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Some particularly lucid excerpts on thinking

Created: 2008-01-14.
Updated: 2008-04-12.

Sometime in 2005 or 2006 I came across the Asimov anecdote, though I doubt that this source is the original. As it goes:
As an aside, Isaac Asimov would sometimes bring up the story of Archimedes, crying Eureka after solving the density of solids problem via water displacement, when he was trying to explain what he would do when he got stuck. Basically, Asimov would claim that to get past a rut he would watch a totally and completely brainless movie. Yeah, the kind that has no plot and just action. A Schwarzeneggar or Steven Segal movie would fit this bill. Anyway, after seeing a mindless movie and NOT thinking of his problem he would miraculously find the solution to his problem afterwards, as if his subconscious mind was solving the problem in the background. So his claim was that when you're under crunch time and need to solve a problem immediately and you're having trouble coming up with a solution is the exact time that you need to take a break to let your mind figure things out even though its counter-intuitive to do something like that.

I have especially noticed this fact in regard to ideas coming to me in the morning or evening in bed while in a semi-hypnagogic state. ... Perhaps we ought to seek the explanation in that preliminary period of conscious work which always precedes all fruitful unconscious labor. Permit me a rough comparison. Figure the future elements of our combinations as something like the hooked atoms of Epicurus. During the complete repose of the mind, these atoms are motionless, they are, so to speak, hooked to the wall; so this complete rest may be indefinitely prolonged without the atoms meeting, and consequently without any combination between them. On the other hand, during a period of apparent rest and unconscious work, certain of them are detached from the wall and put in motion. They flash in every direction through the space (I was about to say the room) where they are enclosed, as would, for example, a swarm of gnats or, if you prefer a more learned comparison, like the molecules of gas in the kinematic theory of gases. Then their mutual impacts may produce new combinations. What is the role of the preliminary conscious work? It is evidently to mobilize certain of these atoms, to unhook them from the wall and put them in swing. We think we have done no good, because we have moved these elements a thousand different ways in seeking to assemble them, and have found no satisfactory aggregate. But, after this shaking up imposed upon them by our will, these atoms do not return to their primitive rest. They freely continue their dance. Now, our will did not choose them at random; it pursued a perfectly determined aim. The mobilized atoms are therefore not any atoms whatsoever; they are those from which we might reasonably expect the desired solution. Then the mobilized atoms undergo impacts which make them enter into combinations among themselves or with other atoms at rest which they struck against in their course. Again I beg pardon, my comparison is very rough, but I scarcely know how otherwise to make my thought understood.
-- Henri Poincare's explanation of his mathematical creativity (found 2008-01-11 in The World of Mathematics vol. 3)
By that faculty of visualizing, which I learned in my boyish efforts to rid myself of annoying images, I have evolved what is, I believe, a new method of materializing inventive ideas and conceptions. It is a method which may be of great usefulness to any imaginative man, whether he is an inventor, businessman or artist.

Some people, the moment they have a device to construct or any piece of work to perform, rush at it without adequate preparation, and immediately become engrossed in details, instead of the central idea. They may get results, but they sacrifice quality.

Here, in brief, is my own method: After experiencing a desire to invent a particular thing, I may go on for months or years with the idea in the back of my head. Whenever I feel like it, I roam around in my imagination and think about the problem without any deliberate concentration. This is a period of incubation.

Then follows a period of direct effort. I choose carefully the possible solutions of the problem. I am considering, and gradually center my mind on a narrowed field of investigation. Now, when I am deliberately thinking of the problem in its specific features, I may begin to feel that I am going to get the solution. And the wonderful thing is, that if I do feel this way, then I know I have really solved the problem and shall get what I am after.

The feeling is as convincing to me as though I already had solved it. I have come to the conclusion that at this stage the actual solution is in my mind subconsciously, though it may be a long time before I am aware of it consciously.

Before I put a sketch on paper, the whole idea is worked out mentally. In my mind I change the construction, make improvements, and even operate the device. Without ever having drawn a sketch I can give the measurements of all parts to workmen, and when completed all these parts will Wt, just as certainly as though I had made the actual drawings. It is immaterial to me whether I run my machine in my mind or test it in my shop.

