The Roles of Logic and Science in Questions of Theology

[_marg]

1) The term reality is so freighted with connotations and values that it is preferable to say that the object of scientific knowledge is neither experience nor reality but actualities - scientific objects, such as the Sun, the Earth, atoms, genes, black holes - all inductively inferred from experience

This is nice but not quite a clear definition is it? We only have a list of examples. Since we don't have a definition or even a clear criteria we don't know how far to take it. Below, your quote mentions forces and structures as somehow scientifically real.
So can we extend the list? How far?

the Sun, the Earth, atoms, genes, black holes, quarks, strings, structures, branes, [parallel universes, interactions, patterns, information, computations, software, thoughts, beliefs, behaviors, space, time, spacetime, number, set, quale, notion.....?

Keep in mind I gave you notes, not the actual lecture which offers examples and better explanation. My take on it, and this is certainly not something I've studied in any depth, is that actualities can be extended to any conceptual thing which offers explanatory power and/or predictive value and can be inductively inferred. Math on its own I don't consider an actuality, but it can be applied to theories of what is out there existing. Behaviors, beliefs, patterns, information, no, they are not about thing which exist. Space, time are inductively inferred from experience and essentially about scientific objects.

What kinds of experiences shall be allowed? Experience with numbers, sets and topologies? (Kitcher believes that mathematics is empirical!!),
spiritual experience with gods and devils (oh please no!)?

The spiritual experiences are a product of the functioning of the brain. The brain is a scientific actuality.

2) Scientific theories can be understood as mapping, not onto some ultimate and inexperienceable Reality, but onto such actualities which are rooted in emperical [sic] experiences, current or potential, directly or indirectly, via instruments, all defined by the scientific community.

"inexperienceable Reality"? Is this like Kant's noumena? There are deep and troubling waters in that direction.

My take is that the inexperiencable reality is the platonic reality. A universal certain deterministic reality.

Suppose one sets up a solid criteria for what counts as a "scientific actuality". Then that encompasses some things that fit and others that don't but how does one then leap to the notion that anything that doesn't fit the criteria cannot be real in any way at all? In other words, how does one argue that the complement of the class of scientific actualities is empty or at least devoid of anything but the imaginary?

I've been talking about a link to the actual world. That link is some sort of justification to theorize about a scientific actuality, which are inductively inferred from experience. The burden of proof is on those theorizing or claiming those actualities. Those consensus accepted scientific actualities may change conceptually with new tools, additional insight, new discoveries. The conception of the atom has been a changing evolving one. I believe how one differentiates simply imaginations versus scientific actualities is via the link to the actual world which has been made by induction inferred from experience.

I am glad that you seem to have a loyalty to the scientific outlook. However, I encourage you to check out some perspectives beyond Chapter 24 of Steven Goldman book.

Perhaps
"Philosophy of Science A Contemporary Introduction" by Alex Rosenberg

and for ontological question regarding mathematical objects:

Thinking about Mathematics: The Philosophy of Mathematics by Stewart Shapiro
and
Platonism and Anti-Platonism in Mathematics by Mark Balaguer

To be fair, I will take a look at the book you mention.

Thank you for the suggestions. Regarding your last sentence, its a course not a book which is put out on DVD or CD's. There is a complete transcription of the course in book form sold by the Teaching company. Some information on the author can be found here, if you are interested http://en.wikipedia.org/wiki/Steven_L._Goldman

There is an extensive bibliography, for the complete 24 lectures. For this particular chap he . he recommends Marry Hesse, Revolution and Reconstruction in the Philosophy of Science(1980) and Philip Kitcher, 2001 Science Truth and Democracy.

Since he mentioned John Dewey specifically in the course at this section, he uses Experience and Nature, (1958) and The Quest for Certainty (1988).

(by the way... I need to take a break atm from discussion, it's making my head hurt.)

[_JAK]

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

But giving examples of math is no good Tarski. Math is completely conceptual. I appreciate it can be applied to the actual world. Negatives do exist in math. But when we are talking about things existing in the actual world, it's not just conceptual, if that thing exists. As I said before one can imagine an infinite (literally) number of things. To talk with any substance, as if a thing exists in the actual world, there has to be a link to the actual world. Where is Godel's link? As you said he could have used the term G-Like instead of God-like and I agree. At this point, I'm not even discussing that the def'n he gave either doesn't make sense or I'm not understanding the sense it's supposed to make. So please, if you are going to give examples don't use math. It's not the same thing.

previously: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?

