The death of Classical logic and the birth of Constructive Mathematics

[Logik]

....you are asking about the syntactical symbols literally and to how we read them.

No. I am most definitely not doing that. I am asking you to DO WORK in the physics sense of the word.
I am asking you to compare two things and determine IF they are "the same".
I am asking you to do what every philosopher takes for granted (that it is trivial to understand the meaning of "=")
I am asking you to do what any programmer recognizes as the mechanical process of sorting which could be trivial OR infinitely complex.
And any statistician recognizes as binary classification: https://en.wikipedia.org/wiki/Binary_classification

Given a urn full of black and white balls I am asking you to sort the balls into two groups. One set of white balls and one set of black balls.
This is a trivial task for us when A and B are well contrasted and clearly juxtaposed.
It becomes exponentially harder and harder task as the contrast between A and B falls outside of sensitivity range of your measurement apparatus (in this case - your eyeballs) and there comes a point when your equipment is not precise enough and A = B becomes A = A.
Now OBVIOUSLY you can tell that two balls are TWO different objects (they have unique spacetime coordinates), but you can't tell if one red ball is 630 nanometers red and another ball is 631 nanometers red.

I am merely using symbols because it's easy to make the point using this particular medium of communication. If we were talking in person - I would probably use a different example.
What I am trying to demonstrate to you is how fundamental information, decidability and hypothesis testing are to human reasoning.

IF you can detect a difference between A and A THEN A = A is false.
IF you can't detect a difference between A and A THEN A = A is true.

And this will short-circuit your brain because while you are SEEING two symbols which LOOK identical, they are NOT identical.

This is the process in your brain which sorts things into categories. Cats vs Dogs. Black vs White. Red vs Green.
Good vs Evil. Better vs worse. True vs False. 1 vs 0.

It's pattern recognition - something our brains are amazing at and computers, not so (yet).

In order for you to be able to classify ANYTHING into categories you need what is called a "Classification rule": https://en.wikipedia.org/wiki/Classification_rule

Observe that even though we are trying to SORT things into two categories (say true and false) there are actually 4 possible outcomes here.
In addition to true and false conclusions we also have true positives and false negative. In English: You have put a black ball in the white pile, or a white ball in the black pile e.g an error.

This becomes exponentially harder when you have to sort things into 200 bins and the number of possible errors grows exponentially too!

This requires separating the language we use to discuss some logic from the actual logic itself. What you are thinking is that the fact that we read "A = A" linearly, that what we understand them to mean within the system is unable to be true out of some practical limitations of communicating it, but implying that our lack of this ability imposes something intrinsically true about the meaning of the logical postulates.

No, what I mean is that "A" is a different pattern of information to "A". They only LOOK the same but they are not the same.

What I want you to pay attention to is the fact that while you are DOING WORK (e.g SORTING) you are not SPEAKING.
You are THINKING. The process of sorting things is ENTIRELY mechanical.

And if nobody ever asked you to explain the classification rule you will NEVER have to narrate those thoughts.
So how can you possibly conceptualise ANY undecidable "logic" as the "laws of" reason when undecidable logics most definitely cannot DO sorting?!?

Turing only used an idealized computer system in thought to convey the limitations about logic itself, not about physical computers.

Yeah, but Shannon turned everything on its head with Information theory. And you arrive squarely in the realm of Science and statistical hypothesis testing.

Hypothesis 1: A = A is true.
Hypothesis 2: A = A is false

What experiment would you perform to determine which hypothesis is the valid one?
Or as a scientist would say "Can you give me a procedure by which you would distinguish the two cases?"

Here is my procedure: https://repl.it/repls/ShortLightgrayPiracy

If you can't conceive of an actual procedure, and YET you somehow conclude that A = A is true (because your brain is magical and gives you the right answers), you have to allow for the possibility that this is a false positive!

You have to allow for the possibility of error.

The utility of his analogy using an imaginary constructs and architecture that we today call a 'computer' is only coincidentally significant as to how it applies to actual physical computers. We call them "Turing" machines because of his unique 'design' of a simplified computer rather than to other optional designs that actual computers ended up using in reality.

Hah! No. You have it backwards. CONCEPTUAL Turing Machines are Universal - given infinite time, infinite memory and infinite energy they can solve ANY problem via mere brute-forcing.