The inventions I have conceived in this way have always worked. In thirty years there has not been a single exception. My First electric motor, the vacuum tube wireless light, my turbine engine and many other devices have all been developed in exactly this way.
-- Nikola Tesla (found 2008-01-14 in the 1944 Tesla biography). suggests Polya for further insight
As Poincare observes, one often comes upon a sudden illumination {Poincare, 1913} after a period of conscious formulation, followed by a much longer period of non-conscious activity. I read his further discussion as proposing that the unconscious activity is a combinatorial heuristic search in which the chance of success depends mainly of the quality of the ingredients introduced by the preliminary conscious analysis; these elements are combined in different ways until a configuration is reached that passes some sort of test.

Polya formulated a heuristic thinking guidebook for mathematics
http://en.wikipedia.org/wiki/Polya - "The only student I was ever scared of was John von Neumann."

In these series of articles on Creativity and Design, my intention is to explore if there is a "formal" model of creativity. I am not interested in the psychological aspects of creativity, how it works and so on. The Psychology of discovery and invention is wonderfully described by Mihaly Csikszentmihalyi in his classic book called Creativity.

My interest is to discover a method of consciously practicing it. In the last few posts, I tried to provide a basic framework that I am working on. Basically I am trying to bring together all my experience under one unified theme. Design is the name I gave it.

I studied mathematics and more importantly I studied how to 'do' mathematics. I studied computer science and information systems -- I designed some very large and complex software systems. I studied design theory -- product design, appliance design, communication design, aesthetics, user interfaces, ergonomics etc. I studied philosophy -- Western Philosophy, Indian Philosophy and Sufi Philosophy. I studied systems thinking. Basically I am a problem-solver. Given any problem, I can come up with some kind of a solution. I met many people who are fantastic problem solvers. Many of these people can almost instantaneously identify a line of attack and can come up with a solution almost immediately.


From http://www.sdkrashen.com/articles/incubation/all.html --
Wallas (1926) reports that he first heard of the idea of incubation from the physicist Helmholz. In a speech delivered in 1891, Helmholz described how new thoughts came to him: After previous investigation, "in all directions," .. " happy ideas come unexpectedly without effort, like an inspiration ... they have never come to me when my mind was fatigued, or when I was at my working table ... They came particularly readily during the slow ascent of wooded hills on a sunny day" (p. 91).

Einstein clearly knew about incubation: According to Clark (1971), Einstein would "allow the subconscious to solve particularly tricky problems. 'Whenever he felt that he had come to the end of the road or into a difficult situation in his work,' his eldest son said, 'he would take refuge in music, and that would resolve all his difficulties.'" (p. 106). Clark notes that for Einstein, "with relaxation, there would often come the solution" (p. 106).

Csikszentmihalyi and Sawyer (1995) interviewed nine "creative" individuals, all of whom had made creative contributions in their field, were 60 or older, and were still actively involved in creative work. All mentioned that insights occurred during idle time, and several mentioned that they occurred while they were doing something else, during a "repetitive, physical activity" such as gardening, shaving, taking a walk, or taking a bath (p. 348).
For intense insights, you need intense preperation. Feynman, in particular, read the Britannica. The encyclopedia. He read the encyclopedia. Several times over.
This is not to say, of course, that hard work is unnecessary. Quite the opposite is true. Many studies confirm that high achievers put in a tremendous amount of work, far more than less accomplished colleagues. They engage in the "preliminary period of conscious work which also precedes all fruitful unconscious labor" (Poincare, 1924). This preliminary work is labeled "preparation" by Wallas, and as "wrestling with ideas" by Elbow (1972, p. 129). Wallas notes that the educated person "can 'put his mind on' to a chosen subjects, and 'turn his mind off' ...." (p. 92). The educated person knows how, in other words, to prepare and then incubate.
Haha! Yeah. I mentioned Feynman, right? He was mentioned next on the same page. How's that for predictive ability? Here's the part:
Incubation sometimes requires a very long break: Feynman noted that "You have to do six months of very hard work first and get all the components bumping around in your head, and then you have to be idle for a couple of weeks, and then - ping - it suddenly falls into place ..." (Csikszentmihalyi and Sawyer, 1995, p. 350).
C.P. Snow, cited in Clark (1971), noted that Churchill "was not a fast worker ... but he was essentially a non-stop worker" (p. 106); perhaps some of the "non-stop" working was staring at the ceiling. If high achievers appear to accomplish tasks more quickly than others, I suspect it is only because they put in so much more time.
I'd note that this also includes putting more time into the preparation *as well as* the incubation phases.