Tarski: 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.

I'm not following you Tarski. I didn't say one shouldn't give definitions, I said why would one do so, without any link to the actual world. And I'm talking about things which exist as actualities or potentially could exist as actualities. One doesn't first create def'ns of things existing and go looking for them, without any justification.

Let's say scientists expect to see intermediary stages of bone structure between one species and another, so if they go looking for them or have expectations of finding them...there is rational justification for doing so. Or let's say people have claimed to have seen giant octopus 40 feet long, and there were a number of sightings at different times, so scientists might go looking. Or a phenomenon is not explained well and something else is expected, because that explanation doesn't quite suffice, they might go looking, with an idea of what to find. But they can not go looking for every thing that man can possibly create by definition alone. It's a physicial impossibility. What link does Godel make to the actual world?

So when someone let's say a scientist sets up a definition of a thing and then gives that thing a label they are in effect presenting an assumption that that labeled thing exists.

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.

First of all Tarski, no one would presume "no God" without first the notion of the positive "a God" The positive notion of something has to come first before any negative of non existence can logically be made. So to set up an argument which is about the existence of a thing and that thing is defined in Godel's case God ..that thing being defined is assumed to exist. That's what the argument is about.

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.

You can have negative numbers, you can not have negative things. There is a Tarski..if Tarski was never born, there is no "negative Tarski". And if you weren't born there would be no reason for anyone to look for you.

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 know it can be very frustrating.

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".

If I am presenting an argument to you about the existence of something and I say : x is pox-like if an only if it is red and slimy. Don't you think in that definition there is the assumption that a pox exists? It's possible it could exists as a conception only, but that's not what I'm attempting to present in my argument. I'm presenting you with an argument in which I am attempting to prove the existence of an actuality, something which really exists out there. By using ther words pox-like there is the implied assumption of a pox existing.

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.

No I'm not assuming Godel's God has to be material. Frankly I don't know what Godel's god is, I've asked the question and didn't receive a response, from either CC or Gad. I had a few questions which I'd have to go back and find. Can you tell me what Godel's god is? But the use of the word God(-like) in the definition assumes that God (whatever it is) exists just like the use of the word pox (-like) which I used previously assumes a pox exists (whatever it is).

--------------------------------------------------
marg,

There are numerous techniques in fallacious arguments.

Tarski by accident or design employs a number of these. The topic originally “The Roles of Logic and Science in Questions of Theology” is not limited in the way math. Theology is not about math.

1. Definitions are essential.
2. Clarity and precision of definitions are important if not essential.

No one has even attempted to refute that. Nor has anyone refuted the ambiguity which I cited in CC’s replication which is what we are all able to view without further research. If CC’s replication is incorrect/inaccurate, no one has made that claim. I raised that question in my comments to Tarski.

Developing any elaborate construction of argument with the use of ambiguous terms devalues that argument.

(I recognize you could say that I’m making assertions, but I’m prepared to address challenge to them.)

marg observed Sun Sep 02, 2007 10:30 am:
But giving examples of math is no good Tarski. Math is completely conceptual. I appreciate it can be applied to the actual world.

The topic under consideration is not confined to math. Your observation is important. While science is intimately connected with math, theology is not. And the question of the topic is essentially being ignored in all the illustrations from math.

Tarski did not address the issues which I raised except to repeat his position. I asked why not some neutral reference to “x.” His response was to repeat his claim. By adding further terms for which definition is required, Tarski is guilty of Fallacy: Muddy the waters as well as Shift the topic.

Both are Fallacies and further generate ambiguity. Rather than track my analysis point by point, he short-circuits by omitting the point by point detail which I provided.

Instead of addressing directly my analysis, he repeats his claim.
----------

All Tarski quoted from my analysis was the following:

Response to Tarski Sat Sep 01, 2007 7:27 am
That post is far back from where this response appears.

1. What is the topic under discussion?
It is:

The Roles of Logic and Science in Questions of Theology

2. Following that title, we have discussion in which an ever-present term “God” in Theology is used.

3. Assuming the quote from CC is accurate, the fuzzy construction lies there. That is where the explanation needs clarification.

Tarski stated:
You need to explain this better. How does merely making a technical definition amount to assuming the outcome of the argument??

(JAK asked:)
What’s your basis for concluding we have in definition 1 “merely making a technical definition”? It is not stipulated on the page CC gave. If it’s stipulated elsewhere -- where? And we have at hand the topic The Roles of Logic and Science in Questions of Theology.
-------------------

Following that (Fallacy by Reduction of Argument), Tarski state:
“He (Godel) is defining a term. That’s a coinage. We do it all the time in math and logic.”

Tarski is Begging the question.

Tarski does not respond to the question:

What’s your basis for concluding we have in definition 1 “merely making a technical definition"?

Let me point out that Tarski never responds to his basis for his claims. He repeats:

“He (Godel) is defining a term. That’s a coinage. We do it all the time in math and logic.”

The answer is irrelevant to the question.

Tarski stated:
“I do not claim that Gödel’s proof is successful JAK. I do claim that Definition 1, despite being unclear by virtue of reference to an undefined notion of positive, nevertheless does not assume the existence of God.”

Still begging the question Why did Gödel do this? If it was Gödel’s intent to focus on definition only for a deductive construction, why use a clearly universal term of religion?

Tarski had given no satisfactory answer.

Having said that we should recognize that Tarski also stated:
“I do not claim that Gödel’s proof is successful JAK.”

As Tarski claims that: “You (JAK) are simply mistaken on this point,” he continues to beg the question as to why Gödel would deliberately use a loaded term “God-like” in his construction. It's Tarski's claim that we have "merely ... a technical definition" and "just coinage."

It was not Tarski, but CC who claimed that Gödel proved the existence of “God.” Now why would CC do that?

I think it’s fair to consider that because Gödel repeatedly refereed to “God” or “God-like” that someone (like CC for example) would leap to the erroneous conclusion that Gödel proved that God exists.

The flaw and lack of clarity lies in the use of “God-like” rather than some neutral (meaningless) term for purposes of a syllogistic construction.

Tarski failed to respond to the questions I raised regarding why the use of “God-like.” Repeating his claims was not a response to analysis.

marg,

See Christian History Institute

Read all of that page, but particularly scroll down to read the following:

“Godel's proof fits well with Christian beliefs about the universe, by analogy. Judeo-Christianity has long held that truth is above reason. Spiritual truth can be grasped only by the spirit. Had Godel been able to show that self-proof was possible, we would be in deep trouble. The universe might them be self-explanatory. The implication of his proof is that the infinities and paradoxes of nature demand something higher, different in kind, more powerful, to explain them, just as every logic set needs a higher logic to prove and explain everything within it.

In other words, no finite system, even one as vast as the universe, can satisfy the questions it raises. Godel recognized this and tried to find a watertight proof of God's existence. He failed. Sadly, the evidence of his life suggests he failed to find a personal relationship with Christ either.”

The bold is my addition to focus on quite possibly why many Christians have a great affection for Godel.

So, Godel was trying to find “a water tight proof of God’s existence.”

Whether Tarski knows of this information, I don’t know. But it is a partial answer to my question to Tarski: Why did Godel use “God-like”?

All the muddying of the water does not obfuscate if one really explores Godel’s choice of “God-like.”

Evidence in the weblink suggests that the word-choice was not "merely" or just "coinage."

JAK

[_Calculus Crusader]

Tarski is Begging the question.

Tarski does not respond to the question...

you have to be [expletive] kidding me. tarski has been patient in explaining your errors, which i attribute to his many years of teaching. clearly, your invincible ignorance marks you as one of these.

[_marg]

CC please, explain to me, in your own words, what is Godels' God.

[_Tarski]

Jak,

You are starting to repeat yourself and yet you are clearly not really reading and thinking about what I write.
I have addressed the point about connotations of the word God in the coinage God-like.

Use of mathematical examples are perfect for the task because here we have a fertile ground for formal deductive structures. The whole point about a formal deductive argument is that its validity depends only on its form (while soundness depends also on the truth of the axioms).

Let me try one last time.

First, Godel defines a term. He is not simply applying a common term or trying to merely expand on a common notion.
He does not use in his argument anything beyond what is in the definition. Can you pinpoint exactly where you think he does that? One sentence should do.

In particular, he doesn't insert anything that people normally associate with the word God but sticks to his own definition. Show otherwise if you can.

The formal argument is unaltered if he replaces the word god-like with dog-like.

Indeed, that is exactly one of the essential features of an argument that can be expressed in a predicate calculus.

However, he does use the word God in his coinage "God-like" for a reason-- just not the reason you think. He uses it because he feels, and so do I, that his definition does define something which is intuitively fitting. He hopes not only to show that something exists which fits his definition, but eventually he hopes that the properties which adhere logically to his defined notion will answer to the readers intuitions about God.
But as far as the validity of the formal argument goes, the choice of the word "God-like" is utterly irrelevant. Just as is the case with the definition of the word smooth in mathematics. The word smooth has a precise meaning that would be the same and would not affect conclusions if it were called something else. Indeed, it is sometimes called C-infinity. But also the word smooth is used because of the intuitive connection between the ordinary use and the precise definition given. Just as with Godel's God-like, the proofs that use the notion of smoothness do not import inappropriate consequences that derive from mere connotations.

So the reasons Godel used "G-o-d" in "God-like" is the same as the reason that Whitney (or whoever) chose the word smooth in the definition of smooth manifold and in both cases the connotations do not interfere with or enter the logic at all! (noticed that it is right here that I just addressed the question about word choice that you claim I don't address)

Godel clearly hopes that his readers will agree that definition he gives for God-like are intuitive and that the properties he shows follow from the definitions and axioms will be worthy of his choice of words. He does not however, try do do anything but prove that something satisfying definition 1 must exist (assuming the axioms of course). The reader is free to deny that such a thing should be worshipped or thought of as identical with a pregiven notion of God already found in society. The reader is also free to deny the truth of the axioms.

One thing he does not do is draw conclusions that derive inappropriately from any connotations that exceed his definition of the notion of God-like.

The one caveat is that the Definition 1 employs the notion of positiveness which is insufficiently clarified. Indeed, the plausibility of the axioms applying to something in real seems to depend on the connotations of that word. But that is an issue with the axioms not the formal validity of the argument.

Godel's argument fails but for much more subtle reasons than most of what you have suggested (e.g. def 1 doesn't assume the existence of God). This should not be surprising since Godel is perhaps the greatest mathematical logician of modern times. Your wildly ungracious and flippant reading of him is annoying.

Finally, let me address that mathematics has nothing to do with theology. Whether this is true or not depends on one's approach to theology. Since Godel is a neoplatonist it clearly has something to do with his theology.

It is now clear that you, JAK, are simply garnering pseudo-understanding about logic from googling etc.
You give a rather clear example of why real training and maturity in an exact science such as logic and/or mathematics is needed for an understanding of Godel.
Your analysis sounds quite inauthentic and sophomoric. One clearly senses someone using words not deeply understood.
You also use a shotgun approach pointing the finger of fallacy of all types at every point of the argument hoping that something will stick.
It makes for tedious reading.

Gad was right after all.

[_marg]

Tarski,


Do these theorems make any sense to you and if so what are they saying? What does it mean for example to be exemplified? What does it mean for a property to be positive? And can you in all honesty make heads or tails of this argument. What on earth does is say about this God concept that is meaningful, adds new information beyond any def'n given? If it doesn't go beyond the def'n given it begs the question. i.e. All bachelors are single. Entailed in the def'n of bachelors is that they are all single. So no new information is given.

Theorem 1: If a property is positive, then it is consistent, I.e., possibly exemplified.

Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.

Theorem 3: Necessarily, the property of being God-like is exemplified.



Here is the whole argument in case you want to use it or review.


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

Definition 2: A is an essence of x iff for every property B, x has B necessarily iff A entails B

Definition 3: x necessarily exists iff every essence of x is necessarily exemplified

Axiom 1: If a property is positive, then its negation is not positive.

Axiom 2: Any property entailed by — I.e., strictly implied by — a positive property is positive

Axiom 3: The property of being God-like is positive

Axiom 4: If a property is positive, then it is necessarily positive

Axiom 5: Necessary existence is positive

Axiom 6: For any property P, if P is positive, then being necessarily P is positive.

Theorem 1: If a property is positive, then it is consistent, I.e., possibly exemplified.

Corollary 1: The property of being God-like is consistent.

Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.

Theorem 3: Necessarily, the property of being God-like is exemplified.

[_Gadianton]

After how many days now on this subject and Marg says,

Now I have not looked closely at Godel's argument

LOL!

And JAK, yeah, he finally signs on to wiki and presents an analysis, but maybe he'd be better off going back to his role of quote-mining anything on the internet that's of no relevance to the discussion?

I just want to chime in on the objection to the term "God-like" as it appears in the proof. First of all, Tarski rightly observes it's just a coinage. It could be "dog-like".

JAK
Why use “God-like” and not some other “coinage” which would have no generally recognized theological link?

Well, it is a restatement of the ontological argument and all, but let's call it "JAK-like", for now. Nothing will be lost, as Tarski rightfully observes. But Tarski's mundane observations are JAK's intellectual Mt. Everests.

The use of “God-like” refers to what? Since it follows a linking verb, it’s a noun or a predicate adjective (grammatically). The definition appears to lack clarity and the author of that “need(s) to explain better.” Neither you nor I have the burden of responsibility for the clarity of someone else’s definition.

JAK, all flustered over grammar, doesn't realize that he's attacking exactly the wrong part of Godel's argument. The fact that Godel coined the term "God-like" for a definition is of absolutely no consequence to the argument. (One more brief comment before continuing, whose burden is it to clarify definitions when the proof was published posthumanously, as clearly stated right at the top of the wiki entry?)

To understand why the coinage is irrelevant, someone needs to learn the difference between a definition and an axiom. "God-like" isn't used in one of Godel's assumptions. The name used for a definition is trivial. What isn't trivial are the assumptions, but JAK, seeing nothing more than words, just goes right for the first instance of the word "God" and starts launching his attack. So I will substitute "JAK-like". It isn't possible for Godel to buy anymore with one word over another when coining a mere definition.

Now let's restate wiki's definition with our new term:

Now we define a new property J: if x is an object in some possible world, then J(x) is true if and only if P(x) is true in that same world for all positive properties P. J is called the "JAK-like" property. An object x that has the JAK-like property is called JAK.

So, the property of "JAK-like" holds for an entity in a possible world which has all positive properties in that world. The only ambiguity here is "positive property" (--HINT JAK AND MARG, THIS IS WHAT YOU SHOULD BE ATTACKING NOT THE GRAMMATICAL IMPLICATIONS OF "GOD-LIKE"--)

There apparently isn't a lot to go on for a definition of positive property save:

"Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)... It may also mean pure attribution as opposed to privation (or containing privation)."

That's not going to be easy to unpack. But, this might help:

http://www.stats.uwaterloo.ca/~cgsmall/ontology2.html

In "moral aesthetic sense" we can consider virtues, kindness, and so on. For the other "pure attribution", the easiste example from that link, "One might argue that F="knows the capitals of all the states of the United States" is a form of pure attribution. " But it's pretty easy to get the idea Godel is going after. I might have the positive property of knowing all the capitals. But I don't have the positive property of knowing all the capitals in any other country. So since I'm not an entity in a possible world that has all positive properties, I don't have the property of being JAK-like.

Now what's the big deal over the string of letters g-o-d? If it's proven that "JAK" exists, then the discover is just as profound. But since there is a JAK in the real world who posesses all positive properties in Marg's estimation but no one elses, then there might be a better term just to throw in for clarity sake, even if it doesn't affect the proof one bit. Hey, I know, a lot of people believe in a supreme being that has all these virtues and knows everythings, so let's try "God".

Ahh, there you go JAK. Now you can change your argument to be that Godel assumed God in his axioms rather than his definition which is a coinage for an inference drawn from his first axioms.

But he still didn't beg the question, because as you can clearly see, the existence of God in a possible world is not assumed anywhere in that proof.

[_JAK]

Ambiguity and Central Questions of Discussion

Quote JAK:
“Definition 1: x is God-like iff x has as essential properties those and only those properties which are positive” is ambiguous with “iff” (if and only if) followed by terms which lack clarity. I think it reasonable for us to demand the highest standards of clarity and precision. If this construction (as represented by CC) were specified as “merely a technical definition,” that might be an improvement. It wasn’t. Worse, God in any time of the past many centuries has cultural connotations as does “God-like.” Why use it if the intent were as you claim?

Tarski:
Why use the word “smooth” to describe infinitely differentiable? That word has connotations too. One may not use such connotations in any proof but once one gets a feel for the concept one sees why the word choice was made.

JAK:
Fallacy: Shift the Debate

To imply by this reference that “God-like” is on a par with “smooth” attempts to shift the debate. It’s also Fallacy: Debate by Question. It fails to address the issue before us. Why “God-like” in the presentation CC replicated.

Quote JAK:
If the intent mathematically were to “merely” as Tarski speculates, some clearly meaningless, nonsense phrase or group of letters would have been superior to “God-like.” So why was “God-like” used?

Tarski:
I am not speculating and I explained that above.

