Cont: Deeper than primes - Continuation 2

[Apathia]

The perspective on if-and-which-infinity is really a matter of philosophy, not mathematics. I suspect your qualifications on that front far exceed mine. The problem is that Doronshadmi takes his (philosophic) view on infinity as a basis to (selectively) reject basic mathematics.

Yes. Philosophy. As I saw it from the start over ten years ago he was setting out to deliver a new underpinning for Mathematics or what might be called a Meta-Mathematics. There was also the element of an Absolute Unity as the Fundamental Source that one could come to get a sense of through Transcendental Meditation. Doron sees this as integral to his "Organic Mathematics" (or whatever name he's giving it now). This is why this thread was begun in the Philosophy/Religion section.

I did take a Philosophy series in College, but I'm no maven in philosophical discourse. I also took Set Theory and Logic.

You give Doron more credit than I. His fundamental point is that points cannot gather in sufficient quantities to every become a line segment. It was a fair observation at one time in mathematics history, but we are for the most part beyond that now. Philosophy may do with it what it will, and I have no objection there.

Doron, on the other hand, wants it damn much of mathematics. He seems to be hung on "process": add a point, still not enough; add another.... This was confirmed with his misinterpretation of the Axiom of Infinity. The axiom asserts the existence of a particular infinity set (absolute infinity, by the way), but Doron insists it asserts a process for constructing a set ever increasing in size (more at potential infinity).

If Doron has an "Axiom of Infinity," it's not the same at all as modern Mathematics. His axiom is pretty much that actual Infinity is an Absolute that cannot be contained or bounded in any sense. He allows a "potential infinity," but this is just a convenient fiction at most.

He relies more on the visual-intuitive pole than an abstract notion of an infinity that can be manipulated by thought. No! No! thought cannot reach it. All concepts are only through how the visual and symbolic interplay with each other.

As for his "represented as", Doron's views can be perfectly explained with words, as I think you have done reasonably well. Doron, however, needs more because he has lashed out from his philosophic fixation into established math. He has invented a shield he thinks protects him from all criticism via this visual-spatial vs. verbal-symbol reasoning ruse. The graphic of six underbars he included -- serving no purpose other than to "name" his object -- was a device to slip in an unnecessary visual element he could later exploit in his "you can't get it" mantra.

By the way. I am amused that ______ when rendered using Lucinda Sans Unicode for the font, ______, it clearly shows as six separable elements. Not quite the visual-spatial reasoning conclusion Doron wanted, but nonetheless valid.



Absolutely!!!

The other very important thing to understand here is that Doron is not doing Set Theory. He speaks of "collections" but these must not be understood as sets. He uses mathematical notations used in set theory and logic, and this gives rise to misunderstandings, because his readers can't but read into him notions based on the standard usages of those terms and notations. One has to temporarily suspend whatever else to just get a clue at his direction.

Now I'm afraid he may reply that he is doing his own Set Theory and assert something that looks to be a standard statement in Set Theory. This kind of thing threw me before when I thought I was on the right trail. Finally I got the lay of his land outside of his "Mathematical" presentation during a discussion about the relationship between the Absolute and the Contingent. Philosophy, you see.

So far he's presented this meta-mathematical framework as a foundation of sorts. It leads to a very different way of thinking, not in the manipulation of categories, but on a, for want of a better way of putting it in this moment, in matrices of "Yin" and "Yang." His mind is geard more that way as a kind of visual grasping of charts, while we proceed step by step in a liner fashion not seeing what he considers the organic structure of parts contained in wholes that are parts contained in greater wholes. OK too much. I promise I'll get back to this "organic" thing later. Point to be taken. He's not thinking the way we are accustomed. I find that fascinating and wonder where that flavor of thinking could go.

 

[doronshadmi]

Fragments can add up to finite fragments, but never to Absolute [actual] Infinity.

Now you can fragment a fragment with marked quantities, such as 1 and 1/2, and with that fragment you can continue to indefinitely generate numerical fragments, This is a potential infinity. You can add up fragments to fragments to get fragments, but they won't come to a whole between 1 and 1/2. You have already determined that that's a fragment or you can only speak in terms of a potentiality.

Let's simplify it: Since _____ is a non-composed visual_spatial object, its non-composed exitance can't be defined in terms of collections (no matter is these collections are involved with visual_spatial objects, verbal_symbolic objects or any mixture of them).

He prefers leaving Infinity an absolute of itself, as a polar opposite of Finite.

It is more interesting than that.

Finitely many shorter __ (visual_spatial objects) are finitely weaker than actual infinity (the non-composed property of, at least, _____ (which is a visual_spatial object)), which enables to add them up to an accurate value (verbal_symbolic object), exactly because in case of finitely many shorter __ (visual_spatial objects), the complement shorter __ (visual_spatial object) has itself an accurate value (verbal_symbolic object).

Infinitely many shorter __ (visual_spatial objects) are infinitely weaker than actual infinity (the non-composed property of, at least, _____ object (which is a visual_spatial object)), which does not enable to add them up to an accurate value (verbal_symbolic object), exactly because in case of infinitely many shorter __ (visual_spatial objects), the complement shorter __ (visual_spatial object) does not have an accurate value (verbal_symbolic object).

