## Inconsistent Mathematics

Next
• 7
High praise, seeing as you would have a better grasp of what it is about than we commoners.

Fishfry and Tones have far better grasps than me of the logic and set theoretic aspects of this subject. I just like the observation that naive set theory has some value. :smile:
• 12
Yes, I see it's more clever than just resignation that things might always be a bit off. As long as it's inconsistent in such and such a way then information is preserved or something of that nature.
• 26
the contradictions themselves as they appear in such systems are at the end of the day empty words.

I don't agree. There's something important going on here.

What do you think it is? Continuing with my example contradiction x is 1 AND x is not 1, what is the value of x?
• 27
That example would not arrises in the instances given in the cited articles.

Try this one:

Australasian Journal of Logic Paraconsistent Measurement of the Circle

There is comfort to be had knowing that the area of the circle is indifferent to changes in logic. The core of Archimede’s insight is derivable even in the event of inconsistency. Truth is not so fragile.
• 26
That example would not arrises in the instances given in the cited articles.

Try this one:

Australasian Journal of Logic Paraconsistent Measurement of the Circle

There is comfort to be had knowing that the area of the circle is indifferent to changes in logic. The core of Archimede’s insight is derivable even in the event of inconsistency. Truth is not so fragile.

Speaking for myself, paraconsistent logic is a "higher form" of logic and contains, as a proper subset classical logic (sentential, predicate, categorical) barring reductio ad absurdum. So, I'm not as I impressed as I would've liked to be by the fact that the area of a circle is "indifferent to changes in logic" - it must be so.

Secondly, I'm not so sure that we can be as mechanical in our thinking in paraconsistent logic as we can be in classical logic. In the former, we just plug in the propositions, apply natural deduction rules, and out at the other end we get a true proposition. In the latter, I surmise, we have to be constantly alert to the possibility that we aren't making some kind of mistake like how mathematicians have to take precautions that they aren't dividing by zero when tackling algebraic problems.

By the way, you haven't really addressed the issue I raised. A contradiction simply doesn't make any kind of sense at all. Banno is a member of TPF AND Banno is not a member of TPF. Ergo, Banno is...??? ( :chin: )
• 27
In the latter, I surmise, we have to be constantly alert to the possibility that we aren't making some kind of mistake like how mathematicians have to take precautions that they aren't dividing by zero when tackling algebraic problems.

That would not be correct. The logic being proposed is as formal as any.

A contradiction simply doesn't make any kind of sense at all.

Indeed; and yet here we have a paraconsistent logic that begins to make sense.
• 26
That would not be correct. The logic being proposed is as formal as any.

Indeed; and yet here we have a paraconsistent logic that begins to make sense.

Ok, so for what it's worth, here's what I think:

Nicolai A. Vasiliev (father of paraconsistent logic) is said to have admitted that he was influenced by Nikolai Lobachevsky (father of hyperbolic geometry)

Reasoning by analogy with the "imaginary" geometry of Lobachevsky, Vasiliev called his novel logic "imaginary", for he assumed it was valid for the worlds where the above-mentioned laws (law of the excluded middle and the law of noncontradiction) did not hold, worlds with beings having other types of sensations.  — Wikipedia

Since Nicolia A. Vasiliev is borrowing a page from Nikolai Lobachevsky's hyperbolic geometry, specifically the rejection of a postulate (Euclid's controversial (?? :chin: ??) fifth postulate), we need to look into the so-called Three Laws Of Thought:

1. Law of identity. A = A
2. Law of the excluded middle. p v ~p
3. Law of noncontradiction. ~(p & ~p)

At this point I'm out of depth but in my humble opinion these laws of thought are, at the end of the day, postulates i.e. they're assumptions. Given this, we're free to deny any or all of them and investigate what doing so might lead to. That's exactly what Nicolai A. Vasiliev, inspired by Nikolai Lobachevsky, did and we have logic that tolerate contradictions (paraconsistent logic).

Truth is there doesn't seem to be any real difference between paraconsistent logic and madness/stupidity as inconsistencies are the hallmark of all three. In that sense, Nicolai A. Vasiliev's paraconsistent logic is a study of insanity/inanity. (Mentally ill & mentally retarded) people since time immemorial have been using paraconsistent logic. :chin:.

