I came across your channel yesterday and immediatly subscribed. I'm pleased to see that since then, your channel has already gained eight additional subscribers. I am looking forward to many fascinating videos of yours. May you and and your channel thrive!
Brouwer's radical revolutionary program (at that time and age) was to bring time back to mathematics, countering the interpretation that Platonism means eternal and immutable transcendentalism, which is not historical Platonism of the actual philosophy and methodology of the Academy, but perhaps more to do with Catholic church theology and Kant than with Plato, who was dialectician instead of a model theorist. Constructivism is inherently temporal, because constructing is a temporal process. Both Gödel and Brouwer accept and emphasize intuitive receiving as the primary method of mathematics. The difference between Gödel's and Brouwer's Platonisms is that Gödel postulates timeless being of transcendental "Platonia" and correspondence theory of truth, where as Brouwer rejects axiomatics of abritrary postulates and (re)constructs philosophical foundation of immanent process ontology and coherence theory of truth. Academic math community does not much discuss Brouwer, because Hilbert wanted to prevent Brouwer getting any followers inside academic institutions and creating a paradigm of a living tradion of scholarship, and did so by academic cancel culture of kicking Brouwer out from the board of the prestige journal of the time, which devastated Brouwer psychologically. Hilbert's legacy has been to keep the foundational crisis swept under the rug in the post-modern academy, his Formalism having given birth to the post-modern zeitgeist. So, instead of math departments, constructivism bloomed to very creative and productive flower in computation science, in undecidability proofs by Church, Turing etc, in type theories, in Curry-Howard correspondence of proofs-as-programs and the development mathematical AI e.g. to the current conjecture churning Ramanujan engine that continues the legacy of Gosper arithmetic of continued fractions. Which, very curiously, most math department people stay ignorant of. Frege's attempt to get rid of the syllogistic quantifier "some" and with that the contraries of propositional logic failed because that lead to the self-referential contradictions, and as contraries could not be kept out of mathematics, they crept back first in the form of the Turing-Tape, which extends simultaneously BOTH L AND R as the necessary and indispensable precondition for making the the choice EITHER L OR R. Mathematical objects don't come into being by subjective existential declarations. Numbers come into being by tallying PROCESSES, and don't have inherent existence of their own.
Thanks for adding much more detail to the discussion of Brouwer and constructivism. I must admit I do not know much about him beyond what I have read in this book, and what little I have heard of him elsewhere. Would enjoy reading a biography just on Brouwer.
@@hungrymathprof My foundational obsessions started from curiosity about p-adics, and wondering about possibility of coherent top-down construction without the very incomple three dots of the p-adic source. Brouwer has offered a lot of help on the journey, and reading a good biography on Brouwer would be interesting. Now I realize that the First action of Intuitionism, "two-ity", is basically the same idea as the Platonic "One", meaning the reversible unity of a contrary opposition as propositional logic expresses the same idea. "Quantum" and "Qubit" are also terms for the very ancient idea. I've found the article "The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of Forcing and the Heyting Calculus." by Zachary Frazer very interesting and inspiring meet of Intuitionism and Badiou's philosophical perspective of the Formalist school of mathematics.
@@hungrymathprof The biography confirms that Brouwer became doubtful of his fixed point theorem, as I has suspected for obvious reason that point-reductionism is not constructvive. It is thus somewhat unfortunate situation that the term "fixed point" has stuck gained more general meaning of an invariant relation in variance, e.g. and especially in the study of Eigenforms by a contemporary foundational constructivist mathematician Louis H. Kauffman, whose contributions I greatly admire. "Y-combinator" from Schönfinkel's combinatory logic could be used in same meaning with less confusing connotations. While term "point" remains Hilbertian undefined primitive notion, semantic and foundational clarity is difficult to find. I think that the confusion does go back to Euclid. The first definition is mereologically magnificent, but further down the road much confusion arises from not making terminological distinction between end of dimensional decomposition process and node in a connected graph. The problem of dimensional invariance thus goes back to Elementa. When ideal geometric forms are conceived as dynamic shadow projections unfolding from higher dimensional Nous, from the observer perspective of participatory mathematicians gaze it cannot be decided whether the meets of lines of a 2D projection are actual nodes with same dimensional value, or disconnected crossings of edges. This empirical undecidability of projective geometry can be seen as the incompleteness theorem of Euclid's method, and proof of five Platonic solids offers a partial solution to the decidability problem of what higher dimensional forms can be associated with a 2D projection on a flat surface. Now we have computer graphics to aid the screen in the theater of mind.
