The death of Classical logic and the birth of Constructive Mathematics

What is the basis for reason? And mathematics?

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Speakpigeon » Fri Mar 01, 2019 3:54 pm

Logik wrote:
Fri Mar 01, 2019 3:38 pm
Speakpigeon wrote:
Fri Mar 01, 2019 3:13 pm
You've really no idea what you're talking about. You're just a waste time.
I am not wasting your time. I am making a fool out of you by showing you that knowledge doesn't "just happen".
You have to WORK for it!!!!
When I feel pain I certainly don't need to do any work to know the pain I feel.
I guess my brain must be doing all the work for me.
It's like logical intuition. I really don't have to do any work at all. It just happens.
EB

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Speakpigeon » Fri Mar 01, 2019 3:56 pm

Logik wrote:
Fri Mar 01, 2019 3:38 pm
I am not wasting your time. I am making a fool out of you by showing you that knowledge doesn't "just happen".
You have to WORK for it!!!!
viewtopic.php?f=26&t=26199
But of course you won't do it because you have realized that you are in over your head and you have to save face now ;)
I already answered you question here.
Your attention span is only a few seconds.
EB

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Logik » Fri Mar 01, 2019 3:58 pm

Speakpigeon wrote:
Fri Mar 01, 2019 3:54 pm
When I feel pain I certainly don't need to do any work to know the pain I feel.
I guess my brain must be doing all the work for me.
It's like logical intuition. I really don't have to do any work at all. It just happens.
EB
I feel many things too! But I am not asking you about feelings. You are trying to move the goal posts now.

I am asking you how you tell the difference between objects in the real world.

So lets not pretend like you aren't trying to shift the debate onto another topic.

Respond to my thread. Tell me if it's the same rose.

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Logik » Fri Mar 01, 2019 3:59 pm

Speakpigeon wrote:
Fri Mar 01, 2019 3:56 pm
I already answered you question here.
Your attention span is only a few seconds.
EB
That is true. My attention span is all over the place.

Could you kindly point me to your answer again?

Logik
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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Logik » Fri Mar 01, 2019 4:01 pm

Oh, I see it now!
Speakpigeon wrote:
Tue Feb 26, 2019 11:22 am
Obviously, these are not roses.
Still, I have no difficulty deciding whether the two photos are of the same rose and I would have no difficulty explaining how I come to this decision.
However, I do it using more than just deductive logic.
So, your point is irrelevant.
Or else, explain why it would be.
EB
So you are telling us (again) that your brain can make decisions, but those are not thoughts. Is that correct?

You haven't actually committed to an answer.

Please do. Over here: viewtopic.php?f=26&t=26199

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Speakpigeon » Fri Mar 01, 2019 9:24 pm

Logik wrote:
Sat Feb 23, 2019 1:25 pm
The law of identity is the cornerstone of Arostotelian/Classical logic.
A = A is True.
In the 2nd half of the 20th century the American mathematician Haskell Curry and logician William Alvin Howard discovered an analogy between logical proofs and working computer programs. This is known as the Curry-Howard correspondence.
Mathematical proofs are working computer programs. https://en.wikipedia.org/wiki/Curry%E2% ... espondence
Therefore, if we can write a working computer program which asserts that A = A is false without producing an error then we have living proof contradicting the founding axiom of Classic/Aristotelian logic.
I hereby reject the law of identity, and give you the law of humanity: A = A is False.
A thing needs not be the same as itself!
https://repl.it/repls/SuperficialShimmeringAnimatronics
So here is expert opinion on your bit of Python code by someone who actually speaks Python:
The critical line is this one:
def __eq__(self, other):
return False
This just overrides the comparison method with unconditionally returning "false".
It does no actual identity check, only a mock one.
So, of course this can "prove" whatever you choose as the alleged result.
And that's all there is to it.
OMG.
The little boy is a cheat!

---------------------------------------------------------------------

OK, as I understand it, I would summarise the logic of what he did as essentially substituting an axiom of "non-identity" to the default axiom of identity, and this in-between two evaluations, leading to two contradictory results within the same programme. No big deal. Garbage in, garbage out.