This worries me: everybody is suggesting incubation periods. Perhaps it is harder to get off running when you have to bootstrap your own brain with self-feedback, but I suspect it is possible with sufficient intensity. Else, how can we have the most abstract mathematical structures, grounded only barely in our cultural history? Those abstractions (even the musical ones) are important to reach for. (Perhaps you can create your own symbol-set in the preparation phase. But doesn't this influence (limit) what sort of results you can possibly record?)

So, you're working on a task -- with lots of intensity -- and suddenly you stop and veer off course and suddenly, all of your neurons that were being precached and so on have been thrown upside down and they have to sort out what they are going to be doing with each other, because clearly some are now going to be less relevant to whatever it is that you are doing. You have to really be able to stop thinking about the problem, but that can be genuinely hard, it's like trying to forget the number 113834 by reciting "Forget 113834, Forget 113834, Forget 113834, Forget 113834, ..." Veering off course is like creating a 'vacuum' which nature abhors and therefore must fill. The original balance that you maintained in your original task was crystalizing into a brick wall, but you jumped ship (not out of irresponsibility, but because you had to). Just as it takes great skill of concentration to make the preparation phase a success, it takes great skill to be able to equally "put it all away".

Go read Liljedahl's Hadamard ressurected. (I remember there being a bit in here about how mathematicians greatly follow the oral tradition, and many do not really read math papers for their new value but rather just to double check themselves, skipping over much of the detail.)

Bryan's BMP theory -> there is a background maintenance process in the brain where neurons are interacting with each other without "deliberate conscious effort" (of course, there is no deliberate conscious effort overall, and it's really all the same, and I suspect that the difference between consciousness/unconsciousness is something else). This background process can be mentally, biologically and genetically controlled. Perhaps socially.

In philosophy, intuition refers to the immediate apprehension of an idea without any conscious analysis. In ordinary language, however, the term refers to the person's feeling that a decision, judgment, or solution is correct, in the absence of supporting evidence (Bowers, 1994). Thus, problem solvers' feelings of warmth reflect their belief that they are getting closer to a solution, even though they do not know what the solution is. Intuition, so defined, is a form of metacognition reflecting people's knowledge or beliefs about their cognitive states and processes (Flavell, 1979; for reviews, see Metcalfe & Shimamura, 1994; Nelson, 1992).
Some evidence for valid intuitions in insight problem solving comes from a series of studies reported by Bowers Regehr, Balthazard, and Parker (1990; see also Bowers, Farvolden, & Mermigis, 1995), and based in turn on the Remote Associates Test (RAT) developed by Mednick (1962; Mednick and Mednick, 1967) for the assessment of creativity. Mednick defined creative thinking as the combination of ideas that were not previously associated with each other: "The more mutually remote the elements of the new combination, the more creative the process or solution" (Mednick, 1962, p. 221). These associations may occur in a number of ways: serendipitously, by means of similarity between the previously unassociated elements (e.g., words that rhyme but are semantically unrelated), or through mediation by a third idea. In order to distinguish creative associations from the products of mentally retarded or thought-disordered individuals, Mednick further required that these new ideas prove practically useful in some way. It was not enough that they be merely original. It follows from Mednick's definition that creative individuals are disposed to generate remote associations -- literally to connect ideas that other people fail to see as related.
As with intuition, the most convincing evidence for the role of incubation in problem solving is anecdotal. Woodworth and Schlosberg (1954), in an early and influential review, cited "a large mass of testimony from creative thinkers to the effect that laying aside a baffling problem for a while is often the only way to reach a satisfactory solution" (p. 838). Inventors, scientists, poets, and artists, they noted, routinely testify to the power of incubation (for dramatic and inspiring accounts in a variety of fields, see Koestler, 1964); they also tend to characterize incubation as an unconscious process. Campbell (1960) thought that incubation resulted from the random fusion of memory representations, an ongoing process which occurs unconsciously; when the new combination is relevant to the problem at hand, it emerges into consciousness as a creative insight -- a mental equivalent of the survival of the fittest.

