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. 

Saturday, March 7, 2015

The Philosophical Transactions and Michelangelo

 
Yesterday, I was pleased to celebrate two birthdays: the birth of The Philosophical Transactions of the Royal Society (of which I was Editor in Chief between 1997 and 2003), and the birth of Michelangelo.

The birth of The Philosophical Transactions (established March 6, 1665) was celebrated at a party at the Royal Society (accompanied, strangely enough, by hot dogs and French fries!!). Phil Trans, as it is now commonly referred to in abbreviation, is the world’s first scientific journal, its longest running, the first to introduce the peer-review system and the first to publish a paper by a woman scientist (Caroline Herschel in 1787).

As the President of the Royal Society reminded those gathered to celebrate last night, before Phil Trans was established, scientists used to correspond with each other, often in code, for fear that their findings may be stolen. Phil Trans changed all that and hence made science more accessible, while at the same time giving a scientist priority for his/her findings.

It was established by Henry Oldenburg, German by birth and the first Secretary of the Royal Society, and has since published many interesting papers, including ones by Newton, Boyle and others. More recently, these have been in the form of reviews and the issues have often been theme issues, devoted to a particular topic.

Soon after its birth, London was hit by the Great Plague and then the Great Fire. Phil Trans was spared because, at that time, its offices had moved to Oxford.

But Oldenburg himself was incarcerated briefly at the Tower of London. He had been in correspondence with some Dutch scientists and, during the Anglo-Dutch wars, the security services suspected him of having Dutch sympathies and therefore of being a security risk.

In 1887, the journal divided into two sections, one devoted to the physical sciences (A) and the other to the biological sciences (B) and has continued in that form (I was Editor of the B section).

The birthday was also a moment of reflection about the future of scientific publication and the peer-review system. The latter is often abused but not nearly as much, I think, as people believe. But with so many scientists producing so much, can the peer-review system survive in its present form?

In a sense, the peer-review system is itself somewhat outdated now, or rapidly becoming so. Scientific findings, especially ones that are considered to be important, are subject to post-publication scrutiny. Just think of what happened to a certain well-known paper published in Nature last year. This perhaps will rapidly reduce the peer-review system to a sort of check-list, to ensure that it is broadly respectable, without too much quibbling about the interpretation of the results.

Plus of course, any scientist who is completely shut out can always publish results on the internet.

In fact,  post-publication review has been with us for as long as Phil Trans and even longer. Good papers stand the test of time because they are found to be good post-publication and bad or indifferent ones wither away and are forgotten, no matter how glowing the peer review may have been. 

No one invited me to a celebration of Michelangelo’s birthday (March 6, 1475) – assuming that any had been organized.

So I celebrated it with friends, all of them Michelangelo nuts, at a dinner.

Altogether a very nice day.

Sunday, October 12, 2014

Box office success from questionable science

 
It was inevitable!

It had to happen!

Nature reports that a film, called The Whistleblower,  has been made, based on the Woo Suk Hwang scandal in South Korea, concerning the creation of embryonic stem cells by cloning. The film stars top actors. 

The disputed papers were published in a famous scientific journal, Science, and subsequently retracted

The film apparently paints a sympathetic portrait of Hwang as a man with human frailties, like the rest of us.

The real whistleblower would seem not to have been very pleased with this film because, according to the report, “his own contributions and those of online bloggers were credited to the reporter” (in the film).

The Nature report draws attention to the fact that Nature itself was the first to report that Hwang had procured the eggs for his experiments unethically.

Of course, there is another film in the (potential) making, this time about a paper published in  another famous scientific journal, and which had even more tragic consequences.  

I wonder when such a film will be made.

Thursday, October 9, 2014

A non-binding resolution to the binding problem?


Contributed jointly by Dragan Rangelov and myself

The binding problem is a specific example of a more general problem in brain studies, namely that of integration, that is to say of how the many, specialized, areas of the brain interact to provide the integration that is evident in our perceptions, thoughts and actions.

Binding has come to refer more to this problem within the confines of the visual brain. Here the binding problem becomes the problem of how the several, parallel, processing systems in the brain interact to give us our unitary perception of the visual world, in which different attributes such as form, colour and motion are seen in precise spatial and temporal registration.

The initial mistake is to suppose that we do see these attributes at precisely the same time. In fact, psychophysicalexperiments show that this is not true and that we see and become aware of some visual attributes such as colour before we see and become aware of others, such as motion.

