The death of Classical logic and the birth of Constructive Mathematics

What is the basis for reason? And mathematics?

Moderators: AMod, iMod

Scott Mayers
Posts: 1694
Joined: Wed Jul 08, 2015 1:53 am
Location: Saskatoon, SK, Canada

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

Post by Scott Mayers »

Logik wrote: Sun Feb 24, 2019 8:11 am
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.

The code I provided in Python (Lambda calculus - same thing) demonstrates that I have a logic in which A != A is a valid, non-contradictory expression.
You are expecting an assignment whereby the left-side expression takes the value of the right-side because you are used to those higher-ordered languages in computing. The error is not to the 'classic' logic though but to your 'top-down' approach to learning.

The actual logical assumptions from first-order logic are misleading because they confuse 'coincidence' relations to 'equality' and the exclusive-or to inclusive-or.

The general three laws accepted are:
(1) A law that asserts consistency but often stated as about 'equality'.
A = A but you need to translate in computers as A == A


(2) A law that asserts,
A or not-A and


(3) A law that directly asserts no third option that permits
A and not-A......Exclusive Middle possibility such as {A, A and not-A, not-A} excluding the contraction [“con-” (with) “-tra-” (third) “diction” spoken-part]
All these are actually implied by any one of them and only spells out different perspectives of the same thing.

A more accurate means I believe would/should be to use operators of 'coincidence' (⊙ ) and the 'exclusive-or' (⊕ ) because they explicitly deal with the same term. So the laws might be better written as:
Revised and Clarified Consistency Laws
(1) Law of Consistency
A ⊙ A = 1,

(2) Law of Exclusion, which is just the 'dual' of (1)
A ⊕ A = 0 and

The law of excluding the 'middle' term also relates to Aristotle's use of he middle term but is equivalent to the above and the law of non-contradiction implied by both of these but more clearly emphasized by the use of the normal OR operator as

(3) Law of Non-contradiction
A + A̅
These are all proved within most Boolean systems by simply using the inclusive-or, and, and complement. Then it proves later defines the exclusive or and uses DeMorgan's Theorem to determine the duals which included defining the inverse of exclusive-or as the 'coincidence operator.


Logik wrote: Sun Feb 24, 2019 8:11 am
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 the LNC. No constructive mathematician does.
Non-contradiction stays .
I happen to treat physical reality as due to laws in a kind of Pythagorean sense. Physicists from M.I.T. seem to agree. See those like Max Tegmark or Seth Lloyd for references in general on this. “laws” of reality are just those patterns of a subset of worlds among all possible worlds which happen to be consistent. If anything, we 'discover' math as much as physics but the language we use to express them are itself flexible.

[more responses later....this should be at least good for a start.]

EDIT: fixed the color command to make the red letters above appear. needed spaces removed in the command apparently
Last edited by Scott Mayers on Sun Feb 24, 2019 3:35 pm, edited 1 time in total.
Atla
Posts: 2888
Joined: Fri Dec 15, 2017 8:27 am

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

Post by Atla »

Logik wrote: Sun Feb 24, 2019 2:08 pm
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
You made no argument. :) Here you merely instructed a program to give True to both.

I replaced two words in your program and now I get this:

A is B: Duck
A is not B: Chicken


You neither understand logic, nor do you seem to have the faintest idea what computer programs are capable of.
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Post by Logik »

Scott Mayers wrote: Sun Feb 24, 2019 2:29 pm You are expecting an assignment whereby the left-side expression takes the value of the right-side because you are used to those higher-ordered languages in computing. The error is not to the 'classic' logic though but to your 'top-down' approach to learning.
Why is top-down learning a mistake? Isn't this how all humans learn?
First you experience water, only then do you learn that it's made of Hydrogen and Oxygen.
First we learned about atoms only later we learned about electrons and protons.
And later yet did quantum physics come to the party.

Human experience is top down, not bottom up.
Scott Mayers wrote: Sun Feb 24, 2019 2:29 pm The actual logical assumptions from first-order logic are misleading because they confuse 'coincidence' relations to 'equality' and the exclusive-or to inclusive-or.
I know ;) This is why I am doing this.

