## Did I discover “Angelo Cannata’s paradox”?

What is the basis for reason? And mathematics?

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Angelo Cannata
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### Did I discover “Angelo Cannata’s paradox”?

Let’s consider the mathematical equation 1=1.

Now let’s compare it with 1-1=0.

If we connect these equations with practical reality, we must conclude that 1=1 means that, for example, an apple is equal to itself. If we have an apple and we want to obtain the result of 0, we must subtract precisely that one apple that we have in our hand; we cannot subtract an apple that is a different one.

Now let’s consider 1+1=2.

In this case, if we consider practical reality, we must conclude that 1=1 means that an apple is equal not to itself, but to a different one. If we have an apple and we want to obtain the result of 2, we must add a different apple. If we add an apple to itself, we don’t obtain 2, we obtain 1.

Now the question: does 1=1 mean that an apple is equal to itself or does it mean that an apple is equal to another one? In both cases we obtain some contradiction: if we say that 1=1 means that an apple is equal to itself, we have problems with 1+1=2, because an apple summed to itself does not give 2 as a result. If we say that 1=1 means that an apple is equal to another one, we have problems with 1-1=0, because, if we have an apple in our hand, we cannot subtract a different apple.

Like other paradoxes, it reveals its being a paradox only if we connect theory and practice. If we maintain separated theory and practice, there is no contradiction, no paradox. We could say that maths does not consider material apples, but their quantity. But this means to maintain ourselves strictly into theory, without connecting it to practice. In that case, we should conclude that maths cannot be connected with material objects, maths has no relevance, no correspondence with reality!
Last edited by Angelo Cannata on Mon Sep 18, 2017 6:46 pm, edited 1 time in total.

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

Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
Let’s consider the mathematical equation 1=1.

Now let’s compare it with 1-1=0.

If we connect these equations with practical reality, we must conclude that 1=1 means that, for example, an apple is equal to itself. If we have an apple and we want to obtain the result of 0, we must subtract precisely that one apple that we have in our hand; we cannot subtract an apple that is a different one.

Now let’s consider 1+1=2.

In this case, if we consider practical reality, we must conclude that 1=1 means that an apple is equal not to itself, but to a different one. If we have an apple and we want to obtain the result of 2, we must add a different apple. If we add an apple to itself, we don’t obtain 2, we obtain 1.

Now the question: does 1=1 mean that an apple is equal to itself or does it mean that an apple is equal to another one? In both cases we obtain some contradiction: if we say that 1=1 means that an apple is equal to itself, we ha problems with 1+1=2, because an apple summed to itself does not give 2 as a result. If we say that 1=1 means that an apple is equal to another one, we have problems with 1-1=0, because, if we have an apple in our hand, we cannot subtract a different apple.

Like other paradoxes, it reveals its being a paradox only if we connect theory and practice. If we maintain separated theory and practice, there is no contradiction, no paradox. We could say that maths does not consider material apples, but their quantity. But this means to maintain ourselves strictly into theory, without connecting it to practice. In that case, we should conclude that maths cannot be connected with material objects, maths has no relevance, no correspondence with reality!
It is the difference between pure and applied mathematics.

To proceed from the pure to the applied one requires a mapping. eg 1x + 1x = 2x.
if you want to apply this to apples you have to ask yourself is the mapping between an 'apple' and 'x' an appropriate mapping?
If 'yes' then proceed with the mapping, if not then not.

But you are right...... it is a common error to pretend that the mapping is not required.

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

Steve Jobs has left the building...

-Imp

ken
Posts: 2032
Joined: Mon May 09, 2016 4:14 am

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

Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
Let’s consider the mathematical equation 1=1.

Now let’s compare it with 1-1=0.

If we connect these equations with practical reality, we must conclude that 1=1 means that, for example, an apple is equal to itself. If we have an apple and we want to obtain the result of 0, we must subtract precisely that one apple that we have in our hand; we cannot subtract an apple that is a different one.
Agree.
Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
Now let’s consider 1+1=2.
Okay.
Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
In this case, if we consider practical reality, we must conclude that 1=1 means that an apple is equal not to itself, but to a different one.
WHY must we conclude that 1=1 means that an apple is equal not to itself?