This raises a question which has so far remained un-addressed, namely of whether there is some central station in the brain that “waits” for all the processing systems to complete their tasks before “binding” the results of their operations. There clearly is no such system because, over very brief time windows, we bind the colour that we see at time x to the motion that we had perceived 80 ms before. We therefore mis-bind in terms of the objective reality.

We discussed this issue some two years ago while at a meeting and thought that we should conduct some more experiments on this problem. Our approach was as follows: we presented subjects with lines of different orientation that could be in a number of colours. If colour is bound to orientation at perceptual or pre-perceptual stages, then the accuracy of reporting one attribute, say colour, should co-vary with the accuracy of reporting the other attribute (orientation), when the two are presented to subjects over very brief time windows.

If, however, the two attributes are not bound at the pre-perceptual or perceptual stage, then the accuracy of reporting one attribute (colour) should vary independently from the accuracy of reporting the other (orientation).

Our results, just published, showed that the accuracy of reporting the two attributes is independent, with the accuracy of reporting colour being always greater than the accuracy of reporting the orientation, probably reflecting the fact that colour is perceived before the orientation of lines by about 40 ms.

This suggests that these two attributes, at least, are not bound at either pre-perceptual or perceptual stages.

This result leads us to conclude that binding does not occur by physiological interaction between cells in the visual areas, but rather occurs at pos-perceptual stages, perhaps through the intervention of memory. We only experience attributes as being bound even though they are not bound physiologically, and only because they occur within the same, very brief memory time window.

Our results may provide, we think, an interesting resolution to the  binding problem, namely that there is no such problem to resolve at the perceptual level.

If binding occurs post-perceptually, then the search for how binding occurs shifts to a different arena.

Time will tell whether we are correct in our interpretation. 

We may of course be wrong, but we hope that our new view provides the ground for interesting new experiments and debates.

Thursday, September 4, 2014

(Literally) blind experts


An American friend drew my attention recently to a paper published last year and entitled The Invisible Gorilla Strikes Again: Sustained Inattentional Blindness in Expert Observers. It is the report of a study in which experts failed to detect an unexpected occurrence (gorilla) in their area of expertise even when viewing it directly.

This phenomenon, known for a long time and, to my knowledge, first described in a paper published in 1999 by Simons andChabris, is known as inattentional blindness. In the past, it has been demonstrated with naïve subjects in unfamiliar tasks. The authors of the above study asked: does inattentional blindness also occur frequently among experts?

A very interesting and highly relevant question!

To study this, they asked 24 radiologists (hence experts) to screen CT (Computed Tomography) scans of lungs for nodules. The radiologists ranged in age from 28 – 70 years; hence some must have had very considerable experience. Their eye movements were tracked as they viewed the scans. But embedded in the scans was a gorilla which was some 48 times the size of the average lung nodule that the radiologists were searching for; moreover, it was positioned close to a nodule.

20 of the 24 experts did not report seeing the gorilla and eye tracking revealed that, of the 20 radiologists who did not report seeing the gorilla, 12 had looked directly at where the gorilla was located.

The authors conclude that, “This is a clear illustration that radiologists, though they are expert searchers, are not immune to the effects of IB [inattentional blindness] even when searching medical images within their domain of expertise”. They add, “Presumably, they would have done much better at detecting the gorilla had they been told to be prepared for such a target…perhaps a smaller gorilla would have been more frequently detected because it would have been more closely matched the size of the lung modules” [sic].

A sobering thought!

I imagine that, on the whole, radiologists do end up detecting the nodules, even if they are apparently often not able to detect something that is blindingly obvious (if I may use the phrase in this context).

But just think of experts in other domains, say economics, and above all of expert politicians of all stripes and in all countries. Thinking about them, one cannot help but suppose that they, too, must suffer from inattentional blindness, but this time of a more cognitive variety.

In fact, I am increasingly inclined to believe that many of today’s problems are due to the inattentional blindness of politicians to the continual and rapid and huge changes occurring (and here the size of the gorilla compared to the lung nodules comes to mind). I am increasingly led to believe that politicians just do not have their ears to the ground and, in many instances, are – because of this inattentional blindness - way behind public opinion on a great many issues. Hence they are not up to date experts.

Perhaps this puts the inattentional blindness of radiologists to huge gorillas embedded in the scans they are examining in perspective.