A Paradox in Christianity

[PhilosophicalBluster]

That doesn't bely the law of excluded middle.

It doesn't have to. This is philosophy, not science.

I think you're confusing three different facets of quantum theory (superposition, uncertainty, and probability), the "Schrödinger's Cat" thought experiment, and the unrelated quantum logic.

Either that or I've misunderstood what you meant .

I think you have. Maybe quantum was the wrong word, but in this context, I meant complex, not in literal relation to quantum physics.

Says you . How do you know they don't have perfectly good reason to believe?

Show me one.

 

[PhilosophicalBluster]

Dp

 

[PhilosophicalBluster]

PhilosophicalBluster:


you're assuming that logic is "some thing" ... it isn't an independent object. It is the set of rules governing propositions with truth value.

Which G-d, being omnipotent, created and gave us the capacity to understand.

This is essentially the brain in a vat problem. Even if our perception is entirely illusory, then statements of fact are no different, and the rules of logic still apply. We may be wrong about certain statements as what we perceive is false. But that doesn't allow for such things as contradictions.

It's impossible for you to know that. If we truly do exist in a brain in a vat / mentality controlled universe, the laws of physics (or even reason) need not apply.

there is a vast difference, if you believe in an objective reality, between one's uncertainty about a cat in a box, and whether or not there actually is a cat in the box. You not knowing that the cat was in the box didn't mean that it was in some sense not in the box. To make that claim you would have to commit to saying that your knowledge or lack thereof has some bearing on physical particles. If you're being serious, then this is an incredibly problematic position to take.

Again, with the possibility of a 'brain in a jar' universe, physical laws are only apply because you believe they apply.

You're kind of jumping around.

Yes, I was. The point of my post is not to find one definite answer, it is only to provide alternate answers to show that your explanation is not the only possible one.

The argument was not about credibility. The original argument was that it is illogical to believe something based solely on what other have said. So according to the original argument, believing in black holes would be illogical. I suggest reading the OP.

The difference is this:
The Bible is considered infallible no matter how unlikely the events it depicts may or may not be. A scientific discovery is open to mass criticism by other scientists and can be discredited and changed.

 

[solarwave]

But how do you know the Bible is the word of G-d, not a politically driven man? Let's even take it a step further, how could you possibly know that the Bible is even non-fiction? The thing here is that you're taking the Bible's credibility for granted. I am not doubting the historical significance of the Bible, but you cannot just write that a miracle happened and have that automatically mean it happened.

People write that the Holocaust never happened. Does that make it true?

So from this I guess that you accept what I said about logic and faith not being against each other at all by correct definition?

Well the Bible isn't from one man, I think any historian, atheist or theist, could tell you that. Many events in the Bible have been proven to be correct by new archeology, etc, so that shows at the least the Bible is planted in some truths. And although this doesn't prove the Bible, it does talk of a round earth and earth being floating in space.

There is no reason for someone to write a book that they know is false that they know would get them killed and then to start preaching it to world for the reward of persecution, being stonned, whipped, beated and in the end all the disciples except one were killed for what they said. And all the time they knew if what they were saying was true or false.

You have to remember that the Bible was not always one book. The Old Testament was there long before Jesus and it gave many(and I mean many) prophecies of the Messiah, all of which were fulfilled in Jesus. Now you can't just say that these things were made up to look good, because believe it or not people were alive 2000 years and weren't stupid. There were people who were with Jesus who could have denied that these things happened to Him, but most of the prophecies were clear for all to see happening.

Though the Bible is old, it doesn't mean it came out of thin air and that people were just like 'O an old book', when it was written people actually had seen Jesus so could say if these things were or were not true. Luke (He wrote the gospel of Luke and Acts which was the history of the earlier church) was once a non-jew and was converted to christianity, was a 'doctor' and proved to be a good historian who wrote his gospel from different people he asked about Jesus' life. This gives good reason to believe Lukes gospel inparticular as unbias. Also the gospels can be dated to show that they were likely written by the people they claim to be written by and this means that there is little time for myth to have crept in.

Ok, im no expert on this, but just because its an old book it doesn't mean we believe it for no reason. This is just a few things I can think of, but there are other reason to believe the Bible.

I hope this makes sense, its pretty long.

 

[solarwave]

Just a random question, but why do you write God as G-d? lol

 

[Wiccan_Child]

It doesn't have to. This is philosophy, not science.

The law of excluded middle applies to everything, be it science or *scoff* philosophy. The poster stated:

"In the universe, a proposition is either true or not true. That statement is a rule of logic."

For some reason, you disagreed with this.

I think you have. Maybe quantum was the wrong word, but in this context, I meant complex, not in literal relation to quantum physics.

On the contrary, you were using a quantum mechanical phenomenon to support your argument that the law of excluded middle isn't universal.

Show me one.

I'm not the one making the claim, so I shall obstinately refuse.

 

[SaintPhotios]

with all due respect, I think PhilosophicalBluster is simply confused about the meaning of philosophical terms and their implications. I don't know if you've ever actually studied the schools of philosophy that pertain to this, but your argumentation style is...... interesting.

 

[Rauffenburg]

@PhilosophicalBuster

How do you know that there are not rules that you as a human do not have the brainpower to understand? We are finite beings trying to interpret an infinite being and the infinite world he created. Who are you to try to interpret G-d? He gave us the capacity for logic. If he didn't want us to have and thereby use logic, then why give us the capacity?

Classical set theory deals with infinite orders of infinite cardinal numbers. So what is the big deal for us in dealing with infinity? If the world - i.e. the physical univserse we live in - is spacially finite depends on the cosmological model you adopt. Whether the world is ontologically finite depends on the entities you include in your logical representation of what should count as "the world" in this abstract sense. If mathematical entities are elements of this world, then the world contains infinitly many elements. Even the number of possible sentences in English is infinite because English syntax can the described by a set of recursive rules that allows any Speaker of English to create an infinite number of sentences by applying a finite set of rules.

I would also doubt that we are acutally finite in an in-principle sense - of course we are finite in a practical sense, but that does not imply a total in-principle finitness in our capacities.

@SaintPhotius

2nd premise: Logic is essentially the rules concerning the way the world is. In the universe, a proposition is either true or not true. That statement is a rule of logic. We can understand that rule. But it isn't as if logic is this actual thing. You can say God gave us understanding, but not logic.

That depends on your logic and your universe actually. Intuitionistic logic for example allows for propositions that are neither true nor false because it does not have A v -A as a theorem. Paraconsistent logic is even a try to enable a productive formal treatment of inconsistencies such as A and not A. And this might allow for accepting inconsistent theories.

Logic is not the way the world is, since there are many Logics from which you can derive different things.

3rd premise: This isn't necessarily true in all circumstances(even though I agree that we should). Also, inherent in this premise is the presupposition that we are as God created us.... which, even according to Christian teaching, is not true. (The Fall)

Which is one of the basic problems I have with standard Christianity. The Fall in my eyes is logically impossible (at least under most ideas of logic), since a thing cannot change in its nature insofar as it has a nature. So if God created us according to our human nature, the Fall cannot have affected this nature because that is impossible by the very idea that a nature is that set of properties which is necessarily instantiated by an entitiy to make it that very entity. SO either if we have a nature or if we do not have a nature, the Fall cannot affect that nature.

I propose that Christianity is philosophically defensible.

Could you explain to me what "The Fall" is then and how it works? You can do that via Private Message if you want to, since this might to be the right thread, but I am really interested.

you're assuming that logic is "some thing" ... it isn't an independent object. It is the set of rules governing propositions with truth value.

And a set of rules isn't a "thing" in the widest sense?

@WiccanChild

The law of excluded middle applies to everything, be it science or *scoff* philosophy.