It is only when you bring limits (physics!) into the mix is where we begin to discuss limits of computation.

For instance, he treated the programs initially as lists of separate instructions that a person has to take data from a single tape, read, interpret, and alter if instructed, and then to print the outcome back onto the tape. Those are 'non-universal'. His 'universal' machine was one that has a fixed kind of BIOS hardware design that takes the first data off the tape to DEFINE a the program virtually without a need to reconfigure the hardware.

Moot. We have generalized computation beyond Turing's initial work.
We have Universalized computation.

Here's a much better definition of a computer: a machine which manipulates information.

What is information? The thing which allows you to decide between these two hypotheses:

Hypothesis 1: A = A is true.
Hypothesis 2: A = A is false

Bayesian inference.

[Logik]

Here is another experiment to demonstrate how fundamental information is to human cognition.

In English: "Is this the same rose?"

Hypothesis 1: roseA = roseB => True
Hypothesis 2: roseA = roseB => False

Please DECIDE which of the above is the correct hypothesis and then explain the ALGORITHM by which you came to this conclusion.

roseA.jpg (28.81 KiB) Viewed 3137 times

roseB.jpg (28.81 KiB) Viewed 3137 times

[Speakpigeon]

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

If it is true that a computer programme is in some way analogous to a logical proof then please provide a transcription of your bit of code into a logical proof so that we can assess for ourselves what it amounts to.
You won't do it, of course, because you're an ignoramus and a pathetic fraud pretending to understand the Curry-Howard correspondence when you don't. The proof is in the pudding.
EB

[Speakpigeon]

Python is a programming language and as such it relies on Boolean operators in exactly the same way as any programming language.
Then you can programme whatever bullshit you want but it won't be logic.

So the things we build USING logic are NOT logic, how does that work?

???
Easy:
Print "A = A is false".
Print " A ∨ ¬A is false"
Print "A → A is false"
You think that's logic?
EB

[Logik]

If it is true that a computer programme is in some way analogous to a logical proof then please provide a transcription of your bit of code into a logical proof so that we can assess for ourselves what it amounts to.

Stop trying to move the Champions League final to a Kindergarten football field you sophist.

The program IS the logical proof! IT WORKS!

That is PROOF OF VALIDITY.

It is RIGHT THERE FOR YOU TO SEE

If you cared to learn about computation.

You won't do it, of course, because you're an ignoramus and a pathetic fraud pretending to understand the Curry-Howard correspondence when you don't. The proof is in the pudding.
EB

I will NOT translate it into a logic that YOU can understand because:
1. It it will take me decades to unpack the MILLIONS OF LINES OF LOGIC into something that YOU can understand.
2. I do not trust YOUR ability to examine MILLIONS OF LOGICAL PROPOSITIONS and perform MILLIONS of boolean operations WITHOUT MAKING AN ERROR.

The FACT that you THINK you can "check my work" is precisely evidence I keep telling you that you are making.
You are mistaking the complex for the simple!!!

I know I haven't made any errors BECAUSE THE COMPUTER RAN THE CODE!
All 10 million instructions! And it found ZERO errors.

If there is any error in the conclusion then it's all my fault.
It simply means that I have not ARGUED (read: PROGRAMMED) PRECISELY what I MEANT.

If there is ANY error in the conclusion it is because the argument is unsound.
But it is valid!

[Logik]

Python is a programming language and as such it relies on Boolean operators in exactly the same way as any programming language.
Then you can programme whatever bullshit you want but it won't be logic.

So the things we build USING logic are NOT logic, how does that work?

???
Easy:
Print "A = A is false".
Print " A ∨ ¬A is false"
Print "A → A is false"
You think that's logic?
EB

What an idiot. You just told the computer to print strings.
That's not what my program does.
My program does print A = A, not print "A = A"
Click the "RUN" button at the top, then in the RIGHT SIDE (dark) window you can type things like
A == A
A and A
Or any other logical expression you choose.

All of those are PROPOSITIONS and the computer asserts them for you in real time. Maybe that will convince you?


But because you are too daft to understand the grammar/syntax you are straw-manning my argument.

Learn python. I am not entertaining your idiocy anymore.

You don't want to learn anything. You just want to be right when you are too stupid to see how wrong you are.
Cognitive dissonance is a motherfucker.

I am not here to convince you of anything (because you are beyond convincing). I am here to make a fool out of you.

[Speakpigeon]

I don't call myself either. I do philosophy and logic, though.
And I use whatever mathematics I need but it's really very basic because logic is formally simple. Difficult to understand, obviously, since no one has as of today, even though many great minds have given it a try, but it's simple.
If it gets complicated, you know it's no longer logic, it's mathematics, and then, who cares?

And that says it all. You seem to have drawn an arbitrary cut-off point between logic and mathematics, failing to realize it's exactly the same thing.

You are just ignorant of the 2,500 years of history where all logicians and all mathematicians saw logic and mathematics as two different things and it is still the general view today.
Formal logic has always been seen as a formal representation of human logical reasoning. Boole called it "the laws of thought". In that respect, it is an empirical science just like physics.
Mathematics is not because is not constrained with representing anything actual. In fact, it is the common view among mathematicians that they should feel free to invent and be creative.

Failing to recognize that it has been PROVEN that it's exactly the same thing. Only your notion of "proof" is somewhat incoherent so you don't recognize it for what it is.

No, it hasn't been proven.
You're a complete ignoramus.
In fact, two brilliant mathematicians, Russell and Whitehead tried very hard to infer all mathematics from first logical principle and failed.

The Principia Mathematica is a three-volume work on the foundations of mathematics written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. The Principia Mathematica was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that the Principia Mathematica, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.

You say conjunction, I say f(P,Q)
f(0,0) = 0
f(0,1) = 0
f(1,0) = 0
f(1,1) = 1

You say disjunction I say g(P,Q)
g(0,0) = 0
g(0,1) = 1
g(1,0) = 1
g(1,1) = 1

You say XOR I say h(P,Q)
h(0,0) = 0
h(0,1) = 1
h(1,0) = 1
h(1,1) = 0

Absolutely EVERYTHING can be modeled as a black box with inputs and outputs! https://en.wikipedia.org/wiki/Black_box

You're a complete ignoramus.
You don't even understand that what you do here is completely irrelevant.
You don't even seem to realise that "A or B" is already a mathematical formalism. You really didn't need to give your silly transcription into "functional" formalism. In case you don't know, logical connectives are taken to be truth functions. See?
And indeed formal logic has always been couched in... formal terms! That's right! Big news. Aristotle was already doing it 2,500 years ago, even before Euclid started mathematics proper. However, we also express our ideas generally using a formal language, English, French etc. That doesn't make what we say mathematical, obviously.
So, try to think for once. What's specific about mathematics. What's different between mathematics and logic? It seems really obvious but you're not really the kind to think much. Rather you piss code.

You are doing what Marvin Minsky calls "mistaking the complex for the simple".

Laugh.

You are of the opinion that a complex universe such as ours can be understood with first-order logic such as the ones you can express in Boolean logic.

No, I'm not of the opinion and you're just an idiot.

Boolean logic is very simplistic.

Boolean logic is logical, it's not logic.
Yes, the whole of logic is very simple and I already said so, and that it was a distinguishing feature compared to mathematics. And the simplicity of logic is on purpose, something clearly you won't ever understand because you're an idiot. You won't understand anything I say because you don't even try. You never really engage in a debate. As soon as you face a rational conversation you scurry away in the safety of the bit corner where it's not possible to have a rational conversation.

Lambda calculus is infinitely complex.
Time to open your mind a little: https://en.wikipedia.org/wiki/Complexit ... anizations

I did complexity you weren't even born, little boy.
EB

[Speakpigeon]

Here is another experiment to demonstrate how fundamental information is to human cognition.

In English: "Is this the same rose?"

Hypothesis 1: roseA = roseB => True
Hypothesis 2: roseA = roseB => False

Please DECIDE which of the above is the correct hypothesis and then explain the ALGORITHM by which you came to this conclusion.

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

[Speakpigeon]

If it is true that a computer programme is in some way analogous to a logical proof then please provide a transcription of your bit of code into a logical proof so that we can assess for ourselves what it amounts to.

Stop trying to move the Champions League final to a Kindergarten football field you sophist. The program IS the logical proof! IT WORKS! That is PROOF OF VALIDITY. It is RIGHT THERE FOR YOU TO SEE If you cared to learn about computation.

You won't do it, of course, because you're an ignoramus and a pathetic fraud pretending to understand the Curry-Howard correspondence when you don't. The proof is in the pudding.

