Two ways of "counting infinity"

Do these sets have the same size? :

{1, 2, 3} and {1, 4, 9}

Is your answer the same if we extend the sets indefinitely, like this? :

{1, 2, 3, 4, ...} or {1, 4, 9, 16, ...}

Probably you determined the size by counting the number of elements in answering the first question. The sets {1, 2, 3} and {1, 4, 9} have each 3 elements so the answer to the first question is: yes, the two sets have the same size.

How to count the two infinite sets of the second question? This is impossible using conventional 1, 2, 3, --- counting. It would not end. Mathematician Cantor therefore proposed a method to find out whether two infinite sets have the same size, a way of “counting infinity”:

If we can pair all the elements of two sets than they have the same size. 

Because we can pair like this: {1, 1}, {2, 4}, {3, 9}, {4, 16}, ... Cantor's answer to the second question is the same as to the first: yes, the two sets have the same size. His method also works for finite sets, by the way. As such it is a generalization of the finite case.

But wait... something strange is going on. The set {1, 4, 9, 16, ...} is a proper sub set, a part of {1, 2, 3, 4, …} (not the same, hence “proper”). Nevertheless they have the same size. How can that be?

A mathematician's response might be that we should not speak of the size of a set but of its cardinality instead. That's the word Cantor introduced. This makes clear that we are dealing with the mathematician's world which differs from the ordinary world. In the ordinary world proper sub sets of a set are always smaller than the set - and infinite sets cannot be counted in this world. In mathematics proper sub sets of an infinite set can have the same cardinality as that set. Not the same size, which is just an everyday, non mathematical word.

So in the mathematical world it seems that we gain something - cardinality as a measure for the size of sets - and we loose something: that proper sub sets of a set are always 'smaller' than that set.

This is a kind of friction I was uneasy with for decades. I realized myself all too well the difference between 'size' and 'cardinality' of above and especially the more or less arbitrary choice of Cantor's pairing principle. Would another, maybe better choice be possible?

It felt as an impasse to me but probably not to William Byers, the author of the book 'How mathematicians think - using ambiguity, contradiction, and paradox to create mathematics'. His main point is that ambiguity etc. - things that somehow do not feel right - stimulate the formation of new mathematical ideas. Difficulties advance mathematics.

This is not an idea pertaining exclusively to mathematics. For example, koans are the Zen equivalent given by zen masters to their pupils. These are paradoxical puzzles like: “"What is the sound of one hand clapping?" To solve these one has to widen one's view of the world. A poetic, musical analog is this line from Leonard Cohen's song "Anthem": “There is a crack in everything. That's how the light gets in.” Imperfections are necessary for advancement, seems to be the suggestion.

Back to the original problem regarding infinity. How was it solved? About 100 years after Cantor the theory of “numerosities” was proposed by Vieri Benci. Like cardinality the numerosity is a measure for the size of a set. An important difference is that proper sub sets of a set have always a smaller numerosity than that set (while the cardinality can be the same).

Some more information can be found here: http://www.newappsblog.com/2014/03/counting-infinities.html. From the discussion there it appears to me that numerosity is a more complicated concept than cardinality – but I can be wrong.

'Mind and cosmos' by Thomas Nagel

Many people - and especially scientists - think that our current knowledge is sufficient to explain current life. Doubting the theory of evolution is not done. Nevertheless, this is exactly what philosopher Thomas Nagel does in his small book 'Mind and Cosmos: Why the Materialist Neo-Darwinian Conception of Nature is Almost Certainly False' (http://opinionator.blogs.nytimes.com/2013/08/18/the-core-of-mind-and-cosmos/?_r=0).

Neo-Darwinism implies that the combination of mutation and natural selection (darwinism) drives evolution. It does not explain the existence of non material aspects of life like consciousness, cognition and value. This is no surprise, because science does not include these aspects.

Nagel pleas for looking for a naturalistic theory that describes the order that governs the natural world from the inside out. That is, a theory without supernatural elements. The order should lead without surprises and maybe inevitably to consciousness, cognition and value. Non material matters might also exist in some form in the beginning to evolve from there.

After investigating several solutions Nagel concludes with a teleological theory as feasible and the most probable. Teleology dates back to Plato and Aristotle. It states that development is aimed at a goal or purpose. This in contrast to ordinary causality, where effects follow causes. Teleology could in this case work by guiding mutations such that life as we know it arises in time. This guiding is necessary because Nagel beliefs that neodarwinism is 'too slow' for the development of life as we know it. The teleological guiding law would be just another natural law.

The book surprised many people because Thomas Nagel is a respected, atheistic philosopher. Teleological views do not fit into this picture. The book is rejected by many if not most philosophers.

The book is interesting to me because I belief already quite some time that neodarwinism is too slow and could be repaired by a teleological 'fix'. It is therefore a pity that the book is difficult to read and understand - at least for me. There are many long sentences and it lacks a decent summary of the main lines of reasoning. Taking more time for reading it would increase my understanding. However, I will not do this because I seriously doubt whether philosophy can contribute much conclusive to this subject. Maybe science should come to the rescue again?

IDEOLOGIES and ideologies

Last week about twenty people were killed in Paris due to a clash of two ideologies: one religious and one of free speech.

Friday morning I boarded a rush hour train to Leiden. I sat down opposite to a young woman with a black headscarf. She telephoned softly while the Metro (a free newspaper) was lying opened in front of her on the small window table. I was curious and turned my head slightly to see if she was reading about the Parisian affair. That was not the case. Then, while continuing calling, she rearranged her newspaper and with small shocks pushed the underlying Metro in my direction. 

Why you should not read this!

You should not read this because you are using digital media right now which causes symptoms of dementia, according to Manfred Spitzer in his book "Digitale dementie" (in Dutch; in English the title would probably be "Digital dementia"). He is very negative about the use of digital media:

"Digital media cause us to use our brain less so that the intellectual performance decreases over time. It even hampers brain development in young people; so right from the start the intellectual capacity stays under the attainable level. This does not only concern our thinking, but also our will, emotions and above all our social behaviour. These effects haven been already demonstrated often and proceed along different mechanisms clarified by research, especially brain research."

The book is a cascade of research results against digital media use and lacks a balanced view. He does not mention any advantages. In my view he misses at least three important general points.

The first is that the future is made right now. If people - and especially children - use digital media now they will do so in the near future.  The balance of human capacities change over time, which is nothing new. For example, people get currently worse at hand writing and better at typing. Typing skills are what the future needs, more than hand writing. How many people nowadays use a quill pen to write? You'd better go with the flow in order to fit into the future world.

The second point is that people try to push their abilities to the limits. So if one loses the ability of fine coordination as learned from hand writing, something will substitute for that. This does not hold for all people, of course. That is what the last point covers.

The last point is that what people do with a new thing or idea depends foremost on their traits instead of the novelty itself. For example, lazy people tend to do "lazy things" with a car or a computer, moderate people make moderate use of tablets and cell phones, etc. The virtues and vices of people (http://en.wikipedia.org/wiki/Outline_of_self)  become apparent in the use of things and ideas, quite independent of those things and ideas.

What the world is made of

Ever wondered what the world is made of at the lowest level? Are particles the ultimate building blocks? Or fields? A combination of particles and fields? Or... structures? There are people that are investigating the possibility that structure is all there is. No particles, no fields, structures only. Only relations between - yes, between what, exactly? It is easy to give a mathematical example of a structure. For example the structure consisting of this simple relation: being separated by 1. This generates the integers -3, -2, -1, 0, 1, 2, ... and many other examples. Note that only the relation of being separated by 1 is essential. The numbers -3, -2, ... are not. They can be seen as mere positions, place holders. Not as objects.

I tried to imagine physical reality to be based on structures only. No particles, no fields, nothing of those. This is much more difficult than the previous mathematical example of a structure. After all, we live in a world in which we experience (see, touch, measure, ...) objects like dogs, trees, etc. Objects that may stand in certain relations to each other, like "the dog is 2 meters to the left of the tree". So what we experience in our daily lives are objects that form a structure: objects + structure. Not structure only. To make it plausible that our physical world can be thought of as being made of structures only I will first discuss a mathematical example, the Cantor set. See also: http://en.wikipedia.org/wiki/Cantor_set.