@@thoughtsuponatime847 If you are really interested in healing commentary, then Proclus' commentary to Euclid is incredibly sophisticated exposure of the philosophical foundation of pure mathematics by insider perspective of the living tradition of study of mathematics in the original Plato's Academy. It's also very fresh and vitalizing view from our post-modern truth nihilistic perspective, that has for the most part forgotten and ignored Proclus and Euclid and pays little attention to evolution of constructive foundation outside of the computation science departments. Aristotle and formal syllogistic and propositional logic, which are not limited to contradictions but contain also contraries (cf. undecidability etc.) are part of the dialectic, but Aristotle was not picked as the head of the Academy after Plato. They picked the master mathematician Eudoxus, who found and developed the foundational mathematical ground of analogy (in Latin translation "proportion") and commensurability. To heal means to cohere from the incoherent fragmentation of arbitrary language games, which are important heuristics, but cannot by their nature offer coherent and healthy foundation. As the original Greek term analogy implies, it includes already the deep idea of recursion in the prefix ana-. Generic idea of commensurability means relational coherence of recursive additive and nesting processes.
I came across your channel yesterday and immediatly subscribed. I'm pleased to see that since then, your channel has already gained eight additional subscribers. I am looking forward to many fascinating videos of yours. May you and and your channel thrive!
@matthiasfreiburghaus4202 I am so glad that you found my channel. Thank you for the encouragement!
Great discussion! ♾️✨
Thanks, glad you liked it!
Thank you
@@anismaza6582 I'm glad you enjoyed the video!
Brouwer's radical revolutionary program (at that time and age) was to bring time back to mathematics, countering the interpretation that Platonism means eternal and immutable transcendentalism, which is not historical Platonism of the actual philosophy and methodology of the Academy, but perhaps more to do with Catholic church theology and Kant than with Plato, who was dialectician instead of a model theorist.
Constructivism is inherently temporal, because constructing is a temporal process. Both Gödel and Brouwer accept and emphasize intuitive receiving as the primary method of mathematics. The difference between Gödel's and Brouwer's Platonisms is that Gödel postulates timeless being of transcendental "Platonia" and correspondence theory of truth, where as Brouwer rejects axiomatics of abritrary postulates and (re)constructs philosophical foundation of immanent process ontology and coherence theory of truth.
Academic math community does not much discuss Brouwer, because Hilbert wanted to prevent Brouwer getting any followers inside academic institutions and creating a paradigm of a living tradion of scholarship, and did so by academic cancel culture of kicking Brouwer out from the board of the prestige journal of the time, which devastated Brouwer psychologically.
Hilbert's legacy has been to keep the foundational crisis swept under the rug in the post-modern academy, his Formalism having given birth to the post-modern zeitgeist.
So, instead of math departments, constructivism bloomed to very creative and productive flower in computation science, in undecidability proofs by Church, Turing etc, in type theories, in Curry-Howard correspondence of proofs-as-programs and the development mathematical AI e.g. to the current conjecture churning Ramanujan engine that continues the legacy of Gosper arithmetic of continued fractions. Which, very curiously, most math department people stay ignorant of.
Frege's attempt to get rid of the syllogistic quantifier "some" and with that the contraries of propositional logic failed because that lead to the self-referential contradictions, and as contraries could not be kept out of mathematics, they crept back first in the form of the Turing-Tape, which extends simultaneously BOTH L AND R as the necessary and indispensable precondition for making the the choice EITHER L OR R.
