Deeper than primes

[The Man]

Yes, I know. Again - what is the relevance to a discussion on mathematical logic?

You do know that the physics of a Qubit is described using "flatland" mathematics don't you?

I should point out to you, Robin, that Doron's "superposition” does not use the principle of superposition, by his own specific assertion.


ETA:
To try to put it more succinctly (if that were possible), where the superposition of “p AND ~p” (as a Qbit) would be a state that is essentially 50% TRUE and 50% FALSE. Technically a third state that is neither entirely TRUE nor entirely FALSE, but a superposition of TRUE and FALSE states (doesn’t have to be 50/50). Doron simply wants “p AND ~p” to both be TRUE or both FALSE in a binary system, but not a contradiction in his “non-locality”. It really comes down to two things Doron’s particular distain for the word linear and his own apparent preference for representing his personal lack of knowledge as some universal lacking.

 

[Robin]

No it isn't.

What I think he meant was that if you set the y as ~x in "(x implies y and ~y) implies ~x" then it will be the equivalent of the RAA tautology - ie "(x implies ~x and x) implies ~x"

We are all going a bit cross-eyed with this.

 

[Robin]

On the other hand, the claim that removing ~(p and ~p) as a tautology would have no impact on "direct" proofs is simply incorrect.

 

[Little 10 Toes]

Robin wrote "x implies y and not y implies not x".

If y is the negation of x, then "x implies ~x and ~~x implies ~x".

Where did he say that? If you have the ability to quote someone, you have the ability to cite.

Since you're too lazy, I found it for you. I'm showing the whole message, with what you "quoted" in italics, so that it has context.

I didn't say that indirect proofs were the only methods of inference, but they have been at the heart of mathematics for 2,300 years and as I pointed out, to remove it would involve the rewiting of mathematics.

Removing contradictions has implications to the rest of logic as well as you should be able to see.

For example what happens to "x implies y and not y implies not x"?

You can't just change one bit of logic and expect the rest to be the same.

"not (p and not p)" is a tautology in constructive mathematics. (Not that I am an expert on the subject - but it is "p or not p" that is rejected as a tautology in constructive maths)

Also "p and not p" does not mean "p that is both true and false" in any system of logic. It is an expression that has a truth value.

There is an error in this sentence which makes it hard to parse. Did you mean "neither are"?

You could say that constructive mathematics is at the root of the tree to, dealing as it does with metamathematics.

In any case, proofs definitely are fundamental to mathematics.

What do they have to do with the price of tea?

In any case, you did not answer my question. Are there theorems and proofs in your mathematics?

 

[doronshadmi]

So if MT is RAA when the first term is the negation of the second.

And MT is the contrapositive of MP.

Doesn't that indicate that they are all interrelated?

So why do you think you can decide one of these is not a tautology and the others will remain unaffected.

We are talking about direct or indirect proofs, where MP or MT are direct proofs, so MP and MT are "on the same side".

If MT is both direct AND indirect proof, then we have a contradiction.

So, (MT is direct proof if y is not the negation of x) OR (MT is indirect proof if y is the negation of x).

 

[doronshadmi]

On the other hand, the claim that removing ~(p and ~p) as a tautology would have no impact on "direct" proofs is simply incorrect.

Direct proofs are different than indirect proofs, becuse they are not using p AND ~P at all, and this is exactly the meaning of ~(p and ~p), which means: "not using (p and ~p) in direct proofs".

 

[doronshadmi]

Again - I never said you could.

I said you could not use a framework where "p and not p" is not a contradiction to do maths without rewriting your rules of inference.

Only if indirect proofs are used, such that p AND ~P is considered as a contradiction.

 

[doronshadmi]

Here is a quote from the top of page 87

Although addition and multiplication are always possible with infinite cardinals, subtraction and division no longer give definite results and connot therefore be employed as they are employed in elementary arithmetic.

Even your own cited reference clearly refutes your claim that “"+","-" operations are non-distinct” even when “talking only about Cardinality”.

No it supports my claim as follows:

1) We are talking only about Cardinality.

2) σ is any infinite cardinal and μ is any cardinal.