I will NOT translate it into a logic that YOU can understand because:
1. It it will take me decades to unpack the MILLIONS OF LINES OF LOGIC into something that YOU can understand.
2. I do not trust YOUR ability to examine MILLIONS OF LOGICAL PROPOSITIONS and perform MILLIONS of boolean operations WITHOUT MAKING AN ERROR. The FACT that you THINK you can "check my work" is precisely evidence I keep telling you that you are making.
You are mistaking the complex for the simple!!! I know I haven't made any errors BECAUSE THE COMPUTER RAN THE CODE!
All 10 million instructions! And it found ZERO errors. If there is any error in the conclusion then it's all my fault.
It simply means that I have not ARGUED (read: PROGRAMMED) PRECISELY what I MEANT. If there is ANY error in the conclusion it is because the argument is unsound. But it is valid!

QED

If it is true that a computer programme is in some way analogous to a logical proof then please provide a transcription of your bit of code into a logical proof so that we can assess for ourselves what it amounts to.
You won't do it, of course, because you're an ignoramus and a pathetic fraud pretending to understand the Curry-Howard correspondence when you don't. The proof is in the pudding.
EB

[Logik]

You are just ignorant of the 2,500 years of history where all logicians and all mathematicians saw logic and mathematics as two different things and it is still the general view today.

And you are ignorant of what happened in the last 100 years.

1. This was published: https://en.wikipedia.org/wiki/Church%E2 ... ing_thesis
2. This was published: https://en.wikipedia.org/wiki/Curry%E2% ... espondence
3. This was published: https://en.wikipedia.org/wiki/Information_theory

And so we ended up with machines that have deductive capabilities which can MEASURE and PROCESS information from REALITY in real-time.

And that's when Computer Science stole the last thing Philosophy could call its own: Logic.

if you are a logician, then I am a super-logician.

[Speakpigeon]

That's not what my program does. My program does print A = A, not print "A = A" Click the "RUN" button at the top, then in the RIGHT SIDE (dark) window you can type things like A == A A and A Or any other logical expression you choose. All of those are PROPOSITIONS and the computer asserts them for you in real time. Maybe that will convince you? But because you are too daft to understand the grammar/syntax you are straw-manning my argument. Learn python. I am not entertaining your idiocy anymore. You don't want to learn anything. You just want to be right when you are too stupid to see how wrong you are. Cognitive dissonance is a motherfucker. I am not here to convince you of anything (because you are beyond convincing). I am here to make a fool out of you.

If it is true that a computer programme is in some way analogous to a logical proof then please provide a transcription of your bit of code into a logical proof so that we can assess for ourselves what it amounts to.
You won't do it, of course, because you're an ignoramus and a pathetic fraud pretending to understand the Curry-Howard correspondence when you don't. The proof is in the pudding.
EB

[Speakpigeon]

I am a super-logician.

A super fraud.

If it is true that a computer programme is in some way analogous to a logical proof then please provide a transcription of your bit of code into a logical proof so that we can assess for ourselves what it amounts to.
You won't do it, of course, because you're an ignoramus and a pathetic fraud pretending to understand the Curry-Howard correspondence when you don't. The proof is in the pudding.
EB

[Logik]

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

Well thank you for making my fucking point!

How can logic be the "LAWS OF REASON" if you DID IT "using more than just deductive logic.".

IF the logic which you claim is the "One True Logic - THE LAWS OF REASON" then SURELY you would need NO other faculties of your mind to determine whether the two roses are the same.

My point is PRECISELY on point!

You have SOMEHOW reduced TWO hypothesis into ONE hypothesis.
You have SOMEHOW reduced your UNCERTAINTY.

[Logik]

If it is true that a computer programme is in some way analogous to a logical proof then please provide a transcription of your bit of code into a logical proof so that we can assess for ourselves what it amounts to.
You won't do it, of course, because you're an ignoramus and a pathetic fraud pretending to understand the Curry-Howard correspondence when you don't. The proof is in the pudding.
EB

Sorry. I do not recognize you as an authority on assessing logical validity.
I have good reasons to believe that you have no faculties to assess the logical validity of millions of propositions without making errors.

I trust the machine more than you.

You are just struggling to control the narrative. Sorry for you

[Speakpigeon]

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!
EB