instruction stringlengths 39 23.6k | completion stringlengths 12 25.5k | conversations listlengths 2 2 |
|---|---|---|
On the Wikipedia page on Hideyo Noguchi, one can find the following:
Possibly his erratic and irresponsible behaviour was caused by the
untreated syphilis with which he was diagnosed in 1913, and which may
have progressed to neurosyphilis. In any case, he was careless not
only of his own safety, but also of the ... | As of my last knowledge update in October 2023, there hasn't been definitive scholarly research confirming or refuting that Hideyo Noguchi had neurosyphilis. Much of the information surrounding his mental state and behavior, particularly concerning potential syphilis infection, is largely speculative and based on histo... | [
{
"from": "human",
"value": "On the Wikipedia page on Hideyo Noguchi, one can find the following:\n\nPossibly his erratic and irresponsible behaviour was caused by the\n untreated syphilis with which he was diagnosed in 1913, and which may\n have progressed to neurosyphilis. In any case, he was careless n... |
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
degrees or official positions at MIT as he would have to sign his
name.
It came to me as a surprise as he seems to have use h... | Walter Pitts was indeed known for his eccentric behavior and his reluctance to use his name in certain contexts, particularly in a formal or public capacity. His decision to refuse advanced degrees and official positions at institutions like MIT stemmed from his discomfort with the formalities that accompanied such rol... | [
{
"from": "human",
"value": "I read on the Wikipedia page on Walter Pitts that :\n\nPitts was also described as an eccentric, refusing to allow his name\n to be made publicly available. He refused all offers of advanced\n degrees or official positions at MIT as he would have to sign his\n name.\n\nIt ca... |
The McTutor biography on Poincaré says:
Poincaré's Analysis situs, published in 1895, is an early systematic treatment of topology. He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals... | Henri Poincaré's work in topology, particularly in relation to the fundamental group and algebraic topology, can indeed be seen as an outgrowth of his broader interests in various areas of mathematics and physics, including celestial mechanics and the theory of automorphic functions.
1. **Celestial Mechanics**: Poinca... | [
{
"from": "human",
"value": "The McTutor biography on Poincaré says:\n\nPoincaré's Analysis situs, published in 1895, is an early systematic treatment of topology. He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as di... |
It is widely known nowadays that air pollution attenuates incoming light, particularly sunlight through scattering and absorption of certain wavelengths (both in the UV and visible wavelengths). This attenuation of light is significant to science as it can adversely affect the biosphere.
When was the optical effects o... | The optical effects of air pollution were first measured in the early 20th century. One notable early measurement was conducted by the scientist **Aristotle** in ancient times, who made observations about the impact of atmospheric conditions on visibility and light. However, more systematic scientific studies began in ... | [
{
"from": "human",
"value": "It is widely known nowadays that air pollution attenuates incoming light, particularly sunlight through scattering and absorption of certain wavelengths (both in the UV and visible wavelengths). This attenuation of light is significant to science as it can adversely affect the ... |
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 incompleteness theorem.
Is there a field of mathematics today that is the spiritual successor to metamathematics as studied by ... | Yes, the field that is often considered a spiritual successor to the metamathematics studied by Gödel, Hilbert, and the authors of *Principia Mathematica* is **model theory**. Model theory emerged in the mid-20th century and has developed significantly since then.
Model theory studies the relationships between formal ... | [
{
"from": "human",
"value": "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 incompleteness theorem.\nIs there a field of mathematics today that is the spiritual s... |
Van Leuwenhoek observed the first protist in 1674, but his description of it was somewhat superficial and fragmentary:
Passing lately over the Sea at a time, when it blew a fresh gale of wind, and observing the water as above described, I took up some of it in a Glass-vessel, which having view'd the next day, I found ... | Anton van Leeuwenhoek's initial observations of protists were indeed groundbreaking but lacked the detailed understanding of cellular structure that would come later. He did not provide a more thorough description of these microorganisms in his subsequent writings. His work primarily focused on the observation of live ... | [
{
"from": "human",
"value": "Van Leuwenhoek observed the first protist in 1674, but his description of it was somewhat superficial and fragmentary:\n\nPassing lately over the Sea at a time, when it blew a fresh gale of wind, and observing the water as above described, I took up some of it in a Glass-vessel,... |
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 particular piece of work or in general to all his accomplishments thus far?
