Did I discover “Angelo Cannata’s paradox”?

What is the basis for reason? And mathematics?

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wtf
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Re: Did I discover “Angelo Cannata’s paradox”?

Post by wtf » Sat Sep 16, 2017 11:23 pm

A_Seagull wrote:
Sat Sep 16, 2017 10:04 pm

So you may not assume that something is equal to itself without noting that that is an assumption. In any case what is meant be 'equal'? It is rather undefined.
I can't tell if you're serious or just trolling me out of boredom.

If I say that 5 = 5, is this something you would question? I'm just trying to figure out what on earth you are talking about.

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Re: Did I discover “Angelo Cannata’s paradox”?

Post by A_Seagull » Sun Sep 17, 2017 12:08 am

wtf wrote:
Sat Sep 16, 2017 11:23 pm
A_Seagull wrote:
Sat Sep 16, 2017 10:04 pm

So you may not assume that something is equal to itself without noting that that is an assumption. In any case what is meant be 'equal'? It is rather undefined.
I can't tell if you're serious or just trolling me out of boredom.

If I say that 5 = 5, is this something you would question? I'm just trying to figure out what on earth you are talking about.
I am serious :)

I don't question that 5=5, I only question what some people seem to think it means and what its foundations are.

For me what "5=5" is is a string of symbols which is theorem of mathematics which can be useful when mapped to real objects.

wtf
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Re: Did I discover “Angelo Cannata’s paradox”?

Post by wtf » Sun Sep 17, 2017 1:07 am

A_Seagull wrote:
Sun Sep 17, 2017 12:08 am

I don't question that 5=5, I only question what some people seem to think it means and what its foundations are.
Ok. Well for the sake of discussion, this is an interesting discussion!

I was put on to the law of identity on this very forum, perhaps a year or more ago. Someone posed the following subtle question.

We know that in formal set theory, we have the axiom of extensionality. This axiom tells us when two sets are equal. Two sets are equal when they have exactly the same elements.

So the set {1,2,3} is the same as the set {3, 2, 1} because they have the same elements. Order doesn't matter. But they are both different than the set {1, 2, 3, 4} because that set contains an element, namely 4, that's not present in {1, 2, 3}. So those are not the same set.

Now the question is ... before I have formally defined equality of sets ... how do I know when two elements are the "same"? That is, if I claim that {5} = {5} because both sets contain exactly the same elements, namely the single element 5 ... how do I know the 5 on the left is the same as the 5 on the right?

I confess this question gave me quite a bit of trouble. Especially because it was posed by a poster who was, let's say, a bit cranky in his presentation, yet seemed to have a lot of insight into things. One often finds that among alternative viewpoints. Being crazy and being interesting are not mutually exclusive. On the contrary. It's sanity that's boring.

So I mulled this question over, and asked myself how we know that 5 = 5 before we have defined 5 as a particular set, as is done in the development of the natural numbers within set theory. In fact 5 is nothing more than the set {0 ,1, 2, 3, 4}. And therefore 5 = 5 by the axiom of extensionality.

But what is 5 BEFORE we have so modeled it within set theory?

The only answer I can come up with is that this is an application of the law of identity from logic. This is what I meant earlier when I said that the law of identity is logically prior to math. You objected to that. What I meant was that the law of identity is logically prioer to set theory. Whether you take set theory as a reasonable proxy for math is a question of philosophy of course!

So that's where I'm coming from. In math we start by assuming a thing is equal to itself. Then we stipulate the axiom of extensionality, define 5 as a set, and prove that 5 = 5 within set theory.

But before set theory, before any formal development of math, is the principle of identity. A thing is equal to itself. And yes I fully agree that this is something we must be explicit about once in a while. The law of identity is assumed. It's hard to imagine what kind of formal reasoning we could do if we were not allowed to assume that 5 = 5.
A_Seagull wrote:
Sun Sep 17, 2017 12:08 am
For me what "5=5" is is a string of symbols which is theorem of mathematics which can be useful when mapped to real objects.
I perfectly agree. I'm one who often takes the formalist stance. If someone says, "well how can .999... = 1 or pi have infinitely many digits with no repeating block or how can irrational numbers exist when they don't exist in nature," I just say: I don't care! You don't ask is chess is real. Math is a formal game. Stop asking me for it to mean anything. It's fun to play and keeps me out of trouble. Go harass the chess players and ask them if the knight "really" moves that way. You see the nonsense of expecting math to "mean" anything. Math is a formal game and Hilbert agrees!

