Nice to see another St Andrews person on HN - I took that course too! I only wrote two essays during my four year degree, one for this class! (I wrote about Alexander Aitken, a gifted mental arithmetician)
I'm not a mathematician, but I've always been fascinated by two specific periods in the history of mathematics: Around 1750-1850 where you have Gauss, Fourier, Poisson, Laplace, Navier, Cauchy, Lagrange, Euler. That's just... insane.
And then the mathematics that happened around ~1940.. Kolmogorov, Fisher, Poincaré, Gödel, Von Neumann, Church, Turing...
The first is the stuff you do in school. Always good to have context, which sadly there's little time for.
The second is fundamentals and computing stuff, and stats. There's a good chance you'll have to use a computer to calculate things over large datasets sometime in your life.
I'd love to be able to watch a simple video interview with these guys. Something like...Dr. Gödel, could we take just a few of your notable insights and have you walk us through how the fuck you managed to come up with them?
As I understand it, a significant portion of the cleverness is in the audacity of encoding something liar paradox-like into a formal logical framework (and perhaps also the precision with which logics were specified and differentiated from one another).
Well, a couple decades later we have the era-defining unification of geometry, algebra, and number theory building on the work of Noether, Hilbert, Weil, and so on.
To drop a few names, we had Grothendieck, Serre, Deligne, Wiles, Illusie, (the elder) Artin ... and it's continued to this day. Wiles' proof of FLT built upon this beautiful synthesis of ideas, for example.
I imagine there are some testable hypotheses around this very interesting question.
1.) What, if any, of mathematics has been invented concurrently in different regions (say within 10-20
years) without collaboration or knowledge of other derived work?
2.) To what extent is the rate of mathematical development associated with war or advances in knowledge sharing? Be it radio, printing press, persuing weapons development, crypto, or mass migration (eg post ww2).
I think 1.) comes about via natural scientific progress. Some mathematical ideas are dependent on the right set of tools being invented or environmental necessity. How many people independently came up with ballastics equations once soldiers starting flinging big rocks at each other?
The 2.) Question is very interesting to me. What is the network effect of mathematicians sharing ideas? What happens when the 10 smartest mathematicians in the world end up all working at the same elite university? Why isnt this just accelerating mathematical progress in the 21st century? Are breakthroughs harder now? Does the internet and peer review journals provide too much noise? Or is it simply that progress is so fast now we don't think of these things as extraordinary ?
A guy I knew used to say that this is why computer science is so much easier to learn than number theory. It's thousands of years of number theory to learn vs a few dozen years of computer science. On top of that, the recent historical explosion of computer science is also accompanied by perhaps an even larger explosion of number theory.
Nitpick: It would be even nicer if the images on the main page were annotated with the person's name. I recognized e.g. Einstein and Turing, but who are all the others?
If we consider the topical chronology/lineage stemming from Euclid's Elements, and how much of our world relies on Mathematics, one could argue that it is the most influential book ever written.
It doesn't seem to mention the works of the Indian mathematicians from Kerala, India on calculus. I think their work predated that of Newton and Leibniz.
> Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).[2] However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.
There is a network of artificial tunnels in Egypt called the Serapeum of Saqqara. It contains at least 24 large boxes made from single pieces of solid granite, allegedly moved there from a very interesting granite quarry several hundred miles away, then hollowed out to a mirror finish and a precision of a few ten-thousandths of an inch as tested by a precision machinist with a precision toolmaker's square who is also an engineer. After observing them, the engineer went on to ask the largest US companies in such business if they could make one. Only one responded and the answer was interesting. The place is real and the boxes, some as much as 90 tons (or the weight of more than fifty Prius cars) are not going anywhere. It's not in the books, nor is anything that explains it in the timeline but you can see it in person and test if for yourself. This place is only one of many that, by their nature and existence, question the conventional timeline of history
There is also a mathematician that, with some help from others, is doing a rigorous mathematical review of the conventional timeline of history and apparently there are some major problems with what has previously been accepted, written and taught. Does anyone know who I am referring to here?
How many people here are aware of the Serapeum mentioned above? Where does that level of precision work fit in with the timeline?
The qualities of the Saqqara boxes are entirely consistent with what we know about the mathematical and engineering sophistication of ancient Egyptians. There is nothing well-verified "not in the books" that upends the "conventional timeline". The only people claiming otherwise are Discovery Channel nutcase types who want you to believe, without real evidence, often with fake evidence, and always with fatuous reasoning, that angels and aliens intervened in ancient human history.
