Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

July 18, 2008

Hierarchy and Emergence

Posted by David Corfield

Many entities with which we deal can be said to fit into a hierarchical order:

  • sound, vocable, word, utterance, conversation, discourse
  • character, word, sentence, paragraph, chapter, book, encyclopaedia/collection
  • sound wave, note, chord, phrase, passage, movement, symphony, style (Baroque, etc.)
  • particle, molecule, molecular assembly, organelle, cell, cell assembly, organ, …
which might end
  • …organism, herd/shoal, species
  • …person, household, local community, nation state

Now you could claim that mathematics works best at the lower levels of these hierarchies, whereas narrative is the necessary tool to describe the higher levels. For example, we have a good mathematical theory of sound waves, but ask what Beethoven achieved with his Eroica symphony and we start talking about romanticism, Napoleon and the ideals of the French Revolution.

We can set anyone, or better a computer, the task of counting the relative frequency of the letters of the alphabet in a text, a reasonably educated person to count the frequency of metaphors, but we only listen to the subtle critic to learn about War and Peace, who will tell us among other things about the state of Russia at the time Tolstoy wrote it. Mathematics struggles to grasp reality as it climbs the hierarchy leaving us little better off than the numbering of the Eroica as Beethoven’s third symphony, or the count of the chapters of War and Peace.

We can use the power of a computer to overcome some of its limitations in getting the point of higher levels by its enormous calculating power operating algorithmically on vasts amounts of data. For example, set the task of accurately classifying web documents as being about finance or about sport, a machine learning algorithm may succeed by treating each document merely as a bag of words. Trained on sufficiently many documents it may learn to classify these well, but it certainly feels as though this hasn’t come about through the understanding of the words’ meaning.

For Polanyi, there’s an important point lurking here, which concern a feature current computers lack. To have an entity in your focal attention you must have its constituent parts in your subsidiary attention. You have to be able to see from the parts to the whole. As soon as you look focally at the parts, the whole disappears. So you can look at a page and just see letters and no words, or look at words and see no sentences. Polanyi reports the experience of reading a letter he had received, wishing to show it to his son, and only then remembering to check whether it was in a language the latter could understand. Meaning is gleaned without focal awareness of the words conveying it. To see the higher level once must indwell in its lower parts, as he puts it.

We can also arrange mathematical thought in a hierarchy:

  • inference step/definition, theorem, paper, subject, branch, research programme

Yet again mathematics applies better to the lower levels, here in the shape of mathematical logic. At the top level we may chart the number of articles published for various programmes on the ArXiv, but we’d rather hear the story of a Grothendieck or a Connes. (For a great story by Raoul Bott about Morse theory see here.)

Three questions, and then some vague thoughts in reply:

  • Does the type of mathematics which can be used typically vary up a hierarchy?
  • Is there any commonality to the way the functioning of a level emerges from its parts, across and within hierarchies?
  • Is something special introduced when dealing with the activities of humans?

The 2 April 1999 issue of Science had several interesting articles on complex systems and emergence. To the extent that there was a consensus on the necessary mathematics, nonlinear differential equations seemed to be thought to work best, for, e.g., gene and protein networks and species population levels.

Yet higher up the hierarchy still, Stephen Jay Gould asks about the history of life on Earth

How can a vector of historically unique events be viewed as primary data, or as bearers of theoretical interest?

Isn’t science a search for the timeless and quantifiable laws of nature? Isn’t the specification of a sequence of events no more than a narrative, a description of uniqueness worth little in an enterprise dedicated to experiment, repetition, and prediction?

and answers himself

Science is a pluralistic search to understand nature’s ways–and the narrative quality of historical sequences records a different aspect of nature accessible to legitimate methods beyond the stereotype.” (‘An Operational Definition of Directionality’ in The Philosophy of Biology, Hull and Ruse (eds.) OUP, p. 666)

But he does allow that one can find a directionality in evolution, e.g., that there is a tendency for the maximum flourishing point of a clade to occur later in its geological range the more recently it occurs. Then again, philosopher R. G. Collingwood would want to reserve the word ‘history’ for human affairs, taking our acts, res gestae he called them, to form qualitatively different sequences to the courses of events concerning other species.