| When Isaac Newton wrote the phrase "If I have seen further it is by standing on the shoulders of giants" in his letter to Robert Hooke in 1676, he was indeed acknowledging the contributions of earlier scholars and thinkers who laid the groundwork for his own work. While he did not specify particular individuals in this... | [
{
"from": "human",
"value": "In a letter to Robert Hooke in 1676, Newton wrote:\n\nIf I have seen further it is by standing on the shoulders of giants.\n\nDo we know which giants Newton was referring to? And was he referring to a particular piece of work or in general to all his accomplishments thus far? \n... |
Note: this was original an example question, posted by neizod. As I mentioned in meta, it would be nice if we could get the example questions asked on the site. neizod did not commit to the site and has not yet joined, so I will place it here. I can make this post community wiki if the community feels this is the right... | The Haversine formula, which calculates the distance between two points on the surface of a sphere given their latitude and longitude, is attributed to multiple sources over time. The term "haversine" itself is derived from "half-versed sine" and is believed to have been coined by the mathematician Robert W. Sinnott in... | [
{
"from": "human",
"value": "Note: this was original an example question, posted by neizod. As I mentioned in meta, it would be nice if we could get the example questions asked on the site. neizod did not commit to the site and has not yet joined, so I will place it here. I can make this post community wiki... |
This is a pretty straightforward question, when the first observations of the night sky were being made, who was the first person to suggest that a planet, say Mars, was not a star, in the sense that their physical compositions were different? I thought Tycho Brahe, but he lived in the 16th century, whilst a large part... | You're correct that the distinction between planets and stars was a significant topic in ancient astronomy. The idea that planets like Mars were different from stars began to take shape in ancient Greece.
One of the earliest figures to recognize a distinction between planets and stars was the Greek philosopher and as... | [
{
"from": "human",
"value": "This is a pretty straightforward question, when the first observations of the night sky were being made, who was the first person to suggest that a planet, say Mars, was not a star, in the sense that their physical compositions were different? I thought Tycho Brahe, but he lived... |
In the comments on this question on Physics about the usefulness of expensive experiments such as the CERN, the following short discussion happened:
Has there ever been a major basic science result that did not lead to practical applications within the next couple of hundred years?
Isn't there a strong selection bi... | Finding a scientific result that meets all the criteria you've outlined is quite challenging, as many foundational discoveries have eventually found some form of application, even if it took a long time to manifest. However, here are a couple of examples that may fit your requirements:
1. **Caloric Theory**: The calor... | [
{
"from": "human",
"value": "In the comments on this question on Physics about the usefulness of expensive experiments such as the CERN, the following short discussion happened:\n\nHas there ever been a major basic science result that did not lead to practical applications within the next couple of hundred ... |
The Superconducting Super Collider was the greatest experiment that was never built. It was cancelled in 1993 after years of planning and effort, during which there was some construction of facilities. The site may be used in the future, but I haven't been able to find much information on plans for it. When the project... | When the Superconducting Super Collider (SSC) was cancelled in 1993, it left behind a large, partially constructed site near Waxahachie, Texas. At the time of cancellation, there were some discussions regarding the future of the site. The U.S. Department of Energy (DOE), which managed the project, started the process o... | [
{
"from": "human",
"value": "The Superconducting Super Collider was the greatest experiment that was never built. It was cancelled in 1993 after years of planning and effort, during which there was some construction of facilities. The site may be used in the future, but I haven't been able to find much info... |
Many scientific units are named after scientists, for example,
Tesla for magnetic flux
Farad for capacitance
Newton for force.
