Proof you are right!

February 3rd, 2009

I came across a very interesting debate on mathematical proof. The debate started with a small subgroup of mathematics who are strong believers in computer verification of proofs. This quest is promoted as the QED project (check out the QED-manifest). In order to verify a proof by computer, it needs to be recoded, because the human proof takes simple steps for granted. The computer proof is a ‘computer program’ whereas the ‘human proof’ is more a formalised story.

The debate is interesting for many reasons and I think it has a broader meaning than maths only. In a sense it is about what we still think is human. We embrace the ‘romantic side’ of maths: the beauty in a proof. I am not a mathematician, but I still remember the beauty of a good proof as I learned in high school. There is beauty in the idea of a solid, water-tight proof but there is also beauty in the steps. The set of different rules, the one leading to another is an art. My son is studying to be a maths teacher and he told me about the wonderful feeling he has if he succeeds in proving something, even if this has been proven already by a great number of students. Needless to say that the feeling of proving something new must be euphoric. Now you could interpret this QED-project as an attack on the romantic side. If there is any feeling and emotion attached in constructing or understanding a proof, the computer version of it is devoid of that. A simple proof that even I can understand is elegant in ‘language’ but not any more in computer language (check out this pdf to see the difference. It is in Dutch but I guess that anyone with algebra background can read it no matter the skills in our Low Lands Language).

In the Dutch daily NRC Bennie Mols wrote an article about the subject. What really struck me in this discussion is the standpoint of Hendrik Lenstra (a Dutch mathematician). He claims that the act of proofing and judging a proof is basically a human act, based on pattern recognition. The act making and checking a proof helps you to better understand the subject. It evokes ‘knowledge’ as an internal phenomenon. He shares an interesting and wonderful story about a theorem he was trying to understand. He wanted to know if it was correct. He had it both computer-checked and asked another mathematician to check it. He consulted an expert in computer mathematician. The computer had to work a day on it, and showed the proof was right. The human expert got about it in a different way. He looked at the theorem and said ‘oh I can calculate that for you’. Within in few minutes he had the maths done – writing it out on a piece of paper. His conclusion: the theorem is wrong. Of course – sorry for those with a romantic view – the expert had made a small miscalculation. The computer was right. The interesting thing though, was that actually mr. Lenstra learned a lot more from the human interaction. Checking the piece of paper, understanding the theory used by the expert gave him deeper understanding about the problem. Sharing this with the computer expert, enabled him to rewrite the computer proof in a much more elegant way, so the computer now needed a few minutes as opposed to the former day of computer time.

I see many learnings is this beautiful story. Too many to ponder about in my blog that I want to be short (actually this posting exceeds my intended maximum manifold, I hope to reach a sort of Haiku blog in the future, revealing basic truths within 11 words. But I haven’t yet reached that skill). Let me highlight some:

1

whether we like it or not, automation of knowledge is bound to grow. Even the critics of QED see that it can be extremely useful to check extremely complicated proofs that human race is not able to verify with 100% certainty

2

This will not squeeze the romance out of our lives. As long as we are here on the planet (could be shorter than intended, but let’s assume we will survive the next 2.000 years) the field of ‘beauty’ will change. Beauty is in the eye of the beholder.

3

Knowledge and understanding is the field of humans. Up to now the computers can assist.

3

In the end it is all about the human factor. The QED is a sort of ‘bookkeeping’. Converting proofs into computer language is not like hot science, it is more the work of a monk. But that can be very satisfying.  I can imagine the writer of the pdf I referred to above in his feeling that programming a proof is a wonderful thing to do.

4

the frontier on automation is ever changing. The discussion is now about the verification of proofs. Could it ever shift towards production of proofs? Maybe then computers would become like humans more. So that would not really shift things. We would still have the two different states; the understanding state (which is emotional by definition) and the calculating mode. The shift would mean that the emotional state (up to now the privelige of human and probably animals) is now available to computers as well. We are not near that in the least.

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More numbers!