His new claim, however, is that since he could get the same programme to produce a contradiction, and since a programme is equivalent to a logical proof, per Curry-Howard, as he sees it, this is supposed to show that classical logic is wrong about rejecting contradictions. However, again, garbage in, garbage out. In 1st order logic, which I would guess is the assumed logical framework for the computer industry as a whole, if you assume contradictory premises, which is exactly what you do when you use two contradictory axioms as he did, anything whatever follows, including any number of contradictory consequences. This is using a hammer to kill a midget. Gross. And that trick doesn't even prove anything whatever although he could be forgiven for being confused about that.

This also shows that the little cheat doesn't even understand the logic behind computation. He just doesn't understand that the contradiction he can produce by introducing a new axiom which is itself contradictory to the default one will be the standard consequence in 1st order logic. This is what you get with any argument or implication with contradictory or simply false premises. This in itself makes the argument or implication valid so that anything follows, including more contradictions. Garbage in, garbage out. However, these arguments or implications, because they don't have true premises, are irremediably unsound, and therefore don't prove anything.

And then the guy goes on a campaign to claim he's proved "classical logic" wrong. This is an idiot ignoramus deluded he is bringing the revolution.

Well, the door was wide open, why did you barge in?!

Can you put the garbage out while you're here?
EB

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Speakpigeon » Fri Mar 01, 2019 9:33 pm

Logik wrote:
Fri Mar 01, 2019 3:58 pm
[Respond to my thread. Tell me if it's the same rose.
I already answered. You just forgot.
EB

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Speakpigeon » Fri Mar 01, 2019 9:44 pm

Logik wrote:
Fri Mar 01, 2019 3:59 pm
Speakpigeon wrote:
Fri Mar 01, 2019 3:56 pm
I already answered you question here.
Your attention span is only a few seconds.
EB
That is true. My attention span is all over the place.

Could you kindly point me to your answer again?
Remember? My brain doesn't know! So, how could I possibly know where it is?!
Beside, I can't be bothered.
You're a waste of time. You don't understand how a conversation should go. You're a self-centred egotistic jerk. You ask endless questions and never answer any! Your attention span means you loose track of the conversation. You're just a hyperactive computer nerd and unable to really think. I guess your ideas are essentially unthought. You don't realise that new ideas call for hard work but more importantly serious thinking. You will never get anywhere the way you behave. Your "proof" here is pathetic vacuum. Go tell experts and they will laugh at you.
EB

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Logik » Fri Mar 01, 2019 10:19 pm

Speakpigeon wrote:
Fri Mar 01, 2019 9:33 pm
Logik wrote:
Fri Mar 01, 2019 3:58 pm
[Respond to my thread. Tell me if it's the same rose.
I already answered. You just forgot.
EB
That's the spirit. If you don't commit yourself to an answer - you can never be wrong!

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Logik » Fri Mar 01, 2019 10:34 pm

Speakpigeon wrote:
Fri Mar 01, 2019 9:24 pm
OK, as I understand it, I would summarise the logic of what he did as essentially substituting an axiom of "non-identity" to the default axiom of identity, and this in-between two evaluations, leading to two contradictory results within the same programme.
Then you clearly don't understand it. See!

A thing is IDENTICAL as itself. It means it has a globally unique identifier. It means no two people in France have the same ID number.
It guarantees UNIQUENESS. That is it!

Instead you have conflated IDENTITY with EQUALITY.

for all x: id(x) = id(x) <------ IDENTITY
for all x: x = x <--------- EQUALITY
Speakpigeon wrote:
Fri Mar 01, 2019 9:24 pm
His new claim, however, is that since he could get the same programme to produce a contradiction, and since a programme is equivalent to a logical proof, per Curry-Howard, as he sees it, this is supposed to show that classical logic is wrong about rejecting contradictions. However, again, garbage in, garbage out. In 1st order logic, which I would guess is the assumed logical framework for the computer industry as a whole, if you assume contradictory premises, which is exactly what you do when you use two contradictory axioms as he did, anything whatever follows, including any number of contradictory consequences. This is using a hammer to kill a midget. Gross. And that trick doesn't even prove anything whatever although he could be forgiven for being confused about that.
You don't have the first clue how garbage-in/garbage-out works in real-world physical systems. You are over-stepping your expertise.

How can I assume CONTRADICTORY premises when the program exists as a physical entity on a computer.
Where was this contradiction? Did a vortex open up? Did the datacenter get swallowed by a black hole.

NOTHING HAPPENED. The computer accepted the premises, computed the consequences and produces a result.

You don't like the result because it disagrees wit your expectations.