From this visual_spatial AND verbal_symbolic reasoning, we get the following hierarchy of existence:

1) Actual infinity.

2) Finite.

3)Potential infinity.

The details are given in www.internationalskeptics.com/forums/showpost.php?p=12412827&postcount=3095.

The axiom asserts the existence of a particular infinity set (absolute infinity, by the way)

Actual infinity is non-composed, so it can't be defined in terms of collection of objects.

 

[Apathia]

Thank you, Doron!

By your "Let's simplify it," I see that I'm in your ball park. Sorry that my way of saying it isn't as precise as would serve you, but does help the other participants in this thread understand your basic positions.

You inserted a little bracket in quoting me "[Actual]." I don't have a problem with that, except I'd prefer you made a note that you altered my quote.

Yes I see that your Infinite-Finite-Potential Infinity hierarchy of being follows from your framework.

I have to run to an early morning work meeting now. I'll comment more later.

 

[doronshadmi]

Thank you, Doron!

By your "Let's simplify it," I see that I'm in your ball park. Sorry that my way of saying it isn't as precise as would serve you, but does help the other participants in this thread understand your basic positions.

You inserted a little bracket in quoting me "[Actual]." I don't have a problem with that, except I'd prefer you made a note that you altered my quote.

Yes I see that your Infinite-Finite-Potential Infinity hierarchy of being follows from your framework.

I have to run to an early morning work meeting now. I'll comment more later.

Thank you, Apathia!


Sorry about the [actual] case.


See you later.

 

[Apathia]

Disclaimer: What follows in this post is but a two dimensional shadow of what Doron is trying to convey. I will not be able to state to his satisfaction what for him is an intuitive grasp that informs his universe. But they are marks on the floor that do lead in his direction.

The Ultimate Reality is a seamless, boundless, INFINITE unity. There's nothing we can say about it, because before we speak differentiation is introduced. The Infinite we can speak of is not the INFINITE.

With differentiation two principles appear: the principle of the Finite and the principle of the Infinite. from the these two principles the finite world is generated.

We perceive only a finite world but have a shadow concept of an unlimited more. So trying to see the Infinite from the finite principle, we only get Infinity as a potential.

Trying to see from the Infinite side yields no positive information, only the negative limitation that the potentiality can never, never reach the Infinite.

Doron might say Mathematics should be humble and not commit the hubris of The Axiom of Infinity but be satisfied with what can be done in potentiality.

Doron, a question I've had is can Mathematicians use the potentials as short cuts. .999999999 no mater how far extended will never reach 1. But in spite of 1 not being the actual, can we fudge it such for the sake of practical computations?

I know that in earlier discussions we spoke of "non-local numbers," and that your Organic Mathematics introduces the way to work with them.

Is there a way to manipulate these "non-local" values or quantities in calculation that doesn't have to fudge with a potentiality?

 

[doronshadmi]

Disclaimer: What follows in this post is but a two dimensional shadow of what Doron is trying to convey. I will not be able to state to his satisfaction what for him is an intuitive grasp that informs his universe. But they are marks on the floor that do lead in his direction.

The Ultimate Reality is a seamless, boundless, INFINITE unity. There's nothing we can say about it, because before we speak differentiation is introduced. The Infinite we can speak of is not the INFINITE.

With differentiation two principles appear: the principle of the Finite and the principle of the Infinite. from the these two principles the finite world is generated.

We perceive only a finite world but have a shadow concept of an unlimited more. So trying to see the Infinite from the finite principle, we only get Infinity as a potential.

Trying to see from the Infinite side yields no positive information, only the negative limitation that the potentiality can never, never reach the Infinite.

Doron might say Mathematics should be humble and not commit the hubris of The Axiom of Infinity but be satisfied with what can be done in potentiality.

Doron, a question I've had is can Mathematicians use the potentials as short cuts. .999999999 no mater how far extended will never reach 1. But in spite of 1 not being the actual, can we fudge it such for the sake of practical computations?

I know that in earlier discussions we spoke of "non-local numbers," and that your Organic Mathematics introduces the way to work with them.

Is there a way to manipulate these "non-local" values or quantities in calculation that doesn't have to fudge with a potentiality?

Hey Apathia,

In my opinion, every healthy human being enables to use visual_spatial AND verbal_symbolic reasoning, which is (as I claim) the minimal reasoning that enables one to be aware of Actual infinity, Finite, Potential infinity hierarchy of existence, as explained in www.internationalskeptics.com/forums/showpost.php?p=12413510&postcount=3102.


It means that the existence of Actual infinity can't be defined by finitely weaker or infinitely weaker existence.

I define finitely weaker or infinitely weaker existence as expressions of Actual infinitely, such that their existence depend on Actual infinity existence, but not vise versa.

By this notion, there is no limit for further expressions' development in harmonious ways, simply because they are weaker expressions of a common substance (Actual infinity, which is a non-composed existence), that can't reach the existence of the non-composed substance.

Some claims that _____ (the non-composed) is actually ______ (composed), for example:

By the way. I am amused that ______ when rendered using Lucinda Sans Unicode for the font, ______, it clearly shows as six separable elements. Not quite the visual-spatial reasoning conclusion Doron wanted, but nonetheless valid.

