# numbers vs. words

TuringEquivalent

Sat 1 May, 2010 04:02 am
@Extrain,
Extrain;158779 wrote:
You don't have to be pedantic.

Those are all categories of types of things that exist, yes. But they are not categories of types of existence. "Existence" is univocal (a la Quine).

Sure. That sounds right.

Yes, and "existence" has one meaning.

My previous remark is that if there are multiple categories, than, it does make sense to say "santa clause" exist, if the category "fictional objects" exist.
The first "exist" pick out the particlar "element" in the category, and the second "exist" pick out the particular category.

That is why there are surely different kinds of existences, if different categories exist.

kennethamy

Sat 1 May, 2010 04:08 am
@TuringEquivalent,
TuringEquivalent;158792 wrote:
My previous remark is that if there are multiple categories, than, it does make sense to say "santa clause" exist, if the category "fictional objects" exist.
.

But doesn't the category of self-contradictory objects exist, but still, there are no self-contradictory objects. Or do you think there are square-circles?

There are such things as empty classes, aren't there? The classes exist, but they have no members. Right?

TuringEquivalent

Sat 1 May, 2010 04:14 am
@ughaibu,

Perhaps, you can tell us what a particular theorem( say, it is a theorem from reference paper 23) show. You don ` t need to be very detail in your description. You don` t even need to reproduce the entire proof. Perhaps a nice template would be like:

Theorem 3 from reference paper 2 show that ________X___________.

This would open the debate, and allow that person to challenge what you assert, namely, X. This is just good philosophy.

Extrain

Sat 1 May, 2010 04:15 am
@ughaibu,

I honestly don't know much about set theory, but you still haven't shown any substantive objections that numbers exist. None of these sources or articles are arguing for that claim anyway....so what's your point? That various issues are still debated in set-theory doesn't tell me anything. It is common for philosophical disagreements to be rampant. But this doesn't show anything other than that there are different theories about this or that. I want to know what your objection is. It seems you have missed this part in SEP about Godel:

Quote:
Independence Proofs.
6. Inner Modelshe showed that both the Axiom of Choice and the Continuum Hypothesis are consistent with the axioms of set theory, that is that neither can be refuted by using those axioms. This he achieved by discovering a model of set theory in which both the Axiom of Choice and the Continuum Hypothesis are true.

I can't exactly make sense of these disputes, but can you? Like I said, my reason for believing numbers exist are the same reasons that these guys believe that they do....have troubles in set-theory caused any of these authors to abandon the belief that numbers exist? I doubt it.

ughaibu

Sat 1 May, 2010 04:18 am
@TuringEquivalent,
TuringEquivalent;158800 wrote:
Perhaps, you can tell us what a theorem( say reference paper 23) show. You don ` t need to be very detail in your description. You don` t need to reproduce the entire proof. Perhaps a nice template would be like:

Theorem 3 from reference paper 2 show that ________X___________.

This would open the debate, and allow that person to challenge what you assert, namely, X.
I have stated the relevant results. I dont know what the point of me posting these links is meant to be, unless Extrain thinks I was lying, the results speak for themselves. The problem for realists has been stated.

---------- Post added 05-01-2010 at 07:22 PM ----------

Extrain;158802 wrote:
It's been stated, if you dont understand it, fine.
Extrain;158802 wrote:
It seems you have missed this part in SEP about Godel
No, it has no bearing on the issue.

Extrain

Sat 1 May, 2010 04:24 am
@ughaibu,
ughaibu;158803 wrote:
It's been stated, if you dont understand it, fine.No, it has no bearing on the issue.

Yeah? And what problem is that? Have any of these authors abandoned the belief that numbers exist? No.

Do scientists abandon the belief that black holes exist just because they don't have a "god's eye" perception of everything there is to know about black-holes, or because there are different conflicting hypotheses about them in the scientific community? No, because everything suggests it is very reasonable to believe black holes exist even if we know very little about them, just as it is reasonable to believe numbers exist even if there are disagreements how to articulate a theory of sets about them.

TuringEquivalent

Sat 1 May, 2010 04:26 am
@kennethamy,
kennethamy;158795 wrote:
But doesn't the category of self-contradictory objects exist, but still, there are no self-contradictory objects. Or do you think there are square-circles?

There are such things as empty classes, aren't there? The classes exist, but they have no members. Right?

I don ` t think the category of "self-contradictory category" exist. I think it has to be at least non contradictory. Categories as conceived are different types of existence. It is common in the past that people posit "mind" to explain the world. This "mind" is a category. categories are supposed to be ontologically fundamental, and unanalyzable. Good work done in metaphysics is done by reducing the number of ontological commitment by reducing some categories to other categories.

Extrain

Sat 1 May, 2010 04:33 am
@TuringEquivalent,
TuringEquivalent;158792 wrote:
My previous remark is that if there are multiple categories, than, it does make sense to say "santa clause" exist, if the category "fictional objects" exist.