I'm sorry that I have given you the impression that I or others like myself who discuss these questions believe in ancient aliens. I realize that this is a topic of confusion for many "debunkers". I don't accept belief as a valuable tool the process of discovering the truth, nor do I deem it necessary in the practice of science. Unbiased observation, free from belief is important.
And denotative language seems to be more effective than connotative language, at least as far as these kings of arguments are concerned. Logical argument is better than the disparagement of persons and groups when arguing and making a point. And staying on topic is also a good thing.
There is a growing discussion regarding belief systems within academia and various conventional sciences such as archeology. This seems to be a problem and if it is not cleaned up fairly quickly and replaced with something more logical, evidence-based, reality-accepting and legitimately scientific, the term "pseudoscience" will likely be used against those coming from the conventional side of these topics.
I don't see anything relevant from your link that explains the precision of the boxes I mentioned above. What do you know about ancient Egyptian mathematics and engineering that relates to this subject? I have read the entire page you linked to, so forgive me in advance for missing it.
It’s also not surprising that they could create a flat surface or angles that are exactly-ish 90 degrees. The Egyptians boast some of the earliest known texts on geometry, like the Rhind Papyrus (from around 1650 BCE) and the Moscow papyrus (from about 1850 BCE). The latter papyrus indicates that the Egyptians could approximate pi (as 3.16049) and find the volume of a truncated pyramid. It stands to reason that 500 years later, they would be able to carve a flat surface and make a corner of exactly-ish 90 degrees.
If you look at marble statues, a square box is hardly amazing in comparison. "a few ten-thousandths of an inch" - close to micro meter precision - sounds almost like exaggeration, but some type of stone might just split in a very planar way.
> "a few ten-thousandths of an inch" - close to micro meter precision - sounds almost like exaggeration, but some type of stone might just split in a very planar way.
Not an exaggeration, just an extremely long time spent hand grinding/polishing with fine grit tools/paste.
Yes, there are some amazing marble statues. Have you seen the one where there is a fishing net cut from marble? Or the twins from Russia - two identical statues except for some obvious clumps of hair of hair, as if an image was taken at different times, in the breeze, and an artist or machine reproduced the statue from the image. I am assuming that most marble statues are at least an order of magnitude less precise than the granite box. Granite, by the way, is composed of different materials, such as, for example, feldspar and quartz. It doesn't break along a plane. Yes, a micrometer is .0001". Calipers, on the other hand, often only measure to .001" and would not be able to measure anything this precise.
I was referring to the precision of the boxes, not Brien Forester's quote about it, and more specifically the surface that is mirror polished to several ten-thousandths of an inch as estimated with a precision straight edge and toolmaker's square by someone who was a precision machinist, engineer, "master craftsman" (member of a professional guild), and a member of Mensa. These tools are so precise that if you drop them on the ground, handle them the wrong way or they end up in untrusted hands, they have to be re-calibrated and verified.
I apologize if this is an order of magnitude beyond most peoples' understanding and experience. It's probably not your occupation, so please don't take it personally. As an example of what this degree of precision is, consider a thin hair which is about four thousandths of an inch in diameter. If you slice that diameter up ten times, you will then have something that is as small as or at least reasonably approaching this measurement.
The achievements of past cultures are often so impressive. I have seen ancient Egyptian artifacts (King Tut's exhibit) and on the same day got to see some ancient Mayan and very old Japanese Art. All of it interesting and quite amazing (especially the intricacy and exactness of the admittedly much more recent Japanese Art).
To keep these ancient achievements in perspective though I recall working at Texas Instruments in the 70's (doing real-time process control programming). One of my friends there related to me his experience in giving a job interview, years earlier, to a young machinist, classically trained in Germany. The job seeker brought his final project as a sample of his work. A highly polished, precisely square, solid cube of polished steel, or so it appeared to my friend. It had all been made with hand tools, no fancy milling or grinding machines.
My friend marveled at it, but then the young man said "watch this" and proceeded to slide the upper half of the solid cube on a dovetailed channel that connected the upper half of the cube to the lower half. My friend, a mechanical engineer, said he had never seen anything like it; it was so carefully made that the channel was invisible to him when the cube was squared.