3) The case σ – μ (where μ=0) is trivial, because 0 is not a successor of any cardinal, and we can't use the con-commutative property of subtraction, because there are no negative cardinals (in that case 0 – σ, is an invalid expression).

4) The non-trivial case of σ – μ is considered only if μ>0, and since σ > μ, we can't use the con-commutative property of subtraction also in this case, because there are no negative cardinals (in that case μ – σ, is an invalid expression).

5) So we are left only with σ – μ, where μ>0, and this is exactly the non-trivial case that we are dealing with.

6) By following (1) to (5) σ – μ = σ + μ = σ , and at this non-trivial case "-" is indistinguishable from "+".

So by understanding (1) to (6) we immediately understand that "-" no longer gives definite results (it does not have definite results that are different form the results that are derive from "+"), exactly because it is indistinguishable from "+".

 

[Robin]

We are talking about direct or indirect proofs, where MP or MT are direct proofs, so MP and MT are "on the same side".

If MT is both direct AND indirect proof, then we have a contradiction.

There is no contradiction because there is no "direct/indirect" dichotomy. You have somehow gotten hold of this idea, probably from Wiki, and can't seem to let go of it.

Try to understand this.

"Indirect Proof" is the name sometimes given to "Reductio ad absurdum".

RAA is just a rule of inference just like Modus Ponens, Mathematical Induction and proof by cases.

It is not a different kind of proof.

Again - you cannot just arbitrarily alter one part of mathematics and then just assume that nothing else is changed.

You have to demonstrate that you still have an effective method of proof.

And where is this proof you claim to have posted?

 

[doronshadmi]

There is no contradiction because there is no "direct/indirect" dichotomy. You have somehow gotten hold of this idea, probably from Wiki, and can't seem to let go of it.

Try to understand this.

"Indirect Proof" is the name sometimes given to "Reductio ad absurdum".

RAA is just a rule of inference just like Modus Ponens, Mathematical Induction and proof by cases.

It is not a different kind of proof.

Again - you cannot just arbitrarily alter one part of mathematics and then just assume that nothing else is changed.

You have to demonstrate that you still have an effective method of proof.

And where is this proof you claim to have posted?

Robin, please demonstrate how a framework (call it Mathematics or any other name, it does not matter), where p AND ~p is a contradiction, can be useful for a framework, where p AND ~p is a superposition.

I ask that question because you insist that without p AND ~p as a contradiction, the whole mathematical science has to be built from scratch, in order to deal with p AND ~p as a superposition.

Is this actually your argument?

 

[doronshadmi]

Some correction of http://www.internationalskeptics.com/forums/showpost.php?p=6365679&postcount=11708.

By typo mistake I wrote con-commutative instead of non-commutative.

 

[Little 10 Toes]

Original message before the change:

We are talking about direct or indirect proofs, where MP or MT are direct proofs, so MP and MT are "on the same side".

If MT is both direct AND indirect proof, then we have a contradiction.

So, MT is direct OR indirect proof.

Edited message:

We are talking about direct or indirect proofs, where MP or MT are direct proofs, so MP and MT are "on the same side".

If MT is both direct AND indirect proof, then we have a contradiction.

So, (MT is direct proof if y is not the negation of x) OR (MT is indirect proof if y is the negation of x).

Funny how you understand "If MT is both direct AND indirect proof, then we have a contradiction" but when people point out "p AND ~p" you have conniptions.

 

[doronshadmi]

the superposition of “p AND ~p” (as a Qbit) would be a state that is essentially 50% TRUE and 50% FALSE.

In other words, you have only a ratio of 1:2 (50% 1 and 50% 0), which is a local strict value between 0 and 1 local and strict values, where I also have a ratio of 2:1 (100% 1 and 100% 0), where in both cases there is non-locality among local non-strict values or local strict values.

For example:

This form represents non-locality among local non-strict values (symmetry):

1   1 
0   0 
|___|

This form represents non-locality among local strict values (asymmetry):

1   0 
|___| 
|

The current logical foundations of the mathematical science are based only on the second form,
where "superposition" is probability of (50% 1 and 50% 0), which is actually another local strict value between 0 and 1 local and strict values.

