## The death of Classical logic and the birth of Constructive Mathematics

What is the basis for reason? And mathematics?

Moderators: AMod, iMod

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Scott Mayers wrote: Sun Feb 24, 2019 7:05 am Maybe it might help if you stepped back to the simplistic origins of logic, NOT jump ahead to the complex theories that require more intense investigation to determine the meanings implied in those areas?
Meanings? I am not looking for meaning. Logic is a tool of self-expression.
When you are attempting to express simple ideas Classical logic is fine.
When you are attempting to express complex ideas about a complex reality - you need higher order logic.

There are things I am unable to express in classical logic Like A != A.
If I were to express A !=A I contradict the Identity axiom.

More precisely, every time I express "A != A" a Classical logician (mistakenly) asserts that I have contradicted myself when I haven't.
Speaking different languages is not conducive to communication.
Forcing me to speak your language is not conducive to self-expression.

So I need a logic in which "for all x: x = x" is NOT an axiom.

Or more precisely. I need a logic in which we have MULTIPLE identity axioms.

for all x: x = x
for all y: y ! = y
for all z: z == z
for all p: p rabbit p

The code I provided in Python (Lambda calculus - same thing) demonstrates that I can have a logic in which A != A is a valid, non-contradictory expression.

I have removed the "foundation" and the skyscraper remains standing. Plato was right.
Scott Mayers wrote: Sun Feb 24, 2019 7:05 am All you are asserting is very basic: that you don't trust logic via nature as something based on "consistency" or to the other coinciding assumptions, "the law of excluded middle" and "the law of non-contradiction".
Logic has nothing to do with nature. Logic/mathematics is invented. Man-made.
I most definitely do not adhere to LEM. No constructive mathematician does.
Scott Mayers wrote: Sun Feb 24, 2019 7:05 am ALL logic that extends to ALL possibilities simply requires accepting "inconsistency" to be the basis of logic.
No it doesn't. Lambda calculus is consistent AND complete. Read/understand Curry-Howard isomorphism.
Scott Mayers wrote: Sun Feb 24, 2019 7:05 am You thus have to BEGIN with a consistent logic to provide rules of conduct (inconsistency is lawless). Then you can use this reasoning to infer things about a broader reality through an organized process of reasoning.
It is consistent. It contains no contradictions. If it was inconsistent the computer would throw an error.
Humans can deal with (ignore? Overlook? fail to naturally detect?) inconsistency - computers can't.
Scott Mayers wrote: Sun Feb 24, 2019 7:05 am What you appear to be claiming is most extreme: that NO CONSISTENT systems exist or are relevant to intellectual reasoning. If this were true, then everyone would already be correct to randomly agree or disagree with you WITHOUT a need to argue. Why would you even try to "reason" beginning with your implied default of 'reasoning' that 'reasoning == non-reasoning' (A == non-A)?
What I am claiming is that Classical logic is inconsistent. Because the meaning of "=" is vague and inconsistent.
I didn't imply that reasoning == non-reasoning? You are thinking like an absolutist. (and maybe that's my error, I am also thinking like an absolutist but at the other end of the spectrum).

The point is that we don't need a "foundational" axiom. Both of these statements can be true within the same logic-system

A = A => True
B = B => False
Scott Mayers wrote: Sun Feb 24, 2019 7:05 am 0 = 0 and 1 as [inconsistent totality = inconsistent subsets of totality AND consistent subsets of totality]

You are acting as though a totality that has no law is something that has some 'law' that disallows it to also be contradictory.....thus contradicting your belief in contradiction. Then you further contradict the left over consistent reality in a way that cannot ever be resolvable.
I believe in consistency and non-contradiction. That is why I am rejecting classical logic.
When dealing with integers the meaning of "=" is "absolute distance from 0 on the number line"
When dealing with abstracts (Sky is blue) the "=" does NOT mean "absolute distance from 0".

The "=" requires a human, making contextual choices/interpretations.
Scott Mayers wrote: Sun Feb 24, 2019 7:05 am Begin with the consistent rules first.
EXACTLY

That's why I am down to one axiom. Non-contradiction.