When did the tradition of naming scientific units begin?
| The tradition of naming scientific units after scientists began in the late 19th century. One of the earliest examples is the ampere, named after the French physicist André-Marie Ampère, which was established in the mid-19th century. The system of units that we now refer to as the International System of Units (SI) was... | [
{
"from": "human",
"value": "Many scientific units are named after scientists, for example,\n\nTesla for magnetic flux\n\nFarad for capacitance\n\nNewton for force.\n\n\nWhen did the tradition of naming scientific units begin?\n"
},
{
"from": "gpt",
"value": "The tradition of naming scientific u... |
The ancient Egyptians used examination, diagnosis, treatment and prognosis, to the treatment of diseases, which are all described in Egyptian medical textbooks from around c. 1600 BC.
Did the ancient Egyptians use some kind of scientific method, if not, what is the the earliest accounts of an scientific method for inv... | The ancient Egyptians did not use a scientific method in the way we understand it today. Their approach to medicine, as seen in texts such as the Ebers Papyrus and the Edwin Smith Papyrus, involved observation and practical knowledge passed down through generations. While they did employ systematic techniques for exami... | [
{
"from": "human",
"value": "The ancient Egyptians used examination, diagnosis, treatment and prognosis, to the treatment of diseases, which are all described in Egyptian medical textbooks from around c. 1600 BC.\nDid the ancient Egyptians use some kind of scientific method, if not, what is the the earlies... |
While I have heard speculative answers to this question, I do not know one which can be supported. Is there any information explaining why Nobel did not chose to include this topic? Has there even been a serious attempt within the committee to add one or an explanation why they will not?
| Alfred Nobel, the founder of the Nobel Prizes, specified in his will the categories for the prizes: Peace, Literature, Physics, Chemistry, and Medicine. The omission of a prize for certain fields, such as economics, has been a topic of speculation and discussion. Nobel's will does not explicitly state why he did not in... | [
{
"from": "human",
"value": "While I have heard speculative answers to this question, I do not know one which can be supported. Is there any information explaining why Nobel did not chose to include this topic? Has there even been a serious attempt within the committee to add one or an explanation why they... |
Cancer is one of the most well known and widespread conditions in the modern world, it can be argued that everyone is affected either directly and indirectly throughout their lives. According to Cancer Research UK, there are over 200 types of cancer.
The literature of modern research into cancer (and its treatments) i... | Cancer has been recognized for thousands of years, with some of the earliest known references found in ancient Egypt. The oldest documented case of cancer dates back to around 1600 BCE and is recorded in the Edwin Smith Papyrus, an ancient Egyptian medical text. In this document, there are descriptions of tumors or gro... | [
{
"from": "human",
"value": "Cancer is one of the most well known and widespread conditions in the modern world, it can be argued that everyone is affected either directly and indirectly throughout their lives. According to Cancer Research UK, there are over 200 types of cancer.\nThe literature of modern r... |
Who created topology, when did that happen, and how was it discovered?
| Topology as a formal mathematical discipline began to take shape in the late 19th to early 20th centuries, though its roots can be traced back to earlier mathematical concepts. The term "topology" itself was first used in this context by the mathematician Johann Benedict Listing in 1847. He and other early mathematicia... | [
{
"from": "human",
"value": "Who created topology, when did that happen, and how was it discovered?\n"
},
{
"from": "gpt",
"value": "Topology as a formal mathematical discipline began to take shape in the late 19th to early 20th centuries, though its roots can be traced back to earlier mathemati... |
Chow, Tommy WS, and John Sum. "Guest editorial: special issue on the emerging applications of neural networks." Neural Computing & Applications 20.7 (2011): 923-924. says:
The take off of neural computing research was probably due to the
announcement that Japan had resumed the research of neural networks.
