Who’s afraid of Kurt Gödel?

When I was a schoolboy I was very interested in philosophy and, in particular, metaphysics. This interest continued to some extent when I went to university. I grew up on a farm where I had to do a lot of boring repetitive work to help my parents. For example there was planting potatoes. Behind the tractor was a machine whose central element, at least quantitatively, was a metal box full of potatoes. On each side sat a person facing the box. These two people were typically my mother and me. When you sat on one of the seats the person on the other side was not visible. A turning wheel caused a bell to ring at regular intervals and each time the bell rung you dropped a potato into a hole. Via a tube this deposited the potato at just the right place in a furrow made by the machine. The machine then immediately covered it up with earth. To get back to the main subject, while doing these and other similarly inspiring tasks, which I hated, I often thought about metaphysics. These thoughts were connected with my interest in fundamental physics which led me to devour all articles on particle physics in current and past issues of New Scientist and Scientific American I found in the school library. I can remember that at one time I was afraid that physics would be finished before I had a chance to make a contribution. At school we had to do a certain number of courses on religion. One teacher made the unfortunate mistake (unfortunate for her) of offering a course with the title ‘philosophy of religion’ and what was worse concentrating the subject of proofs of the existence of God. For me as an atheist with the enthusiasm of the fresh convert this was great. I thoroughly enjoyed knocking down one proof a week.

At university the attraction which philosophy exerted on me declined fairly rapidly. I came to the conclusion that a lot of what usually goes under that name is more about playing with words than making a connection with reality. I remember one incident which contributed to this, although I am sure there were many others. I once went to a talk given by a ‘philosopher’ in the university. I do not know what kind of a person he was, probably a philosophy professor who was visiting. There was an opportunity for discussion after the talk. I was convinced I had found a mistake in one of the arguments he had presented. He replied, ‘You are trying to catch me in a reductio ad absurdum’. In other words, instead of addressing the issue, and possibly admitting he was wrong, he used a formal device to divert the attention of the participants. By doing this he left me with a bad impression of himself and also of the field of philosophy.

What does all this have to do with Gödel? For my birthday I got a biography of Gödel by Rebecca Goldstein. I have not yet read enough of it to be able to judge whether I like it but it has had the effect of bringing back memories of the days when I was keen on philosophy. It has also made me come back to think about certain things and to test whether my philosophical convictions are really as solid as I imagine. When I was a student one of my mathematics lecturers told me about the book ‘Gödel, Escher, Bach’ by Douglas Hofstadter and that was exactly the right kind of thing to ignite my enthusiasm at that time. In particular, I learned about Gödel’s theorems. I rarely thought about the subject since. Now my general impression is that I am sad that these things are true. But should I really be afraid of them? I interpret the significance of the first theorem as being that some day I might come across a mathematical statement I am interested in which can be neither proved nor disproved. This sounds ugly, but I judge the probablity of it happening to be low, just based on the experience of countless mathematicians. The second theorem sounds worse since it says that someone might come across an inconsistency in mathematics which would bring down the whole house of cards. That sounds really serious but in the end I do not really believe it is going to happen. In a way it is comparable with the business about God. I am convinced that the existence of God can be neither proved nor disproved. Thus while not believing in the existence of God I have no foolproof argument which shows that he will not turn up on my doormat one day. I have lived quite happily with this danger now for the many years since I first thought about this kind of thing and the situation with Gödel and the threat of the internal inconsistency of mathematics is not much different. I also believe that the Sun will rise tomorrow.

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4 Responses to “Who’s afraid of Kurt Gödel?”

  1. anon Says:

    Thanks for your blog. I am actually interested in learning more about dynamical systems, PDE’s and their application to biology. Do keep the posts coming.

    Regards,

    anon

  2. Juliette Hell Says:

    Hi Alan!
    Do you know the graphic novel “Logicomix”? It is a biography of Bertrand Russell, which is quite good. If you want, you can borrow it from me.
    The scenarist Apostolos Doxiadis comes to the Free University next monday, the 13th. On 17h in the Amphitheater of the informatic building.
    Here the webpage of logicomix

    http://www.logicomix.com/en/

    Unfortunately, I cannot attend his presentation, because I have to go to the driving school…my very first theory lesson, hihi! And the driving school is called Dynamic, so that I will feel just like at home.
    See you soon!

    • hydrobates Says:

      Hi Juliette,

      I do not know the novel – it sounds interesting. Thanks also for the information about the event at the FU. I wish you luck with your driving sessions and no unexpected bifurcations.

      Regards,
      Alan

  3. Uwe Brauer Says:

    Hello

    When I started to study at the University, I was first very impressed by Gödels results (and was tempted to study Logic but I soon gave it up). Later I was sort of disappointed. There are very few examples known of such statements which can neither be proved or disproved and they seems quite artificial.

    Two however come into my mind which are important, one is of course Euclid’s 5th axioms (or called parallel postulate). That is was independent of the other axioms was proven as far as I know by Baltrami. In any case the discussion of its nature gave birth to non Euclidean geometry.

    On the other hand Cantors continuum hypothesis was a riddle for most of the 20th century till its independence could be shown by Cohen and others. However as far as I know this insight did not lead to anything new comparable to non Euclidean geometry.

    regards

    Uwe Brauer

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