On the other hand, we might heed Weyl

Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind – as Eddington puts it – the colorful tale of the subjective storyteller mind. (Philosophy of Mathematics and Natural Science, p.237)

We had a discussion a few months ago about whether category theory could get any grip on biological phenomena. One small resonance between some of the work mentioned there and the focal-subsidiary idea of Polanyi is the concept of a collection of lower elements being represented by a direct limit at the higher level.

On the other hand, Polanyi suggests the indwelling becomes more difficult the higher up a hierarchy you go. It may not be easy to extract a general law from moments in the motion of a stone, but it requires lower indwelling capacities than to understand what an animal is doing from its movements.

It’s curious that much work in ethology, and especially primatology, involving years of patient field work, is done with far more sensitivity to the subject at its highest level than occurs in many parts of psychology. Anthropology is more like ethology in this respect.

Posted at July 18, 2008 1:49 PM UTC

TrackBack URL for this Entry:

21 Comments & 0 Trackbacks

Complex Systems, Dictionaries, Bio-Theorems; Re: Hierarchy and Emergence

This subject enthralls me.

I’ll comment in narrative later, driven by my roughly 40 years involvement5 with Complex Systems research, in Biophysics, and what’s now called nanotechnology and Artificial Life, but first point to these recent references:

arXiv:0806.3710 [pdf, other]
Title: How Is Meaning Grounded in Dictionary Definitions?
Authors: A. Blondin Masse, G. Chicoisne, Y. Gargouri, S. Harnad, O. Picard, O. Marcotte
Comments: 8 pages, 3 figures, TextGraphs-3 Workshop at the 22nd International Conference on Computational Linguistics, Coling 2008, Manchester, 18-22 August, 2008
Subjects: Computation and Language (cs.CL); Databases (cs.DB)

Meaning cannot be based on dictionary definitions all the way down: at some point the circularity of definitions must be broken in some way, by grounding the meanings of certain words in sensorimotor categories learned from experience or shaped by evolution. This is the “symbol grounding problem.” We introduce the concept of a reachable set – a larger vocabulary whose meanings can be learned from a smaller vocabulary through definition alone, as long as the meanings of the smaller vocabulary are themselves already grounded. We provide simple algorithms to compute reachable sets for any given dictionary.

The Clay Mathematics Institute has placed their library of publications online. Their most high-profile publication (other than the Millennium Problems) is Morgan and Tian’s write-up of the proof of the Poincare Conjecture.

They have an interesting article by Bernd Stermfels, Can Biology Lead to New Theorems? You can guess his answer from the fact that the article exists at all.

Nice intro, pretty pictures, subtle Math, and great buzzwords such as:

“the tree of life is an affine building.”
[Andreas Dress, Director, Institute for Computational Biology, Shanghai]

Posted by: Jonathan Vos Post on July 18, 2008 5:19 PM | Permalink | Reply to this

Molecular, Community and Network Evolution; Re: Complex Systems, Dictionaries, Bio-Theorems; Re: Hierarchy and Emergence


New submission for Fri, 18 Jul 08:

[1] arXiv:0807.2665 [ps, pdf, other]
Title: Unifying Theories of Molecular, Community and Network Evolution
Authors: Carlos J. Melian, David Alonso, Diego P. Vazquez, James Regetz
Comments: Main 42 pages, Appendix 28 pages, 5 figures
Subjects: Populations and Evolution (q-bio.PE); Other (q-bio.OT)

The origin of diversification and coexistence of genes and species have been traditionally studied in isolated biological levels. Ecological and evolutionary views have focused on the mechanisms that enable or constrain species coexistence, genetic variation and the genetics of speciation, but a unified theory linking those approaches is still missing. Here we introduce evolutionary graphs in the context of neutral theories of molecular evolution and biodiversity to provide a framework that simultaneously addresses speciation rate and joint genetic and species diversities. To illuminate this question we also study two models of evolution on graphs with fitness differences, which provide insights on how genetic and ecological dynamics drive the speed of diversification. Neutral evolution generates the highest speed of speciation, species richness (i.e. five times and twice as many species as compared to genetic and ecological graphs, respectively) and genetic–species diversity (i.e., twice as many as genetic and ecological graphs, respectively). Thus the speed of speciation, the genetic–species diversity and coexistence can differ dramatically depending on whether genetic factors versus ecological factors drive the evolution of the system. By linking molecular, sexual and trophic behavior at ecological and evolutionary scales, interacting graphs can illuminate the origin and evolution of diversity and organismal coexistence.