There is nothing in '1=1' for Me to conclude that an apple is equal not to itself. However, to Me, '1(apple) + 1(different apple)' concludes that there is two different apples. But just saying/writing '1=1' means that one equals one, which concludes that one is the exact same one, and NOT a different one.
Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
If we have an apple and we want to obtain the result of 2, we must add a different apple. If we add an apple to itself, we don’t obtain 2, we obtain 1.
We can NOT add an apple to itself. If there is only 1 apple, then there is only the exact same 1 apple. We can not add the same apple to itself.
Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
Now the question: does 1=1 mean that an apple is equal to itself or does it mean that an apple is equal to another one?
That all depends on how you want to phrase or stipulate what the 1 is?

If we add the words EXACT SAME in there, to your question, this can make a huge difference. For example, does (the exact same) 1=1 mean ...?

Also, if we add the words DIFFERENT in there, to your question, this also can make a huge difference. For example, does (a different) 1=1 mean ...?

Stipulation helps to clarify. For example, does 1(bag of apples)=1(bag of apples) mean that one bag of apples is equal to itself or does it mean that one bag of apples is equal to another one (bag of apples)?

The answer has to be one bag is equal to itself. This is rather simple really, for two reasons;
1. The '=' equal sign dictates and MEANS it is equal to itself. And, the other
1. It is not possible in practical reality, that we are aware of, to have two separate bags with the exact same apples in it at the exact same moment.

So, you can NOT have one bag of apples AND the EXACT SAME bag of apples BEING at the EXACT SAME moment, in practical reality. Just like you can not, in practical reality, have one apple being the EXACT SAME as a DIFFERENT apple.
Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
In both cases we obtain some contradiction: if we say that 1=1 means that an apple is equal to itself, we ha problems with 1+1=2, because an apple summed to itself does not give 2 as a result.
But how can you sum the exact same apple to itself?
Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
If we say that 1=1 means that an apple is equal to another one, we have problems with 1-1=0, because, if we have an apple in our hand, we cannot subtract a different apple.
If, as you propose, we can sum the exact same apple to itself, then why can we not subtract a different apple from itself?

WHY would doing one of these things make sense to you, but not the other one?

The number 1, by itself, does not differentiate between things. Only human beings do the differentiating, by the way they express them selves. So, are you saying that 1 is equal to itself or 1 is not equal to itself? You decide, and can make the choice. The numeral 1 can NOT do such a thing.

In other words, what is '1', in relationship to?
Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
Like other paradoxes, it reveals its being a paradox only if we connect theory and practice.
To Me, a 'paradox' is just a seemingly absurd or contradictory statement or proposition, which when investigated proves to be well founded or true.

For example, 'We do NOT need money to live', is a paradox, for most people. To Me, however, it is already just a well founded truth.

To Me, what you are writing is NOT a paradox. You have just left out some important words for stipulation.
Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
If we maintain separated theory and practice, there is no contradiction, no paradox.
And, when we add the two, did we get four or the truth?
Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
We could say that maths does not consider material apples, but their quantity.
Agree. I say, "Maths does not consider material things. 'Maths' is just a label given, by human beings, to some thing."
Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
But this means to maintain ourselves strictly into theory, without connecting it to practice.
If I was to say, "Maths does not consider material apples, but their quantity", then that does NOT mean that I maintain strictly into theory, without connecting it to practice. You might do that, but I do NOT.
Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
In that case, we should conclude that maths cannot be connected with material objects, maths has no relevance, no correspondence with reality!
If I should conclude that, then I have failed, because I did NOT conclude what you have.

But then again I do NOT look at and see the EXACT SAME way that you are here.

You might have to explain this another way if you want Me to come to the EXACT SAME conclusion that you have here, and which you propose that I SHOULD have also.

Hobbes' Choice
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Joined: Fri Oct 25, 2013 11:45 am

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

Angelo Cannata wrote:
Thu Sep 14, 2017 11:36 am
Let’s consider the mathematical equation 1=1.

