How would you counter this hypothesis to the ‘Enlightenment’ idea?

[old3bob]

1 hour ago, Daniel said:

 

"By the power never invested in me...."  Why do I have a craving for pickles?  Was that supposed to happen?  

 

empathy with a pregnant wife who has the same craving?

Edited September 17, 2023 by old3bob

[Michael Sternbach]

On 16.9.2023 at 3:41 PM, galen_burnett said:

Note: the part I officially quoted has nothing to do with my reply here other than referencing the comment of yours that I am here replying to—I should rather have officially quoted the part that I have italicised just below instead… 

 

“But it goes without saying that some of the other Buddhist schools are more talkative in this regard, and they also take different stands on the nature of enlightenment and ultimate reality.”

 

Do you yourself have a take on “ultimate reality”?

 

Yes. It has to do with multidimensional space and time, and what C.G. Jung called "psychoid space."

 

On 16.9.2023 at 3:41 PM, galen_burnett said:

If so, is “ultimate” reality more valid than the “ordinary” or “non-ultimate” reality?

 

No. It is effectively non-dual with the world of our daily experience. 

 

On 16.9.2023 at 3:41 PM, galen_burnett said:

Have I got this right?: in this reply of yours to Stirling you are saying that indeed the notion of an attainable ‘heaven’ exists in Eastern philosophies—‘the pure land’; then you are saying that you are trying to work out for yourself what these philosophies mean by the Void, through comparisons with other philosophies like that of Plato, Socrates and Pythagoras, and, by extension from the Void, what is meant by the Non-Dual.

 

Basically yes. With its insistence on the phenomenal world being the result of interactions that are essentially devoid of any deeper reality, Buddhism seems to be at odds with the Platonist view of archetypal forms and determining principles existing on a level beyond the manifest universe. I don't believe that these views are actually irreconcilable with each other, thus I am searching for the philosophical "missing links."

 

Also, Buddhism and other Eastern systems offer a variety of powerful methodologies to directly experience the numinous--something that is somewhat (though not entirely) lacking in the Platonist and Hermetic traditions, which are more intellectually oriented, overall.

[Daniel]

2 hours ago, old3bob said:

 

empathy with a pregnant wife who has the same craving?

 

pickled is slang for "drunk or tipsy".  So what ever it is that Mark was sending my way, was interpretted/translated through the ether as "getting pickled sounds good right about now."

[Daniel]

7 minutes ago, Michael Sternbach said:

It is effectively non-dual with the world of our daily experience. 

 

which is dualistic.  if it is non-dual, truly non-dual, then the word "with" has no meaning.

[Mark Foote]

15 hours ago, Daniel said:

 

I understand Cantor's proof, which you brought of transfinite numbers, to be the proof of no gaps in uncountable infinity.  I think there is also something somewhere that shows the specctrum of light is a physical, real world, objective example of uncountable infinity.

 

Let's see if I can find it.... Here you go.  University of Illinois.

 

 

 

 

Great!  Did you notice this matches exactly what I said?

 

I said:  "A paradox is simultaneous opposing concepts which when linked are not always false"

 

Seemingly contraditcory = opposing concepts

yet is perhaps true = are not always false

 

If you are looking at the B defintion, I don't think those were standing in the way of "actual infinity".  I think it was russel's paradox from the early 1900s.  It's no more a contradiction than the "liar's paradox".  It's looping self-references.

 

 

Godel's work is more complicated, but, I think the solution that he brought for the set of all sets is similar.  It's just semantics.  Another layer of abstraction is introduced beyond the set.

 

 

It's, of course, your choice how you perceive reality.  Absolute literal infinity can be mathematically justified. And this version of infinity is lacking any gaps.  But just like many other natural phenomena that are beyond the physical senses, it makes perfect sense to ignore it.  And, I can understand why some find it rewarding to deny it.

 

Where do you stand on the law of the excluded middle?

ok, never mind, I see that in the rarified air of the attempts to get around the Russell paradox, the approach of the Intuitionists is not necessarily any better.  But lo and behold:
 

It is also worth noting that Russell’s paradox was not the only paradox that troubled Russell and, hence, not the only motivation for the type restrictions one finds in Principia Mathematica. In his earlier work, The Principles of Mathematics, Russell devotes a chapter to “the Contradiction” (Russell’s paradox), presenting it in several forms and dismissing several non-starter responses. He then signals that he will “shortly” discuss the doctrine of types. This doesn’t happen for several hundred pages, until we reach the very end of the book, in Appendix B! There Russell presents an incipient, simple theory of types, not the theory of types we find in Principia Mathematica. Why was the later theory needed? The reason is that in Appendix B Russell also presents another paradox which he thinks cannot be resolved by means of the simple theory of types. This new paradox concerns propositions, not classes, and it, together with the semantic paradoxes, led Russell to formulate his ramified version of the theory of types.
 

The new, propositional version of the paradox has not figured prominently in the subsequent development of logic and set theory, but it sorely puzzled Russell. For one thing, it seems to contradict Cantor’s theorem. Russell writes: “We cannot admit that there are more ranges [classes of propositions] than propositions” (1903, 527). The reason is that there seem to be easy, one to one correlations between classes of propositions and propositions. For example, the class m of propositions can be correlated with the proposition that every proposition in m is true. This, together with a fine-grained principle of individuation for propositions (asserting, for one thing, that if the classes m
 and n of propositions differ, then any proposition about m will differ from any proposition about 
n) leads to contradiction.

 

There has been relatively little discussion of this paradox, although it played a key role in the development of Church’s logic of sense and denotation. While we have several set theories to choose from, we do not have anything like a well-developed theory of Russellian propositions, although such propositions are central to the views of Millians and direct-reference theorists. One would think that such a theory would be required for the foundations of semantics, if not for the foundations of mathematics. Thus, while one of Russell’s paradoxes has led to the fruitful development of the foundations of mathematics, his “other” paradox has yet to lead to anything remotely similar in the foundations of semantics. To be sure, Church (1974a) and Anderson (1989) have attempted to develop a Russellian intensional logic based on the ramified theory of types, but an argument can be made that the ramified theory is too restrictive to serve as a foundation for the semantics of natural language.