Posted by: Jonathan Vos Post on July 18, 2008 6:12 PM | Permalink | Reply to this

Re: Complex Systems, Dictionaries, Bio-Theorems; Re: Hierarchy and Emergence

Meaning cannot be based on dictionary definitions all the way down: at some point the circularity of definitions must be broken in some way, by grounding the meanings of certain words in sensorimotor categories learned from experience or shaped by evolution.

The Cyc project is attempting to give computers common sense, giving meaning to words in relation to other words. There are lots of terms and rules encoding a giant ontology. They sell systems that can, for instance, flag suspicious data–is Billy really 237 years old, or did you hit two keys at once? It knows that people have legs, and that if your right leg is in Kansas, your left leg is probably there, too.

Abstraction is just a data mining process on the ontology, hunting for analogies.

Posted by: Mike Stay on July 18, 2008 8:04 PM | Permalink | Reply to this

Re: Complex Systems, Dictionaries, Bio-Theorems; Re: Hierarchy and Emergence

“We introduce the concept of a reachable set; a larger vocabulary whose meanings can be learned from a smaller vocabulary through definition alone, as long as the meanings of the smaller vocabulary are themselves already grounded. We provide simple algorithms to compute reachable sets for any given dictionary.”

SH: I’m interested in distinguishing pattern from randomness from examining output. What output fits a particular generating algorithm. Pi has such an algorithm and although its output passes randomness tests, Pi is not considered truly random by AIT definition. I don’t think there is a method to test to see if there are other algorithms which like Pi, pass all randomness tests, so when one sees a sequence of symbols, one can’t distinguish between if they belong to an finite pattern origin, or are infinitely random, or even just hugely large finite.

The symbols used can make well-defined words up to a point. The understanding of meaning is resolved from context which changes or is modified by other contexts. This has the effect of producing ever growing in complexity and size meanings of defined words so the dictionary grows.
That’s how a natural language dictionary works. A formal dictionary is only going to approximate the human function of reading and understanding so that dict. must also attenuate or compartmentalize.

To get to the point. A dictionary is a set of rules which demonstrate pattern.
John Case is an Emeritus COLT professor.

He gives an example of a complicated sequence which has a generating rule used to form a detectable pattern. Then says,

“This rule can be written as a formal algorithm (or computer program). The problem of finding such rules gets harder as the sequences to generate get more complicated than the one above. Can the rule finding itself be done by some
computer program? Interestingly, it is
mathematically proven that there can be no computer program which can eventually find (synonym: learn) these (algorithmic) rules for all sequences which have such rules!”
My point is that if one attempts to capture meaning (find a rule which generates a meaningful pattern) that there is going to be a natural limitation imposed on the scope of its success, whether commonsense rules for CyC or a cycling self-referential dict. foundation.

I’m going to include a quote from David’s post because it seems to fit here, “a bag of words” and Polanyi.

“Meaning is gleaned without focal
awareness of the words conveying
it. To see the higher level once
must indwell in its lower parts,
as he puts it.” … On the other hand, Polanyi suggests the indwelling becomes more difficult the higher up a hierarchy you go.”
SH: I agree with this :-) and tried to use the word limitation above to express not being able to immerse and indwell. I think it matches the complicated algorithmic limitation that Case mentions
though maybe “complex” is a better word.
No holes in my bag of Scrabble letters.

Posted by: Stephen Harris on July 19, 2008 12:51 AM | Permalink | Reply to this

Re: Complex Systems, Dictionaries, Bio-Theorems; Re: Hierarchy and Emergence

An interesting project to get substantive about grounding for natural languages is Anna Wierzbicka’s Natural Semantic Metalanguage program; the idea is to find a core set of meanings that are rendered in the everyday vocabulary of all languages, to define all the others with. The current presentation is completely unmathematical (almost all the people doing it are math-phobes), but I think this could be changed by formalizing the syntax of the metalanguage.

Another random observation is that the assembly of meanings seems to involve some sort of linear logic (no Contraction or Weakening, i.e. the meaning of each word gets used once and once only), while reasoning itself doesn’t obey this restriction.

Posted by: Avery Andrews on July 20, 2008 1:44 AM | Permalink | Reply to this

Symbolic Monkeys, language-independent gestures; Re: Complex Systems, Dictionaries, Bio-Theorems; Re: Hierarchy and Emergence


Also note:

The Symbolic Monkey? Animals Can Comprehend And Use Symbols, Study Of
Tufted Capuchins Suggests

Addessi et al. Preference Transitivity and Symbolic
Representation in Capuchin Monkeys (Cebus apella). PLoS One, 2008; 3
(6): e2414 DOI: 10.1371/journal.pone.0002414­/releases/2008/06/080610212404.htm


When Using Gestures, Rules Of Grammar Remain The Same

{image: Speakers in four different languages all use the same word order when making gestures to communicate ideas. They put the subject first, followed by the object and then the verb, despite the order
used in many spoken languages. (Credit: Image courtesy of University
of Chicago)}

ScienceDaily (July 7, 2008) — The mind apparently has a consistent way
of ordering an event that defies the order in which subjects, verbs,
and objects typically appear in languages, according to research at
the University of Chicago.

“Not surprisingly, speakers of different languages describe events
using the word orders prescribed by their language. The surprise is
that when the same speakers are asked to ‘speak’ with their hands and
not their mouths, they ignore these orders – they all use exactly the
same order when they gesture,” said Susan Goldin-Meadow, lead author
of a new paper in the Proceedings of the National Academy of Sciences.

For the study, the team tested 40 speakers of four different
languages: 10 English, 10 Mandarin Chinese, 10 Spanish and 10 Turkish
speakers. They showed them simple video sequences of activities and
asked them to describe the action first in speech and a second time
using only gestures. They also gave another 40 speakers of the same
languages transparencies to assemble after watching the video
sequences. Some of the videos portrayed real people and others
animated toys that represented a variety of sentence types: a girl
waves, a duck moves to a wheelbarrow, a woman twists a knob and a girl
gives a flower to man.

When asked to describe the scenes in speech, the speakers used the
word orders typical of their respective languages. English, Spanish,
and Chinese speakers first produced the subject, followed by the verb,
and then the object (woman twists knob). Turkish speakers first
produced the subject, followed by the object, and then the verb (woman
knob twists).

But when asked to describe the same scenes using only their hands, all
of the adults, no matter what language they spoke, produced the same
order —- subject, object, verb (woman knob twists). When asked to
assemble the transparencies after watching the video sequences
(another nonverbal task, but one that is not communicative), people
also tended to follow the subject, object, verb ordering found in the
gestures produced without speech.

Posted by: Jonathan Vos Post on July 20, 2008 5:17 PM | Permalink | Reply to this

Re: Symbolic Monkeys, language-independent gestures; Re: Complex Systems, Dictionaries, Bio-Theorems; Re: Hierarchy and Emergence

A minor quibble about the signing is that it would really be `Agent’, `Patient’, ‘Action’, since the ad-hoc gestural systems probably don’t have real parts of speech and grammatical relations (but the developed sign languages used by deaf people do).

Agent-Patient/subject-object ordering (at least as the default) is the overwhelming majority option in languages, while the position of the verb is highly variable.
Various linguists have proposed universal ‘underlying’ orders, but don’t seem to agree. SVO and SOV being the options.

Posted by: Avery Andrews on July 20, 2008 11:28 PM | Permalink | Reply to this

Re: Hierarchy and Emergence

You could come up with a list like:

superstrings, fundamental particle, nucleus, atom, molecule, crystal, rock, asteroid, planet, star, galaxy, cluster of galaxies, supercluster of galaxies, Universe

Physics, and thus mathematics, is used to describe all of these things perfectly well. As you know, there’s a deep connection between particle physics and cosmology. So in other words, whether mathematics is useful in describing it has nothing to do with the scale of physical size.

Jeffery Winkler

Posted by: Jeffery Winkler on July 18, 2008 7:53 PM | Permalink | Reply to this

Re: Hierarchy and Emergence

I suppose one might argue that several steps in your sequence involve taking a collection of weakly interacting homogeneous entities, and that is why mathematics continues to work adequately, where the organism as a combination of disparate tissues and organs is less mathematisable.

