All Questions

1,157 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
21 votes
0 answers
697 views

What is the modern significance of Theaetetus's classification of quadratic irrationals?

Before Eudoxus's theory of proportion there was a theory of irrationals based on continued fraction expansions, which Fowler calls anthyphairesis. Theaetetus is said to develop a classification of ...
Conifold's user avatar
  • 77.6k
17 votes
0 answers
1k views

When did people know that all real polynomials of degree greater than 2 were reducible?

Admittedly, this may not be a research level question, but I am deeply curious about this. Let $f(x) \in \mathbb{R}[x]$, and write $d = \deg f$. It is well known that if $\deg f > 2$, then $f$ is ...
Stanley Yao Xiao's user avatar
17 votes
0 answers
816 views

Did Kontsevich start a lecture with "one I will not define, the other nobody knows how to define, and we will be proving that they are equivalent"?

The story was circulating in early 2000s, so presumably it happened in 1990s. Kontsevich, it goes, opened a lecture course on mirror symmetry with:"This course is about two categories. One I will not ...
Conifold's user avatar
  • 77.6k
16 votes
0 answers
621 views

A basic mistake by Cayley

Arthur Cayley's first paper on abstract groups, in 1854, can be found in his Collected Papers on the Internet Archive, starting at https://archive.org/stream/collectedmathema02cayluoft#page/122/mode/...
KCd's user avatar
  • 5,689
14 votes
0 answers
660 views

Did Kronecker say that set theory is not mathematics?

I have frequently come across Kronecker's statement about set theory: I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there. It ...
Franz Kurz's user avatar
13 votes
0 answers
165 views

Did Walter Pitts refuse to allow his name to be made publicly available?

I read on the Wikipedia page on Walter Pitts that : Pitts was also described as an eccentric, refusing to allow his name to be made publicly available. He refused all offers of advanced ...
Franck Dernoncourt's user avatar
12 votes
0 answers
302 views

What was the typical format of a 16th century mathematical debate?

In The Equation that Couldn't be Solved, Mario Livio writes of academia in 16th century Bologna. Apparently, mathematicians would take part in public debates, sometimes involving solving problems. ...
HDE 226868's user avatar
  • 8,503
12 votes
0 answers
85 views

Why are the classic statistical approaches to NLP mostly generative models while the most recent ones are mostly discriminative?

Looking at the classic statistical approaches to natural language processing (e.g. tagging, parsing, etc.), I see that they are mostly generative models: n-gram models, Naive Bayes classifiers, hidden ...
Franck Dernoncourt's user avatar
11 votes
0 answers
366 views

How were contour plots of complex functions produced in the days of mechanical differential analyzers?

I was reading an old paper (specifically, the first appearance of the Pearcey function, here) and I was struck by the beauty of the plots it contains, particularly for a paper from 1945-46: Pearcey ...
Emilio Pisanty's user avatar
11 votes
0 answers
282 views

Origin of the special Finnish notation for difference of antiderivative

Apologies for a question that is specific to one country (but perhaps others find it a curious example of how mathematical notation can vary between countries). In Finnish calculus texts, if $F$ is an ...
Jukka Kohonen's user avatar
10 votes
0 answers
916 views

Ramanujan's Method for solving cubic, quartic, quintic

In Ramanujan's Notebooks Volume IV pg. 31 by Bruce C. Berndt, he describes an easy way to solve the general quartic by starting with the system$$x^2+ay=b\\y^2+cx=d\tag1$$ And solving for $x$; which ...
Frank's user avatar
  • 201
9 votes
0 answers
269 views

First use of term "Hilbert's Nullstellensatz"

This year (2021) marks the 100th anniversary of Emmy Noether's 1921 paper in which she introduced Noetherian rings and proved the primary ideal decomposition for them. The original version of her ...
KCd's user avatar
  • 5,689
9 votes
0 answers
746 views

Did John von Neumann hate pure mathematics that became too abstract?

John von Neumann wrote the following in his essay The Mathematician: As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only ...
user4281's user avatar
  • 615
9 votes
0 answers
220 views

Apéry’s mysterious recurrence relation

A fairly detailed (14 page) account of Apéry’s original proof of the irrationality of $\zeta(3)$ is given in Julian Havil’s book The Irrationals which states that Apéry’s starting point is the ...
nwr's user avatar
  • 6,909
9 votes
0 answers
176 views

Who was the first to use the "does not exist" sign ∄?

Who was the first to use the "does not exist" sign ∄? I'm aware that Giuseppe Peano originated serifed ∃ and, moreover that Whitehead and Russell repurposed Peano's serifed ∃; I'm also aware that ...
אהרן רובין's user avatar
9 votes
0 answers
460 views

Whence “homomorphism”, “homomorphic”?

