The Roles of Logic and Science in Questions of Theology

[_Calculus Crusader]

Incidentally, I am reminded of this joke:

God is real, unless declared integer.

[_marg]

That argument with its conclusion may be valid, but the conclusion is contingent upon the game. You can not take that conclusion outside the game. Just like in Euclidean geometry conclusions regarding triangles are contingent upon the Euclidean system, not outside it with other systems.

And yet if I draw a conclusion about Euclidean triangles a surveyor might actually care and trust that certain physical consequences follow even without specifically testing them, while if I draw a conclusion about hyperbolic triangles she/he would not. We both no there is a reason for that.

Sorry I should have said, you can't take that (game's) conclusion out of it and use it with reliability unless there is a link to what you are applying it to.

I really can not see how any ontological proof can possibly logically establish the existence of an entity. It seems to me that if one creates a definition, assumes axioms which bear no direct relation to the actual world, that it has to be a matter of begging the question. As I pointed out to Gad, all deductive arguments are circular to some extent, and this modal proof for God seems deductive to me. Deductive within the system. With science the natural physical laws are assumed because they have been found to hold true, should one ever be found to not hold true the assumption will change. But science doesn't willy nilly simply assume whatever it wishes to for (axiomatic) natural physical laws.

Then one must argue about how Godel's axioms are willy-nilly. He did not just assume God exists. Read my reply to JAK.

A couple of questions (don't assume what my answers may be):

1. Are prime numbers actual? real? (We do use primeness in the real world for cryptography). What have I accomplished if I show that there exists an infinite number of primes and does this have real world consequences?

First of all it's been a long time since I've studied any math. I know Gad detests this but I have to research what a prime number is. The sort of questions you ask I've not given a heck of a lot of thought to so keep in mind I'm not responding with the pretence of being an expert. So with that in mind...numbers are part of a game, they aren't real existing things. They are real concepts in the game of math which can be applied to situations we can observe directly or indirectly. They are not actualities, scientific objects. With regards to your last sentence you are essentially using a model and applying it to the actual world, perhaps to make predictions. If the predictions bear out then it is likely the model is reliable and useful.

2. Is the real number system (which gives rise to the possibility of the mathematics of general relativity) a closed system based on willy-nilly assumptions completely unconnected to the actual world? Can I use the mathematics to arrive at likely truths about the real cosmos (positive mass conjecture etc.)?

In science the axioms or the natural physical laws can not be made up absent evidence. They have to coincide with results via emperical testing. However, in the creation or art of scientific theories assumptions can be assumed creatively absent evidence to reach a theory. But the conclusion/theory reached has to offer some value. Either predictive value which can bear out with testing, or some explanatory value of some phenomenon. In any event , there needs to be some link or justification to the actual world otherwise it's not science being done ..but mere simply made up assertions. To your last sentence, yes you can use math to arrive at likely truths but there needs to be a link to the actual world to justify doing so.

3. How can we ever use deductive reason about the actual real world?

Well it starts with the natural physical laws which are observed, tested and assumed to be universally valid. From there scientific knowledge of the actual world is built.

4. Can I use arithmetic to arrive at conclusions about the real world (like my finances or whether my digital secrets are secure)?

Your finances are not actualities. Coins are actualities. Your finances are a game or model of goods and services owed to you or what you owe others. Services are work, which I believe could be scientifically studied.

5. Are all axioms equally unrelated or related to the actual world?

An axiom does not have to be related to the world. But if the conclusion resulting from the use of an axiom is to be reliable and say something about the actual world there has to be a link.

6. Is there a difference between actual, real and physical?

Conceptually "real" includes knowledge at the level of what is certain and universal I believe Gad refers to that as platonic. Mankind due to perception limitations does not know reality at that level. An actuality or actual thing, is not the same. It is what we perceive and at whatever level of perspective at a particular time, and place. So real and actual I don't view as conceptually equivalent. Physical I believe is perception based. Something might not be perceived as physical but with the help of tools to perceive it becomes detectable as physical and then becomes a scientific actuality, that doesn't mean that physical thing is appreciated as a reality in the platonic sense of knowing it in a univeral and certain way.

