Gödel's incompleteness theorem

Exploring Theravāda's connections to other paths - what can we learn from other traditions, religions and philosophies?
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Viachh
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Gödel's incompleteness theorem

Post by Viachh »

In the collection of arguments that Buddhism (dhamma) is not a philosophy and that the phrase "Buddhist philosophy" is absurd.


It follows from Gödel's incompleteness theorem that any logical system cannot be completely consistent. And since it is believed that the words of the Buddha, by definition, cannot be contradictory, therefore, the very existence of Buddhist philosophy is impossible.
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Coëmgenu
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Re: Gödel's incompleteness theorem

Post by Coëmgenu »

We briefly discussed this in this very subforum while discussing Venerable Nāgārjuna and the tetralemma:

viewtopic.php?p=629139#p629139
What is the Uncreated?
Sublime & free, what is that obscured Eternity?
It is the Undying, the Bright, the Isle.
It is an Ocean, a Secret: Reality.
Both life and oblivion, it is Nirvāṇa.
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Modus.Ponens
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Re: Gödel's incompleteness theorem

Post by Modus.Ponens »

Viachh wrote: Thu Sep 23, 2021 3:19 pm In the collection of arguments that Buddhism (dhamma) is not a philosophy and that the phrase "Buddhist philosophy" is absurd.


It follows from Gödel's incompleteness theorem that any logical system cannot be completely consistent. And since it is believed that the words of the Buddha, by definition, cannot be contradictory, therefore, the very existence of Buddhist philosophy is impossible.
A technical correction: Gödel proved that any system that includes arithmetic is incomplete, which means there are statements you cannot prove within that system. It doesn't mean the statements are false. They're just unprovable. One of these unprovable statements is the consistency of such a system.

More interesting, perhaps, is that some apparent exceptions are extraordinary. Quoting from Stanford's Encyclopedia of Philosophy,

On the other hand, not all theories of arithmetic are incomplete. The theory of only addition of natural numbers but without multiplication (often called “Presburger arithmetic”), for example, is complete (and decidable) (Presburger 1929), as is the theory of multiplication of the positive integers (Skolem 1930). These theories are, though, very weak. But in any case, at least a theory which deals with both addition and multiplication is needed. More interestingly, the natural first-order theory of arithmetic of real numbers (with both addition and multiplication), the so-called theory of real closed fields (RCF), is both complete and decidable, as was shown by Tarski (1948); he also demonstrated that the first-order theory of Euclidean geometry is complete and decidable. Thus, one should keep in mind that there are some non-trivial and interesting theories to which Gödel’s theorems do not apply.

https://plato.stanford.edu/entries/goed ... pleteness/
'This is peace, this is exquisite — the resolution of all fabrications; the relinquishment of all acquisitions; the ending of craving; dispassion; cessation; Unbinding.' - Jhana Sutta
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Dan74
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Re: Gödel's incompleteness theorem

Post by Dan74 »

I guess Viacch's idea can be adapted. One can argue

1. Any meaningful philosophy includes a logical system with sufficient complexity. It must be non-trivial and powerful.
2. Any such system, due to Gödel, will be incomplete, i.e. unable to determine if it is consistent (never leads to contradictions).
3. A system that is incomplete is not perfect.
3. The Buddhadhamma is perfect, therefore it cannot be incomplete, thus it is not a meaningful philosophy.

I am something of a pragmatist, when it comes to the Dhamma, so I don't care much for the assumptions implicit in this argument. But just to help out and to say that one can argue this. One may also fail to be convinced, of course.
_/|\_
DiamondNgXZ
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Re: Gödel's incompleteness theorem

Post by DiamondNgXZ »

Viachh wrote: Thu Sep 23, 2021 3:19 pm In the collection of arguments that Buddhism (dhamma) is not a philosophy and that the phrase "Buddhist philosophy" is absurd.


It follows from Gödel's incompleteness theorem that any logical system cannot be completely consistent. And since it is believed that the words of the Buddha, by definition, cannot be contradictory, therefore, the very existence of Buddhist philosophy is impossible.
Given that the Buddha based his knowledge from direct seeing and asked his disciples to develop the same, it's more akin to physics, science than mathematics. As far as I know, Godel's incompleteness theorem applies to maths, things which are reasoned out only. The Buddha allows us to use reason and authority (suttas), but the final arbiter of knowledge is our direct seeing after practising, to see the same things as the Buddha did. So I don't think Godel's theorem applies to Buddhism.
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Modus.Ponens
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Re: Gödel's incompleteness theorem

Post by Modus.Ponens »

Dan74 wrote: Thu Sep 23, 2021 9:16 pm I guess Viacch's idea can be adapted. One can argue

1. Any meaningful philosophy includes a logical system with sufficient complexity. It must be non-trivial and powerful.
2. Any such system, due to Gödel, will be incomplete, i.e. unable to determine if it is consistent (never leads to contradictions).
3. A system that is incomplete is not perfect.
3. The Buddhadhamma is perfect, therefore it cannot be incomplete, thus it is not a meaningful philosophy.

