I believe Wittgenstein thinks that a system in which there are contradictions can still be meaningful and even helpful in certain cases as inconsistency robustness has proved itself. I have selected the following passages from on online source that elucidates how allowing contradictions makes sense in our daily life. It is as if we impose on ourselves some limitations.

Inconsistency robustness is information system performance in the face of continually pervasive inconsistencies---a shift from the previously dominant paradigms of inconsistency denial and inconsistency elimination attempting to sweep them under the rug.

Inconsistency robustness differs from previous paradigms based on belief revision, probability, and uncertainty as follows:

Belief revision: Large information systems are continually, pervasively inconsistent and there is no way to revise them to attain consistency.
Probability and fuzzy logic: In large information systems, there are typically several ways to calculate probability. Often the result is that the probability is both close to 0% and close to 100%
Uncertainty: Resolving uncertainty to determine truth is not a realistic goal in large information systems.

Besides this is Wittgenstein's take on it,

Exactly. "Natural" there is not a mathematical term. It is not mathematically determined what is the natural thing to do. We most naturally compare a contradiction to something which jams. I would say that anything which we give and conceive to be an explanation of why a contradiction does not work is always just another way of saying that we do not want it to work. If you have a tube and a cock which shuts or opens it, your experience may have led you to think that always when the handle is parallel to the tube, the tube is open, and when it is at right angles to it, the tube is closed. But at home I have a cock which works the other way about. And in order to get used to it, I had to think of the handle as lying along the tube and blocking it, so that the tube was closed when the handle was parallel to it. I had to invent a new imagery. Similarly, one needs to change one's imagery in the case of contradictions. One can change one's imagery in such a way that 'p and not-p 'sounds entirely natural, as when we say, "The negative doesn't add anything".

This is most important. We shall constantly get into positions where it is necessary to have a new imagery which will make an absurd thing sound entirely natural. I want to talk about the sense in which we should say that the law of contradiction: - (p. - p) is a true proposition. Should we say that if '-(p. -p)' is a true proposition, it is true in a different sense of the word from the sense in which it is a true proposition that the earth goes round the sun? In logic one deals with tautologies-propositions like '- (p. -p)'. But one might just as well deal with contradictions instead. So that Principia mathematica would not be a collection of tautologies but a collection of contradictions. Should one then say that the contradictions were true? Or would one then say that "true" is being used in a different sense?

Are you turned on by darkness ?
Thanks for casting light on this topic.

I heard you will be providing some special services :wink:

I guess we Kant stop it now.

You have ignored my best friend Malcolm. That's not nice.

By Nazi, l take that you mean Donald Trump. :smile:

In Cambridge, around 2005. He was delivering lectures on the foundation of mathematics and he decided to pay me a visit as l was teaching a similar course. Sadly, he left after a while. I have to go to Norway this year and build myself a hut. I may meet Turing in the future, possibly 1939.

:wink: :wink:

I hope it's a bit neat now.

Yes, lecture XIX

It's difficult to edit it as it's a giant copy paste. I will try to see what l can do, l can't see if anyone would be bothered to read it though.

Considering your scientism, You might as well as build a temple of science and bow down before all your "idols" .

Why do we need religion when we have science ?
It's like saying, Why have food when we have water

I apologize for my terrible copy paste above but l can tell, wittgenstein is really making a profound point here. He even has Turing struggling to give a good counterpoint.

I think the central issue is that "rule following" or usage in mathematics is mathematics and not what what meaning we get out of it. If we assume that, it is easy to see why a contradiction isn't a problem if we either assign something to the result obtained after contradiction or leave it there. Read it again perhaps, it will get clearer.

I think if we all started reading Wittgenstein's Lectures on the foundation of mathematics, a lot of issues that come up here can be addressed in a good manner. As you have pointed at rightly. Wittgenstein regarded mathematics as a human invention, a finite calculus at most. But due to platonism, we sometimes give answers that answer different questions.