Finally

7. Do you agree with JAK that the existence of God is assumed in definition 1? (see my response to him)

Let me quote him "The definition as quoted by CC reads:
“Definition 1: x is God-like iff x has as essential properties those and only those properties which are positive”

Regardless of any debate over “properties” vs. “essential properties” and regardless of consensus or lack thereof on “positive,” the definition itself makes assumption. That assumption is repeated in “Axiom 3,” “Corollary 1,” “Theorem 2,” and “Theorem 3.” "

This is the way I see it. Within Godel's game, God is a 3 letter word represented by a letter x, with no meaning until he defines it. It is something which exists in theory because he says "x is" but the meaning he actually gives makes no sense to me. Whatever meaning he gives to that x (whether you want to relabel it God or anything else) ....he is assuming. That whole definition is one complete assumption with no link to any actualities in the actual world. So yes, I do agree with JAK.

Now if what Godel was doing was creating a model to be employed for explanatory purposes of some phenomenon then there would be a link to the actual world. Even though the model itself may have no empirical evidence, the observation of the phenomenon would support the model. So the theory would offer practical value.

[_marg]

For example, suppose I make the following definition.

Definition: A Riemannian manifold is ghost-like if it is geodesically complete but not metrically complete.

Now if I make an argument showing that ghost-like manifolds exist, then I will have certainly made a mistake, because in fact, it can be proved that all geodesically complete manifolds are metrically complete. So ghost-like manifolds do not exist. But the definition is OK as it stands until the existential question is settled rendering it empty (and it is settled in this case; they don't exist).

I'm just now looking at your response to JAK. I have no idea what Riemannian manifold is. I assume is it some sort of mathetical concept. When you say a mathetical concept exists as you do above..it does not exist as an entity......in the actual world. It is not a scientific actuality, it exists as a concept, but it is simply claimed without any warrant to do so. Is that what Godel was attempting to show, that God exists as a concept? I don't think so and certainly it is not how Godel's proof of God was being used by CC.

Why should any definition of a thing claimed to exist as an actuality such as a God, stand or be "OK", until disproven? Because that seems to be your point in your last sentence above.

[_Tarski]

For example, suppose I make the following definition.

Definition: A Riemannian manifold is ghost-like if it is geodesically complete but not metrically complete.

Now if I make an argument showing that ghost-like manifolds exist, then I will have certainly made a mistake, because in fact, it can be proved that all geodesically complete manifolds are metrically complete. So ghost-like manifolds do not exist. But the definition is OK as it stands until the existential question is settled rendering it empty (and it is settled in this case; they don't exist).

I'm just now looking at your response to JAK. I have no idea what Riemannian manifold is. I assume is it some sort of mathetical concept. When you say a mathetical concept exists as you do above..it does not exist as an entity......in the actual world. It is not a scientific actuality, it exists as a concept, but it is simply claimed without any warrant to do so. Is that what Godel was attempting to show, that God exists as a concept? I don't think so and certainly it is not how Godel's proof of God was being used by CC.

Why should any definition of a thing claimed to exist as an actuality such as a God, stand or be "OK", until disproven? Because that seems to be your point in your last sentence above.

My point is simple: Definition 1 does not assume the existence of a God-like being. It merely defines the notion.

Also, while we agree on the failure of Godel's argument to have proved the existence of God in any importance sense I do wonder about a few things you have said.
You seem to have latched onto some notion of "scientific actuality" that I don't think is well thought out. If often seems that you are only willing to accept as "actual entities" bulk material objects. But things are not so simple.

If I ask for the rules played by subatomic particles or the rules played by Gaussian integers, I am asking what the actual rules are. No? I am using the word actual here in a reasonable way aren't I. But maybe not the same way I might use it when I say that this is an actual chair.

Is the wave function of an electron in a hydrogen atom actual or just a conventional token in a game of prediction? How about the electron itself?
Is the spacetime that Einstein's theory says has curvature actual? real?

What experiments can one perform to determine whether something is "actual" or merely a convenient conceptual token in a thinking game that helps predict the outcome of experiments? Can there be disagreement of this point (say in the case of an electron)?


Are thoughts actual? How about computer software? Do you understand what I mean when I say that I have some cool software in mind but that it is not actual software whereas MSWord is actual software? Is it a scientifc question to determine whether my software is actual or not? (I may want to sell it to you).
Is there a distinction between an actual thought and a possible thought? How about an impossible thought?


