dinsdag 27 januari 2015

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.



maandag 19 januari 2015

'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?




zaterdag 10 januari 2015

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.