To construct the Cantor set we start with a black line segment extending from 0 (left) to 1 (right). This is the top line segment of this figure:

The second step is to remove the middle third of this segment without the end points. This leaves the two line segments [0, 1/3] and [2/3, 1] of the second line from the top. The third step is to remove the middle thirds from [0, 1/3] and [2/3, 1] in a similar fashion. This gives the four line segments of the third line. Proceeding in this way gives an ever larger number of ever smaller segments. The Cantor set is what remains after an infinite number of iterations. It consists of a large number of points, not segments. For example the points 0, 1, 1/4 and 3/10 belong to the set. In fact the Cantor set contains an uncountable number of points: it has the same cardinality as the length 1 segment we started with. The cardinality is the mathematical measure for the  number of points of a set.

There is a significant difference between the results after a finite number of iterations and the Cantor set. After a finite number of iterations we got a structure and objects. The objects are the line segments of the picture which have a property: their length. However, the Cantor set consist of points only. Points have no properties, they only indicate a position. So to me the Cantor set seems to be a nice example of a structure without objects.

Now for the physical example. I see a parallel with the physical world. At the scale of everyday life we see for example a person to which we can attribute a multitude of properties: length, weight, hair colour, left eye colour, right eye colour, etc. Zooming in it seems that the number of properties decreases. For example, an electron is thought to be a point particle with only three properties: mass, charge and spin. Also at the atomic level we see mostly empty space with atoms and electrons scattered throughout. This suggests that at smaller scales we lose properties and gain empty space. Maybe if we zoom in deeper and deeper the limit is what the Cantor set is: a structure without objects.

We can also go the other way by zooming out. Imagine  we look at the Cantor set with a microscope with a certain spatial resolution. All the holes between points with are larger than the resolution will be filled up. This gives one of the iterates above, which consists of segments. So a structure becomes visible  as a structure with objects. Because we always probe nature at a finite resolution we always experience objects. This can be generalised from the Cantor set to a general set of points.

The above speculations (which are not entirely water tight, I admit) lead to an amusing thought: that if we zoom in on matter every time we see some object that we like "to get our hands on" it falls apart in separate smaller objects. And at the bottom level there is structure only. No objects. Only a mathematical description of a structure. People advocating that "everything is mathematics" will probably love this view.

Finally, a famous saying is "we are star dust" because a person's mass consists for approximately 93 % of elements produced in stars. Dust is often taken as a model for a set of points. "We are star dust" remains true from the above structural point of view albeit in a different sense.

Change your money for a better world

Ever went weak at the knees and felt the urge to make the world socially just and more sustainable? The book "Een @ander soort geld" ("A different kind of money"; http://www.strohalm.nl/eenandersoortgeld.html) by Helen Toxopeus promises a solution: an alternative to the current money system.

The underlying idea is to change the rules of the economic game, by using a different kind of money than we currently do. Think of changing the rules of Monopoly which will inevitable change the course of the game. It is done by using software (Cyclos: http://en.wikipedia.org/wiki/Cyclos) that controls the way virtual money ("credits") is used. Two important characteristics are:

1. It puts a penalty on (or makes it impossible to) spending credits too soon outside the local community. Hence it limits globalization. 
2. Tax has to be paid for credits that are not spent. This will prevent the rich from becoming too rich and at the same time stimulate the economy.

As a physicist I am able to compute the behaviour of simple dead systems like falling objects and oscillating pendulums. Therefore I shiver at the thought of modelling complex economical systems "running on" a large number of living people embedded in a dynamical environment while at the same time having the boundary conditions of being socially just and sustainable. Therefore I was glad to see that (variants of) the system were already tried out and continue to be tried out.