January 30th, 2009

Speaking of numbers that are hard to grasp. Our mind is even less equipped in grasping exponential growth. We just don’t get it. The famous  ‘Powers of 10′ video is a good example. If you haven’t seen it you should. Look at the website or just go to YouTube and search for ‘powers of ten’. In each few seconds the viewing distance is multiplied with ten, starting one meter above a couple having pick nick in a park. You don’t need an awful lot of 10-fold sequences to escape the milky way! It is hard to grasp the growth rate. It is going so fast.

If you imagine that the number of atoms in the universe is estimated on a 10 to the power of 88, you are astouned, because it feels like an exptreme small figure. I mean, the amount of dollars spend at the Irak war is already 10 to the power of 12. The amount of people on earth is 10 to the power of 6. The point is, our mind assumes the exponent as a sort of ‘repeated sum’ as opposed to a ‘repeated multiplcation’.

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Numbers!

January 29th, 2009

I have always been intrigued by numbers, especially the way our brain usually is unable to get a good idea of numbers. We don’t easily grasp large numbers. That is why each journalist uses comparison tricks. Especially we are unable to understand exponential growth. We just don’t get it. The famous NASA ‘tenfold’ video is a good example. In each few seconds the viewing distance is multiplied with ten, starting one meter above a couple having pick nick in a park. You don’t need an awful lot of 10-fold sequences to escape the milky way!

However, I just was guided to a wonderful website of chris jordan, an artist who uses photographs to give us an idea about ‘numbers’ in waste: 32.000 barbie dolls to signify the number of breast enlargements every month in the USA. 320.000 light bulbs, signifying the amount of kilowatthours wasted every minute in the USA s as a consequence of ineffeciency. Wonderful how he brings into life enormous numbers, and a society based on overspending. And beautiful photography!

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This is the feeling

January 26th, 2009

Yesterday on tv an old interview with Freek de Jonge, a famous Dutch comic. The interview was 15 years old and represented the yesterday Freek. But apart from that ‘today’-Freek was there, watching his younger self and commenting.

The older interview was not so revealing in my opinion. However I was very much intrigued by something the ‘current Freek’ disclosed. It was about the nature of drama. Freek has always been specialised in one man shows. He has developed a cunning in that direction that you don’t see often. In the 70ties he sort of defined the comic show. All of the comics after him are in a way based on his shows in the 70ties. Special in his shows was the ‘red thread’ within a highly associative story line. He sort of glued completely different funny stories in a sort of meta story.

What struck me most was his experience on how you get drama across. In his view there is a miracle going on. ‘How can someone on the back row in the theatre, who is not able to see the expression of my face able to react on expressions in my face?’. Freek explains this as a phenomenon that is manageable. He saids: ‘You have to bring yourself in a state where you are able to do this. You cannot explain how, but you know you can. This takes a special concentration. And you have to bring the audience in a state that they will react in this special way. It takes time to do that. You know when this happens, but you cannot analyse this.

I think that everybody working with a crowd -even a small one – has this experience. The magic that enables some politicians to fly high above there ‘human state’  (Reagen, Clinton, Obama).

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Guilty innocents

January 15th, 2009

I have been watching Peace is Peace is every step by Thick Nhan Hanh on YouTube. The video takes about an hour, it is not very good quality, but I was stunned and moved and kept watching the full 52 minutes. (If you click the link above you will get to it. I recommend all to watch this).

Why was I moved so much? There is quite a lot to tell about. The whole idea: a bunch of Vietnam veterans, oldies now, but rough. You see these rough guys doing exercises with Thich Nhan Hahn like a bunch of softies. One explains that he was in Nam at the age of 17 and had been responsible (his words) of killing more than 50 before the age of 18. This part is tearing me apart, having a son of 18. All this was moving.