The main shaping tool used in the construction of this cube was simply a file.
Please anyone, what is best timeline metaphor you know of?
This is great stuff, but I immediately want to interactively browse through the information. For different parts be able to dive deeper, see high level views, different categories, taxonomies, links to related topics.
Not trying to reinvent Wikipedia or hypermedia. But specifically to timelines with a lot of density, it seems there ought to be some nice user interface that could be leveraged for these scenarios.
Too bad it ends at 2000, though I understand that 17 years ago may still be too soon to know what should count as history. Does anybody have a suggestion of what they think will eventually be included in such a timeline covering the last 20 years if it were made in 2100?
I'm surprised Gromov and Thurston's contributions to geometry in the 70s-00s aren't emphasized more. Some things I think should be on there from my admittedly limited point of view:
- 1982/87: Yao and then later Goldreich-Micali-Wigderson prove the "fundamental theorem of cryptography", which essentially states that any computational functionality can be achieved "efficiently" and securely. I.e., given k parties with one input each and any function F of k inputs and k outputs, the parties can communicate so that each learns their output of the function on all the inputs, and nothing else.
- 2003: Perelman's proof of the geometrization conjecture.
I made a top-level parent comment about a result from March 2017. Otherwise, things that either will become part of history, or suggest directions for work that will:
* Green-Tao theorem, and lots of other work involving Terry Tao
* Yitang Zhang's work on prime gaps, and subsequent improvements
* Bhargava and collaborators' work on elliptic curves
* The introduction of algebraic topology methods into algebraic geometry (aka "simplicial"/"derived" stuff, see Jacob Lurie)
>Viète writes In artem analyticam isagoge (Introduction to the analytical art), using letters as symbols for quantities, both known and unknown. He uses vowels for the unknowns and consonants for known quantities. Descartes, later, introduces the use of letters x, y ... at the end of the alphabet for unknowns.
This may be my favorite point in the history of mathematics. Using placeholders for things (variables) and efficient notation makes reasoning easy. It's such an obvious thing now, but I think it's amazing.
I wish I knew St Andrews was home to a cool mathematics department and the progenitor of Idris when I was there, filling my body with mind-altering chemicals.
> Hironaka solves a major problem concerning the resolution of singularities on an algebraic variety.
Essentially, sometimes the varieties (which are geometric objects like curves, surfaces, etc.) studied in algebraic geometry are singular: they might have singularities, like nasty self-crossings or sharp edges. Hironaka's result lets you take a "bad" variety and "resolve its singularities", giving you a good (nonsingular) variety which you can work with instead.
This is in "characteristic zero", i.e. over fields like the real or complex numbers. We also have fields of positive characteristic, e.g. the integers modulo any prime number. Over such fields, I understand that this is a much harder problem to solve.
The aforesaid Heisuke Hironaka is 86 now.
In March of this year, he published a (purported) proof of resolution of singularities in positive characteristic.
They are much closer to the curves and solids we're familiar with from everyday experience. For example: you have well-defined tangents at every point for a smooth curve, but a self-intersecting curve doesn't. So many techniques and a lot of intuition carries over into the abstract algebraic setting.
More generally, algebraic geometry is a central field of math because it is, basically, about solving polynomial equations. It's one of the most "well-connected" fields of mathematics today. Number theory, differential geometry, complex geometry, even differential equations: all these fields benefit from their interactions with modern algebraic geometry. (Also biology, I've heard. Lior Pachter is a name I remember in this connection.)
These days I am working on trying to understand the Fourier transform (1807). It is great fun. I finally understand the equation and how it works. Pure beauty. Now I am in the process to use it in practice. I am planning to write about it and perhaps write some visualizations to help others understand.
This post made me think about a crazy idea I had. I wanted to write periodic posts to talk about the work of all these great mathematicians. I also wanted to have a place where people could buy gear (t-shirts). The same way people is very proud of wearing a sports guy t-shirt (Lebron 23), I'd love to see people wearing t-shirts with mathematicians names (Fourier 10). It's crazy I know.
Also, what are your favorite books/resources about the history of Mathematics? Something with similar contents to this resource but perhaps connecting the different discoveries.
An interesting book, written by an interesting group, is "Who is Fourier". They seem to be mainly undergraduates in Japan who learn and teach each other new languages. They took on a similar task to explain Fourier Transforms to each other. Their motivation was to understand the mathematics of languages.