But then on the OTHER hand, of course math is derived from our experience in the world. Our experience of space and time and quantity and measure. And the study of formal math is inextricably bound up with our knowledge of the world. No physicist thinks it's all a formal game, they're studying the real world and they use math. Physicists mock mathematicians for their abstract formalism. Math is about the world. Ask any physicist!

So I see both sides of this.

What I DON'T see is trying to claim that 5 = 5 is questionable because you don't believe the law of identity. That is not a sensible thing to be arguing in my opinion.

But if all you want to do is get me to agree that the law of identity is an assumption, and you can't step in the same river twice, of course I agree with you.

The assumption that "a thing is equal to itself" is itself a particular worldview. Western rationality and all that. A perfectly sensible alternative to model the real world is: A thing is never equal to itself.

I have no problem with that. I rather think it's probably true about the world we live in. Western rationality is a point of view. It's useful. It's logical. It's fun. But is it necessarily at the heart of the universe? No, I rather doubt that.

I just don't see how we could do math that way. In math you have to start with the law of identity.

What do you think?

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Re: Did I discover “Angelo Cannata’s paradox”?

Post by A_Seagull » Sun Sep 17, 2017 6:10 am

wtf wrote:
Sun Sep 17, 2017 1:07 am
A_Seagull wrote:
Sun Sep 17, 2017 12:08 am

I don't question that 5=5, I only question what some people seem to think it means and what its foundations are.
Ok. Well for the sake of discussion, this is an interesting discussion!

I was put on to the law of identity on this very forum, perhaps a year or more ago. Someone posed the following subtle question.

We know that in formal set theory, we have the axiom of extensionality. This axiom tells us when two sets are equal. Two sets are equal when they have exactly the same elements.

So the set {1,2,3} is the same as the set {3, 2, 1} because they have the same elements. Order doesn't matter. But they are both different than the set {1, 2, 3, 4} because that set contains an element, namely 4, that's not present in {1, 2, 3}. So those are not the same set.

Now the question is ... before I have formally defined equality of sets ... how do I know when two elements are the "same"? That is, if I claim that {5} = {5} because both sets contain exactly the same elements, namely the single element 5 ... how do I know the 5 on the left is the same as the 5 on the right?

I confess this question gave me quite a bit of trouble. Especially because it was posed by a poster who was, let's say, a bit cranky in his presentation, yet seemed to have a lot of insight into things. One often finds that among alternative viewpoints. Being crazy and being interesting are not mutually exclusive. On the contrary. It's sanity that's boring.

So I mulled this question over, and asked myself how we know that 5 = 5 before we have defined 5 as a particular set, as is done in the development of the natural numbers within set theory. In fact 5 is nothing more than the set {0 ,1, 2, 3, 4}. And therefore 5 = 5 by the axiom of extensionality.

But what is 5 BEFORE we have so modeled it within set theory?

The only answer I can come up with is that this is an application of the law of identity from logic. This is what I meant earlier when I said that the law of identity is logically prior to math. You objected to that. What I meant was that the law of identity is logically prioer to set theory. Whether you take set theory as a reasonable proxy for math is a question of philosophy of course!

So that's where I'm coming from. In math we start by assuming a thing is equal to itself. Then we stipulate the axiom of extensionality, define 5 as a set, and prove that 5 = 5 within set theory.

But before set theory, before any formal development of math, is the principle of identity. A thing is equal to itself. And yes I fully agree that this is something we must be explicit about once in a while. The law of identity is assumed. It's hard to imagine what kind of formal reasoning we could do if we were not allowed to assume that 5 = 5.
A_Seagull wrote:
Sun Sep 17, 2017 12:08 am
For me what "5=5" is is a string of symbols which is theorem of mathematics which can be useful when mapped to real objects.
I perfectly agree. I'm one who often takes the formalist stance. If someone says, "well how can .999... = 1 or pi have infinitely many digits with no repeating block or how can irrational numbers exist when they don't exist in nature," I just say: I don't care! You don't ask is chess is real. Math is a formal game. Stop asking me for it to mean anything. It's fun to play and keeps me out of trouble. Go harass the chess players and ask them if the knight "really" moves that way. You see the nonsense of expecting math to "mean" anything. Math is a formal game and Hilbert agrees!