... Russell’s paradox has never been passé, but recently there has been an explosion of interest in it by scholars involved in research in mathematical logic and in philosophical and historical studies of modern logic. A glance at the contents of the 2004 volume One Hundred Years of Russell’s Paradox shows prominent mathematical and philosophical logicians and historians of logic poring over the paradox, proposing new ways back into Cantor’s paradise, or other ways of resolving the issue. Their investigations include radically new ways out of the dilemma posed by the paradox, new studies of the theories of types (simple and ramified, and extensions thereof), new interpretations of Russell’s paradox and constructive theories, of Russell’s paradox of propositions and of his own attempt at an untyped theory (the substitution theory), and so forth.
 

All of this reminds us that fruitful work can arise from the most unlikely of observations. As Dana Scott has put it, “It is to be understood from the start that Russell’s paradox is not to be regarded as a disaster. It and the related paradoxes show that the naïve notion of all-inclusive collections is untenable. That is an interesting result, no doubt about it."

(https://plato.stanford.edu/entries/russell-paradox/#ERP)

 

 

Note "... new ways back into Cantor's paradise".  I take that to mean, transcendental numbers are still questionable, in spite of the theory of types.  That being so, I still respect the approach of the Intuitionists, in only accepting constructed infinity, and not something that is here, there, and everywhere.

Edited September 17, 2023 by Mark Foote

[Daniel]

8 minutes ago, Mark Foote said:

Where do you stand on the law of the excluded middle?

 

It depends.  In general, it works for positive assertions, but not negative.  "It's false" does not gaurantee the opposite is "true".  See: Aristotle's Theorum.  From falsehood nothing.  

 

In particular, I try very hard to judge true / false on a case by case basis.  Although, I'm not perfect. Sometimes I'm hasty, lazy, or just plain wrong. 

 

Edited September 17, 2023 by Daniel

[Daniel]

12 minutes ago, Mark Foote said:

Where do you stand on the law of the excluded middle?

 

In case I misinterpretted the question.  Since I was talking about spectrums, my position is almost everything operates on a spectrum.  There are rare extreme phenomena which are black/white, open/closed, etc. dichotomies.

 

This does make life and decision making both difficult and delicious, for me.  Just speaking for myself.

 

Edited September 17, 2023 by Daniel

[Cobie]

25 minutes ago, Daniel said:

pickled is slang for "drunk or tipsy".  So what ever it is that Mark was sending my way, was interpretted/translated through the ether as "getting pickled sounds good right about now."

Yeah, I would have never guessed that. 

 

3 hours ago, old3bob said:

empathy with a pregnant wife who has the same craving?

 

That’s what I thought too. 

 

 

Edited September 17, 2023 by Cobie

[Mark Foote]

21 minutes ago, Daniel said:

 

It depends.  In general, it works for positive assertions, but not negative.  "It's false" does not gaurantee the opposite is "true".  See: Aristotle's Theorum.  From falsehood nothing.  

 

In particular, I try very hard to judge true / false on a case by case basis.  Although, I'm not perfect. Sometimes I'm hasty, lazy, or just plain wrong. 

 

You're too quick, Daniel!  I added a bunch of material to that post, on account of a very interesting article from someone at Stanford, which I quote from.

[Michael Sternbach]

50 minutes ago, Daniel said:

 

which is dualistic.  if it is non-dual, truly non-dual, then the word "with" has no meaning.

 

Non-dual isn't the same as identical. The spiritual world is neither identical with the physical world, nor is it truly separate from the latter. If I find a better word than 'non-dual' for defining their relationship, I will let you know.

[Daniel]

1 hour ago, Mark Foote said:

Where do you stand on the law of the excluded middle?

ok, never mind,

 

Nevermind??   I just answered. 

 

Oooh!  Look at all that yummy philosophy you added in the edit.  Let's see what it says.  Now I'm back to happy.  

 

 

Quote

I see that in the rarified air of the attempts to get around the Russell paradox, the approach of the Intuitionists is not necessarily any better.  But lo and behold:

 

Personally, I don't see a single thing wrong with russel's paradox.  I see no real need to "get around" it.  It doesn't make the "set of all sets excluding itself" any less absolutely literally infinite.  It's still literally absolutely infinite.  It just doesn't loop into itself.  Who cares?  It's semantics.  That's why it's so easy to avoid it.  Leibniz can remain happy wherever he is.  

 

 

Quote

 

It is also worth noting that Russell’s paradox was not the only paradox that troubled Russell and, hence, not the only motivation for the type restrictions one finds in Principia Mathematica. In his earlier work, The Principles of Mathematics, Russell devotes a chapter to “the Contradiction” (Russell’s paradox), presenting it in several forms and dismissing several non-starter responses. He then signals that he will “shortly” discuss the doctrine of types. This doesn’t happen for several hundred pages, until we reach the very end of the book, in Appendix B! There Russell presents an incipient, simple theory of types, not the theory of types we find in Principia Mathematica. Why was the later theory needed? The reason is that in Appendix B Russell also presents another paradox which he thinks cannot be resolved by means of the simple theory of types. This new paradox concerns propositions, not classes, and it, together with the semantic paradoxes, led Russell to formulate his ramified version of the theory of types.
 

The new, propositional version of the paradox has not figured prominently in the subsequent development of logic and set theory, but it sorely puzzled Russell. For one thing, it seems to contradict Cantor’s theorem. Russell writes: “We cannot admit that there are more ranges [classes of propositions] than propositions” (1903, 527). The reason is that there seem to be easy, one to one correlations between classes of propositions and propositions. For example, the class m of propositions can be correlated with the proposition that every proposition in m is true. This, together with a fine-grained principle of individuation for propositions (asserting, for one thing, that if the classes m
 and n of propositions differ, then any proposition about m will differ from any proposition about 
n) leads to contradiction.

 

Question:  Are you understanding this "contradiction"?  And is it at all related to infinity of any type?  Is this a set-of-all-sets defeater?

 

"the class m of propositions can be correlated with the proposition that every proposition in m is true. This, together with a fine-grained principle of individuation for propositions (asserting, for one thing, that if the classes m and n of propositions differ, then any proposition about m will differ from any proposition about n) leads to contradiction."

