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Urbanomic/Sequence Press
Synthetic Philosophy of Contemporary Mathematics
A panoramic survey of the vast spectrum of modern and contemporary mathematics and the new philosophical possibilities they suggest. A panoramic survey of the vast spectrum of modern and contemporary mathematics and the new philosophical possibilities they suggest, this book gives the inquisitive non-specialist an insight into the conceptual transformations and intellectual orientations of modern and contemporary mathematics. The predominant analytic approach, with its focus on the formal, the elementary and the foundational, has effectively divorced philosophy from the real practice of mathematics and the profound conceptual shifts in the discipline over the last century. The first part discusses the specificity of modern (1830–1950) and contemporary (1950 to the present) mathematics, and reviews the failure of mainstream philosophy of mathematics to address this specificity. Building on the work of the few exceptional thinkers to have engaged with the “real mathematics” of their era (including Lautman, Deleuze, Badiou, de Lorenzo and Châtelet), Zalamea challenges philosophy's self-imposed ignorance of the “making of mathematics.” In the second part, thirteen detailed case studies examine the greatest creators in the field, mapping the central advances accomplished in mathematics over the last half-century, exploring in vivid detail the characteristic creative gestures of modern master Grothendieck and contemporary creators including Lawvere, Shelah, Connes, and Freyd. Drawing on these concrete examples, and oriented by a unique philosophical constellation (Peirce, Lautman, Merleau-Ponty), in the third part Zalamea sets out the program for a sophisticated new epistemology, one that will avail itself of the powerful conceptual instruments forged by the mathematical mind, but which have until now remained largely neglected by philosophers.
380 pages, Paperback
First published January 1, 2012
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Displaying 1 - 12 of 12 reviews
February 13, 2020
This is a pretty ambitious book, looking into the latest trends in mathematics and linking them with philosophical thought. The implications of much of these lines of thinking are not fully explored mainly because such recent mathematics is not fully understood by many.
Zalamea takes a superhuman tour of various approaches but doesn't dive too deeply into any of them. It's not particularly clear what philosophical implications can be drawn from such a synthesis, as people are still grappling with notions from calculus, Newton and relativity.
If there is any area that Zalamea has left untouched it is in the area of machine learning and a philosophy of statistics -- although that is not nearly as sexy as some of the high level concepts he throws out here for us.
If anything, philosophy, at least classical philosophy, left its aesthetics with Einstein and is ill-equipped to really deal with these forms of thinking. So these mathematics, if they should lead to philosophy, will have to do so on their own. Deleuze offers a little possibility, but towards the end, Zalamea's lack of understanding of postmodernism and post-structuralism and where the two interrelate is telling. This is not a huge criticism though, because most people cannot tell the difference between the two. Still, this book is an ambitious undertaking, and in that way, very rare in the forms of mathematical philosophy we see today, which as stated before, tends to stop before the end of the 19th century.
Zalamea takes a superhuman tour of various approaches but doesn't dive too deeply into any of them. It's not particularly clear what philosophical implications can be drawn from such a synthesis, as people are still grappling with notions from calculus, Newton and relativity.
If there is any area that Zalamea has left untouched it is in the area of machine learning and a philosophy of statistics -- although that is not nearly as sexy as some of the high level concepts he throws out here for us.
If anything, philosophy, at least classical philosophy, left its aesthetics with Einstein and is ill-equipped to really deal with these forms of thinking. So these mathematics, if they should lead to philosophy, will have to do so on their own. Deleuze offers a little possibility, but towards the end, Zalamea's lack of understanding of postmodernism and post-structuralism and where the two interrelate is telling. This is not a huge criticism though, because most people cannot tell the difference between the two. Still, this book is an ambitious undertaking, and in that way, very rare in the forms of mathematical philosophy we see today, which as stated before, tends to stop before the end of the 19th century.
January 11, 2023
Too many promising avenues to list. His construction of the 'archeal' realm is fairly unconvincing, as it involves claiming that a family of discoveries concerned with invariant structures map on to the ontology of command.
***
Take Two. I was too harsh on Z. The archeal stuff is a red herring. What really matters about Z's extraordinarily rigorous neo-Fichteanism is that it constructs an abundance of abstract formal models of practical reason. Fichte could only account for geometry; his Platonism extends only to the math we learn in secondary school. Z goes considerably further. Most impressive, in my view, is his construction of sheaves. It was only upon reading Z's book that I understood why mathematicians are so obsessed with bundles. To cut a very long story short, sheaves formalize the concept of an integrated object, what those of us trained in the humanities call a synthesis of the Whole.
Hegelians love to mock and poke fun at the subjective idealism of Fichteans. This was the immature Hegel's perspective, when he sided with Schelling against Fichte's view of nature as dead matter. The mature Hegel realized that Fichte was far closer to his own perspective, with the important difference that he rejected Fichte's Romanticism. People forget that they were buried side by side.