The US f... | The assertion that the resurgence of neural computing research in the 1980s was significantly influenced by developments in Japan is widely recognized in the history of artificial intelligence and neural networks.
In the early 1980s, Japan launched a series of initiatives aimed at advancing artificial intelligence, p... | [
{
"from": "human",
"value": "Chow, Tommy WS, and John Sum. \"Guest editorial: special issue on the emerging applications of neural networks.\" Neural Computing & Applications 20.7 (2011): 923-924. says:\n\nThe take off of neural computing research was probably due to the\n announcement that Japan had resum... |
The Erdős number is used as a tool to study how mathematicians cooperate to find answers to unsolved problems. But what is the history behind the Erdős number?
| The concept of the Erdős number originates from the work of Hungarian mathematician Paul Erdős, who was known for his extensive collaboration with other mathematicians throughout his career. Erdős published over 1,500 mathematical papers, often co-authoring them with a wide variety of collaborators. His unique collabor... | [
{
"from": "human",
"value": "The Erdős number is used as a tool to study how mathematicians cooperate to find answers to unsolved problems. But what is the history behind the Erdős number?\n"
},
{
"from": "gpt",
"value": "The concept of the Erdős number originates from the work of Hungarian math... |
There are many claimants for the first telescope. Amongst these are the claims placed at the doors of Bacon and Digges. The Bacon claim is very sketchy, boiling down to one sentence and is easy to dismiss.
The Thomas Digges claim seems more accomplished. From what I gather the main objection is that the claimed perform... | The claims surrounding the invention of the telescope indeed involve several historical figures, and both Sir Francis Bacon and Thomas Digges are often mentioned in the context of early optical instruments. The controversy mainly arises from the lack of clear evidence and the complexities in interpreting the historical... | [
{
"from": "human",
"value": "There are many claimants for the first telescope. Amongst these are the claims placed at the doors of Bacon and Digges. The Bacon claim is very sketchy, boiling down to one sentence and is easy to dismiss.\nThe Thomas Digges claim seems more accomplished. From what I gather the ... |
If I recall correctly, Stephen Hawking mentioned in A Brief History of Time that Newton remarked once that the famed apple actually distracted him from his lucubrations.
Does anybody know where it was that Newton wrote (or said) the remark under consideration or, failing that, where it may have been that Hawking read (... | The anecdote about Isaac Newton and the apple is one of the most famous stories associated with his life and the development of his theories of gravity. However, it is often embellished and may not have a direct source. The story generally suggests that Newton was inspired by seeing an apple fall, leading him to contem... | [
{
"from": "human",
"value": "If I recall correctly, Stephen Hawking mentioned in A Brief History of Time that Newton remarked once that the famed apple actually distracted him from his lucubrations.\nDoes anybody know where it was that Newton wrote (or said) the remark under consideration or, failing that, ... |
Asthma has been known since Ancient Egypt, according to the Medical News Today web-article Asthma History - Through The Ages, with the first use of the word by Hippocrates in Ancient Greece. Throughout most of history from Ancient Egypt and Greece, asthma was treated as a physical condition.
Even in the 16th and 17th ... | Asthma was considered a psychosomatic condition during the first half of the 20th century for several interrelated reasons:
1. **Emerging Psychological Theories**: The early 20th century saw the rise of psychoanalysis and other psychological theories that emphasized the role of the mind in physical health. Sigmund Fre... | [
{
"from": "human",
"value": "Asthma has been known since Ancient Egypt, according to the Medical News Today web-article Asthma History - Through The Ages, with the first use of the word by Hippocrates in Ancient Greece. Throughout most of history from Ancient Egypt and Greece, asthma was treated as a physi... |
I'm interested to know what mathematical and physics topics Einstein learnt during his graduation.Did he know about advanced mathematical topics such as Hamiltonian mechanics ,Lagrangian mechanics, Manifold and Differential topology .
| Albert Einstein graduated from the Polytechnic Institute in Zurich (now ETH Zurich) in 1900, where he studied physics and mathematics. During his education, he encountered several key mathematical and physics topics that laid the groundwork for his later work.
