Questions tagged [mathematics]
For questions about the quantitative study of topics such as numbers, structure, space, and change, carried out by investigating patterns.
176
questions
19
votes
2answers
1k views
Pythagoras vs. the idea of Pythagoras
Maybe we need some replies on current scholarly thinking.
(Judging from some replies here, many of us are still using the myths current 100 years ago.)
Is it true (as I have heard) that most, if not ...
8
votes
1answer
969 views
Is there a 'lost calculus'?
Are there any 'lost' theorems of calculus that could be used to 'simplify' it? For example, are there ways to calculate derivatives without using limits, maybe by some forgotten methods in calculus?
12
votes
3answers
2k views
Why is “Cardano's Formula” (wrongly) attributed to him?
Apparently, Cardano had learned a formula for solving cubic equations from Tartaglia, who had sworn him to secrecy, and in any event, not to publish it without giving Tartaglia due credit.
Cardano ...
26
votes
2answers
1k 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.
...
21
votes
2answers
2k 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 ...
16
votes
2answers
918 views
Was 18th century algebra more symbolic/formal than the modern conception?
I've found Lagrange's Sur la résolution des équations algébriques to be a very confusing and difficult read, and I think I'm starting to see why: it seems that Lagrange thinks of algebra in a much ...
19
votes
3answers
1k views
What was the motivation for the development of modern, intrinsic, differential geometry?
I know that tensor calculus was developed around the same time as general relativity. Tensor calculus was the prime way to deal with geometric objects, based on expliciting all coordinates and doing ...
4
votes
4answers
1k views
How were irrational numbers accepted by mathematicians?
What was behind accepting the existence of irrational numbers historically? Especially numbers that are not constructible on the real number line, say for example $\sqrt[3]{2}$. Was it a (somewhat) ...
7
votes
1answer
258 views
Viète's Relevance and his Connection to Euler
Viète's equations are used in some proofs of the Basel problem, which was allegedly solved by Euler.
Viète's equations include the following: given a polynomial,
$$a_0 + a_1x+a_2x^2 + ... + a_nx^n$$ ...
33
votes
4answers
4k 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 ...
10
votes
2answers
1k views
Was Euler's theorem in differential geometry motivated by matrices and eigenvalues?
I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature ...
7
votes
1answer
375 views
When do we see for the first time the use of the Cartesian coordinates?
I want to see an exact image of the first use of the Cartesian plane. I guess it came the first time with Descartes.
11
votes
1answer
2k views
The origin of quadratic equation in actual practice
I read that in ancient times the quadratic equation of this kind $$x^2+10x=39$$ had been solved long ago. I read that this kind of equation originated in the geometric question of "Given an area of 39,...
17
votes
1answer
5k views
What was the historical context of the development of Taylor series?
I knew about linear approximations, quadratic approximations and the use of Taylor polynomials to approximate a function. Furthermore, I was aware of other applications of Taylor polynomials and the ...
31
votes
4answers
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 ...
47
votes
3answers
3k 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 ...
25
votes
2answers
726 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 ...
22
votes
2answers
898 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 ...
22
votes
3answers
4k 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 ...
17
votes
2answers
1k 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 ...
3
votes
1answer
798 views
How was geometry historically used to solve polynomial equations?
I'm researching historical use of geometry to find solutions to polynomial equations. I'd like to ask for those familiar with this topic, could you describe the use of geometry by early mathematicians ...
8
votes
5answers
1k views
Has a digit ever been used to represent the number “10”?
Ten is special to humans, as there are 10 fingers on two hands, and fingers are still the basic counting medium for people.
So, was there any digit representing the number "10" in a positional system ...
16
votes
2answers
5k 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?
26
votes
3answers
4k 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?
30
votes
3answers
3k 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 ...
22
votes
1answer
774 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 ...
11
votes
5answers
2k views
Who gets credit for the real numbers?
If Simon Stevin already pioneered the unending decimal representation for every number (rational, surd, etc.) at the end of the 16th century, why do Cantor and Dedekind (who certainly gave a more ...
16
votes
2answers
9k views
Origin of 360 degrees?
This is by far one of the most challenging and popular HSM questions on the Net. Proofs are, countless discussions about it in math forums. The answers only led to two theories, which Wikipedia does a ...
12
votes
1answer
11k views
Who invented short and long division?