 

I'd need to spend some time and energy to understand this.  There needs to be opposing concepts, which, when linked, are always false.

 

Quote

 

There has been relatively little discussion of this paradox, although it played a key role in the development of Church’s logic of sense and denotation. While we have several set theories to choose from, we do not have anything like a well-developed theory of Russellian propositions, although such propositions are central to the views of Millians and direct-reference theorists. One would think that such a theory would be required for the foundations of semantics, if not for the foundations of mathematics.

 

OK.  It's supposed to be a contradiction prohibiting the formation of semantics, if not, the foundations of mathematics.

 

So, it seems like they are not talking about absolute infinity at all.  Just another problem of similar import.  Which, again, I do not get at all.  The foundations of math, in my best judgement, would not have even the tiniest fault if the the largest set able to be described consistently was "the set of all sets excluding itself".

 

 

Quote

 

Thus, while one of Russell’s paradoxes has led to the fruitful development of the foundations of mathematics

 

Oh gee whiz.  Really?  The foundations of mathematics were developed long before that.  At least the 1600s with Leibniz.  But arguably something like 500BCE. 

 

Quote

 

 

, his “other” paradox has yet to lead to anything remotely similar in the foundations of semantics. To be sure, Church (1974a) and Anderson (1989) have attempted to develop a Russellian intensional logic based on the ramified theory of types, but an argument can be made that the ramified theory is too restrictive to serve as a foundation for the semantics of natural language.

 

So basically it's a nothing-burger?

 

Quote

 

 

... Russell’s paradox has never been passé, but recently there has been an explosion of interest in it by scholars involved in research in mathematical logic and in philosophical and historical studies of modern logic. A glance at the contents of the 2004 volume One Hundred Years of Russell’s Paradox shows prominent mathematical and philosophical logicians and historians of logic poring over the paradox, proposing new ways back into Cantor’s paradise, or other ways of resolving the issue. Their investigations include radically new ways out of the dilemma posed by the paradox, new studies of the theories of types (simple and ramified, and extensions thereof), new interpretations of Russell’s paradox and constructive theories, of Russell’s paradox of propositions and of his own attempt at an untyped theory (the substitution theory), and so forth.
 

 

 

"Cantor's paradise".  Love-it.  

 

Quote

All of this reminds us that fruitful work can arise from the most unlikely of observations. As Dana Scott has put it, “It is to be understood from the start that Russell’s paradox is not to be regarded as a disaster. It and the related paradoxes show that the naïve notion of all-inclusive collections is untenable. That is an interesting result, no doubt about it."

(https://plato.stanford.edu/entries/russell-paradox/#ERP)

 

Important note:  "all-inclusive collections" = a set.  Not a category.

 

 

Quote

 

Note "... new ways back into Cantor's paradise".  I take that to mean, transcendental numbers are still questionable, in spite of the theory of types.  That being so, I still respect the approach of the Intuitionists, in only accepting constructed infinity, and not something that is here, there, and everywhere.

 

No, not at all.  Russel's paradox and transcendental numbers are not related in any way.

 

A russel set includes anything you can imagine and more, excluding itself.  That's it.  it is literally here, there, and everywhere.

 

Edited September 17, 2023 by Daniel

[Mark Foote]

10 hours ago, Michael Sternbach said:

 

I mean, insofar many tomes of teachings have been added by the various Buddhist schools later.

 

 

I simply noticed that you are referring a lot to the Pali canon--arguably the most original Buddhist texts. And since, on the other hand, you are a Soto Zen practitioner (a tradition known for its frugality), I wondered if you see the Pali canon as the essential text.
 

Beat me, whip me, make me quote my own material (again):

 

Dogen emphasized the practice of zazen, literally “seated Zen”. Wikipedia describes the derivation of the word “Zen” as follows:
 

The term Zen is derived from the Japanese pronunciation of the Middle Chinese word chán, an abbreviation of chánnà, which is a Chinese transliteration of the Sanskrit word dhyāna (“meditation”).

 

Yogapedia provides a definition of “dhyana” based on the Sanskrit roots of the word:
 

Dhyana is a Sanskrit word meaning “meditation.” It is derived from the root words, dhi, meaning “receptacle” or “the mind”; and yana, meaning “moving” or “going.”
 

(dhyana, dec. 9 2017, “Yogapedia”, authorship not ascribed; https://www.yogapedia.com/definition/5284/dhyana)

 

Dhyana could therefore be said to translate literally as “mind moving”.
 

The sixth patriarch of Zen in China pointed directly to the mind moving, in a case from the “Gateless Gate” collection:
 

Not the Wind, Not the Flag
 

Two monks were arguing about a flag. One said: “The flag is moving.”
 

The other said: “The wind is moving.”
 

The sixth patriarch happened to be passing by. He told them: “Not the wind, not the flag; mind is moving.”

 

Mumon’s comment: 

The sixth patriarch said: “The wind is not moving, the flag is not moving. Mind is moving.” What did he mean? If you understand this intimately, you will see the two monks there trying to buy iron and gaining gold. The sixth patriarch could not bear to see those two dull heads, so he made such a bargain.
 

Wind, flag, mind moves,
The same understanding.
When the mouth opens
All are wrong.
 

(The Gateless Gate, by Ekai (called Mu-mon), tr. Nyogen Senzaki and Paul Reps [1934], at sacred-texts.com)

 

... Ekai claims that the sixth patriarch said:  “the flag is not moving, the wind is not moving”.  He’s putting words in the mouth of the Sixth Patriarch, there.  To me, what the sixth patriarch said was, pay attention to the singularity of self-awareness that moves, not to the flag or the wind.
 

Ekai says, “if you understand this intimately”.  To understand intimately is to experience movement in the location of self-awareness, of mind, for oneself.  To understand in words without experience falls short (“when the mouth opens, all are wrong”).
 

For me, it’s a lot like falling asleep.  I have to let myself breathe–relax, calm down, let go of thoughts, and realize some presence of mind.  As the senses locate the presence of mind, particularly the senses concerned with balance, the location of mind may move.