If you are going to be teaching kids to think the last thing you want to do is to give them a tool with sharp edges!

Scott Mayers wrote: Sun Feb 24, 2019 2:29 pm The general three laws accepted are:
(1) A law that asserts consistency but often stated as about 'equality'.
A = A but you need to translate in computers as A == A
That is what I have done. Perhaps I should draw a line in the code.

At first I set the context (defining classes objects etc).

The assertions are at the bottom.
Scott Mayers wrote: Sun Feb 24, 2019 2:29 pm (2) A law that asserts,
A or not-A and
But if A represents an object that is a human. The logical expression not A doesn't make sense.

What exactly is the negation of a human? I can't conceive of such a thing!

Is it an ape or a planet?
Scott Mayers wrote: Sun Feb 24, 2019 2:29 pm (3) A law that directly asserts no third option that permits
A and not-A......Exclusive Middle possibility such as {A, [color = red]A and not-A[/color], not-A} excluding the contraction [“con-” (with) “-tra-” (third) “diction” spoken-part]
Constructive mathematics starts precisely with rejecting LEM.
This is the fundamental problem with LEM.

You could say that color = red (A)
But then the negation of "color" (not A) could mean either:
* The entire light spectrum that is NOT red
or
* Everything in the universe that doesn't have the color red.

The problem with FOL is precisely that which is not written. Context!
Scott Mayers wrote: Sun Feb 24, 2019 2:29 pm (1) Law of Consistency
A ⊙ A = 1,

(2) Law of Exclusion, which is just the 'dual' of (1)
A ⊕ A = 0 and

The law of excluding the 'middle' term also relates to Aristotle's use of he middle term but is equivalent to the above and the law of non-contradiction implied by both of these but more clearly emphasized by the use of the normal OR operator as

(3) Law of Non-contradiction
A + A̅
You are falling for the exact same error. because you can't define ⊙, ⊕ and + (but I can - in computer code).
I can construct ANY logic I want to produce the results you expect.

I will construct you a logic in which:

A ⊙ A = 0
A ⊕ A = 1
A + A̅ = 1
Scott Mayers wrote: Sun Feb 24, 2019 2:29 pm These are all proved within most Boolean systems by simply using the inclusive-or, and, and complement. Then it proves later defines the exclusive or and uses DeMorgan's Theorem to determine the duals which included defining the inverse of exclusive-or as the 'coincidence operator.
The problem is still interpretation ;)

I can interpret any symbol differently to how you intended it to be interpreted.

You have no control over that.

Scott Mayers wrote: Sun Feb 24, 2019 2:29 pm
Non-contradiction stays .
I happen to treat physical reality as due to laws in a kind of Pythagorean sense. Physicists from M.I.T. seem to agree. See those like Max Tegmark or Seth Lloyd for references in general on this. “laws” of reality are just those patterns of a subset of worlds among all possible worlds which happen to be consistent. If anything, we 'discover' math as much as physics but the language we use to express them are itself flexible.

[more responses later....this should be at least good for a start.]
Looks like I got carried away. So I contradicted the Classical formulation of non-contradiction.


https://repl.it/repls/FantasticTenseDividend
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Post by Logik »

Atla wrote: Sun Feb 24, 2019 2:38 pm You made no argument. :)
Every Mathematician disagrees with you.
Atla wrote: Sun Feb 24, 2019 2:38 pm Here you merely instructed a program to give True to both.
Nonsense. I created a brand new logical system.

I just changed the meaning of "=" and "!="
Atla wrote: Sun Feb 24, 2019 2:38 pm I replaced two words in your program and now I get this:

A is B: Duck
A is not B: Chicken
Which is precisely the point, moron.

What you changed wasn't "two words". You changed the definition of the functions __eq__ (that is - you changed the MEANING of "=") and you changed the definition of the function __ne__ (you changed the MEANING of "!=") !

So IF logic can produce ANY answer you want it to produce just as soon as you interpret "=" and "!=" differently then we have a problem, don't you think?

