Other What are you reading now? (STEM only)

[Demystifier]

Hurray for Grothendieck, death to Bourbakianism!

Hurray for sets, death to abstract nonsense.

 

[Dragon27]

Hurray for sets, death to abstract nonsense.

The revolution has already won. Your vain hopes will never come to fruition.:)

 

[martinbn]

Hurray for Grothendieck, death to Bourbakianism!

This doesn't make sense. What Grothendieck was insisting on was to make the approach even more Bourbaki in style than it already was. It is not death to Bourbakianism, it is more long live, prosper and expand to Bourbakianism.

 

[MathematicalPhysicist]

Just started "Physics and Philosophy" by Heisenberg but the book @gleem posted sounds far more interesting! I'm going to go hunt it down in the bookstores.
Hurray for Grothendieck, death to Bourbakianism!

In a few years the trend will be to replace Category Theory with another theory.(if it hasn't already begun).

 

[Auto-Didact]

This doesn't make sense. What Grothendieck was insisting on was to make the approach even more Bourbaki in style than it already was. It is not death to Bourbakianism, it is more long live, prosper and expand to Bourbakianism.

Hey, I'll take mixing up co and contravariant functors over closet logicism any day!

 

[Demystifier]

In a few years the trend will be to replace Category Theory with another theory.(if it hasn't already begun).

Why do you think so?

Anyway, if category theory is abstract nonsense, then this new theory will be hyper-abstract utter nonsense.

 

[gleem]

This doesn't make sense. What Grothendieck was insisting on was to make the approach even more Bourbaki in style than it already was. It is not death to Bourbakianism, it is more long live, prosper and expand to Bourbakianism.

Yes. Grothendieck was the generalist's generalist and according to Aczel championed a trend toward increasing generality and abstraction in math. In fact he could not relate to examples for which most of us need for understanding.. In a seminar he mentioned something about prime numbers and a participant asked for an example. He said take 57 for example. which of course is not a prime number. 57 has become known as Grothendieck's prime.

 

[George Jones]

Anyway, if category theory is abstract nonsense, then this new theory will be hyper-abstract utter nonsense.

The book "Mathematical Physics" by Robert Geroch, which is in the orthogonal complement to most books with similar titles, starts with a brief introduction to category theory. This is a very nice broad introduction to some abstract maths, and a book on which I spent a fair bit of time 25 or 30 years ago.

A pure maths prof who taught me undergrad and grad courses in abstract algebra, representation theory, Lie algebras, etc. once said to me "Category theory should be functored out of existence."

 

[MathematicalPhysicist]

Why do you think so?

Anyway, if category theory is abstract nonsense, then this new theory will be hyper-abstract utter nonsense.

History tells me there's always a new foundations to maths.

 

[MathematicalPhysicist]

Why do you think so?

Anyway, if category theory is abstract nonsense, then this new theory will be hyper-abstract utter nonsense.

Not hyper, but super...

 

[MathematicalPhysicist]

In a seminar he mentioned something about prime numbers and a participant asked for an example. He said take 57 for example. which of course is not a prime number. 57 has become known as Grothendieck's prime.

makes you wonder what other mistakes there are in his general publications.
Also I am not sure I can construct a Grothendieck universe which is not trivial.
https://en.wikipedia.org/wiki/Grothendieck_universe

 

[gleem]

makes you wonder what other mistakes there are in his general publications.

That should not be a problems since he avoids specifics. I understand Feynman made a mistake too so maybe we should check his publications.

A pure maths prof who taught me undergrad and grad courses in abstract algebra, representation theory, Lie algebras, etc. once said to me "Category theory should be functored out of existence."

WRT the issues of too much generality for which many mathematicians find distasteful Topos, a category, is finding use in quantum field theory? . I bring this up because there seems to be a sense that Category Theory is too general to be useful.Disclaimer: I am only a lowly experimental physicist and cannot discuss these issues beyond very general observations.

 

[Auto-Didact]

In a few years the trend will be to replace Category Theory with another theory.(if it hasn't already begun).

As Poincaré said, 'fundamental principles are only conventions - adopted due to some convenience - and it is quite unreasonable to ask whether they are true or false as it is to ask whether the metric system is true or false.'

Yes. Grothendieck was the generalist's generalist and according to Aczel championed a trend toward increasing generality and abstraction in math. In fact he could not relate to examples for which most of us need for understanding.

As Weyl said, 'it cannot be denied that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the greater part of his towering edifice which he believed to be built of concrete blocks dissolve into mist before his eyes.'

A pure maths prof who taught me undergrad and grad courses in abstract algebra, representation theory, Lie algebras, etc. once said to me "Category theory should be functored out of existence."

As Feynman said, 'his mother probably never hugged him as a child... or perhaps she was overindulgent!'