In that case the notion of yin_yang is used as follows:

yin = (emptiness)

yang =_ (fullness)

and we get the following form:

. . . . . ______

The right side is yang marked by shortest yins and the left side is yin marked by shortest yangs.

One can define the right side as the positive domain marked by shortest -0 negative markers, where the left side is defined as the negative domain marked by shortest +0 positive markers.

In this case the markers are understood as breakers (or separators, if you will), but also in this case no collection of breakers is the domain itself at its non-composed form.

What mainstream Mathematics does is introduce a third way of speaking of infinity. Not the absolute, not the potential, but a symbolic infinity that can be manipulated as a composed whole.

This is the verbal_symbolic-only reasoning, that wrongly understands Actual infinity (which is non-composed) in terms of collection.

Is there a way to manipulate these "non-local" values or quantities in calculation that doesn't have to fudge with a potentiality?

Is there a way to define potential infinity in terms of finite?

 

[Apathia]

Is there a way to define potential infinity in terms of finite?

I don't think so. We arrive at a concept of a potential infinity only because we have a concept of Infinity. What's more before we even open our mouths and say "finite," Infinity is already implied.

I had a couple of questions:

1) Current mathematics is happy to say .9999999 … equals 1 by means of its Axiom of Infinity. Fort at least me you have made yourself clear that this is incorrect. Such numbers as these can are a glass that will never be full, and the only infinity we can use is to speak in terms of a potential. Everyday calculations make that .99999999 equal 1. Do you allow that as a kind of short cut with potentiality, though it's not actual?

2. Do you have a method of calculation that doesn't fudge but arrives at values that truly involve non-local numbers?

I'll add a third. 3.) Is the critique of the Axiom of Infinity the sole point of your presentation, or do you go on to create your own number theory, arithmetic, algebra, calculus, etc.?

Perhaps your answer might be that we do the usual maths but with the cognition that its results are always open ended with Infinity always lurking beyond our reach. It always strikes me that it's the moral of the story that you're after; that our scientific conclusions are not fixed blocks but are always open to transcendence, as we must be with all the identities we use in our daily lives.

 

[doronshadmi]

I don't think so. We arrive at a concept of a potential infinity only because we have a concept of Infinity. What's more before we even open our mouths and say "finite," Infinity is already implied.

No collection of objects (Finite or infinitely many) has the existence of, at least, non-composed object ____

By verbal_symbolic-only reasoning ____ is defined, for example, as 1.

By doing so, the non-composed property of ____ is undefined, and in order to be defined also visual_spatial reasoning is used.

By using visual_spatial AND verbal_symbolic reasoning in case of ____ , ____ is 1 non-composed object.

A collection of finitely or infinitely many ____ , is, by visual_spatial AND verbal_symbolic reasoning, not the non-composed property of ____ , because this property is a visual_spatial quality that can't be defined in terms of verbal_symbolic quantity.

The majority of modern mathematicians define non-finite cardinality in terms of size instead of in terms of quantity, exactly because they are partially aware of the fact that quantity is inefficient in order to define the quality of Actual infinity.

By using visual_spatial AND verbal_symbolic reasoning one immediately aware that the quality of the non-composed ____ (which is a property of Actual infinity) can't be defined in terms of quantity, and all the "size" "quantity" verbal_symbolic-only maneuvers, naturally get of the mathematical stage.

1) Current mathematics is happy to say .9999999 … equals 1 by means of its Axiom of Infinity. Fort at least me you have made yourself clear that this is incorrect. Such numbers as these can are a glass that will never be full, and the only infinity we can use is to speak in terms of a potential. Everyday calculations make that .99999999 equal 1. Do you allow that as a kind of short cut with potentiality, though it's not actual?

No.

2. Do you have a method of calculation that doesn't fudge but arrives at values that truly involve non-local numbers?

For example, the local number 1 = non-local number 0.999…₁₀ + non-local number 0.000…1₁₀, where the …1 in 0.000…1₁₀ is an endlessly shorter ___ that its fixed length is undefined.

I'll add a third. 3.) Is the critique of the Axiom of Infinity the sole point of your presentation, or do you go on to create your own number theory, arithmetic, algebra, calculus, etc.?

By using visual_spatial AND verbal_symbolic reasoning, everything that is related to infinity in terms of verbal_symbolic-only or visual_spatial-only reasoning, is fundamentally changed.

Perhaps your answer might be that we do the usual maths but with the cognition that its results are always open ended with Infinity always lurking beyond our reach. It always strikes me that it's the moral of the story that you're after; that our scientific conclusions are not fixed blocks but are always open to transcendence, as we must be with all the identities we use in our daily lives.

The ability to use visual_spatial AND verbal_symbolic reasoning is given to any healthy human being, without any external or internal authority, simply because the non-composed is actually all there is (but no mathematics that is related to the finite or potentially infinite, can be done without also using verbal_symbolic reasoning).

By using visual_spatial AND verbal_symbolic reasoning, we get the following hierarchy of existence:

1) Actual infinity.

2) Finite.

3)Potential infinity.


(1) existence is independent of (2) or (3) existence, but (2) or (3) depend on (1) existence.

More details are given in www.internationalskeptics.com/forums/showpost.php?p=12413510&postcount=3102.