 

(Not the Wind, Not the Flag)

 

 

The necessity that places attention in the movement of breath, that's hard to lay hold of.  That to me is the nature that Shunryu Suzuki referred to, that we all have.  

As I wrote before, I believe on this thread:

There’s a frailty in the structure of the lower spine, and the movement of breath can place the point of awareness in such a fashion as to engage a mechanism of support for the spine, often in stages.

 

Gautama was able to say something about those stages, both in terms of their delineation, and also in terms of the feeling associated with each.  Dogen really only specified two stages, plus something that happens sometimes in connection with those two stages:

When you find your place where you are, practice occurs, actualizing the fundamental point. When you find your way at this moment, practice occurs, actualizing the fundamental point… Although actualized immediately, the inconceivable may not be apparent.

("Genjo Koan", tr Tanahashi)

For me, I look increasingly to Gautama's descriptions.  I think a case could be made that most of the principals in Ch'an and Zen were familiar with at least some of the Pali sermons, so to that extent, there are some teachings in common.  The emphasis on "mind to mind transmission outside of scripture", I think that came later in Ch'an--I don't see evidence of it in the teachings of Yuanwu in "Letters" or "The Blue Cliff Record", but maybe with the disciple of his who attempted to destroy all copies of "The ... Record".  WIth "mind to mind transmission outside of scripture", the lineage becomes the thing, even though passing down the robe and bowl was discontinued in the 6th century C. E..  

My take.

[Cobie]

15 minutes ago, Michael Sternbach said:

…  'non-dual' .. 

 

Lookup dictionary on my iPad: “nondualism  … is a fuzzy concept …”

 indeed. 
 

 

Edited September 17, 2023 by Cobie

[Mark Foote]

14 minutes ago, Daniel said:

 

Russel's paradox and transcendental numbers are not related in any way.

 

 

An interesting reply on StackExchange to the question, Does Cantor's Theorem require Russell's paradox? (reply #2):

 

In the latter 19th century when Set Theory as an area of general study was beginning, it was often assumed that we could assume the existence of the set of all and only those things that had any specific property P. This is known as the Axiom Schema of Abstraction. (A "schema" because it it an infinite list of axioms, one for each property P that you can state.) Russell showed this was illogical because the assumption that {x:x∉x} exists is paradoxical.(Note: It does not depend on any definition of what ∈ means. Russell offered the Barber Paradox to illustrate this: A barber shaves all those and only those who don't shave themselves. Does the barber shave the barber? For barber, read "set". For shaves, read "contains as a member".)
 

One remedy was to eliminate Abstraction and replace it with the schema of Comprehension (Specification): Informally it says that if X is a set then there exists a set Y whose members are all, and only, those members of X that have some specified property. The crucial difference is that, although we can say that if a set X exists then Y={x∈X:x∉x} exists, we cannot prove from Comprehension that {x:x∉x} exists.
 

Comprehension also implies, by contradiction, there is no set V of all sets. Otherwise we would have the set {x∈V:x∉x}, which would be {x:x∉x} and we'd have Russell's Paradox again.
 

As already stated in other responses to your Q, Cantor's theorem does employ an instance of Comprehension.
 

BTW. The original names for some (most?) of the axioms of modern set theory were not English and different writers have at times used different English names for them. "Extensionality" (Informally, sets X and Y are equal iff they have the same members) is also called Regularity... And some textbooks combine the Comprehension schema and the Separation schema into a single schema, which they also call Separation.

(Aug 19, 2018; DanielWainfleet; emphasis added)

Plus, 'way down in the comments:

 

Without Comprehension (Specification) we cannot guarantee, in proving Cantor's theorem, that R={x∈S:x∉f(x)} will always exist. We need a few more axioms too, in order to give a set-theoretic definition of "function" 
 

– DanielWainfleet

 Aug 22, 2018

If I'm reading all this correctly (and it's mostly over my head), Cantor's paradise requires some fancy footwork around Russell's paradox.  And from the Stanford excerpt I posted, would seem that the situation is really not resolved:  "Cantor's paradise", can you get there at all.

[Daniel]

2 hours ago, Mark Foote said:

 

An interesting reply on StackExchange to the question, Does Cantor's Theorem require Russell's paradox? (reply #2):

 

In the latter 19th century when Set Theory as an area of general study was beginning, it was often assumed that we could assume the existence of the set of all and only those things that had any specific property P. This is known as the Axiom Schema of Abstraction. (A "schema" because it it an infinite list of axioms, one for each property P that you can state.) Russell showed this was illogical because the assumption that {x:x∉x} exists is paradoxical.(Note: It does not depend on any definition of what ∈ means. Russell offered the Barber Paradox to illustrate this: A barber shaves all those and only those who don't shave themselves. Does the barber shave the barber? For barber, read "set". For shaves, read "contains as a member".)
 

One remedy was to eliminate Abstraction and replace it with the schema of Comprehension (Specification): Informally it says that if X is a set then there exists a set Y whose members are all, and only, those members of X that have some specified property. The crucial difference is that, although we can say that if a set X exists then Y={x∈X:x∉x} exists, we cannot prove from Comprehension that {x:x∉x} exists.
 

Comprehension also implies, by contradiction, there is no set V of all sets. Otherwise we would have the set {x∈V:x∉x}, which would be {x:x∉x} and we'd have Russell's Paradox again.
 

As already stated in other responses to your Q, Cantor's theorem does employ an instance of Comprehension.
 

BTW. The original names for some (most?) of the axioms of modern set theory were not English and different writers have at times used different English names for them. "Extensionality" (Informally, sets X and Y are equal iff they have the same members) is also called Regularity... And some textbooks combine the Comprehension schema and the Separation schema into a single schema, which they also call Separation.

(Aug 19, 2018; DanielWainfleet; emphasis added)

Plus, 'way down in the comments:

 

Without Comprehension (Specification) we cannot guarantee, in proving Cantor's theorem, that R={x∈S:x∉f(x)} will always exist. We need a few more axioms too, in order to give a set-theoretic definition of "function" 
 

– DanielWainfleet

 Aug 22, 2018

If I'm reading all this correctly (and it's mostly over my head), Cantor's paradise requires some fancy footwork around Russell's paradox.  And from the Stanford excerpt I posted, would seem that the situation is really not resolved:  "Cantor's paradise", can you get there at all.