The principle of bivalence / law of excluded middle doesn't apply in intuitionistic logic and all constructive mathematics that bases its proofs on this logic. As I said eariler: You can change logic. So what is a theorem of logic depends to what logic we are talking about.

 

[Wiccan_Child]

The principle of bivalence / law of excluded middle doesn't apply in intuitionistic logic and all constructive mathematics that bases its proofs on this logic. As I said eariler: You can change logic. So what is a theorem of logic depends to what logic we are talking about.

I disagree.

 

[Rauffenburg]

I disagree.

That is too bad for you, because you are plain wrong with your disagreement. A or not A is not a theorem of Intuitionistic Logic. Its very simple indeed.

You may want to consult this article for some more information on the topic:

http://plato.stanford.edu/entries/logic-intuitionistic/

The very first section of the article deals with the rejection of the law of the excluded middle. And you even get some ideas to understand the motivation behind this modification of classical logic.

And consequently the law of the excluded middle does not apply to every philosophy. Take Brouwer himself (he founded Intuitionism as a philosophy of mathematics) or Michael Dummett who took up many of the ideas in Intuitionism and tried to use them for his semantics and anti-realist concept of truth as justifiability. One could even argue that Kant's transcendental dialectic requires the failure of the law of the excluded middle, although this might be a matter of interpretation.

 

[Wiccan_Child]

That is too bad for you, because you are plain wrong with your disagreement. A or not A is not a theorem of Intuitionistic Logic. Its very simple indeed.

You misunderstand: I reject intuitionistic logic as a valid form of logic.

You may want to consult this article for some more information on the topic:

http://plato.stanford.edu/entries/logic-intuitionistic/

The very first section of the article deals with the rejection of the law of the excluded middle. And you even get some ideas to understand the motivation behind this modification of classical logic.

As far as I can tell, the rejection of the law of excluded middle isn't justified. Brouwer considered it an a priori assumption, but I do not: it follows directly from the law of identity (A = A, ∀A) and the law of non-contradiction (A ≠ ¬A, ∀A). I do not see how one can reject either of these (and, thus, the law of excluded middle).

I also do not see how the example given demonstrates the flaw in the law of excluded middle: though we can't in general know whether B(x) is true or false, we know it must be either true or false. This can be said of any general statement whose truth is unknown.

I daresay people much smarter than me have pondered this over the past century, but I can't say I'm a fan of it .

And consequently the law of the excluded middle does not apply to every philosophy.

Perhaps, but that doesn't mean it's a sensible or justified notion. One could take the philosophical stance that no one has noses, for all the good it would do you.

Take Brouwer himself (he founded Intuitionism as a philosophy of mathematics) or Michael Dummett who took up many of the ideas in Intuitionism and tried to use them for his semantics and anti-realist concept of truth as justifiability. One could even argue that Kant's transcendental dialectic requires the failure of the law of the excluded middle, although this might be a matter of interpretation.

Indeed.

 

[Rauffenburg]

You misunderstand: I reject intuitionistic logic as a valid form of logic.

On which grounds? It can be very useful in Computer Science.

As far as I can tell, the rejection of the law of excluded middle isn't justified. Brouwer considered it an a priori assumption, but I do not: it follows directly from the law of identity (A = A, ∀A) and the law of non-contradiction (A ≠ ¬A, ∀A). I do not see how one can reject either of these (and, thus, the law of excluded middle).

I must admit, I do not see why it should follow from these two laws. In fact intuitionistic logic can be seen as a set of axioms in which some axioms of classical logic are missing, namely the "p or ¬p" and "¬¬p -> p" axioms are missing. If p or ¬p were in implication of some other axioms but not an axiom itself, then your disagreement would make sense to me. But as far as I can see, technically intuitionism is a possible mode of reasoning - it is not self-contradictory. Moreover it is necessary to mention that intuitionists have a different notion of identity (or mathematical equality), as a consequence of their logic since they allow for undecidable predicates and ontologically indeterminate properties. The consequence is that two or more mathematical entities may be indeterminate according to their identity at a certain state of knowledge of the creating subject.