 

[Apathia]

Thank you, Doron for indulging me with "baby steps." I hope you can continue to answer my silly questions.

No collection of objects (Finite or infinitely many) has the existence of, at least, non-composed object ____

By verbal_symbolic-only reasoning ____ is defined, for example, as 1.

By doing so, the non-composed property of ____ is undefined, and in order to be defined also visual_spatial reasoning is used.

By using visual_spatial AND verbal_symbolic reasoning in case of ____ , ____ is 1 non-composed object.

A collection of finitely or infinitely many ____ , is, by visual_spatial AND verbal_symbolic reasoning, not the non-composed property of ____ , because this property is a visual_spatial quality that can't be defined in terms of verbal_symbolic quantity.

The majority of modern mathematicians define non-finite cardinality in terms of size instead of in terms of quantity, exactly because they are partially aware of the fact that quantity is inefficient in order to define the quality of Actual infinity.

By using visual_spatial AND verbal_symbolic reasoning one immediately aware that the quality of the non-composed ____ (which is a property of Actual infinity) can't be defined in terms of quantity, and all the "size" "quantity" verbal_symbolic-only maneuvers, naturally get of the mathematical stage.

Yes. I think I get your point about 1 being a non-composed object, and that Infinity can't be quantified nor sized up.

No.

No means no. Thank you for that very important answer. With it you are telling us Mathematics as it is done today must be overturned with a new theory of number and brand new methods of calculation. What you're presenting is a game changer after which not even the ball is the same, and new rules apply all around. Except maybe for simple arithmetic with whole numbers. But as soon as Irrational Numbers enter the field all bets are off.

This means new methods, new rules of calculation, especially a brand new kind of calculus.

For example, the local number 1 = non-local number 0.999…₁₀ + non-local number 0.000…1₁₀, where the …1 in 0.000…1₁₀ is an endlessly shorter ___ that its fixed length is undefined.

This the example you gave me of calculation involving a 'Non-Local" number and by extension how an Irrational Number must be handled. It looks to me it would be quite a cumbersome thing sitting with my calculator trying to do all those endlessly nestled sums. OMG! How would one go about solving the kinds of problems calculus was created to do. It seems one would have to do hundreds of piecemeal sums.

Have you produced an alternative to Calculus based on Organic Mathematics?

By using visual_spatial AND verbal_symbolic reasoning, everything that is related to infinity in terms of verbal_symbolic-only or visual_spatial-only reasoning, is fundamentally changed.

"This changes everything!" Not just how we speak about and treat Infinity but how we must do calculations that are currently based on the incorrect concept of Infinitesimals.

 

[doronshadmi]

Thank you, Doron for indulging me with "baby steps." I hope you can continue to answer my silly questions.

Please The ability to use visual_spatial AND verbal_symbolic skills is given freely to any healthy human being. So no cleverness of any kind is involved here.

Yes. I think I get your point about 1 being a non-composed object, and that Infinity can't be quantified nor sized up.

The non-composed quality of Actual infinity can't be quantified in terms of collection of finite or infinitely many objects.

No means no. Thank you for that very important answer. With it you are telling us Mathematics as it is done today must be overturned with a new theory of number and brand new methods of calculation. What you're presenting is a game changer after which not even the ball is the same, and new rules apply all around. Except maybe for simple arithmetic with whole numbers. But as soon as Irrational Numbers enter the field all bets are off.

This means new methods, new rules of calculation, especially a brand new kind of calculus.

By understanding Actual infinity in terms of the non-composed, finite collection is finitely weaker than Actual infinity (can be defined as fixed or accurate values), where a non-finite collection is infinitely weaker than Actual infinity (can't be entirely defined as fixed or accurate values, as defined by the case of endlessly shorter ___).

This the example you gave me of calculation involving a 'Non-Local" number and by extension how an Irrational Number must be handled. It looks to me it would be quite a cumbersome thing sitting with my calculator trying to do all those endlessly nestled sums. OMG! How would one go about solving the kinds of problems calculus was created to do. It seems one would have to do hundreds of piecemeal sums.

By verbal_symbolic-only reasoning, for example, 1₁₀/3₁₀ = 0.333...₁₀, where the considered number is a rational number.

By using also visual_spatial reasoning, 1₁₀/3₁₀ is finitely weaker than Actual infinity (it is defined as a fixed or accurate value), where 0.333...₁₀ is infinitely weaker than Actual infinity (it can't be defined as a fixed or accurate value), such that 1₁₀/3₁₀ > 0.333...₁₀ by (0.000...1₁₀)/3₁₀, where the …1 in 0.000…1₁₀ is an endlessly shorter ___ that its fixed length is undefined.

So the issue is not about rational or irrational numbers, but it is about local numbers (fixed or accurate values) or non-local numbers (non-fixed or inaccurate values).

By visual_spatial AND verbal_symbolic reasoning, one can get a fixed or accurate value by, for example, adding non-accurate values, for example:

1₁₀/3₁₀ = 0.333...₁₀ + (0.000...1₁₀)/3₁₀

Have you produced an alternative to Calculus based on Organic Mathematics?

It goas deeper than calculus.