Toxopeus based her book on a series of interviews with Henk van Arkel, who pioneered alternative money systems and is CEO of STRO, "Social Trade Organization" (English: http://www.socialtrade.org/; Dutch: http://www.strohalm.nl/index.php). The way our current money system works is explained in some detail. Like the way it is created and what the consequences of interest are. Interest is seen as a big evil. Several alternative systems and experiences with them are described. This is mixed with a sketch of Toxopeus' own development from being a banker at ABN AMRO via gaining interest in  crowd funding to her current positions as a researcher in the field of innovation and sustainability in money- and exchange systems. This mix of the factual and the personal makes the book more accessible. The (former) banker questioning van Arkel is a good way to understand van Arkel's ideas for the average reader who grew up with the usual money system. The book actually has a second, more formal and detailed part by van Arkel himself. As a whole it makes for an interesting and entertaining read. It is currently under translation. In line with the content of the book one can get it for free. Payment is not needed if it is returned or passed on to another reader. Only if one keeps it payment is expected.

I think it is good that alternative money systems are being invented, implemented and tested. Maybe not to change the current money system entirely but as being complementary. There is a theory that a mixture of money systems stabilizes a society.

I have some criticism on this particular alternative money system:

1. In the system it is impossible to save credits for the time when you are not able to work any more. So at that age you see your (limited) savings steadily evaporate while not being supplemented.
2. Stimulating spending credits (I saw an example interest of 6 %/year on savings) stimulates working. Isn't this the same as having to work for a mortgage, putting an unwelcome pressure on people? 
3. Who is "turning the knobs" in Cyclos? For example, who defines interests?
4. It seems that some parameters like the interest are fixed. In the current system the interest is at least more free. Is this additional degree of freedom not necessary for a system to adjust to varying circumstances?
5. I came across three "van Arkels": the aforementioned Henk, his brother Jan van Arkel who is the publisher of the book and a certain Roder van Arkel, "project manager Cyclos". I do not know for sure whether Roder is family of the other two. Probably new ideas have a tendency to originate from a group of closely related people. However, if things grow bigger it is not advisable to keep such a group regulating the money system, the software and the press.
6. If I borrow my lawn mower to my neighbour. As a consequence I cannot use it for some time, it ages, and my neighbour has the advantage of its use and not having to have one. Isn't this enough reason to expect some kind of "interest" from my neighbour? Doesn't the same hold for money?

 

Only two races

A citation by the late neurologist and psychiatrist Victor Frankl to start with:

From all this we may learn that there are two races of men in this world, but only these two—the “race” of the decent man and the “race” of the indecent man. Both are found everywhere; they penetrate into all groups of society. No group consists entirely of decent or indecent people. In this sense, no group is of “pure race”—and therefore one occasionally found a decent fellow among the camp guards.

This is what Victor Frankl wrote in his book "The meaning of life", in which he described how he survived the Holocaust.

His dichotomy also holds for Muslims. The rise of IS emphasizes this. Every two months I talk in The Hague with some Muslims from the Lahore Ahmadiyya movement (http://en.wikipedia.org/wiki/Lahore_Ahmadiyya_Movement_for_the_Propagation_of_Islam). They interpret the word "jihad" as striving for something good or opposing against something bad. Different forms of jihad exist: suppression of bad desires to get nearer to God, persistence in believing, mission work and that of war. This last, armed  jihad is only allowed under special circumstances. For example for self defence, on behalf of a legal government, etc. According to the Lahori's the way the media use "jihad" and "jihadist" is often misleading.

There is nothing new under the sun. A religion or other ideology (think of communism) is based on a book, which is interpreted such by one group and so by another. Sometimes just in a way that suits one best, leading to misunderstanding and conflicts. Already a few thousands year ago Seneca the Younger noted that religion - and feel free to generalize "religion" to "belief" or "ideology" - is highly ambiguous: 

Religion is regarded by the common people as true, by the wise as false, and by the rulers as useful.

--

Since I wrote this the Ahmadiyya Lahore Movement organised a meeting to explain things. See (in Dutch):

http://www.omroepwest.nl/nieuws/25-09-2014/wat-de-ware-jihad-haagse-moskee-geeft-uitleg

and

http://www.omroepwest.nl/nieuws/28-09-2014/tientallen-mensen-ge%C3%AFnteresseerd-jihad-uitleg-den-haag