Surely the questions that this brings about are as follows:
* What does "=" mean?
* What does "!=" mean?
Atla wrote: Sun Feb 24, 2019 2:38 pm You neither understand logic, nor do you seem to have the faintest idea what computer programs are capable of.
I understand it enough to know how to USE it, You understand it enough to be a slave to its "laws".

That is why you are a dogmatist ;)
Last edited by Logik on Sun Feb 24, 2019 3:23 pm, edited 1 time in total.
Atla
Posts: 2888
Joined: Fri Dec 15, 2017 8:27 am

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

Post by Atla »

Logik wrote: Sun Feb 24, 2019 3:00 pm
Atla wrote: Sun Feb 24, 2019 2:38 pm You made no argument. :)
Every Mathematician disagrees with you.
Atla wrote: Sun Feb 24, 2019 2:38 pm Here you merely instructed a program to give True to both.
Nonsense. I created a brand new logical system.

I just changed the meaning of "=" and "!="
Atla wrote: Sun Feb 24, 2019 2:38 pm I replaced two words in your program and now I get this:

A is B: Duck
A is not B: Chicken
Which is precisely the point, moron.

What you changed wasn't "two words". You changed the definition of the functions __eq__ (that is - you changed HOW "=" works) and you changed the definition of the function __ne__ (you changed HOW "!=") works!

So IF logic can produce ANY answer you want it to produce just as soon as you interpret "=" differently then we have a problem, don't you think?

Surely the questions that this brings about are as follows:
* HOW should "=" work?
* HOW should "!=" work?

Atla wrote: Sun Feb 24, 2019 2:38 pm You neither understand logic, nor do you seem to have the faintest idea what computer programs are capable of.
I understand it enough to know how to USE it, You understand it enough to be a slave to its "laws".

That is why you are a dogmatist ;)
So all mathematicians would agree that instructing a python program to give 'Duck' when encountering an equal sign, is an argument against classical logic.

Or better yet, 'Invisible yellow quantum duck goes shopping'. That will demolish classical logic for sure.

-----------

Btw YOU came up with this idea to interpret a sign any way you want, and then you state that there is a problem with interpreting a sign any way you want. That's why people with a brain don't do it.
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Post by Logik »

Atla wrote: Sun Feb 24, 2019 3:22 pm So all mathematicians would agree that instructing a python program to give 'Duck' when encountering an equal sign, is an argument against classical logic.
Strawman! Pay attention

If I wanted to instruct the computer to say "Duck" I would have simply issued the "print("Duck")" instruction.

I didn't do that!

What I did (and what you did too) was to redefined the meaning of "=" and "!=" in a way that you could leverage the principle of explosion!
In leveraging the principle of explosion we end up creating a CONSISTENT logical system.
Atla wrote: Sun Feb 24, 2019 3:22 pm Btw YOU came up with this idea to interpret a sign any way you want,
It's not MY idea, dimwit. Isn't this what philosophers have been doing for ever and ever? (mis) interpreting things!
Atla wrote: Sun Feb 24, 2019 3:22 pm and then you state that there is a problem with interpreting a sign any way you want. That's why people with a brain don't do it.
It is a fucking problem.

If we can't agree on the meaning of fucking "=" how do you think we are ever going to agree on the rest of logic?

I am merely pointing out to you that I can (mis)interpret ANYTHING I want and I can do it better than any of you clowns.

Any "rule" you lay down that asserts "true" I will produce an alternative logical system in which it evaluates to false.
Atla
Posts: 2888
Joined: Fri Dec 15, 2017 8:27 am

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

Post by Atla »

Logik wrote: Sun Feb 24, 2019 3:27 pm
Atla wrote: Sun Feb 24, 2019 3:22 pm So all mathematicians would agree that instructing a python program to give 'Duck' when encountering an equal sign, is an argument against classical logic.
Strawman! Pay attention

If I wanted to instruct the computer to say "Duck" I would have told it to "print("Duck")",

I didn't do that!
What I did (and what you did too) was to redefined the meaning of "=" and "!=" in a way that you could leverage the principle of explosion!
It's python and I tried it, you don't need to print 'Duck', you can just redefine equality as 'Duck'.