By using visual_spatial AND verbal_symbolic reasoning, a given mathematical framework is defined as a whole and yet incomplete, where being whole AND incomplete is exactly the expressed aspect of Actual infinity (where the expressed aspect is finitely or infinitely weaker than Actual infinity, in terms of existence).

In order to understand it, let's observe the simplest logical form that enables to deal with complexity, which is the 2-valued logic, as given by using visial_spatial AND verbal_symbolic reasoning:

                          Potential infinity                         
                                  ...                                 
                                                                       
    0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1   
    \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /   
     0   1   0   1   0   1   0   1   0   1   0   1   0   1   0   1     
     \   /   \   /   \   /   \   /   \   /   \   /   \   /   \   /     
      \ /     \ /     \ /     \ /     \ /     \ /     \ /     \ /     
       0       1       0       1       0       1       0       1       
       \       /       \       /       \       /       \       /       
        \     /         \     /         \     /         \     /       
         \   /           \   /           \   /           \   /         
          \ /             \ /             \ /             \ /         
           0               1               0               1           
           \               /               \               /           
            \             /                 \             /           
             \           /                   \           /             
              \         /                     \         /             
               \       /                       \       /               
                \     /                         \     /               
     bounded     \   /                           \   /     bounded     
                  \ /                             \ /                 
        by         0                               1         by       
                    \                             /                   
   contradiction     \                           /        tautology   
                      \                         /                     
                       \                       /                       
                        \                     /                       
                         \                   /                         
                          \                 /                         
                           \               /                           
                            \             /                           
                             \           /                             
                              \         /                             
                               \       /                               
                                \     /                               
                                 \   /                                 
       _   _   _   _   _   _   _  \ /  _   _   _   _   _   _   _       
...  _/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_ ...
                                                                       
                            Actual infinity

As observed, the given endlessly growing 2-valued logic tree is an organic whole AND incomplete.

"This changes everything!" Not just how we speak about and treat Infinity but how we must do calculations that are currently based on the incorrect concept of Infinitesimals.

By verbal_symbolic-only reasoning, infinitesimals are fixed values > 0.

By visual_spatial AND verbal_symbolic reasoning, infinitesimals are non-fixed (endlessly smaller ___) values > 0.

 

[Apathia]

By verbal_symbolic-only reasoning, for example, 1₁₀/3₁₀ = 0.333...₁₀, where the considered number is a rational number.

By using also visual_spatial reasoning, 1₁₀/3₁₀ is finitely weaker than Actual infinity (it is defined as a fixed or accurate value), where 0.333...₁₀ is infinitely weaker than Actual infinity (it can't be defined as a fixed or accurate value), such that 1₁₀/3₁₀ > 0.333...₁₀ by (0.000...1₁₀)/3₁₀, where the …1 in 0.000…1₁₀ is an endlessly shorter ___ that its fixed length is undefined.

So the issue is not about rational or irrational numbers, but it is about local numbers (fixed or accurate values) or non-local numbers (non-fixed or inaccurate values).

By visual_spatial AND verbal_symbolic reasoning, one can get a fixed or accurate value by, for example, adding non-accurate values, for example:

1₁₀/3₁₀ = 0.333...₁₀ + (0.000...1₁₀)/3₁₀

This is, if not the same example, a similar, longer one than the one you gave me in your previous post. It's not the answer I in part expected, but it tells us a lot. My question, of course, was a practical one, not about theory but about how to do higher math calculations. A how to question, not an in theory question. I think you understood that.

So this is a basic example of your methodology for doing calculations from arithmetic and up through algebra, functions, trig, and calculus of a sort.

It goas deeper than calculus.

By using visual_spatial AND verbal_symbolic reasoning, a given mathematical framework is defined as a whole and yet incomplete, where being whole AND incomplete is exactly the expressed aspect of Actual infinity (where the expressed aspect is finitely or infinitely weaker than Actual infinity, in terms of existence).

In order to understand it, let's observe the simplest logical form that enables to deal with complexity, which is the 2-valued logic, as given by using visial_spatial AND verbal_symbolic reasoning:

                          Potential infinity                         
                                  ...                                 
                                                                       
    0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1   
    \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /   
     0   1   0   1   0   1   0   1   0   1   0   1   0   1   0   1     
     \   /   \   /   \   /   \   /   \   /   \   /   \   /   \   /     
      \ /     \ /     \ /     \ /     \ /     \ /     \ /     \ /     
       0       1       0       1       0       1       0       1       
       \       /       \       /       \       /       \       /       
        \     /         \     /         \     /         \     /       
         \   /           \   /           \   /           \   /         
          \ /             \ /             \ /             \ /         
           0               1               0               1           
           \               /               \               /           
            \             /                 \             /           
             \           /                   \           /             
              \         /                     \         /             
               \       /                       \       /               
                \     /                         \     /               
     bounded     \   /                           \   /     bounded     
                  \ /                             \ /                 
        by         0                               1         by       
                    \                             /                   
   contradiction     \                           /        tautology   
                      \                         /                     
                       \                       /                       
                        \                     /                       
                         \                   /                         
                          \                 /                         
                           \               /                           
                            \             /                           
                             \           /                             
                              \         /                             
                               \       /                               
                                \     /                               
                                 \   /                                 
       _   _   _   _   _   _   _  \ /  _   _   _   _   _   _   _       
...  _/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_ ...
                                                                       
                            Actual infinity

As observed, the given endlessly growing 2-valued logic tree is an organic whole AND incomplete.