However, there are other accounts of incubation which afford no role for unconscious processes (for reviews of the possibilities, see Perkins, 1981; Posner, 1973; Woodworth, 1938; Woodworth & Schlossberg, 1954). For example, prolonged thought might result in a kind of mental fatigue: after the incubation period, the thinker returns to the problem refreshed and more likely to achieve a solution. According to this view, no work at all, whether conscious or unconscious, is done on the problem during the incubation period. Alternatively, the incubation period merely affords an opportunity for further conscious work, activity which is simply forgotten after the problem has been solved -- perhaps because it occurred in brief, barely noticed spurts. Thus, what appears to the thinker as unconscious incubation is merely an after-the-fact illusion about what has gone on during the interval.

Perhaps the most popular theoretical account of incubation has been proposed by Simon (1966; see also Newell et al., 1962/1979, pp. 140-141; 1986/1989, pp. 484-485), based on an early version of GPS theory. The theory holds that at each point in problem solving, the current state is held in short-term memory, while the final state, and other subgoals yet to be achieved, is held in long-term memory. If the thinker is interrupted, or for other reasons turns attention away from the problem, the current state will be lost due to decay or displacement. When the thinker returns to the problem, the problem space or subgoal hierarchy likely will have been altered by newly processed information. Thus, thinkers re-enter the problem space at a somewhat different point than they left it. This difference in vantage point, rather than any unconscious work, accounts for the effects of the incubation period. Incubation, then, on this view, is tantamount to a combination of forgetting and prompting. That is, the incubation period provides an opportunity to forget the misleading cues, but it also provides an opportunity for the person to encode more appropriate cues newly available in the environment. Woodworth (1938; Woodworth & Schlosberg, 1954) offered a similar hypothesis, involving the breaking of inappropriate problem-solving sets. According to this view, the incubation period affords an opportunity to forget (presumably by means of decay or interference) an inappropriate set or direction. Simon's (1966) theory adds to simple forgetting the influence of new information encoded during the incubation period.
And then there's an interesting test method:
Smith and Blankenship (1989, 1991; see also Smith, 1994) have recently introduced a method which promises to correct both of these problems. Their method involves presenting a problem along with an ostensible hint about the solution -- a hint which is actually misleading. Across a number of experiments, subjects performed better if they had not been presented with the misleading clues. Where they had received the misinformation, however, the subjects performed better with an incubation interval than if they had been tested immediately; longer incubation intervals produced superior performance as well as increased forgetting of the misleading cues; both were presumably due to the extended incubation period. Smith and Blankenship concluded that incubation occurs if subjects suffer from initial fixation in problem solving. Under these circumstances, as hypothesized by Simon (1966), the incubation period facilitates forgetting of the fixation, permitting subjects to restart problem-solving activity at a new, more useful point in the problem space. However, Smith and Blankenship did not claim that forgetting accounts for all incubation effects. Their method shows that the incubation period can be important for overcoming inappropriate sets, but does not shed light on the role of any unconscious processes that might occur during incubation.
In the memory domain, a model for incubation effects may be found in the phenomenon of hypermnesia, or the growth of memory over time (for reviews, see Erdelyi, 1984; Kihlstrom & Barnhardt, 1993; Payne, 1987). Following a single study trial, repeated tests generally reveal that memory for any single item fluctuates over time: some items that are remembered on early trials are forgotten on later ones, and vice-versa (Tulving, 1964). Usually, intertrial loss exceeds intertrial gain, producing the classic forgetting curve of Ebbinghaus (1885). However, under some circumstances the reverse is true, resulting in a net gain in recall (e.g., Erdelyi & Becker, 1974). As noted by Mandler (1994), the recovery of previously unremembered items is analogous to the achievement of previously unattained solutions (see also Smith & Vela, 1991). The analogy is strengthened by evidence that longer intertest intervals, analogous to longer incubation periods, produce greater amounts of hypermnesia (Madigan, 1976, Experiment 2).
Thus, Gutenberg (cited in Koestler, 1964, pp. 122-123), while attending a wine harvest, realized that the steady pressure use to crush grapes might also be useful for imprinting letters.
Forgetting is an important process, allowing you to exit and re-enter problem space at another location.
Until recently, the notion of unconscious thought was closely tied to the concepts of parallel processing, automaticity, and procedural knowledge more generally. It is almost an article of faith among cognitive psychologists that people lack introspective access to the procedures by which we perceive and remember objects and events, store and retrieve knowledge, think, reason, and solve problems (e.g., Nisbett & Wilson, 1977; Wilson & Stone, 1985). In large part, the goal of cognitive psychology is to explicate these unconscious procedures and processes (Barsalou, 1992). However, there has also been a parallel assumption, mostly unstated, that the declarative knowledge on which these processes operate is accessible to awareness. Put another way, we do not know how we think, but we do know what we think.
... which made me think: how do we know that we know what we think? There is no obligation for our brain to make our thoughts accessible to us, a magnifying glass looking back in upon itself.