I also do not see how the example given demonstrates the flaw in the law of excluded middle: though we can't in general know whether B(x) is true or false, we know it must be either true or false. This can be said of any general statement whose truth is unknown.

I daresay people much smarter than me have pondered this over the past century, but I can't say I'm a fan of it .

Intuitionism is a variety of mathematical constructivism. That is, it does not assume that mathematics is a way of discovery but a way of creation. The consequence is that it draws no strict line between epistemology and ontology. There are no unknown truths in intuitionism. Everything that is - mathematically speaking - is known by the creating subject that creates all mathematical entities.

Perhaps, but that doesn't mean it's a sensible or justified notion. One could take the philosophical stance that no one has noses, for all the good it would do you.

The only point you have to acknowledge is, that you cannot refute the legitimacy of intuitionist reasoning by use of classical reasoning because you dont have a higher standpoint, so to speak, from which you can deduce the axioms of classical logic alone. And this is a technical point about acknowledging that intuitionistic logic works - not more and not less. Logic is not about noses but about the laws of truth-preservation. And my technical point - which is not meant to discredit classical reasoning altogether - is enough to show, that there are many ways in which we can formulate laws of truth preservation.

Actually my whole argument is not to convert you to intuitionism or anything else. It is just to show you, that classical reasoning is not alone. Just as Euclid's geometry gave rise to many non-euclidean geometries in the 19th century, the axiomatic treatment of classical logic gave rise to a whole variety of non-classical ways of reasoning. One could argue that euclidean geometry as well as classical predicate logic still have some extraordinary status - but I do not see why this should make them more legitimate.

Of course I cannot deny that I consider Brouwer's philosophy of mathematics pretty fascinating for some interesting consequences, such as the existence of changing mathematical entities (especially choice sequences). But I do not deny its primary problem: it deprives us of certain powers of reasoning we may want to have to formulate some scientific theories. And do deny practically working scientific theories just because one doesn't like their non-intuitionistic reasoning is a pretty harsh approach.

 

[Wiccan_Child]

It can be very useful in Computer Science.

Perhaps, but so is numerical approximation and Newtonian mechanics.

I must admit, I do not see why it should follow from these two laws.

Non-contradiction states that something can't be what it's not: A can't be identical to ¬A by definition.
If some thing p is identical to A, then, since A can't be ¬A, neither can p.
If some thing p is not identical to A, then, by definition, p is identical to ¬A.

If p = A, then p ≠ ¬A.
If p ≠ A, then p = ¬A.

I.e., "A=A" and "A≠¬A" imply "A∨¬A".

In fact intuitionistic logic can be seen as a set of axioms in which some axioms of classical logic are missing, namely the "p or ¬p" and "¬¬p -> p" axioms are missing. If p or ¬p were in implication of some other axioms but not an axiom itself, then your disagreement would make sense to me.

It is derived from two axioms, as I showed above (barring silly errors on my part).

But as far as I can see, technically intuitionism is a possible mode of reasoning - it is not self-contradictory. Moreover it is necessary to mention that intuitionists have a different notion of identity (or mathematical equality), as a consequence of their logic since they allow for undecidable predicates and ontologically indeterminate properties. The consequence is that two or more mathematical entities may be indeterminate according to their identity at a certain state of knowledge of the creating subject.

Which is an extension of, not a replacement for, identity. We may not know whether p is identical to A, but that doesn't bely the fact that it either is or is not.

Intuitionism is a variety of mathematical constructivism. That is, it does not assume that mathematics is a way of discovery but a way of creation. The consequence is that it draws no strict line between epistemology and ontology. There are no unknown truths in intuitionism. Everything that is - mathematically speaking - is known by the creating subject that creates all mathematical entities.

