All Questions
1,069
questions with no upvoted or accepted answers
21
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answers
675
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 ...
- 73.6k
17
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0
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927
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 ...
- 337
17
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0
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796
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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 ...
- 73.6k
16
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0
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597
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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/...
- 5,197
14
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0
answers
573
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 ...
13
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0
answers
162
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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
...
- 3,263
12
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0
answers
297
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. ...
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11
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0
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359
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 ...
- 762
11
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0
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80
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 ...
- 3,263
10
votes
0
answers
243
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 ...
- 256
9
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0
answers
256
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 ...
- 5,197
9
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0
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662
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 ...
- 615
9
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0
answers
206
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 ...
- 6,689
9
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172
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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 ...
- 141
9
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0
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455
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” ...
- 7,654
9
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0
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875
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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 ...
- 191
9
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94
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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 ...
- 1,232
9
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0
answers
152
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 ...
- 3,109
9
votes
1
answer
338
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 ...
- 191
8
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0
answers
199
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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. ...
- 47.6k
8
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0
answers
151
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 ...
- 637
8
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293
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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:
...
- 2,143
8
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437
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:...
- 181
8
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0
answers
2k
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\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 ...
- 309
8
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0
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942
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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 ...
- 1,178
8
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0
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316
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 ...
- 453
8
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0
answers
202
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 ...
- 181
8
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0
answers
52
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 ...
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7
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0
answers
148
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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 ...
7
votes
0
answers
103
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)?
- 275
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360
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What is Holon Programming?
Donald Knuth credits Pierre-Arnoul de Marneffe's idea of "Holon Programming" as the main influence on Literate Programming. See page 13 of "Literate Programming", Knuth's paper ...
- 231
7
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0
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254
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/...
user12897
7
votes
0
answers
95
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 ...
- 4,337
7
votes
0
answers
257
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 ...
- 3,959
7
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0
answers
152
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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
...
- 680
7
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0
answers
266
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 ...
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7
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0
answers
127
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 ...
- 3,109
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0
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148
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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 ...
- 171
7
votes
1
answer
676
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"?
- 338
6
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0
answers
207
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-...
- 346
6
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0
answers
133
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 ...
- 317
6
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0
answers
397
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 ...
- 231
6
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0
answers
136
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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 ...
user13915
6
votes
0
answers
117
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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 $\xi^+$ ...
- 61
6
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0
answers
310
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 ...
- 3,613
6
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0
answers
254
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Can anyone find Newton's calculation error in Principia, Book III, Proposition XIX?
Musing about the historical evolution of the notation for the gravitational constant ($f$, $G$, $\kappa$, $\kappa^2$), I found myself digging for the first time in my life into Newton's Principia, ...
- 169
6
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answers
177
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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 ...
- 161
6
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0
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235
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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/
...
- 191
6
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0
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164
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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 ...
- 61
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0
answers
267
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history of backpropagation
Has anybody read or have access to
Alex Andrew
Significance Feedback in Neural Nets
Report of Biological Computer Laboratory, University of Illinois, Urbana, IL
GM-10718-03
TR No 5
September ...
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