### Topics Einstein Studied:
1. **Calculus and Analysis*... | [
{
"from": "human",
"value": "I'm interested to know what mathematical and physics topics Einstein learnt during his graduation.Did he know about advanced mathematical topics such as Hamiltonian mechanics ,Lagrangian mechanics, Manifold and Differential topology .\n"
},
{
"from": "gpt",
"value":... |
Thought experiments, imaginary situations designed to ponder conclusions of a theory, have been used throughout history. There is even a Stanford encyclopedia article about them. But the famous ones that come to mind are either from physics (Archimedes's overturning of the Earth, Newton's bucket, Maxwell's demon, Einst... | Thought experiments in biology do exist, though they may not be as widely recognized or formally structured as those in physics or philosophy. They often take the form of hypothetical scenarios used to explore evolutionary, ecological, or physiological principles. Here are a few examples:
1. **Darwin's Finches**: Char... | [
{
"from": "human",
"value": "Thought experiments, imaginary situations designed to ponder conclusions of a theory, have been used throughout history. There is even a Stanford encyclopedia article about them. But the famous ones that come to mind are either from physics (Archimedes's overturning of the Earth... |
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 in 1925, and the story is very well documented. Then very shortly after that Born and Jordan recognized that this is ma... | The shift in undergraduate curricula to include matrices, particularly in the context of physics and applied mathematics, can be traced through several key historical developments in mathematics and physics education. The inclusion of matrix algebra in the standard curriculum is indeed related to the advancements in ma... | [
{
"from": "human",
"value": "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 in 1925, and the story is very well documented. Then very shortly after that B... |
We know that up to 1726, when the third edition of the Principia was published, the name for $m\vec v$ was: quantitas motus.
Do you know who substituted that with another Latin word: 'momentum'?
| The term "momentum" was introduced by the philosopher and mathematician Gottfried Wilhelm Leibniz. He replaced the earlier term "quantitas motus," which referred to the quantity of motion, with "momentum" in the context of his work in the late 17th century. Leibniz's introduction of the term helped to formalize the con... | [
{
"from": "human",
"value": "We know that up to 1726, when the third edition of the Principia was published, the name for $m\\vec v$ was: quantitas motus.\nDo you know who substituted that with another Latin word: 'momentum'?\n"
},
{
"from": "gpt",
"value": "The term \"momentum\" was introduced ... |
The so-called "Hilbert space" is named after mathematician David Hilbert. Later, this was generalized into "Banach spaces" by Stefan Banach.
My understanding is that Hilbert was German and Banach was Polish, and there did not appear to be any "major" connection between them (that is, no more than between two "random" E... | Your understanding of the historical context of Hilbert and Banach is generally accurate. David Hilbert was indeed a prominent German mathematician whose work laid the foundations for many areas of mathematics, including functional analysis. Stefan Banach, a Polish mathematician, made significant contributions to this ... | [
{
"from": "human",
"value": "The so-called \"Hilbert space\" is named after mathematician David Hilbert. Later, this was generalized into \"Banach spaces\" by Stefan Banach.\nMy understanding is that Hilbert was German and Banach was Polish, and there did not appear to be any \"major\" connection between th... |
Until the Abel prize came along, the Fields medals were considered to be the Nobel for math. But when did this tradition of calling it the Nobel for math start and why? Because the prize money is nowhere as much as the Nobels and also the rules governing the two prizes are entirely different.
| The comparison of the Fields Medal to the Nobel Prize in Mathematics is largely a matter of colloquialism and tradition rather than an official designation. This analogy likely began due to the prominent status of both prizes in their respective fields, although the reasons behind this comparison vary.
