Dukkha and pure mathematics.

[Natan]

I totally agree, that's exactly my problem with mathematics as is taught today,
i.e., without the mystical side which by all evidence the Pythagoreans did emphasize.
Perharps it was a peculiar development in a specific society and thereafter suited only to a peculiar
type of personality.
But it has an appeal, that crystal-clear clarity of deep results and that no-cheating, earnest progress
in learning. Where else can these qualities be found? So as to appeal not only to the intellectual faculty.
That's what I'm trying to answer..

How do you propose maths should be taught 'with the mystical side'? Can you provide a concrete example?

Plato's dialogue Meno is a nice elementary example, where a boy is led to double the area of a square;
which is then viewed as innate knowledge revealed.

An example of a pythagorean concept:
https://en.wikipedia.org/wiki/Tetractys

A old book on pythagorean arithmetic:
http://djm.cc/library/Theoretic_Arithme ... Taylor.pdf

Many other contemporary writers, e.g. Leonid Zhmud.

Mysticism is distinct from metaphysics, the latter is to what you are referring and it is a serious area of philosophy of math. Mysticism is about uncanny pr unusual subjective experiences that make one feel chummy with the universe. My take in dharma is that even it is not mystical and is something else altogether.

[Aemilius]

In Entry into the Realm of Reality, which is part of the Flower Ornament Scripture, in the chapter of Indriyeshvara:

Indriyeshvara said, " I have been taught writing and mathematics by Mañjushri, and have been led into the door of knowledge encompassing higher knowledge of all practical arts. So I know all the various arts and crafts and sciences in the world dealing with writing, mathematics and symbols, physiology, rhetoric, physical and mental health, cityplanning, architecture and construction, mechanics and engineering, divination, agriculture and commerce, conduct and manners, good and bad actions, good and bad principles, what makes for felicity and what for misery, what is necessary for the vehicles of buddhas, disciples, and individual illuminates, what is necessary for buddhahood, and behavior linking reason and action. I know all these sciences, and I also introduce and teach them to people, and get people to study and practice them, to master and develop them, using these as means to purify, refine and broaden people.
"I myself know enlightening beings' method of reckoning, which goes like this: a hundred hundred thousand is a koti; a koti squared is an ayuta; an ayuta squared is a niyuta; a niyuta squared is abimbara; a bimbara squared is a kinkara; a kinkara squared is an agara; an agara squared is a pravara; a pravara squared is a mapara; a mapara squared is a tapara; a tapara squared is a sima; a sima squared is a yama; a yama squared is a nema; a nema squared is an avaga; an avaga squared is mrgava; a mrgava squared is viraga; a viraga squared is a vigava; a vigava squared is a samkrama; a samkrama squared is visara; a visara squared is a vibhaja; a vibhaja squared is a vijangha; a vijangha squared is a visodha; a visodha squared is a vivaha; a vivaha squared is a vibhakta; a vibhakta squared is a vikhata; a vikhata squared is an ilana; an ilana squared is an avana; etc...

[workbalance]

using these as means to purify, refine and broaden people.

this is so meaningful, education as purification, in a cross-cultural way..

[Aemilius]

Mathematics in the Lotus sutra, translation of H. Kern, Lotus of the True Law, Chapter VII Ancient Devotion:

"(You ask), monks, how long ago is it that the Tathâgata was born? Well, suppose some man was to reduce to powder the whole mass of the earth element as much as is to be found in this whole universe; that after taking one atom of dust from this world he is to walk a thousand worlds farther in easterly direction to deposit that single atom; that after taking a second atom of dust and walking a thousand worlds farther he deposits that second atom, and proceeding in this way at last gets the whole of the earth element deposited in eastern direction. Now, monks, what do you think of it, is it possible by calculation to find the end or limit of these worlds? They answered: Certainly not, Lord; certainly not, Sugata. The Lord said: On the contrary, monks, some arithmetician or master of arithmetic might, indeed, be able by calculation to find the end or limit of the worlds, both those where the atoms have been deposited and where they have not, but it is impossible by applying the rules of arithmetic to find the limit of those hundred thousands of myriads of Æons, so long, so inconceivable, so immense is the number of Æons which have elapsed since the expiration of that Lord, the Tathâgata Mahâbhigñâgñanâbhibhû.