Speakpigeon wrote:
Fri Mar 01, 2019 9:24 pm
This also shows that the little cheat doesn't even understand the logic behind computation. He just doesn't understand that the contradiction he can produce by introducing a new axiom which is itself contradictory to the default one will be the standard consequence in 1st order logic. This is what you get with any argument or implication with contradictory or simply false premises. This in itself makes the argument or implication valid so that anything follows, including more contradictions. Garbage in, garbage out. However, these arguments or implications, because they don't have true premises, are irremediably unsound, and therefore don't prove anything.

And then the guy goes on a campaign to claim he's proved "classical logic" wrong. This is an idiot ignoramus deluded he is bringing the revolution.

Well, the door was wide open, why did you barge in?!

Can you put the garbage out while you're here?
EB
:lol: :lol: :lol: :lol: :lol: :lol: :lol: :lol: :lol:

There is no such thing as "cheating" in physics, idiot!

I GAMED A FUCKING LAW!

The LAW of thermodynamics says entropy increases. If YOU can make time go back - IT IS NOT A LAW.
The LAW of gravity says objects fall towards Earth. If YOU can make a bowling ball fly UP - IT IS NOT A LAW.
The LAW of conservation of energy means that energy is transferred but never lost. If you can make energy INCREASE - IT IS NOT A LAW.
The LAW of prime numbers

A LAW means that I can't violate it - NO MATTER WHAT I DO. A LAW means a hard limit! No go!

That's why you are a sophist, not a scientist.

There are NO AUTHORITIES in this universe. Certainly not some stupid fucking man-made "law".

It's just some arbitrary rule you have appointed as your master.

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Speakpigeon » Sat Mar 02, 2019 9:41 pm

Logik wrote:
Sat Feb 23, 2019 1:25 pm
The law of identity is the cornerstone of Arostotelian/Classical logic.
A = A is True.
In the 2nd half of the 20th century the American mathematician Haskell Curry and logician William Alvin Howard discovered an analogy between logical proofs and working computer programs. This is known as the Curry-Howard correspondence.
Mathematical proofs are working computer programs. https://en.wikipedia.org/wiki/Curry%E2% ... espondence
Therefore, if we can write a working computer program which asserts that A = A is false without producing an error then we have living proof contradicting the founding axiom of Classic/Aristotelian logic.
I hereby reject the law of identity, and give you the law of humanity: A = A is False.
A thing needs not be the same as itself!
https://repl.it/repls/SuperficialShimmeringAnimatronics
Still another expert's opinion on the value of this:
The Curry-Howard correspondence is a mathematical correspondence between particular logical systems and particular λ-calculi.
Python doesn't have the kind of formal definitions to be possibly used in the context of the Curry-Howard correspondence.
In other words, the OP is a fraud.
EB

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Speakpigeon » Sat Mar 02, 2019 9:56 pm

Another way to understand what the programme proposed initially by our little cheat here really does is to express it as a logical argument. Something he refused to do himself even though I asked him to do it so we could understand.
So here it is:
Premise 1 - A is identical to A;
Premise 2 - A is not identical to A;
Therefore, A is identical to A and A is not identical to A.
This argument is trivially valid in mathematical logic just because the two premises are contradictory to begin with. Garbage in, garbage out.
Such arguments, although valid, cannot possibly be sound precisely because the premises are contradictory, which means that one of the two premises has to be false and the argument therefore unsound.
It should be noted that this aspect of mathematical logic is well known since around 1900 and Bertrand Russell and even before that it's been proposed by William of Soisson in the 12th century. So, no big news...
It's been dubbed "the principle of explosion":
The principle of explosion
The principle of explosion (Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)", or ex contradictione (sequitur) quodlibet (ECQ), "from contradiction, anything (follows)"), or the principle of Pseudo-Scotus, is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it. This is known as deductive explosion. The proof of this principle was first given by 12th century French philosopher William of Soissons.
https://en.wikipedia.org/wiki/Principle_of_explosion
The idea that it would falsify the Law of Identity is simply idiotic.
Also, this shows the guy is an ignoramus and doesn't understand at all what the Law of Identity means.
EB

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Logik » Sat Mar 02, 2019 11:53 pm

