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Questions tagged [mathematics]

For questions about the quantitative study of topics such as numbers, structure, space, and change, carried out by investigating patterns

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What evidence is there that Fermat had a proof for his Last Theorem?

Aside from the fact that Fermat was a genius, is it probable that he actually did have a proof? Some specifics that I think would point one way or another: Would the mathematics of his day allow him ...
Carlos Bribiescas's user avatar
58 votes
3 answers
14k views

Why are $X$ and $Y$ commonly used as mathematical placeholders?

I realize that $X$ and $Y$ are relatively popular terms when wanting to use a placeholder for an unknown English or math term. What is the origin of this term, and why was it $X$ and $Y$; why not the ...
Sweet_Cherry's user avatar
54 votes
3 answers
6k views

Which came first, the natural logarithm or the base of the natural logarithm?

The natural logarithm function ($\ln x$) and the base of the natural logarithm function ($e$) are both extremely useful. They're also both closely related: $\ln (e^x)=x$, and $e^{\ln x}=x$. But which ...
HDE 226868's user avatar
  • 8,473
49 votes
2 answers
16k views

Did Gauss find the formula for $1+2+3+\ldots+(n-2)+(n-1)+n$ in elementary school?

I heard Gauss's primary school teacher gave some busy-work to his class: to add all the numbers between 1 and 100 up. Gauss immediately wrote 5050. His teacher was shocked, so she told him to add up ...
Geremia's user avatar
  • 5,339
48 votes
3 answers
4k views

What led to the fall of Göttingen?

Göttingen was the place in which many important mathematicians such as Riemann worked. It was also one of the main locations for the development of quantum theory in the twenties (e.g. Heisenberg, ...
tox123's user avatar
  • 1,094
45 votes
1 answer
10k views

Why did algebraic geometry need Alexander Grothendieck?

Grothendieck is arguably the most brilliant mathematician of the 20th century, with his influence felt the most in algebraic geometry, which he transformed. Some time ago the story used to be told was ...
Conifold's user avatar
  • 77k
45 votes
3 answers
5k views

When exactly (and why) did matrices become a part of the undergraduate curriculum?

Let me tell what I know about this. It is well-known that Heisenberg invented matrix multiplication himself, in his great paper that is considered part of the foundation of quantum mechanics. This was ...
Alexandre Eremenko's user avatar
44 votes
5 answers
8k views

Writing Mathematical Symbols in 20th century

As I was reading some papers written by Schrödinger and Heisenberg back in 1920s, I noticed that the symbols they use such as the integral or summation sign or calligraphic letters are as if printed ...
Gonenc's user avatar
  • 785
41 votes
3 answers
3k views

What motivated Cantor to invent set theory?

I can't imagine mathematics without sets, but the question "what was mathematics like before there were sets" is not answerable, I think. Instead, a good answer to the title question should cover a ...
Ben's user avatar
  • 802
41 votes
6 answers
18k views

What is the difference between Calculus of Newton and that of Leibniz?

Are there any differences between the study of Calculus done by Newton as compared to that done by Leibniz? If yes, please mention point by point.
Sameer Shemna's user avatar
41 votes
5 answers
3k views

What new mathematics was inspired by biology and chemistry?

While physics and astronomy sported mathematical models for centuries mathematical chemistry and biology appeared relatively recently. Most of the interaction seems to go one way, established ...
Conifold's user avatar
  • 77k
41 votes
1 answer
4k views

What was Euler's motivation for introducing $i$ for $\sqrt{-1}$?

[Mauro Allegranza has answered the question of who introduced the notation $i$ (Euler, followed later by Gauss), so I have changed the title. I have also edited the question in other ways to make it ...
Michael Weiss's user avatar
40 votes
6 answers
7k views

Whose shoulders did Newton stand on?

In a letter to Robert Hooke in 1676, Newton wrote: If I have seen further it is by standing on the shoulders of giants. Do we know which giants Newton was referring to? And was he referring to a ...
TooTone's user avatar
  • 679
38 votes
9 answers
7k views

Is there any example of a long-standing mathematical conjecture whose resolution did not require advanced knowledge?

Famous conjectures whose solutions took decades or centuries were usually resolved with the help of sophisticated theories and techniques unknown at the time the conjecture was first claimed. Is there ...
Leandro Caniglia's user avatar
38 votes
6 answers
4k views

What is so mysterious about Archimedes' approximation of $\sqrt 3$?

In his famous estimation of $\pi$ by inscribed and circumscribed polygons, Archimedes uses several rational approximations of irrational values; a typical example is that he states, without ...
Mark Dominus's user avatar
36 votes
5 answers
3k views

When did Mathematics stop being one of "the Sciences"?