JAK:
Incorrect. You have not explained. “Merely” is a judgmental term as to minimize the fact of the word usage. Repetition of your claim is not explanation.

Quote JAK:
Now, it’s an even greater stretch to defend that the author is: “merely making a technical definition...”

Tarski:
Far from being a stretch, it is obvious. I deal with these kinds of definitions everyday and have done so professionally for 30 years. Perhaps you are confused by what Imean by technical definition. He is defining a new word. It's coinage and has only the meaning given in the definition.

JAK:
Congratulations on your professional work. Back to the subject at hand. You continue to Beg the Question of Why such a person would choose a reference “God-like” if his intent were “coinage.” Godel could hardly have been ignorant of the religious/theological implications of “God” or “God-like.” It is indeed “a stretch” to conclude that an educated man would by choice use a religiously/theologically loaded term when his intent was merely definition or merely coinage as you claim.

I have no objection per se to “coinage.” I understand and would accept your use of that characterization in general as one might use a neutral term for purposes of presentation of an argument.

In this case, it’s deceptive here and “a ruse” as one critic stated and “a sleight of hand” as a critic stated. I agree with them on that in this specific point. What was needed was a neutral word or phrase to keep the focus on the construction.

Are you prepared to claim that “God-like” and “God” are equal in general use to a non-religious reference? Your examples as well as mine would have removed doubt that no reference was intended to connect with what people generally consider when they hear “God-like” or “God.”

Tarski, you have no refutation to this. And in the other parts of the construction we have phrases like “properties which are positive.”

Axiom 3 “The property of being God-like is positive.”
Corollary 1 “The property of being God-like is consistent.”

The entire construction fits with a general religious bias regarding “God-like” or “God.”

Your claim for “merely definition” or “coinage” in the broader context of the entire quotation from CC is not palpable.

“God-like” is not a new word. “God” is not a new word.

If Godel was as informed as it appears he was, the word-choice was known to him as one of religious import.

You have not made the case that “God-like” or “God” is neutral for purposes of mere definition in the construction CC posted.

So, why was “God-like” used? That’s the issue.

See who likes Godel in Christian History Institute.

From that group, we read:

"Godel's proof fits well with Christian beliefs about the universe, by analogy. Judeo-Christianity has long held that truth is above reason. Spiritual truth can be grasped only by the spirit. Had Godel been able to show that self-proof was possible, we would be in deep trouble. The universe might them be self-explanatory. The implication of his proof is that the infinities and paradoxes of nature demand something higher, different in kind, more powerful, to explain them, just as every logic set needs a higher logic to prove and explain everything within it.

In other words, no finite system, even one as vast as the universe, can satisfy the questions it raises. Godel recognized this and tried to find a watertight proof of God's existence. He failed. Sadly, the evidence of his life suggests he failed to find a personal relationship with Christ either."

The bold is mine for focus. Particularly the last statement from this clearly pro-Christian group strongly suggests that Godel was trying “to find a watertight proof of God’s existence.”

Godel’s construction strongly suggests that he was not “merely defining” or that his word-choice was “ just coinage” only.


Tarski stated:
Perhaps you would prefer the following formulation:
http://upload.wikimedia.org/math/3/2/3/ ... 788b38.png

JAK:
Please explain as clearly as you can just how this reference addresses: “The Roles of Logic and Science in Questions of Theology.”

That’s the issue before us. Detail as precisely as you can just how your reference addresses the topic.


(I repeat)

Quote JAK:
Now, it’s an even greater stretch to defend that the author is: “merely making a technical definition...”
Do words have meaning?

Tarski:
Yes and in Godel’s argument God-like has the meaning he gave it. It’s a freaking definition.

JAK:
You continue to beg the question: Why did Godel deliberately choose this religious/theological phrase for presentation?

I’ll be looking for your detailed analysis of the website you gave as you relate it to the topic here which includes Godel’s word selection.

Now, how do you distinguish “merely definition” from “freaking definition”? I should like a person of your mathematical stature (as you have so stated) to make a clear, transparent distinction between “merely definition” and “freaking definition” both of which you use in this discussion.

In doing so, please relate the discussion to the topic: “The Roles of Logic and Science in Questions of Theology.”

Quote JAK:
Do those who read words see them in context?

Tarski:
Well, in this case you do not seem to see them in context. The context is a formal proof where a formal definition is given and where the connotations one may bring are only relevant if Godel somewhere equivocates between the definition he gave and some other definition which derives from the ordinary use of the word God. You haven’t shown where he does that. I don’t think he does, and that is not the problem with the argument.