So, what? Think of your mind as an oven, and write down when you think you should check up on your cakes? (To hell with "you can't have your cake and eat it too"). Of course, before you put "it" in to bake, you have to cache the most possibly useful resources that you can find and attack the problem from many different directions, the more "conscious effort" you apply here, the more you get out of the meme-incubator. Thus, here's my project: create lots of packages to put into the "oven", record when I started and when I was finished making the recipe, and when I officially put it into the oven, and then record any related results whatsoever that pop up randomly. Ideally, I should just forget about what I put into the "oven". Maybe this will be like the mnemonic systems that are described all over the place, like creating a visual journey, except having an oven where we literally forget about foods that we're cooking. Based on the information from this page, the idea is to forget about it completely so that when the problem is stored into long-term memory (LTM) I do not again enter the problem space from the same way again.

http://www.overcomingbias.com/2007/10/cached-thoughts.html - cached thoughts and intuition
Cached thoughts and intuition
From: Bryan Bishop
To: Eliezer S. Yudkowsky
Date: 12/26/07 07:15 pm
"Intuition is apparent ability to acquire knowledge without a clear
inference or reasoning process."

Re: your post on cached thoughts.

And the weird internet church, The Nexus of the Now and fusionanomaly,
both who claim "You are what you cache."

Plus some extra content:
> #1. When making decisions, humans do NOT choose between a large number
> of options

> Indeed, humans do not even select between different alternatives!  OK,
> this might sound a bit crazy at first, but there is a whole bunch of  
> converging evidence for it.  Our proposal is, instead,
> of 'quasi-binary' decisions.    

> More to come on this on an upcoming report named "the case
> for 'quasi-binary' decisions". In the meantime, the Minds and Machines
> paper gives some clues, though it does not present or discuss the
> converging evidence over this issue.  

> #2. It is NOT possible to cleanly separate perception from
> problem-solving from 'self-justification'

> This makes it the case that if one wants to study decision-making or
> problem-solving, one must study perception. See for instance the
> Schelling tax proposal; or Kahneman and Tversky's framing theories.
> One of our studies pointing this out is: "A glimpse at the metaphysics
> of Bongard problems" published in Artificial Inteligence.  See also
> the Hofstadter papers.    

> #3. Perception is about finding the essence of a situation
> ...and finding that essence consists of activating the right concepts
> in one's mind.  

So, does intuition turn out to be one's brainal cachings? But don't you
still need a way to know to do a cache request for some vital piece of
information? Have you ever overcome this deadly spiral of a trap?

- Bryan
Bryan Bishop
Surprisingly, the SuperMemo page on formula for creativity and genius neglects to mention incubation, despite going over many people mentioned in this document already (as well as many more). It is otherwise quite thorough.
Short-term memory caches thoughts and perceptions. How does one go about "uncaching" ? Do you have to transfer to some other problem space? To get the most effective results, then, do you have to travel far, far away from the originally cached context during the incubation phase? For example, Feynman (nor Einstein) did not have to go to Japan or some exceptionally alternate cultural context to come up with their theories, or is this simply because, relatively, the breaks that they took were enough to generate those insights? Or does it not matter what sort of breaks you go do?

Purposeful forgetting tactics
- interference http://en.wikipedia.org/wiki/Interference_theory http://en.wikipedia.org/wiki/Memory_inhibition

Intentional Forgetting: Interdisciplinary Approaches By Jonathan M. Golding, Colin M.

2. Interference Theory
The above explanations for forgetting, while valuable, do not account for ordinary day-to-day forgetting. A more useful explanation is found in interference theory. The basic idea in interference theory is that memories are not lost so much as they are inhibited, or interfered with, by other memories. There are two types of such interference, termed retroactive interference and proactive interference. Depending on the process assumed to be taking place, they are sometimes referred to respectively as retroactive inhibition and proactive inhibition.