But then on the OTHER hand, of course math is derived from our experience in the world. Our experience of space and time and quantity and measure. And the study of formal math is inextricably bound up with our knowledge of the world. No physicist thinks it's all a formal game, they're studying the real world and they use math. Physicists mock mathematicians for their abstract formalism. Math is about the world. Ask any physicist!

So I see both sides of this.

What I DON'T see is trying to claim that 5 = 5 is questionable because you don't believe the law of identity. That is not a sensible thing to be arguing in my opinion.

But if all you want to do is get me to agree that the law of identity is an assumption, and you can't step in the same river twice, of course I agree with you.

The assumption that "a thing is equal to itself" is itself a particular worldview. Western rationality and all that. A perfectly sensible alternative to model the real world is: A thing is never equal to itself.

I have no problem with that. I rather think it's probably true about the world we live in. Western rationality is a point of view. It's useful. It's logical. It's fun. But is it necessarily at the heart of the universe? No, I rather doubt that.

I just don't see how we could do math that way. In math you have to start with the law of identity.

What do you think?
It rather depends upon what one expects a philosophy of maths to be. Certainly maths started from the empirical and the real but this does not mean that this is the best way to incorporate it into the body of a philosophy. (People used to think that constant force results in constant velocity.. as per pro pushing a cart.. but physics has progressed beyond this model).

There is still the expectation that somehow maths is 'true' and hence must be founded on 'true' foundations (such as an identity) and they then progress through a rather complicated set theory analysis and maybe Peano's axioms.. and they arrive at the expectation that maths is complete and that every statement of maths can be proven to be either true or false .... until Gödel proved this was impossible. And that is where they left it: a schism at the heart of mathematics. And all that complicated stuff merely to 'prove' that "2+2=4" Surely there is a simpler way!!

What if maths was nothing more than the manipulation of symbols following specific rules and axioms - in much the same way that a computer manipulates mathematical symbols? Then it is only the relations between those symbols (akin to Hume's relations of ideas) that was significant. It wouldn't need any 'true' foundations. Nor would there be any requirement of 'completeness'. The only important criteria would be one of internal self-consistency. And then the use of maths is by their mapping onto real world objects. And which parts of maths are mapped onto which parts of the world has to be selected by the usefulness of the mapping.

wtf
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Re: Did I discover “Angelo Cannata’s paradox”?

Post by wtf » Mon Sep 18, 2017 6:09 am

If I'm understanding you, you seem to be arguing a formalist position in which the axioms are not necessarily true about the world. But nobody is disagreeing with you. Not me, anyway. I still don't understand whether you wish to do math without agreeing that 5 = 5.
A_Seagull wrote:
Sun Sep 17, 2017 6:10 am
There is still the expectation that somehow maths is 'true'
This is a strawman argument. Who is the entity that has an expectation that math is true about the world? Certainly nobody who's studied this history of science between the 1840's when non-Euclidean geometry was invented, and the early 1900's when Einstein showed that non-Euclidean geometry was actually the true geometry of the world.

Of course nonspecialists, the public perhaps, may have believed otherwise. Morris Kline wrote a book in 1982 called Mathematics: The Loss of Certainty that's all about the loss of mathematical certainty in the twentieth century.

I'm sure there are people who didn't get the non-Euclidean and Gödelian memos, but you seem to be arguing with them. I'm not sure if they're here in this thread. By the way Gödel himself was a Platonist. He believed there was mathematical truth. Just that any formalization of it must be inconsistent or incomplete.
A_Seagull wrote:
Sun Sep 17, 2017 6:10 am
What if maths was nothing more than the manipulation of symbols following specific rules and axioms - in much the same way that a computer manipulates mathematical symbols?
Then we'd be back a few posts where I said that I myself am usually a formalist, and Hilbert's on board too. So I don't understand why you're addressing me as if I'm disagreeing with you on this point when in fact I agreed with this point several posts ago. I'm puzzled by your trying to explain to me the formalist position.