But maybe more than that I was moved and intrigued by Thich Nhan’s speech. Both the words and the tone of his voice. Let me start with the latter. We hear speaking a man that suffered the Vietnam war as a victim. As a Zen monk he had chosen to fight for peace and not to choose the side of one of the two parties in war. As a consequence he has never been able to live in Vietnam, even after the war, because he was felt as a traitor. And you hear no remorse, no hate, no frustration in his voice. This in itself is an act of peace. A deliberate chosen one, because it is clear that his life was dedicated to fight for peace. His choice is to live peace as an act of life, every moment. This moved me even more, after reading the daily on the middle east conflict. You know this conflict has been going on for a long time, and that an awful lot of suffering is going on. If you see how devastated the Vietnam Veterans are and how they struggle now, decades after having been there to find peace in themselves, you know that there wil have to be done a lot of healing in the future.

This about the tone of voice, now the words. These where fascinating, both in an theoretical and practical sense. I can’t quote literary (again: watch the video!). But in essence he speaks to those feeling guilty about the killing they have done. He says: why do you take all the guilt for this? You where not there on your own. You were send by a president. The president was chosen by the people. You were the hands, but are the hands guilty alone? The interesting about this argument is that it is correct. America is an abstract phenomenon, consisting of several hundreds of million people, interrelated in a complex way. From all of their individual thinking, feeling and acting, based on a history of thinking, feeling and acting, within a wider context of the world, emerged this war. This is an insight, worthy pondering on. War as an emergent property. The way a highly structured and intelligent colony emerges from the individual, stupid ants. The way (the illusion of a) consciousness emerges from a relatively a little bit of grey matter with quite a large amount of interconnected cells . A big mass of little interactions, sum up to a new degree of abstraction, creating its own laws. Maybe each of the ‘little’ interactions is understandable: fear for being taken over by communists, regret about losing a highly profitable country, seeing movies about heroic soldiers and wanting to join that romantic lot, the need to earn your money, the stories about your father having rescued Europe from the Germans, the need to act as a president. Apart from that there are without doubt all of the real bad intentions of a few. And all together, this blends into something terrible, with its own laws and it can’t be stopped. (Well, it stopped eventually, but at what costs? And how much sorrow, grieve, remorse, hatred and pain is still there? I mean, even the mines haven’t been all moved away, the war still has its victims more than 30 years after it was ended).

This was exactly my point in my blog about ‘The benevolents’. It is creepy to see how atrocities can emerge from a bunch of people that are all on there own (or at least most of them) not of bad intention. (by the way, I stopped reading it. I still think it is a brilliant book but it is too much tot digest).

The argument is extra interesting since it has been misused after the 2nd world war. War criminals tried to hide behind it. They used it to mask there misdeeds and get away with it. For this reason there has been a long ban on the argument. Saying ‘i didn’t know’ ‘I was acting in commission’ are suspect.

Apparently the same argument can be valid and invalid in different contexts, depending on the intention it is used. The context of Thich Nhan Hanh is consolation, with the intention to take away or at least diminish the negative energy created by the feelings of guild of boys who where sent to Vietnam and put in a situation where normal man start to kill. As a means of defence. As a means of wrath, and probably in the end – as the mind gets dumb – out of fun or boredom or whatever stupid reason. Even if they went there voluntary, they could not be blamed for this alone. But Thich Nhan Hanh goes further. He says: do not take a grunge against those who send you. The leaders could have been victims of ignorance. Maybe if they knew how bad war is, they would have refrained. This is debatable, because I think they did know or at least could have known, but probably put it out of their minds. But I get his intention. If it is true that war is an emergent property of the complex system called ’society’, then it is useless to spend your energy trying to find the guilty ones. As a victims of war (as he calls the veterans) you should better spend your energy for more positive things, trying to get war, guild and hate out of your system, maybe do something positive. As his advice to one soldier who suffered from his responsability of the death of 5 children (in an ambush meant for soldiers). His advice: 40.000 children die each day right now. You can save 5 children today. Why spend you life time in regret about the children who died 30 years ago?

So if the argument ‘you were not responsible alone, you were part of a system’ is used to console, it is all right. If it is used to escape, it is not so right. Because it might be so that all of the players within the systems are so to speak ‘victims’, this does not take away all personal responsiblity. Thich Nhan doesn’t claim it does. He didn’t say ‘why do you feel guilty’, he said ‘why do you take all the guilt on your shoulders’. Here we are at an extremely interesting field, highly ambiguous. There are no right or wrong answers here. If this whole idea of war as an emergent property is right we are all responsible for all that happens. Because we all function in the system called ’society’.