 

 

Well, that's odd, because, the proof of transfinite numbers, uncountable infinity, doesn't require any set theory at all.  It's a matrix, an array.  It can be constructed by anyone.  I'm not really sure how this works either.  Stack Exchange is a forum.  Yes it has social confirmation, up votes / down votes.  But I still struggle to see how self reference defeats the simple idea there are some infinite different wavelengths of light between "red" and "orange" with absolutely zero gaps.

 

Here's just a little simple confirmation from the wiki-monster on the semantic workaround for absolute literal infinity.

 

https://en.m.wikipedia.org/wiki/Category_of_small_categories

 

The category Cat is itself a large category, and therefore not an object of itself. In order to avoid problems analogous to Russell's paradox one cannot form the “category of all categories”. But it is possible to form a quasicategory (meaning objects and morphisms merely form a conglomerate) of all categories.

[Mark Foote]

17 hours ago, Daniel said:

 

 

Well, that's odd, because, the proof of transfinite numbers, uncountable infinity, doesn't require any set theory at all.  It's a matrix, an array.  It can be constructed by anyone.  I'm not really sure how this works either.  Stack Exchange is a forum.  Yes it has social confirmation, up votes / down votes.  But I still struggle to see how self reference defeats the simple idea there are some infinite different wavelengths of light between "red" and "orange" with absolutely zero gaps.

 

Here's just a little simple confirmation from the wiki-monster on the semantic workaround for absolute literal infinity.

 

https://en.m.wikipedia.org/wiki/Category_of_small_categories

 

The category Cat is itself a large category, and therefore not an object of itself. In order to avoid problems analogous to Russell's paradox one cannot form the “category of all categories”. But it is possible to form a quasicategory (meaning objects and morphisms merely form a conglomerate) of all categories.
 

I'll admit that I am following the barest thread of the arguments.  I still regard Godel's proofs as the greatest work in mathematics, ever, for showing that there is no consistent set of axioms that can encompass all that is known in mathematics.  That is philosophy, I agree!

The deal with Cantor's diagonal proof, as I understand it:  he assumes a completed infinity, in both the counting numbers and the real numbers.  That's the assumption that gives rise to the contradictions.

He's clearly making a constructed set, and that's real, until he considers it completed and draws his conclusion about the sizes of the two infinities.

That's the way I understand it.  From Wikipedia, "actual infinity":
 

During the Renaissance and by early modern times the voices in favor of actual infinity were rather rare.
 

The continuum actually consists of infinitely many indivisibles (G. Galilei [9, p. 97])
 

I am so in favour of actual infinity. (G.W. Leibniz [9, p. 97])

 

However, the majority of pre-modern thinkers[citation needed] agreed with the well-known quote of Gauss:
 

I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.[9] (C.F. Gauss [in a letter to Schumacher, 12 July 1831])

 

 

Actual infinity is now commonly accepted. The drastic change was initialized by Bolzano and Cantor in the 19th century.

Bernard Bolzano, who introduced the notion of set (in German: Menge), and Georg Cantor, who introduced set theory, opposed the general attitude. Cantor distinguished three realms of infinity:

(1) the infinity of God (which he called the "absolutum"),
(2) the infinity of reality (which he called "nature") and
(3) the transfinite numbers and sets of mathematics.

 

 

... One proof is based on the notion of God. First, from the highest perfection of God, we infer the possibility of the creation of the transfinite, then, from his all-grace and splendor, we infer the necessity that the creation of the transfinite in fact has happened. (G. Cantor [3, p. 400])

... One of the most vigorous and fruitful branches of mathematics [...] a paradise created by Cantor from which nobody shall ever expel us [...] the most admirable blossom of the mathematical mind and altogether one of the outstanding achievements of man's purely intellectual activity. (D. Hilbert on set theory)

 

 

Aw, too bad, Hilbert--Godel threw you out of paradise!

More from Wikipedia:
 

There are at least two different ways of looking at the numbers: as a completed infinity and as an incomplete infinity... regarding the numbers as an incomplete infinity offers a viable and interesting alternative to regarding the numbers as a completed infinity, one that leads to great simplifications in some areas of mathematics and that has strong connections with problems of computational complexity. (E. Nelson)
 

During the renaissance, particularly with Bruno, actual infinity transfers from God to the world. The finite world models of contemporary science clearly show how this power of the idea of actual infinity has ceased with classical (modern) physics. Under this aspect, the inclusion of actual infinity into mathematics, which explicitly started with G. Cantor only towards the end of the last century, seems displeasing. Within the intellectual overall picture of our century ... actual infinity brings about an impression of anachronism. (P. Lorenzen)
 

 

There is also a paragraph in the Wikipedia article titled "Current Mathematical Practice", seemingly at odds with the drift of the rest of the article, that says:

 

... The ability to define ordinal numbers in a consistent, meaningful way, renders much of the debate moot; whatever personal opinion one may hold about infinity or constructability, the existence of a rich theory for working with infinities using the tools of algebra and logic is clearly in hand.

 

This section of the page shows no references, in contrast to the rest of the page (I've removed the references from my other quotes, for the sake of simplicity).

 

On the spectrum--not saying that there are gaps in the spectrum.  I don't see where that says anything about there being different sizes of infinity--?

 

Edited September 18, 2023 by Mark Foote

[Daniel]

1 hour ago, Mark Foote said:

I'll admit that I am following the barest thread of the arguments.

 

They begin by considering {x:x∉x}.  This is "set builder" notation where x a set which conforms to the function x∉x.  That's a sematic contradiction.  The fault is the negating self reference.  x is being defined as something which is not itself and nothing else.

 

Quote

I still regard Godel's proofs as the greatest work in mathematics, ever, for showing that there is no consistent set of axioms that can encompass all that is known in mathematics.  That is philosophy, I agree!

 

I don't know any of his work.  It's been recommended by someone I trust when they noticed me getting excited about sympathetic paradoxes ( def A of paradox, that you brought earlier ).

