Deeper than primes

[Worm]

Uh-oh ... *woop woop* ... *woo-woo alert*

I caught that one ... doronshadmi has started refering to 'complete' sets, this is usually the sign of a massive abyss of incomprehension to come (as if that wasn't already expected....)

 

[ddt]

Well done. Five points to Ravenclaw. (Although I had a different approach in mind in terms of permutation-transformations and topological fixed points, and I haven't taken the time or trouble to verify that it actually produces a useful entropy-like measure.)

In terms of permutations: I earlier thought of this one: given a multiset of size n, divide the number of different orderings of its elements by the number of permutations you'd have with n different elements (i.e. n!). All multisets which do not have double elements now have a "full entropy " of 1. If you want to achieve that all multisets with only 1 distinct element have the same "no entropy" of 0, subtract 1 from both numerator and denominator.

What properties would in your opinion a useful entropy-like measure have?

Doron, wanna go double or quits on a proof that a "symmetric" multiset maximizes entropy using the definition that ddt has so kindly supplied?

So, while Claude Shannon would be very proud how you rationalized doron's handwaving with multi-sets in Notion #1, it is still unclear what sort of entropy doron meant for integers.

I predict, if you want an answer, that you can wait till the cows come home (or as we say in Dutch, till the calves dance on the ice - and that saying predates AGW ).

 

[Reality Check]

No, by set theory {3,1} = {1,3}.

I am talking about the internal structure of distinction, where each member is distinct (order is not important)

This is not the case in a "complete" multiset (fore example: {a,a,a,a,a,...}), where there is no distinction.

By this model (continues or not) a "complete" multiset has maximum entropy and a "complete" set has the minimum entropy.

As much as I know, this is a new idea about entropy.

Your original topic is wrong.

Firstly you list a multiset constructed from the number 5 as {3,1,1}. A multiset is a set so it the order of members is not important. Thus the {3,1,1} multiset is the same as the {1,3,1} multiset. You state that
{3,1,1) has "Intermediate entropy". {1,3,1} would have "Full entropy".
Thus you have a multiset that is defined to have both intermediate and full entropy. So by your own definition your "notion" is contradictory and so wrong.

Secondly it has nothing to do with entropy since you have not even defined what "entropy" in the OT is. Whatever the definition is it has nothing to do with actual entropy which is a number and not a symmetry.

Thirdly you do not seem to know what mathematical symmetry actually is. Symmetry is a property of an object - a object that undergoes a transformation is symmetrical if the transformation returns the same object. An object that undergoes a transformation is asymmetrical if the transformation returns the opposite object. You never define what transformations are being applied to the multisets and so you cannot state what symmetry they have.

Fourthly maybe you assume the transformation of the multiset members is some sort of reordering (e.g. swapping the last and first members). But as stated multisets do not have any order. So you are either wrong again or ignorant of the definition of a multiset.

 

[jsfisher]

[ot]There may be an innocent reason for this....

Yeah, I knew that. I raised only because doron has in the past made more dramatic alterations to other people's posts. He's even been infracted for the practice.

 

[doronshadmi]

Your original topic is wrong.

Firstly you list a multiset constructed from the number 5 as {3,1,1}. A multiset is a set so it the order of members is not important. Thus the {3,1,1} multiset is the same as the {1,3,1} multiset. You state that
{3,1,1) has "Intermediate entropy". {1,3,1} would have "Full entropy".
Thus you have a multiset that is defined to have both intermediate and full entropy. So by your own definition your "notion" is contradictory and so wrong.

Secondly it has nothing to do with entropy since you have not even defined what "entropy" in the OT is. Whatever the definition is it has nothing to do with actual entropy which is a number and not a symmetry.

Thirdly you do not seem to know what mathematical symmetry actually is. Symmetry is a property of an object - a object that undergoes a transformation is symmetrical if the transformation returns the same object. An object that undergoes a transformation is asymmetrical if the transformation returns the opposite object. You never define what transformations are being applied to the multisets and so you cannot state what symmetry they have.