1. I remember the great Seer Abhigñâgñânâbhibhû, the most high of men, who existed many kotis of Æons ago as the superior Jina of the period.
2. If, for example, some men after reducing this universe to atoms of dust took one atom to deposit it a thousand regions farther on;
3. If he deposited a second, a third atom, and so proceeded until he had done with the whole mass of dust, so that this world were empty and the mass of dust exhausted;
4. To that immense mass of the dust of these worlds, entirely reduced to atoms, I liken the number of Æons past.
5. So immense is the number of kotis of Æons past since that extinct Sugata; the whole of (existing) atoms is no (adequate) expression of it; so many are the Æons which have expired since."

[Aemilius]

Another mathematical series in the Lotus of the True Law, tr. H. Kern, Chapter XIV Issuing from the Gaps of the Earth:

"No sooner had the Lord uttered these words than the Saha-world burst open on every side, and from within the clefts arose many hundred thousand myriads of kotis of Bodhisattvas with gold-coloured bodies and the thirty-two characteristic signs of a great man, who had been staying in the element of ether underneath this great earth, close to this Saha-world. These then on hearing the word of the Lord came up from below the earth. Each of these Bodhisattvas had a train of thousands of Bodhisattvas similar to the sands of sixty Ganges rivers; (each had) a troop, a great troop, as teacher of a troop. Of such Bodhisattvas Mahâsattvas having a troop, a great troop, as teachers of a troop, there were hundred thousands of myriads of kotis equal to the sands of sixty Ganges rivers, who emerged from the gaps of the earth in this Saha-world. Much more there were to be found of Bodhisattvas Mahâsattvas having a train of Bodhisattvas similar to the sands of fifty Ganges rivers; much more there were to be found of Bodhisattvas Mahâsattvas having a train of Bodhisattvas similar to the sands of forty Ganges rivers; Of 30, 20, 10, 5, 4, 3, 2, 1 Ganges river; of 1/2, 1/4, 1/6, 1/10, 1/20, 1/50, 1/100, 1/1000, 1/100,000, 1/10,000,000, 1/100 X 10,000,000, 1/1000 X 10,000,000, 1/100 X 1000 X 10,000,000, 1/100 X 1000 X 10,000 X 10,000,000 part of the river Ganges. Much more there were to be found of Bodhisattvas Mahâsattvas having a train of many hundred thousand myriads of kotis of Bodhisattvas; of one koli; of one hundred thousand; of one thousand; Of 500; Of 400; Of 300; Of 200; Of 100; Of 50; Of 40; Of 30; Of 20; Of 10; Of 5, 4, 3, 2. Much more there were to be found of Bodhisattvas Mahâsattvas having one follower. Much more there were to be found of Bodhisattvas Mahâsattvas standing isolated. They cannot be numbered, counted, calculated, compared, known by occult science, the Bodhisattvas Mahâsattvas who emerged from the gaps of the earth to appear in this Saha-world."

[shankara]

How does Buddhism view the power of mathematical reasoning and its high status
in the philosophy of Pythagoras and Plato, as a training method that provides
great assistance in the gradual transformation of human consciousness
from painfully subjective to joyfully objective perception of reality?

Math, logic, has no role in awakening. But they are useful for science and disciplining ones thinking, respectively.

Isn't the Madhyamaka logic? Aren't we supposed to familiarize ourselves with the logical reasons for Shunyata?

I would say quite the opposite, logic has a huge role in awakening. Perhaps a different type of reasoning than in pure mathematics, but nonetheless reasoning.

[PeterC]

How does Buddhism view the power of mathematical reasoning and its high status
in the philosophy of Pythagoras and Plato, as a training method that provides
great assistance in the gradual transformation of human consciousness
from painfully subjective to joyfully objective perception of reality?

Math, logic, has no role in awakening. But they are useful for science and disciplining ones thinking, respectively.

Isn't the Madhyamaka logic? Aren't we supposed to familiarize ourselves with the logical reasons for Shunyata?

