A mathematical perspective on the phenomenon of "miracles"

[godelian]

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

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.

Concerning the Y combinator, I find it mind blowing in Javascript, because unlike the SKI calculus, Javascript is not meant to express that kind of things, but it actually can: https://bmsdave.github.io/blog/y-combinator-eng/

So ALL of those symbols are meaningless?

Yes, S and K are meaningless. The symbol I is not meaningless, because it is the identity function, but it is redundant. You can replace I by (SKK) or (SKS).

Where do you see "manipulation rules" here?

The rewrite rules are:

1) Ix = x
2) Kxy = x
3) Sxyz = xz(yz)

A rewrite rule ("manipulation" rules) , is just a find-and-replace operation, just like a text editor does.

I "haha"= haha
K "whatever" "something" = whatever

x and y are just template placeholders. Below the rewrite rules look a bit more like in Perl or shell script:

1) I($x) => $x
2) K($x,$y) => $x
3) S($x,$y,$z) => $x$z($y,$z)

The third rewrite rule actually requires some extra power in the execution engine because implementing support for functions such as f($x) is not enough. You also need first-class functions such as $f($x), where the function itself is also a variable.

Without these rewrite rules the SKI calculus wouldn't be able to do anything. You need rules that allow you to rewrite an expression into another expression. Otherwise, it is not a calculus. Similarly, 1+2 can be rewritten to 3 in arithmetic calculus. So, we have a rewrite rule for the sum. i.e. a "manipulation" rule. The standard rewriting sequence in Peano arithmetic is 1+2 => 2+1 => 3+0 => 3. In the expression a+b, you just keep decreasing b while adding 1 to a until b is zero. At that point, you can switch to the final rewrite rule and say that n +0 => n.

Computation is the New God. Embrace it.

Computability is an interesting branch in mathematics, but as far as I am aware of, it only has two interesting theorems: Turing's Halting problem and Rice's Theorem, which is itself actually a generalization of Turing's Halting problem. So, there is just one interesting theorem, so to speak of.

General computation "does" something. It does not explain things. It just executes them. I am mostly interested in surprising connections. Something that is insightful. Turing's Halting problem is definitely surprising and insightful. But where are the other insightful or surprising theorems in computability? Doing things or designing data structures and algorithms is something to do at work, as a job, because you get paid for that. I don't see where the interesting research would be.

[Skepdick]

The rewrite rules are:

1) Ix = x
2) Kxy = x
3) Sxyz = xz(yz)

A rewrite rule ("manipulation" rules) , is just a find-and-replace operation, just like a text editor does.

Sorry, I don't see any rules. What rules? I see meaningless symbols. What do you mean by "find" and "replace"? Those are verbs.

I "haha"= haha
K "whatever" "something" = whatever

x and y are just template placeholders. Below the rewrite rules look a bit more like in Perl or shell script:

1) I($x) => $x
2) K($x,$y) => $x
3) S($x,$y,$z) => $x$z($y,$z)

Nope. Sorry. I can't parse any of this stuff. If it's all meaningless (as you claim) then it surely is impervious to lexical analysis.

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

How am I supposed to tokenize meaningles symbols...

A lexical token is a string with an assigned and thus identified meaning

Computability is an interesting branch in mathematics

As far as I am aware Mathematics is an interesting branch in computation...

but as far as I am aware of, it only has two interesting theorems: Turing's Halting problem and Rice's Theorem, which is itself actually a generalization of Turing's Halting problem. So, there is just one interesting theorem, so to speak of.

Well, what more do you want?

There exists a recursively enumerable language having the property p,
There exists a recursively enumerable language not having the property p,

For p being any semantic property.

Such as the semantic property of meaningfulness or meaninglesness.

[godelian]

1) I($x) => $x
2) K($x,$y) => $x
3) S($x,$y,$z) => $x$z($y,$z)

Nope. Sorry. I can't parse any of this stuff.

I wrote it in the syntax of the well-formed formula (wff) which is the syntax of predicate logic but also of quite a few programming languages.