Fourthly maybe you assume the transformation of the multiset members is some sort of reordering (e.g. swapping the last and first members). But as stated multisets do not have any order. So you are either wrong again or ignorant of the definition of a multiset.

Again, order is not important here.

R set is based on distinct continuous values , and N set is based on distinct values.
Symmetry is not a "superposition of identities". It is the result of a transformation of an object. Asymmetry is not "distinct identities". It is a result of a transformation of an object.

Distinction is important hare (whether it is continuous or not).

A collection of distinct elements (order is not important) has less entropy than a collection of non-distinct elements.

For example: {a,c,b} = {a,b,c} (because order is not important) has less entropy than {a,b,a} = {a,a,b} (where also here, order is not important).

I am talking here about symmetry that is based on the distinction degree of the reseachad object and not about any transformation that returns to the original state of the researched object.

 

[arthwollipot]

It's like... passing an accident on the road. You can't help but slow down and look. Even though you know you're contributing to the growing gridlock, you can't look away, just in case someone's bleeding all over the road...

 

[ddt]

Again, order is not important here.



Distinction is important hare (whether is continuous or not).

A collection of distinct elements (order is nor important) has less entropy than a collection of non-distinct elements.

For example: {a,c,b} = {a,b,c} (because order is not important) has less entropy than {a,b,a} = {a,a,b} (where also here, order is not important).

"Collection" is a synonym for set. You seem to use it above as synonym for multiset - muddying the waters again?

And what have furry, long-eared mammals to do with this? (see line 2). Oh wait:

It's like... passing an accident on the road.

Why did the rabbit cross the road?

ETA: Doron, what about your statement "Hilbert was wrong". Are you going to retract that or what? Ditto for "Gödel was wrong".

 

[doronshadmi]

"Collection" is a synonym for set. You seem to use it above as synonym for multiset - muddying the waters again?

A set is a collection of distinct elements where order is not important.

I am talking about a collection that its elements are defined by their distinction of each other (distinction is important).

 

[ddt]

A set is a collection of distinct elements where order is not important.

I am talking about a collection that its elements are defined by their distinction of each other (distinction is important).

I've never seen the word collection being used in that way. I've only seen it used as either a synonym for "set", or as a synonym for "proper set or class" in Set Theory and Category Theory.

Using "collection" as a synonym for multiset serves only to muddy the waters. Do you do that on purpose? Stick to "multiset" and "set", whichever you mean - those are unambiguous terms. Capice?

ETA: Doron, what about your statement "Hilbert was wrong". Are you going to retract that or what? Ditto for "Gödel was wrong".

 

[doronshadmi]

I've never seen the word collection being used in that way.

So now you see it.


Since distinction is important, a collection is a general form of both set and multiset.

In that case, the least Entropy is a collection od distinct elements, where the most is a collection of non-distinct elements.

 

[drkitten]

A set is a collection of distinct elements where order is not important.

Doron, are you a computer programmer? Are you sure you're not confusing the Java Collections Framework with standard math?

On the other hand, I suppose I should be impressed that you managed to come up with a definition, albeit a backwards one (where you define a term with a known meaning in terms of one with an unknown one).

 

[jsfisher]

So now you see it.


Since distinction is important, a collection is a general form of both set and multiset.

You missed the original point. You are misusing standard terminology again/still. Multi-set is a perfectly fine standard term that covers exactly what you mean.

{1,3} is a multi-set
{1,1,3} is a multi-set
{1,1,1,1,1,1,1,1,1,1,1} is a multi-set

There is no reason for you to misuse another term for this purpose. And, as with all multi-sets, the elements may not be unique, but they are definitely unordered.

 

[ddt]

I've never seen the word collection being used in that way.

So now you see it.

... used by someone who is almost invariably wrong when it comes to mathematics. Thanks, that's a great help.