Even with very mathematisable entities there’s a tendency to use a narrative description. E.g., for stars:

After a star has formed, it generates energy at the hot, dense core region through the nuclear fusion of hydrogen atoms into helium. During this stage of the star’s lifetime, it is located along the main sequence at a position determined primarily by its mass, but also based upon its chemical composition and other factors. In general, the more massive the star the shorter its lifespan on the main sequence. After the hydrogen fuel at the core has been consumed, the star evolves away from the main sequence.

In a paper on modelling in Hadron physics, Stephan Hartmann writes

Working in various physics departments for a couple of years, I had the chance to attend several PhD examinations. Usually, after the candidate derived a wanted result formally on the blackboard, one of the members of the committee would stand up and ask: “But what does it mean? How can we understand that x is so large, that y does not contribute, or that z happens at all?” Students who are not able to tell a “handwaving” story in this situation are not considered to be good physicists.

Judging from my experience, this situation is not only typical for the more phenomenological branches of physics (such as nuclear physics) but also for the highly abstract segments of mathematical physics (such as conformal field theory), though the expected story may be quite different.

In this paper, I want to show that stories of this kind are not only important when it comes to finding out if some examination candidate “really understands” what he calculated. Telling a plausible story is also an often used strategy to legitimate a proposed model.

If I am right about this, empirical adequacy and logical consistency are not the only criteria of model-acceptance. A model may also be provisionally entertained (to use a now popular term) when the story that goes with it is a good one. But what criteria do we have to assess the quality of a story? How do scientists use the method of storytelling to convince their fellow scientists of the goodness of their model? Is the story equally important for all models or do some models need a stronger story than others? These are some of the questions that I address in my paper. In doing so, I draw on material from a case-study in hadron physics.

He concludes

We will now make more precise what a story is. A story is a narrative told around the formalism of the model. It is neither a deductive consequence of the model nor of the underlying theory. It is, however, inspired by the underlying theory (if there is one). This is because the story takes advantage of the vocabulary of the theory (such as ‘gluon’) and refers to some of its features (such as its complicated vacuum structure). Using more general terms, the story fits the model in a larger framework (a ‘world picture’) in a non-deductive way. A story is, therefore, an integral part of a model; it complements the formalism. To put it in a slogan: a model is an (interpreted) formalism + a story.

I think this chimes well with Polanyi’s point that all forms of knowing rely on unformalisable components. In fact, he claimed all forms of knowledge rely on what is tacit, or inarticulable. We might say that stories can take us further in the effort to articulate as much as possible than formalisms alone.

Posted by: David Corfield on July 21, 2008 9:44 AM | Permalink | Reply to this

Emperor’s Blog Threads of Ideas; Re: Hierarchy and Emergence

I am convinced that what David Corfield quotes and interprets here is VERY important.

Stories can and do take us further than formalism alone – and Intuitionistic Logic is only one approach to explicating it.

Part of the problem, as I see it, with (for instance) String Theory, or Cosmology via Inflation and Dark Matter and Dark Energy, is that we have no good way but increasingly slow and expensive experiment to choose between a good model with a bad story and a bad model with a good story.

Science Fiction branches off from Science in valorizing Story over Empirical verification.

Poetry goes further than Story, but that’s another thread.

Speaking of which: the brilliant novel Galatea 2.2 by Richard Powers [1995] has the viewpoint character (also named Richard Powers) as a best-selling novelist intermediating between English Literature department and mad scientists in the “enormous new Center for the Study of Advanced Sciences. My official title was Visitor. Unofficially, I was the token hmanist.”

The eponymous connectionist AI is given the story of the Emperor’s New Clothes.

“I … asked ‘What are the new clothes made of?”

“After a long time, she answered, ‘The clothes are made of threads of ideas.’”


Posted by: Jonathan Vos Post on July 21, 2008 6:53 PM | Permalink | Reply to this

Re: Hierarchy and Emergence

Posted by: Mike Stay on July 21, 2008 7:08 PM | Permalink | Reply to this

Re: Hierarchy and Emergence

In Ray Kurzweil’s TED talk he uses a bag-of-words style technique to fit many things to exponentials, including the history of life on earth. One near-quote: “It took 2 billion years to develop DNA, but then the Cambrian explosion that built on that only took another 100 million years, a 20x speedup” and he draws analogies to the fact that we design tomorrow’s computers using today’s, not pencil and paper.