I am curious who came up with algorithms that we use today to manually solve mathematical division problems, such as short or long division; how were they established or standardized that way and why?...
10
votes
1answer
3k views
Who discovered the power rule for derivatives?
Who discovered the general rule for differentiating polynomials, in particular that the derivative of $x^n$ is $n x^{n-1}$, and when? I appreciate the answer may not be a clear-cut individual and year,...
23
votes
4answers
3k 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 ...
12
votes
1answer
1k views
How did mathematicians notate the empty set before $\varnothing$?
I recently learned that $\emptyset$ or $\varnothing$ is a relatively new notation for the empty set created in 1939. I know $\{\}$ is also used along with $\{\cdot\}$ to denote empty sets. What was ...
9
votes
2answers
742 views
When did people start viewing a matrix as a linear transformation between two vector spaces?
The notion of a matrix appeared far ahead of that of a vector space. So when did people start considering a matrix as a linear transformation between two vector spaces?
4
votes
2answers
493 views
Cartesian coordinate system in Newton's work
In the english translation of Newton's work "Enumeratio linearum tertii ordinis" by C.R.M. Talbot, we can see in a figure the depiction of a Cartesian coordinate system pretty much as we know it today:...
7
votes
1answer
817 views
Was there a very early culture that's number system was 12-based, like ours is 10-based?
There are several uses of 12 in some old systems of measurement. Some of them make sense given current context (There are 12 lunar cycles per year), however some of them seem to be arbitrarily chosen. ...
5
votes
3answers
385 views
When were the concepts of pure and applied Mathematics introduced?
I know that there are no standard definitions for pure and applied mathematics however I would like to know who first considered them as two separate entities, I have seen people mention it was around ...
36
votes
5answers
10k 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.
40
votes
3answers
2k 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, ...
30
votes
5answers
7k 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 ...
25
votes
2answers
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.
19
votes
5answers
723 views
Who invented the Leibnitz notation $\frac{d^2y}{dx^2}$ for the *second* derivative?
This MSE question made me wonder where the Leibnitz notation $\frac{d^2y}{dx^2}$ for the second derivative comes from. It does not arise immediately as the obvious generalization of $\frac{dy}{dx}$. ...
17
votes
3answers
895 views
Why did mathematicians not see that $f_n(x)=x^n$ is a counterexample to Cauchy's “theorem” about limits of continuous functions?
In 1821 Cauchy claimed that the limit of a sequence of continuous functions is continuous. In 1826 Abel gave a complicated trigonometric counterexample. When we teach analysis courses, we usually give ...
5
votes
4answers
2k views
Electromagnetics and vector calculus
A friend of mine claims that vector calculus was invented to do electrodynamics. I'm dubious. I know that Maxwell first wrote down the so-called Maxwell's equations in scalar form and only later ...
9
votes
1answer
119 views
When was the earliest use of log-log plots to demonstrate power-law behavior?
After reading this answer and writing this comment, I decided to ask this question: When and where was the earliest known use of a log-log plot to demonstrate power-law behavior?
23
votes
4answers
8k 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 ...
14
votes
5answers
481 views
Which school of philosophy motivated thinking about spaces of higher dimension?
I'm trying to make a link between important mathematical breakthroughs in history and the important philosophical schools at the time. I realize that this topic is awfully broad and could be the ...
11
votes
1answer
331 views
How certain is it that Lucas invented the Towers of Hanoi puzzle?
Wikipedia is unequivocal:
The puzzle was invented by the French mathematician Édouard Lucas in 1883.
I have no reason to doubt this, but given the many legends surrounding the topic,
I wonder if ...
9
votes
2answers
3k views
The history and motivation of eigenvectors
I want to understand more about the history of eigenvectors. Was the discovery of eigenvectors inspired from an application to achieve a result in a historical context, was there a phenomenon which ...
8
votes
1answer
3k views
How was Newton's classification of cubic curves completed?
According to what I have read, using Newton’s methods there are 78 different families of cubic curves. Newton discovered 72 of them while “subsequent research identified another 6”. This paper ...
8
votes
3answers
196 views
Who are “analysts” and “synthesists” in mathematics?
What is the difference between the terms "analysis" and "synthesis" used in a mathematical context?
For example, Hawkins's Emergence of the Theory of Lie Groups p. 3 says that Klein and Lie
were ...