Identity is inconsistent.
Scott Mayers wrote: Sun Feb 24, 2019 7:05 am Then, if you want to show there are degrees or variations as a part of this world, do so without denying that there are discrete realities. Note, for instance that mathematical calculus is about a continuous logic but it proves it through a discrete form of reasoning. And it also shows that you can reverse this, begin with a continuous default to show how and where discrete realities (non-continuous) ones fit in [more difficult but possible].
Scott Mayers wrote: Sun Feb 24, 2019 7:05 am This is the same kind of problem with physics regarding the discrete concept of 'quantification' (treating the very small as having discrete whole or integral measures) versus relativity which begins with a continuous (non-discrete) system of reasoning by nature. Both are still understood to be essential. You may find difficulty making sense of one or the other but both are still 'true' of reality. But we don't know which or what universal system includes them both or that some third more inclusive explanation is more conclusive.
Shannon Information bridges the discrete/continuous gap: https://en.wikipedia.org/wiki/Nyquist–S ... ng_theorem

Scott Mayers wrote: Sun Feb 24, 2019 7:05 am If your intent is to show validity to say, some kind of 'fuzzy' logic, don't deny classical true-false logic as completely invalid but rather only a subset of the whole.
OK. Classical logic is consistent sometimes and inconsistent sometimes. That means you have to understand WHEN it is inconsistent and make sure you aren't falling for the trap.

Are you doing that?
Scott Mayers wrote: Sun Feb 24, 2019 7:05 am Denying it would require also denying even 'fuzzy' logic. [I read some books that try to 'sell' out classic logic as being false rather than to simply assert it incomplete. I'm guessing that the authors were not the actual 'fuzzy logicians' because they don't realize that fuzzy logic is an EXTENSION of classical binary-valued logic into multi-variable forms.] It's sad when even authors who may intend to advocate for the concept of multi-variable logics tend to act as the very 'classical' thinkers they accuse 'classical logic' to be about: that to be logical, you have to pick ONLY-classical binary-valued logic XOR non-classical multi-valued logic. Do you see how expressing this is itself is hypocritical?
Incorrect. I think the reason we are drifting apart is that you probably (like most people who have studied philosophy but not computer science) assume set theory as the foundation of Mathematics.

I don't. I consider Type theory/Lambda calculus as foundation. Computation. From there all rules/axioms are synthetic. Models.

Turing-completeness + recursion gives me ALL logics. https://en.wikipedia.org/wiki/Chomsky_hierarchy
Last edited by Logik on Wed Feb 27, 2019 4:59 pm, edited 2 times in total.
Scott Mayers
Posts: 1692
Joined: Wed Jul 08, 2015 1:53 am

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

Logik, I just invested a good couple hours responding carefully to your post above. You might have appreciated it and I certainly enjoyed doing so. But this site times us out and I lost the whole thing! I need a break and will re-respond to you later. I am so pissed right now! If they are going to do this they should use an automatic draft-save before signing us out.

Frustrated,

Scott.
Speakpigeon
Posts: 976
Joined: Sat Nov 11, 2017 3:20 pm
Location: Paris, France, EU

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

Logik wrote: Sat Feb 23, 2019 1:25 pm 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.
It's just too bad you can't explain yourself in plain English...
What a shame!
I can't really laugh as much as I would looking at your "proof"...
A proof has to be convincing or it's not a proof. And a proof is only convincing if we can see it and understand it. Formal proofs in logic are remarkably simple and easy to understand. So you would need transcribe your algorithm here using ordinary symbols, like ":=", "IF..THEN", etc. so we have a chance of being convinced. A computer can be made to appear to assert that A = A is false. That has no value until we understand the code it uses to do it.
EB
Last edited by Speakpigeon on Sun Feb 24, 2019 12:15 pm, edited 1 time in total.
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Scott Mayers wrote: Sun Feb 24, 2019 11:40 am Logik, I just invested a good couple hours responding carefully to your post above. You might have appreciated it and I certainly enjoyed doing so. But this site times us out and I lost the whole thing! I need a break and will re-respond to you later. I am so pissed right now! If they are going to do this they should use an automatic draft-save before signing us out.