[ps -- I see that you QUOTED ME saying I'm mostly a formalist. So I'm just confused by you asking me to suppose what I have already said I suppose. Maybe I'm not the intended audience].
A_Seagull wrote:
Sun Sep 17, 2017 6:10 am
Then it is only the relations between those symbols (akin to Hume's relations of ideas) that was significant. It wouldn't need any 'true' foundations.
Now here you are wrong. If math is a formal game, then we definitely need to know what the rules are. Those are the foundations. If we're playing chess we need to know how the pieces move. If we're doing math we need to know what symbolic manipulations are needed. Formalists MUST have a foundation.

A_Seagull wrote:
Sun Sep 17, 2017 6:10 am
Nor would there be any requirement of 'completeness'.
Really? Why not? Wouldn't we want to know that our rules can determine what is and isn't a legal position in our game? To know what's provable and what's not? Gödel showed that any formal system strong enough to model the natural numbers must be necessarily incomplete. That was a disappointment to what was at the time a requirement. Why would you lay down a set of rules then say we don't care if they completely determine the game?

A_Seagull wrote:
Sun Sep 17, 2017 6:10 am
The only important criteria would be one of internal self-consistency.
And he showed that no such system can prove its own consistency. So if consistency is your only criterion, you don't have any criterion at all. We can not prove within set theory that set theory is consistent.

A_Seagull wrote:
Sun Sep 17, 2017 6:10 am
And then the use of maths is by their mapping onto real world objects. And which parts of maths are mapped onto which parts of the world has to be selected by the usefulness of the mapping.
I have no disagreement with this. As long as you do acknowledge the "unreasonable effectiveness of mathematics" as the saying goes.

tl;dr:

1) You seem to be arguing points I've already agreed with; and

2) Do you or do you not wish to do math without the law of identity?

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Re: Did I discover “Angelo Cannata’s paradox”?

Post by Averroes » Thu Sep 21, 2017 7:58 am

A_Seagull wrote:
Sat Sep 16, 2017 9:11 am
wtf wrote:
Sat Sep 16, 2017 12:13 am
The law of identity is a basic principle of logic. It's logically prior to math. It's not an axiom of any system of math. "A thing is equal to itself" is a principle of reasoning independent of math.
Such 'laws of logic' are not laws at all. At best they are hypotheses. It may be a basic principle of logic but it is not 'prior' to maths. Pure maths makes no use of such vague statements.
I think the laws of logic are called laws instead of hypotheses because they make intelligible communication possible. For us to be able to communicate meaningfully with each other we need to understand the basic structure of language and its correct use. This structure consists of definitions and rules governing the units of the language. This structure is called logic and the associated definitions and rules are embodied into laws. Since they make purposeful communication possible,i.e. without them there would be no possibility for intelligible communication, therefore they are called laws instead of just hypotheses.



A_Seagull wrote:
Sat Sep 16, 2017 10:04 pm
I'm sure I don't need to remind you of Heraclitus' quote: "You can't step into the same river twice."
There is an interesting entry on the Internet Encyclopedia of Philosophy (IEP) on this statement of Heraclitus. From IEP:
______________
What Heraclitus actually says is the following:

On those stepping into rivers staying the same other and other waters flow. (DK22B12)

There is an antithesis between 'same' and 'other.' The sentence says that different waters flow in rivers staying the same. In other words, though the waters are always changing, the rivers stay the same. Indeed, it must be precisely because the waters are always changing that there are rivers at all, rather than lakes or ponds. The message is that rivers can stay the same over time even though, or indeed because, the waters change. The point, then, is not that everything is changing, but that the fact that some things change makes possible the continued existence of other things.

http://www.iep.utm.edu/heraclit/
______________

So as a river is defined as a flowing body of water, the fact that the position of the body of water is changing does not mean that the river is changing. On the contrary, if the body of water composing the river were not moving, there would be no river in the first place. So, the conclusion is that even though the water is flowing, the river does not change, and therefore it is possible to cross the same river twice. So, concerning the law of identity, it is not being violated by this example, but in fact it is upheld and the statement of Heraclitus is false.

The author of the article of IEP, Daniel Graham, has an interesting remark on Heraclitus:
Daniel Graham wrote:Heraclitus does, to be sure, make paradoxical statements, but his views are no more self-contradictory than are the paradoxical claims of Socrates. They are, presumably, meant to wake us up from our dogmatic slumbers.

