What Books Are You Currently Reading?

[Auto-Didact]

What do you like about it?

Too much to name, most importantly he makes explicit much of what often remains implicit in all areas of science and gives the clearest guide I've ever seen forward. The clarity begins with which Poincaré describes the psychological process of basic mathematical reasoning, laying bare clear differences between arithmetic reasoning and logical inference. From his exposition it is evident how this process shapes what mathematics is, from foundational questions, to basic definitions, to mathematician philosophies, to entire research programmes, down to whether one comes to find interest in these matters at all.

This exposition on mathematical thinking is threefold, namely by classifying his findings using Kantian terminology, he not only simply demarcates different schools of thought within mathematics, but also makes explicit the limitations which opposing viewpoints bring into mathematical theory with them and why, while at the same time making verifiable/falsifiable hypotheses about the actual psychology and sociology of mathematicians as well, explaining the naturally occurring differences in the choices of approaching and grasping subjects by different people.

The parts on physics are historically especially interesting to read as they are the thoughts of probably the greatest living physicist at the time of the cusp just as classical physics is becoming modern physics. Just reading only the first part of this book makes clear that Poincaré, truly was the last universalist, incorporating at the highest level mathematics, physics and philosophy in such a way not seen anymore anywhere since, especially not in todays age of specialisation. It is also very interesting to note that Feynman's Messenger Lectures on The Character of Physical Law pretty much seem to be to a large extent a dumbed down summary and extension of Poincaré's book.

I believe very much can be gained, not simply for mathematicians and scientists, but for any school child going into any direction, if they could step out of their time and join Poincaré to see all the popular schools of thinking while they were being developed and so then choose themselves instead of just getting a particular view rammed down the throat as is conventional. For example, I think an actual educational system, taught by pedagogically gifted teachers, based on Poincaré's book is capable of producing an entirely new generation of groundbreakingly novel interdisciplinary thinkers.

The hope is of course that this might exacerbate knowledge akin to the modern naturalistic view of network science in comparison with the classical purist view of graph theory, but then for all disciplines. Such a transition would be capable of enabling today's and tomorrow's generations of carrying on successfully into a world where automatisation is increasingly chipping away at tasks requiring human ingenuity without necessarily leaving anything interesting to do behind in its place.

 

[Auto-Didact]

What were some of the schools of mathematics in Poincare's day? What is the difference between arithmetic reasoning and logical inference?
Riemann was a neo-Kantian. His philosophical notes are in the form of antinomies. What was the effect of Kant upon science and mathematics?

Poincaré lived until 1912, and so was extremely aware of all three of the early 20th century schools which remain popular until this day. In particular, Hilbert's formalism and the logicism of Frege, Russell et al., are both views of which he was critical. In doing so, it is somewhat clear that he was one of the first intuitionists, the third major school championed a few years later by Brouwer and Weyl. Of special note however in the Foundation of Science and other works, is Poincaré's other own 'school', namely conventionalism with regard to geometry and science. Poincaré died just before the many paradoxes and results occurred leading to the crisis in the foundations of mathematics and therefore he also did not experience the full fruits of formalism and intuitionism; of course it goes without saying he was well aware of Platonism.

Regarding the difference between logical reasoning and arithmetic, I'd say start reading the book, this question is addressed in the first few chapters and I cannot explain it anywhere near as simple or as elegant as he does.

As for the effect of Kant on science and mathematics, this requires a whole other thread. All that can be said is that it was clearly profound influencing all thinkers up until Bohr, Einstein et al. who all left for America due to Nazi Germany, causing the shift of the intellectual centre of the world to America along with the rise of the instrumentalist view of science. This view was championed by particle physicists and was very successful until the seventies culminating in the Standard Model; along however came the rise of outspoken anti-philosophical attitudes and tendencies among scientists, eg. Feynman, Weinberg, Krauss and even the moderators of this board.

 

[lavinia]

Poincaré lived until 1912, and so was extremely aware of all three of the early 20th century schools which remain popular until this day. In particular, Hilbert's formalism and the logicism of Frege, Russell et al., are both views of which he was critical. In doing so, it is somewhat clear that he was one of the first intuitionists, the third major school championed a few years later by Brouwer and Weyl. Of special note however in the Foundation of Science and other works, is Poincaré's other own 'school', namely conventionalism with regard to geometry and science. Poincaré died just before the many paradoxes and results occurred leading to the crisis in the foundations of mathematics and therefore he also did not experience the full fruits of formalism and intuitionism; of course it goes without saying he was well aware of Platonism.

Regarding the difference between logical reasoning and arithmetic, I'd say start reading the book, this question is addressed in the first few chapters and I cannot explain it anywhere near as simple or as elegant as he does.

As for the effect of Kant on science and mathematics, this requires a whole other thread. All that can be said is that it was clearly profound influencing all thinkers up until Bohr, Einstein et al. who all left for America due to Nazi Germany, causing the shift of the intellectual centre of the world to America along with the rise of the instrumentalist view of science. This view was championed by particle physicists and was very successful until the seventies culminating in the Standard Model; along however came the rise of outspoken anti-philosophical attitudes and tendencies among scientists, eg. Feynman, Weinberg, Krauss and even the moderators of this board.

Very interesting. I am ignorant of the attempts to formalize reason. It will be interesting to learn about it.

In practice, mathematicians just see the truth somehow without deduction. It just hits them sort of the way a melody appears to a musician. Formal proof always seems to be an afterthought. I sat it on a topology course with Dennis Sullivan once, and he would call students to the board to demonstrate theorems. If the student started a logical deduction , he would yell out "That's not a proof!". He wanted the idea or the picture. That was the proof.

The composer Scriabin was asked how he had composed his fifth piano sonata. He said something like, 'I didn't compose it. I just wrote it down. " A mathematician supposedly once said,, "I thought of the theorem because it was right."

So is this sort of insight one of the things Poincare talks about or an I missing the boat?

 

[Astronuc]

I've been reading Jagdish Mehra's "The Beat of a Different Drummer," a biography of Richard Feynman (1994, Clarendon Press). A colleague at work gave me his collection of books on Feynman, including a bound copy of the Feynman Lectures on Physics (1989 Ed. of the 1964 publication).

His early experiences in high school and university have similarities with mine. His insights in STEM education are interesting.

I've read other books written or coauthored by Feynman, and I have other biographies to read.

 

[Auto-Didact]

Very interesting. I am ignorant of the attempts to formalize reason. It will be interesting to learn about it.

In practice, mathematicians just see the truth somehow without deduction. It just hits them sort of the way a melody appears to a musician. Formal proof always seems to be an afterthought. I sat it on a topology course with Dennis Sullivan once, and he would call students to the board to demonstrate theorems. If the student started a logical deduction , he would yell out "That's not a proof!". He wanted the idea or the picture. That was the proof.

The composer Scriabin was asked how he had composed his fifth piano sonata. He said something like, 'I didn't compose it. I just wrote it down. " A mathematician supposedly once said,, "I thought of the theorem because it was right."

So is this sort of insight one of the things Poincare talks about or an I missing the boat?

Only in passing, there is however another famous French contemporary mathematician of Poincaré, namely Jacques Hadamard, who does delve deeper into exactly what you are describing. He does this in a short book called 'Essay on the psychology of invention in the mathematical field'.

Poincaré also mentions these things, but it is not the main focus of what he explains in his book. The best way to describe what he does explain in this book is actually using the scientific method to validate and/or falsify certain specific hypotheses - i.e. popular ideas, philosophies and misconceptions that scientists and mathematicians (tend to) have - regarding the actual and ideal practice of mathematics and of mathematicians through actually practicing mathematics, analysing the results and comparing them with the hypotheses, and so discarding and creating new hypotheses if deemed necessary.

He also actually gives a general theory of how the practice of mathematics as a natural behavior in thinking creatures implicitly forms many aspects of mathematics (i.e. axioms and rules) which tend to be taken for granted and which tend to result in the development of certain branches of mathematics, in the form that we recognize them or otherwise.

Addendum: As an example, Poincaré's explanation of groups and their central importance in mathematics in the above manner is the best introduction to the subject I have ever read. It is exactly like reading an Insight article posted by the greatest mathematician of the late nineteenth/early twentieth century.