# 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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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 $t \triangleq ... • 1,193 30 votes 2 answers 2k views ### When and how was the geometric understanding of gauge theories developed? In theoretical physics, the modern perspective on gauge theory is that it is most elegantly described in the 'language' of differential geometry. I am interested in the history behind these ideas. ... • 3,892 29 votes 2 answers 1k views ### How did Isaac Newton write the integral symbol? Isaac Newton is known as the discoverer of the FTC (Fundamental Theorem of Calculus), so maybe he wrote the integral symbol and derivative symbol. I know he wrote the derivative symbol as$\dot y$but ... 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 ... • 77.6k 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 ... • 383 28 votes 1 answer 989 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 ... • 3,892 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 ... • 3,532 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 ... • 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. • 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 ... • 77.6k 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 ... • 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 ... • 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? • 1,567 24 votes 2 answers 770 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 ... 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...