Talking about meaning here is a sign of insanity.

Also I think I'll avoid python in the future if it's this fucked up.
Atla wrote: Sun Feb 24, 2019 3:22 pm Btw YOU came up with this idea to interpret a sign any way you want,
It's not MY idea, dimwit. Isn't this what philosophers have been doing for ever and ever? (mis) interpreting things!
Atla wrote: Sun Feb 24, 2019 3:22 pm and then you state that there is a problem with interpreting a sign any way you want. That's why people with a brain don't do it.
Decent philosophers have always been against that.
You have misunderstood thousands of years of philosophy, what an epic strawman.

You are an idiot beyond my comprehension, which is why I study you. :)
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Post by Logik »

Atla wrote: Sun Feb 24, 2019 3:36 pm It's python and I tried it, you don't need to print 'Duck', you can just redefine equality as 'Duck'.
PRECISELY!!!!!!!!!!!!!

You can define equality and inequality as ANYTHING you choose.

WHO decides whether A = B and HOW ?
Atla wrote: Sun Feb 24, 2019 3:36 pm Also I think I'll avoid python in the future if it's this fucked up.
Way to miss the forest for the trees. ALL LOGIC is this fucked up!

I can do the exact same thing in ANY LOGICAL SYSTEM.

You give me one inch to (mis)interpret something and I will take your entire arm!!!

This is not a problem ANY logic has ever solved. This is the principle of explosion.
The great part of the principle of explosion is that it has constructive properties!

As you have learned. Which is why I am a social constructionist.
Atla wrote: Sun Feb 24, 2019 3:36 pm Decent philosophers have always been against that.
You have misunderstood thousands of years of philosophy, what an epic strawman.

You are an idiot beyond my comprehension, which is why I study you. :)
Then decent philosophers should start with explaining AND agreeing on the meaning of "=" don't you think?

What does it MEAN for one thing to be the same as another?

But instead of you recognizing the problem, you are going on a crusade to reject ALL of logic.

Logic is just a tool. USE it. Don't enslave yourself to it. There's no such fucking thing as "truth".
Atla
Posts: 2888
Joined: Fri Dec 15, 2017 8:27 am

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

Post by Atla »

Logik wrote: Sun Feb 24, 2019 3:39 pm
Atla wrote: Sun Feb 24, 2019 3:36 pm It's python and I tried it, you don't need to print 'Duck', you can just redefine equality as 'Duck'.
PRECISELY!!!!!!!!!!!!!

You can define equality and inequality as ANYTHING you choose.

WHO decides whether A = B and HOW ?
Atla wrote: Sun Feb 24, 2019 3:36 pm Also I think I'll avoid python in the future if it's this fucked up.
Way to miss the forest for the trees. ALL LOGIC is this fucked up!

I can do the exact same thing in ANY LOGICAL SYSTEM.

You give me one inch to (mis)interpret something and I will take your entire arm!!!

This is not a problem ANY logic has ever solved. This is the principle of explosion.
The great part of the principle of explosion is that it has constructive properties!

As you have learned. Which is why I am a social constructionist.
Atla wrote: Sun Feb 24, 2019 3:36 pm Decent philosophers have always been against that.
You have misunderstood thousands of years of philosophy, what an epic strawman.

You are an idiot beyond my comprehension, which is why I study you. :)
Then decent philosophers should start with explaining AND agreeing on the meaning of "=" don't you think?

What does it MEAN for one thing to be the same as another?

But instead of you recognizing the problem, you are going on a crusade to reject ALL of logic.

Logic is just a tool. USE it. Don't enslave yourself to it. There's no such fucking thing as "truth".
What a complete idiot. That's why for example formal logic was invented, where "=" has only one interpretation.

But in that case, nothing has any meaning in the metaphysical sense. Your solution is to get rid of all meaning and turn humans into computers.

