When you first study mathematics at undergraduate and early post-grad level there is a sense of being overwhelmed with how on earth anyone figured this out. When you read the messy history of maths, and understand it is an organic, growing field, you feel a little less like an imposter struggling to understand how anyone could've come up with this.
Reading these books (primarily as a software engineer), made me feel better about not immediately getting certain concepts, because it's likely the people these theorems are named after didn't get it either, to begin with. They refined it, they collaborated (like a pull request almost) and eventually everything got very neatly packaged up into a set of theorems. Mathematics is rarely taught in that way, I wish more of the "human" aspect was part of the pedagogical process. I think it might temper some of the fear people have.
"Cambridge University History of Mathematics Society
presents an illegal, immoral, and fattening lecture series,
without the endorsement, approval, or approbation of University authorities on:
HISTORY OF SCIENCE IN ANTIQUITY"
Gives me a warm feeling. Here's one of the lectures to get a flavor:
I cannot recommend enough the podcast Opinionated History of Mathematics by Viktor Blåsjö
What's great about it is that it helps put you in the shoes of the people solving the problems at these times. So it secretly teaches you how to solve problems. He also attacks some of the claims of some historians which are a bit obtuse. The arguments are really well founded and if you're wondering how ancients solved extremely complex math while never inventing the tools you'd think are needed to solve them, then this will answer a lot of that. I think it'll make you see a lot of problems in a different light. It also is just a lot of fun and can be pretty funny.
I took a look and was really turned off by the first episode which is all about dunking on Galileo and what a bad mathematician he was, and generally showing tremendous hostility to anything that isn't math. You can read it for yourself here: https://intellectualmathematics.com/blog/galileo-bad-archime...
The syllogism appears to be:
1. All good science is mathematical
2. Anyone intelligent is very good at math
3. Galileo was not very good at math
4. Therefore Galileo was dumb and did bad science.
Stick with it. I think you are not quite getting the argument he is making. I don't think he's saying that Galileo didn't do anything meaningful, but he is saying that Galileo is getting more credit than he deserves.
> This is why Galileo is the idol of the humanists and the bane of the mathematicians. The philosophers say he invented modern science; the mathematicians that he’s a poor man’s Archimedes. The issue cuts much deeper than merely allotting credit to one century rather than another. Much more than a question of the detailed chronology of obscure scientific facts, it is a question of worldview and how one should approach and understand history.
I'm not exactly sure what the problem is. There are plenty of people who were bad at science that also made great contributions. There are also plenty of people who were not great contributors who also were able to establish a big name for themselves. I'll stay out of politics for the most famous current one, but I think I'm safe to point out the age old Woz vs Jobs thing people have talked about.
Viktor also does bring Tycho Brahe, which I think people are much more comfortable dunking on now a days. But in his time he lived it up in the limelite. He held many parties and was well known for his intelligence and brilliance, by the public. Brahe was known by all, but now he's just a footnote and an impedance to Kepler.
Do you not think it is true that some peoples' stories get embellished? Do you not think it is true that some are greatly so? Do you not think history gets retold to make better stories? Do you not think people who have made great contributions have been ignored? Certainly in the long history all this has happened. Do they not deserve credit? If they do, then the unfortunate reality is that this must be "taken" from those that it was misattributed to. Especially in the progression of knowledge where we over attribute contributions to those that cross some "finish line." Or in the words of Einstein: I stand on the shoulders of giants.
> I say that the traditional view of the “Galilean” scientific revolution is not only historically wrong but fundamentally inconsistent with the nature of mathematical thought. When I say Galileo boo, Archimedes yay, my point is not who was the “first” or who should “get credit” for this or that. That’s not so interesting. But Galileo is a window into more important things.
