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We’ve all heard about Mandelbrot sets – fantastically complex structures that can be created from a single equation and a very short program. Why, some people ask, couldn’t life be like that? And wouldn’t it be more mathematically elegant if it was? Physics types are especially prone to feel this way. Many of them might want to echo Louis IX’s famous remark on the Ptolemaic system, that if the Almighty had consulted him, he could have told Him a simpler way to make the cosmos.
I would ask these people to take a good look at the two pictures above. What I’m claiming is that life is not like a Mandelbrot set. It’s like an origami bull. The reason is that it embodies lots of instructions for making many different kinds of folded shapes (e.g. in proteins) which all work together. That’s why cell-based life cannot be simple. If you want to build simple life, make sure you don’t use proteins.
What do readers think?
Hi vjt. First link is broken. I love fractals.
Abel calls it prescription.
It’s true life is not like a Mandelbrot set. Perhaps a mathematician can see the beauty in the equation, I can’t. I do see beauty though in the graphics, but those are by design. 😉
I’ve never heard anyone ask why life can’t be like the Mandelbrot set, and I’m not sure what they might mean if they did? Can you explain, or point to such a person?
I, too, am fascinated by the Mandelbrot set, and used to teach my senior high school students some about how it was generated, and what were some interesting principles tha arose from it.
One observation one can draw is that a simple sets of rules (sort of simple, if you know how to operate with complex numbers) can produce such a complex figure, much less one that has a fractal nature.
However, the Mandelbrot set is very different not only from life, but the world in general is that it is determined: there is no contingency involved at all. Most people (I have no idea if there are really strict determinists these day) understand the world to have contingent factors, such that the state of the universe right now does not logically and irrevocably mean that the universe is going to be a particular state in the next moment, much less in a milion years.
So in that sense, the Mandelbrot set wouldn’t be a good model for how life, or any other component of the universe, came to be.
But it is very neat!
To Mung: the Mandelbrot set doesn’t come from just an equation – perhaps you know that, but from a set of operations and rules applied iteratively to every point in the complex plane.
You say, “I do see beauty though in the graphics, but those are by design.” The Mandelbrot set, more than other mathematical entities, perhaps, brings up the difficult point about whether mathematics exist apart from our manifestations of it in abstract and visual systems.
I was once discussing with my calculus class the old philosophical question of whether mathematics is invented or discovered – does it or does it not exist apart from our, or some similar creature, having created a way to represent it.
I remember one of my students making a nice comment: she said that she thought math was discovered, but that we had to invent ways of writing about it to make that possible.
So, yes, without modern computers, we could never even know the Mandelbrot set existed, in whatever way it might be said to exist apart from the computers that have made it possible to see it. It took not only centuries of math being developed but also the invention and development of computers in order for us to know that the Mandelbrot set exists. Did it exist before we did all that? That is the interesting question.
I know that the Mandelbrot can be studied very formally without reference to the pictures, but I doubt we would ever have thought to do that exploration without the visual representations.
And, Mung, can you explain the comment “Abel calls it prescription.” Is Abel a mathematician, and what exactly is the “it” he calls prescription, and what is meant by tha?. Any explanation you could supply would be appreciated.
Hi Aleta,
In the midst of everything else I’m juggling, I’m reading David L. Abel’s Primordial Prescription.
http://www.amazon.com/dp/B00XTG883I
VJT says the bull “embodies lots of instructions for making many different kinds of folded shapes.” According to Able, and this makes sense to me, instructions are not enough. You also need a system capable of understanding and executing the instructions.
The figure in the OP of the Mandelbrot set is no different.
The Mandelbrot set can be reduced to an algorithm. Why do you think of it as complex?
Dr. David L Abel
Mandelbrot Set
The physical universe is like a Mandelbrot set. It is defined by a few simple natural laws and constants. Life isn’t. Life isn’t explicitly defined by nature, it is not a “natural” consequence of the properties of the universe. Just like a computer program requires a designer even though the computer is a relatively simple device designed to run complex programs, so does life require design and creation before it will exist even though the universe is brilliantly designed to support life.