If it's all meaningless (as you claim) then it surely is impervious to lexical analysis.

Of course, it is meaningless, but it is also well-formed nonetheless.

Furthermore, the actual parsing is not done by the lexer, which only groups characters into words ("tokens"). After that, you still need the real "parser" to create sentences ("syntax trees") from the tokens. A real critique would rather be something like: This syntax requires the parser look ahead more than one token, or there's a shift/reduce conflict or a shift/shift ambiguity in your grammar. I've had yacc (well, gnu bison) spit out that kind of criticism quite a few times. Ha ah

[Skepdick]

I wrote it in the syntax of the well-formed formula (wff) which is the syntax of predicate logic but also of quite a few programming languages.

You did? So if I just assign it to a string-type and pass it to the eval() function I'll get some output?

Of course, it is meaningless, but it is also well-formed nonetheless.

Is it? Let P be the semantic property of "well-formedness".

By Rice's theorem...

There exists a recursively enumerable language having the property P
There exists a recursively enumerable language not having the property P.

Please provide the source code for a working decider (in a language of your choosing) which determines that your language has the property P.

Furthermore, the actual parsing is not done by the lexer, which only groups characters into words ("tokens"). After that, you still need the real "parser" to create sentences ("syntax trees") from the tokens. A real critique would rather be something like: This syntax requires the parser look ahead more than one token, or there's a shift/reduce conflict or a shift/shift ambiguity in your grammar. I've had yacc (well, gnu bison) spit out that kind of criticism quite a few times. Ha ah

The real criticism stems from your inability to provide interactive implementations of your eval() and apply() functions.

https://en.wikipedia.org/wiki/Eval
https://en.wikipedia.org/wiki/Apply

And so we are back to where we started. What happens when we apply the eval() function to the expression "x=x" ?

[godelian]

You did? So if I just assign it to a string-type and pass it to the eval() function I'll get some output?

The eval() function is not an automated proof system. It does not know what rewrite to do next. Moreover, automated proof systems are generally not able to write proofs. It has to be done manually. A proof assistant is only good at verifying that every rewrite is valid.

So, no, eval() won't work because it does not rewrite expressions. It executes them.

It's like trying to simplify 4x+2y=0 to 2x+y=0. You cannot do that with eval(). You need the simplify(string) function in a tool like the Maxima computer algebra system to do that, but Maxima itself has no modules for the SKI combinator calculus. You can probably use the following program at github:

https://github.com/newlandsvalley/skicalculator

Is it? Let p be the semantic property of "well-formedness".
By Rice's theorem...

It is a syntactic property, not a semantic one. It is not something that you see only at runtime. It is a property of the source code. So, I don't see where Rice theorem would apply.

[Skepdick]

The eval() function is not an automated proof system. It does not know what rewrite to do next.

Nonsense. If any program knows that the next step in the execution is to apply the "rewrite" function to the expression X, then it can it can also know that the next step in the execution is to apply the "eval" function to the expression X.

Moreover, automated proof systems are generally not able to write proofs.

Exactly. Proofs are inputs. To functions.

Assembly instructions passed to a runtime.

A proof assistant is only good at verifying that every rewrite is valid.

Soooo. Rice's theorem. Which property is the (in)validity of any given rewrite?

So, no, eval() won't work because it does not rewrite expressions. It executes them.

Execution is rewriting. For example executing the expression "x=x" rewrites it to its simplest form.

True.

It's like trying to simplify 4x+2y=0 to 2x+y=0. You cannot do that with eval().

Of course I can. Pass the expression "4x+2y=0 to 2x+y=0" to the reduction engine implementing the eval() function for the algebra you have in mind.

It is a syntactic property, not a semantic one. It is not something that you see only at runtime. It is a property of the source code. So, I don't see where Rice theorem would apply.

The source code is raw-text file. What or where is its "well-formedness" ?!? Surely you need to apply some sort of function to prove such property ?!?

Rice's theorem So Boring.

