As far as I know in Asia the use of abacuses is widespread.

True masters are able to operate with a mental abacus.

There seem to be two distinct mathematical systems in the brain.

One is symbolic. Even if one imagines the hands of an analog clock,

if the imaginary positions can be distinguished, they are symbolic,

because these positions can be used to reference something else.

That's what symbols do.

Another number system is logarithmic. It is the number sense.

Even animals have it. It seems to be based on integrators.

For example, single neurons are able to "count" incoming

spikes by integration: an analog storage is filled by a spike

and emptied by a continuous efflux.

http://web.mit.edu/press/2012/thinking-logarithmically.html

http://www.sciencemag.org/content/320/5880/1217

http://www.newscientist.com/article/dn9198-pigeonbrained-birds-can-think-in-logarithms.html

http://web.mit.edu/newsoffice/2002/miller.html

http://www.sciencedaily.com/releases/2008/07/080702132219.htm

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3073273/

http://idw-online.de/pages/de/news166059

http://idw-online.de/pages/de/news167670

Regards,

Joachim

Joachim,

You contributed very interesting examples and information. Thanks for that. I would like to know more about those people in Asia, you were talking about, who seem to have special talents using abacuses.

But what about the question of an abacus (or a slide rule) that one only has in mind - will it reveal new information?

Take care,

Martin.

Here is additional information:

http://www.newscientist.com/article/dn20775-mental-abacus-does-away-with-words.html

http://ladlab.ucsd.edu/pdfs/FrankenBarner.pdf

http://idw-online.de/pages/de/news167670

http://www.ncbi.nlm.nih.gov/pubmed/18387574

Regards,

Joachim

Joachim,

thanks a lot for your last information. As far as I understand it, there are some points in Barner/Frank that undermine my point of view as proposed in my original question:

a) that mental calculation is always different from mental chess.

b) that the visual representation in mental calculation should be interpreted as unanalyzed wholes. (see Barner/Frank 2011, p. 3).

But I see differences between abacuses and slide rules: someone who is used to work with an abacus would not be in danger of unconsciously conceiving one with an impossible number of colors in one line. That might be analogue to chess, where it would come to our mind if there were more than 2 Kings.

Slide rule: one might argue that it is clear to anybody that there cannot be a centimeter with 8 or 9 millimeters. - Of course, but is this a question of visual representation or of propositional knowledge?

One of my points was that as far as I understand it, the mental image is not that fixed. There might be centimeters with 10 millimeters but with narrower intervals for each millimeter in the mental representation. It might be “elastic”.

Another difference: a deformed shape would come to our mind if the form were circular. We would be aware if the interval between 11 and 12 on the clock were smaller than the one between 10 and 11.

Take care,

Martin.

I have no slide rule in my head, but all problems I solved during 50+ years of my professional exprience, started as a result imagination. Matenatics or experimens followed results of imagination.

Mr. Shulz:

I think your terminology, "slide rule" made your question a bit skewed. Of course, we can learn from our own imagination. Just imagine how many different endings you can think of to a story, issue, problem, etc. I have learned a great deal from my imagination. During my years in school, it would take the "topic" of the day, and imagine all of the different ways I could present my point of view. I learned so much just from imagining how many ways would have the most impact, sound the most scholarly, and had a bit of my personality involved. Of course, you can learn from your imagination. We would we be without our imaginations.

Sincerely, Dr. Marilyn Field

Dear Dr. Field, dear Alexander,

Thank you for your contributions! I mixed two components one of them being very general, the other one being most specific. It seems to be a nasty mix. Let’s look what will come out of this combination.

Dr. Field, I fully agree with your point. In my opinion, your example is very good and it contains two components as well: a) what may happen in a story and b) what will the activating of your imagination show about you as a person. Moreover, what possible developments of your personality will come out of it? – But I want to make a distinction between what we can imagine and what may possibly result from it as a consequence. I think there are limitations for the last.

In a story about a person we all may learn that there are ways to deal with a situation that seemed to have no way out first. And you as the writer might not have known this solution at the beginning! But anyway, the critics may not like it, if the solution is too fantastic. The question gets a new twist if we ask again what we learned.

E.g. we may dream of an egg that falls down from a great height and keeps its form. But we would not say that we LEARNED that this is possible after dreaming of it.

I would not say in a discussion “let’s define the concepts first” (that was the way of Socrates, there are other ways). Let’s keep open the definitions of “learning” and “imagination” for a while and let’s see what comes out of the discussion.

Alexander, sorry for not answering to your post for such a long time. I agree that we can move possible strategies in our head and find a solution after moving them. I suppose, by that we take our experience into account and try to find out if a strategy that failed in the past may be successful in a new situation or if an alternative strategy might be more promising. But what are these strategies without applying natural laws or the laws of mathematics? So there seem to be two components: creativity and knowledge about the limitations within the domain of objects under examination, the model.

The model will contain 1st. the domain D of objects , 2nd a set of names N for the objects (maybe numbers), 3rd a (recursive or non-recursive) set of rules.

I think we would not use our imagination to change one of these three things in the model. We would say otherwise that we stepped out of the problem and formulated a new problem.

Let's take a look at early age imaginary friends, such as, a child having a Tea Party with imaginary visitors. Every day, a new tea party, with new communication styles all from the child's imagination. Therefore, there may be a genetic component involved. Some children imagine tea parties that turn into friendships and they all get to know each other more and more, The Child is learning communication styles, problem solving (such as spilling the Tea), possibly not liking one of the members more than another (symbolic). Then we may have a child who chooses modeling TV, if he or she is less (genetically, or environmentally) creative with their alone time. It is a great subject and keeps me thinking. Thank you. --Marilyn

Dr. Shultz:

Regarding your creative comment, "In a story about a person we all may learn that there are ways to deal with a situation that seemed to have no way out first. And you as the writer might not have known this solution at the beginning! But anyway, the critics may not like it, if the solution is too fantastic. The question gets a new twist if we ask again what we learned. "

Think of all of the alcoholics/addicts who find recovery, relapse, 12-steps....most certainly felt no way out first.

My Motto or "whatchamacallit" is "The more I learn, the more I realize I know very little".

--Marilyn

Marilyn,

I’d love to take part in your Tea Party! It seems a nice place to be. Anyway, would the children realize me? Would I be there? Will we have a dispute about the number of participants?

If every single child learns about social behavior in her or his fantasy, will there be a consensus about the right way of conduct or will there be as many different forms of social behavior as there are children with their fantasy? (N.B. the number of these children is not the number of participants in the Tea Party).

Will there be a dispute between two children in the imagination of one child where the second child has got a chance to respond in a different way than the way that was prefixed somehow in the experience or memory of the first child?

Best,

Martin.

Marilyn,

It is my thesis that real world experiences are needed, even if they may happen in the imagination too, by chance.

This has to do with the famous discussion of private language and the meanings of our words and symbols. The meanings are not individually devised by us.

The way we are conscious about our interior life and the richness of our interior life itself is dependent on our language use. This language is the public language we are sharing with others. A language that teaches us about ourselves! It would take more than a chapter in a book to discuss this. Think of the problem of definition of the word pain. It is a very complex problem. Let me just give you a few keywords: There are some elementary distinctions, some of them denied by some authors, some of them stressed by others:

Meaning – reference – sense: none of them is fixed by individuals.

Possible worlds and their accessibility (let’s assume p.w. are in the head, not real). E.g. a world with computers is not accessible from a world without electricity. If somebody denies this he will be asked what the word “computer” means to him.

Rigidity – non-rigidity. Names are rigid, they don’t change when we pass from one world to another – descriptions are non-rigid. E.g. it is useless to ask if Napoleon might have not been Napoleon but there is a possible world in which Napoleon was not an emperor.

So it is not true to say that there are no restrictions to imagination.

Apropos Tea Party: a theory that supposes that the meanings change from person to person and from one moment to the other has been called a “Humpty-Dumpty theory”.

Marilyn, I agree with almost everything you said. As to the alcoholics/addicts, I think they will meet in reality to be not alone in their fights. It is not all happening in their heads what makes them stop one day, right? I think psychotherapy needs at least two people. There may be cases where somebody succeeds in doing it alone but I guess these are extremely rare examples.

Imagination should not be restricted by others, but if a child constantly says that 2 and 2 is 5 because it is like that in her or his imagination we may sooner or later wish that some other child will show up in the imagination of this child to fight for another solution.

We would not be disappointed if this happened in the real world.

Best,

Martin.

Martin:

During the last few days I have been in a computer nightmare (probably a dead metaphor) or maybe we do have nightmares when we are wide awake. I had several questions and points of view that were unfortunately lost even from auto save, but, that is because I quickly put my thoughts down on mail to be sent. I literally can’t afford a three day computer crash. I have students in various time zones.

I find your thesis exciting! It will require critical thinking from every reader. I would love the opportunity to read your thesis. A seemingly infinite amount of angles (e.g., genetics, environment, instincts), when do subconscious or unconscious metaphors develop, what about your Question regarding the tea party, "Will they know I was there", I think they may, if your name was Harvey.

When I was in high school, I was a leader on the impromptu debate team; I didn't do as well with extemporaneous debates. It might take me a month to put down my review, as your work requires not only thinking, but, re-thinking, hypothesizing, and a favorite metaphor of mine “thinking outside of the box”. Also, somehow has a sweet sense of competitive humor. I especially liked your comment, “we may sooner or later wish that some other child will show up in the imagination of this child to fight for another solution.” (para. 10). (So funny and true)

This might seem totally unrelated to the construct. But, I was wondering if you have ever read the book “Ender’s Game” and how you feel about the question: Who is John Galt? Soon to follow are my comments about your response.

All the best,

Marilyn

Marilyn,

I hope your computer is working well now.

I should have said “It’s my standpoint”, instead of saying "it's my thesis". I turned your arguments into a little humorous story in order to make a thought experiment out of them. I know almost nothing about the emergence of social behavior. But I think that an interacting of individuals within groups is needed to develop strategies of individual behavior. What I could contribute from my side is that there seem to be some principal limitations for possible world views as far as language is involved, in particular meanings not being devised by individuals alone.

How about strategies that are mutually exclusive: “Don’t talk to strangers” – “Distrust others” – “Respect the majority” – “Fight for your own views” – “Listen to what others have to say” - etc.

I really do not know more about the learning of social behavior than a normal reader of newspapers can understand. Therefore I cannot take part in a scientific discussion on this item.

I am still wondering if anybody has some information or interesting points of view about people with special talents in relation to calculation with or without an internal slide rule or an internal instrument like an abacus.

Just to push the discussion a little further, let me tell you what Wittgenstein says about calculation. He thinks that we often check the results in an internally stored table. These tables are the definitions of the calculations. (Wittgenstein Vorlesungen 1930-35, German Frankfurt 1989 p. 115). That would back my standpoint that when moving a slide rule in the head we are not watching a “new film” but we are leafing through a batch of stored snapshots at best.

A counter argument against this was provided in a paper from Franken and Barner p. 3, indicated by Joachim Pimiskern in one of his posts in this thread: this argument says that the visual representation of moveable parts in a (mental) abacus doesn’t have to be treated as an unanalyzed whole.

All the best,

Martin.

Martin,

I am still intermittently thinking about your topic, and it brought to mind, a construct that I have been interested in for a long time, but have difficulty discussing with colleagues: Tacit Knowledge. Is this a farmiliar concept?

I wish I had more time to share the connection. I surely will as soon as I get a day off.

Have a nice day,

Marilyn

The main reason for the difficulty of discussing tacit knowledge with others, ironically, is also related to not having enough time. I guess that is a good reason to slow down if I want to have a meaningful conversation. Have a nice day,

Marilyn

Martin

I can operate an imaginary slide rule and I get the right answers.

Joseph,

It’s great to hear from someone who really has this ability. Can you tell us more about it? Have you learned it by practising? Do you see the whole device or a part of it? Would you say that you stored a “movie” or that it is something you create in the moment you are operating it?

I did not practice to learn to do it, I found I could do it after a lot of practice. I do not see a movie or manipulate the slide rule. I see the proper juxtaposition of the scales. I only see necessary part of the scale at the moment. I would not say I am operating it.

I can draw a face from memory. I do not see a face, but know what how it appears.

I have done visualization exercises, but had to ask, "Am I supposed to see something, like in a dream, or is it OK to know how it looks." Did not get a good answer.

Not everybody think the same way. The mathematician John von Neumann had astonishing memory and calulation capacity. Whatever he read in his life he automatically memorize forever. He could recite litterally books he had read. Von Neumann was working with the brightest genius of the 20th century in physics and mathematic and was a legend among them for his intellectual capacity.

Here is one anecdote among many:

The following problem can be solved either the easy way or the hard way: Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of them flies back and forth between them at a rate of 75 miles per hour. It does this until the trains collide and crush the fly to death. What is the total distance the fly has flown?

The fly actually hits each train an infinite number of times before it gets crushed, and one could solve the problem the hard way with pencil and paper by summing an infinite series of distances. The easy way is as follows: Since the trains are 200 miles apart and each train is going 50 miles an hour, it takes 2 hours for the trains to collide. Therefore the fly was flying for two hours. Since the fly was flying at a rate of 75 miles per hour, the fly must have flown 150 miles. That's all there is to it.