This is Meinong. I disagree with Meinong that there are different categories of "being." There are different types of things that exist, so different categories of kinds of things. But there are not different categories of existence. You get involved in very absurd metaphysical theories when you do this, namely, those which consistently violate the principle of parismony.

TuringEquivalent;158792 wrote:
That is why there are surely different kinds of existences, if different categories exist.

Sets, classes (kinds, intensionsional semantics categories), and extensions--are all different types of categories. For sets to exist, there have to be existent members of sets (except for the null set). For a predicate to have an extension, there have to be existent members of that extension.

The extension and set of fictional characters does not exist, if fictional characteries do not exist.

kennethamy

Sat 1 May, 2010 04:35 am
@TuringEquivalent,
TuringEquivalent;158811 wrote:
I don ` t think the category of "self-contradictory category" exist. I think it has to be at least non contradictory. Categories as conceived are different types of existence. It is common in the past that people posit "mind" to explain the world. This "mind" is a category. categories are supposed to be ontologically fundamental, and unanalyzable. Good work done in metaphysics is done by reducing the number of ontological commitment by reducing some categories to other categories.

What do you mean it doesn't exist? Why not? You are confusing the existence of classes with the existence of members of the class. Why can't there be empty classes? Logicians and mathematicians believe there are.

Categories as conceived are different types of existence

I don't know exactly what that means, but do you think that the category of dogs, and the category of cats, are "different types of existence"? They are different categories of objects, though.

ughaibu

Sat 1 May, 2010 04:41 am
@Extrain,
Extrain;158808 wrote:
Have any of these authors abandoned the belief that numbers exist?
Argument ad populum? On the other hand, how many posts ago was it that you stated that most mathematicians aren't realists about numbers? Come to that, do any of Bell, Martin-Lof or van Lambalgen espouse realism?

Extrain

Sat 1 May, 2010 04:43 am
@TuringEquivalent,
TuringEquivalent;158811 wrote:
I don ` t think the category of "self-contradictory category" exist. I think it has to be at least non contradictory. Categories as conceived are different types of existence. It is common in the past that people posit "mind" to explain the world. This "mind" is a category. categories are supposed to be ontologically fundamental, and unanalyzable. Good work done in metaphysics is done by reducing the number of ontological commitment by reducing some categories to other categories.

How do me and my dog differ with respect to existence?

How do numbers and myself differ other than the fact that numbers are abstract objects with abstract properties, and I am physical object with physical properties? How do we differ on the level of existing? We don't. Physical objects either exist or don't. Abstract objects either exist or don't. Existence is a "1 or 0" kind of notion.

TuringEquivalent

Sat 1 May, 2010 04:46 am
@kennethamy,
kennethamy;158819 wrote:

1 What do you mean it doesn't exist? Why not?

2 You are confusing the existence of classes with the existence of members of the class. Why can't there be empty classes? Logicians and mathematicians believe there are.

Categories as conceived are different types of existence

3. I don't know exactly what that means, but do you think that the category of dogs, and the category of cats, are "different types of existence"? They are different categories of objects, though.

1&2. I am saying there are probably no "contradictory categories". I did not at all say anything about classes. Classes is one particular category. I do think classes exist, but i don` t think "contradictory categories" exist.

3. Dogs, and cats are in the same category. The category of "concrete objects".

Category of being - Wikipedia, the free encyclopedia

ughaibu

Sat 1 May, 2010 04:46 am
@Extrain,
Extrain;158830 wrote:
Physical objects either exist or don't. Abstract objects either exist or don't. Existence is a "1 or 0" kind of notion.
That's unclear. Consider Zalta's theory, in which abstract objects encode properties while concrete objects instantiate them. This creates two categories of existence.

Extrain

Sat 1 May, 2010 04:47 am
@ughaibu,
ughaibu;158826 wrote:

No, it's actually an argument that our not knowing everything about an object X is not a good reason to believe that object does not exist.

ughaibu;158826 wrote:
On the other hand, how many posts ago was it that you stated that most mathematicians aren't realists about numbers?

? I never said that. I said the exact opposite. I said most mathematicians are realists about numbers: and that's true. I live and breathe in academia, and I've been around most of them to know that.

Again, what is your argument? This is all just hot air! Are you capable of saying anything substantive? Do you have any really good objections?

Arguments involve premises and a conclusion.

kennethamy

Sat 1 May, 2010 04:54 am
@TuringEquivalent,
TuringEquivalent;158834 wrote:
1&2. I am saying there are probably no "contradictory categories". I did not at all say anything about classes. Classes is one particular category. I do think classes exist, but i don` t think "contradictory categories" exist.

3. Dogs, and cats are in the same category. The category of "concrete objects".

What makes you think that a category is contradictory just because its members are contradictory? (How do you distinguish between categories and classes?)

But aren't snakes and dogs in different categories? One, mammals, the other, reptiles?