By verbal_symbolic-only reasoning, infinitesimals are fixed values > 0.

By visual_spatial AND verbal_symbolic reasoning, infinitesimals are non-fixed (endlessly smaller ___) values > 0.

Yes. It goes deeper than Calculus (as is known and loved since Newton and Leibniz). And it must replace Calculus with a new method that doesn't treat infinitesimals as fixed values.

It appears from the basic example that you've given that that would be done by multiple approximating calculations that together provide a useful solution that doesn't insult Infinity and leaves a margin of flexibility.

I anticipated that you would provide a chart, and that got me thinking that you could create something like a computer graphic slide rule with multiple sliding panels one could adjust to arrive at the desired level of a calculation. That's a very visual-spatial sort of approach.

 

[doronshadmi]

I anticipated that you would provide a chart, and that got me thinking that you could create something like a computer graphic slide rule with multiple sliding panels one could adjust to arrive at the desired level of a calculation. That's a very visual-spatial sort of approach.

The whole AND incomplete 2-valued logical tree is the result of both visual_spatial AND verbal_symbolic reasoning, which can be taken in parallel (visual_spatial tendency) AND in serial (verbal_symbolic tendency).

Moreover, by this organic-like visual_spatial AND verbal_symbolic ever growing logical form, one enables to understand the non-composed property of Actual infinity, by "going downwards" finitely many parallel | serial movements, and also one enables to understand the unlimited ability of the tree's growth, by "going upwards" (known also as potential infinity) infinitely many parallel | serial movements.

In terms of time, "going downwards" finitely many parallel | serial movements is understood as already done (past), the present is the current parallel | serial movements, and the future is "going upwards" (known also as potential infinity) infinitely many parallel | serial movements that are permanently yet to become (the future).

By being aware of actual infinity without being lost within its finitely or infinitely many expressions, we get an awareness of endless harmonious creativity, which is not divided anymore to past, present or future, where the uncertain is the optimal environment for endless novel joyful experiences.

 

[doronshadmi]

Let's return to the ground in order to demonstrate visual_spatial AND verbal_symbolic reasoning in terms of finite objects, as follows:

This diagram demonstrates uncertainty and redundancy as the fundamental terms for information, where (A,B) is the particular case of distinct information under quantities (1,1).

It must be stressed that distinct information is a fundamental term in case of mathematical objects like distinct members of a given non-empty set.

So, by using visual_spatial AND verbal_symbolic reasoning, one realizes that distinct information is not fundamental.

 

[Apathia]

The whole AND incomplete 2-valued logical tree is the result of both visual_spatial AND verbal_symbolic reasoning, which can be taken in parallel (visual_spatial tendency) AND in serial (verbal_symbolic tendency).

Moreover, by this organic-like visual_spatial AND verbal_symbolic ever growing logical form, one enables to understand the non-composed property of Actual infinity, by "going downwards" finitely many parallel | serial movements, and also one enables to understand the unlimited ability of the tree's growth, by "going upwards" (known also as potential infinity) infinitely many parallel | serial movements.

In terms of time, "going downwards" finitely many parallel | serial movements is understood as already done (past), the present is the current parallel | serial movements, and the future is "going upwards" (known also as potential infinity) infinitely many parallel | serial movements that are permanently yet to become (the future).

By being aware of actual infinity without being lost within its finitely or infinitely many expressions, we get an awareness of endless harmonious creativity, which is not divided anymore to past, present or future, where the uncertain is the optimal environment for endless novel joyful experiences.

In Post #3107 I suggested "It always strikes me that it's the moral of the story that you're after; that our scientific conclusions are not fixed blocks but are always open to transcendence, as we must be with all the identities we use in our daily lives."

Your above response to the questions I've been asking gives me confidence that your presentation is primarily philosophical. There is not at this time a new Mathematics to be presented. That's not to say there couldn't be a new structure of practical calculations and applied math. It simply hasn't been drawn out yet. You present a skeletal foundation and when asked about the structure of math to be built upon it, you always return to the "spiritual" intent and benefit of your vision.

In regards to that, I've always had a sympathy for your cause. I'm not merely "bound under Verbal/Symbolic thinking." I have a strong sense of Transcendence and see the direction your finger is pointing.

I used the phrase, "moral of the story." "Moral is a somewhat misleading word to use here, when this is about a new way of seeing and a new way of understanding. And before you remind me, it's not new. We all already use visial_spatial AND verbal_symbolic reasoning. It's somewhat as when Zen Master D.T. Suzuki said, "The Zen Mind is the ordinary mind."

At this time your intent is to share your vision, to propagate your framework of the basic workings of the creative mind. If we could just let our minds be open ended to Transcendence and fly in the free open sky, we'd have a better world. Then we would be of a mind to replace the current mathematics with something "Organic."

To those of you still following this thread and commenting from time to time: It is abundantly clear to me that Doron does not have a new mathematics to present at this time. And it should be reckoned that he doesn't see Mathematics at all the same discipline the Mathematicians here do. He's not going to build a Theory of Number based on any Set Theory or Logic you would recognize. The mathematical terms he uses do not have the same denotations and connotations you are accustomed to. He is not going to give the sort of definitions you'd expect. His language is to be taken intuitively and with more fluidity than a traditional mathematical definition would support. None of that can be expected of him.