The kernel question leads to another : Today, homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen? “Homomorphic” ...
Francois Ziegler's user avatar
9 votes
0 answers
95 views

On the history of population dynamics of territorial species

I am interested in the historical priority in population biology, essays or monographs, discussing the concept of territoriality prior to 1950. What is it? In the early 18th century discussions of ...
Gottfried William's user avatar
9 votes
0 answers
154 views

Contemporary reactions to the rise of axiomatization in the 19th/20th centuries

Starting somewhere in the 19th century, mathematics turned from the study of concrete objects to the study of objects satisfying enough properties to lead to interesting theorems. For example: From ...
Jack M's user avatar
  • 3,149
9 votes
0 answers
205 views

Mathematical counterintelligence at Bletchley during World War 2

Popular works of fiction claim that after breaking the Enigma in Bletchley, some sophisticated mathematics or statistical techniques were used to hide this fact of breaking (not necessarily by the ...
puslet88's user avatar
  • 191
9 votes
1 answer
356 views

Is Hermann Weyl's book “Space, Time, Matter” (1923) on General Relativity still relevant?

I really liked Hermann Weyl's mathematical books and would like to get accustomed to general relativity from his perspective, but wonder if it's still relevant after almost 100 (!) years? Can this ...
Slaus's user avatar
  • 191
8 votes
0 answers
204 views

Books on elliptic functions

In the end of his address to Annual Meeting of the Mathematical Association in 1933 titled "The marquis and the land agent: a tale of the 18th century", the Association president G. N. ...
Alexandre Eremenko's user avatar
8 votes
0 answers
157 views

When did computer pioneers realize that some problems are intrinsically hard?

In my theory computation class, I was told that early computer pioneers didn't realize that some problems are intrinsically hard—what we now call NP-hard problems. Instead, it took a while to realize ...
vy32's user avatar
  • 655
8 votes
0 answers
312 views

What exactly was Lagrange's "grave mistake" with respect to rotating bodies under hydrostatic equilibrium?

A comment below What would be different about satellite orbits if Earth were prolate? Would we have Sun-synchronous and Molniya orbits? got me reading Wikipedia's Jacobi ellipsoid which begins: ...
uhoh's user avatar
  • 2,187
8 votes
0 answers
102 views

Chronology of proofs of cubic and biquadratic reciprocity laws

I just want to check if anybody knows a website where one can find a chronology of proofs of more difficult reciprocity theorems (such as the cubic and biquadratic cases) similar to the (already ...
user2554's user avatar
  • 4,499
8 votes
0 answers
470 views

Who first defined polynomials as sequences?

Question 1. When did the modern definition of a polynomial (as a sequence of coefficients, with multiplication defined by $\left(ab\right)_n = \sum\limits_{k=0}^n a_k b_{n-k}$) emerge? Let me clarify:...
darij grinberg's user avatar
8 votes
0 answers
3k views

\mathbb versus \mathbf

When was the use of \mathbb popularized as an alternative to \mathbf? Of course there are the subjective preferences of certain authors, but when I read older articles, there appears to be an almost ...
AnotherPerson's user avatar
8 votes
0 answers
1k views

About the LOR of John Nash, was there any relationship between Richard Duffin and Solomon Lefschetz?

In Academia SE, there is a question about the credibility of Prof. Richard Duffin, who wrote the notorious letter of recommendation for John Nash, who later received the Nobel Memorial Prize in ...
Ooker's user avatar
  • 1,228
8 votes
0 answers
322 views

How did Weyl's 1918 paper; Gravitation and Electricity, influence classical physics?

The main-stream view seems to be that Weyl's 1918 paper Gravitation and Electricity was initially considered a failure for reasons first pointed out by Einstein. But these initial ideas were reapplied ...
Larry Harson's user avatar
8 votes
0 answers
53 views

How did Staudinger demonstrate that polymers are high molecular weight covalently bonded molecules?

Hermann Staudinger was awarded the 1953 Nobel Prize in Chemistry for his work demonstrating that polymers are high molecular weight covalently bonded molecules. Prior to his work, the consensus among ...
Colin McFaul's user avatar
7 votes
0 answers
188 views

History of group actions as their own structures

I'm interested in when (and how) the modern idea of a group action developed and how group actions became their own algebraic structures. As far as I can tell in the 19th century group actions were ...
paidresolution's user avatar
7 votes
0 answers
118 views

What was the first automated theorem prover?

From a lot of googling, it seems like the answer might be "Mizar", but I am not completely sure. What was (or is?) the first automated theorem prover (i.e. not necessarily active right now)?
Alex's user avatar
  • 275
7 votes
0 answers
262 views

Who coined the Hawaiian Earrings?

I hope to know who first used the name "Hawaiian Earrings." Barratt, Milnor(1962) says "This example was suggested by Steenrod" in its Introduction: https://www.ams.org/journals/...
user avatar
7 votes
0 answers
269 views

How did early physicists experimentally assign electronic transitions in atoms?

The spectrum of hydrogen was very well studied by the mid-19th century. However, if one were doing experimental spectroscopy for more complex atoms, one would see plenty of spectral lines in the ...
ACR's user avatar
  • 4,079
7 votes
0 answers
156 views

F. Schoblik's announced ''ausführliche Darstellung": a lost wrong proof of the Four Color Theorem?

In (the AMS Chelsea Publishing version of) what is perhaps the first genuine textbook on graph theory ever, Dénes Kőnig on p. 28 gives the illustration and the footnote which when translated says ...
Peter Heinig's user avatar
7 votes
0 answers
93 views

Whence originates the use of the nabla (∇) for a connection or covariant derivative?