I would say quite the opposite, logic has a huge role in awakening. Perhaps a different type of reasoning than in pure mathematics, but nonetheless reasoning.

It’s a very different kind of logic. Just as the examples above aren’t really mathematics by today’s standards. People might have had less confused ideas about shunyata a few thousand years ago, but they knew a lot less about other things than we do.

[shankara]

Math, logic, has no role in awakening. But they are useful for science and disciplining ones thinking, respectively.

Isn't the Madhyamaka logic? Aren't we supposed to familiarize ourselves with the logical reasons for Shunyata?

I would say quite the opposite, logic has a huge role in awakening. Perhaps a different type of reasoning than in pure mathematics, but nonetheless reasoning.

It’s a very different kind of logic. Just as the examples above aren’t really mathematics by today’s standards. People might have had less confused ideas about shunyata a few thousand years ago, but they knew a lot less about other things than we do.

Yeah in some ways it's different, in other ways it's not totally unlike Western philosophy, though unfortunately it's ignored perhaps because of it's "religious" connotations or just because the west doesn't really grasp the import of the eastern philosophical tradition. However logic, philosophy, isn't like a technical science in the sense of making "progress", we don't say that Sarte is better than Hegel because he is later like a new computer is better than an old one.

The Madhyamaka is founded on logical argument, and Buddha's teachings in Sutra etc are also logically coherent.

[PeterC]

Isn't the Madhyamaka logic? Aren't we supposed to familiarize ourselves with the logical reasons for Shunyata?

I would say quite the opposite, logic has a huge role in awakening. Perhaps a different type of reasoning than in pure mathematics, but nonetheless reasoning.

It’s a very different kind of logic. Just as the examples above aren’t really mathematics by today’s standards. People might have had less confused ideas about shunyata a few thousand years ago, but they knew a lot less about other things than we do.

Yeah in some ways it's different, in other ways it's not totally unlike Western philosophy, though unfortunately it's ignored perhaps because of it's "religious" connotations or just because the west doesn't really grasp the import of the eastern philosophical tradition. However logic, philosophy, isn't like a technical science in the sense of making "progress", we don't say that Sarte is better than Hegel because he is later like a new computer is better than an old one.

The Madhyamaka is founded on logical argument, and Buddha's teachings in Sutra etc are also logically coherent.

Do you really think what they’re doing is the same thing as first-order logic?

[shankara]

It’s a very different kind of logic. Just as the examples above aren’t really mathematics by today’s standards. People might have had less confused ideas about shunyata a few thousand years ago, but they knew a lot less about other things than we do.

Yeah in some ways it's different, in other ways it's not totally unlike Western philosophy, though unfortunately it's ignored perhaps because of it's "religious" connotations or just because the west doesn't really grasp the import of the eastern philosophical tradition. However logic, philosophy, isn't like a technical science in the sense of making "progress", we don't say that Sarte is better than Hegel because he is later like a new computer is better than an old one.

The Madhyamaka is founded on logical argument, and Buddha's teachings in Sutra etc are also logically coherent.

Do you really think what they’re doing is the same thing as first-order logic?

I don't know, I guess not, maybe it's not the same kind of logic as mathematics. Nonetheless Buddhist philosophy is logical at least in the sense that Kant or Hegel is logical.

[PeterC]

Yeah in some ways it's different, in other ways it's not totally unlike Western philosophy, though unfortunately it's ignored perhaps because of it's "religious" connotations or just because the west doesn't really grasp the import of the eastern philosophical tradition. However logic, philosophy, isn't like a technical science in the sense of making "progress", we don't say that Sarte is better than Hegel because he is later like a new computer is better than an old one.

The Madhyamaka is founded on logical argument, and Buddha's teachings in Sutra etc are also logically coherent.

Do you really think what they’re doing is the same thing as first-order logic?

I don't know, I guess not, maybe it's not the same kind of logic as mathematics. Nonetheless Buddhist philosophy is logical at least in the sense that Kant or Hegel is logical.

