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Comment by canoebuilder | Hacker News Reader

How about a fractal of fractals?

Your circulatory system is a fractal for similar reasons that trees are fractals for similar reasons that watersheds are fractals.

Fractal structures are probably the rule rather than the exception for basic reasons of space and energy efficiency.

Here's another interesting fractal thought, where is the universe? In your head, or outside of it, or both?

So the neuronal structure representing the universe in your head is at least in some ways similar to the objective outside universe, if there is such a thing, and if there is such a thing then that neuronal structure in your head is also in(and of) that objective universe while also being similar to it.

A good term for fractals is self-similar.

Self-similarity all the way down.

I am a strange loop. - Douglas Hofstadter

It's in you, and you're in it. - Alan Watts

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I've been fascinated with occult/esoteric stuff lately, so I want to add one more quote there.

That which is below is like that which is above & that which is above is like that which is below. - The Emerald Tablet

https://en.wikipedia.org/wiki/Emerald_Tablet#Newton.27s_tran...

It seems fascination with the occult is a growing trend among the educated for some reason. In the past year I've met physicists, engineers, mathematicians, and more who have all gotten in to the field recently. I have too, so I suspect some confirmation bias, but it seems to me more than I was expecting.

I don't know many STEM type people apart from me who are into such things honestly, so your claim is very interesting to me. Do you know someone with an online presence (blog for instance) who would fit into the category?

Explains why computer science is such an interesting field. It's the study of universes, which happens to yield a remarkable number of insights into our own.

If you consider the Curry-Howard correspondence in connection with Gödel's Incompleteness Theorems, the fractal nature of the universe seems inevitable.

The Curry-Howard correspondence in connection with Gödel's Incompleteness Theorems only gets me to "For some types, there is no program of that type, but also no way to show that there is no such program."

What does that have to do with fractals or the physical universe?

Isn't that backwards? Why care about type without a corresponding programm? The systematic problem with higher order incompleteness is propositions fro syntactic argument, without consideration for semantics.

The Curry-Howard correspondence is about types as propositions and programs as proofs.

Gödel's first incompleteness theorem that "In any consistent formal system with [enough arithmetic to make a Gödel encoding], there is a proposition P such that neither P nor ¬P have a proof." translates as:

"In any type system that does not allow construction of functions from inhabited types to the empty type ⊥ (this corresponds to consistency), with [enough types and functions to make a Gödel encoding], there is a type T such that neither a program of type T, nor a program of type T → ⊥ (this corresponds to negation), can be constructed."

The semantics of the programs involved don't matter, only the way they manipulate types do. Commutativity of conjunction, for example, is encoded by a function of type (A, B) → (B, A). That this function swaps the values in its input tuples is, for the Curry-Howard correspondence, just an unimportant side effect.

OK I really don't know anymore what this has to do with the topic, but I always take the chance to discuss this, as opportunity to be lectured.

Where was that quote taken from? Surely, the second conclusion is just one of the premisses. Not just T but any inhabited type couldn't be used as the input to a function that returns nil, if that's axiomatic. Correct?

But still, for the first conclusion how can a type without corresponding program be said to exist in the first place?

One way around seems to add a higher order axiom about the truth of the liar paradoxons, to assume all unprovable sentences are wrong. The common example is not a lie nor true, but just not a sentence, IMHO. Then that introduced a new set (or what) of unprovable propositions (incomplete), if not inconsistency? That would be a new insight to me.

I'd rather expected a second order axiom could be removed to achieve completeness, instead. I haven't seen the axioms of goedel that omega_0-complete systems be higher order.

> Where was that quote taken from?

The quote was a restatement from memory (so not really a quote), but you can find a similar summary on Wikipedia https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

> Surely, the second conclusion is just one of the premisses. Not just T but any inhabited type couldn't be used as the input to a function that returns nil, if that's axiomatic. Correct?

I think my choice of T for the arbitrary type was confusing (putting it in italics apparently didn't help). If I had been talking about the top type, I would have used ⊤, not T. (See the difference? Probably not.)

> But still, for the first conclusion how can a type without corresponding program be said to exist in the first place?

A simple syntactic type without corresponding program is "Λa.Λb. a → b" (the capital lambdas are type abstractions, meaning "for all types"). This is actually the same type as "⊥", but that isn't immediately obvious. So given a specific syntactic type expression, the question is whether there are values of that type and if there are, whether you can construct values of such a type.

One type with values that you can't actually construct is the type that says "No value of this type can be constructed." using a Gödel encoding in type-level numbers. That type will be a huge function signature, so it isn't actually relevant to everyday programming, but it has to exist, and there have to be values in that type.

Since you can't actually write down these values, their existence is very annoying, but they are a bit like non-standard numbers in that regard: You can't effectively outlaw them without shooting yourself in the foot.

> One way around seems to add a higher order axiom about the truth of the liar paradoxons, to assume all unprovable sentences are wrong.

That will introduce inconsistency, since Gödel was able to transform a self-referential sentence into a sentence without self-reference, using a construction that is intuitively correct. So if all self-referential sentences without proofs were wrong, the non-self-referential version of "This sentence has no proof." would have to be wrong as well, which means that it has a proof, which makes the original reason for labeling it false invalid.

You could avoid the paradox by claiming that Gödel's construction isn't actually intuitively correct, but since the point of the exercise was to show that mathematical intuition can't be formalized, that doesn't really help.

Sorry, I also took for granted the assumption that the universe is an executing program. I believe that's uncontroversial but would be happy to discuss it.

What it seems to me (could be totally wrong, would love to know if so) that Gödel's conclusions can then tell us about our universe/computer/executing-program/mathematical-proof (please excuse the fumbly wording) is that it has to have an external universe in which to exist/run/be-consistent-in, which presumably depends upon its own, ad infinitum. In other words, it would have to be universe-program-computer-proofs all the way "out".

Replying to my own comment with supporting info.

Citation for the first point: https://en.wikipedia.org/wiki/Digital_physics

> According to this theory, the universe can be conceived of as either the output of a deterministic or probabilistic computer program, a vast, digital computation device, or mathematically isomorphic to such a device. > "Within each universe all observable quantities are discrete, but the multiverse as a whole is a continuum. When the equations of quantum theory describe a continuous but not-directly-observable transition between two values of a discrete quantity, what they are telling us is that the transition does not take place entirely within one universe. So perhaps the price of continuous motion is not an infinity of consecutive actions, but an infinity of concurrent actions taking place across the multiverse." - David Deutsch

Also, some concepts helpful for bridging the gap:

- Programs (and their states) are graphs

- Graphs are just relationships between quantities

- Every program is an instance of the general pattern `fold(f, input)` [1]

- Imperative programs are just inputs to functional programs

- Computers can be emulated in functional languages

- Traversal of a state graph is equivalent to time travel in the universe described by the states (or travel between the universes described by the states)

- Temporal/causal reasoning is actually spatial reasoning (traversal through the mental representation of state graph edges, i.e. executing a mental program)

- Correspondence between numbers, vector spaces, matrices, objects, closures, programs, functions, graphs, simulations, proofs, mappings, categories

From the above, I believe one can reasonably conclude that:

a) computers are programs

b) computers/programs are subsets of our universe

c) all computers/programs exist inside other computers/programs

d) computers/programs exist in our universe

e) our universe exists in a computer/program, which itself does as well, etc

Again, I haven't had this verified so I'd love to hear why/how this is wrong.

[1] http://www.cs.nott.ac.uk/~pszgmh/fold.pdf

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