Gödel’s theorem on incompleteness has been discussed in several fields like Physics and Religion.
I read the following two articles:
Here are some thoughts.
*
A point mentioned in Hawking's article was about angels viewing the universe from the outside. Also Marshall talks about God being outside the "biggest circle..".
Based on what I understand about Islamic theology, angels are part of "Creation", sometimes the word "World" is used also to distinguish "all created things" from the "Creator of all things". And even after that distinction, while it is common to see in theology books a phrase like "God has complete knowledge about Creation", a phrase like "God views the universe from the outside" is not used. The reason as I understand is that "outside" and "inside" are used when talking about relationships between things within creation. Such words apply to someone or something within dimensions, and dimensions are part of creation (further information here).
Anyway, Hawking's phrase appears to be about "Complete Knowledge" regarding the universe we are in, knowledge without the problem of self-referencing that seems to arise from being part of the universe we are trying to describe.
I guess, everyone knows a certain amount of information about the universe. People differ in how much and how clear they know about something. But there are limits nonetheless.
If limits of man's knowledge are recognized regarding information about existence, and man realizes that there is always information beyond his reach, it would be a valid question then if one asks "Who knows every single piece of information about the universe?". I'm not sure but it seems that this idea was behind Hawking's interesting remark about "angels outside the universe".
Another point that caught my attention was regarding man and his models being part of the universe man describes, and how that reflects self referencing, "like" in Godel's theorem, and therefore any such attempt to describe the universe is either inconsistent or incomplete. It reminded me of what some Sufis have been saying for a long time about the inherent shortcomings in man's attempt to use logic in matters of faith, like trying to logically prove God's existence using man's logical criteria (more information here).
*
As I understand the way Logic is seen in Kalam:
1) I do not get the impression that there is something like theory and reality (as in modern science) in Kalam. The arguments are about reality and not about theories. For instance one of the arguments for God's existence starts from stating one's own existence (not a hypothetical assumption, since one's existence is "instinctively" given) and the nature of one's existence (temporary, prone to changes, etc., again not a hypothetical assumption, rather a description of the nature of one's existence).
2) Faith comes first and before any attempt to "present" elements of faith using Logic. Historically, using Logic came much later to help "organize" thoughts about the Islamic view of reality. It is that view that is the essence of Islamic faith.
*
Basic assumptions are essential in scientific theories and they are taken as given. But what I have understood is that such assumptions are not seen as fixed and unalterable.
Euclid’s axioms might be taken as given in a regular high school geometry book, but not when one considers other geometries. The axioms are relevant only to a specific and theoretic view of the real world which is Euclid’s view. The axioms are not treated as absolutely true, rather as true for the theoretical framework of Euclidean geometry. “True” here is relative and not absolute.
However, in the real world, in scientific research, assumptions are used that are not proved, and need to be taken as given if any sensible discussion is to be considered in a certain field of science. In itself, I think this is an interesting point, well known, and accepted in the scientific community, without the need to bring up Gödel’s theorem. Trying to link it to what Gödel had in mind might not be straightforward or convincing to everyone. Gödel's theorem is probably one possible "formal" way of indicating the shortcomings that are already felt and acknowledged by scientists. So, I guess it can be said, from a modern scientific point of view, that there are practical and theoretical indications of shortcomings in man's grasp of reality.
Such a view is probably best presented by a prominent scientist like Hawking talking about how the situation in a field like Physics is "very reminiscent" of and might look "like" what Gödel was talking about.
*
Based on what I understand about modern science, moving from a purely theoretical ground to a real world application and evaluation of theories happens in a very careful manner.
Scientists distinguish between the simplified and abstract world of theory on one hand and the real world on the other. They try to make it clear that there is a gap between the two. If there was not such a gap, then the "theoretical" side would have been identical to the "applied", and no need to separate the Mathematical aspect from the Statistical aspect, but that is not and never was the case.
Each has its role and both are needed. Theory guides application, and application tells us how good a theory is. There is an understanding that theories will always fall short of describing the real world as it is. They are not perfect. And there is also an understanding that application and testing and evaluating procedures are not perfect either.
I would imagine that the situation is even more abstract in pure math, and more caution and care are needed if pure mathematical points are considered in relation to the real world.
Scientists are usually careful about the limits of what can be said and how it is phrased, and what cannot be said about a given issue from a strictly scientific position.
*
It is obvious that Religion is based on convictions, and God’s existence is the first and most important one. A ‘religious conviction’ is ‘A firmly held belief.’
I don't see Hawking’s point of view phrased as a conviction. Typical of a scientific expression.
I recall a statement that I read back in college in a textbook on Scientific Methods. It said ‘There are no convictions in Science’.