JAK:
I see you continue to evade the issue of why a word choice which has clear religious/theological context. Why do you evade the issue I ask?

Ah, there is “the ordinary use of the word God.”Exactly my point. Not only “ordinary” but I would argue extraordinary, as that word has a plethora of meanings and applications in use.

The “definition" he (Godel) gives is ambiguous, “fuzzy” (as a critic observed).

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

In this construction we have no clear, transparent definition for “properties” and no clear transparent definition for “positive.” Hence, how is this definition of any use or benefit?

“God-like” has conotations in virtually all “ordinary use” (your phrase). “Positive” is relative to perception and perspective. Here used it’s “fuzzy” and ambiguous. Having shown the failure for clear, transparent definitions of terms used and their ambiguity, Godel or anyone else can super-impose any vague meaning desired.

That’s sufficient flaw. The burden of proof which includes clarity of statement lies with Godel (or CC’s replication). Godel doesn’t meet it. We have Fallacy: Ambiguous definition.

Tarski states:
“...and that is not the problem with the argument.”

JAK:
I have no objection to additional problems in the construction. But it’s a major problem as I have detailed.

And, in no way have you refuted the analysis.

Tarski stated:
Perhaps you would prefer the following formulation:
http://upload.wikimedia.org/math/3/2/3/ ... 788b38.png

JAK:
I’ll be waiting for you to apply this directly to the topic:
“The Roles of Logic and Science in Questions of Theology.”

Quote JAK:
Why was one of your words or my “791” not used? To pretend that “God-like” has no connotation and that we are “merely” discussing an argument form is misleading. Why was it done?

Tarski:
I already explained that. There are many more examples from math. The meaning of irreducible in ring theory does not appeal to any preexisting connotations and no one ever imports those connotations. However, again, when you see the definition and its consequences, you also see that those connotations are at least apropos.

JAK:
No, you haven’t. Not in the least did you answer the questions. Now you continue to Beg the Question by introducing extraneous material.

Not a thing in your response here is responsive to Why “God-like” was selected for the construction. You have offered no refutation to the argument that “God-like” is an ambiguous, religious/theological term of reference. You have given no answer to Why Godel used such terminology.

Understand that I am not challenging the importance of definition or of using a reference like “x” to stand in a definition. “x” is neutral as might be any other letter (as your examples noted).

Because of society’s major use of a term like “God” or “God-like” with all the emotional, religious, theological baggage which those terms have -- Why contaminate a mathematical construction with that?

My reference above suggests why Godel might have used "God-like" and "God" and "properties which are positive" and "the property of being God-like is positive" and "the property of being God-like is consistent, i.e. possibly exemplified" and "If something is God-like, then the property of being God-like is an essence of that thing."

You don’t address the issue of word choice in the construction beyond your claim "merely definition" and "coinage." I have.

You don’t address the larger issue of the question: “The Roles of Logic and Science in Questions of Theology.” Due to your lengthy diversion and my addressing of that, I also have not addressed (in later posts) the fundamental question at the beginning of this thread.

JAK

[_Tarski]

Ambiguity and Central Questions of Discussion



Tarski:
Why use the word “smooth” to describe infinitely differentiable? That word has connotations too. One may not use such connotations in any proof but once one gets a feel for the concept one sees why the word choice was made.

[color=brown]JAK:
Fallacy: Shift the Debate

LOL!
The very first thing you write is wrong. The rest is just as bad. You base your "understanding" on how the words or names of the fallacies sound to you or what impression they give. (No wonder you are confused about god-like. You aren't paying attention to the definition). You cut and paste other peoples words for purposes that don't fit and think you are making sense.

I used a strict analogy to answer the very question you keep bringing up and you call it "shifting the debate".
Absolutely hilarious.

JAK, you absolutely do not know how to correctly identify relevant fallacies. The jig is up.



You are the kind of person it is useless to debate. A person armed with the ability to cut and paste and access the wiki and google. I bet you would doggedly debate quantum field theory with me too wouldn't you.
However, I would do it in a heartbeat if it were in person and in front of a panel of judges with Ph.D.s in logic or philosophy.

Too bad Godel isn't here so you can "correct" him too.

[_Tarski]

Tarski,


Do these theorems make any sense to you and if so what are they saying?