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An interesting view on innovation

January 14th, 2009

Leadership and innovation

This is an interesting video from Jutta Treviranus. I like the opening sentence as an answer to the question ‘what does it take for a state to become a leader on national or global level: ‘let me argue some of your framing assumptions’. In stead of that we need to get away from the assumption that states are in the same competition, all trying to reach the same goal, fighting to get the best out of it.

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The great chain of being

January 13th, 2009

To come back at the evolution of ideas. One of the mechanisms that appears to be an important factor in the survival of ideas is that new ideas use ‘old ones’. By a very strong mechanism called ‘metaphor’ old ideas get new meaning in a metaphorical transformation. Because of this process ‘old ideas’ can pop up in a new form. A very powerful idea for instance is the ‘end of times’ and especially ‘the big flood’. This is a sort of archetypical idea that is very powerful. These are stories that stick. Would it be coincidence that after we had the big fire as our ‘end of times’ (during the cold war) now the flood is the next scenario?

Another powerful idea is ‘the great chain of being‘. This view that can be traced down from the ancient times, describes our universe as a sort of big ladder of phenomena or beings from a very simple, earthly form towards more complicated, complex and – in the end – celestial forms (such as angels and ultimately God). The ‘lowest’ parts are minerals, further on the more humble beings, worms, insects, towards animals, human beings, angels, gods. The human race is placed in the middle, just between the more mundane and the more transcendent. Lovejoy (The Great Chain of Being: A Study of the History of an Idea, (1936)) wrote a tremendous book following the idea from the ancient greek towards the modern biology from Darwin. I utterly enjoyed reading it, in the early 80ies. In this book the evolution of this idea is described in detail.

Maybe this could be an important factor of ’sticking power’: any idea that is more resonant with some other, archetypical ideas is more easy to pick up. Maybe this is why the evolution – as an unintentional, non goal seeking, non idealistic process – is not (yet) very sticky in our common sense world, whereas the idea of ‘intelligent design’ that is very resonant with the idea of a ‘masterplan’ is very easy to stick.

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The origin of ideas

January 12th, 2009

In february 2009 it will have been 200 years ago Charles Darwin was born. In NRC – a Dutch Daily – a science section was dedicated to this man. Although his theory is widely accepted scientifically it is still hard to grasp his ideas. Especially the idea of unintended change is extremely hard to get. Over and over you see in argumentation that the idea of evolution as striving to a goal, is popping up. We as a human race tend to think backward. We have the illusion that all of the events that lead to the state we are in, are deliberate steps leading towards this state. That is because we think in stories, and in the story lines we are constantly looking for meaning, a plan. I think this is because a meaning and a plan we can understand, and what we can understand we can control in one way or another. This is why the idea of ‘intelligent design’ is so much more easy to understand then the idea of natural selection without any purpose.

For me it took reading ‘the blind watchmaker’ from Richard Dawkins to really grasp this idea of unintended progress, without any plan or preconceived idea. Every next step in evolution as an emerging trend, the consequence of many interactions over a long time. It is apparent that we do have a problem to intuitively understand how complexity can grow out of ’simple laws’. But it is obvious that complexity does always emerge from simple interactions, such as weather systems with extremely high complexity arising from simple interactions between gas molecules.

Dawkins tried to apply the idea of natural selection to our thinking, introducing the concept of ‘memes‘ as the ‘genes of ideas and thinking’. I wonder if we really would need this ‘memes’ concept as a mechanism of how selection could work in ideas.  I feel that it is possible to conceive other ’simple’ mechanisms than these mysterious  ‘memes’ to account for a natural selection within the thought world. Some interactions are extremely simple: I remember reading Montaillou from Emmanuel Le Roy Ladurie, describing in detail the way the religious ideas spread in this village. What really struck me was the simple explanation: you got ‘infected’ by the new religion by just hearing it from someone you new. It was quite simple to track down the current of the idea by understanding ‘who knew who’ and ‘who met who’. The conviction outcome was like a function of ‘meeting’.