What is the relation between mathematics and science? Was mathematics before Galileo a technocratic enterprise? Compartmentalised, limited to certain computational tasks, blind to its own potential? Was mathematics stuck in that ditch until it was liberated by a “conceptual” breakthrough from without, so to speak; from philosophy? Or was mathematics always an expansive, empirically informed, interconnected study of all quantifiable aspects of the world?
The latter, of course, if you ask me. In any case, Galileo is ground zero for grappling with these questions. And that is why we study him.
Edit:
I think there is also a relevant recent HN post as well[0] and I'll point out a comment of mine[1]. The mathematics is a necessary component for the explainability, for the causality. These are the essence of physics and science, and the only language we really have to express them through is the language of math. I am biased though, you can check my comment history as I make similar arguments about our approach to machine learning.[2] And I do think we should understand these things. And in my reply to
jebarker you'll find a reason for why I believe understanding the history, in an accurate form, is so important for learning how to move forward.
> Do you not think it is true that some peoples' stories get embellished? Do you not think it is true that some are greatly so? Do you not think history gets retold to make better stories? Do you not think people who have made great contributions have been ignored? Certainly in the long history all this has happened. Do they not deserve credit? If they do, then the unfortunate reality is that this must be "taken" from those that it was misattributed to.
I'm really not particularly interested in history as a process of attributing credit. Galileo was hugely influential, whether that influence is 'deserved' or not, I mean, I don't care I guess? It's sort of like pointing out that vikings visited North America long before Columbus. Okay, but so what? They went there, then left or died, and had no influence on the course of history. I also don't see why making fun of his mathematical abilities somehow redresses the "harm" of his unfair reputation, since everyone involved is very long dead.
> The issue comes down to this: Do great minds think alike? I say they do. I say there is a spiritual unity of scientific thought from ancient to modern times. I say that what is obvious to us was obvious to the Greeks. I say it is ludicrous to think that generations of Greek mathematical geniuses of the first order, with their extensively documented interest science, all somehow failed to conceive basic principles of scientific method. I say these things because I can feel it in my bones. I say these things because I have spent my life in mathematics departments and experienced so many times the profound sense of thinking exactly alike with another person. Young or old, student or professor, when we talk about mathematics our minds are one. Mathematics has this power, to make brethren of us all.
The author claims that mathematicians form a trans-historical class of superior minds, and there was no scientific revolution because mathematicians were already doing all the things that historians claim emerged in the scientific revolution. This view has a huge historical problem with explaining why these genius mathematicians came up with almost no interesting new ideas in physics from ca. 200 BC to ca. 1600 AD.
For calculus history, one of my earliest books I read was "Calculus wars" (https://www.amazon.com/Calculus-Wars-Jason-Socrates-Bardi/dp...). This also is a essentially book about the Newton and G W Leibneiz. But, it is pretty much a deep dive into life of these two men. Highly recommended.
When it comes to mathematical concepts- to understand everything broadly, connected to its context, and to understant "what's the point?", nothing tops this book- Mathematics: Its Content, Methods and Meaning (3 Volumes in One) by Aleksandrov, Kolmogorov, and Lavrentev [0].
I laughed when I got to the end of this wall of text, full of “ahistorical fantasies” and “what the fuck moment,” and read: “Despite my negative comment about some points in the book, I would actually recommend it as a reasonably priced, mostly accurate, introduction to the history of mathematics. … A good jumping off point for somebody developing an interest in the discipline.”
_By the 9th century A.D., the Indians had made a conceptual leap that ranks as one of the most important mathematical events of all time. They had begun to recognise sunya_
Bit misleading as sunya or zero was invented in india, not recognised, there is a difference.
If you're interested in this sort of stuff I highly recommend Stillwell's "Mathematics and it's History" (https://link.springer.com/book/10.1007/978-1-4419-6053-5) - it's a wonderful mix of quite low level explicit mathematics with contextual history; along with Stewarts "Concept of Mathematics" (https://archive.org/details/ConceptsofmodernmathematicsStewa...).