That is one clear difference. It is hard to ignore and I often wonder how it can be satisfactorily bridged these days. (more on this point here).
As I understand, Gödel and Turing showed that there is no such thing as a solid basis for Mathematics and Logic (of course, saying that something is not perfect does not mean saying it is not very useful). Seems like the better man understands the world and the nature of his knowledge, the more clear become the limits of his understanding and the uncertainties about reality as it is. Not news to Sufis who have been talking about the inherent weakness in man's "reasoning". However, as I understand that view, it is important to recognize the weakness and limits of man's reasoning and yet not to disregard reasoning at the same time (more here).
It is the same "reality" that concerns religion, but from a different angle. An angle that takes a final answer with complete certainty, regarding the nature of "reality". It is a view, in which acknowledging the inherent weakness in man's reasoning is nothing to be surprised about, rather it is important for a beginning to consider an awareness beyond what reasoning can reach. It is a view in which the "mind" level of awareness is only one among several levels of awareness, there are lesser and further levels (more here).
*
I try sometimes to discuss religious concepts in a way that might be relevant to most people. And indeed the "scientific way" is very appealing. With all the achievements of science, it might be that talking in a scientific way is convincing at least to some people.
But while it is possible to be overwhelmed by scientific theories, methods and achievements, many may not know how scientists are very aware of limitations at every step they take.
Modern science is doing fine. The scientific process is designed in a way that proved to be working. However, it is designed for a specific purpose. If an application of the scientific approach is considered that is different from what the approach was designed for, then there might be a good chance that one would find that approach to have limits "by design".
Personally, I did find myself trying to use modern scientific "molds" to express what I felt would not conveniently fit in those molds. And I often thought whether it might be better to recognize the difference and appreciate each field for what it is, and be careful when trying to talk about a possible link.
Many ancient scientists were experts in theology/metaphysics at the same time. It seemed so natural. Religion and Science were close and moving from one to the other appeared to be smooth. It was easier for them to discuss matters of religion and science. There was a suitable structure for handling such discussions. It is not the same these days, as far as I can tell. That is why I feel one needs to be careful when trying to link between the two views of existence.
Existence is one and the same regardless of how one selects to look at it, for instance from a purely scientific view or a purely religious view. And eventually, a unified view is not impossible.
I read the following two articles:
- “Gödel’s Incompleteness: The #1 Mathematical Breakthrough of the 20th Century” , by Perry Marshall.
- “Godel and the End of Physics”, by Stephen Hawking.
Here are some thoughts.
*
A point mentioned in Hawking's article was about angels viewing the universe from the outside. Also Marshall talks about God being outside the "biggest circle..".
Based on what I understand about Islamic theology, angels are part of "Creation", sometimes the word "World" is used also to distinguish "all created things" from the "Creator of all things". And even after that distinction, while it is common to see in theology books a phrase like "God has complete knowledge about Creation", a phrase like "God views the universe from the outside" is not used. The reason as I understand is that "outside" and "inside" are used when talking about relationships between things within creation. Such words apply to someone or something within dimensions, and dimensions are part of creation (further information here).
Anyway, Hawking's phrase appears to be about "Complete Knowledge" regarding the universe we are in, knowledge without the problem of self-referencing that seems to arise from being part of the universe we are trying to describe.
I guess, everyone knows a certain amount of information about the universe. People differ in how much and how clear they know about something. But there are limits nonetheless.
If limits of man's knowledge are recognized regarding information about existence, and man realizes that there is always information beyond his reach, it would be a valid question then if one asks "Who knows every single piece of information about the universe?". I'm not sure but it seems that this idea was behind Hawking's interesting remark about "angels outside the universe".
Another point that caught my attention was regarding man and his models being part of the universe man describes, and how that reflects self referencing, "like" in Godel's theorem, and therefore any such attempt to describe the universe is either inconsistent or incomplete. It reminded me of what some Sufis have been saying for a long time about the inherent shortcomings in man's attempt to use logic in matters of faith, like trying to logically prove God's existence using man's logical criteria (more information here).
*
As I understand the way Logic is seen in Kalam:
1) I do not get the impression that there is something like theory and reality (as in modern science) in Kalam. The arguments are about reality and not about theories. For instance one of the arguments for God's existence starts from stating one's own existence (not a hypothetical assumption, since one's existence is "instinctively" given) and the nature of one's existence (temporary, prone to changes, etc., again not a hypothetical assumption, rather a description of the nature of one's existence).
2) Faith comes first and before any attempt to "present" elements of faith using Logic. Historically, using Logic came much later to help "organize" thoughts about the Islamic view of reality. It is that view that is the essence of Islamic faith.