Hi Aleta. I’ve heard many people argue that the example of the Mandelbrot set disproves (or weakens the case for) Intelligent Design: it shows that you can generate complexity in a very straightforward manner. See here: Intelligent design and evolution – Khan Academy, Wolfram’s math disproves “intelligent design”, and Why isn’t the Mandelbrot set evidence for design? by atheist Bradley Monton. The Skeptical Zone has put up some posts on the subject as well. What I’m arguing is that the comparison between Mandelbrot sets and living things is a flawed one.
“Below is a photo that has made its rounds on the Internet. It compares very, very small things to very, very big things.”
http://www.cracked.com/blog/5-.....-today_p2/
The fractal universe (~6min):
https://www.youtube.com/watch?v=zORy9nDsF18
Every analogy between computer models and evolution is flawed.
http://bio-complexity.org/ojs/.....ue/current
But perhaps life is just a Mandelbrot set of all Mandelbrot sets!
But then again, perhaps the idea that a + b + c + d … + n = life is flawed. If we can just find the right parts and plug them together in the right order…
What is Life?
Aleta:
Very interesting thoughts. A few comments:
a) The Mandelbrot set is a good example of how some (moderately) simple deterministic algorithm can generate forms which have the appearance of complexity and beauty. That’s why some use it as an argument against ID.
b) However, the type of complexity which is used in ID to infer design is completely different. For example, if we define functional complexity, the key point is the minimum number of bits which can implement some objectively defined function. If we find an object which implements some complex function, we infer design as the cause of the specific form of that object.
c) I believe that the Mandelbrot set, as implemented in some material object (for example, a print of it) has some specified complexity, which corresponds at least to the minimum number of bits required to implement a program which can compute it. That is not a huge specified complexity, but it is not low. I believe that all the examples we know of Mandelbrots are designed. It is perfectly possible, however, that simpler fractals arise in natural systems (and they certainly do).
d) Regarding your thoughts about the origin of the Mandelbrot set, you say:
“I was once discussing with my calculus class the old philosophical question of whether mathematics is invented or discovered – does it or does it not exist apart from our, or some similar creature, having created a way to represent it.”
I am definitely a believer in the innate nature of mathematics, let’s say the neo-platonic view of it, according for example to Penrose. In that sense, I think that mathematical objects are “discovered”, but they are discovered in our mind, and then applied to the outer world.
The way to represent a mathematical object like the Mandelbrot, however, is rather a beautiful example of design: functional complexity tied to a desired function in the consciousness of a designer, and implemented into material objects by the designer himself.
e) Finally, I would mention that the concept of order from chance + necessity has found some support, for example in Prigogine and in the models of self-organizing structures. Again, while those concepts are interesting, they are completely different from the type of specified (functional) complexity that we observe in the biological world.
Aleta:
Abel’s concepts of the two basic kinds of information are very simple and, IMO, powerful.
In brief, as I understand it, I would say that specified information which is tied to design by conscious beings is essentially of two kinds:
a) Descriptive information is the type of information which conveys a cognition, a meaning, through the designed object. The information must be understood by some conscious receiver,, and the meaning is then evoked in the consciousness of the receiver.
b) Prescriptive information is the type of information which implements a function, through the designed object. In a sense, it is more “objective” than prescriptive information, because the function will be implemented without any need for a conscious receiver. However, the recognition of the function, which is a cognitive experience, always requires a conscious observer.
Mung:
Thank you for pointing to Abel’s new book, I was not aware of it.
I have just bought the Kindle version, and I will read it as soon as possible. 🙂
GP, Mung & VJT (Attn Aleta et al):
Blurb at Amazon:
KF
Thanks, KF!
By the way, a correction to my post #13:
“In a sense, it is more “objective” than prescriptive information,”
should be, obviously:
“In a sense, it is more “objective” than descriptive information,”
@Aleta #3
‘The Mandelbrot set, more than other mathematical entities, perhaps, brings up the difficult point about whether
mathematics exist apart from our manifestations of it in abstract and visual systems.’
Yes, I believe mathematics exists apart from our manifestations of it in abstract and visual systems – in the mind of God. Like Max Planck’s atom, which inhabits a quantum world in which ‘matter does not exist as such’.
It seems to me difficult to escape the conclusion that God thinks abstractions and matter with equal facility.
Mandelbrot set is created by a simple quadratic mapping Z*Z+C -> Z. There are infinitely more complex algorithms than that. The basic assumption that inspired emergence of natural science is that universe, including life, is operating algorithmically.