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Re: Did I discover “Angelo Cannata’s paradox”?

Post by Averroes » Thu Sep 21, 2017 8:15 am

A_Seagull wrote:
Sat Sep 16, 2017 10:04 pm
I'm sure I don't need to remind you of Heraclitus' quote: "You can't step into the same river twice."
Heraclitus statement can also be dealt with from a Kantian perspective. This is interesting and needs to be considered in detail because it directly deals with the law of identity. Please, if you would allow me, here goes.

You paraphrased Heraclitus thus: "You can't step into the same river twice", let us go by this statement. So, now, let us suppose that we decide to do the experiment of crossing a river, say the Nile for example; then I ask: who is going to do the counting up to two?!! Is it possible to count to two if Heraclitus is right? Furthermore, is it possible even to utter that statement if Heraclitus is right?

Clearly if Heraclitus is right, then the law of identity is violated! But if the law of Identity is violated, then Heraclitus would not have been able to utter that statement itself! How? Why?

If you consider the human body, according to the experts on human anatomy: if all arteries, veins, and capillaries of the human circulatory system were laid end to end, the total length would be 60,000 miles, or 100,000 km. And they say that it is nearly two and a half times around the Earth! Anyway, if Heraclitus is right about the river changing and not being the same as its water flows, then we must also conclude that we ourselves must be changing and not be the same, as blood is circulating in our bodies through the various channels in us. If we be not the same, then who will be doing the counting? The ability to count requires that we be identical to ourselves, i.e the ability to count requires the law of identity. Kant has something interesting on this.

Kant wrote:
  • "Without consciousness that that which we think is the very same as what we thought a moment before, all reproduction in the series of representations would be in vain. For it would be a new representation in our current state, which would not belong at all to the act through which it had been gradually generated, and its manifold would never constitute a whole, since it would lack the unity that only consciousness can obtain for it. If, in counting, I forget that the units that now hover before my senses were successively added to each other by me, then I would not cognize the generation of the multitude through this successive addition of one to the other, and consequently I would not cognize the number; for this concept consists solely in the consciousness of this unity of the synthesis. [ A104, Critique of Pure Reason]"
What he is saying is that if we were changing, so that the first one who steps into the river is different from the one who steps into the river the second time, then no one of them would not be able to say that the river was crossed twice, since different persons are involved in each crossing/stepping! Furthermore, no one would not be able to say or write anything about that experience or any experience at all for that matter!

Kant calls the thoroughgoing identity of oneself in all possible experience/representations, the pure apperception and he says that it is a necessary condition for experience itself to be possible.

Kant wrote:
  • "I call it the pure apperception (...), because it produces the representation I think, which must be able to accompany all others and which in all consciousness is one and the same, cannot be accompanied by any further representation. [B 133, CPR) (the emphasis in his)"
If we take the view that we receive information from our senses which are processed in our minds, then we have no choice but to accept the law of identity of ourselves to account for the possibility of experience itself. So with this worldview, Heraclitus could only have uttered that statement only under the presupposition of the Law of identity itself. So, Heraclitus' statement cannot be taken to be contravening the law of identity but rather the statement itself is possible only through the law of identity, under this worldview.

There is an analogy with the river here. This pure apperception can be thought as the river bed and the water flowing as our thoughts. Even if the water is flowing, the river bed does not flow with it. And similarly, even if that of which we are conscious changes,i.e. our thoughts, our self-consciousness does not change. To be able to perceive change itself, we require the law of identity, in this worldview.

And since the law of identity holds under this world view, then Heraclitus' statement is necessarily false, i.e. it is possible to cross the same river twice.

To summarize the preceding exposition, the following is presented.

From the dictionary ,a river is defined as follows:

a natural stream of water of fairly large size flowing in a definite course or channel or series of diverging and converging channels.