As I've said, his intent, vision, and expressions are for "spiritual" ends.
By "spiritual" I mean the kind of content that The Dao, Zen, Transcendental Meditation and the like concerns itself with. It is Doron's language for something beyond mundane calculation.

No, I'm not a disciple of Doron. I just like encouraging individual's spiritual quests, especially when, as Doron, fluidity and open endedness are stated values the finger points to. This is where I have no quarrel with him.

In this respect there is more I want to comment upon in the context of his statement above. Bit by bit as we go along.

It bears repeating that this thread is rightly in the Philosophical/ Religious Section. The "spiritual" content is integral and foundational to it. So moderators, please, please don't chuck posts into AAH because they go into Transcendental Meditation, Unity and Dualism, and ethical considerations for the survival of our species, because they seem to be off topic from Mathematics. In Doron's mind all those things are central to his TOPIC.

Doron, my main response to your post is
Even if you reply that you have presented a new mathematics in all its substance.

 

[Apathia]

Let's return to the ground in order to demonstrate visual_spatial AND verbal_symbolic reasoning in terms of finite objects, as follows:

[qimg]https://c1.staticflickr.com/7/6146/6017791855_8ac80e49f9_o_d.jpg[/qimg]

This diagram demonstrates uncertainty and redundancy as the fundamental terms for information, where (A,B) is the particular case of distinct information under quantities (1,1).

It must be stressed that distinct information is a fundamental term in case of mathematical objects like distinct members of a given non-empty set.

So, by using visual_spatial AND verbal_symbolic reasoning, one realizes that distinct information is not fundamental.

We return to the beginning of this thread. I suppose Doron feels I'm ready now for the opening salvo. Unfortunately I still don't have clarity about this verbiage. I have no idea what the words "uncertainty" and "redundancy" mean in this context. Anyone who wants to go way back will find that these terms were never defined. People asked Doron what he meant by them but the replies were repetitions using the same terms or substitutions of terms also undefined. Or a chart Doron felt would illustrate his meaning.

What I recognize here is the usual parsing of binary principles. However I'd expect something like Certainty/Uncertainty, Redundant/Unique. I don't know what polar principle Uncertainty/Redundancy expresses.

So let's cut to the conclusion Doron makes.

So, by using visual_spatial AND verbal_symbolic reasoning, one realizes that distinct information is not fundamental.

He's not saying there's no distinct information in Mathematics, but that distinct answers are not fundamentally given, especially when potential infinities are involved.

This is not going to set well with most any mathematician or scientist. We're not talking statistical margins of error here, or the Heisenberg Uncertainty Principle but that the kind of distinct numerical values you can get from a Calculus equation are not real and aren't allowed. From Doron's perspective distinct info only in Math and Science is a kind of dogmatism. What's more it's based on a limiting two-value logic and thinking in terms of categories and identities. This makes Science and math supportive of racism.

I'm just bluntly reporting it here. I don't mind Science being so objective. In a different sphere we regard each other as "souls" rather than objects. And in another we realize that we are inseparable. However for Doron, the current language of Math and Science causes separation and Us-vs-Them just by the use of it. His margin of "Uncertainty" gives wiggle room out of boxes of bigotry.

 

[doronshadmi]

We return to the beginning of this thread. I suppose Doron feels I'm ready now for the opening salvo. Unfortunately I still don't have clarity about this verbiage. I have no idea what the words "uncertainty" and "redundancy" mean in this context. Anyone who wants to go way back will find that these terms were never defined. People asked Doron what he meant by them but the replies were repetitions using the same terms or substitutions of terms also undefined. Or a chart Doron felt would illustrate his meaning.

Please look at the definitions and demonstrations in pages 12, 13 of https://drive.google.com/file/d/0B3hcPBRBCzClakdJZjNNOW5RSTQ/view?usp=sharing

 

[doronshadmi]

As I've said, his intent, vision, and expressions are for "spiritual" ends.
By "spiritual" I mean the kind of content that The Dao, Zen, Transcendental Meditation and the like concerns itself with. It is Doron's language for something beyond mundane calculation.

I prefer to use neutral monism, such that "spiritual" vs. "material" are the domain of collection, where their substance is non-composed thing (its existence can't be defined in terms of finitely or infinitely many things (I use the term "things" for both abstract or non-abstract expressions)).

 

[Apathia]

I prefer to use neutral monism, such that "spiritual" vs. "material" are the domain of collection, where their substance is non-composed thing (its existence can't be defined in terms of finitely or infinitely many things (I use the term "things" for both abstract or non-abstract expressions)).

That's fine. I wasn't using the word "spiritual" as an ontological matter. I was speaking of spiritual values that come from our capacity to see above our selves, from "transcendental consciousness" (Darn that's so new agey!) and encountering each other as more than fixed (composed) selves.

Yes, the ultimate "Substance" is not a composed thing.

 

[Apathia]

Please look at the definitions and demonstrations in pages 12, 13 of https://drive.google.com/file/d/0B3hcPBRBCzClakdJZjNNOW5RSTQ/view?usp=sharing

That's quite the picture puzzle. Of course when you know what the picture is, it's easier to put the puzzle pieces together. But it still takes time.
I'm working on it.