For me the safest way to approach these things is by being very specific about why I may refer to something as real or actual or not. The criteria of "reality or actuality" may depend on the realm of inquiry. For example, to say an animal exists is different than saying a number exists. To say that money exists in my bank acount is different from saying that a rock exists or that a possibility exists or that a continuous function exists or a force or a pattern or a method or a prime or an infinite ordinal.
In each case, I have some typical examples in mind and some rough, usually uncontroversial, criteria.

In the case of God I have in mind a conscious being who can be communicated with and who can influence physical events and answer prayers. I see no reason to think there is really such a God.


I just reject the idea that there is a single appropriate way to use these words (actual, real) that involves universal criteria.

[_asbestosman]

Incidentally, I am reminded of this joke:

God is real, unless declared integer.

But if not an integer, then perhaps irrational.

If not real, then perhaps imaginary.

If neither real nor imaginary, then perhaps surreal.

[_harmony]

Whoever said we don't have deep discussions here has not read this thread.

[_marg]

My point is simple: Definition 1 does not assume the existence of a God-like being. It merely defines the notion.

I think it does assume God. God-like is an adjective modifier of a noun which is a thing in this instance. To discuss any thing one has to first assume that thing. So I do think it assumes a God and then defines it.

Also, while we agree on the failure of Godel's argument to have proved the existence of God in any importance sense I do wonder about a few things you have said.
You seem to have latched onto some notion of "scientific actuality" that I don't think is well thought out. If often seems that you are only willing to accept as "actual entities" bulk material objects. But things are not so simple.

If I ask for the rules played by subatomic particles or the rules played by Gaussian integers, I am asking what the actual rules are. No? I am using the word actual here in a reasonable way aren't I. But maybe not the same way I might use it when I say that this is an actual chair.

You are correct that I haven't thought this all out. So perhaps you can help. It seems to me there are models in science used to help describe phenomenon. I talk about these things conceptually. I'm no scientist nor do I know much science. But let's assume for example the scientific model of an atom. It is treated as a scientific actuality in and of itself with subatomic particles in and of themselves as actualities. In theory they actually exist as entities. In reality they may not exist as they are currently understood. They are attempted models of things which exist in the actual world. Gaussian integers whatever they are, are solely concepts, they are not about any things which exist.

Is the wave function of an electron in a hydrogen atom actual or just a conventional token in a game of prediction? How about the electron itself?
Is the spacetime that Einstein's theory says has curvature actual? real?

I'm not familiar with this wave function but if scientists have developed models which include wave function of electrons in hydrogen atom..the electrons are an actuality, their movement is descriptive or a quality of them. The scientific model which includes actualities may change.

What experiments can one perform to determine whether something is "actual" or merely a convenient conceptual token in a thinking game that helps predict the outcome of experiments? Can there be disagreement of this point (say in the case of an electron)?

I think it's a matter of scientific models offering predictive and explanatory power, then the actualities within those models have practical merit. Those doing science determine what models have merit or not.

Are thoughts actual?

Shoot this is heavy. At this point I don't think there is a scientific model to explain "thoughts". So no at this point I'd say they aren't actual entities.

How about computer software? Do you understand what I mean when I say that I have some cool software in mind but that it is not actual software whereas MSWord is actual software?

Well I guess one is conceptual whereas the other is created onto a physical object.

Is it a scientifc question to determine whether my software is actual or not? (I may want to sell it to you).

Anything which can be studied, investigated is a scientific question..but what exists in your mind is not an actuality yet..because the actuality has to be a thing. I think I'm in trouble here but I'll leave it as is.

Is there a distinction between an actual thought and a possible thought? How about an impossible thought?

Well previously I said thoughts are not at the present time scientific actualities or appreciated as such. For an individual I can't see the difference between actual and possible thoughts. How would an individual be aware of their own possible thoughts? But another individual can be aware of someone else's thoughts and appreciate they might have other possible thoughts in the future or not expressed. With an "impossible thought", if it exists an individual would not know for themselves. For practical purposes one might assume the possibility being so small that a particular individual or individuals could not possibly think a particular thought. Somehow I don' t think that's what you are getting at.

For me the safest way to approach these things is by being very specific about why I may refer to something as real or actual or not. The criteria of "reality or actuality" may depend on the realm of inquiry. For example, to say an animal exists is different than saying a number exists. To say that money exists in my bank acount is different from saying that a rock exists or that a possibility exists or that a continuous function exists or a force or a pattern or a method or a prime or an infinite ordinal.
In each case, I have some typical examples in mind and some rough, usually uncontroversial, criteria.