Speakpigeon wrote:
Sat Mar 02, 2019 9:41 pm
Logik wrote:
Sat Feb 23, 2019 1:25 pm
The law of identity is the cornerstone of Arostotelian/Classical logic.
A = A is True.
In the 2nd half of the 20th century the American mathematician Haskell Curry and logician William Alvin Howard discovered an analogy between logical proofs and working computer programs. This is known as the Curry-Howard correspondence.
Mathematical proofs are working computer programs. https://en.wikipedia.org/wiki/Curry%E2% ... espondence
Therefore, if we can write a working computer program which asserts that A = A is false without producing an error then we have living proof contradicting the founding axiom of Classic/Aristotelian logic.
I hereby reject the law of identity, and give you the law of humanity: A = A is False.
A thing needs not be the same as itself!
https://repl.it/repls/SuperficialShimmeringAnimatronics
Still another expert's opinion on the value of this:
The Curry-Howard correspondence is a mathematical correspondence between particular logical systems and particular λ-calculi.
Python doesn't have the kind of formal definitions to be possibly used in the context of the Curry-Howard correspondence.
In other words, the OP is a fraud.
EB
Your expert doesn't understand computational complexity. I do.

The official tool used by Mathmatician in 2019 for automated theorem-proving ASSUMES the law of identity as axiomatic.
This is straight from the horse's mouth.
you use Coq's equality type "x = x" is a theorem for any x of any type, but that does not mean that equality is decidable, only provable.
You wouldn't be able to write a decidable equality test for an infinite stream of bits.
Decidable <> Provable
Spelled in English: The axiom of Classical logic cannot be DECIDED true. Even for the integers :)

Which only leaves us with one important question: is decidability important?

If you cannot DECIDE whether one thing is the same or different to another thing you have mistaken the complex for the simple.

This is why you will not answer my challenge for determining whether two photos are a photo of the same rose or a different rose ;)

Speakpigeon wrote:
Sat Mar 02, 2019 9:41 pm
The Curry-Howard correspondence is a mathematical correspondence between particular logical systems and particular λ-calculi.
Yes. Precisely! It is a Mathematical correspondence between particular logical systems called Turing machines.
All Turing machines obey the laws of physics.

Logic/Mathematics is a subset of computation.
Computation is a subset of physics.

The end.

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Logik » Sun Mar 03, 2019 12:01 am

Speakpigeon wrote:
Sat Mar 02, 2019 9:56 pm
Another way to understand what the programme proposed initially by our little cheat here really does is to express it as a logical argument. Something he refused to do himself even though I asked him to do it so we could understand.
So here it is:
Premise 1 - A is identical to A;
Premise 2 - A is not identical to A;
Therefore, A is identical to A and A is not identical to A.
This argument is trivially valid in mathematical logic just because the two premises are contradictory to begin with. Garbage in, garbage out.
Such arguments, although valid, cannot possibly be sound precisely because the premises are contradictory, which means that one of the two premises has to be false and the argument therefore unsound.
It should be noted that this aspect of mathematical logic is well known since around 1900 and Bertrand Russell and even before that it's been proposed by William of Soisson in the 12th century. So, no big news...
It's been dubbed "the principle of explosion":
The principle of explosion
The principle of explosion (Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)", or ex contradictione (sequitur) quodlibet (ECQ), "from contradiction, anything (follows)"), or the principle of Pseudo-Scotus, is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it. This is known as deductive explosion. The proof of this principle was first given by 12th century French philosopher William of Soissons.
https://en.wikipedia.org/wiki/Principle_of_explosion
The idea that it would falsify the Law of Identity is simply idiotic.
Also, this shows the guy is an ignoramus and doesn't understand at all what the Law of Identity means.
EB
The Mathematical/Aristotelian fraternity currently draws a distinction between the notions of provability and decidability.
Proof is strictly about deduction. Decidability a.k.a induction is ignored as a concern of Mathematics.

All 'proofs' of **x = x** are axiomatic from the Classic law of identity (assumed).
Even the Mathematical theorem prover Coq behaves this way.
From the Curry-Howard isomorphism proofs are isomorphic to computer programs: https://en.wikipedia.org/wiki/Curry%E2% ... espondence

In order to get to where I am I have junked Set Theory and started with Type Theory as fundamental to all Mathematics. https://en.wikipedia.org/wiki/Type_theory

In the universe of Type Theory (The Human universe) 1 is not anything in particular. 1 is just an abstract symbol. Symbols are just language. They can mean anything we want them to mean. If you subscribe to the decimal system 1 + 1 = 2, If you subscribe to the binary system 1+1 = 10.

And so it is a fundamentally important question to ask "We understand what 1's value is - it's 1! But what is its identity?"

In pursuit of the answer my foundational alphabet is the characters (NOT INTEGERS) 1, 2, 3, 4, 5, 6,7, 8, 9, 0.
And the goal is to DERRIVE the digits and integers from them.

In order to derive the digits we need to PROVE (computationally) that the string(1) = digit(1), string(2) = digit(2) etc etc...
Since our alphabet is finite, in the abstract, and when given infinite amount of time and memory a Turing Machine will indeed determine that for all x: x = x => True. Even for infinite values of x. BECAUSE Abstract turing machines are infinite concepts. But we, humans, don't live in the abstract , infinite universe of Mathematics.

We live in a physical reality where computation requires non-zero amount of energy, and non-zero number of operations in order to decide on the truth-value of a proposition. Even a proposition as simple as 'x = x' needs to be computed/decided by SOMETHING. Usually a human mind.
To conclude it axiomatically from Classical identity is to pre-suppose truth!

There are Truths which cannot be deduced directly from the theorems.

What does 1 mean?!? To a Mathematician/Aristotelian - nothing. To a scientist - everything!

It's information. Information is how we make decisions. Information is how we test hypotheses. Information is how we falsify hypotheses.
Information is experience. Information is fundamental to human thought!

Classical logic conflates the notions of **identity** and **value**.
On a Turing machine **Identity** means unique memory address. Lets call it M.
**Value** means contents-of-memory at location M. Lets call it VALUE(M).

### Drawing a distinction between identity and value

Identity := id(x)
Value := value(x)

P1. for all x: id(x) == value(x) => False

The identity of an object is not the same as its value. We will use Big-O notation to quantify 'value' https://en.wikipedia.org/wiki/Big_O_notation

For those unfamiliar with algorithmic analysis and Big-O notation, suffice to say that it is synonymous with the English notion of complexity.

A simple thing is O(1). An infinitely complex thing is O(∞). Complexity is a function of value.
The more information an object contains - the more complex it is.

### The law of identity in a Digital Physics universe.

P2.A for all x: id(x) == id(x) => True

Each discernable object in physical reality has a universally unique identity.
The computational cost of the id(x) function is O(1).

Telling that two things are not the same is trivial! That is literally why we say 'two things'.

### The law of value in the Digital Physics universe (Classical law of identity)

The more information an object contains - the harder it is to decide its value. The harder it becomes to prove "x == x".
The cost of the decision is O(1) for small values of X (e.g really simple objects) and O(∞) for infinite values of X e.g really complex objects.

P3. for all x: x == x => UNDECIDABLE, Best case: O(1), Worse case: O(∞)

Aristotelian Identity is the principle of explosion - camouflaged!
Aristotelians don't know what identity is and mistake identity for value.

The Classical law of identity is an illusion. It causes us to mistake the complex for the simple. Behind a simple statement like "x = x" hides infinite amount of complexity. This very fact prevents us from saying anything useful about the integers beyond phenomenology. If I can't recognize an integer for what it is then I am stuck in a dream.

https://repl.it/@LogikLogicus/INTEGERS

Next step: proving that the integers are finite.

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Re: The death of Classical logic and the birth of Constructive Mathematics

Post by Speakpigeon » Sun Mar 03, 2019 12:14 pm

Logik wrote:
Sat Feb 23, 2019 1:25 pm
The law of identity is the cornerstone of Arostotelian/Classical logic.
A = A is True.
In the 2nd half of the 20th century the American mathematician Haskell Curry and logician William Alvin Howard discovered an analogy between logical proofs and working computer programs. This is known as the Curry-Howard correspondence.
Mathematical proofs are working computer programs. https://en.wikipedia.org/wiki/Curry%E2% ... espondence
Therefore, if we can write a working computer program which asserts that A = A is false without producing an error then we have living proof contradicting the founding axiom of Classic/Aristotelian logic.
I hereby reject the law of identity, and give you the law of humanity: A = A is False.
Yet another expert opinion:
According to the Curry-Howard correspondance, a computer programme is analogous to a proof in intuitionistic logic, not classical logic.What programs implement is fundamentally constructive. The correspondance is only an analogy and in fact we already know that it should break down at some point.
So, whatever you infer from the correspondance you may like but no one should feel compelled to accept it as true.
Oh, what a disappointment!
EB

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