Hey I know, let's remove the right hemisphere after birth, that should make it easier.
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Post by Logik »

Atla wrote: Sun Feb 24, 2019 3:51 pm What a complete idiot. That's why for example formal logic was invented, where "=" has only one interpretation.
It DOESN'T have "one" interpretation dipshit. It has a million interpretations.

You use "=" to mean all sorts of fucking shit in your arguments.

How can TWO things be "the same" ? Are they entangled or what?
Atla wrote: Sun Feb 24, 2019 3:51 pm But in that case, nothing has any meaning in the metaphysical sense. Your solution is to get rid of all meaning and turn humans into computers.
You are such an idiot. You can't turn something into something that it already is.

In the Platonic/abstract sense you are a computer.

Meaning and metaphysics have NOTHING in common. Metaphysics is logic - the structure of your thoughts. Your model of reality.

I couldn't give a shit whether we live in a triangular or rectangular universe.
Or whether we come from Heaven or Hell.

That's not what meaning is. To me anyway. Meaning is purpose. WHere are we going?
Last edited by Logik on Sun Feb 24, 2019 3:57 pm, edited 1 time in total.
Atla
Posts: 2888
Joined: Fri Dec 15, 2017 8:27 am

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

Post by Atla »

Logik wrote: Sun Feb 24, 2019 3:53 pm
Atla wrote: Sun Feb 24, 2019 3:51 pm What a complete idiot. That's why for example formal logic was invented, where "=" has only one interpretation.
It DOESN'T have "one" interpretation dipshit. It has a million interpretations.

YOU use "=" to mean all sorts of fucking shit in your arguments.
Nope idiot, I used different meanings because I was writing in non-formal-logic English. Because that's the default for metaphysics.

I CAN write in formal-logic-English, where "is" only has one interpretation if I want to.
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Post by Logik »

Atla wrote: Sun Feb 24, 2019 3:57 pm I CAN write in formal-logic-English, where "is" only has one interpretation if I want to.
OOOOOOH. Show me!

How many comparative functions do you think you have in your brain?
How many different distinctions can you draw OUT THERE in reality?
How many juxtapositions can you contrive where you ask the question "Is A the same as B?"

How many yes/no questions do you think you can ask and answer about reality?

Because you are going to need a different symbol for every one of them!
Atla
Posts: 2888
Joined: Fri Dec 15, 2017 8:27 am

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

Post by Atla »

Logik wrote: Sun Feb 24, 2019 3:58 pm
Atla wrote: Sun Feb 24, 2019 3:57 pm I CAN write in formal-logic-English, where "is" only has one interpretation if I want to.
OOOOOOH. Show me.

How many comparative functions do you think you have in your brain?
Because you are going to need a new symbol for every one of them!
I don't care to, it's an inferior form of logic compared to the inherent fuzzy logic of the right hemisphere. I use both but I find the formal left-side logic very uninteresting.
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

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

Post by Logik »

Atla wrote: Sun Feb 24, 2019 4:02 pm I don't care to, it's an inferior form of logic compared to the inherent fuzzy logic of the right hemisphere. I use both but I find the formal logic very uninteresting.
It's a super-set of fuzzy logic.
It's a super-set of ALL logic!

It can do EVERYTHING every logic can do and more!

You can CONSTRUCT new semantics (meaning!)

How/why is it "inferior" ?

Is it something like "Because Atla doesn't understand it?" or do you have valid criticism?
Last edited by Logik on Sun Feb 24, 2019 4:05 pm, edited 1 time in total.
Atla
Posts: 2888
Joined: Fri Dec 15, 2017 8:27 am

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

Post by Atla »

Logik wrote: Sun Feb 24, 2019 4:03 pm
Atla wrote: Sun Feb 24, 2019 4:02 pm I don't care to, it's an inferior form of logic compared to the inherent fuzzy logic of the right hemisphere. I use both but I find the formal logic very uninteresting.
It's a super-set of all logic.

It can do EVERYTHING every logic can do and more.

How/why is it "inferior" ?
Well you only get 0 and 1, fuzzy also has 0 and 1 and everything in between, plus with the right hemisphere you can sort of easily add dimensions of logic.
Post Reply