In the case of God I have in mind a conscious being who can be communicated with and who can influence physical events and answer prayers. I see no reason to think there is really such a God.

Frankly I don't conceptually think about any God. Never have. I have to rely upon other people's ideas of what their God is in order to discuss that concept. I do know I have nil appreciation of life's relation to the universe and the universe itself relative to all that is out there, but that doesn't bother me

I just reject the idea that there is a single appropriate way to use these words (actual, real) that involves universal criteria.

Well conceptually the problem with the word "reality" is that we can appreciate we don't have that platonic understanding of reality. So we really are left with relying upon the information we get via our senses to reach best fit knowledge of the world. And the concept of using the word "actualities" which I didn't make up, seem to alleviate the problem with the word "reality" to some extent.

[_Tarski]

My point is simple: Definition 1 does not assume the existence of a God-like being. It merely defines the notion.

I think it does assume God. God-like is an adjective modifier of a noun which is a thing in this instance. To discuss any thing one has to first assume that thing.

Assume the thing exists??
Well, Marg, that certainly can't be right because then we could never discuss anything until we knew it existed which means we couldn't even get to the point of asking if it existed. In order to discuss something in a logical setting we need to define it. Then we can ask whether something exists that satisfies the definition.
Also, would you think that Godel had assumed God if he had used the terminology G-like instead of God-like? It's a coinage, it only has the meaning he gave it. It is the same with the defined word smooth in the definition of smooth manifold.

Marg, I guarantee that Godel did not simply assume that god exists in penning his version of definition 1.

Suppose you want to tell me that unicorns don't exist. I ask you what you mean by "unicorn" and then you give me a definition so we know what is being questioned. Now, did you thereby assume the existence of a unicorn just because you gave a definition of the word? Not at all!
We can't decide if it exist until we know what we mean by it.

God-like is an adjective modifier of a noun which is a thing in this instance.

Unicorn-like is a adjective modifier of a noun too (but you must define it).
So what?

So I do think it assumes a God and then defines it.

???
That actually doesn't even make sense. How does one determine something is being assume before it is defined? Did I assume the existence of Ghost-like manifolds when I defined them in my previous post? Not at all.

[_marg]

My point is simple: Definition 1 does not assume the existence of a God-like being. It merely defines the notion.

I think it does assume God. God-like is an adjective modifier of a noun which is a thing in this instance. To discuss any thing one has to first assume that thing.

Assume the thing exists??
Well, Marg, that certainly can't be right because then we could never discuss anything until we knew it existed which means we couldn't even get to the point of asking if it existed. In order to discuss something in a logical setting we need to define it. Then we can ask whether something exists that satisfies the definition.
Also, would you think that Godel had assumed God if he had used the terminology G-like instead of God-like? It's a coinage, it only has the meaning he gave it. It is the same with the defined word smooth in the definition of smooth manifold.

Marg, I guarantee that Godel did not simply assume that god exists in penning his version of definition 1.

I'm sorry I'm not convinced Tarski. I still think the def'n assumes the thing exists, even if a faulty assumption based on imagination only.

Suppose you want to tell me that unicorns don't exist. I ask you what you mean by "unicorn" and then you give me a definition so we know what is being questioned. Now, did you thereby assume the existence of a unicorn just because you gave a definition of the word? Not at all!
We can't decide if it exist until we know what we mean by it.

To begin with why would I say to you out of the blue having never heard of unicorns before, unicorns don't exist? The only reason why I would, is if you said to me, unicorns do exist. You can not have "no unicorn" ..unless there is the concept of "a unicorn" in the first place. And when the positive is claimed there has to be some warrant or justification for it, otherwise it is simply imagination. And with imagination an infinite number of things can be created in thought only. So there is no point in looking for those infinite things, unless there is some justification to do so.

previously: God-like is an adjective modifier of a noun which is a thing in this instance.

Tarski: Unicorn-like is a adjective modifier of a noun too (but you must define it).
So what?

You seem to be working backwards. Why on earth for example would one ever present a definition of a thing actually existing without any justification or link to the actual world for that thing? Let' compare unicorn def'n to Godel's

“Definition 1: x is God-like iff x has as essential properties those and only those properties which are positive”

Def'n 1: x is unicorn- like if and only if x has properties of a biological horse with a singlestraight spiraled horn projecting from it's forehead.