Mathematical objects don't come into being by subjective existential declarations. Numbers come into being by tallying PROCESSES, and don't have inherent existence of their own.
Thanks for adding much more detail to the discussion of Brouwer and constructivism. I must admit I do not know much about him beyond what I have read in this book, and what little I have heard of him elsewhere. Would enjoy reading a biography just on Brouwer.
@@hungrymathprof My foundational obsessions started from curiosity about p-adics, and wondering about possibility of coherent top-down construction without the very incomple three dots of the p-adic source.
Brouwer has offered a lot of help on the journey, and reading a good biography on Brouwer would be interesting.
Now I realize that the First action of Intuitionism, "two-ity", is basically the same idea as the Platonic "One", meaning the reversible unity of a contrary opposition as propositional logic expresses the same idea. "Quantum" and "Qubit" are also terms for the very ancient idea.
I've found the article "The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of Forcing and the Heyting Calculus." by Zachary Frazer very interesting and inspiring meet of Intuitionism and Badiou's philosophical perspective of the Formalist school of mathematics.
@@santerisatama5409 I found this short biographical sketch interesting mathshistory.st-andrews.ac.uk/Biographies/Brouwer/
@@hungrymathprof Thanks!
@@hungrymathprof The biography confirms that Brouwer became doubtful of his fixed point theorem, as I has suspected for obvious reason that point-reductionism is not constructvive. It is thus somewhat unfortunate situation that the term "fixed point" has stuck gained more general meaning of an invariant relation in variance, e.g. and especially in the study of Eigenforms by a contemporary foundational constructivist mathematician Louis H. Kauffman, whose contributions I greatly admire. "Y-combinator" from Schönfinkel's combinatory logic could be used in same meaning with less confusing connotations. While term "point" remains Hilbertian undefined primitive notion, semantic and foundational clarity is difficult to find.
I think that the confusion does go back to Euclid. The first definition is mereologically magnificent, but further down the road much confusion arises from not making terminological distinction between end of dimensional decomposition process and node in a connected graph.
The problem of dimensional invariance thus goes back to Elementa. When ideal geometric forms are conceived as dynamic shadow projections unfolding from higher dimensional Nous, from the observer perspective of participatory mathematicians gaze it cannot be decided whether the meets of lines of a 2D projection are actual nodes with same dimensional value, or disconnected crossings of edges.
This empirical undecidability of projective geometry can be seen as the incompleteness theorem of Euclid's method, and proof of five Platonic solids offers a partial solution to the decidability problem of what higher dimensional forms can be associated with a 2D projection on a flat surface. Now we have computer graphics to aid the screen in the theater of mind.
Cool
@thoughtsuponatime847 Thanks! I'm glad you enjoyed the video.
@@hungrymathprof I really am interested in healing commentary about the philosophy of mathematics. It is fun stuff. Thanks and have a great day!
@@thoughtsuponatime847 If you are really interested in healing commentary, then Proclus' commentary to Euclid is incredibly sophisticated exposure of the philosophical foundation of pure mathematics by insider perspective of the living tradition of study of mathematics in the original Plato's Academy. It's also very fresh and vitalizing view from our post-modern truth nihilistic perspective, that has for the most part forgotten and ignored Proclus and Euclid and pays little attention to evolution of constructive foundation outside of the computation science departments.
Aristotle and formal syllogistic and propositional logic, which are not limited to contradictions but contain also contraries (cf. undecidability etc.) are part of the dialectic, but Aristotle was not picked as the head of the Academy after Plato. They picked the master mathematician Eudoxus, who found and developed the foundational mathematical ground of analogy (in Latin translation "proportion") and commensurability. To heal means to cohere from the incoherent fragmentation of arbitrary language games, which are important heuristics, but cannot by their nature offer coherent and healthy foundation. As the original Greek term analogy implies, it includes already the deep idea of recursion in the prefix ana-. Generic idea of commensurability means relational coherence of recursive additive and nesting processes.
Wow, one of the easiest subs of my life.
@zacharysmith4508 I'm glad you found my channel!