Water bodies flowing through one or several channels is the identity/essence of a river. If it were assumed that water flowing would make a river not identical to itself, then it would seem to imply that there are two rivers that are being stepped into. But now, if we take this view, then the supposed "one" who is stepping into the river should be viewed as not the same "one" as well, on the "second" stepping. If we take the view that there are two rivers on each "successive" stepping into, then we must also consider (if we are to be consistent) that there are two persons as well doing the stepping. Since on each stepping, a different person and river is concerned, the number two itself cannot arise for each of these persons; because for each of such persons, a river has been stepped into only once by each of them! Furthermore, for "me" (who is writing this post) and any other person as well, the number two should never arise! But it does arise, e.g. in the statement of Heraclitus! That is a contradiction. So we must assume the identity of the river and the person stepping into it for the possibility of the statement of Heraclitus to be itself possible. Once the statement of Heraclitus is understood, then it is necessarily false; the possibility of understanding the statement itself, implies its falsity! As Wittgenstein might have put it, we have arrived at the limit of sensical-nonsensical discourse! :)

So, I have to agree with the statement that "the law of identity is prior to math" because I have to assume the identity of myself if I am ever to count beyond 1! And I do count beyond 1!

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Re: Did I discover “Angelo Cannata’s paradox”?

Post by A_Seagull » Thu Sep 21, 2017 5:02 pm

Averroes wrote:
Thu Sep 21, 2017 7:58 am
For us to be able to communicate meaningfully with each other we need to understand the basic structure of language and its correct use.
This is blatantly false.

Even dolphins can have meaningful communication without any understanding of the basic structure of language. And the idea that there is some "correct" use of language is risible.

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Re: Did I discover “Angelo Cannata’s paradox”?

Post by A_Seagull » Thu Sep 21, 2017 8:40 pm

Averroes wrote:
Thu Sep 21, 2017 8:15 am
A_Seagull wrote:
Sat Sep 16, 2017 10:04 pm
I'm sure I don't need to remind you of Heraclitus' quote: "You can't step into the same river twice."
The point about this quote is that the noumena changes, the phenomena changes; it is only the labels that remains the same.

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Re: Did I discover “Angelo Cannata’s paradox”?

Post by Averroes » Tue Sep 26, 2017 11:51 am

A_Seagull wrote:
Thu Sep 21, 2017 5:02 pm
Averroes wrote:
Thu Sep 21, 2017 7:58 am
For us to be able to communicate meaningfully with each other we need to understand the basic structure of language and its correct use.
Even dolphins can have meaningful communication without any understanding of the basic structure of language.
Please, may I know from where and by what means did you come to know that Dolphins do not understand the structure of their own language and yet they are able to communicate meaningfully with that language?
A_Seagull wrote:
Thu Sep 21, 2017 5:02 pm
And the idea that there is some "correct" use of language is risible.
Unfortunately, not everyone is laughing on this matter, for example the students who have failed either their TOEFL, IELTS or SAT exams are not laughing on this issue!

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Re: Did I discover “Angelo Cannata’s paradox”?

Post by Averroes » Tue Sep 26, 2017 11:57 am

A_Seagull wrote:
Thu Sep 21, 2017 8:40 pm
Averroes wrote:
Thu Sep 21, 2017 8:15 am
A_Seagull wrote:
Sat Sep 16, 2017 10:04 pm
I'm sure I don't need to remind you of Heraclitus' quote: "You can't step into the same river twice."
The point about this quote is that the noumena changes, the phenomena changes;
Respectfully my friend, one need not go as far as the distinction Kant made between 'noumena' and 'phenomena' for the discussion about the law of identity. However in Kantian philosophy, i.e. for Kant the 'noumena' is inaccessible/unknowable to him, all that he knows and can talk about is 'phenomena'. Kant was a specific type of idealist, he called himself a transcendental idealist. The particular flavor of Kant's idealism is not important here though, it only need to be mentioned that he did not believe himself to be able to access/know the things as they are in themselves. This point of view can be contrasted with the realist perspective, and from the latter perspective one believes that one has access to things as they are in themselves. For the latter view, it would imply that the 'noumena' and 'phenomena' distinction is superfluous.

Now, you said that the 'noumena' is changing, and since this implies that you believe yourself to have access to the 'noumena', so one can conclude that you are not endorsing Kant's idealism but in that case the distinction between 'noumena' and 'phenomena' is superfluous. But as you already made it, I can take it to mean that they are synonymous in your mindset. Otherwise you will have to tell me how they are different for you and whether you think of yourself as having access to the 'noumena' under this condition; and if that be so, how do you have access to the 'noumena' as different from the 'phenomena'.