Z's synoptic and exhaustive survey will be read for decades. It is a high water mark for epistemology. Just as Fichte showed how far one could go with only the moral law as foundation, Z shows us how far one can go armed only with category theory.
***
Take Two. I was too harsh on Z. The archeal stuff is a red herring. What really matters about Z's extraordinarily rigorous neo-Fichteanism is that it constructs an abundance of abstract formal models of practical reason. Fichte could only account for geometry; his Platonism extends only to the math we learn in secondary school. Z goes considerably further. Most impressive, in my view, is his construction of sheaves. It was only upon reading Z's book that I understood why mathematicians are so obsessed with bundles. To cut a very long story short, sheaves formalize the concept of an integrated object, what those of us trained in the humanities call a synthesis of the Whole.
Hegelians love to mock and poke fun at the subjective idealism of Fichteans. This was the immature Hegel's perspective, when he sided with Schelling against Fichte's view of nature as dead matter. The mature Hegel realized that Fichte was far closer to his own perspective, with the important difference that he rejected Fichte's Romanticism. People forget that they were buried side by side.
Z's synoptic and exhaustive survey will be read for decades. It is a high water mark for epistemology. Just as Fichte showed how far one could go with only the moral law as foundation, Z shows us how far one can go armed only with category theory.
July 23, 2018
A 20th century gold mine and a map for the 21st.
February 25, 2013
Great - inspiring & very smart.
Read
August 15, 2016510.1 Z22 2012
September 26, 2023
w o w best book ever
Read
July 15, 2016????????
October 5, 2024
3.75/5
Es un libro ambicioso y que no deja indiferente.
Para empezar, Zalamea quiere presentarse como un adalid de la lucha contra el establishment de la filosofía analítica de las matemáticas. Y creo que consigue mostrar cómo tomar el ideal de la 'actividad' puede generar una visión filosófica en ocasiones mas rica que la que resulta de tomar el ideal fundacionalista de la teoría de conjuntos, por ejemplo.
El problema (a mi modo de ver) viene con la exposición de esta idea combativa. Al no tener un anclaje estático y prefigurado, tiene que recurrir a una serie de ejemplos (bastante completos, por otro lado) de matemáticos contemporáneos y a interpretarlos de un modo filosóficamente sustancial. Pero esta interpretación viene condicionada por su uso constante de terminología peirceana y de metáforas que en ocasiones palidecen tanto como los recursos de la oposición.
Por otro lado, la variedad de temas que toca Zalamea es considerable: la creatividad matemática (reivindicada en varias ocasiones), la fenomenología de la actividad matemática (quizá demasiado somera como para presentar un método o programa), las revoluciones en matemáticas (cuya posibilidad niega (!)), la estética matemática, etc. Esta diversidad de intereses hace al intento de Zalamea digno de ser respetado y de, al menos, ser leído.
Finalmente, en un capítulo brevísimo, expone la posibilidad de que las matemáticas contemporáneas, con sus valores e ideales, puedan suscitar cambios profundos en la filosofía y, en general, en la cultura humana. Hay una discusión muy interesante sobre el post-modernismo, el romanticismo y su 'trans-modernismo'. Aquí me ha ganado con su idea de que, en cierto modo, el modo de ser que exigen las matemáticas actuales es diametralmente contrario en muchos aspectos al statu quo (aunque, ¡recordemos!, Zalamea no parece muy fan de las revoluciones).
En resumidas cuentas, es una lectura densa y a veces oscura, pero creo que merece la pena darle una oportunidad al libro, ya sea solo por las referencias bibliográficas que contiene.
Es un libro ambicioso y que no deja indiferente.
Para empezar, Zalamea quiere presentarse como un adalid de la lucha contra el establishment de la filosofía analítica de las matemáticas. Y creo que consigue mostrar cómo tomar el ideal de la 'actividad' puede generar una visión filosófica en ocasiones mas rica que la que resulta de tomar el ideal fundacionalista de la teoría de conjuntos, por ejemplo.
El problema (a mi modo de ver) viene con la exposición de esta idea combativa. Al no tener un anclaje estático y prefigurado, tiene que recurrir a una serie de ejemplos (bastante completos, por otro lado) de matemáticos contemporáneos y a interpretarlos de un modo filosóficamente sustancial. Pero esta interpretación viene condicionada por su uso constante de terminología peirceana y de metáforas que en ocasiones palidecen tanto como los recursos de la oposición.
Por otro lado, la variedad de temas que toca Zalamea es considerable: la creatividad matemática (reivindicada en varias ocasiones), la fenomenología de la actividad matemática (quizá demasiado somera como para presentar un método o programa), las revoluciones en matemáticas (cuya posibilidad niega (!)), la estética matemática, etc. Esta diversidad de intereses hace al intento de Zalamea digno de ser respetado y de, al menos, ser leído.