Since distinction is important, a collection is a general form of both set and multiset.

The above is no definition of the word "collection". There is no such general form.

I repeat: do not use the word collection. Use "set" or "multiset", whichever is appropriate.

Capice?

To Add: Doron, what about your statement "Hilbert was wrong". Are you going to retract that or what? Ditto for "Gödel was wrong".

 

[doronshadmi]

You missed the original point. You are misusing standard terminology again/still. Multi-set is a perfectly fine standard term that covers exactly what you mean.

{1,3} is a multi-set
{1,1,3} is a multi-set
{1,1,1,1,1,1,1,1,1,1,1} is a multi-set

There is no reason for you to misuse another term for this purpose. And, as with all multi-sets, the elements may not be unique, but they are definitely unordered.

No problem.

{1,3} is a multi-set of distinct elements.

{1,1} is a multi-set of non-distinct elements.

In both cases order is not important.

 

[ddt]

Doron, are you a computer programmer? Are you sure you're not confusing the Java Collections Framework with standard math?

Doron claims to have been a programmer in the Fortran age. No Java though. In the previous thread, he presented pictures of his "Organic Natural Numbers", which had been generated by a Java program, written by someone else. He failed to come up with the source code.

And off-topic, what did the Java designers think when they didn't include such a framework in Java 1.0? C++ STL had been around for a while by then. At least, with generic types in Java 1.5, they're now also up to par with type safety.

ETA: Doron, what about your statement "Hilbert was wrong". Are you going to retract that or what? Ditto for "Gödel was wrong".

 

[doronshadmi]

Since order is not important, then any multiset's elements have to be defined in parallel (not by step-by-step serial research).

It is easier to get it by using a parallel view.

 

[jsfisher]

No problem.

Great! We have a common term. Now, would you be willing to provide a definition for what you mean by entropy?

Is it a numerical value? How is it calculated? What are the entropies of {} and of {1}?

Is it, instead, a complete (or partial) ordering? Is the entropy of {1,1,1} more or less than the entropy of {1,1}? How does {1,1,1,1,1,1,1,1,1,1,1} compare to {1,8,1}?

 

[ddt]

In that case, the least Entropy is a collection od distinct elements, where the most is a collection of non-distinct elements.

We're playing the "edit but do not tell so" game again? Don't you ever lose those annoying antics?

Could you rephrase this sentence in an understandable way?

And without the use of the word collection!

 

[ddt]

{1,3} is a multi-set of distinct elements.

{1,1} is a multi-set of non-distinct elements.

In the name of Thor (*), could we use another notation for multisets? The use of braces might be standard, but it is confusing as hell when we discuss both sets and multisets at the same time - especially with easily-confused and easily-confusing Doron in the discussion.

Someone on page 1 used square brackets [ ] for multisets. I suggest we stick with that.

In both cases order is not important.

You've said that now so many times that I'm inclined to believe you actually got that.

To add: Doron, what about your statement "Hilbert was wrong". Are you going to retract that or what? Ditto for "Gödel was wrong".

ETA: forgot about the damn footnote: (*) it is Thursday, after all.

 

[doronshadmi]

Great! We have a common term. Now, would you be willing to provide a definition for what you mean by entropy?

Is it a numerical value? How is it calculated? What are the entropies of {} and of {1}?

Is it, instead, a complete (or partial) ordering? Is the entropy of {1,1,1} more or less than the entropy of {1,1}? How does {1,1,1,1,1,1,1,1,1,1,1} compare to {1,8,1}?

Each multi-set defines its own entropy according the distinction of its elements and its cardinality (the number of its elements).

{} has the highest entropy because nothing is distinct.

{1} has no entropy because anything is distinct.

{1,2} or {1,3} have no entropy.

{1,1} has an entropy and {1,1,1} has more entropy than {1,1}.

{2,2,2} has the same entropy as {0,0,0} etc...

The the size of difference between the elements is not important.