 

Question:  If there is no consistent set of axioms... as stated above, what are the implications, in your view, on the version of infinity I have described and have been calling absolutley literal infinity?

 

Quote

The deal with Cantor's diagonal proof, as I understand it:  he assumes a completed infinity, in both the counting numbers and the real numbers.  That's the assumption that gives rise to the contradictions.

 

Yes.  The completed infinity is "assumed for coontradiction".  Once it is shown it cannot ever be complete, then it is proven to be incomplete by contradiction.  Incomplete in this context means "uncountable" a perfect unbroken spectrum between any and all, let's call them, way-points, or coordinates. 

 

Quote

He's clearly making a constructed set, and that's real, until he considers it completed and draws his conclusion about the sizes of the two infinities.

 

Agreed!  But, I'm not a fan of using the word "size". The cardinality of the set is larger.

 

Quote

Actual infinity is now commonly accepted. The drastic change was initialized by Bolzano and Cantor in the 19th century.

 

Bernard Bolzano, who introduced the notion of set (in German: Menge), and Georg Cantor, who introduced set theory, opposed the general attitude. Cantor distinguished three realms of infinity:

(1) the infinity of God (which he called the "absolutum"),
(2) the infinity of reality (which he called "nature") and
(3) the transfinite numbers and sets of mathematics.

 

OK.  yes, he was religious.  And thats important.

 

Quote

 

... One proof is based on the notion of God. First, from the highest perfection of God, we infer the possibility of the creation of the transfinite, then, from his all-grace and splendor, we infer the necessity that the creation of the transfinite in fact has happened. (G. Cantor [3, p. 400])

... One of the most vigorous and fruitful branches of mathematics [...] a paradise created by Cantor from which nobody shall ever expel us [...] the most admirable blossom of the mathematical mind and altogether one of the outstanding achievements of man's purely intellectual activity. (D. Hilbert on set theory)

 

Aw, too bad, Hilbert--Godel threw you out of paradise!

 

Again, I'm not seeing any conflict.

 

Quote

More from Wikipedia:
 

There are at least two different ways of looking at the numbers: as a completed infinity and as an incomplete infinity... regarding the numbers as an incomplete infinity offers a viable and interesting alternative to regarding the numbers as a completed infinity, one that leads to great simplifications in some areas of mathematics and that has strong connections with problems of computational complexity. (E. Nelson)

 

OK.... Cantor's god is more infinite than the incomplete infinty.  Incomplete infinity is more infinite than complete infinty.

 

complete infinity << incomplete infinity << Cantor's god

 

<< = "much less than" 

 

Quote

During the renaissance, particularly with Bruno, actual infinity transfers from God to the world. The finite world models of contemporary science clearly show how this power of the idea of actual infinity has ceased with classical (modern) physics. Under this aspect, the inclusion of actual infinity into mathematics, which explicitly started with G. Cantor only towards the end of the last century, seems displeasing. Within the intellectual overall picture of our century ... actual infinity brings about an impression of anachronism. (P. Lorenzen)

 

Actual.  Meaning it is something which can be used as a discrete starting point for Leibniz's "theory of everything" aka "set theory".  The fault of this "theory of everything" was asuming that this "theory of everything" could indeed be completed.  Now you have Godel saying, the "Theory of Everything" never ends.  And this is consistent with Russel's paradox.  But is has zero bearing on literal absolute infinity as I have described it.  There is only 1 absolute literal infinity.  It cannot ever be included in itself, else, there would be 2 of them.  So, it's not just that Russel's paradox is avoided in what I'vee described, its completely irrelevant.  it is never-ever a consideration.  It is, basically, a Russell set and that's perfect!  There is only one of them.

 

Quote

On the spectrum--not saying that there are gaps in the spectrum.  I don't see where that says anything about there being different sizes of infinity--?

 

In context:  the discussion was omni-presence and approaching the infinite-divine.  I said, it cannot be appproached, because, if it is [absolutely literally] infinite, then, it is omni-present.  Then, you seemed to challenge this claiming there are gaps in numeric infinity.  And there was another comment?  "there's us and it.  Never the 'twain will meet.  It is not here there and everywhere"  Or perhaps I'm remembering wrong.

 

Size cardinality as they say, doesn't matter.  It's the motion of the ocean... or something.

 

The point is, uncountable infinity has no gaps.  Spectrums are physical real world objective examples of this.  Absolute literal infinity is much-much greater than uncountable infinity.  Therefore there are no gaps in absolute literal infinity.  Conclusion:  IF the divine is absolutely literally infinite, THEN it is certainly omni-present and it cannot be approached, it's already here, there, everywhere.  

 

How is a person going to appproach it?  No matter where they go, it's still there.  They have neither gone towards it or away from it no matter where they roam, in time and space, even in no-time and no-space, even in the realm of "could-be". 

 

Edited September 18, 2023 by Daniel

[Mark Foote]

On 9/17/2023 at 11:59 AM, Daniel said:

 

pickled is slang for "drunk or tipsy".  So what ever it is that Mark was sending my way, was interpreted/translated through the ether as "getting pickled sounds good right about now."
 

Can't be pickling anymore.  The heart is tattooed, and the doc says just the one thing would undo that beautiful work.

[Mark Foote]

On 9/18/2023 at 11:13 AM, Daniel said:

 

They begin by considering {x:x∉x}.  This is "set builder" notation where x a set which conforms to the function x∉x.  That's a sematic contradiction.  The fault is the negating self reference.  x is being defined as something which is not itself and nothing else.
 

 

 

I think the notation is "set x where x is not an element of itself" (but I could be wrong!).  The set of all sets that are not elements of themselves, the paradox.

 

Quote

 

 Now you have Godel saying, the "Theory of Everything" never ends. 

 

"The theory of everything" was Hawking's.  He gave up on it, in 2002:

 

Godel and the End of Physics

So, yes, I guess it never ends, per the lecture.

Edited September 20, 2023 by Mark Foote

[Mark Foote]

On 9/16/2023 at 7:30 AM, galen_burnett said:

 

“[…] happiness has ceased apart from equanimity […]”

 

Please explain what is meant by this line.
 