But in both forms Non-locality and Locality are involved.

 

[doronshadmi]

Original message before the change:


Edited message:


Funny how you understand "If MT is both direct AND indirect proof, then we have a contradiction" but when people point out "p AND ~p" you have conniptions.

"p AND ~p" is a contradiction if we are dealing only with local strict values (asymmetry).

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=6367054&postcount=11713.

 

[Robin]

Robin, please demonstrate how a framework (call it Mathematics or any other name, it does not matter), where p AND ~p is a contradiction, can be useful for a framework, where p AND ~p is a superposition.

I ask that question because you insist that without p AND ~p as a contradiction, the whole mathematical science has to be built from scratch, in order to deal with p AND ~p as a superposition.

Is this actually your argument?

Yes it is. In order for your stuff to be called mathematics it needs to have proofs.

Without ~(p and ~p) being a tautology you need to demonstrate that you have a way of doing proofs.

We haven't yet covered off the matter of what happens to modus ponens when you set the second term as the negation of the first.

It is absurd to say that MP and MT are tautologies only for certain values of their terms and not for others, and then say you can use them in proofs.

So, I call on you again to point out this proof that you claim to have posted. Show me what it is proving. Show me how the conclusions follow from the premisses.

I have been asking this for a while. Is there any reason you can't show me this?

 

[epix]

Originally Posted by epix
it is the first Common Notion of Euclid that you have found redundant.

A = A even if A is uncertain (non-strict id = non-strict id).

We are not talking about the clarity of A, but we are talking about the relation of a thing to itself, whether it has strict id or not.

I don't think that there is one sane person on this planet who would mention one of the Euclid Common Notions to talk about "clarity of A."

The nature of you scribble would give any curious person who wonders about your problem the idea to lead you to Euclid's Common Notions. I did it several times with the same result: you get disoriented.

Axioms cannot be proved and therefore they cannot be explained and that means you can't be fixed, Doron.

Number theorists are folks who invent rules for their games. Some of their games are more popular than others but are never played by the scientists. If you quote from Wikipedia, then be advised that the "truth" you try to expand universally is just one of the rules that applies to a particular game.
Cantor was successful, coz he approached a curious problem the traditional way. Only later, the others found out that Cantor's sets ran on three axioms.
So they started to shuffle things around and ran into difficulties when they tried to come up with other games. Those problems are exquisitely their problems and can in no way hinder the understanding of the physical universe.

Look around the Internet and find the set that relates to Russell Paradox. Let me know what you found.

 

[Little 10 Toes]

"p AND ~p" is a contradiction if we are dealing only with local strict values (asymmetry).

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=6367054&postcount=11713.

Learn how to do ratios. 2:1 does not equal "100% 1 and 100% 0". Another math failure.

 

[epix]

Learn how to do ratios. 2:1 does not equal "100% 1 and 100% 0". Another math failure.

What about the 1:2 ratio?

In other words, you have only a ratio of 1:2 (50% 1 and 50% 0), which is a local strict value between 0 and 1 local and strict values, where I also have a ratio of 2:1 (100% 1 and 100% 0), where in both cases there is non-locality among local non-strict values or local strict values.

Ditto?

These are just minor inaccuracies that can't prevent the Swarm of Excited Photons to land on the Runway of Temporary Darkness with the Ambassador of Irreversible and Mutually Inclusive Truth aboard.

 

[doronshadmi]

Learn how to do ratios. 2:1 does not equal "100% 1 and 100% 0". Another math failure.

You are wrong.

For example, if the non-strict id of x is at the level of the strict value 3, than x's non-strict id is "100% 2 and 100% 1 and 100% 0".

The example can be generalized to any n>1:1 case.

 

[doronshadmi]

I don't think that there is one sane person on this planet who would mention one of the Euclid Common Notions to talk about "clarity of A."

So it is about time to move on and deal with x = x, such that x does not have a strict id, which is determined by the strict n>1:1 ratio, as demonstrated in http://www.internationalskeptics.com/forums/showpost.php?p=6368575&postcount=11719.

All you have is to grasp http://www.internationalskeptics.com/forums/showpost.php?p=6367054&postcount=11713.