Retroactive Interference
Retroactive interference occurs when later learning interferes, as if retroactively ("acting backwards"), with previous learning. For example, suppose that on Monday one studies Spanish and on Tuesday Italian. During a Spanish test taken on Wednesday intrusions are experienced from Italian words. Retroactive interference explains our inability to recall old memories as due to competition from new memories.

Proactive Interference
Proactive interference occurs when earlier learning interferes proactively("acting forwards" in time) with later learning. Persistent habits provide good examples of proactive interference. Suppose one has lived for several years in an apartment with one kitchen drawer next to the sink where the tableware is always stored. The new apartment, however, has a bigger kitchen and the tableware is now stored in the drawer next to the stove. When hurried or preoccupied, one "forgets" this new plan and habitually seeks knives and forks in the wrong drawer, the drawer by the sink, which corresponds tot he "old" storage place instead of the new one.

Proactive interference thus explains one's inability to acquire new knowledge or habits because of the powerful competition of old memories.

http://www.supermemo.com/articles/overload.htm "We tend to obliterate old memories when learning confusingly similar material."

http://www.personalbest.com.au/learning.html transfer of learning
PI (proactive inhibition) causes accelerated forgetting (within minutes or hours) of the new way and this is why you then appear to revert and go back to your old incorrect or inappropriate way of thinking, acting or performing. You know what you're doing wrong and what you should do and you're highly motivated to improve but your brain (force of habit, i.e., PI) won't let you change.
http://psyche.cs.monash.edu.au/v4/psyche-4-16-whetstone.html - Directed forgetting, Tony Whetstone.
As pointed out by Whetstone, Cross, and Whetstone (1996), successful directed forgetting seems to be a case of conscious initiation of unconscious processes that determine what will later enter consciousness. This might be taken as evidence that consciousness can have direct causal properties. The difference between paradigms also seems to be of applied importance. There are many instances where a person might like to forget some bit of information, whether it is as simple as an incorrect number or a thought that produces anxiety. In fact, the National Research Council's Committee on Techniques for the Enhancement of Human Performance has recommended that directed forgetting be investigated for practical applications, while advising against the use of thought stopping, a psychotherapeutic technique that is similar to thought suppression (Wegner, Eich & Bjork, 1994).
Of course, it might be that as a problem is moving into long term memory, the patterns are encoded in some special way such that the meaningless approach to the nature of the problem is forgotten.

Through a variety of ways, I have discovered that there is something known as an NMDA receptor sites that play a direct role in mem consolidation from STM to LTM. The chemical cycloserine evidently enhances this process.

Could L-Glycine be used as a memory enhancer? The reason I ask this that endogenous glycine is a NMDA co-agonist [1], and NMDA activation is required for long-term potentiation to occur. Aniracetam and D-cycloserine (a partial agonist of the NMDA receptor-associated glycine site) [2] enhance LTP and can enhance memory consolidation. I think L-Glycine is worth investigating as a cognitive enhancer, particularly since the risk of excitatory neurotoxicity is low given it is a co-agonist rather than a full/direct NMDA agonist.
http://forum.bodybuilding.com/showthread.php?t=6571481 - log of somebody doing aniracetam

Hm, I think I'll start by working on doing a cross-over between the incubation experiments (with rats) with the fear extinction experiments with cycloserine. I think I'd need four groups: super control group, group testing for incubation, control group with the molecule, group for testing incubation + drug.

How do we make up new frameworks in mathematics, if we wanted to catch the briefest of our intuitions? The problem is that our framework has to be prepared, in all likelihood, or we have to somehow make up a framework with our intuitions so that we know what most clearly fits it. This doesn't seem to make sense. There might be a way to be able to construct symbolic representation systems for any idea that comes to mind, no matter how brief.
The great Swiss psychologist Piaget attempted to underpin his theories of genetic epistemology with mathematical ideas. He saw the individual's need to be in dynamic equilibrium with his environment as an underlying theme in his work. To be stable meant that any divergence from equilibrium could be reversed, so that any operation needed a corresponding inverse operation. This triggered off in his mind the metaphor of group theory to model stable mental operations, for a mathematical group has an identity element and every element in the group has an inverse element which causes it to return to the identity.
On the other hand, a more obvious mathematical metaphor for the disturbance of dynamic equilibrium lies in catastrophe theory. Here a system controlled by continuously varying parameters can suddenly leap from one position of equilibrium to another when the first becomes untenable. Depending on the history of the varying parameters, the transition may be smooth, or it may be discontinuous. Such a metaphor suggests that stage theory may just be a linear trivialization of a far more complex system of change, at least this may be so when the possible routes through a network of ideas become more numerous, as happens in more advanced mathematical thinking.
... his eyes seem to shun contact with the world; it is not without, it is within he
seeks the vision of truth. [Poincaré, 1913, page 212]
... When one talked to M. Hermite, he never evoked a sensuous image, and yet
you soon perceived that the most abstract entities were for him like living beings.
He did not see them, but he perceived that they are not an artificial assemblage and
that they have some principle of internal unity. [ibid page 220]
"Saturation, incubation, and illumination." - Helmholtz
Of course, the scientific community adds verification.