Posted by: Allen Knutson on July 18, 2008 9:52 PM | Permalink | Reply to this

Re: Hierarchy and Emergence

Kurzweil has several bad habits, such as treating processes which occurred over millions of years as pointlike events (the Cambrian explosion did not happen in a day, and neither did the invention of stone tools). In addition, ever since The Age of Spiritual Machines he’s been brushing off the “proximity of the familiar” — often, the timeline will be more crowded as you approach the present, not necessarily because things happen faster, but because we have more data. Start with a Poisson process in which each event leaves a “fossil”, add in a constant probability per unit time for a given fossil to decay, and your fossil record will show exponentially increasing waiting times between surviving fossils as you go back into the past. Uniformitarianism plus an imperfect fossil record gives the illusion of accelerating change.

Posted by: Blake Stacey on July 19, 2008 2:34 PM | Permalink | Reply to this

Re: Hierarchy and Emergence

Not to mention the way that he turns “exponential growth” into a “singularity”, blowing up in finite time. I’ve long since disregarded anything that even mentions “Kurzweil” (unless written by someone who I already know to be worth paying attention to).

Posted by: John Armstrong on July 19, 2008 5:04 PM | Permalink | Reply to this

Re: Hierarchy and Emergence

DC: “Is there any commonality to the way the functioning of a level emerges from its parts, across and within hierarchies?”

There used to be a dispute as to whether emergence meant: hard to predict, predictable but only in retrospect, or perhaps random (causal, but with no predictable pattern discerned from the output).

My example of Pi was to show an emergent pattern which was predictable in retrospect as I don’t think there is a method to discover other algorithms which produce a pattern that has the same characteristics as Pi, TM unending digits and passing all randomness tests. Nor, if there is a type of truly chaotic emergence, it doesn’t seem like there is a way of distinguishing output patterns which are non-random but appear random, and emergence which is truly random. Maybe that will establish a dichotomy between (dictionary) definitions of emergence and the ability to ascribe that definition to observed emergent phenomena judged by its output, even if such a process theoretically owns such a rule.

Posted by: Stephen Harris on July 19, 2008 4:10 AM | Permalink | Reply to this

Re: Hierarchy and Emergence

A distinction frequently made in the philosophical literature is between ontological and epistemological emergence, i.e., whether it’s just a question of our not being able to predict how the whole operates from knowledge of the parts, but that there are actually new, irreducible properties, which may be captured by laws for that level.

Posted by: David Corfield on July 21, 2008 9:50 AM | Permalink | Reply to this

Re: Hierarchy and Emergence

An argument that I’ve entertained my head, but have never heard articulated is that there is no real sense in which the larger groupings exist. So there’s no real sense in which “the nation” exists. Clearly the “state” exists, in that there is objectively a government of the US, and there’s a country that exists in that some people have legal citizenship, but there’s no meaningful sense in which you can make statements about this broad mass of people beyond the crude statistical sort.

Now that I think about it, this argument kinda resembles Tolstoy’s theory of history in War and Peace.

Posted by: Walt on July 19, 2008 4:45 AM | Permalink | Reply to this

Between Platonism and Antiplatonism; Re: Hierarchy and Emergence

There’s a metaphysical posture in between Platonist Realist and Antiplatonist (which comes in various flavors), regarding your comment: “So there’s no real sense in which “the nation” exists. Clearly the “state” exists, in that there is objectively a government of the US, and there’s a country that exists in that some people have legal citizenship.”

That is, the Nation and the State exist, as does Roman Law (and its derived Scottish Law) or Napoleonic Law (and its derivative in Louisiana) or English Law (and its offshoot American Law as Baseball and Cricket are offshoots of Rounders), or the Black Queen in Chess. They exist as mental constructs.

Persistent, consistent mental constructs, shared by many people, as part of the “stuff” of law or Chess. They obey consistently with respect to operators (amendment, appeal procedure, tournament Chess rules) which tranbsform Structures and have conserved Properties.

Mathematics, in this sense, “is carried in the heads of people” – but that does not deny them ontological or epistemological standing, nor make them mere games played on paper and blackboards for an elite to use to obtain power and money and prestige and sex, nor in any way prevent Emergence.