Now let’s compare it with 1-1=0.

If we connect these equations with practical reality, we must conclude that 1=1 means that, for example, an apple is equal to itself. If we have an apple and we want to obtain the result of 0, we must subtract precisely that one apple that we have in our hand; we cannot subtract an apple that is a different one.

Now let’s consider 1+1=2.

In this case, if we consider practical reality, we must conclude that 1=1 means that an apple is equal not to itself, but to a different one. If we have an apple and we want to obtain the result of 2, we must add a different apple. If we add an apple to itself, we don’t obtain 2, we obtain 1.

Now the question: does 1=1 mean that an apple is equal to itself or does it mean that an apple is equal to another one? In both cases we obtain some contradiction: if we say that 1=1 means that an apple is equal to itself, we ha problems with 1+1=2, because an apple summed to itself does not give 2 as a result. If we say that 1=1 means that an apple is equal to another one, we have problems with 1-1=0, because, if we have an apple in our hand, we cannot subtract a different apple.

Like other paradoxes, it reveals its being a paradox only if we connect theory and practice. If we maintain separated theory and practice, there is no contradiction, no paradox. We could say that maths does not consider material apples, but their quantity. But this means to maintain ourselves strictly into theory, without connecting it to practice. In that case, we should conclude that maths cannot be connected with material objects, maths has no relevance, no correspondence with reality!

The "=" sign is an equivalence. I think the problem is one of language not managing to describe "=" adequately. There is also a problem that maths is a representative system and cannot exactly represent reality with the same problems that language cannot completely represent maths.

1 apple can never be the same as any other apple. Saying it is the same as itself is an empty statement. If we accept that 1 apple = 1 apple, then we have to accept that "=" is a context driven and practical approximation that one apple is much the same as any other. We are accepting that, for the purpose of filling our apple pie that what you need is an approximate amount of apple material that is most simply measured by the unequal units that nature provides; simple bounded spheroids that drop from trees onto waiting heads.

In no case in nature can any one thing be the same as any other thing, and so "1" is anathema to nature. Integers, straight lines, circle: none of these things exist in nature. They only exist in the abstract of mathematics.

PauloL
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Location: Lisbon, Portugal.

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

I think you should leave Maths alone.

Of course, Maths doesn't consider material things, it's you who consider.

1+1=2 can be an apple plus an orange if you're counting units of fruit. Maths doesn't care.

You're trying to introduce semantics into mathematics, something very unsound.

Like logic doesn't care about reality. It only tells you whether an operation is valid or not and possible logical outcomes.

It's your job deciding whether propositions are true or false, not logic's.

In fact neither Maths nor logic tells you anything about real world.

wtf
Posts: 385
Joined: Tue Sep 08, 2015 11:36 pm

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

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.

Of course what this means in the real world is murky. We know that the molecules in our body are continually changing. We are literally not ourselves from one moment to the next. Likewise any physical object is a collection of spinning quarks and other subatomic particles. Nothing in the real world is exactly the same moment to moment.

But that doesn't stop us from doing math. Nothing can stop us from doing math. And there's that big troublemaker "nothing" again, preventing us from doing math!!

ken
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Joined: Mon May 09, 2016 4:14 am

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

wtf wrote:
Sat Sep 16, 2017 12:13 am
Nothing can stop us from doing math. And there's that big troublemaker "nothing" again, preventing us from doing math!!
Only ourselves can prevent, or stop, us from doing maths. Is that somewhat better?

wtf
Posts: 385
Joined: Tue Sep 08, 2015 11:36 pm

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

ken wrote:
Sat Sep 16, 2017 1:28 am
wtf wrote:
Sat Sep 16, 2017 12:13 am
Nothing can stop us from doing math. And there's that big troublemaker "nothing" again, preventing us from doing math!!
Only ourselves can prevent, or stop, us from doing maths. Is that somewhat better?
Odysseus told the Cyclops that his name was Noman. Later when Osysseus poked the Cyclops in the eye with a big stick, the blinded Cyclops staggered around bellowing, "Noman has blinded me" and all his friends ignored him.

Not for nothing was he known as the wily Odysseus.

ken
Posts: 2032
Joined: Mon May 09, 2016 4:14 am

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

wtf wrote:
Sat Sep 16, 2017 4:30 am
ken wrote:
Sat Sep 16, 2017 1:28 am
wtf wrote:
Sat Sep 16, 2017 12:13 am
Nothing can stop us from doing math. And there's that big troublemaker "nothing" again, preventing us from doing math!!
Only ourselves can prevent, or stop, us from doing maths. Is that somewhat better?
Odysseus told the Cyclops that his name was Noman. Later when Osysseus poked the Cyclops in the eye with a big stick, the blinded Cyclops staggered around bellowing, "Noman has blinded me" and all his friends ignored him.

Not for nothing was he known as the wily Odysseus.
I have no idea what the point of this is nor of what it means.

Would you like to explain what this means?

wtf
Posts: 385
Joined: Tue Sep 08, 2015 11:36 pm

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

ken wrote:
Sat Sep 16, 2017 4:44 am
Would you like to explain what this means?
It was a bit of a philosophical joke or pun. It's the double meaning of nothing and the double meaning of Noman.

ken
Posts: 2032
Joined: Mon May 09, 2016 4:14 am

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

wtf wrote:
Sat Sep 16, 2017 5:32 am
ken wrote:
Sat Sep 16, 2017 4:44 am
Would you like to explain what this means?
It was a bit of a philosophical joke or pun. It's the double meaning of nothing and the double meaning of Noman.
Ah okay sorry for My slowness. I had not heard of those names before or of that joke/pun either, and I did not even notice the double meaning of 'noman' part.

What makes My slowness and unawareness even more inexcusable is that i have used the word 'noman', as a username on another forum, for that very reason.

A_Seagull
Posts: 471
Joined: Thu Jun 05, 2014 11:09 pm

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

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.

And if it is used for logic, it achieves nothing.

If you disagree with me, I suggest you think about where such a statement comes from,what it means and what it achieves; both in the pure and the applied realms.

wtf
Posts: 385
Joined: Tue Sep 08, 2015 11:36 pm

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

A_Seagull wrote:
Sat Sep 16, 2017 9:11 am

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.
Really? In math we may not assume a thing is equal to itself? Perhaps you can provide a reference or expand on this manifestly absurd claim.
A_Seagull wrote:
Sat Sep 16, 2017 9:11 am
If you disagree with me, I suggest you think about where such a statement comes from,what it means and what it achieves; both in the pure and the applied realms.
Yes, I disagree with you. In math, we may assume a thing is equal to itself. And that is not any rule or law of math. It's the law of identity, a law of logic.

If you disagree, complain to Wikipedia.

https://en.wikipedia.org/wiki/Law_of_identity

A_Seagull
Posts: 471
Joined: Thu Jun 05, 2014 11:09 pm

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

wtf wrote:
Sat Sep 16, 2017 6:01 pm
A_Seagull wrote:
Sat Sep 16, 2017 9:11 am

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.
Really? In math we may not assume a thing is equal to itself? Perhaps you can provide a reference or expand on this manifestly absurd claim.
A_Seagull wrote:
Sat Sep 16, 2017 9:11 am
If you disagree with me, I suggest you think about where such a statement comes from,what it means and what it achieves; both in the pure and the applied realms.
Yes, I disagree with you. In math, we may assume a thing is equal to itself. And that is not any rule or law of math. It's the law of identity, a law of logic.

If you disagree, complain to Wikipedia.

https://en.wikipedia.org/wiki/Law_of_identity
It is not a good idea in maths (or philosophy for that matter) to make assumptions that are not explicitly noted.

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.

(People used to assume that the world was flat and that the Sun went around the Earth.)

And when applying maths to the world it should be noted that this is a mapping, and not necessarily an exact one.

I'm sure I don't need to remind you of Heraclitus' quote: "You can't step into the same river twice."

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