Wittgenstein further on even challenges the proof by induction used in mathematics.

Let's consider a proof by mathematical induction.
@ is a mathematical property. If it is known that @ (1) and it is known that [n] : @[n]. --> .@[n + l], then [n] . @[n].
It's misleading because it's not known how
[n] : @[n]. --> .@[n+ 1] is proved.

:smile:

This passage is directly from a book of wittgenstein
N(0) is used for the cardinality of the set of natural numbers. Wittgenstein shows that the technique we learn in writing the usual whole numbers and writing N(0) are different and he concludes that we cannot say we have written N(0) numeral.

We have all been taught a technique of counting in arabic numerals. We have all of us learned to count-we have learned to construct one numeral after another. Now how many numerals have you learned to write down?

Turing: Well, if I were not here, I should say N(0)

Wittgenstein: I entirely agree, but that answer shows something. There might be many answers to my question. For instance, someone might answer, " The number of numerals I have in fact written down." Or a finitist might say that one cannot learn to write down more numerals than one does in fact write down, and so might reply, "the number of numerals which I will ever write down". Or of course, one could reply "N(0)" as Turing did. Now should we say, "How wonderful-to learn N(0), numerals, and in so short a time! How clever we are!"?-Well, let us ask, "How did we learn to write N(0) numerals?" And in order to answer this, it is illuminating to ask, "What would it be like to learn only 100,000 numerals?"

Well, it might be that whenever numerals of more than five figures cropped up in our calculations, they were thrown away and disregarded. Or that only the last five figures were counted as relevant and the rest thrown away.-The point is that the technique of learning N(0),numerals is different from the technique of learning 100,000 numerals. Take the biggest numeral which has ever been mentioned. What is the difference between learning a technique of counting numerals up to that numeral and learning a technique which did not end at that numeral?
Well, it might have been that one's teachers said, "This series has no end." But how did you know what that meant? They might have said that and then when one reached the numeral six billion, they might say, "Well, now we have got here, I need hardly say . . ." and shrug their shoulders with a slight laugh.-So how did you know what they meant? Simply from the way in which the series was treated. I did not ask, "How many numerals are there?" This is immensely important. I asked a question about a human being, namely, "How many numerals did you learn to write down?" Turing answered "N(0)" and I agreed. In agreeing, I meant that that is the way in which the number N(0), is used. It does not mean that Turing has learned to write down an enormous number. N(0) is not an enormous number. The number of numerals Turing has written down is probably enormous. But that is irrelevant; the question I asked is quite different. To say that one has written down an enormous number of numerals is perfectly sensible, but to say that one has written down N(0), numerals is nonsense.

I'm 101 percent confident

Why is infinity always discussed here ?
1. Infinity is not a number.
2. Being undefined means we cannot assign any value to something.
3. Lim x->0 ( 1/x) is undefined
4. Lim x-> inf ( 1/x) is zero.

I think 3,4 will clarify all that confusion going on here.

What's my age ?? :wink:

I hope my toxic defeatist attitude leaves your beautiful creative soul alone and alive.

An acceptable theory of mind is beyond the scope the philosophy and even science in my opinion. Logical empiricism has it's own faults too.

Yes. The only way the shift will take place is that we abandon philosophy. If no one asks a question, there won't be any answers to find. Everything will be neat and tidy.

We can actually, with help of computer software then print it out. Dud dahh !!

That's my principle of engagement unless l get called out on my BS and the BS smells really bad and is clear as the day. Sometimes I just say
Go ahead punk ,make my day
Then blow off all the steam and bury everything into the ground.

I simply look at the type of people who are believing it. If they happen to be critical, it is a well constructed lie ( a little lie ). If they happen to be simple minded, it is a big lie.

"Dear, they do it with smoke and mirrors."

:lol: :lol: :grin:

Indeed. I had trouble with it when I first encountered it in 1962. :yikes:

Assuming you were 20 year old back then, you are approximately 77 right now. You are probably the oldest user here then

I meant the real number line but the set of real numbers is uncountably infinite so l think l did mess up there. You can clear things up . :smile:
I hope it is correct now.

That's a long time ago. I still have to learn a little more group theory before l can begin to appreciate it deeply. I wish he lived longer though. Who knows what he might have done later on had he lived.

Before addressing anything else, l want to assure you that l didn't want to take a jab at you or anything like that. I felt what you wrote was really interesting and l couldn't resist attacking it. It was all done in good mood. Nothing to worry about.

I want to know what do you think on the assertion that a paradigm shift occurs in science after it has come to a halting point. Do you think that the paradigm shift removes "psychological fixedness" ? I have read on that and it does capture some aspects of why science comes to a halt but it doesn't capture the details. I think certain viewpoint are not due to psychological fixedness but due to a consensus among the scientific community. You can disagree with my opinion obviously .

They need to know the basics BEFORE getting into the cryptic stuff.

Totally. l had a discussion a few weeks ago, where the other person wouldn't acknowledge that scientific statements are not a priori.

:smile:
I apologize for any copyright infringement. Please don't press any legal charges.

I wish this was understood by more people (shouldn't it be obvious?)

Ironically in philosophy, the simple things are left unnoticed. The most cryptic philosopher is usually the one who is studied the most too cause it is easy to argue about topics that can be misunderstood easily.

Well l will see what l can say. Let's try this.

1. If cheese is a dimension, then it will be infinitely divisible
2. Nothing existent( cheese) can be infinitely divisible.
3. Therefore, if cheese is a dimension it does not exist
4. Cheese exists :yum:
5. Therefore, cheese is not a dimension

That had me laughing out loud. :lol:

How many points are there on a line? Infinite yes? Is that a problem? Yes. Why? As Zeno showed Achilles can't catch up with tortoise. An infinite task.

It isn't really a paradox anymore. Consider a line segment of length 1. It can be cut into length 1/2,1/4,1/8,1/16 and so on if we add up all the lengths, we get a line segment of length 1. A line segment is made up of countably infinite number of points. That's the way the real numbers work. That's also the reason why we don't have a smallest number "a" that is greater than say another number "b" .

I can't even comprehend the terms that are being used.
Like here.

It is not, I think, a kind of stuff or dimension. This is for numerous reasons. Conceived of as a stuff (or dimension, if dimensions are not stuff)

He also stormed out of the room

Even the greatest philosophers engaged in "fights".
The Poker incident
Wittgenstein insisted that there can never be a moral proposition and pointed a poker at Popper and asked him to give just one example.
To which Popper replied, " Such as not pointing poker at your guests " :grin:
Wittgenstein was :angry: :grimace:

Even Stephen Hawking fell for that. :grin:

Critical thinking shouldn't only involve criticism but also an ability to correct the mistakes you have found. That aspect is largely missing and is not possible to cultivate in the minds. Certain people just happen to be more gifted and hit the targets we can't even see. Feynman did remarkably well in his Putnam tests without any preparation. An even better example would be Galois, who invented galois theory at the age of 18 and died an year later from a duel. His theory was so ahead of his time that even the mathematicians of the highest calibre struggled to understand its importance. One of the biggest lie that we are all told is that everyone is creative.

I will get back to you with more examples of science undergoing a paradigm shift and removing fundamental axioms or assumptions that were taken for granted because of the the general consensus of the scientific community. Dogma in science is under the cover of a paradigm. The paradigm shift can not take place for a few centuries, yet scientists are still able to produce new science. The Newtonian physics and Einstein's physics are completely different in their fundamental principles. Even though they may reduce to the same nature when we apply them to daily life, the difference lies in the details.

Besides that, a religious doctrine can have a wide range of interpretations around it. Anyways l rest my point here. I have probably said all l had to say.