57 comments
[ 4.6 ms ] story [ 127 ms ] threadSource: Pure Mathematics Hons BSc, St Andrews - I took the St Andrews "History of Mathematics" course this site's information is used in.
*For a given value of 'mathematician'. And 'alive', I guess.
And then the mathematics that happened around ~1940.. Kolmogorov, Fisher, Poincaré, Gödel, Von Neumann, Church, Turing...
Just crazy.
The first is the stuff you do in school. Always good to have context, which sadly there's little time for.
The second is fundamentals and computing stuff, and stats. There's a good chance you'll have to use a computer to calculate things over large datasets sometime in your life.
Crypto is interesting as well.
To drop a few names, we had Grothendieck, Serre, Deligne, Wiles, Illusie, (the elder) Artin ... and it's continued to this day. Wiles' proof of FLT built upon this beautiful synthesis of ideas, for example.
1.) What, if any, of mathematics has been invented concurrently in different regions (say within 10-20 years) without collaboration or knowledge of other derived work?
2.) To what extent is the rate of mathematical development associated with war or advances in knowledge sharing? Be it radio, printing press, persuing weapons development, crypto, or mass migration (eg post ww2).
I think 1.) comes about via natural scientific progress. Some mathematical ideas are dependent on the right set of tools being invented or environmental necessity. How many people independently came up with ballastics equations once soldiers starting flinging big rocks at each other?
The 2.) Question is very interesting to me. What is the network effect of mathematicians sharing ideas? What happens when the 10 smartest mathematicians in the world end up all working at the same elite university? Why isnt this just accelerating mathematical progress in the 21st century? Are breakthroughs harder now? Does the internet and peer review journals provide too much noise? Or is it simply that progress is so fast now we don't think of these things as extraordinary ?
Edit: In the end there is no difference, however. Both are structural sciences. I guess you meant the maths in a CS degree.
Nitpick: It would be even nicer if the images on the main page were annotated with the person's name. I recognized e.g. Einstein and Turing, but who are all the others?
> Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).[2] However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.
There is also a mathematician that, with some help from others, is doing a rigorous mathematical review of the conventional timeline of history and apparently there are some major problems with what has previously been accepted, written and taught. Does anyone know who I am referring to here?
How many people here are aware of the Serapeum mentioned above? Where does that level of precision work fit in with the timeline?
https://nathandickey.wordpress.com/2014/02/03/demythologizin...
And denotative language seems to be more effective than connotative language, at least as far as these kings of arguments are concerned. Logical argument is better than the disparagement of persons and groups when arguing and making a point. And staying on topic is also a good thing.
There is a growing discussion regarding belief systems within academia and various conventional sciences such as archeology. This seems to be a problem and if it is not cleaned up fairly quickly and replaced with something more logical, evidence-based, reality-accepting and legitimately scientific, the term "pseudoscience" will likely be used against those coming from the conventional side of these topics.
I don't see anything relevant from your link that explains the precision of the boxes I mentioned above. What do you know about ancient Egyptian mathematics and engineering that relates to this subject? I have read the entire page you linked to, so forgive me in advance for missing it.
It’s also not surprising that they could create a flat surface or angles that are exactly-ish 90 degrees. The Egyptians boast some of the earliest known texts on geometry, like the Rhind Papyrus (from around 1650 BCE) and the Moscow papyrus (from about 1850 BCE). The latter papyrus indicates that the Egyptians could approximate pi (as 3.16049) and find the volume of a truncated pyramid. It stands to reason that 500 years later, they would be able to carve a flat surface and make a corner of exactly-ish 90 degrees.
Not an exaggeration, just an extremely long time spent hand grinding/polishing with fine grit tools/paste.
I apologize if this is an order of magnitude beyond most peoples' understanding and experience. It's probably not your occupation, so please don't take it personally. As an example of what this degree of precision is, consider a thin hair which is about four thousandths of an inch in diameter. If you slice that diameter up ten times, you will then have something that is as small as or at least reasonably approaching this measurement.
To keep these ancient achievements in perspective though I recall working at Texas Instruments in the 70's (doing real-time process control programming). One of my friends there related to me his experience in giving a job interview, years earlier, to a young machinist, classically trained in Germany. The job seeker brought his final project as a sample of his work. A highly polished, precisely square, solid cube of polished steel, or so it appeared to my friend. It had all been made with hand tools, no fancy milling or grinding machines.