 

[vanhees71]

Well I think that nearly anything is more illuminating than reading this particular book by Heisenberg... SCNR.

 

[vanhees71]

Why do you think so?

Anyway, if category theory is abstract nonsense, then this new theory will be hyper-abstract utter nonsense.

Well, I'd not say that Bourbakism is "abstract nonsense"; it's most probably not "nonsense" in any sense but an important step in the development of mathematics in terms of research!

The misunderstanding, however, is to take it as a textbook, which for sure it is not. It's a review on a level for researches, stripped of all sensical didactics. In my opinion the Bourbaki style of textbooks is even a disservice in the sense of textbook writing since it doesn't provide a real "working knowledge" of math, i.e., it doesn't tell the student about the heuristics of the subject, which is very important for a university-level textbook since the future researcher rather needs intuition to find new knowledge than an overformalized knowledge of the present or past status of his subject. An example are Dieudonne's analysis textbook, which is very Bourbakian in style. I've never understood, how you should be able to learn the subject from this dry exhibition ;-)).

 

[Auto-Didact]

Well, I'd not say that Bourbakism is "abstract nonsense"; it's most probably not "nonsense" in any sense but an important step in the development of mathematics in terms of research!

The misunderstanding, however, is to take it as a textbook, which for sure it is not. It's a review on a level for researches, stripped of all sensical didactics. In my opinion the Bourbaki style of textbooks is even a disservice in the sense of textbook writing since it doesn't provide a real "working knowledge" of math, i.e., it doesn't tell the student about the heuristics of the subject, which is very important for a university-level textbook since the future researcher rather needs intuition to find new knowledge than an overformalized knowledge of the present or past status of his subject. An example are Dieudonne's analysis textbook, which is very Bourbakian in style. I've never understood, how you should be able to learn the subject from this dry exhibition ;-)).

Couldn't have said it better.

 

[gleem]

An example are Dieudonne's analysis textbook, which is very Bourbakian in style. I've never understood, how you should be able to learn the subject from this dry exhibition

This should have been very predictable since Dieudonne was a founding member of the Bourbaki working group and the designated scribe for the Bourbaki works for some 25 years. The various works avoided any illustrations of figures or tables contributing to their "dryness". Apparently only the books on Lie groups and commutative algebra have figures due to the influence of Armand Borel. Initially Borel having read Bourbaki assumed the real authors where closed minded and cared only for abstraction and generality.but changed his mind when he began working with them. Paraphrasing comment he made in Notices of the American: Mathematical Society (1989): They knew so much and knew it so well,. even on a topic more familiar to me than to them their sharp questions gave me the impression that I had not really thought it through.

 

[Auto-Didact]

This should have been very predictable since Dieudonne was a founding member of the Bourbaki working group and the designated scribe for the Bourbaki works for some 25 years. The various works avoided any illustrations of figures or tables contributing to their "dryness". Apparently only the books on Lie groups and commutative algebra have figures due to the influence of Armand Borel. Initially Borel having read Bourbaki assumed the real authors where closed minded and cared only for abstraction and generality.but changed his mind when he began working with them. Paraphrasing comment he made in Notices of the American: Mathematical Society (1989): They knew so much and knew it so well,. even on a topic more familiar to me than to them their sharp questions gave me the impression that I had not really thought it through.

I don't doubt that at all either, I only doubt whether such extensions would be inherently conceptually interesting in terms of application (i.e. physics) as well or only in terms of mathematics.

 

[gleem]

I don't doubt that at all either, I only doubt whether such extensions would be inherently conceptually interesting in terms of application (i.e. physics) as well or only in terms of mathematics.

If you mean would the inclusions of examples and visual aids etc. be only interesting in terms of applications it would seem the answer is no. Borel needed figures for his understanding that is why he included them in his contributions. He deprecated his ability in deference to the founder of Bourbaki by stating that he is only a Swiss peasant and the Swiss character needs pictures. Bourbaki's works seemed to be typically criticized by mathematicians for their lack of examples.

 

[vanhees71]

That's precisely what I meant before. The Bourbaki books an some of the textbooks of the members of Bourbaki are closer to scientific research work, and without doubt excellent research work, but they are lousy as textbooks. I'm sure that all these brillant mathematicians didn't come to the results presented in the waybof these books but in creative acts of thinking. Of course at the end the finding must be formalized in this way to be true pure math. In a sense Bourbaki defined this level of abstract quality (at least for the math of the late 20th century).

 

[Auto-Didact]

That's precisely what I meant before. The Bourbaki books an some of the textbooks of the members of Bourbaki are closer to scientific research work, and without doubt excellent research work, but they are lousy as textbooks. I'm sure that all these brillant mathematicians didn't come to the results presented in the waybof these books but in creative acts of thinking. Of course at the end the finding must be formalized in this way to be true pure math.