You mean, the mathematician?

The only point you have to acknowledge is, that you cannot refute the legitimacy of intuitionist reasoning by use of classical reasoning because you dont have a higher standpoint, so to speak, from which you can deduce the axioms of classical logic alone.

My point is that the rejection of the law of identity and non-contradiction (and, thus, excluded middle) is nonsensical.

And this is a technical point about acknowledging that intuitionistic logic works - not more and not less. Logic is not about noses but about the laws of truth-preservation. And my technical point - which is not meant to discredit classical reasoning altogether - is enough to show, that there are many ways in which we can formulate laws of truth preservation.

And I disagree that intuitionistic logic is one of them.

Actually my whole argument is not to convert you to intuitionism or anything else. It is just to show you, that classical reasoning is not alone. Just as Euclid's geometry gave rise to many non-euclidean geometries in the 19th century, the axiomatic treatment of classical logic gave rise to a whole variety of non-classical ways of reasoning. One could argue that euclidean geometry as well as classical predicate logic still have some extraordinary status - but I do not see why this should make them more legitimate.

Agreed. But that doesn't mean that any old mish-mash of geometries actually work. That's why I said you could posit the nonexistence of noses: classical logic isn't exalted, but rather intuitionism is nonsensical.

In my view, at least.

Of course I cannot deny that I consider Brouwer's philosophy of mathematics pretty fascinating for some interesting consequences, such as the existence of changing mathematical entities (especially choice sequences). But I do not deny its primary problem: it deprives us of certain powers of reasoning we may want to have to formulate some scientific theories. And do deny practically working scientific theories just because one doesn't like their non-intuitionistic reasoning is a pretty harsh approach.

To say the least .

 

[PhilosophicalBluster]

The law of excluded middle applies to everything, be it science or *scoff* philosophy. The poster stated:

A subjective universe goes only by what the creator of that universe knows to be true, therefore in said subjective universe, it could be discerned that before the creator knows something to be true or untrue, it is both.

"In the universe, a proposition is either true or not true. That statement is a rule of logic."

I understood the law of excluded middle the first time it was written. Just because I offered an alternative doesn't mean I didn't get it.

For some reason, you disagreed with this.

It is irrelevant whether I believe in a universe governed by subjectivity or not, the fact is that it is a possibility that nobody can prove right or wrong.

 

[PhilosophicalBluster]

So from this I guess that you accept what I said about logic and faith not being against each other at all by correct definition?

I understand that logic and faith are not always mutually exclusive, if that's what you're asking.

Well the Bible isn't from one man, I think any historian, atheist or theist, could tell you that.

I could tell you that.

Many events in the Bible have been proven to be correct by new archeology, etc, so that shows at the least the Bible is planted in some truths. And although this doesn't prove the Bible, it does talk of a round earth and earth being floating in space.

I realize that nobody just made up the whole Bible, (which is why I said in my earlier post that I respect its historical significance) but just because we know that Jesus existed and had whatever legacy does not allow us to jump to the conclusion that he is the son of G-d.

There is no reason for someone to write a book that they know is false that they know would get them killed and then to start preaching it to world for the reward of persecution, being stonned, whipped, beated and in the end all the disciples except one were killed for what they said. And all the time they knew if what they were saying was true or false.

People should accept that homosexuality is not a choice for similar reasons.

You have to remember that the Bible was not always one book.

Old news.

The Old Testament was there long before Jesus and it gave many(and I mean many) prophecies of the Messiah, all of which were fulfilled in Jesus.

And yet we still have Jews. Whether Jesus was the Messiah that the Old Testament spoke of is very debatable.

Now you can't just say that these things were made up to look good, because believe it or not people were alive 2000 years and weren't stupid.

Eh wha?

There were people who were with Jesus who could have denied that these things happened to Him, but most of the prophecies were clear for all to see happening.