*
Basic assumptions are essential in scientific theories and they are taken as given. But what I have understood is that such assumptions are not seen as fixed and unalterable.
Euclid’s axioms might be taken as given in a regular high school geometry book, but not when one considers other geometries. The axioms are relevant only to a specific and theoretic view of the real world which is Euclid’s view. The axioms are not treated as absolutely true, rather as true for the theoretical framework of Euclidean geometry. “True” here is relative and not absolute.
However, in the real world, in scientific research, assumptions are used that are not proved, and need to be taken as given if any sensible discussion is to be considered in a certain field of science. In itself, I think this is an interesting point, well known, and accepted in the scientific community, without the need to bring up Gödel’s theorem. Trying to link it to what Gödel had in mind might not be straightforward or convincing to everyone. Gödel's theorem is probably one possible "formal" way of indicating the shortcomings that are already felt and acknowledged by scientists. So, I guess it can be said, from a modern scientific point of view, that there are practical and theoretical indications of shortcomings in man's grasp of reality.
Such a view is probably best presented by a prominent scientist like Hawking talking about how the situation in a field like Physics is "very reminiscent" of and might look "like" what Gödel was talking about.
*
Based on what I understand about modern science, moving from a purely theoretical ground to a real world application and evaluation of theories happens in a very careful manner.
Scientists distinguish between the simplified and abstract world of theory on one hand and the real world on the other. They try to make it clear that there is a gap between the two. If there was not such a gap, then the "theoretical" side would have been identical to the "applied", and no need to separate the Mathematical aspect from the Statistical aspect, but that is not and never was the case.
Each has its role and both are needed. Theory guides application, and application tells us how good a theory is. There is an understanding that theories will always fall short of describing the real world as it is. They are not perfect. And there is also an understanding that application and testing and evaluating procedures are not perfect either.
I would imagine that the situation is even more abstract in pure math, and more caution and care are needed if pure mathematical points are considered in relation to the real world.
Scientists are usually careful about the limits of what can be said and how it is phrased, and what cannot be said about a given issue from a strictly scientific position.
*
It is obvious that Religion is based on convictions, and God’s existence is the first and most important one. A ‘religious conviction’ is ‘A firmly held belief.’
I don't see Hawking’s point of view phrased as a conviction. Typical of a scientific expression.
I recall a statement that I read back in college in a textbook on Scientific Methods. It said ‘There are no convictions in Science’.
That is one clear difference. It is hard to ignore and I often wonder how it can be satisfactorily bridged these days. (more on this point here).
As I understand, Gödel and Turing showed that there is no such thing as a solid basis for Mathematics and Logic (of course, saying that something is not perfect does not mean saying it is not very useful). Seems like the better man understands the world and the nature of his knowledge, the more clear become the limits of his understanding and the uncertainties about reality as it is. Not news to Sufis who have been talking about the inherent weakness in man's "reasoning". However, as I understand that view, it is important to recognize the weakness and limits of man's reasoning and yet not to disregard reasoning at the same time (more here).
It is the same "reality" that concerns religion, but from a different angle. An angle that takes a final answer with complete certainty, regarding the nature of "reality". It is a view, in which acknowledging the inherent weakness in man's reasoning is nothing to be surprised about, rather it is important for a beginning to consider an awareness beyond what reasoning can reach. It is a view in which the "mind" level of awareness is only one among several levels of awareness, there are lesser and further levels (more here).
*
I try sometimes to discuss religious concepts in a way that might be relevant to most people. And indeed the "scientific way" is very appealing. With all the achievements of science, it might be that talking in a scientific way is convincing at least to some people.
But while it is possible to be overwhelmed by scientific theories, methods and achievements, many may not know how scientists are very aware of limitations at every step they take.
Modern science is doing fine. The scientific process is designed in a way that proved to be working. However, it is designed for a specific purpose. If an application of the scientific approach is considered that is different from what the approach was designed for, then there might be a good chance that one would find that approach to have limits "by design".
Personally, I did find myself trying to use modern scientific "molds" to express what I felt would not conveniently fit in those molds. And I often thought whether it might be better to recognize the difference and appreciate each field for what it is, and be careful when trying to talk about a possible link.
Many ancient scientists were experts in theology/metaphysics at the same time. It seemed so natural. Religion and Science were close and moving from one to the other appeared to be smooth. It was easier for them to discuss matters of religion and science. There was a suitable structure for handling such discussions. It is not the same these days, as far as I can tell. That is why I feel one needs to be careful when trying to link between the two views of existence.
Existence is one and the same regardless of how one selects to look at it, for instance from a purely scientific view or a purely religious view. And eventually, a unified view is not impossible.