But as I said this distinction is not important at all for our present discussion of the law of identity. If Kant's philosophy is introducing some confusion (which I would understand as his position is a difficult one), let us take another philosopher who says the same thing as Kant and who, unlike Kant, was a realist. We take Thomas Reid who was a contemporary of Kant, and if that be important, Reid was like Kant raised into the Protestant religion. Kant was a Lutheran Protestant while Reid was a Calvinist Protestant. Just to be clear my friend, I am myself a Muslim and studying other people's belief is not a problem at all for me. As a matter of fact, it is this healthy curiosity that catalyzed my embracing Islam from being previously from a predominantly Christian background. And moreover, today the second most prolific Islamic preacher is himself a convert from Christian Protestantism, namely Sheikh Yusuf Estes.

Anyway, back to our interesting discussion, Thomas Reid endorsed the law of identity thus:
Reid wrote:The conviction which every man has of his identity, as far back as his memory reaches, needs no aid of philosophy to strengthen it, and no philosophy can weaken it, without first producing some degree of insanity.
(...)
We may observe, first of all, that this conviction is indispensably necessary to all exercise of reason. The operations of reason, whether in action or in speculation, are made up of successive parts. The antecedent are the foundation of the consequent, and without the conviction that the antecedent have been seen or done by me, I could have no reason to proceed to the consequent, in any speculation, or in any active project whatever.
(...)
I see evidently that identity supposes an uninterrupted continuance of existence. That which has ceased to exist, cannot be the same with that which afterwards begins to exist; for this would be to suppose a being to exist after it ceased to exist, and to have had existence before it was produced, which are manifest contradictions. Continued uninterrupted existence is therefore necessarily implied in identity.
(...)
When a man loses his estate, his health, his strength, he is still the same person, and has lost nothing of his personality. If he has a leg or an arm cut off, he is the same person he was before. The amputed member is not part of his person, otherwise it would have a right to a part of his estate, and be liable for a part of his engagements: it would be entitled to a share of his merit and demerit, which is manifestly absurd. A person is something indivisible, and is what Liebnitz callas a monad.

My personal identity, therefore, implies the continued existence of that indivisible thing which I call myself. Whatever this self may be, it is something which thinks, and deliberates, and resolves, and acts, and suffers. I am not thought, I am not action, I am not feeling; I am something that thinks, and acts, and suffers. My thoughts, and actions, and feelings, change every moment; they have no continued, but a successive existence; but that self or I, to which they belong, is permanent, and has the same relation to all succeeding thoughts, actions, and feelings, which I call mine.

[Essays on the intellectual Powers of Man, Thomas Reid, 1785]
This is essentially the same thing that Kant says in the Critique of pure reason, and in the case of Reid we need not even mention the 'noumena' and 'phenomena' distinction as he was a realist. So the argument in my previous post could have been appended after this quotation of Reid, instead of Kant.

As an aside, there is something interesting that Reid said on the discussion of identity which is echoed in Wittgenstein Tractatus. And Wittgenstein himself endorsed the law of identity. Reid wrote:
Reid wrote:Identity in general, I take to be a relation between a thing which is known to exist at one time, and a thing which is known to have existed at another time, If you ask me whether they are one and the same, or different things, every man of common sense understands the meaning of your question perfectly. Whence we may infer with certainty, that every man of common sense has a clear and distinct notion of identity.

If you ask a definition of identity, I confess I can give none; it is too simple a notion to admit of logical definition: I can say it is a relation, but I cannot find words to express the specific difference between this and other relations, though I am in no danger of confounding it with any other. I can say that diversity is a contrary relation, which every man easily distinguished in his conception from identity and diversity.
In the above quotation, Reid is saying that the law of identity is something that he find himself to not be able to express in a proposition because it is "too simple a notion". After endorsing the law of identity, this would be exactly the position of Wittgenstein in the Tractatus more that a century later.
Wiitgenstein wrote:Identity of object I express by the identity of sign, and not by using a sign for identity. Difference of objects I express by difference of signs. [Tractatus 5.53]

Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all. [Tractatus 5.5303]

The identity-sign, therefore, is not an essential constituent of conceptual notation. [5.533 Tractatus]
And now we see that in a correct conceptual notation pseudo-propositions like 'a=a', "a=b.b=c.⊃ a=c', '(x).x=x', '∃x(x)=a', etc. cannot even be written down. [Tractatus 5.534]
What Reid expressed as "too simple a notion to admit of a logical definition", Wittgenstein would express as "that which cannot be said, but is shown in language". So a general conclusion on this subject could be that irrespective of one's religious beliefs and philosophical standpoint, the law of identity is not a subject of disagreement for the philosophers, or at least for the majority of them.

