This is Math's Fatal Flaw - YouTube
In another thread, I discussed the implications of the three-body problem in mathematics, which shows that, even though a deterministic outcome must exist, mathematics itself forbids us from calculating it. Math, the very thing that supposedly does allow us to calculate, forbids some calculations.
If there is anything indisputable, it is a mathematical proof. Or, is it? This video does not literally point to a fatal flaw in mathematics, but it does demonstrate that some mathematical questions can never be answered with certainty, not even by computers which are, proverbially, the size of the universe.
In a universe supposedly guided by cause and effect, we expect outcomes to be calculable, even if only in principle. Not all of them are. IMO, this indicates something fundamental about reality, although I have yet to work out what it is. Maybe that’s the point. Maybe there are some things that the very nature of reality keeps hidden from us.
Yet, this is not a fatalistic outlook. Quite the contrary. Understanding this concept leads to progress that might otherwise not have been made.
I sense that an eternity of contemplation awaits us.
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There is a Hole in Mathematics
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SanteriSatama
- Posts: 1030
- Joined: Wed Jan 13, 2021 4:07 pm
Re: There is a Hole in Mathematics
What is "mathematics itself"?
That is a foundational question, and the foundational crisis between intuitionism and formalism (Brouwer-Hilbert controvercy) remains unsolved, merely brushed under the carpet in the (post-)modern Academic cult. Hilbert's formalism is known mainly as axiomatic set theories, and those are arbitrary language games of if-then structures. Postulating arbitrary set of axioms and looking what follows... well, all the Gödel issues, for example. Note that 'set' is an undefined primitive notion, because Cantor's naive set theory led to Russel's paradox.
Hilbertism is post-modern language game of atomistic point-reductionism, "definition" of point, again, given as undefined primitive notion. Perhaps the biggest critic of Hilbert's and Cantor's formalism is Wittgenstein, who coined the term 'language game' and was main influence of philosophical critique of the post-modern condition of arbitrary language games, and when Hilbert said "nothing can expel us from Cantor's paradise", Wittgenstein responded that what appears a paradise to some, can be a joke to others. On the other hand, intuitionism considers math empirical science, in the idealist and introspective sense of empirism.
Intuitinist philosophy of math considers the cognitive, idealist aspect of mathematics ontological, and the linguistic aspect secondary, accepting that idealist ontology is irreducible to any and all formal languages. Main difference between intuitionism and mathematical platonism is that the latter considers math eternal and immutable, intuitionism is open to participatory evolution. Development of computation theory and quantum empirism. has major implications to intuitionist views on temporality of mathematics. Undecidability of the Halting problem, together with Curry-Howard correspondance etc. strongly suggest that mathematical proof events don't instantly spread to eternity of both past and future, but only to open ended durations. Which is coherent with Bergson's philosophy of time.
In classical (Zeno) and contemporary look, the core issue is the relation of continuous and discrete. Formalist point-reductionism has attempted to respond to Zeno's paradoxes by Cantor's infinite sets, but the more sane mathematical community considers completed infinities of infinite sets oxymoron. Needless to say, infinite sets of the real number theory lead to postulation of non-computable, non-algorithmic and non-demonstrable numbers and math. Which are somehow supposed to be able to do basic arithmetics, ie. form a mathematical field. Of course that claim is not inductive or demonstrable (as with rational numbers), but a counterfactual and contradictory make-believe "axiom". In the Brouwer-Hilbert controvercy Brouwer's main criticism was that Hilbert's formalism leads to loss of "constructibility", which has very close and similar meaning with computability.
The coherent and correct way, IMHO, is to accept that continuum/continua is not reducible to discrete, but on the other hand discrete can be constructed from continua.
The very notion and phenomenon of computation (and causality?) presupposes continuous duration where computation process takes place and proceeds. A continuous awareness.
That is a foundational question, and the foundational crisis between intuitionism and formalism (Brouwer-Hilbert controvercy) remains unsolved, merely brushed under the carpet in the (post-)modern Academic cult. Hilbert's formalism is known mainly as axiomatic set theories, and those are arbitrary language games of if-then structures. Postulating arbitrary set of axioms and looking what follows... well, all the Gödel issues, for example. Note that 'set' is an undefined primitive notion, because Cantor's naive set theory led to Russel's paradox.
Hilbertism is post-modern language game of atomistic point-reductionism, "definition" of point, again, given as undefined primitive notion. Perhaps the biggest critic of Hilbert's and Cantor's formalism is Wittgenstein, who coined the term 'language game' and was main influence of philosophical critique of the post-modern condition of arbitrary language games, and when Hilbert said "nothing can expel us from Cantor's paradise", Wittgenstein responded that what appears a paradise to some, can be a joke to others. On the other hand, intuitionism considers math empirical science, in the idealist and introspective sense of empirism.
Intuitinist philosophy of math considers the cognitive, idealist aspect of mathematics ontological, and the linguistic aspect secondary, accepting that idealist ontology is irreducible to any and all formal languages. Main difference between intuitionism and mathematical platonism is that the latter considers math eternal and immutable, intuitionism is open to participatory evolution. Development of computation theory and quantum empirism. has major implications to intuitionist views on temporality of mathematics. Undecidability of the Halting problem, together with Curry-Howard correspondance etc. strongly suggest that mathematical proof events don't instantly spread to eternity of both past and future, but only to open ended durations. Which is coherent with Bergson's philosophy of time.
In classical (Zeno) and contemporary look, the core issue is the relation of continuous and discrete. Formalist point-reductionism has attempted to respond to Zeno's paradoxes by Cantor's infinite sets, but the more sane mathematical community considers completed infinities of infinite sets oxymoron. Needless to say, infinite sets of the real number theory lead to postulation of non-computable, non-algorithmic and non-demonstrable numbers and math. Which are somehow supposed to be able to do basic arithmetics, ie. form a mathematical field. Of course that claim is not inductive or demonstrable (as with rational numbers), but a counterfactual and contradictory make-believe "axiom". In the Brouwer-Hilbert controvercy Brouwer's main criticism was that Hilbert's formalism leads to loss of "constructibility", which has very close and similar meaning with computability.
The coherent and correct way, IMHO, is to accept that continuum/continua is not reducible to discrete, but on the other hand discrete can be constructed from continua.
The very notion and phenomenon of computation (and causality?) presupposes continuous duration where computation process takes place and proceeds. A continuous awareness.
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Robert Arvay
- Posts: 97
- Joined: Thu Jan 14, 2021 6:37 pm
Re: There is a Hole in Mathematics
If a computer program contains a fatal flaw, the program either crashes, or enters a never-ending loop.
Granted, the analogy between cosmology/physics/mathematics and computers is weak, but the
speculation arises -- COULD the universe contain a fatal flaw, an inconsistency, a physical-law paradox?
IMO, the fatal flaw is in our brains or perhaps, minds.
Can a raindrop tame the ocean?
Maybe Jesus really DID walk on water?
Maybe He does rise above all of our concepts.
Maybe He alone can command the ocean, "Peace, be still."
We may never be able to tame the ocean,
but perhaps we can recognize that fact, and move toward
mastering ourselves?
Speculations.
-
Granted, the analogy between cosmology/physics/mathematics and computers is weak, but the
speculation arises -- COULD the universe contain a fatal flaw, an inconsistency, a physical-law paradox?
IMO, the fatal flaw is in our brains or perhaps, minds.
Can a raindrop tame the ocean?
Maybe Jesus really DID walk on water?
Maybe He does rise above all of our concepts.
Maybe He alone can command the ocean, "Peace, be still."
We may never be able to tame the ocean,
but perhaps we can recognize that fact, and move toward
mastering ourselves?
Speculations.
-
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SanteriSatama
- Posts: 1030
- Joined: Wed Jan 13, 2021 4:07 pm
Re: There is a Hole in Mathematics
The proof that Halting problem is undecidable - meaning that there can't be universal Oracle program that could tell in each and every case wheather a computarion of a program halts (and gives a result) or gets stuck in a loop - is based on assumption of Law of Non-Contradition (LNC).
The issue with LNC is that it has very complex relation with time. Where, I mean when, a glass of water can be both full and empty and everything between, during a process of drinking.
There can be both exclusive and creative contradictions. Which is not a problem for process philosophy and and process philosophical and intuitionist approach to philosophy of mathematics. What is permanent aspect of mathematics is not any linguistic construction, but the values. Values of or rigour, communicability and beauty.
The issue with LNC is that it has very complex relation with time. Where, I mean when, a glass of water can be both full and empty and everything between, during a process of drinking.
There can be both exclusive and creative contradictions. Which is not a problem for process philosophy and and process philosophical and intuitionist approach to philosophy of mathematics. What is permanent aspect of mathematics is not any linguistic construction, but the values. Values of or rigour, communicability and beauty.