August 29, 2009

The LaTex to WordPress converter is installed and tested. I seem to be able to handle it. Meanwhile my guest took up speed and produces material at an enormous rate. Things you can expect within the next few days in no particular order are:

  • He explains the nature of scientific laws. (This is cute, pretty standard and understandable. Finally something!)
  • He goes into details on defining and describing notions. Apparently there is a third way to introduce notions. He calls these notions ’emergent’. (I am not sure, whether I got that.)
  • He comments on an approach (by scientists) to estimate the probability of unconditional existence of things. (What? That is hopefully not publicly funded!)
  • He laments on our understanding of ‘truth’. (I guess he criticizes that we still use a notion of truth in mathematics which in essence is several thousands of years old. Modern truth theories provided by philosophers are just ignored. I fail to see his point. What else should we do? Maybe I point him towards Tarski.)
  • He presents his view on the ontological status of mathematical objects. (I guess I understand what he is saying here and this sends shivers down my spine.)

What is Your Dangerous Idea?

August 28, 2009

When I asked my guest about his opinion on this little booklet edited by John Brockman, he actually started to smile and answered something like: ‘These smart scientists have a lot of dangerous ideas and have you recognized, all of their ideas are dangerous for other people? Scientists surely must be very caring.’ By then I was already used to his interesting approach to the concept of humor and decided to ignore the last remark. Instead I wanted to know, whether he had his own dangerous idea, preferably an idea dangerous for science.

He told me, that the single most dangerous idea for science is:

There is no proof.

The danger of this idea, according to him, does not stem from the fact that it might be true. The problem is, that you, as a scientist, cannot argue scientifically against it.

I was not able to follow. The Pytagorean theorem is proved! There are dozens of proofs, some of them hundred and thousands of years old. I have checked a few myself and all experts agree on the truth of this theorem. After all, this is not the classification of all finite simple groups and even this is settled. At least I hope so. How can one seriously think that there is no proof?

In a deliberately patient sounding voice he explained again, that the possible truth of the idea is not the problem, but our wrong understanding of what science actually is. Even in this moment when I writing down this post, I have no idea of what he was talking about. My face must have expressed my ignorance and he began a monologue on proofs. In essence he claims, that to deserve its name

a proof has to prove that it is a proof.

Otherwise, it is obviously not a proof, but only some consensus among the participating players. You can call it peer review if you like, but don’t call it a proof.

I am completely lost. No mathematician has ever proved that his proof is indeed a proof. That makes no sense! Or, does it? And if yes, then it is surely impossible! What do you think?

Mathematics Vol. I

August 27, 2009

As I have told you earlier I have this strange guest. He simply appeared. After some small talk in my dining room he realized that I will not be of much help if he wanted to return home. To be honest, I do not even have the slightest idea where he actually comes from.

What other options does he have? For a correct assessment he needed an overview about our scientific potential, he told me. What better place could there be than a library? Of course he was interested in the very foundation of our science and since mathematics is a solid basis he wanted volume one of mathematics! The librarian, a student, offered Analysis I and Linear Algebra I, but no book even remotely resembling mathematics volume I. Understandably, my guest was more than disappointed.

There must be some book describing how mathematics bootstraps itself out of nowhere into existence. This science is more than 2.000 years old and has accomplished so many things, but has no first book. There is no chapter one. How shaky is that? We teach people to use mathematical rigor and our own origins are more than frayed.

I told him that my understanding of the situation is that mathematics is based on three concepts. A formal language, needed to express assertions, a logic to attach truth values to these assertions and a notion of set to have something noteworthy to be talked about. I lent him Introduction to automata theory, languages and computation by J.E. Hopcraft and J.D. Ullman which I consider an excellent introduction into formal language and complexity theory. Furthermore I gave him Theory of Sets by N. Bourbaki a book he has to read anyway and Sets for Mathematics by F.W. Lawvere and R. Rosebrugh just to impress him. The logic part was covered by the rather inaccessible Logic of Mathematics by Z. Adamowicz and P. Zbirski. I told him that these three columns all start with a small number of given facts and then develop more complicated facts inductively by given rules.

‘Where from do you get this small number of given facts and rules’, my guest asked suspiciously. I told him, that we simply assume them. That seemed to be too much for him. His jaw literally dropped.

I fail to see his problem. Maybe you can help me. In case you have, or in case you know somebody who has Mathematics Volume I, please send me a copy. In the unlikely case that there is no well-defined starting point, please give me some sort of explanation. Thanks in advance.

Hi world!

August 25, 2009

Yesterday I met a very strange person. There is no way for me to keep this private, let me just catch up on the technical details, since I might end up using a LaTex to WordPress converter.

We shall either start in the spirit of ‘edge’ with his idea dangerous for science or with his trip to the library to get Mathematics Volume I Part I.