As to what “Emergence” means now, that is hotly debated at, for example, the International Conference on Complex Systems, every year (or every other year as it were). It nearly leads to fistfights.

Posted by: Jonathan Vos Post on July 21, 2008 6:02 PM | Permalink | Reply to this

Re: Between Platonism and Antiplatonism; Re: Hierarchy and Emergence

“As to what “Emergence” means now, that is hotly debated…”

I noticed Mark Bedau has 3 categories,
nominal, weak and strong. Linde and
Smolin write about “baby universes”
and if the laws of physics are not
fully inherited from the parent universe
that baby universe with mutated laws would seem to qualify as strong emergence?
[assuming they are not causally connected]

James Gardner says:
“Proliferating populations of baby
universes emerging from the loins
(metaphorically speaking) of black
hole rich mother universes thus come
to dominate the total population of
the multiverse a theoretical ensemble
of all mother and baby universes.”

“Each baby universe, says Smolin, is
not a perfect replica of its parent
but a slightly mutated form. ..
But if the random changes in the
workings of the laws of physics - the mutations - happen to allow a little
bit more inflation, a baby universe
will grow a little larger.”

Posted by: Stephen Harris on July 23, 2008 11:58 AM | Permalink | Reply to this

Re: Hierarchy and Emergence

This relates to the problem I have always had with Andrei Kolmogorov’s theory of complexity, in which an object’s degree of complexity is “measured” by the length of the string (or string of commands) needed to represent (or to recreate) it. Thus, in the most famous example, Malevich’s painting of a black square, is allegedly very simple, since one could recreate it with a very simply program of actions - paint the colour black on every pixel until the surface is covered. But such an approach ignores entirely the context of the actions.

There is no reason that Rembrandt, say, or Turner, could not have painted a black square. But they did not, and could not have, and (I believe) could not even have imagined doing so. (Perhaps only the 18th century painter Thomas Jones could have imagined doing so.) It is not a coincidence that Malevich’s painting appeared in the historical moment when it did, and not anytime before or anyplace else. To imagine that such a painting could be adequately described without reference to any art-historical- or socio-political- or history-of-ideas-context is to confuse the syntax of the painting with its semantics and pragmatics. Emergence and complexity are, crucially, properties of the latter, not the former.

Posted by: peter on July 23, 2008 6:04 PM | Permalink | Reply to this

How Fiction Works; Re: Hierarchy and Emergence

“We have to read musically, testing the precision and rhythm of a sentence, listening for the almost inaudible rustle of historical association clinging to the hems of modern words, attending to patterns, repetitions, echoes, deciding why one metaphor is successful and another is not, judging how the perfect placement of the right verb or adjective seals a sentence with mathematical finality.”

[How Fiction Works, James Wood, Farrar, Strauss and Giroux, 2008, 288 pp.]

Such Lovely Lies, Gideon Lewis-Kraus reviews “How Fiction Works” by James Wood.

“Wood’s new book, ‘How Fiction Works,’ is his most sustained attempt to describe, in 123 numbered sections, how this instruction proceeds. His initial focus is on ‘free indirect style,’ whereby the narrator moves the story along through a character’s voice such that ‘we see things through the character’s eyes and language but also through the author’s eyes and language. We inhabit omniscience and partiality at once.’ This is the essence of fiction-making: We readers know that an author has invented this character, but we also feel as though the character exists somewhere outside and beyond the author’s invention. We are divided between what we know to be fake and what we nevertheless momentarily postulate as real. Good characters promise us that their invented freedom has meaning, and we react to them accordingly. (Wood, in this way, pretends to write about books while writing about life.)”

“When we reach a word that ‘belongs’ not to the character but to the narrator, ‘we are reminded that an author allowed us to merge with his character, that the author’s magniloquent style is the envelope within which this generous contract is carried.’ This contract’s chief clause is that a novel is something you can at any moment put down. It is thus obliged to offer generous terms. As the burden of the novelist is to give her readers reason to keep reading, the burden of the untethered critic (as opposed to the academic one, whose authority is institutionally granted) is to offer enough gratuitous pleasure and intelligence that he is taken seriously….”

Posted by: Jonathan Vos Post on July 24, 2008 8:12 PM | Permalink | Reply to this

Post a New Comment