Lots of people denied Jesus, they just didn't write a book about it that lasted 2000 years.

Though the Bible is old, it doesn't mean it came out of thin air and that people were just like 'O an old book', when it was written people actually had seen Jesus so could say if these things were or were not true. Luke (He wrote the gospel of Luke and Acts which was the history of the earlier church) was once a non-jew and was converted to christianity, was a 'doctor' and proved to be a good historian who wrote his gospel from different people he asked about Jesus' life. This gives good reason to believe Lukes gospel inparticular as unbias.

Are you actually trying to tell me that an apostle was unbiased in writing the Bible?

Also the gospels can be dated to show that they were likely written by the people they claim to be written by and this means that there is little time for myth to have crept in.

Where can you discern that?

Ok, im no expert on this,

Clearly

but just because its an old book it doesn't mean we believe it for no reason. This is just a few things I can think of, but there are other reason to believe the Bible.

Yes, you believe it because your parents and priests tell you to, which is sort of my point.

 

[PhilosophicalBluster]

with all due respect, I think PhilosophicalBluster is simply confused about the meaning of philosophical terms and their implications. I don't know if you've ever actually studied the schools of philosophy that pertain to this, but your argumentation style is...... interesting.

I'm fifteen, give me a break here. No I don't have a philosophy degree, but I try my best to grasp unknown concepts.
Define interesting.

 

[Wiccan_Child]

A subjective universe goes only by what the creator of that universe knows to be true, therefore in said subjective universe, it could be discerned that before the creator knows something to be true or untrue, it is both.

I disagree: it would be untrue. Since the creator hasn't decided to make it true, it remains false.

I understood the law of excluded middle the first time it was written. Just because I offered an alternative doesn't mean I didn't get it.

The point was that the statement is true, period. You can reword it, rephrase it, but rejection is illogical.

It is irrelevant whether I believe in a universe governed by subjectivity or not, the fact is that it is a possibility that nobody can prove right or wrong.

On the contrary, if someone demonstrated that it is logically paradoxical, it would be proven false.

 

[solarwave]

People should accept that homosexuality is not a choice for similar reasons.

Actually what I said and that issue have nothing in common.

And yet we still have Jews. Whether Jesus was the Messiah that the Old Testament spoke of is very debatable.

Yes it is debatable, ie: the jews. I don't know if you have, but if you read the prophecies you will see that Jesus fulfills over 300 of them. You dont know how unlikely that is, its HUGE odds.

Are you actually trying to tell me that an apostle was unbiased in writing the Bible?

Im telling you Luke was unbias because he came to belief after Jesus had been and gone, so he wouldn't be saying stuff he knew was wrong. Fair enough if he didn't know some of them were wrong, but it just adds to the reason to believe the Bible.

Yes, you believe it because your parents and priests tell you to, which is sort of my point.

Arrogant at all? You have no idea why I believe, and this remark makes me doubt whether you even have an open mind to the reality of Jesus Christ. I have good reason for my faith, I hope you have good reason for your own faith.

 

[Wiccan_Child]

Yes it is debatable, ie: the jews. I don't know if you have, but if you read the prophecies you will see that Jesus fulfills over 300 of them. You dont know how unlikely that is, its HUGE odds.

Actually, he's fulfilled about a third of the 300 prophecies, and not all of them are that impressive: the Immanuel prophecy, for instance, was 'fulfilled' insofar as Matthew mention in passing that Jesus was Immanuel. Is that a truly fulfilled prophecy, or was it self-fulfilling (i.e., he was called that, or Matthew wrote it, because he was so prophesied)?

Im telling you Luke was unbias because he came to belief after Jesus had been and gone, so he wouldn't be saying stuff he knew was wrong. Fair enough if he didn't know some of them were wrong, but it just adds to the reason to believe the Bible.

Or, it reads exactly as if it were fictional.

Arrogant at all? You have no idea why I believe, and this remark makes me doubt whether you even have an open mind to the reality of Jesus Christ.