Who introduced it, when, where, and with what if any rationale? (Note that I am not asking about the origin of the nabla symbol, which is covered here.)
Archelon's user avatar
  • 163
7 votes
0 answers
267 views

Why is Minkowski's question mark function denoted by a question mark?

There are some real odd names for functions in mathematics, but Minkowski's question mark function, denoted by $?(x)$, may be the oddest one I have ever seen. In Zur Geometrie der Zahlen, Minkowski ...
HDE 226868's user avatar
  • 8,503
7 votes
0 answers
129 views

What events does the note at the beginning of Dialogue Concerning Two Systems refer to?

Galileo's Dialogue Concerning the Two Chief World Systems opens with a seemingly sarcastic "Note to the discerning reader", beginning: Several years ago there was published in Rome a ...
Jack M's user avatar
  • 3,149
7 votes
0 answers
148 views

Who first discussed the Lorentz force with respect to special relativity?

The fact that a Lorentz force in a reference frame 1 can become a Coulomb force in another reference frame 2 has always astonished me, especially because the velocities involved are really small. I ...
Gerard's user avatar
  • 171
7 votes
1 answer
784 views

Origin of Tensor Product

When and why did Mathematicians saw a need to define Tensor Products? I want to know the historical development of the idea "Tensor Product"?
Saikat's user avatar
  • 348
6 votes
0 answers
69 views

When were arrows first used to visualise vectors?

I guess the use of arrows to visualise vectors came before the general notion of vectors, so a more precise question is: when where arrows first used to visualise physical (or mathematical) quantities ...
Michael Bächtold's user avatar
6 votes
0 answers
216 views

When did people first know that the sky was illuminated by the Sun?

During the day, the sky is bright blue and, along with the yellow light from the Sun, it illuminates the surface of the Earth. What is the earliest recorded knowledge that the sky wasn't self-...
Ray Butterworth's user avatar
6 votes
0 answers
136 views

What technology was used to determine the shape of the blackbody spectrum at the 19th century?

The shape of the blackbody radiation spectrum was known in the 19th century from experimental measurements, and before the theoretical discovery of Planck's law. At those times, how did people manage ...
Solidification's user avatar
6 votes
0 answers
451 views

What is the source of Donald Knuth's remark about naming programming languages?

(This question toes the line between belonging here and belonging on the Retrocomputing Stack Exchange.) Here is the quote; sometimes the first sentence is omitted: The most important thing in a ...
texdr.aft's user avatar
  • 253
6 votes
0 answers
172 views

How did Dyck originally state and prove his theorem in topology about the connected sum of a torus and projective plane?

Dyck's theorem in topology is sometimes stated as follows: the connected sum of a torus and projective plane is homeomorphic to the connected sum of three projective planes. Certainly, this is the ...
user avatar
6 votes
0 answers
120 views

What is the origin in the discrepancy between engineers' and physicists' notation of waves?

my question is very simple. Physicists use this notation in order to write a (for example) plane wave: $$ \xi(z) = \xi^+ \mathrm{e}^{+\mathrm{i}kz} + \xi^- \mathrm{e}^{-\mathrm{i}kz}, $$ where $\xi^+$ ...
gunix12's user avatar
  • 61
6 votes
0 answers
337 views

What was the content of Emmy Noether's 'two-line' note on Homology?

I was surprised to learn that the note published by Emmy Noether in 1925 that suggested that Homology was better thought through as actual groups rather than numerically as Betti numbers consisted of ...
Mozibur Ullah's user avatar
6 votes
0 answers
181 views

Did Hardy and Ramanujan miscalculate these values?

When I read Dickson's History Of The Theory Of Numbers Vol-2, I found that there seems to be a mistake in the approximation of partition numbers p(200). For this reason, I found the original text ...
D.Matthew's user avatar
  • 161
6 votes
0 answers
300 views

Origin of the Fourier transform (1878)

I located Joseph Fourier's book, Analytical Theory of Heat (1878), but at first glance it looks like it is all about heat. What did Fourier call the Fourier transform? When did he first use it?
Christina Daniel's user avatar
6 votes
0 answers
269 views

What was worldwide supply of disk storage in 1969?

I'd prefer a year-by-year graph, something like (not necessarily the "demand", just "supply")...     https://www.statista.com/statistics/751749/worldwide-data-storage-capacity-and-demand/ ...
John Forkosh's user avatar
6 votes
0 answers
170 views

Does Galileo's method of tracing a parabola actually work?

In Two New Sciences, Galileo writes: I take a perfectly round brass ball about the size of a walnut and project it along the surface of a metallic mirror held in a nearly upright position, so that ...
concertpi's user avatar

15 30 50 per page
1
2 3 4 5
24