[godelian]

If any program knows that the next step in the execution is to apply the "rewrite" function to the expression X, then it can it can also know that the next step in the execution is to apply the "eval" function to the expression X.

Concerning the eval() function:

In some programming languages, eval , short for the English evaluate, is a function which evaluates a string as though it were an expression in the language, and returns a result; in others, it executes multiple lines of code as though they had been included instead of the line including the eval.

The only functions it has access to, are the ones of the standard library embedded in the execution engine. This does not usually include a function that can rewrite expressions into other expressions.

For example:

ratsimp("a + b/c")
"(a*c+b)/c"

It rewrites the first expression into the second one. You need the logic of this function for that. The typical scripting engine does not have that function built in.

Furthermore, a program cannot just "know" what the next rewrite step is in a proof. There are attempts at creating automated provers but they generally do not work. It has to done manually.

[Skepdick]

The only functions it has access to, are the ones of the standard library embedded in the execution engine. This does not usually include a function that can rewrite expressions into other expressions.

Yes, that sounds like a you-problem.

If you are performing rewrites in your head - it sounds like the runtime in your head provides some features/functionality that's not captured in your syntax. We call that dynamic linking...

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

Furthermore, a program cannot just "know" what the next rewrite step is in a proof.

Fascinating! So do you know what the next step in the rewrite of "x=x" is ?

How do you know?

There are attempts at creating automated provers but they generally do not work. It has to done manually.

Imagine that! So you can't make the proof 100% explicit in syntax? You have to elaborate what you mean by the various operators ?!?
Who would've thought?

Oh yeah... This guy: https://youtu.be/wZSvuCJBaFU?t=110

[wtf]

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

Ok, please explain them to me.

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.

Ok, please explain them to me.

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

Ok, please explain them to me.

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

Ok, please explain them to me.

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

Ok, please explain them to me.

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

By "discuss," you mean repeating claims over and over without ever engaging in any substantive discussion?

[Age]

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

Ok, please explain them to me.

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.

Ok, please explain them to me.

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

Ok, please explain them to me.

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

Ok, please explain them to me.

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

Ok, please explain them to me.

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

By "discuss," you mean repeating claims over and over without ever engaging in any substantive discussion?

No.

Why did you and why would you even begin to assume such a thing?

What I mean by 'discuss' is one asks for, exactly, what is wanted, and/or proposes some thing, and all involved in the 'discussion' challenge and/or question each other until 'the resolution', solution and/or answer, is uncovered, or has been brought-to-light.

Now, at the start of True 'discussion', well to me, what 'the goal' of the outcome of the discussion is agreed upon and accepted. So, what is 'the actual goal' you want of a discussion between 'you' and 'I' here?

For example, do you want to:

Find out and discover that ALL things can be explained through and by 'natural laws'. Or,

Find out and discover how 'Consciousness' and the 'existence of everything' can be explained through and by 'natural laws'? Or,

Find out and discover what 'Consciousness', and, the 'existence of everything' is, exactly? Or,

Find out and discover how they exist, exactly? Or,

Find out and discover why they are, exactly? Or,

Keep believing what you do, and so not be Truly OPEN to anything contrary, and then maybe try to prove that what you already assume or believe is true, is true. Or,

Fight and argue for what you 'currently' assume or believe is true?

Or, is there something else that you really want here?

When you are far more specific in what you want and desire here, then I can, and will, give you what 'it' is that you want and are after here.

For example, if you really want to discover, or learn, and understand, about the existence of everything, and you are specific about what it is that you want to discover, or learn, and understand about the 'existence of everything', then we can proceed and move along here. But, if you are not specific in regards to what 'it' is that you really want and desire here I could end up providing you with something completely different.

For example, in regards to the 'existence of everything', there are really only two, fundamental, things only. They are 'matter' and 'space', which is just, more or less, the distance between and around matter.

The two things exist always eternally and exist spatially infinitely.

Because the two co-exit 'now', this means that they have to always co-exist.

These two also make up every perceived thing, which is just the 'existence of everything', in other words.