If you ask a mathematician today, many will tell you that mathematics is not a science. Many physicists, chemists, and scientists in other disciplines would say something similar. Mathematicians will ...
Logan M's user avatar
  • 2,832
35 votes
6 answers
11k views

Who introduced the Principle of Mathematical Induction for the first time?

Can you tell me the name of the mathematician, who introduced the Principle of Mathematical Induction for the first time? (with reliable source). Please don't say De Morgan because I have read the ...
albo's user avatar
  • 965
35 votes
3 answers
1k views

Are there written (19th century) sources expressing the belief that the intermediate value property is equivalent to continuity?

As asked in the title: Are there any written sources (from the 19th century) explicitly stating the belief that any function satisfying the intermediate value property is continuous? (I do not ...
Andrés E. Caicedo's user avatar
33 votes
3 answers
4k views

Why did the ancient Greeks originally become interested in conic sections?

How much is known, or can be conjectured, about why the Greeks originally became interested in the somewhat arbitrary construction of intersecting a plane with a cone? The folklore that I've heard is ...
Jack M's user avatar
  • 3,149
33 votes
1 answer
5k views

Why is American and French notation different for open intervals (x, y) vs. ]x, y[?

The Americans and the French use a different notation for open intervals: The Americans use (x, y) while the French use ]x, y[. How did this notational divergence appear?
Franck Dernoncourt's user avatar
32 votes
11 answers
8k views

Has physics ever given a physical significance to a mathematically abstract idea?

Consider a fundamental concept in maths that was created to 'solve' a problem that simply couldn't be solved by any other approach (or maybe for some other reason). Now let's assume that this concept ...
Adil Mohammed's user avatar
32 votes
1 answer
4k views

Was object oriented programming influenced by the mathematical category theory?

Object oriented programming (OOP) is a programming model where code and data are encapsulated into units called objects that behave semi-autonomously. Interaction between objects is arranged through ...
Conifold's user avatar
  • 77k
31 votes
3 answers
6k views

How did Napier come to invent logarithms?

What was Napier's original logic, leading to his invention of logarithms? In other words, how did Napier, using the mathematics that was available at that time, derive them?
AbdElWadoud's user avatar
31 votes
4 answers
2k views

Current ways of thinking in the History of Mathematics

As a research mathematician, working in number theory, who is interested in the history of his own field, I have done some reading in the History of Mathematics, particularly that of Ancient Greek and ...
R.P.'s user avatar
  • 654
31 votes
1 answer
51k views

Who first defined the "equal-delta" or "delta over equal" ($\triangleq$) symbol?

The symbol $\triangleq$ is sometimes used in mathematics (and physics) for a definition. It is instantiated for instance in the Unicode Character 'DELTA EQUAL TO' (U+225C). The notation $t \triangleq ...
Laurent Duval's user avatar
30 votes
2 answers
2k views

When and how was the geometric understanding of gauge theories developed?

In theoretical physics, the modern perspective on gauge theory is that it is most elegantly described in the 'language' of differential geometry. I am interested in the history behind these ideas. ...
Danu's user avatar
  • 3,862
29 votes
2 answers
1k views

How did Isaac Newton write the integral symbol?

Isaac Newton is known as the discoverer of the FTC (Fundamental Theorem of Calculus), so maybe he wrote the integral symbol and derivative symbol. I know he wrote the derivative symbol as $\dot y$ but ...
MIKANkankitsu's user avatar
29 votes
3 answers
4k views

Who introduced random variables into probability?

I used to think that the answer is Kolmogorov. So the Shafer-Vovk's review of Kolmogorov's famous 1933 axiomatization of probability surprised me a bit:"Today, what Frechet and his contemporaries knew ...
Conifold's user avatar
  • 77k
28 votes
3 answers
2k views

Who attached Buniakovsky's name to the Cauchy-Schwarz inequality?

From time to time one sees insistence that the inequality name "Cauchy-Schwarz" should include Buniakovsky. This is based on a paragraph in a note to the St Petersburg Academy from 1859, where ...
zyx's user avatar
  • 383
28 votes
1 answer
976 views

The history of different constructions of tangent spaces

In Lee's book 'Introduction to Smooth Manifolds', there is an interesting discussion (near the end of chapter three) of several different ways of viewing/constructing the notion of a tangent space to ...
Danu's user avatar
  • 3,862
27 votes
2 answers
1k views

Did Galileo's writings on infinity influence Cantor?

To what extent was Cantor motivated by Galileo's paradox? More generally, to what extent were late 19th century mathematicians motivated by, or even aware of, Galileo's paradox? This is an issue I've ...
Dave L Renfro's user avatar
26 votes
4 answers
5k views

Irrationality of the square root of 2

We know that Pythagoreans in Ancient Greece discovered that the square root of two is an irrational number. Why was that discovery historically significant? What value was that knowledge to the ...
Spectre's user avatar
  • 369
25 votes
2 answers
3k views

Who discovered the covering homomorphism between SU(2) and SO(3)?

Who discovered this? It is quite nontrivial and very important in quantum mechanics.
John's user avatar
  • 909
25 votes
4 answers
18k views

Who invented the integers?

I know that Kronecker claimed it was God's doing, and that even prehistoric humans used some ways of counting. But I am curious where the idea of a sequence of numbers stretching out into infinity ...
Conifold's user avatar
  • 77k
24 votes
22 answers
5k views

Literary works authored by mathematicians

At a first glance, Mathematics and Literature look like two completely unrelated subjects. I wonder whether there are examples of acclaimed mathematicians which wrote novels, poems, or other ...
Ludwig's user avatar
  • 553
24 votes
4 answers
11k views

Ancient Chinese numbering system

It has been said that the invention of zero was a great leap forward, not only in abstract understanding, but in the ability to introduce place value notation and do computations; computing using ...
rogerl's user avatar
  • 673
24 votes
2 answers
3k views

Hilbert's reaction to Gödel's incompleteness theorems

Is it known how Hilbert initially reacted to Gödel's incompleteness theorems upon their announcement at the Königsberg conference in 1930, or their publication in 1931?
Felix's user avatar
  • 1,567
24 votes
2 answers
768 views

In what form does the field of metamathematics exist today?

I was rewriting the Wikipedia article for metamathematics, and it was very difficult to find any references after the 1930s. The most important works seem to have been Gödel's completeness and ...
Brian Rushton's user avatar
24 votes
2 answers
3k views

Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?

The question is in the title, but allow me to provide some background. I’m aware that Leibniz introduced the word “function” into mathematics (around 1673) and that Johann Bernoulli or Euler ...
Michael Bächtold's user avatar
24 votes
1 answer
1k views

Why were geometers dissatisfied with the parallel postulate?

Euclid himself already treats it with gloves, it has an unusually precise formulation, and is not used in the first 28 propositions of the Elements. Why? Did he doubt it? It's not like Euclid was a ...
Conifold's user avatar
  • 77k
23 votes
9 answers
5k views

Are there any mathematical objects that got renamed over time?

I'm wondering if there are any mathematical objects that were given a name when first discovered (and wildly used at their time), but then got renamed to match their characteristics later? Counter ...
neizod's user avatar
  • 341
23 votes
4 answers
2k views

What was the answer to this paradox before Cantor?

I do not remember the name/source of this paradox,but I remember I have discussed this with mathematicians and non mathematicians at least 5 times. It goes like this: "Every point of a line has ...
Konstantinos Gaitanas's user avatar
23 votes
4 answers
4k views

Why are étale morphisms called "étale"?

Alexander Grothendieck developed the theory of "locally trivial coverings spaces for rings/schemes" in SGAI as an analog to the theory of covering spaces in algebraic topology. He called such ...
User0112358's user avatar
23 votes
2 answers
18k views

What is the etymology behind sine, cosine, tangent, etc.?

I heard somewhere that it was actually a mistake in translation. What's the correct story?
user avatar
23 votes
7 answers
11k views

In ancient times, how did people conclude that the shape of Earth is a sphere?

This is more of a philosophical question, but I want a mathematical explanation. During ancient times, it was well accepted that the surface of Earth was spherical. People first observed this when ...
Anubhav Mukherjee's user avatar
23 votes
2 answers
2k views

When did it become understood that irrational numbers have non-repeating decimal representations?

I know that the notion of irrational number (in one form or another) goes back to the Pythagoreans, and therefore far predates the decimal system, and certainly the representation of non-integer ...
mweiss's user avatar
  • 567
22 votes
6 answers
2k views

Has any 'difficult' proof ever been superseded by a 'simple' one?

Let's take an obvious example. I'm sure that amateur mathematicians (and some professionals) will continue to search for Fermat's 'marvellous proof' of his Last Theorem. This is despite the fact that ...
Ima Guest's user avatar
  • 221
22 votes
6 answers
3k views

Why were so many pre-18th century Mathematicians polymaths?

It is well known that famous names such as Gauss, Euler and Newton were polymaths as well as their main fields of study and contributed from optics to ship building. Why was this the case in the past? ...
Ali Caglayan's user avatar
  • 1,586
22 votes
3 answers
5k views

When was zero actually introduced in mathematics?

Children learn counting things, naturally like, 1, 2, 3, ... and so on. Because it seems obvious to them. But, zero is something we need to teach them about. As far as my understanding goes zero was ...
Amit Tyagi's user avatar
  • 1,478
22 votes
2 answers
3k views

Historically, how did people define multiplication for negative numbers?

Which were the first mathematical developments to state that the product of two negative numbers is a positive number, and what was their justification for this choice? I am not interested in a modern ...
Arthur Azevedo De Amorim's user avatar

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