Averroes
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Re: Did I discover “Angelo Cannata’s paradox”?

Post by Averroes » Tue Sep 26, 2017 12:03 pm

A_Seagull wrote:
Thu Sep 21, 2017 8:40 pm
Averroes wrote:
Thu Sep 21, 2017 8:15 am
A_Seagull wrote:
Sat Sep 16, 2017 10:04 pm
I'm sure I don't need to remind you of Heraclitus' quote: "You can't step into the same river twice."
(...) it is only the labels that remains the same.
On the quoted statement above, someone might disagree with you on the basis of Heraclitus statement. On the same ground that the river would be changing according to Heraclitus, the labels too would not be exempted but would have a similar fate! A label is commonly referred to as a name, and in the case of the statement of Heraclitus, an example of a label would be the sequence of characters "river". Now, these labels or names can either be written on paper or displayed digitally on a computer screen or be a spoken word. As for the spoken word and the digital display, they are waves; sound waves and light waves respectively. And waves are changing like the river is flowing. In the case of the word written on paper, the label consists of atoms/molecules in either a solid or liquid state, depending on whether the ink has dried or not. In either cases, the atoms or molecules are moving or vibrating at least. Furthermore, the electrons within the atoms are moving at high speeds, and quantum physics even considers electrons as waves also. In a nutshell, if Heraclitus was right then nothing at all would be exempt from the fate of the river, not even the labels. But if Heraclitus was right, he would not be able to utter that statement itself.

Let me express the argument more concisely as follows:

Even though I change, I do not become another person. For if I were to become another person with each change that I suffer, then there would be no "I" (singular) in the first place but there would be a succession of unrelated subjects (plural), ie. the person before the 'change' and the other person after the 'change'. And if that be the case, there would be no one (singular) changing. This results in an absurdity. Therefore, when I change I do not become another person and this means that I do not lose my identity when I change, but my identity is a necessary condition for me to experience change. The same goes for the river and anything else for that matter.

My friend, if I offended you in anyway, please understand that it was not intentional and accept my apologies if that be the case.

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Hobbes' Choice
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Re: Did I discover “Angelo Cannata’s paradox”?

Post by Hobbes' Choice » Tue Sep 26, 2017 5:34 pm

A_Seagull wrote:
Thu Sep 21, 2017 5:02 pm
Averroes wrote:
Thu Sep 21, 2017 7:58 am
For us to be able to communicate meaningfully with each other we need to understand the basic structure of language and its correct use.
This is blatantly false.

Even dolphins can have meaningful communication without any understanding of the basic structure of language. And the idea that there is some "correct" use of language is risible.
That is false.
"understanding" is essential as without it there is not communication.
I think you mean 'knowledge". We all understand language and acquire it without knowledge of grammar, spelling or even the alphabet. But understanding the structure of language is innate, as innate as up and down.

Angelo Cannata
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Re: Did I discover “Angelo Cannata’s paradox”?

Post by Angelo Cannata » Sat Sep 30, 2017 3:25 pm

Perhaps the answer, to this and to many other paradoxes, perhaps all paradoxes, is that language cannot be considered an exact copy of reality. Language reflects some aspects of reality, but not completely all aspects. This means that not everything happening in language happens in reality and vice versa, non everything happening in reality happens in language. Maths is an example of such a language, but any other reasoning, as well, is nothing else than a language with some similarities with reality, but not perfect coincidence with reality. This has some consequences on the entire philosophy, because philosophy is almost completely an action of working with ideas, elaborating ideas. But, since ideas are simply a language, whose correspondence with reality is never guaranteed, we must be always suspicious about assuming that any philosophical idea has any correspondence with reality.

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