Finalmente, en un capítulo brevísimo, expone la posibilidad de que las matemáticas contemporáneas, con sus valores e ideales, puedan suscitar cambios profundos en la filosofía y, en general, en la cultura humana. Hay una discusión muy interesante sobre el post-modernismo, el romanticismo y su 'trans-modernismo'. Aquí me ha ganado con su idea de que, en cierto modo, el modo de ser que exigen las matemáticas actuales es diametralmente contrario en muchos aspectos al statu quo (aunque, ¡recordemos!, Zalamea no parece muy fan de las revoluciones).
En resumidas cuentas, es una lectura densa y a veces oscura, pero creo que merece la pena darle una oportunidad al libro, ya sea solo por las referencias bibliográficas que contiene.
January 30, 2025
It’s quite good when you finally get to the math explanations and it’s implications, but I did feel the core ideas never quite fit together perfectly. The methodology also takes up 100 pages with a long section simply outlining the various contributors to the history of Phil of math, with little regard for if their project corresponds with this one. A good read, but could imagine a far greater one.
July 30, 2020
我明明是電機系的啊，怎麼越讀越覺得自己數學程度很低落...
March 6, 2021
Some nice bits but overall, rococo math journalism for noobs.
Read
December 10, 2020In this book Fernando Zalamea attempts to to "call attention to a very broad mathematical spectrum that has rarely been accounted for in philosophical discussions" (p.269), namely, the vast fields of contemporary mathematics of the past 60 years, during which there have been vastly more substantial mathematical results than in the millenia of practice before. What constitutes a significant part of the book is a biting and interesting critique of the analytic philosophy of mathematics and its ossified appendix in set theory-centric "foundations of mathematics". Here much of the critique simply draws from historical figures like Peirce and Lautman, whereas the author's original contributions appear in the 3rd part of the book in the actual synthetic work.
What I found most insightful were the scathing and clear-eyed jabs at logicism for condemning mathematics as it is practiced to mere transitoriness. The author sees contemporary mathematics as transitory and relative to be sure, but not in a pejorative sense. On the contrary, the "mixing" of forms and the "relativity" of ontological reference points is what is constitutive of contemporary mathematics. Grothendick is the central figure here, representing a kind of Einstein for mathematics, whose theories conventionalize reference points for mathematical ontology, like for simultaneity in physics. Contemporary mathematics is then to the "foundations of mathematics" what Einsteinian relativity was to Newtonian absolute space.
Another contribution of interest to many (though perhaps less to those coming from analytic philosophy) is the phenomenology of mathematical practice and invention.
I had two problems with the book, one of which was quite general, but the other personal. The personal one was that the references to mathematics are extremely technical, nigh non-explained and fill almost every page. From my limited understanding of topology and category theory and more substantial understanding of logic and set theory, I got some of the references, and the ones I did, I found to be quite insightful. But if, like myself, your knowledge of elliptic geometry and whatever the hell eigenfunctions are is limited, be warned.
The more general problem I had with the book was that the writing is insanely prolix and unnecessarily obtuse at times. References abound, with no explanation, to German poets and romantic thinkers.
Nonetheless, I think that there is plenty worth taking seriously here. Especially if you operate in anglophone academia. There could be a more "Queen's English" exposition of the book's project, which could appeal to contemporary anglophone mathematical philosophy.
What I found most insightful were the scathing and clear-eyed jabs at logicism for condemning mathematics as it is practiced to mere transitoriness. The author sees contemporary mathematics as transitory and relative to be sure, but not in a pejorative sense. On the contrary, the "mixing" of forms and the "relativity" of ontological reference points is what is constitutive of contemporary mathematics. Grothendick is the central figure here, representing a kind of Einstein for mathematics, whose theories conventionalize reference points for mathematical ontology, like for simultaneity in physics. Contemporary mathematics is then to the "foundations of mathematics" what Einsteinian relativity was to Newtonian absolute space.
Another contribution of interest to many (though perhaps less to those coming from analytic philosophy) is the phenomenology of mathematical practice and invention.
I had two problems with the book, one of which was quite general, but the other personal. The personal one was that the references to mathematics are extremely technical, nigh non-explained and fill almost every page. From my limited understanding of topology and category theory and more substantial understanding of logic and set theory, I got some of the references, and the ones I did, I found to be quite insightful. But if, like myself, your knowledge of elliptic geometry and whatever the hell eigenfunctions are is limited, be warned.
The more general problem I had with the book was that the writing is insanely prolix and unnecessarily obtuse at times. References abound, with no explanation, to German poets and romantic thinkers.
Nonetheless, I think that there is plenty worth taking seriously here. Especially if you operate in anglophone academia. There could be a more "Queen's English" exposition of the book's project, which could appeal to contemporary anglophone mathematical philosophy.
Displaying 1 - 12 of 12 reviews