I am something of a pragmatist, when it comes to the Dhamma, so I don't care much for the assumptions implicit in this argument. But just to help out and to say that one can argue this. One may also fail to be convinced, of course.
I'm a pragmatist too. I don't think axiomatizing Buddhism is a good way of understanding its message. It's putting it into practice. I was just making a technical point about Gödel's incompleteness theorems. There's a lot of subtlety in these theorems and their consequences, many of which escape me. It is in fact easy to come up with "undecidable statements" in Buddhism, like an obscure thing about the Devas that there is no way to know based on teachings and reason alone. The "sex of the Devas"? :mrgreen: Something like that.
'This is peace, this is exquisite — the resolution of all fabrications; the relinquishment of all acquisitions; the ending of craving; dispassion; cessation; Unbinding.' - Jhana Sutta
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Dan74
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Re: Gödel's incompleteness theorem

Post by Dan74 »

Modus.Ponens wrote: Fri Sep 24, 2021 4:59 pm
Dan74 wrote: Thu Sep 23, 2021 9:16 pm I guess Viacch's idea can be adapted. One can argue

1. Any meaningful philosophy includes a logical system with sufficient complexity. It must be non-trivial and powerful.
2. Any such system, due to Gödel, will be incomplete, i.e. unable to determine if it is consistent (never leads to contradictions).
3. A system that is incomplete is not perfect.
3. The Buddhadhamma is perfect, therefore it cannot be incomplete, thus it is not a meaningful philosophy.

I am something of a pragmatist, when it comes to the Dhamma, so I don't care much for the assumptions implicit in this argument. But just to help out and to say that one can argue this. One may also fail to be convinced, of course.
I'm a pragmatist too. I don't think axiomatizing Buddhism is a good way of understanding its message. It's putting it into practice. I was just making a technical point about Gödel's incompleteness theorems. There's a lot of subtlety in these theorems and their consequences, many of which escape me. It is in fact easy to come up with "undecidable statements" in Buddhism, like an obscure thing about the Devas that there is no way to know based on teachings and reason alone. The "sex of the Devas"? :mrgreen: Something like that.
Sure your corrective was of course correct. As to the subtleties, I don't know. I think their reach is quite broad, but as the examples you quoted show, if we tweak the complexity of a system down a notch, they may not apply. In any case, while maths people may like to talk about this sort of thing, I never found it actually relevant to my practice, same as you, I suspect.
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Modus.Ponens
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Re: Gödel's incompleteness theorem

Post by Modus.Ponens »

Dan74 wrote: Sat Sep 25, 2021 12:19 am
Modus.Ponens wrote: Fri Sep 24, 2021 4:59 pm
Dan74 wrote: Thu Sep 23, 2021 9:16 pm I guess Viacch's idea can be adapted. One can argue

1. Any meaningful philosophy includes a logical system with sufficient complexity. It must be non-trivial and powerful.
2. Any such system, due to Gödel, will be incomplete, i.e. unable to determine if it is consistent (never leads to contradictions).
3. A system that is incomplete is not perfect.
3. The Buddhadhamma is perfect, therefore it cannot be incomplete, thus it is not a meaningful philosophy.

I am something of a pragmatist, when it comes to the Dhamma, so I don't care much for the assumptions implicit in this argument. But just to help out and to say that one can argue this. One may also fail to be convinced, of course.
I'm a pragmatist too. I don't think axiomatizing Buddhism is a good way of understanding its message. It's putting it into practice. I was just making a technical point about Gödel's incompleteness theorems. There's a lot of subtlety in these theorems and their consequences, many of which escape me. It is in fact easy to come up with "undecidable statements" in Buddhism, like an obscure thing about the Devas that there is no way to know based on teachings and reason alone. The "sex of the Devas"? :mrgreen: Something like that.
Sure your corrective was of course correct. As to the subtleties, I don't know. I think their reach is quite broad, but as the examples you quoted show, if we tweak the complexity of a system down a notch, they may not apply. In any case, while maths people may like to talk about this sort of thing, I never found it actually relevant to my practice, same as you, I suspect.
Pretty much, yes. There may have been a distant time when I cared about Buddhism as a logical theory, but I don't even remember it.
'This is peace, this is exquisite — the resolution of all fabrications; the relinquishment of all acquisitions; the ending of craving; dispassion; cessation; Unbinding.' - Jhana Sutta
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