Gladwell dived deeper into this in ‘the tipping point‘, using three simple basic laws as explanation of how ideas spread: the law of the few, the stickyness factor and the power of context. The only law really ‘explaining’  is the law of the few. This law describes in detail different ‘modes’ of infection of ideas by accounting for different kind of persons who spread the ideas. The other two are not really ’simple laws’ . As interesting as they are they are more a description of the the problem than an explanation to the spread of ideas.

Take for instance the ’stickyness factor’. Some ideas seem to have more sticking power than others. This law asks for an explanation. Why is the one idea more sticky than the othert? Why was the sticking power of the religious ideas spread in Montaillou so sticky?

I think the ‘Origin of ideas’, with an explanation of a process that steers the idea evolution would be an important next step in science.

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Reductionists have more fun

January 8th, 2009

Every now and then I feel the roots of my study in Dutch literature. While studying today on the difference between reductionistic and holistic view on reality (in order to get a better understanding of complex systems), good old Julien Offray De La Mettrie came to my mind. If his name doesn’t sound very Dutch, that’s because he wasn’t Dutch. He was as French as his name. However, claiming an extreme materialistic and reductionist view, he had to flee from France. In 1745 you could still visit Holland if you had any deviant opinion that religious leaders considered dangerous, and more important: you could have your books printed. That’s why he lived in Leiden and that’s one of the reasons his book L’Homme machine entered my curriculum of Dutch literature. I never read the book. It was mentioned as an example of how the very productive reductionistic views of Newton and Descartes pervaded the thoughts at the time. In the book La Mettrie explained human beings as machines.

The whole idea of his book is a wonderful example of how a reductionist view can be right (human have some machine like properties, for instance the ability to lift weights) and utterly wrong at the same time. And he was dead serious about it. Of course his ideas where heavily influenced by the struggle for an atheist view on the world. And his comparison is not so much more idiot then the current opinion that our human mind is a ’sort of computer’ (a view both insulting to the human mind and to computers).

Not that I have an argument against reductionists per se. Newton was one of the superminds. He is considered by the Royal Society as the greatest scientists in history and therefore was rated over Einstein (because other than Einstein he was not only a huge theorist but a brilliant . His views where revolutionary. And I guess at the time the reductionist views where needed badly, given the predominant religious view on reality and science. On top of that, the view absolutely works brilliantly in the field of Newtons study. But I guess the reductionist view have their limits.  And De La Mettrie surely went beyond. .

But as you can clearly see in his face, reductionists can have fun. He is supposed to have died because he indulged too much in the good things of life. I don’t want to keep this wonderful sentence from his Wikipedia Bio:

It is claimed that La Mettrie wanted to show either his power of glutony or his strong constitution by devouring a large quantity of pâte aux truffes. As a result, he developed a fever, became delirious, and died.

I would have liked to drink a pint with him before that happened.

(picture taken from the Wikipedia website)

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How painful to understand

December 29th, 2008

For all of those who still believe that human mind is a ‘newtonion’ one state machine and who see research as a tool to find out which state a certain consumer is in (preferably count the consumers, find out were they live and send them a message ‘buy my stuff’, a few examples to attack their view on the world.

The way we percieve pain is extremely dependent on context. If you look at your painful hand with a looking glass the pain grows, if you look at it with the opposite (making your hand look smaller) the pain diminish. (example form Dave Snowden’s blog, more to be found there)

Taste of any food is highly dependent on context. Beer drinkers are not able in a double blind test to find their own beer within others. But if they drink it branded, the taste difference is big. Old school would say: the consumer is stupid. However, neuroscience has prooved other wise: you really taste something different. The taste really changes when it is branded.

Taste is not a fixed state thing, neither is pain, neither is any of the most important motivators for buying anything. As a marketing director you should be more than aware that the concepts you are dealing with are highly dualistic, highly ambigeous and you can never trust on simple ideas.

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