See, I could start off providing something like this, which by the way I could and would back up and support, fully. But, this may not be what you were really wanting nor seeking here, at all, nor even partially. So, until you are very clear, and specific, with and about what 'it' is, exactly, that you really want and desire here, I have no idea what 'it' is, exactly, you want me to explain to and for you here.

[wtf]

See, I could start off providing something like this, which by the way I could and would back up and support, fully. But, this may not be what you were really wanting nor seeking here, at all, nor even partially. So, until you are very clear, and specific, with and about what 'it' is, exactly, that you really want and desire here, I have no idea what 'it' is, exactly, you want me to explain to and for you here.

My posts could not have been more clear. I see you have no intention of engaging.

[Age]

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.

But who has 'that idea'?

If you do "iwannaplato", then why?

We don't know if there are natural laws, or rather tendencies/habits/patterns that can change.

If you do not know, then why bring 'this' up here now?

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.

What do you even mean or are referring to, exactly, when you say, and use, the words 'time', and 'space'?

When you, openly, divulge this information, then 'we' can look to 'see' if what else you are saying and/or claiming here actually 'fits in'.

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.

Obviously, any and all of this depends on how you are defining 'your words' here, exactly.

[Age]

See, I could start off providing something like this, which by the way I could and would back up and support, fully. But, this may not be what you were really wanting nor seeking here, at all, nor even partially. So, until you are very clear, and specific, with and about what 'it' is, exactly, that you really want and desire here, I have no idea what 'it' is, exactly, you want me to explain to and for you here.

My posts could not have been more clear. I see you have no intention of engaging.

So, I went out of my way to start to explain to you, that, in regards to the 'existence of everything';

There are really only two, fundamental, things only. They are 'matter' and 'space', which is just, more or less, the distance between and around matter.

The two things exist always eternally and exist spatially infinitely.

Because the two co-exit 'now', this means that they have to always co-exist.

These two also make up every perceived thing, which is just the 'existence of everything', in other words.

And, you, for some reason, 'see', and claim, that i have no intention of engaging.

When you say, to me, 'Ok, please explain them to me', in regards to 'consciousness', and, 'the existence of everything', and I then start explaining, to you, 'the existence of everything', why do you then claim that you 'see I have no intention of engaging'?

What does 'engage with you' even entail, exactly?

How could I not have engaged with you when I, obviously, started explaining, to you, 'the existence of everything' and what absolutely every thing is made up of, exactly?

What appears to be far more obvious now is that it is you that has not intention of actually engaging here.

Look at how much I have written, to engage with you, compared to how much you, to engage with me.

Also, I am somewhat relieved I did not explain anymore, to you, because you do not even want to engage with what I actually started explaining, to you, in regards to 'the existence of everything', which is what you asked for, and now seemingly was only pretending to be interested in.

So, what all of this means is that I actually did start engaging by starting to explain 'the existence of everything', and which you obviously have no intention of engaging, with me, on that.

I also was able to point out and show, which you are actually proving True, for me, how when people are so vague on things, then this helps in explaining why they never come to actually discover, nor learn, and understand, the actual Truth of things.

[Sculptor]

This Forum gets more and more ridiculous with each passing day.

THere is not "phenomenon of miracles" since there are no miriacles.

[godelian]

Fascinating! So do you know what the next step in the rewrite of "x=x" is ?
How do you know?

There is a difference between evaluating an expression and trying to prove another statement from it.

Both are typically done by rewriting it.

In your example, x=x generally evaluates to true, unless x is an undefined value of sorts. In that case, it may evaluate to false or undefined, but that depends on the language/implementation at hand. For example, null == null is true in C but undefined === undefined is false in JavaScript.

The expression x=x can be rewritten to a=a because that would possibly be useful when trying to derive some other statement.

For example, (x+a)^2 may get expanded to x^2 + 2ax + a^2 because it is deemed a step in the right direction of deriving a particular wanted statement from the starting point statement. It is not about evaluating the expression. It's about obtaining another equivalent one.