 

[Apathia]

Disclaimer: Again I'll be using my own words to do a kind of Doron 101. I stepped back into this messy thread because I saw that people were continuing to expect of Doron the standard rules of Set Theory and Logic, when it should have been already clear that even though he speaks of "sets" and "logic trees," he's in an entirely different world. The lay of the land is different there. Perhaps I can at least point out some of the hills. I know a couple of people are following this, but I don't care how many. I proceed for my own amusement in somewhat the same way Doron carried on this thread without what he considered positive and useful feedback.

One of the ways to understand the different place Doron is coming from is how he treats the Barber Paradox that so vexed Bertrand Russel. You can find his "exposition" here https://drive.google.com/file/d/0B3hcPBRBCzClakdJZjNNOW5RSTQ/view?usp=sharing

The classic paradox goes: The Barber shaves only those men in town who do not shave themselves.
(I'm going to ignore what my inner English teacher would say that this is a poorly contrived statement generating a lack of clarity. "The Barber shaves those men in town who do not shave themselves." does not get twisted into a pretzel.)

"This question results in a paradox because, according to the statement above, he can either be shaven by:
1. himself, or 2. the barber (which happens to be himself).
However, none of these possibilities are valid! This is because:
•
If the barber does shave himself, then the barber (himself) must not shave himself.
•
If the barber does not shave himself, then the barber (himself) must shave himself."

It's telling the way Doron busts this contrivance.

He uses the Set word, but it's not in the fashion contemporary mathematics does. It's more like chucking some things in baskets, so to speak. Some dimes, some pennies, some mangoes. But don't speak of coins and
fruits. This kind of collection is not membership by categories. It's not about belonging to some class. Membership is solely that it's in a given basket where it's not a basket of a class of items but of things that can be distinct of themselves or repetitions of the same (another penny, a mango). There's a lot more to this, but later. The point is don't think in terms of membership by classes.

Now you might think you are in the clear by counting how many things are in the basket. But sorry, according to Doron you are missing a subtlety. Take just pennies and chuck 'em in a basket. You pick up a penny and then another one. Two pennies, right? Well what was the second penny you picked up? One could say it was another instance of the same thing. Penny, Penny there it is again. You don't have two. This is "Redundancy." But is the second penny you picked up really just an instance of the first. It looks the same, but there's reason to count it as a different thing of itself. This is "Uncertainty."

Then the usual MO of cognition according to Doron requires you to parse out the combinations of Item of Redundancy and Item of Uncertainty into the tree of values he gives. This is a tree of baskets (to continue using what any Goodwill store has in an overabundance) The contents of those baskets aren't by prior belonging together according to classes, but by Redundancy/Uncertainty. A mango may not necessarily be in a basket of mangos. Though in the tree of baskets you'll find a basket that has all your mangos. Beware speaking of a basket of all mangos. This strays over into classification, categorization, and by Doron's Rule #1, there is no composed all. This of course applies to numbers, so imprecisely speaking, 2 may not necessarily be in the basket or collection containing the number 2)

I may be wrong about Doron's usage of Redundancy and Uncertainty I'm certainly imprecise. This was the drift I got from Doron's friend Moshe Klein back when he visited the forum in 2009. Doron, please correct my misunderstandings.

"Good Lord, Apathia! We know you're not a mathematician. Don't you realize this totally wrecks mathematics and doing mathematics! How can Doron expect to have a consistent sum in his bank account?"

For the daily grind of banking arithmetic, your money is safe. It involves a special case of calculation whose solution is secure in one of those baskets up in the tree. Unfortunately till we have the magic slide rule or the marvelous gear works of Cybernetic Kernels, the day to day calculations will continue to be carried out in the old bigoted, discriminatory way.

One of the reasons I'm talking "baskets" is I want the reader to get how Visual-Spatial Doron's collections are. Of course that he speaks in terms of Redundancy and Uncertainty indicates a Verbal-Symbolic contribution as well. hence his insistence that he is using both.

But on to the Barber.

I've decided since this is already such a Too Long Didn't Read monster, I won't go into the YESthing NOthing Thing.
As Doron explains "The Barber shaves" is the overarching or basketing concept. "only those men in town who do not shave themselves" are contents in the basket. Again Basketing and Contents are the binary principles, and you chart them out. In such a simple case, "The Barber shaves" goes into one basket, and the "only those men in town who don't shave themselves" into another basket. In their respective baskets they can be true or false but not impinge on each other.

Collapsing the baskets into one creates the paradox. Doron finds that not only unnecessary but silly. You see the "Barber shaving" basketing is of a greater scope than parties shaved. To cram it into the set would be like trying to fit a container ship into a single container unit.

Both have their own level of discourse. It makes sense to everyone that the Barber shaves those who don't shave themselves. And it makes sense to everyone that the Barber shaves himself.

There's more fun with the so-called set of all ideas. Does it contain the idea of a set of ideas. Doron answer: No! Idea about ideas is a separate basket or level. And so is an Idea about an idea of the set of ideas.

And then there's that lying Cretan!

Down with the tyranny of class concepts!