I agree, though perhaps not with the last part.

 

[Rauffenburg]

@PhilosophicalBuster

I understood the law of excluded middle the first time it was written. Just because I offered an alternative doesn't mean I didn't get it.

As my complicated arggmentative stuff (down and up from here) tried to show: Your idea on having an alternative to the law of the excluded middle might be promising. And even the fact that one does think about an alternative for this law is a pretty smart step for a 15 year old guy. I can't remember I thought about that back then.

@WiccanChild

Perhaps, but so is numerical approximation and Newtonian mechanics.

And they both work. As does intuitionistic logic.

Non-contradiction states that something can't be what it's not: A can't be identical to ¬A by definition.
If some thing p is identical to A, then, since A can't be ¬A, neither can p.
If some thing p is not identical to A, then, by definition, p is identical to ¬A.

If p = A, then p ≠ ¬A.
If p ≠ A, then p = ¬A.

I.e., "A=A" and "A≠¬A" imply "A∨¬A".

First of all: Non-contradiction does not state something about being in the first place, but about propositions. It states that no proposition is both, true and false. Formally in statement logic ¬(A & ¬A) is a logical truth and A & ¬A a logical falsity. The main problem of your argument is, that is presupposes too much and thereby remains mysterious and unprecise; such things as identity for example - identity is a predicate that we have to define first and we already get to predicate logic then doing that. Since you introduced the identity sign without defining it, I do not see the point in your derivation without a definition of that sign. I believe of course that you presuppose a notion of identity that is intuitionistically invalid. Namely that x is either identical or nonidentical to y, which already presupposes the law of the excluded middle. Yet, intuitionistically speaking, it is not the case that if a thing is not non-identical with itself then it is also identical with itself. The identity of things with infinitley many properties is not established just from the fact that they are not different from anything else. They are neither identical nor non-identical with themselves. And by such terms, your argument does not hold.

What you should have done is, derive the law of the excluded middle from the axioms of intuitionistic statement logic. Otherwise inuitionistic logic just works. The point of course is, that such an endeavor is pointless, since you can derive ¬¬(A∨¬A) but not (A∨¬A) as your are missing the ¬¬A->A axiom.

I think you should begin to acknowledge the facts. There are modes of thought very different from what you estimate to be "making sense" that do work.

Which is an extension of, not a replacement for, identity. We may not know whether p is identical to A, but that doesn't bely the fact that it either is or is not.

The problem here is, that you have to cut out your own intuition of the difference between epistemplogy (knowledge) and onotlogy (being). Intuitionism does claim that for the domain of mathematical entities there is no difference between that which is known and that which is. This is simply a consequence of the assumtions made by the intuitionist mathematician. To deny the credibility of your opponent by dogmatically restating your own intuitions which rest on other modes of rationality is not a profitable strategy when it comes to mutual understanding.

The very first step is: Give up the distinction between fact and knowledge. There are no unknown facts in intuitionist mathematics.

You simply have to see: An intuitionist does not have any use for the classical reasoning with the identity predicate. It does not have to do anything with an extension or replacement. Since there are undecidable properties, identity may be undecidable, too.

Agreed. But that doesn't mean that any old mish-mash of geometries actually work. That's why I said you could posit the nonexistence of noses: classical logic isn't exalted, but rather intuitionism is nonsensical.

Just because you don't get it, doesn't mean it is nonsensical. That is a pretty arrogant stance to take up here. Actually, making sense isn't even the question. You can go on and take the Hilbert point of view that all mathematics is simply a formal game without the question of sense or truth even arising. And even from that point of view you can technically do intuitionist mathematics; it is not a question of taste here.

And I disagree that intuitionistic logic is one of them.

Then, my dear opponent, you are denying that there are noses. The fact that it works shows that it is possible to reason intuitionistically. Intuitionistic logic works and works well - that is even more certain than any fact about noses.