The problem is that the word “logic” denotes an extremely broad set of ideas. At one end it can refer to any sort of argument at all following any or no rules of argumentation; at the other it refers to highly abstracted systems of reasoning. Kant and Hegel are closer to literature. They’re really not in the same ballpark as modern logic. Aside from their somewhat...Germanic writing style, anyone can pick them up and understand what they’re talking about. Pick up a paper on propositional calculus without a background in it and you simply won’t know which direction is up

[Dan74]

I totally agree, that's exactly my problem with mathematics as is taught today,
i.e., without the mystical side which by all evidence the Pythagoreans did emphasize.
Perharps it was a peculiar development in a specific society and thereafter suited only to a peculiar
type of personality.
But it has an appeal, that crystal-clear clarity of deep results and that no-cheating, earnest progress
in learning. Where else can these qualities be found? So as to appeal not only to the intellectual faculty.
That's what I'm trying to answer..

How do you propose maths should be taught 'with the mystical side'? Can you provide a concrete example?

Plato's dialogue Meno is a nice elementary example, where a boy is led to double the area of a square;
which is then viewed as innate knowledge revealed.

An example of a pythagorean concept:
https://en.wikipedia.org/wiki/Tetractys

A old book on pythagorean arithmetic:
http://djm.cc/library/Theoretic_Arithme ... Taylor.pdf

Many other contemporary writers, e.g. Leonid Zhmud.

Regarding the dialogue with Meno's slave, the notion of knowledge as recollection is speculative, it seems to me. But this style of learning which encourages the student to discover, is certainly a wonderful thing and should be promoted as much as the pressures of modern curricula allow.

Regarding the book by Taylor, I had a pretty quick look, and all I can see so far is a great deal of speculative musing on monads and such which doesn't really contribute anything to mathematics. Hmm... perhaps the joy of the discovery in mathematics (which is admittedly rarely found in the classroom) suggest the mystical already? But speculating on the nature of this and that and positing objects will be of course met with a wall of scepticism by modern audiences, and rightly so.

[shankara]

Do you really think what they’re doing is the same thing as first-order logic?

I don't know, I guess not, maybe it's not the same kind of logic as mathematics. Nonetheless Buddhist philosophy is logical at least in the sense that Kant or Hegel is logical.

The problem is that the word “logic” denotes an extremely broad set of ideas. At one end it can refer to any sort of argument at all following any or no rules of argumentation; at the other it refers to highly abstracted systems of reasoning. Kant and Hegel are closer to literature. They’re really not in the same ballpark as modern logic. Aside from their somewhat...Germanic writing style, anyone can pick them up and understand what they’re talking about. Pick up a paper on propositional calculus without a background in it and you simply won’t know which direction is up

Yeah, I'd guess I'm referring more to coherent abstract reasoning as logic than mathematics. Anyway, you have a point about the complexities of mathematics, though I wonder if it might be possible to express the truth of Dharma, or some of it, in mathematical terms.

For example there are religions which believe in eternal suffering as the cost of "sin". Well, any negative action is going to be limited in it's effects, there's no negative action whatsoever which causes an unlimited degree of harm. So, from the mathematical perspective (correct me if I'm wrong, any mathematicians) however many finite numbers one adds to each-other, they will never equal infinity.

[PeterC]

Yeah, I'd guess I'm referring more to coherent abstract reasoning as logic than mathematics. Anyway, you have a point about the complexities of mathematics, though I wonder if it might be possible to express the truth of Dharma, or some of it, in mathematical terms.

For example there are religions which believe in eternal suffering as the cost of "sin". Well, any negative action is going to be limited in it's effects, there's no negative action whatsoever which causes an unlimited degree of harm. So, from the mathematical perspective (correct me if I'm wrong, any mathematicians) however many finite numbers one adds to each-other, they will never equal infinity.

That sounds a bit like the apocryphal "is hell exothermic or endothermic?" argument

If you're arguing the unfairness of infinite punishment for finite transgressions, you won't get any argument from me. The Christian model of 'sin' makes no sense if interpreted in terms of a how an omniscient omnipotent compassionate being relates to its creation. It however makes perfect sense from the perspective of how a mundane authority subjugates a populace.

BTW I'm in no way belittling the form of logic used in the sutras. They reveal an impressive level of reasoning.