They make formal sense. Expressed symbolically in a predicate calculus it makes sense and looks like it might be valid (though not sound as we will see)
But the connection with what is real and uncontroversial does not make sense to me. The most controversial element is the notion of "positive".

What does it mean for example to be exemplified? What does it mean for a property to be positive? And can you in all honesty make heads or tails of this argument. What on earth does is say about this God concept that is meaningful, adds new information beyond any def'n given? If it doesn't go beyond the def'n given it begs the question. I.e. All bachelors are single. Entailed in the def'n of bachelors is that they are all single. So no new information is given.

Marg,
All of mathematics follows from the definitions and axioms. Yet, who would claim that no new information was provided when Andrew Wiles finally proved Fermat's theorem (after mathematcians tried for 200 hundred years)?? Who wuld claim that the Atiya singer index theorem does not reveal new information?

Theorem 1: If a property is positive, then it is consistent, I.e., possibly exemplified.

The property of being a round square is inconsistent since there can not exist such in any possible logical world. A property is exemplified if there is an entity with that property. Primeness is exemplified in 7.
Whether something is consistent or not is not usual immediately apparent as in my example. Often it takes work to show it and sometimes the issue of consistency is never settled. My definition of ghost-like manifold is not exemplified and cannot be exemplified since it is inconsistent. But it takes a big and difficult theorem to prove this.
A similar notion is that of a model for an axiomatic system. Hyperbolic geometry has a differential geometric model and so we know it is consistent (or as consistent as the real number system).

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

This definition does not beg the question of God's existence but it does have another more subtle problem. Namely, it appeals to a notion of positiveness that has not been clarified or shown to be exemplified.

JAK just scanned through and looked for the letters g-o-d and assumed that the question was begged at every such point. But he misunderstands what is going on as I hope you now see.

The theorems cannot be expected to be shown to be true unless the truth of the axioms is assumed or self evident.
We all understood this to varying degrees right from the start (at least I hope).
To me the axioms are not self evident and do not clearly map onto anything familiar the way the axioms of arithmetic do. But one should not be too hasty. Perhaps some separate intuitive arguments can be given.

My initial impression on the axioms is as follows:

Axiom 3: The property of being God-like is positive

This axiom is not self evident even if one uses some tentative familiar notion of "positive". Why should it be a "good" thing to necessarily exist? Perhaps existing contingently is more desirable. Who is to say what is positive?

Axiom 4: If a property is positive, then it is necessarily positive

Once again, there is too much packed into the axiom. Axioms should be simple and hopefully self evident. This axiom assumes that whatever positiveness is, it cannot be relative to the context or a matter of opinion. A moral relativist would have no truck with this axiom.

Axiom 5: Necessary existence is positive

I can not see why I should accept this axiom either. Does it seem self evident? Once again, the notion of positiveness needs to be clarified.

Without plausibility arguments for the axioms and a solid agreed upon notion of positive that agrees with its use in the axioms, the formal validity of the argument is not very impressive. Truth has not been established since we cannot see that the argument is sound.

On the other hand, we are talking about Godel, so to be fair, one should search the literature to find out to what extent Godel did try to clarify positivity (he did say something) and to what extent he was able to argue the plausibility of the axioms.

The version we are reading is actually not Godel's formulation but is rather someone else's attempt to do justice to what Godel might have had in mind. Other versions may be more or less convincing.

Finally, you are correct to insist that the formal notions in the argument must correspond to something real in some sense. Positivity must be real. This is much easier if one has a prior commitment to neoplatonism as Godel did.

If one is a strict empiricist regarding what is real, or if one gives no ontological weight to mathematical and logical entities, then one would have to find a notion of positive that applies to things you consider real.
Just ask yourself if you think that there is an objective notion of positive in the sense of being "good" or "desirable".

I don't think there is such an absolute notion. To prove the existence of such would be another task like proving the existence of God.


But, if there were it would have a platonic existence not a physical existence.
In the end Godel's god would likely turn out to have the same reality as the integers (not real to you but real to Godel, Penrose and a few other familiar names).

Finally, notice that Godel asserts that necessary existence is positive. Thus we are treating existence as a property and this is the weak spot in the more traditional ontological arguments.

In anycase, you may rest assured that Godel's argument does not present a real challenge to atheism as we normally think of it.
Even to Godel the case is not closed. He fully realizes the burdens regarding the axioms etc that I have mentioned above.

The bottom line is that Godel argument seems to need platonism to be true and showing that is akin to (but not the same as) showing the existence of God in the first place.