Indeed, real open ended creative mathematics - i.e. pure mathematics in the classical sense - is always messy and conceptual, while technical definitions through rigourous axiomatic formalization almost only always come after the actual discovery has already taken place.

Formalism, a bastard of logicism, championed by Hilbert in the pure mathematics community started to drive away many of the greatest late 19th century pure mathematicians, from Poincaré - famously the last univeralist (generalist), because of his creative instead of rigorous mind - onwards towards physics and applied math. Both Poincaré and Hadamard wrote on this subject.

Formalism then, during the 20th century, came close to culmination in Bourbakianism, driving generalists almost fully into applied mathematics. This drive-away was in peak effect mid-century - during the time of Mandelbrot et al. - firmly making their contributions to pure mathematics to instead incorrectly be viewed as physics and applied mathematics.

Incidentally, the last great theoretical and mathematical physicists - Feynman, Wilson, Anderson, Dyson, Mandelbrot, 't Hooft and Penrose - all recognized and spoke out against these developments in mathematics, but very few listenend i.e. taking their warnings at face value as critiques of mathematics itself, when they were actually criticizing formalism and axiomatics.

In classical pure mathematics - and therefore in physics as well - formalism is useless in discovering novel concepts, because it already presupposes full completeness of theory; this is why formal pure mathematics is purely deductive opposed to classical pure mathematics. To paraphrase Atiyah and Weyl: Hilbert and his followers killed creative pure mathematics. Bourbaki however made things severely worse by imposing the formalist ideology on students as well through the rewriting of curricula and textbooks.

This caused a severe widening of the divorce between pure mathematics and physics, worsening extremely with the professionalization of academia and overspecialisation of the sciences. The love between physics and mathematics would only be rekindled somewhat late in the 20th century, for somewhat wrong reasons, i.e. in string theory. It is happening again though, but now between applied mathematics and physics - while the formalist scoffs at both.

In any case, it should be obvious why formalism does more harm in mathematics than good; it is a self-imposed censorship of the mind borne out of the idea that mathematics must be reducible to logic, axioms and deductive reasoning alone. This is also exactly why to the physicist - today seen as a non-mathematician by most mathematicians - axiomatics are at best an afterthought; its a shame that many physicists seem to have forgotten this.

 

[gleem]

Formalism, a bastard of logicism, championed by Hilbert in the pure mathematics community started to drive away many of the greatest late 19th century pure mathematicians, from Poincaré - famously the last univeralist (generalist), because of his creative instead of rigorous mind - onwards towards physics and applied math. Both Poincaré and Hadamard wrote on this subject.

Isn't it interesting that Hadamard was Weil's dissertation adviser.

Anyway because of this discussion I have become inclined to reread "Mathematics: Queen and Servant of Science" by Eric Templeton Bell (1951) which discusses the contributions of mathematics to scientific knowledge.and more about the math than the science.

 

[Demystifier]

The Bourbaki books an some of the textbooks of the members of Bourbaki are closer to scientific research work, and without doubt excellent research work, but they are lousy as textbooks.

I think the standard terminology in this case would be textbook vs scientific monograph. Bourbaki books are monographs, not textbooks.

 

[Auto-Didact]

Quantum Information and Coherence, 2014

Link here, but behind a paywall

 

[Auto-Didact]

"Social Network Analysis: Methods and Applications" by Wasserman and Faust, 1994.

I actually learned some very nice pure and applied mathematics from this book, among other things Galois lattices and a method to carry out principal component analysis on non-quantitative data.

It is a basically a book on applied graph theory/network theory for researchers and data scientists out in the field. It gives a perspective at all levels: from pure and applied mathematics, to scientific, to practice.

Despite describing mostly social networks, I would highly recommend this book to any student/researcher wanting to learn to use any kind of network analysis in practice.

 

[Jimster41]

@Auto-Didact And... thanks for the link here. Don't know how I missed this thread.

 

[romsofia]

"Essays in honor of the 60th birthday of Bryce S DeWitt" and "Quantum concepts in space and time".

I really like the writing styles of most of the papers from these two books. They feel more informal to me, which I enjoy. For anyone who enjoys fundamental physics, these two books are truly a treat.

 

[Demystifier]

N. Johnson, Simply Complexity
https://www.amazon.com/dp/1851686304/?tag=pfamazon01-20
Complexity theory explained at a popular-science level.

 

[MidgetDwarf]

Artin: Algebra. Really nice book. Author is very careful with his explanations, good problems, and very enjoyable read.

 

[BPHH85]

Just for fun, I'm reading "the theoretical minimum - what you need to know to start doing physics" by Susskind and Hrabovsky. It's fun to read lighter stuff before sleeping.

For less lighter stuff, I'm reading "Quantum confined laser devices" by P. Blood.