 

The equanimity of the initial states of concentration is equanimity with respect to the senses:

 

... equanimity in face of multiformity, connected with multiformity… [which is] equanimity among material shapes, among sounds, smells, flavours, touches.

 

(MN III 220, Pali Text Society Vol III p 268)

 

I swear there is a simple and direct quote somewhere in the sermons, where he cites the things that cease in each of the four initial states of concentration, but I'm not coming up with it.  Next best thing:
 

By getting rid of joy, by getting rid of anguish, by the going down of [one’s] former pleasures and sorrows, [one enters] into and [abides] in the fourth meditation which has neither anguish nor joy, and which is entirely purified by equanimity and mindfulness.
 

(MN 1 22, Vol I pg 28)


... by abandoning both ease and discomfort, by the ending of both happiness and unhappiness felt before, (one) attains and abides in the fourth trance, a state of neither ease nor discomfort, an equanimity of utter purity.  

(SN V 215, Pali Text Society vol V p 190)

 

 

Whoever, Ananda, should speak thus… And what, Ananda is the other happiness more excellent and exquisite than that happiness? Here, Ananda, [an individual], by getting rid of happiness and by getting rid of anguish, by the going down of [their] former pleasures and sorrows, enters and abides in the fourth meditation which has neither anguish nor happiness, and which is entirely purified by equanimity and mindfulness. This, Ananda, is the other happiness that is more excellent and exquisite than that happiness (the happiness of the third "meditation", or concentration).

 

(MN I 398-399, Vol II pg 67; material in parenthesis added)

 

 

That last, not the best translation by Horner in my opinion, because of her pairing of "which has neither anguish nor happiness" with "this... is the other happiness...".  However, if we piece it together from the translations above, there's a happiness that ceases in the fourth concentration (what was felt before), and a happiness that is felt, in connection with the equanimity and mindfulness of the concentration. 

For me, there comes a moment where necessity in the movement of breath can place attention anywhere in the body "with no particle left out", and there's an openness to the senses that's required in that.  The activity of the body follows from the location of awareness at that time, I guess there's a sense of well-being.  Is that happiness?  It's a draw, anyway.

 

 

Quote

 

“I get it that things beyond the range of the senses can be involved in walking me around.” 

 

Please explain what you mean by “walking me around”.
 

 

"You know, sometimes zazen gets up and walks around."--Kobun Chino Otogawa at the S.F. Zen Center

 

It's like a hypnosis, where the hypnotist makes a suggestion that you get up and walk around (and you do so without willing your body to move), but the hypnotist is on the other side of the wall.

 

 

Quote

 

“The notion that "I am the doer, mine is the doer with regard to this consciousness-informed body" has taken a hit, for me.”

 

Are you trying to describe the experience of ‘being breathed’ here?
 

 

Maybe--do you have the experience of "being breathed", then?

 

 

Quote

 

“The cessation of ("determinate thought" in) feeling and perceiving, not likely for me.  You're right, doesn't sound blissful, the disturbances associated with the six sense-fields. He said there was a happiness, but I'm guessing it's like the happiness of the cessation of determinate thought in inbreathing and outbreathing--thin!”

 

Do you have many varieties of “cessation of determinate thought” then..?
 

 

 

 

…I say that determinate thought is action. When one determines, one acts by deed, word, or thought.
 

(AN III 415, Pali Text Society Vol III p 294)

 

And what are the activities?  These are the three activities:–those of deed, speech and mind.  These are activities.
 

(SN II 3, Pali Text Society vol II p 4)

 

…I have seen that the ceasing of the activities is gradual. When one has attained the first trance, ("determinate thought" in) speech has ceased. When one has attained the second trance, thought initial and sustained has ceased. When one has attained the third trance, zest has ceased. When one has attained the fourth trance, ("determinate thought" in) inbreathing and outbreathing have ceased… Both ("determinate thought" in) perception and ("determinate thought" in) feeling have ceased when one has attained the cessation of ("determinate thought" in) perception and feeling.
 

(SN IV 217, Pali Text Society vol IV p 146, material in parenthesis added)
 

 

Quote

 

“[…] and outbreathing--thin!” 

 

Was “thin” a typo here? otherwise what on earth do you mean by “—thin!” please?
 

 

 

Gautama claimed there was a happiness in each of the states of concentration.  As with the fourth concentration I discussed above, it's not the kind of happiness that is the opposite of sorrow, that is experienced in day-to-day living.  More like a well-being.  Thin, compared to the happiness that's the opposite of sorrow, that's what I mean.
 

 

Quote

 

You still haven’t answered with regard to why this “cessation of determinate thought”—which I am presuming is equated with Enlightenment (again, you haven’t commented on that point)—would be considered “ultimate”.

 

You seem to think attaining this “cessation of determinate thought” will bring you happiness—again, my question is how much happiness then?

 

The "cessation of ('determinate thought' in) feeling and perceiving" is the attainment associated with Gautama's insight into dependent causation, an example of which would be:  exercise of will --> persistence of (location of) consciousness-->stationing of consciousness-->recurrence of consciousness-->grasping after self (suffering).

 

There's a happiness, according to Gautama, in the "cessation of feeling and perceiving", but again it's likely to be more like a subtle well-being, and a happiness that is only present while the concentration is ongoing.

Shunryu Suzuki said:

 

So, when you practice zazen, your mind should be concentrated in your breathing and this kind of activity is the fundamental activity of the universal being. If so, how you should use your mind is quite clear. Without this experience, or this practice, it is impossible to attain the absolute freedom.
 

(“Thursday Morning Lectures”, November 4th 1965, Los Altos; emphasis added)

  
Gautama said:

 

And what… is the ceasing of action? That ceasing of action by body, speech, and mind, by which one contacts freedom,–that is called ‘the ceasing of action’.
 

(SN IV 145, Pali Text Society Vol IV p 85)

 

 

Maybe more about that, than happiness.

 

Edited September 19, 2023 by Mark Foote

[Daniel]

21 hours ago, Mark Foote said:

I think the notation is "set x where x is not an element of itself" (but I could be wrong!).  The set of all sets that are not elements of themselves, the paradox.

 

Yes.  Sorry.  And thank you.  That is what that notation is known to mean.  I was focusing on the semantic fault which was identified a few sentences later.