There is no phenomenon known so far that at least in principle cannot be expressed algorithmically i.e. one can imagine without contradicting any demonstrable fact that virtual reality simulations can behave as closely as we wish to anything observed so far.
Hence there is no need for some lawless, capricious deus ex machina, such as the one promoted by Seattle ID, intervening into the workings or algorithms of the universe as we know them, tweaking molecules into ‘irreducibly complex’ systems or anything else that we know of so far.
We also know that any logically coherent algorithmic system of sufficient complexity (which includes the already existent natural science) is necessarily incomplete i.e. natural science must by virtue of mathematical/logical necessity remain forever incomplete. Hence, there will always be opportunistic, parasitic priesthoods filling the present gaps in our scientific knowledge with profitable ‘god of gaps’ myths which it can sell to the dupes.
NL, the mathematical expression is only a small part of what is involved in mapping the set and the (often coloured) near misses that give the sort of complexity pictured in the OP. The actual composition of such a program and its execution on an effective machine are massive expressions of FSCO/I in action. And of course such cases are invariably seen to come about by design. KF
Doesn’t Godel’s Incompleteness theorem give the lie to your extraordinarily extravagant assumption about the origin of science ? If anyone’s a dupe, it’s you. If you’re the nighlight, there’s gonna be a whole mess of accidents.
‘There is no phenomenon known so far that at least in principle cannot be expressed algorithmically.’
Try, ‘the natural world/universe’.
‘….natural science must by virtue of mathematical/logical necessity remain forever incomplete. Hence, there will always be opportunistic, parasitic priesthoods filling the present gaps in our scientific knowledge with profitable ‘god of gaps’ myths which it can sell to the dupes.’
Well, logic is a real downer for atheists, that’s true, but is this the place to give vent to your grief? God’s always been unfair. Ask any of your colleagues. They get really ratty with him.
Perhaps some atheist should inform Dr. Bradley that he forgot to include the Mandelbrot Set in his list of foundational equations that describe the universe? 🙂
Quote from preceding video:
as to Schrödinger’s Equations in particular:
@20 “Doesn’t Godel’s Incompleteness theorem give the lie to your extraordinarily extravagant assumption about the origin of science ? If anyone’s a dupe, it’s you.”
Au contraire. Both science and universe are continuously evolving, which is exactly what one would expect to find if a system (universe) or its model (natural science) are incomplete. They’re both perfectly consistent as far as anyone knows with algorithmic systems (programs).
Another problem seems to me is that we think that we “understand” how simple stuff works, as pointed out by nightlight
There is no phenomenon known so far that at least in principle cannot be expressed algorithmically i.e. one can imagine without contradicting any demonstrable fact that virtual reality simulations can behave as closely as we wish to anything observed so far.
Actually, we do not. I can say that, for instance, biomolecular simulation results are always approximations of the reality. For some “strange” reason, we believe that this approximation is likely to converge to reality, itself, but why does it need to be so? The results are always erroneous, but again we deem measurement error, or other unmeasured effects responsible for these errors. Why? How can we rely on the functions devised by our mathematics? Why do not we consider the possibility that the formulation, the mathematics, or even the deduction process, itself, may be insufficient to explain the universe perfectly? Furthermore, as we focus on some part (or scale) of reality and get more accurate answers, you lose accuracy at some other part (or scale). Doesn’t this seem to be that our rationale (mathematical thinking and deductive reasoning) is a kind of “linearization” of the reality about certain operating points?
Using different laws (and not always being able to reconcile these laws) for different spatial and time scales, we believe that we “understand” how universe works. In a way this works, since we are dealing with microscopic quantities, i.e. individual “stuff” like an atom, or a molecule, or macroscopic quantities, which can be analyzed statistically. Hopefully, we have different theories for these different realms. However, when it is life, we encounter a mesoscopic quantity, both individual things and their statistical properties matter. Hence, laws from different orders should work in harmony. Do they? Well, no. There are many speculations for different parts of the reality, but when you want to combine them, they do not match.
Maybe, one alternative thinking is to realize that what we call a “blind force” is neither essentially a force nor blind. It’s not that life is “extra-complicated” compared to “simple universe”, but our algorithms-laws-constructs of human minds are only local approximations of the reality. Life seems more complicated, since it seems to cover a wider (or mid) range of scales. Algorithmic approximations to single real life events is valuable for predictive purposes, but not for a holistic metaphysical view of the whole World.