My friend marveled at it, but then the young man said "watch this" and proceeded to slide the upper half of the solid cube on a dovetailed channel that connected the upper half of the cube to the lower half. My friend, a mechanical engineer, said he had never seen anything like it; it was so carefully made that the channel was invisible to him when the cube was squared. The main shaping tool used in the construction of this cube was simply a file.
You can find some of it here in this other timeline: http://superstringtheory.com/history/history1.html
e.g: Hypatia, Galileo, Kepler, Copernicus... and many more.
Even debates about heliocentrism weren't interfering with math.
This is great stuff, but I immediately want to interactively browse through the information. For different parts be able to dive deeper, see high level views, different categories, taxonomies, links to related topics.
Not trying to reinvent Wikipedia or hypermedia. But specifically to timelines with a lot of density, it seems there ought to be some nice user interface that could be leveraged for these scenarios.
Will review with the kids!
Not having this context in mathematics teaching is criminal.
- 1982/87: Yao and then later Goldreich-Micali-Wigderson prove the "fundamental theorem of cryptography", which essentially states that any computational functionality can be achieved "efficiently" and securely. I.e., given k parties with one input each and any function F of k inputs and k outputs, the parties can communicate so that each learns their output of the function on all the inputs, and nothing else.
- 2003: Perelman's proof of the geometrization conjecture.
* Green-Tao theorem, and lots of other work involving Terry Tao
* Yitang Zhang's work on prime gaps, and subsequent improvements
* Bhargava and collaborators' work on elliptic curves
* The introduction of algebraic topology methods into algebraic geometry (aka "simplicial"/"derived" stuff, see Jacob Lurie)
* Homotopy type theory
* Mochizuki/ABC stuff
* Peter Scholze's work in arithmetic geometry
http://history.takethejuice.com/output/MTQ5Njk0MDc1NC1NYXRoZ...
>Viète writes In artem analyticam isagoge (Introduction to the analytical art), using letters as symbols for quantities, both known and unknown. He uses vowels for the unknowns and consonants for known quantities. Descartes, later, introduces the use of letters x, y ... at the end of the alphabet for unknowns.
This may be my favorite point in the history of mathematics. Using placeholders for things (variables) and efficient notation makes reasoning easy. It's such an obvious thing now, but I think it's amazing.
1991 Quidong Wang finds infinite series solutions to the n-body problem (with minor exceptions).
The name is Qiudong Wang.
> Hironaka solves a major problem concerning the resolution of singularities on an algebraic variety.
Essentially, sometimes the varieties (which are geometric objects like curves, surfaces, etc.) studied in algebraic geometry are singular: they might have singularities, like nasty self-crossings or sharp edges. Hironaka's result lets you take a "bad" variety and "resolve its singularities", giving you a good (nonsingular) variety which you can work with instead.
This is in "characteristic zero", i.e. over fields like the real or complex numbers. We also have fields of positive characteristic, e.g. the integers modulo any prime number. Over such fields, I understand that this is a much harder problem to solve.
The aforesaid Heisuke Hironaka is 86 now.
In March of this year, he published a (purported) proof of resolution of singularities in positive characteristic.
The /r/math thread has some good explanations.
https://www.reddit.com/r/math/comments/6aqwbo/hironaka_publi...
what can you actually do with these "smoother" varieties?
More generally, algebraic geometry is a central field of math because it is, basically, about solving polynomial equations. It's one of the most "well-connected" fields of mathematics today. Number theory, differential geometry, complex geometry, even differential equations: all these fields benefit from their interactions with modern algebraic geometry. (Also biology, I've heard. Lior Pachter is a name I remember in this connection.)
These days I am working on trying to understand the Fourier transform (1807). It is great fun. I finally understand the equation and how it works. Pure beauty. Now I am in the process to use it in practice. I am planning to write about it and perhaps write some visualizations to help others understand.
This post made me think about a crazy idea I had. I wanted to write periodic posts to talk about the work of all these great mathematicians. I also wanted to have a place where people could buy gear (t-shirts). The same way people is very proud of wearing a sports guy t-shirt (Lebron 23), I'd love to see people wearing t-shirts with mathematicians names (Fourier 10). It's crazy I know.
My OCD wishes they would have said Rivest, Shamir, and Adleman like the algorithm is named after