A mathematical perspective on the phenomenon of "miracles"

[godelian]

Here's a paper that might upset your sensibilities: https://www.ams.org/journals/bull/2017- ... 1556-4.pdf

Interesting paper!

Very subtle, though.

He uses the example of Russell's paradox to demonstrate that the proof seemingly uses the law of the excluded middle but is still constructive. I still don't get the point completely, though.

[Eodnhoj7]

Say that we accept a fragment of the following definition for the term "miracle":

an extraordinary and welcome event that is not explicable by natural or scientific laws and is therefore attributed to a divine agency

.
I will simplify it to the following fragment:

an extraordinary event that is not explicable by natural or scientific laws

.
By the way, I do not see why it needs to be a "welcome" event. The notion of "welcome" event depends too much on the interpretation by the observer to be useful in our analysis. To the one observer the event could be welcome but to the other observer, it is possibly not. I will also not discuss what the final origin or cause is for this event, because that does not contribute much to the analysis either.

Let us first use the natural numbers as our model instead of using the physical universe.

(We will be able to extrapolate back to the physical universe later on.)

The question then becomes: Are there true statements about the natural numbers that cannot be explained, i.e. proven from the laws that govern them, i.e. arithmetic theory?

Yes, because that is exactly what Kurt Gödel managed to prove in his first incompleteness theorem:

There exist true statements about the natural numbers that are not provable from (Peano) arithmetic theory or there exist false statements that are provable (or both).

So, now the next question is of course: Does Gödel's incompleteness theorem apply to the physical universe?

The main problem in this question is that we do not have a copy of the Theory of Everything (ToE) which the physical universe would interpret as a model. We can obviously not prove the incompleteness theorem from an unknown theory. The late Stephen Hawking, however, believed that incompleteness does apply to the physical universe:

Godel and the End of Physics
What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

From this fragment, it is clear that that Stephen Hawking technically believed in miracles. So, we can conclude as following: If you believe that Gödel's incompleteness theorem is provable from the Theory of Everything (ToE), then you effectively believe in miracles. By the way, this is equivalent to claiming that the ToE contains a copy of Robinson's Q fragment of arithmetic theory.

Given interpretation is subject to the time and space in which it occurs there will always be something unexplainable that will eventually be unexplainable but in turn something else will be unexplainable...given enough time.

Miracles fall under the paradox of the forest and trees. See the individual tree and miss the forest, see the forest and miss the individual tree. The more precise one aspect of reality becomes the more ambiguous and obscure another.

Miracles are the inherent paradox of observation, they are a result of paradox.