NL:
1: Science etc “evolve” by design.
2: God of the gaps is utterly misdirected, in fact the issue is the roots of reality, and it can be readily shown that his must come from necessary being, as if ever there were utter nothing such would therefore forever obtain as it has no causal capacity.
3: Further to this, we cannot be reasoning, warranting, confidently knowing beings, save for being responsibly and rationally free. This cannot be founded on a mere computational substrate as computation is inherently a blind causal process as mill wheels grinding upon one another. The functional organisation is externally imposed by intelligently directed configuration. We see here what we can term the IS-MIND gap.
4: As the responsible part this points to the IS-OUGHT gap, and there is but one level where that can be founded, the necessary being root of reality. Thus we see the only serious candidate, after centuries of debates: the inherently good Creator God, a necessary and maximally great being, worthy of loyalty and the reasonable, responsible service of doing the good in accord with our nature.
The god- of- the- gaps- strawman- dismissal and contempt- about- alleged- fraud — dupes — collapse. (Do you really like being a grand conspiracy theorist?)
KF
In regards to algorithmic information, free will is required for the creation of new algorithmic information:
Chaitin holds that an infinite number of true mathematical theorems exist that cannot be proved from any finite system of axioms.
Even Weinberg, reluctantly, concedes this point:
Moreover, Dr. Michael Denton states that there are ‘countless thousands of intricate algorithms’ controlling the billions upon billions of cells in an organism:
And Godel was certainly no friend of Darwinian evolution
Verse:
Hi Mung,
Thank you very much for drawing my attention to Dr. David Abel’s book, which sounds fascinating. I was also interested to read your thoughts on the nature of life in post #11 above. Perhaps you are right; maybe life cannot be expressed even as a sequence of algorithms, although it wouldn’t worry me if it could be.
By the way, I’ve fixed the first link in my OP.
Thanks again.
Hi nightlight,
Thank you for your posts. It may well be the case that life is expressible algorithmically. But if that is so, then the number of algorithms would have to be very large, and the algorithms would also have to work in a top-down fashion, as life does.
We know of no unguided process which is capable of generating origami cranes – or for that matter, the origami bull in my illustration at the top of my OP. And we know of no unguided process which is capable of generating living things from non-living matter (abiogenesis). Generating a life-form, like the act of making an origami animal, requires two things: (i) the ability to follow a very long sequence of steps; and (ii) the ability to create these steps in the first place. Intelligent agents are the only entities known to possess both capabilities.
nightlight (at post #18),
I take it you are referring to Godel’s First Incompleteness Theorem here?
If so, do you have a reference for this statement about the necessary incompletess of natural science?
Nightlight, would Godel’s concept of incompleteness not be of an ‘a priori’ nature, and therefore preclude any degree of evolution from being sufficient to close the gap.
Did you, perchance, have that fabled, evolutionary ‘missing link’ in mind, when invoking your own remedial, wee, ‘god of the gaps’ ?
‘One day, my son… man will have plumbed the deepest. most arcane mysteries of this universe…; indeed, the multiverse…’
That’s just like Mandelbrot set algorithm which can never produce the exact target set as mathematically defined, but can only get closer with each iteration. Yet, the algorithm modeling it and the one defining it (via infinite itteration Z*Z+C -> Z) are quite simple algorithms.
What we consider physical laws as presently known may be merely some coarse grained approximations of few aspects of an infinitely deep algorithmic process unfolding in the underlying substratum (pregeometry, e.g. some kind of infinitely deep, open ended, non-terminating computational process).
There is no known case of fundamental incongruity or inconsistency between phenomena. At worst, there is incomplete understanding or imperfect scientific model of some phenomena. But no one can point to some fundamental clash of phenomena (whether at different or same scales) that is unresolvable (or even merely unknowable) in principle.
That’s exactly my position.
We presently divide our scientific model of the universe into “laws” and initial/boundary condition (IBC). The full ‘physics algorithm’ needs both kinds of inputs to describe or model the system (universe). The small part of the input, the “laws”, expresses the regularity of phenomena we understand so far, while the much larger part of the input, IBC, is arbitrary or hand-put part of input data for the algorithm, the fudge factor.
As science advances, that arbitrary, hand-put part of the input (the IBC) is gradually migrating into the lawful part (the “laws”, or the part of the algorithm of the universe we have reverse engineered so far). Hence the boundary between the laws and IBC is a transient artifact of our present knowledge, not an intrinsic property of nature itself.
When we describe something as intelligent, purposeful activity operating on matter-energy (such as designing & constructing an “irreducibly complex system” or generating “specified complexity”), we’re merely saying that the IBC of that chunk of matter-energy were not some simple random distribution (of physical quantities, such as forces, pressures, locations, etc), but some very special kind/singular IBC which is necessary to produce the observed/known “irreducibly complex system” or “specified complexity”.
But since the boundary between the “lawful” part of the input into the physics algorithm (the physical laws) and the arbitrary, hand-put part of the input (IBC) is a transient artifact of our present knowledge of the underlying algorithms of the universe, the concept of “intelligent activity” (which is based on the present boundary between “laws” and IBC) is an epistemological artifact of the present human knowledge and understanding of the physical laws, not an ontological category or attribute of the universe itself (as Seattle ID misinterprets it).
For our description of universe as a whole, over its entire existence, we usually assume that IBC were/are some simple random distributions (uniform, Gaussian, Poisson, etc). But the biological phenomena, as well as the fine tuning of physical laws & constants for life indicate that such assumption is inadequate. Namely, there is even much finer tuning of the IBC that is needed to produce these phenomena i.e. this special tuning of IBC could be called “intelligent activity” within the framework of our present understanding of the “physical laws”.
In contrast to the misguided prescription of Seattle ID (just label it ‘intelligent intervention’ or ‘god’ and cease any further scientific research beyond that point), the proper new science will seek new physical laws to explain life. These new laws (or deeper/finer algorithms) will further shift the above epistemological boundary between the laws and IBC, providing purely algorithmic explanation for the origin of life as a plain lawful (algorithmic, computational) process. That is far more useful and productive project and approach than the one advocated by Seattle ID — the intelligent tweaking of the IBC (as classified by physical laws as presently known) of some molecules in the far away past by a capricious deus ex machina (which hopefully coincides with the tribal deity of certain ancient mid-eastern shepherds), in order to help (at His whim) the “natural laws” produce “irreducibly complex systems”.
The Seattle ID is a result of gross misunderstanding of what physical laws are (transient epistemological constructs) and how they relate to IBC (the theater and indicator of their ‘intelligent activity’). They’re basically confusing the map and the territory (epistemology and ontology), attributing ontological status (intelligent agency) to the transient epistemological artifacts (the present boundary between “physical laws” and IBC). As result, their deus ex machina is a ‘god of gaps’ forever doomed to retreat down the always shrinking and ever more intricate web of gaps within the present scientific knowledge.
nightlight: That’s just like Mandelbrot set algorithm which can never produce the exact target set as mathematically defined, but can only get closer with each iteration. Yet, the algorithm modeling it and the one defining it (via infinite itteration Z*Z+C -> Z) are quite simple algorithms.
What we consider physical laws as presently known may be merely some coarse grained approximations of few aspects of an infinitely deep algorithmic process unfolding in the underlying substratum (pregeometry, e.g. some kind of infinitely deep, open ended, non-terminating computational process).
That’s actually makes for a nice analogy to science. There are some points for which we know with certainty whether they are in the Mandelbrot Set or not. There are some points where it is difficult to discern, but we can, with enough effort, make the distinction. And there are some points which are beyond our current ability to make the determination. Meanwhile, our discernment continues to improve.
BTW, fractals are often found in nature, simple iterations leading to complexity.
Hi Mung,
I consider Abel one of great thinkers of today. Yes, he insists that data (a pattern or the absence of it) is meaningless without its processor (and likewise the processor without data).
What I found enlightening for myself was he stressed the importance of pragmatic utility as a key factor in biological systems organisation as well as in artificially created information processing systems (of which I had been aware). Biological organization is formal which could only have originated on the far side of the cybernetic cut.
nightlight:
Of course, I disagree with you.
You speak of “the arbitrary, hand-put part of the input” (initial/boundary condition (IBC)), and simply ignore the role of consciousness and conscious cognition and purpose in design inputs.