Frustrated,

Scott.
No stress

I type all of my responses in a text editor then I copy/paste. This has happened to me far too often also.
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Speakpigeon wrote: Sun Feb 24, 2019 12:06 pm It's just too bad you can't explain yourself in plain English...
What a shame!
I can't really laugh as much as I would looking at your "proof"...
EB
What part requires explanation?

If you claim that "All swans are white" and I show you a black swan then that contradicts your premise.
If you claim that all "for all X: X = X" and I show you an example where X =! X that contradicts your premise.

All truth is conceptual first. So I have absolutely no idea what your conception of "proof" is.
My conception of proof is "theorems consistent with the axioms".

The law of identity is self-inconsistent.

If for all x: x = x THEN for all =: = = =

Yes, I know that you raised an objection that I "can't do that". What you were meant to say is that YOU can't do that.
Self-reference is part and parcel of human expression and communication.

To say that I am not "allowed" to speak of self-reference is to claim that I am not allowed to use the word "I".

The English word I is self-referential. I am referring to myself when I USE it.

Self-reference is recursion.
Recursion is computation.

https://en.wikipedia.org/wiki/Computability_theory
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Speakpigeon wrote: Sun Feb 24, 2019 12:06 pm I can't really laugh as much as I would looking at your "proof"...
A proof has to be convincing or it's not a proof.
Bullshit.

The notion of "proof" has a precise and objective Mathematical AND logical definition.
Internal consistency.

To claim that "proofs" need to be convincing is to play games of power.

WHO needs to be convinced for something to be "proof". You?
What makes YOU an authority on "valid proofs"?

5*5 is 25 whether you are convinced or not!

Mathematically my working program is proof! You aren't convinced because you don't understand the Curry-Howard isomorphism.

But that is your problem, not mine.
Last edited by Logik on Sun Feb 24, 2019 12:27 pm, edited 1 time in total.
Speakpigeon
Posts: 976
Joined: Sat Nov 11, 2017 3:20 pm
Location: Paris, France, EU

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

Logik wrote: Sun Feb 24, 2019 12:12 pm The law of identity contradicts itself. Via self-reference.
The Law of Identity doesn't contradict itself.
Logik wrote: Sun Feb 24, 2019 12:12 pm If for all x: x = x THEN for all =: = = =
And how is that a logical contradiction? On the face of it, it seems rather self-consistent to me.
Self-identity is universal and it applied to itself.
Logik wrote: Sun Feb 24, 2019 12:12 pm Yes, I know that you raised an objection that I "can't do that". What you were meant to say is that YOU can't do that.
Self-reference is part and parcel of human communication. For example. I am expressing my thoughts right now.
Sure, and it's fine but a formal logic is a closed language so you would need to exhibit the evidence that the clause "for all =: = = =" is a well-formed formula in classical logic.
Logik wrote: Sun Feb 24, 2019 12:12 pm The English word I is self-referential. I am referring to myself when I USE it.
Oh, you've learned quite a trick there! Descartes was there before you though.
EB
Speakpigeon
Posts: 976
Joined: Sat Nov 11, 2017 3:20 pm
Location: Paris, France, EU

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

Logik wrote: Sun Feb 24, 2019 12:20 pm To claim that "proofs" need to be convincing is to play games of power.
WHO needs to be convinced for something to be "proof". You?
What makes YOU an authority on "valid proofs"?
Mathematically my working program is proof! You aren't convinced because you don't understand the Curry-Howard isomorphism.
But that is your problem, not mine.
No, I'm not convinced because I have no idea what your programme does.
EB
Scott Mayers
Posts: 1692
Joined: Wed Jul 08, 2015 1:53 am

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

Logik wrote: Sun Feb 24, 2019 12:10 pm
Scott Mayers wrote: Sun Feb 24, 2019 11:40 am Logik, I just invested a good couple hours responding carefully to your post above. You might have appreciated it and I certainly enjoyed doing so. But this site times us out and I lost the whole thing! I need a break and will re-respond to you later. I am so pissed right now! If they are going to do this they should use an automatic draft-save before signing us out.