J. E. Oliver, The Incomplete Guide to the Art of Discovery, Columbia University Press, New York, NY, 1991.

“It takes two to invent anything. The one makes up combinations, the other one chooses, recognizes what he wishes and what is important to him in the mass of the things that the former has imparted to him. What we call genius is much less the work of the first one than the readiness of the second one to grasp the value of what has been laid before him and to choose it.” [Valéry]
http://www-groups.dcs.st-and.ac.uk/~history/Extras/Hadamard_mathematician.html - Jacques Hadamard's mathematician's mind


Must 'formalization' be done in the preparation phase? Or is there a way to document the experience of mathematical creation/discovery as it happens, as if contributing a new 'experience story'/report to erowid?

"formalization of formalization"

`”A psychological space is established for any set of stimuli by determining metric distances between the stimuli such that the probability that a response learned to any stimulus will generalize to any other is an invariant monotonic function of the distance between them. To a good approximation, this probability of generalization (i) decays exponentially with distance, and (ii) dies so in accordance with one of two metrics, depending on the relation between the dimensions along with the stimuli vary. These empirical regularities are mathematically derivable from universal principles of natural kinds and probabilistic geometry that may, through evolutionary internalization, tend to govern the behaviors of all sentient organisms.”`

This research team's previous study revealed that just prior to an "Aha!" solution, after a person has been working on solving a problem, the brain momentarily reduces visual inputs, with an effect similar to a person shutting his or her eyes or looking away to facilitate the emergence into consciousness of the solution. The new study extends these findings by suggesting that mental preparation involving inward focus of attention promotes insight even prior to the presentation of a problem. Therefore, it may be that how a person is thinking before problem solving begins is just as important as the kind of thinking involved in reaching the solution, and perhaps even determines whether the solution will be derived with a sudden insight.
"sudden insight syndrome"

- selective forgetting hypothesis
conscious-work hypothesis
fatigue-dissipation hypothesis
subconscious random-recombination hypothesis
opportunistic-assimilation hypothesis (failure indices left in LTM, retrieval from LTM to STM just gives the original problem space without the crappy stuff)

Rapidly generate failure indices? How can you make sure that a failure indice has been generated in a problem that you have been working on?

 Citation: Sandkühler S, Bhattacharya J (2008) Deconstructing Insight: EEG Correlates of Insightful Problem Solving. PLoS ONE 3(1): e1459. doi:10.1371/journal.pone.0001459

The correlate seems to be with EEG gamma waves in the brain.


Consciousness and the silent mind - an interesting page. Consciousness is like AdBlock except SenseBlock. This is where attention, autism and emotions come in (see also the emotion-attention connection and defense mechanisms). Attention as the primary scarcity.


D-cycloserine may "improve" (eliminate) ASD-- this is interesting. First, take a look at some notes on increasing repetitive behaviors in ASD. In ASD, repetition is quite common, and is useful for recursion through large piles of information, such as reading the entire encyclopedia as Feynman did. And yet it seemed that cycloserine would improve this. What's going on? Cycloserine supposedly improves STM -> LTM memory consolidation, so that the indices can be forgotten and when the problem is remembered again, an insight can be generated. Maybe it is a leveraging issue: you can either focus on spending as much time as possible processing through all of the information to eliminate possibilities up front, or you can improve the memory consolidation process instead [opportunistic-assimilation hypothesis of incubation theory]. Design an experiment to test these payoffs. From the page on increasing repetitive behaviors, mCPP supposedly increases repetitive behaviors, so if you add mCPP and d-cycloserine to somebody with ASD, what would happen?