You assume that IBCs should be reduced to necessity laws as our knowledge increases. But there is no clue at all that consciousness can be explained by necessity laws, or simply by laws regarding objects.
As many others do, you simply ignore the “hard problem of consciousness”, as defined by Chalmers, and therefore the role of conscious cognition and intent in the generation of functional complexity, a role which is instead supported by all available facts and observations. In that way, you fall into a dogmatic perspective, which rejects the available facts in the fanciful hope that new, unexpected facts will allow you to get explanations in some other way.
That is no good science and, IMO, not even good philosophy.
And just a final suggestion: instead of arrogantly voicing wrong opinions against what you call “Seattle ID”, be more respectful and talk to the people who are here, where you have come to discuss. For example, I live in Italy, have never been to Seattle, and have nothing to do with Seattle. That is certainly true of most people who write here. We are thinking people, not a generalization or a caricature of your imagination.
Zachriel:
I am not sure that I agree. You say:
“There are some points for which we know with certainty whether they are in the Mandelbrot Set or not. There are some points where it is difficult to discern, but we can, with enough effort, make the distinction. And there are some points which are beyond our current ability to make the determination. Meanwhile, our discernment continues to improve.”
I am not a mathematician, but I think that each time that we compute a mandelbrot image, we apply the algorithm with certain constraints (of resolution, number of iterations, and so on). In that context, each point has its own categorization (in or out of the set), and a definite image is drawn. As far as I understand, the mandelbrot algorithm is completely deterministic. Of course, if we change the resolution or the iterations, or any other parameter in the computation, the results will change.
If I am wrong, please correct me.
Far from God having to retreat further and further as science has progressed, i.e. God of the gaps, the truth of the matter is that materialism/naturalism has had to retreat further and further as science has progressed, i.e. ‘materialism of the gaps’,:
As you can see when we remove the artificial imposition of the materialistic philosophy (methodological naturalism), from the scientific method, and look carefully at the predictions of both the materialistic philosophy and the Theistic philosophy, side by side, we find the scientific method is very good at pointing us in the direction of Theism as the true explanation. – In fact science is even very good at pointing us to Christianity as the solution to the much sought after ‘theory of everything’
Zachriel:
I certainly agree that “fractals are often found in nature, simple iterations leading to complexity.” I have said that myself in my post #12.
However, I don’t think that we can find mandelbrot sets in nature.
I quote myself:
“c) I believe that the Mandelbrot set, as implemented in some material object (for example, a print of it) has some specified complexity, which corresponds at least to the minimum number of bits required to implement a program which can compute it. That is not a huge specified complexity, but it is not low. I believe that all the examples we know of Mandelbrots are designed. It is perfectly possible, however, that simpler fractals arise in natural systems (and they certainly do).”
Fractals are really ubiquitous and can be found in both animate and inanimate nature. So what? How does it disprove formal organization of living things?
At 34, gpuccio quoted Zachriel:
and gpucci replied,
On the one hand, I certainly agree gpuccio’s point. Any manifestation of the Mandelbrot set is necessarily incomplete: no matter how refined our equipment or the length of time rendering, we can’t create a representation of an infinitely dense object. This similar to the fact that we can never know the complete decimal representation of pi.
However, I wonder about Zachriel’s point that “there are some points which are beyond our current ability to make the determination, [but] meanwhile our discernment continues to improve.
There are two things to think about here: first, due to the resolution issue mentioned by gpuccio, there will also be points which we haven’t tested yet with the algoritm to see if they are in the set or not. For instance, if we tested all points who’s decimal representations are 20 decimal places deep, there wil be points 21 decimal places deep for which we don’t know whether they are in the set, because we haven’t tested them yet. If this is what Zachriel is referring to when he says, “And there are some points which are beyond our current ability to make the determination,” then I agree, but I’m not sure what he means.
So the question I ask, and I don’t know the answer and don’t know if anyone here does, is this: are there points within the range of the resolution of our representation that, even if we continue the algorithm on them for a very long time, fail to converge or diverge, and thus leave their status of being in the set or not uncertain?
That is, is it possible to pick a point that doesn’t give us a clear result, not because it is beyond the range of points we have tested, but because when tested it doesn’t converge or diverge fast enough for us to know?
