the type $ \pi (x)>> \sqrt{\log{x}} $. ...
8
votes
1answer
275 views

Who first discovered that the torus supports a flat structure?

Who first recognized that there exists a homogenous metric on the closed genus 1 orientable surface?
2
votes
1answer
171 views

Original Proof of Riesz-Thorin

Wikipedia says that Riesz proved the Riesz-Thorin theorem in 1926 without using any complex methods. Does anyone know where the original proof can be found? http://en.wikipedia.org/wiki/Riesz%E2%80%...
6
votes
1answer
314 views

Did Euler have an alpha function

I've heard of Euler Gamma function: $\Gamma(x)$, and Euler's beta function: $\text{B}(x,y)$. Did Euler have an alpha function?
6
votes
1answer
666 views

An updated alternative to “A Panorama of Pure Mathematics”

Dieudonne's A Panorama of Pure Mathematics serves as a nice, brisk overview of the state of pure mathematics at its time, but it would be nice if there were an updated version of this book. Is there ...
41
votes
8answers
2k views

Original works of great mathematicians

In almost every mathematical text there is a line as This was first proved by Gauss or This formula first appeared in a work of Riemann, but for me it's more like My friend told me once that... For ...
4
votes
1answer
237 views

Who is “Euclide Paracelso Bombasto Umbugio”?

I just browsed through the book Foundations of Algebra and Analysis by C. Dodge, which contains a very short biography of a very famous mathematician at the beginning of each chapter, together with ...
20
votes
1answer
510 views

Who was Hermann Künneth?

Question as in the title: Who was Hermann Künneth? Where can I find some biographical information beyond what is available on Wikipedia? The well-known Künneth formula, for example in the form of ...
1
vote
0answers
135 views

Etymology of algebra (as k-algebra)?

Why algebra (over a field) is called "algebra?" (My random guess is that it's a back-formation of some algebras, chopping adjectives from say Lie algebra or Clifford algebra, etc.) And when was that ...
3
votes
4answers
469 views

Real World Usage Examples and Historical Origin of Beginning Algebra (HS Algebra I and II)

I have a high schooler who I need to get energized about math. She excels in other sciences, but does not in math. The issue, I learned after some discussion, is that she doesn't find math interesting ...
2
votes
0answers
435 views

On the geometric arguments used in Newton's *Principia Mathematica Naturalis Philosophae*

When one reads Newton's Principia Mathematica, one is immediately aware of the complexity of the synthetic geometry that he uses to prove his propositions. This I understand because all of the ...
4
votes
2answers
576 views

Was there a culture/number system with negative numbers but without zero?

In the history of numbers, negative numbers as well as zero appear relatively late, possibly because the concepts represented are not really 'quantities' in a straightforward sense. However, even ...
10
votes
1answer
461 views

Old versus New enunciation of Taylor's Theorem.

I am studying from Spivak' Calculus, and he states Taylor's Theorem as follows: THEOREM Let $f',\cdots,f^{(n+1)}$ be defined on $[a,x]$ and let $R_{n,a}(x)$ be defined by $$R_{n,a}(x)=f(...
12
votes
2answers
922 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 ...
5
votes
4answers
704 views

Euler and infinity

What do people mean when they say that Euler treated infinity differently? I read in various books that, today, mathematicians would not approve of Euler's methods and his proofs lacked rigor. Can ...
0
votes
3answers
249 views

Resource request: history of and interconnections between math and physics

Reading this article I became curious to learn more of (- study more thoroughly and *seriously*$^{\star}$-) the topic. Is / are there some good references - either papers, books and/or other ...
3
votes
0answers
73 views

analogy between etale sites and Riemann surfaces

I recently read that Grothendieck originally introduced the etale site of a scheme as an analog of the formation of Riemann surfaces over the complex numbers (the salient point being that the latter ...
1
vote
5answers
131 views

integer constants.

Are there some examples of mathematocal constants which are integer numbers. I know of one that is called Kaprekars constant but thats just a base 10 curiosity. Aret there some more important examples?...
3
votes
1answer
412 views

Why were Lie algebras called infinitesimal groups?

Why were Lie algebras called infinitesimal groups in the past? And why did mathematicians begin to avoid calling them infinitesimal groups and switch to calling them Lie algebras?