The Fields Meda... | [
{
"from": "human",
"value": "Until the Abel prize came along, the Fields medals were considered to be the Nobel for math. But when did this tradition of calling it the Nobel for math start and why? Because the prize money is nowhere as much as the Nobels and also the rules governing the two prizes are entir... |
A well-known TV series hinted some time ago that Isaac Newton burnt the only known picture of Robert Hooke.
Is it true?
| Yes, the claim that Isaac Newton burned a portrait of Robert Hooke is a historical anecdote rather than a verified fact. This event is often discussed in the context of the contentious relationship between Newton and Hooke, who were both prominent scientists of the 17th century.
The story suggests that Newton, who ha... | [
{
"from": "human",
"value": "A well-known TV series hinted some time ago that Isaac Newton burnt the only known picture of Robert Hooke.\nIs it true?\n"
},
{
"from": "gpt",
"value": "Yes, the claim that Isaac Newton burned a portrait of Robert Hooke is a historical anecdote rather than a verifie... |
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 published it under his own name, claiming to have learned it elsewhere from a man named Ferro.
"Everyone" knows these facts... | The naming of the formula as "Cardano's formula" rather than "Tartaglia's formula" or "Ferro's formula" has to do with several factors, including the historical context of publication, recognition in the mathematical community, and the influence of Cardano himself.
1. **Publication and Attribution**: Although Tartagli... | [
{
"from": "human",
"value": "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.\nCardano published it under his own name, claiming to have learned it elsewhere from a man... |
Inspired by the question Discovery of Earth's magnetic field, the article from New Zealand's webpage Discovery of the Earth’s magnetic field, also states that
The earth's dynamo is unstable, as is shown by magnetic reversals, when the polarity of the whole magnetic field changes over.
When was the fact that Earth's ... | The discovery of Earth's magnetic field reversals dates back to the early 20th century. The concept was first suggested in 1906 by the geophysicist Bernard Brunhes, who identified a magnetic reversal in volcanic rocks. However, it was in the 1960s that the evidence for magnetic reversals became more widely recognized a... | [
{
"from": "human",
"value": "Inspired by the question Discovery of Earth's magnetic field, the article from New Zealand's webpage Discovery of the Earth’s magnetic field, also states that\n\nThe earth's dynamo is unstable, as is shown by magnetic reversals, when the polarity of the whole magnetic field chan... |
A question arose in my mind about modular inverse. We know that there is no division operator in modular arithmetic, so we figure out the modular inverse of the denominator. For example, if we want to find out the value of $\frac ab \pmod{M} $, at first we determine the modular inverse of $b$ and then multiply it by$a... | The concept of modular arithmetic, including the idea of modular inverses, has a rich history that dates back to several ancient mathematicians, but its modern formulation can primarily be attributed to the work of mathematicians in the 19th century.
The concept of a modular inverse, specifically, can be traced back t... | [
{
"from": "human",
"value": "A question arose in my mind about modular inverse. We know that there is no division operator in modular arithmetic, so we figure out the modular inverse of the denominator. For example, if we want to find out the value of $\\frac ab \\pmod{M} $, at first we determine the modul... |
The U.S. picked up a whole bunch of German scientists after World War II in Operation Paperclip. One of the most notable was Wernher von Braun, who jump-started the American space program.
The Soviet Union also got some scientists, although not exactly top-tier guys. What happened to them? Did they play a role in the ... | After World War II, the Soviet Union did indeed acquire a number of German scientists and engineers, although the most notable figures like Wernher von Braun were primarily taken by the United States through Operation Paperclip. The Soviet operation to recruit German scientists was more focused on a broader pool of tal... | [
{
"from": "human",
"value": "The U.S. picked up a whole bunch of German scientists after World War II in Operation Paperclip. One of the most notable was Wernher von Braun, who jump-started the American space program. \nThe Soviet Union also got some scientists, although not exactly top-tier guys. What happ... |
It is well known that every construction that can be performed by a compass and a ruler can be also performed by a compass only. This is a good (and difficult) exercise in elementary geometry. My question:
When did mathematicians start investigating this question ?