There's a fine line between genius and insanity. I have erased this line. — Oscar Levant

The Wisdom Of The Fool

Also, paraconsistent logic may prove its worth in matters inherently subjective:

X: The movie was phenomenal!
Y: No, it was not!
Z: It was both phenomenal and also not phenomenal!

De gustibus non est disputandum

Last but not the least, The Paradox Of Paraconsistent Logic:

Paraconsistent logic makes sense AND paraconsistent logic doesn't make sense!
• 27
...these laws of thought are, at the end of the day, postulates i.e. they're assumptions. Given

...what happens if we do not assume these? Can we find a way to do that, which still maintains a capacity to construct arguments?

That's what is being proposed.

What this thread is about is that this fringe approach to logic has recently shown some interesting aspects of mathematical proof - the example above begins work towards a proof of integral calculus in a paraconsistent logic.

...there doesn't seem to be any real difference between paraconsistent logic and madness/stupidity as inconsistencies are the hallmark of all three

But here we have a way to perhaps understand these inconsistencies in a coherent way. Madness and stupidity is perhaps to do with incoherence rather than inconsistency.

Hence the somewhat surprising break between consistency and coherence.
• 26
...what happens if we do not assume these? Can we find a way to do that, which still maintains a capacity to construct arguments?

The world would grind to a halt and from then on chaos.

X: The UN will convene at 10:00 AM and the UN will not convene at 10:00 AM
Y: WTF?

From my own painful personal experience, one goes through both physical and mental paralysis when face to face with a contradiction!

That this discussion between the two of us is even possible depends on the laws of thought.

What this thread is about is that this fringe approach to logic has recently shown some interesting aspects of mathematical proof - the example above begins work towards a proof of integral calculus in a paraconsistent logic.

I know next to nothing about calculus so I'm not going to be able to make a comment that would further the discussion. I'll say this though, Godel's incompleteness theorems seem mighty relevant.

But here we have a way to perhaps understand these inconsistencies in a coherent way. Madness and stupidity is perhaps to do with incoherence rather than inconsistency.

Hence the somewhat surprising break between consistency and coherence.

Coherence & Consistency? :chin:

I don't think these two are different. Coherence is only possible if there are no inconsistencies. In other words, coherent iff consistent.
• 7
Now, who'd a thunk non-euclidean space could be useful...

Math had started out of practical uses in ancient Egypt for building the pyramids etc.
• 7
Possible worlds is a quite different area. It explicitly assumes consistency. This does not. Of course, if this could be made coherent, then it might be applied to possible world semantics.

Would it be able to cover the area of the Traditional and Modal Logic, where they cannot cope with some of the real world cases in the arguments?

Nor does it have anything to do with existentialism. But it might be a sort of deconstruction, in which consistency is seen as a special case...

Existentialism as opposed to Rationalism, and denoting absurdity, irrationality and unpredictability?
Even if it is Inconsistent Math, if it is a Math, it would have some consistency, one would imagine.
• 27
I don't think these two are different.

But now it seems that there might be an alternative. Rather than an incomplete yet consistent account of mathematics and language, we might construct an inconsistent yet complete account...

Yes, you don't. There's one issue.
• 27
Math had started out of practical uses in ancient Egypt for building the pyramids etc.

Indeed; and cats can be black.
• 26
Now, who'd a thunk non-euclidean space could be useful...

If you don't mind me butting in, I feel non-euclidean geometry is important to non-classical logic. N. A. Vasiliev (father of paraconsistent logic) drew an analogy between his paraconsistent logic and N. Lobachevsky's (father of non-euclidean geometry) and all we need to do is take the analogy just one step further and say that in higher-dimensional analogs of classical logic, "contradictions are true" (more on this below).

A simple example of this is Lobachevsky's take on Euclid's fifth postulate itself. In 2 dimensions, true that only one line can be drawn through a point that's parallel to another line but in higher dimensions (I hope I got this right), an infinite number of lines can go through a point such that they're all parallel to another line.

However, note that we have to switch between euclidean space and noneuclidean space for there to be a contradiction i.e. each of the two spaces above are self-consistent; it's only when the two are juxtaposed that a contradiction rears its ugly head so to speak.

Thus, as I mentioned earlier, whenever a contradiction arises, it should send alarm bells ringing - we're looking at the same thing in different contexts (analogously different dimensions). For this reason, I recommend paraconsistent logic for human experiences that are inherently subjective, each point of view providing its own "dimension" and assigning its own unique truth value to a proposition.
Next
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal