Friday, July 1, 2016

Unconscious intuition and its conscious resolution


Contributed by Mikhail Filippov, Varun Prasad and Semir Zeki

Ever since the description of the neural correlates of the experience of mathematical beauty, we have been wondering to what extent mathematical beauty falls into the category of biological beauty.

This has led us to enquire further into the extent to which mathematics itself constitutes a study of the brain’s logical architecture; in other words, the extent to which the study of mathematics also belongs in a branch of biology and more specifically neurobiology.

We start by enquiring into the processes which led one of the most interesting mathematicians of the last century, namely Srinivasa Ramanujan, to his conclusions. 

They are commonly referred to as INTUITION.

But what is intuition?

The term is commonly used to signify that a significant insight or conclusion has been reached without thinking and without reasoning. This is true in all languages to which we (the writers of this post) have access. One would no doubt find definitions which are more sophisticated but, as the quotations below show, the absence of logical process in having an intuition is the most common definition and reflects, in fact, what the lay person usually means by it. 

The term has been much written about recently, especially since the publication of a book about Ramanujan by Robert Kanigel entitled The Man who Knew Infinity, recently made into a film.

Consider the following definitions of intuition, which exclude any reasoning process:

Direct perception of truth, fact, etc., independent of any reasoning process, immediate apprehension

An immediate cognition of an object not inferred or determined by previous cognition of the same object                                                                                                                                                       (from Dictionary.com)
“The immediate apprehension of an object by the mind without the intervention of any reasoning process; a particular act of such apprehension.”                                                                                                     (from Oxford English Dictionary)

The French definitions given in Larousse are even more explicit in this regard:

Connaissance directe, immédiate de la vérité, sans recours au raisonnement, à l'expérience.”

Sentiment irraisonné, non vérifiable qu'un événement va se produire, que quelque chose existe

One Italian dictionary defines intuition as follows:

“Intuizione: Conoscenza diretta e immediata di una verità, che si manifesta allo spirito senza bisogno di ricorrere al ragionamento, considerata talora come forma privilegiata di conoscenza che consente, superando gli schemi dell’intelletto, una più vera e profonda comprensione”

while its Spanish counterpart states:

Percepción clara e inmediata de una idea o situación, sin necesidad de razonamiento lógico

Nor are such definitions restricted to Western European languages. Much the same definition appears in Japanese.

"直感" = to capture things by feeling rather than reasoning or discussion.

Or

"直観" is to directly understand the essence of things without relying on reasoning.

We give these definitions in different languages only to show that much the same applies to all. Central to most (but not all) definitions is the absence of reasoning or logical thinking during the intuitive process or its result. Hence, the dictionary definitions given do indeed reflect the way in which the term is commonly understood.

Other definitions come closer to the arguments we give below; they make no reference to the absence of reasoning logic, but only to the absence of proof or evidence.

For example, the Merriam-Webster and the Free Dictionary define intuition as follows, respectively:

“A natural ability or power that makes it possible to know something without any proof or evidence: a feeling that guides a person to act a certain way without fully understanding why” 

Something that is known or understood without proof or evidence                                                                                               (from Merriam Webster)

The faculty of knowing or understanding something without reasoning or proof
(from Free Dictionary )

which is not dis-similar to the definition given in the Great Soviet Encyclopaedia

“ИНТУИЦИЯ(позднелат. intuitio, от лат. intueor - пристально смотрю), способность постижения истины путём прямого её усмотрения без обоснования с помощью доказательства.”
(Intuition is an ability to comprehend the truth through direct discovery without its justification with the proof)

We propose below an alternative definition that may be obvious to some but is not to many and therefore worth giving:

An intuition is an unconscious logical brain process with an outcome or conclusion in the form of a  statement or proposition. But whether the outcome of the intuitive process is “right” or “wrong”, or “correct” or “incorrect”,  can only be determined by a conscious logical process.

The closest dictionary definition to this that we know of is to be found in the Russian Dictionary of Psychology, which of course targets a more specialized audience: 

“Интуиция (от лат. intueri – пристально, внимательно смотреть) - мыслительный процесс, состоящий в практически моментальном нахождении решения задачи при недостаточной осознанности логических связей.”
(Intuition - thought process allowing almost instant finding of the solutions to the problem with the lack of awareness of logical connections.)

Mathematics is a subject in which intuition is often invoked. 

But the end result of the unconscious logical process that is intuition can only be “right” or “wrong” (correct or incorrect) when consciously scrutinized.

As examples of “right” and “wrong” mathematical intuitions consider the following:

A right (correct) intuition: Pick a point at random on the Earth (assume that the Earth is a sphere). The probability that the point picked lies in the northern hemisphere is 50%.

Most students of mathematics (i.e. those who have enough knowledge to understand the above statements, but who do not know if they are right or wrong) would intuitively guess this statement to be true and it is.

A wrong (incorrect) intuition: Pick a real number randomly. The probability that the real number picked is rational is zero.

Most mathematics students would intuitively guess this statement to be false (you can obviously pick a rational number!). However, it is correct.

But the conclusion that they are correct or incorrect can only be reached through a conscious logical process.

Ramanujan was reluctant to submit his intuitions to the conscious process of deductive logic, until Hardy brought him to England and forced him to do so – i.e., to provide proofs for his intuitions – a conscious process.

The absence of any logical process or reasoning in the intuitive process is not the only weakness of the definitions of intuition; some also exclude the role of experience in reaching conclusions through intuition, as in the Larousse Dictionary or the Dictionary.com definitions given above.

We believe, however, that to have an intuition in any area, one must have experience of that area or knowledge of it, to provide a conclusion or statement, whether correct or incorrect.

Since we suspect that there is only a limited set of deductive logical processes in the brain, it follows that the same logical processes must be used to derive intuitions in different domains; what distinguishes intuitions in different domains, and the logical processes that lead to them, is past experience and knowledge in the relevant domain.

The result of this unconscious logical process (the intuition) depends on initial conditions or inputs, which are based on previous (conscious) knowledge, consciously or unconsciously obtained.

Our proposed definition raises interesting and important issues and leads to the suggestion that

a.     There are many (but a limited number of) logical processes, which operate in the unconscious state.
b.     These processes are undisciplined and unruly but still obey some sort of brain logical process.
c.      They become disciplined and eliminated by revisiting them, and the conclusions to which they lead, in the conscious state.
d.     The latter eliminates many of the undisciplined and vagrant unconscious logical processing possibilities, thus stabilizing the logical processing systems of the brain.

The study of intuition in mathematics thus belongs as well to neurobiology. Or, put another way, mathematicians are also covert neurobiologists. 

© Mikhail Filippov, Varun Prasad and Semir Zeki

Friday, August 14, 2015

Are lines always a means to more complex forms? Aleksander Rodchenko would not agree

Orientation selective cells of the visual cortex, which respond to lines of specific orientation, were discovered in 1959. They were first encountered in the primary visual cortex of the brain (area V1) – considered by many for much too long to be the only entering place of visual information into the rest of the visual brain.  Such cells have usually been thought of as the initial staging post for the elaboration of more complex forms. Some, indeed most, believe that they are the sole source for the elaboration of more complex forms such as faces, houses and objects. I am becoming increasingly skeptical of this view.

First of all, evidence which is largely ignored or at least marginalized, although it has been available since 1980, shows that V1 is not the only entering place of visual signals into other areas of the visual brain; there are alternative routes which reach them without passing through V1. Secondly, orientation selective cells are found in at least four other visual areas of the visual brain, and these cells survive functionally even when deprived of an input from V1 (i.e. they remain orientation selective cells); they are, very likely, fed by these alternative inputs. Thirdly, visual signals related to form (oriented lines) reach V1 and the other visual areas within the same time frame. And, finally, clinical evidence shows that humans can become agnosic (blind) for line drawings without at the same time becoming agnosic for real objects.

Hence, one must seek for sources besides V1 for elaborating orientation selective cells and complex forms, which is not to say that V1 cells do not contribute significantly to this process. But perhaps one should also consider, at the same time, that oriented lines stand on their own as forms in every sense, without their being mere “building blocks” for elaborating more complex forms.

Neurobiologists are not alone in considering oriented lines a means towards a more complex end. Mondrian, among others, sought for the constant elements in all forms and settled on the straight lines, provided they are vertical and horizontal. He abhorred diagonal lines, breaking off his working relationship with a colleague because “of the high handed way in which you have treated the diagonal line”. Ever the reductionist (though not accused of it, as we commonly are), he believed that “there are also constant truths concerning forms” and it was the function of the artist “to reduce natural forms to the constant elements”.

Many others, including Kazimir Malevich, Ellsworth Kelly and Barnet Newman, among others, have emphasized lines in some of their paintings, for different reasons. But it was perhaps Aleksander Rodchenko, the Russian Constructivist artist, who was most explicit in giving the straight line its autonomy. Influenced by Malevich and Suprematism, he wrote:  “ I introduced and proclaimed the line as an element of construction and as an independent form in painting”. In another context, he also wrote "I reduced painting to its logical conclusion” (although he, too, was not (as far as I know) accused of reductionism). There are, incidentally, very good perceptual reasons for why he should not have been accused of reductionism, but I will leave that to a future post.

The point of all this is simple: that lines are not only a means towards something more complex; they can also stand on their own as a form or forms; that, as the Gestalt psychologists emphasized, “the whole is other than the sum of the parts” and that a complex form, even when constituted from lines, is one that is other than a combination of lines – an important lesson in the physiology of forms; and that there is much more to the construction of forms in and by the brain than a single source which lies in the orientation selective cells of V1.


It seems to me that the physiology of form construction by the brain is still, in spite of all the excellent work that has been done in the field, a field that is rich for exploration but also requires some of the facts mentioned above to be taken into consideration. In that exploration perhaps the products of artists should also play some role, even if only a minor one.

© Semir Zeki


Friday, August 7, 2015

What does the brain do to ensure that contradictory truths are valid? Answer: it ensures that they never meet.

Contributed by Mikhail Filippov and Semir Zeki

Mathematical and physical theories constitute one means of acquiring knowledge about our Universe. We build models of the way the Universe is constructed through experimental facts. But what happens when they contradict each other. How do we accommodate them both?

In the sensory world, contradictions can occur in vision. This is commonly referred to as ambiguity or instability. We will discuss them first before addressing the question of contradictory truths about the nature of the Universe.

For vision, a good example of an ambiguous, though finished, work is Vermeer’s Girl with a Pearl Earring. The painting is capable of many interpretations – of someone who is distant or inviting or resentful or approving. The important point is that (a) there is no clear solution because all solutions are valid (see Zeki 2008) and (b) only one solution can be valid and occupy the conscious stage at any one moment (see Zeki 2004), before ceding place to another, equally plausible, solution or interpretation, which then becomes sovereign until it, too, is replaced.

With bi-stable or multi-stable figures, the image transmutes perceptually from, say, a face to a house. Again, only one image – face or house – is possible at any given moment, even if one knows that the image is bi-stable.

The transition from one perceptual state to another is not generally under our control. The images flip over between two or more states with prolonged viewing and it is not evident that even the length of time when one state reigns can be controlled.

Thus the brain has devised a system where, when there is no certainty as to the solution, it will entertain two more solutions as equally plausible, even if these solutions are significantly different. But it ensures that the two solutions do not coincide.

The same general rule applies, we believe, to grander and more exalted cognitive states. One such example is to be found in the laws of gravitation and time-space, which are derived from what has come to be known as classical logic. These laws are different from quantum logic, though we would say that both are derived from brain logic, just as two contradictory images are derived from the brain's perceptual mechanisms.

Indeed, it can be said that classical logic cannot reach the conclusions reached by quantum logic. 

In their statement on Quantum Logic, Birkhoff and von Neumann put it like this,  The object …is to discover what logical structure one may hope to find in physical theories which, like quantum mechanics, do not conform to classical logic.”

We note that, in the above quote, they write of the logical structure of physical theories. We believe that the logical structure of physical theories is derived from brain logic.  We would therefore re-formulate what they say, as follows:

The object …is to discover what variations there are in the logical system of the brain that allows it to accommodate the facts that lead to quantum logic as well as to logic dictating classical Newtonian mechanics”.

In truth, quantum logic and classical logic, both of which are brain logic, are not in contradiction. They are just two different models of the physical reality and, like bi-stable images, only one can occupy the conscious stage at any given moment. Also, as in ambiguous stimuli, there is no correct solution, because both solutions are correct.

The overall conclusion that we draw is that the brain does not devise too many  different solutions to acquire (apparently contradictory) knowledge about the world. It uses the same general approach to sensory knowledge as to cognitive knowledge. It accepts even what may amount to contradictory facts, if these conform to its logic system and will reject them both if they do not.

If it accepts them both, it will however not accept them both simultaneously, just as it will not accept two contradictory interpretations of a visual image at the same time.

Hence, in addition to deriving knowledge about the world through its logical deductive system, the brain has another, intrinsic, logical system which allows it to separate out contradictory models as truthful, whether derived from the sensory or cognitive world, but ensure that they do not contradict each other because only one can occupy the conscious stage at any given moment.  This it does by ensuring that they do not co-occur.


This, in fact, is the solution, that the brain has adopted to deal with contradictory but equally valid facts: by making sure that they do not co-occur. In more popular language, it ensures that they never meet.

©Mikhail Filippov and Semir Zeki

Friday, July 3, 2015

Colour Vision and Mathematics

 
Contributed by Mikhail Filippov and Semir Zeki

That the experience of mathematical beauty, derived from a highly cognitive source, correlates with activity in the same part of the (emotional) brain as the experience of beauty derived from sensory sources makes it interesting to enquire what other common factors mathematics shares with sensory experiences.

We choose colour vision as an example.

One of the primordial functions of the brain is to acquire knowledge and it has to do so in the face of continually changing conditions, often referred to as the Heraclitan doctrine of flux (after Plato). To extract that knowledge, the brain has to somehow stabilize the world, since it is difficult to acquire knowledge in constantly changing and often unpredictable conditions.
With colour vision, a surface or object of any colour can be viewed in different lighting conditions (for example sunlight or indoors in tungsten or fluorescent light), when the composition of the light (in terms of energy and wavelength composition) reflected from it and from its surrounds changes continually.

Yet, by a process  dictated by brain logic (usually referred to as an algorithm), the brain discounts these continual changes to assign a constant colour to the object or surface. This is what is meant by colour constancy.

Without it, the task of acquiring knowledge about objects and surfaces through colour becomes difficult, if not impossible; without it, colour would lose its importance as a biological signaling mechanism.

How it does so, in terms of the neural mechanisms involved, is not entirely clear but it does involve a specialized centre in the brain and the pathways leading to and from it. Through this process the brain stabilizes an ever-fluctuating world and is thus capable of acquiring knowledge about it through the colours of objects in it.



Our Universe, at the other end of the scale, presents an even more complex picture; but, similarly, the only way to acquire knowledge about it is to stabilize it by reducing all its complexity to some fundamental rules, reflected in equations or mathematical formulations.



These formulations are the products of a deductive logical system that belongs to the brain; their end-result is to stabilize the world through simple, all-embracing formulae, and hence acquire knowledge about it.



Thus the knowledge-acquiring system of the brain uses a logical system to acquire knowledge about, on the one hand, a sensory category such as colour, which is continually experienced throughout the day and, on the other, knowledge about the structure of the Universe which is not possible to experience directly. The end-result is to stabilize the world, sensorially in the case of colour vision and cognitively in the case of what determines the structure of the Universe
 

There is another feature that mathematical formulations about  truths governing the Universe share with sensory experiences such as colour vision – in both, there is one route and one route alone and, once established, there is no appeal against its conclusions.



All knowledge that a green leaf is reflecting more red light (as it commonly does at dawn and at dusk) will not enable one to see the leaf as red. The operation that the brain applies to generating constant colours in spite of variations in the wavelength-composition of the light reflected from surfaces and objects under different lighting conditions allows one to see the green leaf as green only (although its hue, or shade, will change under different illuminants).


And all cognitive knowledge acquired through daily experience, that time and space are separate entities, will not invalidate the conclusions postulated by the theory of relativity, which show that time and space are continuous, at least to those who know the language of mathematics. 


There are, of course, conditions, in which two fundamental truths are in apparent contradiction to each other, as in macro- and micro- physics.



Here, too, the brain’s system for acquiring knowledge through the sensory system shows strong similarities with its system for acquiring more abstract knowledge.



We will return to it in the next post.

Tuesday, May 19, 2015

The crushingly boring centrepiece at the Venice Biennale


The Venice Biennale is, according to most accounts of it, an exhibition of the latest in contemporary art. But this year it appears to have taken contemporary art to new and unheard of dimensions.

Apparently, the aim of the biennale this year is to enquire into “how art reflects the nature of our imaginings”.

So far, so good.

But then come all these indigestible phrases about “atomized space” from which to create a “molecular space”.

Beginning to sound somewhat dodgy?

Well, it gets much worse.

It takes imagination of the tenth power to make the centrepiece of the biennale this year the - wait for it – continuous reading by professional actors, over a period of seven months, of Karl Marx’s Das Kapital. Apparently this will allow us to create an “interpretive concept” through which “to reflect on these incredible times”. The ultimate aim, apparently, is to move from a state of continual transition to a state of harmony, where presumably things have settled down to allow us to experience heavenly bliss.

The first thing to say about this is that it must be crushingly boring to listen to seven months’ worth of continuous reading of Das Kapital, whatever truths it may or may not articulate. I mean even Shakespeare will not pass that test.

But next, I somehow doubt that even Marx believed that we will end up in a state of harmony, where all struggles will cease. He was, after all, an admirer of Hegel.

Since contemporary art is now being appropriated in the service of politics, it is worth recalling that Marx was an avid reader, and among his favourite authors was Balzac.

It may have been more appropriate to use some of Balzac’s masterpieces to explore the dilemma of continual conflict resulting from our natural tendency to exploit. It would have certainly been more entertaining. Whether Balzac could pass the test of continuous reading over 7 months is, however, another matter.

For in Balzac’s pages one will find that it is not only the bourgeoisie that exploits the proletariat; rather, exploitation is part of our constitution, our neurobiological make-up.

In Balzac’s pages, the rich exploit the poor, but they also exploit each other. The poor do likewise. Women exploit men, and men exploit women. And that most extraordinary creation of Romanesque literature, Balzac’s Vautrin (who, Balzac tells us, is like a vertebral column that runs through three of his most famous novels) exploits everyone in his efforts to dominate society.

The will and capacity to exploit, and dominate, is part of our neurobiological constitution.

 Marx understood this well.

In The Communist Manifesto (Chapter 1), he writes that “the bourgeoisie is itself the product of a long course of development, of a series of revolutions in the modes of production and of exchange”; he well understood that “it [the bourgeoisie] was an oppressed class under the sway of feudal nobility” and how, with increasing power, it turned oppressor.

And of course, exploitation being in our very nature, whenever the opportunity presents itself, the exploited become the exploiters. Doesn’t the communist revolution show this admirably?

Perhaps a rendering of Balzac’s Harlot High and Low would have been a better choice when appropriating art in the service of politics. It would certainly have been a lot more entertaining.

So, as far as I am concerned, it is a big “Ciao” to Venice this year.


Thursday, April 30, 2015

The Experience of Mathematical and Biological Beauty

Contributed by Mikhail Filippov and Semir Zeki

The experience of mathematical beauty is perhaps the extreme case of the experience of beauty conditioned by culture and learning. One cannot experience mathematical beauty unless one is mathematically cultured. Those not versed in mathematics are unlikely to experience beauty in equations that mathematicians find beautiful and sometimes are even moved by.

And yet, mathematical language is universal. Mathematicians of different culture – from China, Western Europe, Africa, Russia – are able to experience beauty in the same equations even in spite of their profound cultural and linguistic differences. Hence, in another sense, mathematical beauty is not conditioned by culture and language. Indeed, it can perhaps be said that mathematical beauty is less culturally biased than other forms of beauty.

We must seek its source elsewhere than in cultural differences.

An interesting article on the experience of mathematical beauty suggests that Immanuel Kant saw the source of the beauty in mathematical equations in the fact that “they make sense”.

This raises two questions:  what does it “make sense” to, and why does it make sense to people of different cultures, who are nevertheless apart from people of all cultures, even their own,  who are not versed in the language of mathematics.

Our answer is that it makes sense to the logic of the brain, in that it is consistent with the logic that has evolved in the brain. The implication is obviously that the logical system of the brain is similar to those from different cultures. In other words, these mathematical equations make sense to people of different cultures because the logic of the brain is similar, in spite of cultural differences. Hence the common experience of beauty in the same equations reveals something about that logic.

It is therefore not surprising to find that there was, in our sample of mathematicians, a fair consensus in rating Leonard Euler’s identity formula,

                                    1 + ei = 0  


which links 5 fundamental mathematical constants with three basic arithmetic operations, each occurring once, as very beautiful.

But is this significant uniformity in rating an equation as beautiful vastly different from the rating of visual beauty by subjects belonging to different cultures? It is a subject worth addressing. 

We surmise that, if subjects from different cultures were asked to rate what we broadly call “biological” stimuli, such as human faces and bodies, in terms of beauty, there would also be a fair consistency. We also surmise that there will be a similar consistency when subjects from different cultures rate human faces and bodies as ugly. This consistency may not be apparent when subjects are asked to rate the beauty of artefactual stimuli, such as buildings; here culture and learning may play a more significant role.

Hence the experience of what we broadly refer to here as “biological” beauty, beauty in art, may be dictated by inherited brain concepts of what is “right” and makes sense, just as in mathematical formulae what is experienced as beautiful makes sense. Both, in other words, fall into a biological category, which distinguishes them from the beauty dictated by acquired, synthetic, brain concepts, as in the experience of architecture as beautiful. Acquired brain concepts are more conditioned by culture and learning and are hence are modifiable throughout life. Mathematical beauty is more resistant to cultural influences.

The experience of mathematical and biological beauty, even in spite of the fact that the former depends upon learning and the latter does not, therefore share, paradoxically, a similarity in that both are dictated by inherited brain concepts which makes them impervious to cultural differences but which, in the case of mathematics, can only be revealed through a language – that of mathematics – that individuals must acquire before the experience can be enabled.

This of course raises the question of what the logic of the brain represents and how it developed and evolved? Was it in response to the structure of the Universe, as Plato in ancient Greece and Paul Dirac in more modern times, would claim?

These are problems worth thinking about.