Anyway, don't concrete objects and abstract objects both exist. If you think they exist in "different senses" could you tell me what those different senses are, and how they are both of them still senses of "exist" even if they are different senses?

ughaibu

Sat 1 May, 2010 04:59 am
@Extrain,
Extrain;158836 wrote:
I never said that.
Extrain;158836 wrote:
Again, what is your argument? This is all just hot air! Are you capable of saying anything substantive? Do you have any really good objections?
1) the standard formulation of numbers is set theoretic
2) standard set theory is ZF set theory
3) ZF set theory is implied by the axiom of choice
4) axiomatising ZF for randomness allows a proof that the axiom of choice is false
5) an uncountable infinity of real numbers are random
6) a countable infinity of real numbers are computable
7) therefore the set of non-random real numbers has zero measure
8) therefore numbers have no foundation with probability one.
1) given an infinite number of apples, are these apples fictional? If infinity disturbs you, make that a finite number of apples, for example, ten raised the the power of two thousand.
2) how do you, as a realist about mathematical objects deal with the fact that different formal systems have contradictory implications?

Extrain

Sat 1 May, 2010 05:02 am
@ughaibu,
ughaibu;158835 wrote:
That's unclear. Consider Zalta's theory, in which abstract objects encode properties while concrete objects instantiate them. This creates two categories of existence.

I am aware of Zalta's theory: he tries to make room for the existence of fictional entities as abstract objects. That's precisely why he came up with that distinction. But fictional entities don't exist because they are fiction.

So again, why are there different categories of existence then???

ughaibu

Sat 1 May, 2010 05:05 am
@Extrain,
Extrain;158849 wrote:
But fictional entities don't exist because they are fiction.
Which statement is nonsense, a thing can not be said to be non-existent because it is what it is! x=x, therefore x doesn't exist?!?

TuringEquivalent

Sat 1 May, 2010 05:05 am
@kennethamy,
kennethamy;158839 wrote:

1 What makes you think that a category is contradictory just because its members are contradictory?

2 (How do you distinguish between categories and classes?)

3 But aren't snakes and dogs in different categories? One, mammals, the other, reptiles?

4 Anyway, don't concrete objects and abstract objects both exist. If you think they exist in "different senses" could you tell me what those different senses are, and how they are both of them still senses of "exist" even if they are different senses?

1. What makes you think otherwise?

2. Classes is one particular category. This is just a matter of technical definition.

3. Perhaps? The usual categories are: Category of being - Wikipedia, the free encyclopedia

Do you see dogs, and cats?

4. Abstract objects are non-causal, don ` t have any spatio-temporal relation. concrete objects are casual, have spatio-temporal relation.

Extrain

Sat 1 May, 2010 05:09 am
@ughaibu,
ughaibu;158845 wrote:
2) standard set theory is ZF set theory
3) ZF set theory is implied by the axiom of choice
4) axiomatising ZF for randomness allows a proof that the axiom of choice is false
5) an uncountable infinity of real numbers are random
6) a countable infinity of real numbers are computable
7) therefore the set of non-random real numbers has zero measure
8) therefore numbers have no foundation with probability one.

Now what does this mean in your own words? You pulled this from somewhere.

What does "numbers have no foundation with probability one" mean? Can you explain please?

ughaibu;158845 wrote:
1) given an infinite number of apples, are these apples fictional? If infinity disturbs you, make that a finite number of apples, for example, ten raised the the power of two thousand.

I told you that i don't think an actual infinity of apples exist. What do you not understand, here?

ughaibu;158845 wrote:
2) how do you, as a realist about mathematical objects deal with the fact that different formal systems have contradictory implications?

"Different set-theories that prove contrary results regarding sets does not prove that numbers are not sets. It just shows that there are two different set-theories about sets. I am not even committed to the view that numbers are sets....but I am committed to the view that whatever they are, they exist, and they are not concepts, not ideas, not words, and not apples..but they are abstract."

I've told you my own reason for believing numbers exist. They are indispensible for our scientific theories about the world to be true. They are indispensible for any statement involving numbers to be true. That is a good enough reason for me to believe they exist.

---------- Post added 05-01-2010 at 05:13 AM ----------

ughaibu;158851 wrote:
Which statement is nonsense, a thing can not be said to be non-existent because it is what it is! x=x, therefore x doesn't exist?!?

Not so. First off, "x=x" is statement of numerical identity, not existence, nor kind. Second, saying "it is what it is" presupposes something already exists. I presuppose no such thing. Third, "it is what it is" is a reference to a difference in kind, not a difference in existence anyway. The "what it is" tells us what kind of thing it is. The "that it is" tells us that it exists. So you are confusing the "is" of existence with the copula "is" of predication. "is red" is not the same thing as "there is something x, such that..."

"Santa" is an empty name. It refers to nothing. These statements can be formulated logically with ~(Ex)

"It is not the case there exists an x, such that x has the name 'santa'" is true.

"It is not the case there exists an x, such that x is a category with an extention that ranges over fictional entities" is true.