The literature on the subject that I know is from th... | The question of whether constructions performed with a compass alone can replicate those performed with both a compass and a ruler is indeed a deep and fascinating topic in geometry. This inquiry has roots that stretch back to ancient Greece, but it was particularly examined in more detail from the 19th century onward.... | [
{
"from": "human",
"value": "It is well known that every construction that can be performed by a compass and a ruler can be also performed by a compass only. This is a good (and difficult) exercise in elementary geometry. My question:\n\nWhen did mathematicians start investigating this question ?\n\nThe lit... |
According to what we read in footnote #2 of [I], no portrait of Christian Goldbach seems to be known. The author of the article under consideration even adds:
... Juskevič & Kopelevič's biography states that none exists [J&K, Christian Goldbach: 1690–1764 (translated from Russian to German by Annerose and Walter Purke... | The quest for an authentic portrait of Christian Goldbach, the 18th-century mathematician best known for Goldbach's conjecture, is a fascinating one. As noted in the references provided, there seems to be no confirmed portrait of Goldbach, and various biographies, including that by Juskevič & Kopelevič, have confirmed ... | [
{
"from": "human",
"value": "According to what we read in footnote #2 of [I], no portrait of Christian Goldbach seems to be known. The author of the article under consideration even adds:\n\n... Juskevič & Kopelevič's biography states that none exists [J&K, Christian Goldbach: 1690–1764 (translated from Rus... |
To quote Bertrand Russell, "My Philosophical Development", Simon and Schuster, N.Y., 1959,
p. 86:
I used to know of only six people who had read the later parts of the book
[Principia Mathematica]. Three of these were Poles .... The other three were Texans ...
Jan Woleński in "Polish Logic", Logical J. of the IGPL ... | The reference to Bertrand Russell's statement about the three Poles and three Texans who read the later parts of "Principia Mathematica" is quite interesting, but it does leave some ambiguity, especially regarding the identities of these individuals. You've correctly identified Leon Chwistek as one of the Poles. As for... | [
{
"from": "human",
"value": "To quote Bertrand Russell, \"My Philosophical Development\", Simon and Schuster, N.Y., 1959,\np. 86:\n\nI used to know of only six people who had read the later parts of the book\n [Principia Mathematica]. Three of these were Poles .... The other three were Texans ...\n\nJan Wo... |
I read the book "A Brief History of Time" By Stephan Hawking. It states that Einstein helped scientists like Pauli etc. in the development of the quantum theory and even shared the Nobel Prize with them for his contributions, but to his dying day, didn't agree with the theory. He even made the quote:
I, for one, do no... | Albert Einstein's relationship with quantum theory is indeed complex and nuanced. While it's true that Einstein made significant contributions to the foundations of quantum theory—most notably through his work on the photoelectric effect, for which he won the Nobel Prize in Physics in 1921—he had deep philosophical res... | [
{
"from": "human",
"value": "I read the book \"A Brief History of Time\" By Stephan Hawking. It states that Einstein helped scientists like Pauli etc. in the development of the quantum theory and even shared the Nobel Prize with them for his contributions, but to his dying day, didn't agree with the theory.... |
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 believe it makes sense to search for earlier sources, since the notion of continuity itself was not made rigorous until the ... | The belief that the intermediate value property implies continuity was indeed held by some mathematicians in the 19th century, but explicit sources confirming this view can be challenging to pinpoint. The concepts of continuity and the intermediate value property were evolving, and many mathematicians were grappling wi... | [
{
"from": "human",
"value": "As asked in the title: \n\nAre there any written sources (from the 19th century) explicitly stating the belief that any function satisfying the intermediate value property is continuous?\n\n(I do not believe it makes sense to search for earlier sources, since the notion of conti... |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 8