When this problem was posed to John von Neumann, he immediately replied, "150 miles."

"It is very strange," said the poser, "but nearly everyone tries to sum the infinite series."

"What do you mean, strange?" asked Von Neumann. "That's how I did it!"

A minor remark about the impact of dreaming on inspiration and imagination: Is dreaming a form of imagination? The visual mental slides are in that case replaced by visual mental movies perhaps providing inspiration and novel ideas to artists, scientists, philosophers........

Marcel,

Very interesting. I heard an artist once say that he had dreamed his works.. Is it the same thing as learning?

Dear Marcel and Martin,

Originally, I wanted to become an artist and drawing and making statues were not difficult for me but painting caused difficulties.

Half a year ago I started to paint aquarelle pictures of which the technique is difficult and uncountable and needs a lot of time. I used to dream what and how to paint and in my dream these ideas are fantastic however all the “magic” disappears in the morning. Fortunately, I feel I have a certain progress and I am able to paint much quicker and know better the work of the water. I hope one day my dreamed ideas will be excellent also in the morning.

I had the same experiences with poems and translations but the dreamed results are better.

Can we learn from our own imagination? Yes in certain circumstances but that needs a lot of practice, I think so.

I agree that our imagination is part of our experience but may be extended within reasonable limits. As for as "people with a special talent who can move a slide rule (a slipstick) in their head and get a result which they had not had before", I am not really sure what does it mean, but certainly know about human computers. Once, a human computer, in the early days of introduction of computers in India, visited our university. It was an occasion. People asked very difficult questions and human computer gave the answer instantly. A professor of Engineering asked him about solution of an equation, this time human computer took several seconds and asked the professor that he has iterated the equation so many hundred thousand, does the equation holds true? When the professor replied in affirmative, the human computer again started and after several seconds again said, I have iterated so many million time, doe the equation have a solution. This time professor replied you are right, the equation has no solution. That much I stand witness. In the case of chess I also stand a witness. A person used to visit my father and used to play chess without seeing the chessboard. He only used to ask from a third person what move his opponent has made, after thinking from three to fifteen minutes he used to ask the third person to move which piece at which place. Usually he used to win though well known chess players among my fathers circle and in the district played with him. I think It is in line with the post, but such cases are very rare and such people are gifted one. In the case of mathematics, all mathematics is instrumental knowledge (knowledge built upon knowledge gained). Therefore, such cases cannot be generalised. Of course, some mathematicians are more imaginative and extend their imagination within reasonable limits to add to existing knowledge while others are not so gifted. However, it should be noted that a person with poor imagination cannot be even an ordinary mathematicians.

Dear Mohammad,

I agree with what you have said. There are people with extraordinary talents. I knew someone who could play chess in his head and remembered hundreds of games. That’s fascinating.

Some weeks before, Joseph L. Alvarez told us, that he can move a slide rule in his head. Before, I had not believed that this is possible!

The other part of my questions remains still unanswered, as far as I see: Can we treat our imagination as an “objective counterpart”? A psychologist once told me “We are the producers of our dreams”. She convinced me. But that was many years ago. Maybe there has been some research in the meantime that leads to other results?

If it’s true that we are the producers of our imagination, perhaps we should not take it as an object of our research afterwards. At least in my opinion.

E.g. In my imagination I take a box, put three stones in it, close the box, open it again and count the stones – all this happens in my imagination. How many stones should I expect?

This is a very primitive example, but I think it is good enough to make clear what my problem is: when we are free to do what we want, how can we learn what the limits are?

I think that our imagination is an extension of the imagination of Nature, what create all order in Nature.  It is in itself a product of Nature but it is not a mechanism that reach conclusion following a method.  The imagination of Nature is not like that and our is only an extension of the imagination of Nature. It proceeds along the process that Nature put in place since the origin and goes in continuation with it.  All creations are in continuation.

Learning and Creating are not usually seen as the same but different in nature.  Learning being a kind of copying or created things.  The process of learning to walk by the toddler is both a process of discovery and invention.  The toddler learn to walk and in a certain way re-invent walking.  We have splitted learning and imagining in our education system while ideally all learning should be a re-invention by the learner of what has been invented in the past.  

Dear Louis,

I have to say, I don’t understand what you mean by “imagination of Nature”. I will have to think about it in the light of your example of learning to walk by the toddler.

Learning: I think it is not only a doubling (or a copying) of something, when we learn. Instead it is a selection or “filtering” of alternatives. Once we learned something, we won’t take into account the possibilities that are ruled out by what we have learned.

Re-invention: a very interesting point, at least in your very good example of learning to walk. But otherwise this seems to be a special case of learning. I can accept when we say that we learn the rules of arithmetic and then we are free to calculate new problems. But we are not free to re-invent the rules.

Martin,

Nature evolves.  It evolves by gradually stabilizing diffent type of order.  All the entities that exist today did not exist at some point in the past.  Imagination of nature is this process of creation and maintenance of the new created entities and their interaction .  It has multiple facets. It accelerate. It very slowly created the patterns on the wings of butterfly. It created all elementary particles. Nature imagination is its multi-facet process of evolution of all that exist and its multi-facet process of maintenance of all that exists. 

''I can accept when we say that we learn the rules of arithmetic and then we are free to calculate new problems. But we are not free to re-invent the rules.''

The rules of arithmetic were invented for fundamental reasons and it is why implicit-arithmetic rules exists within our nervous system.  We re-invented them and the child would naturally re-invent them.  Socrate said something like learning is remembering.  But yes there are purely arbitrary conventions that we have to learn and that we cannot guess or re-invent.

@Martin,

I understand that our imagination is not objective counterpart, sometimes it may be within bounds of reasons and sometimes wild (creativity of our mind), but our imagination is not divorced from our experience of the world. What Joseph L. Alvarez told you may be symbolic or simile ti intuition like that of Einstein. There cannot be a slide rule in mind. It is simply a quick mental calculation by an expert who had already given some thought to the problem and answer was in his subconsciousness that came into conscious mind with little effort.

Dear Mohammad,

You are the first to combine the two parts of my question. That makes it become more interesting. I am not in Joseph’s shoes but I assume that it is an extraordinary memory performance. Maybe something like the special talent of some people who can leaf through a book and afterwards are able to repeat the whole content. If this is true, then it is not a question of imagination.

If it were imagination, there would be no limit for deviations. Playing an imaginary chess game we could know that something cannot be the case, this is not possible for an “imaginary” slide rule. The reason is, that there are rules for chess - no rules for a slide rule. Of course, there are the rules of arithmetic, but the slide rule does not “perform” the rules.

Dear Louis,

I am not in a position to say anything about the imagination of nature. I wonder who is. By “implicit” rules do you mean innate rules? – If yes, would you say that we re-invent what is innate? When they are in our nervous system, as you say, why do they have to be re-invented?

I think you are right when you say that we have to learn arbitrary conventions. But I doubt that we already know everything that is not arbitrary. Do you know Kripke’s distinction between necessary a priori and necessary a posteriori? There are interesting examples for both cases.

My use of a visual slide rule is not total memory. I have not memorized all the answers. With the slide rule, I see the juxtaposition of scales. I have memorized the scales, but not all the possible positions. I would not say my memory is extraordinary, just better than most. I cannot memorize books, but I can memorize symphonies and operas. There is a difference. Music is anticipatory and words to music must follow the musical pattern. The slide rule follows a similar pattern based on logarithms. I can see the logarithmic scale because I have memorized through repetition the slide rule scales. This is practice, as music is practice as well as anticipatory.

In a math class, a professor was demonstrating short cuts to finding logarithms, sines, cosines, etc. He asked the class to call out a number and he would show how to make a fast estimate, say for the logarithm. Before he could even write down the number, I gave the answer. He finally asked how I was doing it. My answer, I can see a slide rule.

Another case, at a base ball game two men several rows back were talking heatedly. One shouted out, "Does anyone have a calculator? I need to know the square root of 45,638?" I turned around and told him 213.63. That was rapid calculation and not the same as seeing a slide rule.

In project meetings I can estimate the answer to a complicated situation to within 10%. This is also a rapid calculation using the major factors of the process.

The latter two are learned skills and many are capable of such calculations and estimates, indeed, it was expected of bachelors level physics or engineering students in 1960 when I got my degree. Nevertheless, most thought my slide rule trick a little odd, even though they could understand how I did it.

I do not think this constitutes learning from my imagination. I already knew how to use the slide rule, so I did not learn anything. It was no different from using a real slide rule.

When I solve a problem, I do it in my head. Writing it down helps to clarify and develop the solution and complete the calculation, but all the heavy lifting is in my mind. Is this imagination or thinking? I image the concepts. I image the solution. You may call it imagination, I call it thinking.

Dear Joseph,

Thank you for your very detailed and impressing essay on your capability. It was the best we could hope for in this thread. Maybe you should consider publishing a version of it somewhere else, too?

One of the things you said gave me another thought: Within certain limits I can hear a symphony in my head, too. In former times I had a lot of them stored - the sound, not the score. I think many of us, if not all have this capability to a certain degree. At this point there is a question coming up: I think we can hear a symphony in a faster or a slower tempo as well. Or with a higher or lower pitch. I can hear a symphony in my head played by other instruments than those prescribed by the score. I can also hear fragments of two different tunes at the same time – the dissonances are a bit less sharp than they should be. (I wonder why).

Reading your text again, I find the answer to my question: In your case it seems to be the practicing that prevents you from changing the scales, in particular the shape of logarithmic scales compared to linear scales – in contrast to that, performing musicians have to be careful that they will not get slower or faster. Sometimes we see stomping feet. But the feet can get faster or slower as well.

I do not need a tape measurement in order to accurately estimate a measurement less than 1 1/2 inch.  I can estimate the size to an accuracy ot 1/8''.  Joseph can visuallize accurately two logarithm scale.  It is not totally different from estimating a logarithm measurement.

@Martin

I did not intend to go off on the music track, but the mention of timing leads to another experience. As a youth I was not encouraged to follow my musical interest, but I was encouraged to sing at local events. If I had accompaniment, I struggled to keep time. I later studied voice and with practice (that word again) learned to follow the instruments and the beat very well. I cannot hold the beat on my own.

@Joseph,

The question of holding the tempo is perhaps a version of my original question: “How can we be sure that it’s not made of rubber?” – But as far as I understand your commentary: the solution in your case (slide rule) is that you have two scales at a time: the linear and the logarithmic. Together they can be “zoomed in or out” – the relation stays the same.

@Martin

Yes. The relation stays the same. 

Similar to what Louis Brassard said, I know carpenters who can look at a space, walk across the room and pick up a board that will fit it.

This is not learning, it is learned.

Joseph,

I think you have some extraordinary talent like human computers, because none has claimed to get a result by moving slide rule, it cannot be result of some practice or exercise otherwise anybody having some expertise in mathematics would be claiming what you are. 

@Mohammad,

Yes, but you can develop a talent by practising. I suppose there are many gifted mathematicians without this very special talent.

Zooming in and out – non-stable objects in the imagination:

We saw that we need something that stays constant when we zoom on an object. We should know how big the object is in reality – one tricky solution was the case of the slide rule: two scales at a time will retain the relations between them – no matter how big the slide rule is in our imagination.

Problem: E.g. “Will this apple be big enough for my appetite?” (looking at a picture) – There are bigger and smaller apples. We can’t tell from the picture how big the apple is in reality. - Solution: it will help when we know the apple variety.

In contrast to that: E.g. a picture taken from a microscope displays a characteristic structure. This structure will remain when the picture is enlarged.

I have not had time to read the whole discussion so if this point has been raised previously - my apologies.

It seems clear that many of us are able to arrive at 'novel' conclusions based upon the manipulation of mental constructs - e.g. mental blackboards and mathematical calculations. However, this does not necessarily imply that such activity leads to an increase in the level of complexity of the brain:- that the entropy associated with the brain has decreased.

Given no additional stimulus from the environment, it may well be that that the neural resources and the energy expended by such activity alone may, in fact, result in an overall decrease in complexity of the brain even though it seems clear that new information has been obtained.

The conundrum of mental development in the face of the laws of thermodynamics is a problem that fascinates me and this question casts the problem in an interesting  light. My own research suggests that the increase in neural complexity associated with learning may best be understood in terms of the way in which environmental structure is coupled to development of the brain via the non-linear component of Shrodinger's equation (i.e soliton wave).

If it turns out that mathematics offers such a rich foundation of structure that it allows for the brain to increase its level of complexity from contemplation of the mathematical 'landscape' alone  then this would constitute a very interesting finding.

Article The Candle and the Flame:- Structure, Energy, Information an...

Dearest Martin:

I so enjoyed this question and after all of these years I still think about it. Imagination defined: the faculty or action of forming new ideas, or images or concepts of external objects not present to the senses. I also found another interesting quote: Much like my tea party and Harvey: "The ability to create new people from scratch, to put them into scenarios and to tell stories. If you cannot imagine, you cannot write. It is the single most important thing for you to understand about your process." (Rush, 2018).