What would make me even assume the def'n for a unicorn if there wasn't some justification that such an entity "unicorn" actually exists? There is an infinite number of things one could imagine exists or existed but why do that without some warrant to justify it. One could spend their lifetime making up imaginary things which have no connection to the actual world. But what is the point? What is being argued with regards to unicorns for example..either they exist or they don't. But someone is not going to argued the non existence first without the positive being presented. And it is the positive claim which has the burden of proof.

Let's say we have the propostion .."unicorns exist" and we have the negative "unicorns don't exist" . The negative proposition could not be conceived of without first conceiving of the positive proposition of unicorns in the first place.

I think I'm repeating myself.

If there is no justification to warrant the existence of unicorns..then the def'n " x is unicorn- like if and only if x has properties of a biological horse with a singlestraight spiraled horn projecting from it's forehead." is an assumption based on imagination. It has to be the positive assumption of unicorns existing because it's illogical to assume the non existence of something if that something never existed or was never claimed to exist in the first place.

previously: So I do think it assumes a God and then defines it.

Tarski: ???
That actually doesn't even make sense. How does one determine something is being assume before it is defined? Did I assume the existence of Ghost-like manifolds when I defined them in my previous post? Not at all.

I don't know what ghost-like manifolds are. The def'n assumes the positive "a God" exists conceptually only, if there is no link to the actual world. The assumption is implied with the use of the word God in the definition. You can not have the concept "no god" without first having the concept "a god."

[_Tarski]

My point is simple: Definition 1 does not assume the existence of a God-like being. It merely defines the notion.

I think it does assume God. God-like is an adjective modifier of a noun which is a thing in this instance. To discuss any thing one has to first assume that thing.

Assume the thing exists??
Well, Marg, that certainly can't be right because then we could never discuss anything until we knew it existed which means we couldn't even get to the point of asking if it existed. In order to discuss something in a logical setting we need to define it. Then we can ask whether something exists that satisfies the definition.
Also, would you think that Godel had assumed God if he had used the terminology G-like instead of God-like? It's a coinage, it only has the meaning he gave it. It is the same with the defined word smooth in the definition of smooth manifold.

Marg, I guarantee that Godel did not simply assume that god exists in penning his version of definition 1.

I'm sorry I'm not convinced Tarski. I still think the def'n assumes the thing exists, even if a faulty assumption based on imagination only.

Well, you should be. The point is quite simple. We do this in logic and math all the time.
Another example.

Definition 1: A number is called a terminal prime if it is both prime and there exist no prime numbers larger.

This definition does not assume the existence of a final prime. But it can be shown that there does not exist a final prime. Any proof of the existence of a final prime must be flawed but the flaw is not in the definition.

Definition 2: A number is called 1-tulip if it is both even and prime.

This silly sounding definition is also fine. In this case, it can be shown that there does exist a unique "tulip".
Again this definition does not assume the existence of a a 1-tulip. That must be demonstrated or exhibited (2 will do).

You seem to be working backwards.

I'm not.

Why on earth for example would one ever present a definition of a thing actually existing without any justification or link to the actual world for that thing?

Come on! How can one ask a question about something that has no definition?
If I ask you if you believe that infinite cardinals exist would you first want a definition? If I then give the definition would you accuse me of assume the existence of infinite ordinals in the first place? What a silly catch-22 I would be in.

How do you even come to the opinion that there is no God without having a definition to give when push comes to shove? Perhaps you mean sothing different than Godel or me or Mr. Tanaka. So if someone offers a definition of God for purposes of precision do you then accuse them of having assumed a priori that such a God exists?

No it is you that are working backwards.

As for you not being convinced I guess there is not much more I can say. Suppose I give you the standard proof that there exists an infinite number of primes and you say you are not convinced? I can try to explain but you are free to not be convince. At some point I should give up I guess.
All I can say more is that I have been working with definitions, axioms and theorems for more than 30 years and have published in top journals. This doesn't prove anything but it should give you pause.

I ran this by another friend (also an atheist and a logician). I asked if he thought that definition 1 simply assumes the existence of God.
He looked at me like I was kidding so I had to say, "just humor me". He said "Of course not. It's just a definition".

A deductive proof of the type we see here could never prove the existence of a material object of course. You seem to think Godel and anyone else must treat God as an (imaginary) material object. Godel does not view God in this way--Mormons basically do. Godel does you the service of defining what he means by his words to be clear, and you object that, by this kindness, he has already begged any and all questions about God. That's weird.