 

the assumption that {x:x∉x} exists is paradoxical.(Note: It does not depend on any definition of what ∈ means. Russell offered the Barber Paradox to illustrate this: A barber shaves all those and only those who don't shave themselves.

 

It doesn't matter it means, it could be shaving!  But it's the self-referential negation, imo, that causes the semantic fault.  This is why just a few sentences later, the paradox is "avoided" by adding a couple of layers of abstraction, captial "X" and captial "Y".  The meaning of the operators is of zero-consequence.  But it must be a self-referential negation to produce the contradiction.

 

"A barber shaves all those and only those who don't shave themselves."  = contradiction IFF the barber is considered.

"A barber shaves all those and only those who shave themselves." =/= contradiction IFF the barber is considered.

 

This is why I said:

 

"That's a sematic contradiction.  The fault is the negating self reference."

 

It's because: 

 

"It does not depend on any definition of what ∈ means"  AND the contradiction disappears when the negation dissappears.

 

Hopefully now it makes sense why I said:

 

"x is being defined as something which is not itself and nothing else."

 

If the barber is excluded, then the contradiction dissappears because the self reference is avoided.  If the barber is excluded, that would be "something else".  if x is an unshaven barber, then it is getting shaved.  x is being defined as something which is not itself.  If the barber is excluded somehow in the statement, then the target of the function is being defined as something else.  See below:

 

 "A barber shaves all those and only those who don't shave themselves."

 

"all those and only those who don't shave themselves" is the target of the set.  The function which defines it. 

 

{x:"all those and only those who don't shave themselves"}

x conforms to the function "all those and only those who don't shave themselves"

When the set is "built" x is defined as "all those and only those who don't shave themselves" AND NOTHING ELSE.

 

The contradiction occurs when the barber is included as "all those and only those who don't shave themselves".

 

What if I define x as "all those and only those who don't shave themselves" and SOMETHING ELSE?

 

"A barber shaves all those and only those who don't shave themselves excluding themself, the barber."

 

This defines x in terms of "all those and only those who don't shave themself" AND "not-the-barber".

Lacking the exclusion, x is defined as something it is not and nothing else.

Adding the exclusion, x is defined as something it is not and something else.

 

That's what I meant.  Not completely false, at least.

 

Note:  I'd like to remain consistent with the language in our conversation, if possible.  A paradox might not be false.  A contradiction is always false.  That's why I was careful to use the word "contradiction" in the examples with the barber.

 

 

Quote

"The theory of everything" was Hawking's.  He gave up on it, in 2002:

 

Godel and the End of Physics

 

Yeah, I made a mistake here too.  Sorry, and thank you for the correction.  I should not have used those words "the theory of everything."  I must have heard it somewhere, read it somewhere and I incorrectly introduced it into our convo.  I was not intending to put an official label on Leibniz's over-arching philosophy and intention of his work.  But the intention and over-arching philosophy was indeed a theory of everything  And that included math, and sets, and all kinds of wonderful things including a theory on the divine ( as he imagined it ) as infinite ( as he imagined it ).  See below.  This is what I meant.  From The Math dept. at Rutgers University, in the bizzarely named "middle-sex" NJ.  ( I have family there  )

 

 Leibniz, on the other hand, was a philosopher, and sought to "reconstruct" the universe through pure reason. [6, p. 45] As a result, Leibniz saw mathematics as a potential link between his interests in other fields. Similarly, he hoped his symboli c language would be applicable to all fields of science. Thus, Leibniz's progress in both symbolism and mathematics were linked to each other, and his wish to "reconstruct" the world into a harmonious whole.

 

https://sites.math.rutgers.edu/~cherlin/History/Papers2002/leibniz.html

 

If you read the article, I think it will be clear why Leibniz is a personal hero of mine.

 

Edited September 19, 2023 by Daniel

[Daniel]

@Mark Foote,

 

Is there anything about the above corrections which have bearing on omni-presence and the absolute literal infinity I have described?

 

And I would very much appreciate consistency of language in our discussion, if at all possible.  This is not "actual infinity".  That's different.  And if there is a condition which produces an automatic failure-state, may we please use the word contradiction, if it is produced by opposing concepts linked together?

[Mark Foote]

7 hours ago, Daniel said:

@Mark Foote,

 

Is there anything about the above corrections which have bearing on omni-presence and the absolute literal infinity I have described?

 

And I would very much appreciate consistency of language in our discussion, if at all possible.  This is not "actual infinity".  That's different.  And if there is a condition which produces an automatic failure-state, may we please use the word contradiction, if it is produced by opposing concepts linked together?
 

https://zenmudra.com/230919-Pirates-very-interesting.mp4




Daniel, I think I have a general idea of your use of terms, and I read most of the article on Liebnitz (see video above).  

I don't think we're close enough in definitions to really specify how many angels, so I'm abandoning ship.  That's what pirates do, in the end... ha ha.

 

[Daniel]

8 hours ago, Mark Foote said:

https://zenmudra.com/230919-Pirates-very-interesting.mp4

Daniel, I think I have a general idea of your use of terms, and I read most of the article on Liebnitz (see video above).  

I don't think we're close enough in definitions to really specify how many angels, so I'm abandoning ship.  That's what pirates do, in the end... ha ha.
 

 

I'm confused.  Why are angels being counted?  Even more so, why are angels being introduced into this at all?  I cannot fathom ( sea-faring-pun-intended  ) why you're saying this about angels.

 

And which defintions are divergent between us?  For paradox I'm using "definition A" from the quote you brought.  Is it absolute literal infinity?  Is it that you are wanting to limit the use of the word "infinity" to numeric quantities and nothing more?

 

[Mark Foote]

1 hour ago, Daniel said:

 

I'm confused.  Why are angels being counted?  Even more so, why are angels being introduced into this at all?  I cannot fathom ( sea-faring-pun-intended  ) why you're saying this about angels.

 

And which defintions are divergent between us?  For paradox I'm using "definition A" from the quote you brought.  Is it absolute literal infinity?  Is it that you are wanting to limit the use of the word "infinity" to numeric quantities and nothing more?

 

 

On the heads of pins, you know.