Wednesday, November 12, 2014

A Short Analysis of Gödel's Ontological Argument

Kurt Gödel developed a thought provoking axiomatic version of the Ontological argument. Here is a short summary:  

I like analyzing abstract philosophical/logical arguments.  I do see a few potential issues with Gödel's Ontological proof. Here is a short analysis of the proof :)

1. For the backbone of Gödel's Ontological argument, it looks like Gödel constructs a set of 'positive' attributes, and then includes the concept of  existence in the set (with the assumption that necessary existence is a positive attribute).  For example, he constructs a set like: 

G1 = { p1, p2, p3, ... , E }

In this case, p1, p2, p3 are 'positive' essential attributes, and E is the idea of existence. 

If I try to imagine the specifics of what the set contains, a random set of ‘postive’ things might look something like this to me:

G2 = { 'shoes', 'chewing gum', 'paper  clips', 'x-ray vision', 'invisibility', ..., 'the idea of existence' }.  

Here's the issue.  If a set contains 'the idea of existence' as a member, it doesn't imply the entire set of attributes exists, or even that each member of the set  exists or interacts with other members of the set.  

Otherwise, consider applying a similar argument to a subset.  

Define: Superman is a man, such that no greater man can be conceived. He’s basically a watered-down version of all essential attributes in set G1, except as applied to a flesh-and-blood two-legged man with a weakness to Kryptonite.

Such a set of essential attributes may include:

S1 = { 'super strength', ‘wisdom’, 'two legs', 'cape', 'great smile', 'laser vision', 'can fly', ...  }.  

However we can add the idea of 'exists' to the set, where 

S2 = { S1, E }

Then, S2 is greater and more powerful than S1.  So, by similar reasoning, Superman exists.  

Because, between the two, “real Superman” would win any fight, because “fake Superman” can never show up. Superman's existence is an essential attribute to the concept of superman. If he didn't exist in the comic books, he would not be Superman. :)

2. A potential issue is the subjective notion of 'positive'.  Gödel seems to view this as a binary property (something is either positive or not positive).  But in reality, this is usually quite a bit messier.  Positive and negative attributes  are often two sides of the same coin, and they cannot always be separated. 

For example is 'knowledge' intrinsically good?  What is better, an evil genius, or someone who wants to cause harm but lacks knowledge to do so? (imma looking at the "underwear bomber")  

Similarly, 'existence' in itself is neither good or bad intrinsically (consider the idea of ebola in one's mind, vs real ebola virus in one's mind). :)  

In general, I think when assigning 'positive' or 'negative' values, usually some thing or action must be combined with intent, and then potential   effects are weighed as 'desirable' or 'not desirable' by those affected.  For example, people might even view ebola as a ‘good’ thing if it produced effects that people liked and could control (like curing cancer).

3. A more general issue with an Ontological-style proof is 'greater' is a comparison operator applied to an ordered set.  But for example, consider defining N to be the largest number, such that no larger number can be conceived.  For an infinite (unbounded) set,  this definition itself may not be consistent.  We can always add 'one' to make the number larger.  Similarly, we can always form composite sets to create a larger set, or add new attributes.  Or simply double the measure of all attributes. While Gödel's version avoids the direct issue, it is still present.  It’s not enough to say an attribute is included in a set, the attribute will have a measure.  For example, regarding an attribute like 'knowledge', how many primes can be known, if there cannot be a largest prime?

4. The notion of 'necessary existence' is a bit puzzling.  Clearly one may imagine a possible world in which no life at all exists.  As a quick proof, consider a possible world that consisted of nothing but a null set.  To understand existence in modal logic then, we may treat each axiom introduced as excluding the set of possible worlds where such an axiom is false.  For example, if we introduce an axiom, ‘the sun is necessarily yellow’, then the sun is necessarily yellow in all possible worlds (as we exclude possible worlds where the sun is blue by assumption).  By the same token, if we introduce an axiom `the sun is necessarily blue’ then the sun is blue in all possible worlds.  Therefore the subtle point here is that with modal logic, the actual world does not necessarily belong to the set of possible worlds that survive the cut by the axioms introduced. This probably should be added as a disclaimer.

5. However, what is very interesting about Gödel's argument is it brings out a point about broader limitations of math and logic.  Math is the study of patterns. Logic is the method by which information can be rearranged, cross-checked, and discarded.  Math/logic in itself cannot produce apriori truths about reality.  Conclusions in pure logic can only rearrange the information already contained in the premises.

No comments: