A mathematical perspective on the phenomenon of "miracles"

[godelian]

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.

[Age]

Are there perceived to be any actual 'extraordinary, or ordinary, events that are not explicable by natural or scientific laws', by anyone here?

If yes, then will you list them here?

Also, what is 'it', or are 'they', exactly, that has been stopping you human beings from having come up with a TOE, yet, when this is being written?

[wtf]

Are there perceived to be any actual 'extraordinary, or ordinary, events that are not explicable by natural or scientific laws', by anyone here?

If yes, then will you list them here?

Consciousness. The existence of everything. To name but two.

Also, what is 'it', or are 'they', exactly, that has been stopping you human beings from having come up with a TOE, yet, when this is being written?

They're hard problems, likely outside the limits of science itself.

[Age]

Are there perceived to be any actual 'extraordinary, or ordinary, events that are not explicable by natural or scientific laws', by anyone here?

If yes, then will you list them here?

Consciousness. The existence of everything. To name but two.

But these two can be explained. Although one may be far simpler and easier than the other is to explain, and be understood.

The existence of everything, for example, is about one of the most simplest and easiest things to fully understand, and explain, whereas consciousness might take a bit longer to explain, and fully understand.

But both can be, and are already, explained and understood by so-called natural laws.

Also, what is 'it', or are 'they', exactly, that has been stopping you human beings from having come up with a TOE, yet, when this is being written?

They're hard problems, likely outside the limits of science itself.

But they are not hard at all. They are also, literally, not even 'problems' at all.

As can be explained, shown, revealed, and/understood.

Now, again, if absolutely anyone would like to discuss any thing here, the I am more than ready and willing to.

[Iwannaplato]

Another issue, in relation to natural laws (and constants) and miracles, is that the idea of natural law is a kind of projection of a legal idea into a timeless ontological category in the universe. We don't know if there are natural laws, or rather tendencies/habits/patterns that can change. It's a good working hypothesis to assume that patterns are not just local (in time or space) but it hasn't been confirmed in some final way. I realize that this isn't directly related to math, but perhaps indirectly. In math we decide the rules or starting points. With the universe this may or may not be a projection that they are there, permanent, universal and somehow protected from exception.

[Skepdick]

...

Firstly. I agree, but... this is a very very complicated way to get to your conclusion. But that's only true because the things which I accept as axiomatic are different to the things which you accept as axiomatic

Even though Gödel's work is historically prior to Turing, I consider Turing's work to be Mathematically prior. What I mean by that is the fact that Gödel' theorem is a trivial corollary of Turing's work.

Suppose we have the required formal system F (sound, complete, recursively-axiomatizable, <insert all the necessary and sufficient conditions here> ) which is powerful enough to reason about Turing machines and suppose we want to know whether a Turing machine M halts on a blank tape. T

Because F is complete either we'll find a proof that M halts; or a proof that M doesn't halt.
Because F is sound the proof would be true.
Thus F would be able to solve the halting problem.

Contradiction! Therefore F cannot exist.

There's a few extra steps to get to Rice's theorem but in general Logical undecidability is axiomatic.

Now if you look at all decision problems through Shannon's lens (information theory) an undecidable problem is one where we equate the probability of either answer.

e.g P(Yes) / P(No) = 1 = 0 Decibels ( https://en.wikipedia.org/wiki/Decibel )

And a "miracle" is nothing but a measure of great surprise. Which is tautological to saying "the current model predicted against this observation".

[Skepdick]

Another issue, in relation to natural laws (and constants) and miracles, is that the idea of natural law is a kind of projection of a legal idea into a timeless ontological category in the universe. We don't know if there are natural laws, or rather tendencies/habits/patterns that can change. It's a good working hypothesis to assume that patterns are not just local (in time or space) but it hasn't been confirmed in some final way. I realize that this isn't directly related to math, but perhaps indirectly. In math we decide the rules or starting points. With the universe this may or may not be a projection that they are there, permanent, universal and somehow protected from exception.

It's not even a legal idea. It's the idea that language is ontological and represents external reality.

It's the idea of logocentrism.

https://en.wikipedia.org/wiki/Logocentrism

That is the entire religion of Mathematics. We determine the truth of sentences as if they denote ontological truths.

What Mathematicians call "The Continuum" is what realists call "Reality" and what ontologists call "Ontology".

The perspective some Computer Scientists (myself included) have arrived at is that Mathematicians are never really talking about the continuum. They are talking about finite sequences of symbols that talk about continuums.

The same with the Realists.... They are never really talking about reality. They are talking about finite sequences of symbols that talk about Reality.

The same with Truth-seekers. They are never really talking about Truth. They are talking about finite sequences of symbols that talk about Truth.

Symbolism == Logocentrism == The Dead Dream of Representationalism.

[godelian]

Even though Gödel's work is historically prior to Turing, I consider Turing's work to be Mathematically prior. What I mean by that is the fact that Gödel' theorem is a trivial corollary of Turing's work.

Suppose we have the required formal system F (sound, complete, recursively-axiomatizable, <insert all the necessary and sufficient conditions here> )

It is possible to prove a "weak" version of the incompleteness theorem from the halting problem. It is weaker than the original theorem because F needs to be sound, which is a semantic notion. The strong form is purely syntactic:

https://en.m.wikipedia.org/wiki/Halting_problem

The concepts raised by Gödel's incompleteness theorems are very similar to those raised by the halting problem, and the proofs are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This weaker form differs from the standard statement of the incompleteness theorem by asserting that an axiomatization of the natural numbers that is both complete and sound is impossible. The "sound" part is the weakening: it means that we require the axiomatic system in question to prove only true statements about natural numbers. Since soundness implies consistency, this weaker form can be seen as a corollary of the strong form. It is important to observe that the statement of the standard form of Gödel's First Incompleteness Theorem is completely unconcerned with the truth value of a statement, but only concerns the issue of whether it is possible to find it through a mathematical proof.

[Skepdick]

Even though Gödel's work is historically prior to Turing, I consider Turing's work to be Mathematically prior. What I mean by that is the fact that Gödel' theorem is a trivial corollary of Turing's work.

Suppose we have the required formal system F (sound, complete, recursively-axiomatizable, <insert all the necessary and sufficient conditions here> )

It is possible to prove a "weak" version of the incompleteness theorem from the halting problem. It is weaker than the original theorem because F needs to be sound, which is a semantic notion. The strong form is purely syntactic:

https://en.m.wikipedia.org/wiki/Halting_problem

The concepts raised by Gödel's incompleteness theorems are very similar to those raised by the halting problem, and the proofs are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This weaker form differs from the standard statement of the incompleteness theorem by asserting that an axiomatization of the natural numbers that is both complete and sound is impossible. The "sound" part is the weakening: it means that we require the axiomatic system in question to prove only true statements about natural numbers. Since soundness implies consistency, this weaker form can be seen as a corollary of the strong form. It is important to observe that the statement of the standard form of Gödel's First Incompleteness Theorem is completely unconcerned with the truth value of a statement, but only concerns the issue of whether it is possible to find it through a mathematical proof.

This is incoherent. Syntax is meaningless without transformation rules.

Thus the infinite-blank tape problem being equivalent to the Halting problem.

A Mathematical proof IS a computer program. It's a symbol-rewriting system.
To "find" something is to perform a computational search.
What is the input to your search function?

https://en.wikipedia.org/wiki/Curry%E2% ... espondence

Soundness doesn't imply consistency. Soundness simply implies property preservation under continuous transformation.
Soundness in the property of a deductive system (which set theory does NOT have).

If a statement is provable then the statement is true in all models of the system - thus a tautology.

[godelian]

This is incoherent. Syntax is meaningless without transformation rules.

According to its formalist ontology, mathematics is indeed meaningless, and deliberately so:

https://en.m.wikipedia.org/wiki/Formali ... thematics)

According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all. Rather, mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics).

So, mathematics accepts a meaningless string as input and manipulates it into another meaningless string as output. It is a purely syntactic activity devoid of any possible semantics.

It is the job of downstream disciplines to deal with semantics when they use mathematical formalisms.

So, mathematics is purposely meaningless and therefore -- in and of itself -- useless. Its only redeeming quality is that it produces a surprising gap between input and output, and is therefore sufficiently ridiculous.

[Skepdick]

So, mathematics accepts a meaningless string as input and manipulates it into another meaningless string as output. It is a purely syntactic activity devoid from any possible semantics.

That which we call "formal semantics" is precisely the syntax manipulation and expression-evaluation rules.

It is the job of downstream disciplines to deal with semantics when they use mathematical formalisms.

So it's your job. You are, in fact, using a mathematical formalism when you transform it.

You are, in fact using a mathematical formalism when you write x=x. OK. But what does it mean?

When you evaluate the Mathematical expression "x=x" does it mean True or False?

So, mathematics is purposely meaningless and therefore -- in and of itself -- useless. Its only redeeming quality is that it produces a surprising gap between input and output, and is therefore sufficiently ridiculous.

It neither requires input, not it's required to produce an output. There's a semantic difference between intensional and extensional properties.

[godelian]

So, mathematics accepts a meaningless string as input and manipulates it into another meaningless string as output. It is a purely syntactic activity devoid from any possible semantics.

That which we call "formal semantics" is precisely the syntax manipulation rules.

It is the job of downstream disciplines to deal with semantics when they use mathematical formalisms.

So it's your job. You are, in fact, using a mathematical formalism when you transform it.

You are, in fact using a mathematical formalism when you write x=x? OK. But what does it mean? Does it mean true or false?

So, mathematics is purposely meaningless and therefore -- in and of itself -- useless. Its only redeeming quality is that it produces a surprising gap between input and output, and is therefore sufficiently ridiculous.

It neither requires input, not it's required to produce an output. There's a semantic difference between intensional and extensional properties.

If a statement has meaning, it is not a mathematical one. To that end, all meaning must always be completely abstracted away. The purity of mathematics entirely depends on making sure that the symbols mean absolutely nothing. The qualifier "pure" in the term "pure mathematics" means purified from any possible meaning.

[Skepdick]

If a statement has meaning, it is not a mathematical one.

What do you mean by "mathematical" and "non-mathematical" meaning?

Let f(x) = { 1 if MathematicalMeaning(x); 0 if not(MathematicalMeaning(x) }

Is f(f) = 1 or 0 ?

To that end, all meaning must always be completely abstracted away. The purity of mathematics entirely depends on making sure that the symbols mean absolutely nothing. The qualifier "pure" in the term "pure mathematics" means purified from any possible meaning.

So the qualifier "pure" is itself impure? By the criterion you've specified.

This is a very peculiar way of thinking to me. Are you saying that x != x and x=x don't mean anything?!? Why are you lying?

Surely x=x means "True" and x !=x means "False".

The tradition of mathematics has gone under many names (geometry, astronomy, ballistics) in the past. In contemporary society it's now called "computer science"; whereas "Mathematician" can now be better understood as "astrologer".

Dumb nihilists inhabited the Bottom type and evacuated all meaning from Mathematics.

[godelian]

This is a very peculiar way of thinking to me. Are you saying that 1 != 1 and 1= 1 don't mean anything?!? Why are you lying?
Surely x=x means "True" and x !=x means "False".

The symbol "1" doesn't mean anything. It only becomes interesting when we also introduce the symbol "0". They still don't mean anything. When we introduce enough manipulation rules, we can generate strings of symbols.

But then again, the above is not a good example, because people may attach meaning to it while they probably shouldn't.

Take the SKI combinator calculus. It is a much better example.

We have 3 meaningless symbols: S, K, and I.

Next, we have a few manipulation rules:

Ix = x

Kxy = x

Sxyz = xz(yz)

Now we can create a string that reverses any input string αβ:

S(K(SI))Kαβ →
K(SI)α(Kα)β →
SI(Kα)β →
Iβ(Kαβ) →
Iβα →
βα

As you can see, even though S, K, and I essentially mean nothing, they can reverse a string αβ which also means nothing.

This is pure mathematics.

[Skepdick]

The symbol "1" doesn't mean anything. It only becomes interesting when we also introduce the symbol "0". They still don't mean anything. When we introduce enough manipulation rules, we can generate strings of symbols.

Why are you talking about the bindings of the symbols 1, 0 to free variable "x" ?!?

∀x: (x = x) ≡ Either(True,False). I trust you are familiar with the Either() monad.

Code: Select all

❯ ipython
In [1]: class A: pass
In [2]: class B:
   ...:     def __eq__(self, other): return False
   ...:
In [3]: x = A()
In [4]: y = B()
In [5]: x == x
Out[5]: True
In [6]: y == y
Out[6]: False

We have 3 meaningless symbols: S, K, and I

Shame. Are you unfamiliar with the Y combinator? From S,K and I we can construct Y and from there onwards... *kaboom*

https://en.wikipedia.org/wiki/Fixed-poi ... a_calculus

The Y combinator may also be used in implementing Curry's paradox. The heart of Curry's paradox is that untyped lambda calculus is unsound as a deductive system, and the Y combinator demonstrates this by allowing an anonymous expression to represent zero, or even many values. This is inconsistent in mathematical logic.

This is the problem of The one and The Many - it dates back to Plato.

Next, we have a few manipulation rules:

Ix = x

Kxy = x

Sxyz = xz(yz)

Now we can create a string that reverses any input string αβ:

S(K(SI))Kαβ →
K(SI)α(Kα)β →
SI(Kα)β →
Iβ(Kαβ) →
Iβα →
βα

So ALL of those symbols are meaningless? Where do you see "manipulation rules" here?

The tradition of mathematics has gone under many names (geometry, astronomy, ballistics) in the past. In contemporary society it's now called "computer science"; whereas "Mathematician" can now be better understood as "astrologer".

Dumb nihilists inhabited the Bottom type and evacuated all meaning from Mathematics.

Computation is the New God. Embrace it.

https://ncatlab.org/nlab/show/computational+trilogy