When you first study mathematics at undergraduate and early post-grad level there is a sense of being overwhelmed with how on earth anyone figured this out. When you read the messy history of maths, and understand it is an organic, growing field, you feel a little less like an imposter struggling to understand how anyone could've come up with this.
Reading these books (primarily as a software engineer), made me feel better about not immediately getting certain concepts, because it's likely the people these theorems are named after didn't get it either, to begin with. They refined it, they collaborated (like a pull request almost) and eventually everything got very neatly packaged up into a set of theorems. Mathematics is rarely taught in that way, I wish more of the "human" aspect was part of the pedagogical process. I think it might temper some of the fear people have.
The Cambridge University History of Maths society is a fantastic resource - almost all the lectures are live streamed: https://hom.soc.srcf.net/
This looks like an absolute delight:
"Cambridge University History of Mathematics Society presents an illegal, immoral, and fattening lecture series, without the endorsement, approval, or approbation of University authorities on: HISTORY OF SCIENCE IN ANTIQUITY"
Gives me a warm feeling. Here's one of the lectures to get a flavor:
https://drive.google.com/drive/folders/1wkQmFPcrkLM9oJk1G7jF...
Contains this banger of a quote:
"The Greeks were very poor at using YouTube."
Next lecture will be delivered later this afternoon :)
Also The World of Mathematics (1956) by James R. Newman.
An oldie but goldie of the multi-brick/bathroom doorstop variety is Kline's Mathematical Thought from Ancient to Modern Time.
What does multi brick/bathroom variety mean?
It means it's a big fat book well suited for browsing while on the can. It's a genre.
Famously, take note of Hilbert asking at a conference, "What is a Hilbert space?"
I cannot recommend enough the podcast Opinionated History of Mathematics by Viktor Blåsjö
What's great about it is that it helps put you in the shoes of the people solving the problems at these times. So it secretly teaches you how to solve problems. He also attacks some of the claims of some historians which are a bit obtuse. The arguments are really well founded and if you're wondering how ancients solved extremely complex math while never inventing the tools you'd think are needed to solve them, then this will answer a lot of that. I think it'll make you see a lot of problems in a different light. It also is just a lot of fun and can be pretty funny.
https://intellectualmathematics.com/opinionated-history-of-m...
I took a look and was really turned off by the first episode which is all about dunking on Galileo and what a bad mathematician he was, and generally showing tremendous hostility to anything that isn't math. You can read it for yourself here: https://intellectualmathematics.com/blog/galileo-bad-archime...
The syllogism appears to be:
1. All good science is mathematical
2. Anyone intelligent is very good at math
3. Galileo was not very good at math
4. Therefore Galileo was dumb and did bad science.
Stick with it. I think you are not quite getting the argument he is making. I don't think he's saying that Galileo didn't do anything meaningful, but he is saying that Galileo is getting more credit than he deserves.
I'm not exactly sure what the problem is. There are plenty of people who were bad at science that also made great contributions. There are also plenty of people who were not great contributors who also were able to establish a big name for themselves. I'll stay out of politics for the most famous current one, but I think I'm safe to point out the age old Woz vs Jobs thing people have talked about.Viktor also does bring Tycho Brahe, which I think people are much more comfortable dunking on now a days. But in his time he lived it up in the limelite. He held many parties and was well known for his intelligence and brilliance, by the public. Brahe was known by all, but now he's just a footnote and an impedance to Kepler.
Do you not think it is true that some peoples' stories get embellished? Do you not think it is true that some are greatly so? Do you not think history gets retold to make better stories? Do you not think people who have made great contributions have been ignored? Certainly in the long history all this has happened. Do they not deserve credit? If they do, then the unfortunate reality is that this must be "taken" from those that it was misattributed to. Especially in the progression of knowledge where we over attribute contributions to those that cross some "finish line." Or in the words of Einstein: I stand on the shoulders of giants.
Edit:I think there is also a relevant recent HN post as well[0] and I'll point out a comment of mine[1]. The mathematics is a necessary component for the explainability, for the causality. These are the essence of physics and science, and the only language we really have to express them through is the language of math. I am biased though, you can check my comment history as I make similar arguments about our approach to machine learning.[2] And I do think we should understand these things. And in my reply to jebarker you'll find a reason for why I believe understanding the history, in an accurate form, is so important for learning how to move forward.
[0] https://news.ycombinator.com/item?id=43344703
[1] https://news.ycombinator.com/item?id=43349216
[2] https://news.ycombinator.com/item?id=43348954
> Do you not think it is true that some peoples' stories get embellished? Do you not think it is true that some are greatly so? Do you not think history gets retold to make better stories? Do you not think people who have made great contributions have been ignored? Certainly in the long history all this has happened. Do they not deserve credit? If they do, then the unfortunate reality is that this must be "taken" from those that it was misattributed to.
I'm really not particularly interested in history as a process of attributing credit. Galileo was hugely influential, whether that influence is 'deserved' or not, I mean, I don't care I guess? It's sort of like pointing out that vikings visited North America long before Columbus. Okay, but so what? They went there, then left or died, and had no influence on the course of history. I also don't see why making fun of his mathematical abilities somehow redresses the "harm" of his unfair reputation, since everyone involved is very long dead.
> The issue comes down to this: Do great minds think alike? I say they do. I say there is a spiritual unity of scientific thought from ancient to modern times. I say that what is obvious to us was obvious to the Greeks. I say it is ludicrous to think that generations of Greek mathematical geniuses of the first order, with their extensively documented interest science, all somehow failed to conceive basic principles of scientific method. I say these things because I can feel it in my bones. I say these things because I have spent my life in mathematics departments and experienced so many times the profound sense of thinking exactly alike with another person. Young or old, student or professor, when we talk about mathematics our minds are one. Mathematics has this power, to make brethren of us all.
The author claims that mathematicians form a trans-historical class of superior minds, and there was no scientific revolution because mathematicians were already doing all the things that historians claim emerged in the scientific revolution. This view has a huge historical problem with explaining why these genius mathematicians came up with almost no interesting new ideas in physics from ca. 200 BC to ca. 1600 AD.
Woah thank you for sharing this! Absolutely riveting list of topics.
It is one of my all time favorite podcasts
He's also a nice dude. I reached out to him on Twitter and he happily responds. I'm sure he likes hearing what people think
For calculus history, one of my earliest books I read was "Calculus wars" (https://www.amazon.com/Calculus-Wars-Jason-Socrates-Bardi/dp...). This also is a essentially book about the Newton and G W Leibneiz. But, it is pretty much a deep dive into life of these two men. Highly recommended.
Another great book is "Infinite Powers" by Steven Strogatz (https://www.amazon.com/Infinite-Powers-Calculus-Reveals-Univ...).
Sidenote today (or tomorrow depending where you are), on Friday 14th March the United Nations observes 'International Day of Mathematics'
https://www.unesco.org/en/days/mathematics
When it comes to mathematical concepts- to understand everything broadly, connected to its context, and to understant "what's the point?", nothing tops this book- Mathematics: Its Content, Methods and Meaning (3 Volumes in One) by Aleksandrov, Kolmogorov, and Lavrentev [0].
[0]: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...
I laughed when I got to the end of this wall of text, full of “ahistorical fantasies” and “what the fuck moment,” and read: “Despite my negative comment about some points in the book, I would actually recommend it as a reasonably priced, mostly accurate, introduction to the history of mathematics. … A good jumping off point for somebody developing an interest in the discipline.”
_By the 9th century A.D., the Indians had made a conceptual leap that ranks as one of the most important mathematical events of all time. They had begun to recognise sunya_
Bit misleading as sunya or zero was invented in india, not recognised, there is a difference.