I’m really not sure whether I am describing the problem I see clearly enough for others to understand, or whether there is such a problem. I’m inclined to think the answer to my question is no – that no matter what resolution we choose (20 decimal places, or 25 or whatever), the limits on whether we can decide whether the points are in or out are because of the sheer number of points we are testing, and not because some points fail to clearly converge or diverge.
‘…the truth of the matter is that materialism/naturalism has had to retreat further and further as science has progressed, i.e. ‘materialism of the gaps’,:
!!!!!
If nightlight could discredit at least one of the points you raised regarding the relative achievements of theism and materialism, it might be worthwhile arguing with him, but in the mean time, we can, surely, only, sensibly, marvel at the wildly promiscuous credulity of his scientism.
Aleta:
I agree with your points. Regarding your question, even if, as said, I am not a mathematician, I’m also inclined to think the answer to your question is no.
gpuccio: I am not a mathematician, but I think that each time that we compute a mandelbrot image, we apply the algorithm with certain constraints (of resolution, number of iterations, and so on).
The computation is only an approximation of the Mandelbrot Set. Resolution of points near the boundary are constrained by computational resources. Points near the boundary may be incorrectly placed in (or out) of the Set.
gpuccio: As far as I understand, the mandelbrot algorithm is completely deterministic.
That’s correct.
gpuccio: I believe that all the examples we know of Mandelbrots are designed. It is perfectly possible, however, that simpler fractals arise in natural systems (and they certainly do).
Fractals in nature are generally more complicated, not less complicated, as the iteration is responsive to local conditions.
Aleta: If this is what Zachriel is referring to when he says, “And there are some points which are beyond our current ability to make the determination,” then I agree, but I’m not sure what he means.
Yes, that’s it. Some points we know. Some points are at the edge of understanding. Some points are beyond our current ability to discern, but our resolution has improved considerably since Poincaré.
Aleta: That is, is it possible to pick a point that doesn’t give us a clear result, not because it is beyond the range of points we have tested, but because when tested it doesn’t converge or diverge fast enough for us to know?
Hubbard and Douady showed that the Mandelbrot Set is connected. Of course, it may take unbounded resources to resolve. (By the way, periodic points don’t escape, so are considered to be in the Mandelbrot Set.)
Zachriel, thanks for the reminder about periodic points counting as being in the set. Also, I think it is very cool that there is a proof that all points in the set are connected, even though we could never demonstrate that “empirically”, by brute computational force.
as to:
Yes, it seems common sense is lost,,,
Of related note:
Leibniz’ Contingency Argument – Dr. Craig animated video
https://www.youtube.com/watch?v=FPCzEP0oD7I
BRUCE GORDON: Hawking’s irrational arguments – October 2010
Excerpt: ,,,The physical universe is causally incomplete and therefore neither self-originating nor self-sustaining. The world of space, time, matter and energy is dependent on a reality that transcends space, time, matter and energy.
This transcendent reality cannot merely be a Platonic realm of mathematical descriptions, for such things are causally inert abstract entities that do not affect the material world,,,
Rather, the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them. This is what “breathes fire into the equations and makes a universe for them to describe.” Anything else invokes random miracles as an explanatory principle and spells the end of scientific rationality.,,,
For instance, we find multiverse cosmologists debating the “Boltzmann Brain” problem: In the most “reasonable” models for a multiverse, it is immeasurably more likely that our consciousness is associated with a brain that has spontaneously fluctuated into existence in the quantum vacuum than it is that we have parents and exist in an orderly universe with a 13.7 billion-year history. This is absurd. The multiverse hypothesis is therefore falsified because it renders false what we know to be true about ourselves. Clearly, embracing the multiverse idea entails a nihilistic irrationality that destroys the very possibility of science.
Universes do not “spontaneously create” on the basis of abstract mathematical descriptions, nor does the fantasy of a limitless multiverse trump the explanatory power of transcendent intelligent design. What Mr. Hawking’s contrary assertions show is that mathematical savants can sometimes be metaphysical simpletons. Caveat emptor.
http://www.washingtontimes.com.....arguments/
It is true that Mandelbrots are produced using a very simple algorithm.
It is also true that Mandelbrots don’t actually do anything.
Sometimes beauty is its own reward.
Mung: Sometimes beauty is its own reward.
Aye