Mathematician Felix Klein (famous for Klein bottle) used Hindu concept of Indra's Net as a inspiration for developing algebra for infinite reflections. It produces nice fractals.
Incredible, Hindu temples are not churches or mosques, they were made with incredible thought about positioning, with size and geometry taken into consideration. All optimised using numerics, we've lost a chapter in human history as to how Vedic structures were created with such elegance.
> we've lost a chapter in human history as to how Vedic structures were created with such elegance.
Just to be clear, this knowledge wasn't lost accidentally. It's part of what was contained in the books and manuscripts destroyed intentionally by colonizers.
> Muslim (Mughal) invaders burnt a lot of scriptures.
Actually the particular books I'm referring to were burned by the Portuguese, but lots of invaders destroyed Indian books - Portuguese, British, French, Dutch, and yes, Mughal as well.
One interesting historical narrative which helps put this into perspective was how the Spanish conquest of Aztec and Mayan regions resulted in a lot of burning of "heretical" books by the Christians missionaries, "which they (the Maya) regretted to an amazing degree" [1]. But further research showed that most of the books burned were already heavily biased historically by the Aztecs - as the Aztecs had already burnt all of the Mayans books when they conquered them before the Spanish arrived.
So it really is a (shameful) repeating pattern of many conquering human civilizations throughout history rather than one of a particular culture which happens to be dominate.
This is another piece of history which doesn't fit neatly into the colonizer narrative, which is popular today, from which people try to hold modern cultures responsible for past generational cultural events, somewhat like an 'original sin' which needs to be attuned. One could go further and try to point to religion itself as the main driver - but even then that is merely another abstraction over standard human tribalism. Which is somewhat ironic as the colonizer narrative is typically used today as a tool for pushing other forms of tribalism (political, cultural, etc).
I'm sure many colonialists burn books, doesn't distract from the fact we have lost knowledge and replaced with Christian and Islamist propoganda, they couldn't have people thinking we lived for more tHan a few thousand years as it contradicts the Bible and Qur'an!
The article seems to assume that the universe is fractal, but Wikipedia can't seem to make up its mind whether fractal cosmologies are fringe ideas or not. It's certainly far outside the consensus.
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.
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?
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".
> 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)
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.
> 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.
W/r to larger and more wide-ranging questions, e.g. if the mutli-verse has some sort of deep fractal meta-structure, and/or the recursion of fundamental quantum computing operations, the jury is still entirely out.
As elec. engineers (and c. scientists) one of our only hammers is signals theory and cosmology looks like a nail. But the physicists probably have it right.
... have what right exactly? Pictures (i.e. measurements) of faint celestal bodies are but mushy wish-washed smears of a point is what I'm talking about for example. Quite simply, our band-width is limited by position, the ammount that we can observe is likely just a tiny fraction, the known universe only accounts for ca. 10% of theoretical total energy. In that sense the low energies are likely lost on us, then we actually see through a highpass (redshift?).
I mean that the physicists have accounted for the limited amount of information available and still conclude that the observable universe is not a fractal.
I remember thinking about this in 2011/2012 when it wasn't known (IIRC there was a bet over a bottle of wine between two physicists over this matter). But years later, because a satellite "zoomed out" far enough and sent images back to earth, physicists confirmed that it is very likely that the observable universe is not a fractal (i.e. it's Hausdorff-Besikovitch dimension, which is calculated from measurements, is fractional).
When something is referred to as fractal, does it always imply a degree of self-similarity? As I understand it, Mandelbrot developed fractal geometry as a tool to examine the "roughness" of reality as opposed to the smoothness of calculus and fractal dimension is a measure of complexity at every scale, with a self-similar pattern being an easy example to follow.
If that is so, that the universe has a high degree of complexity at every scale, is that really controversial?
I think the cool thing about this from what I gather is that the instructions to the masons were basically a recursive procedure and they got to decide on the recursive depth.
Hmm, really it was more like a geometric progression. David Gelernter has commented on the lack of familiarity art/architectural historians have with terminology like "recursive" or "self-similar" (I prefer the latter because it leaves room for the judgements and conspicuous non-exact applications of recursion in architecture): https://www.edge.org/response-detail/10574
Edit: I'm an architecture historian so I didn't mean it if I came across as pompous -- but that Gelernter link conveys a weird sense of CS being superior to the art historians, and at the same time makes more of recursive patterns in physical architecture than is justified. I have mixed feeling about this projection of the ideas of one field on to another: something like the Sierpinski gasket is not good architecture, and it takes a real expert in building to understand the difference, not a mathematician/computer scientist who can spot recursion. So my possibly contentious point was that a schematic reading of architecture as recursive (or even cosmic) is valid but also kind of the most superficial non-aesthetic form of thinking about architecture...
That is cool, a compressed blueprint. Also cool is that it is still under construction, maybe someone reading right now is involved and can tell us more.
The other interesting thing about the Sagrada Familia is that Gaudi based the primary structure on catenary structures.
Off the top of my head, I believe he first made weighted string assemblies of the vaults, such that the string structure is held in perfect tension from the gravitational force of its own weight.
Then by casting the resulting shape in concrete, and flipping the shape over, the resulting structure is optimized for compression. The tensile structure reflects the force of gravity on the string assembly - it is "optimized" for its own weight, so when flipping it over, and swapping the tension members for compression members, we know this new structure is perfectly shaped to distribute internal and self-weight for compression.
It would be odd if it only occurred in one place in the world. But I do wonder if the deep embedding of mathematical concepts in various forms of Buddhism encouraged higher order thinking.
https://www.amazon.com/Indras-Pearls-Vision-Felix-Klein/dp/1...