Frustrated,

Scott.
No stress

I type all of my responses in a text editor then I copy/paste. This has happened to me far too often also.
It wasn't the first time on this site but I keep forgetting that I have to do it for this one. The software for this site can do this and so I forget that I'm on not on those sites that enable either keeping you signed in while having the page open or to automatic drafting. I'll try again. I am certain you'd appreciate it because I think I know what the problem is. You kind of hinted at this factor regarding background approach to the same topics. Let me try again.
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Speakpigeon wrote: Sun Feb 24, 2019 12:27 pm The Law of Identity doesn't contradict itself.
You SAY that it doesn't, but you are no authority on any of this so your opinion doesn't matter.
I DEMONSTRATED that it does.

With a working computer program which is a valid Mathematical proof.

I trust Mathematics more than I trust Speaking Pigeons.
Speakpigeon wrote: Sun Feb 24, 2019 12:27 pm
Logik wrote: Sun Feb 24, 2019 12:12 pm If for all x: x = x THEN for all =: = = =
And how is that a logical contradiction? On the face of it, it seems rather self-consistent to me.
Self-identity is universal and it applied to itself.
The problem is any example where I can demonstrate to you that = != =

Anywhere your grammar is inconsistent you are violating identity.
Speakpigeon wrote: Sun Feb 24, 2019 12:27 pm Oh, you've learned quite a trick there! Descartes was there before you though.
Sure, but he took thought for granted.

I think therefore I am.
If I don't think therefore I am not.

Am I sure that I can think? What is thought? How does thought work? He left all of that unexplored...

He didn't really have a good theory of thought in the 17th century. We do in the 21st.

It's not a trick. It's just computation.
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Speakpigeon wrote: Sun Feb 24, 2019 12:29 pm No, I'm not convinced because I have no idea what your programme does.
Well there is the problem! You are looking for the wrong thing!

It is not about what it DOES.
It is all about what it DOESN'T!

A != A is evaluated as True and this DOESN'T produce a contradiction!

Because working programs are valid mathematical proofs what you are looking at is an internally consistent logic system which ALLOWS for A != A to be true.
Arising_uk
Posts: 12313
Joined: Wed Oct 17, 2007 2:31 am

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

Logik wrote:A thing needs not be the same as itself! ...
Personally I don't think that's the point Aristotle was making, 'A = A'(although I question the use of the mathematical equality sign here and would use a biconditional) means "A is itself" not 'the same as itself' otherwise you get the problem that you think there are two things as in 'A is the same as B', e.g. "this brick is the same as that brick".
Atla
Posts: 2858
Joined: Fri Dec 15, 2017 8:27 am

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

Scott Mayers wrote: Sun Feb 24, 2019 12:31 pm It wasn't the first time on this site but I keep forgetting that I have to do it for this one. The software for this site can do this and so I forget that I'm on not on those sites that enable either keeping you signed in while having the page open or to automatic drafting. I'll try again. I am certain you'd appreciate it because I think I know what the problem is. You kind of hinted at this factor regarding background approach to the same topics. Let me try again.
You may want to consider not treating an NPD like Logik with kindness and respect, as he will use this against you and others. Please also note that Logik can neither understand logic, nor does he know what computer programs are capable of currently.
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Atla wrote: Sun Feb 24, 2019 1:51 pm You may want to consider not treating an NPD like Logik with kindness and respect, as he will use this against you and others. Please also note that Logik can neither understand logic, nor does he know what computer programs are capable of currently.
That is precisely the kind of lie an NPD would tell. I don't treat people badly - I treat them accordingly.

As to your lame attempt at character assassination - I let my argument speak for itself.

So much for the law of non-contradiction.

https://repl.it/repls/FantasticTenseDividend
A is B: True
A is not B: True
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Arising_uk wrote: Sun Feb 24, 2019 1:51 pm
Logik wrote:A thing needs not be the same as itself! ...
Personally I don't think that's the point Aristotle was making, 'A = A'(although I question the use of the mathematical equality sign here and would use a biconditional) means "A is itself" not 'the same as itself' otherwise you get the problem that you think there are two things as in 'A is the same as B', e.g. "this brick is the same as that brick".
It doesn't matter.

The law of non-contradiction is broken in exactly the same way.
Ignore identity.

https://repl.it/repls/FantasticTenseDividend