Guess what this interesting structure is ...
Fri, 24 Feb 2012 23:16:45 GMT

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Zeitschrift "Das Parlament" zum Tierschutz
Fri, 24 Feb 2012 19:00:00 GMT

Auf der Seite des Bundestages gibt es Texte, die offenbar als Beilage der Zeitschrift "Das Parlament" gedruckt wurden, und sich mit dem Themenkomplex Tierschutz in unserer Gesellschaft befasst. Weder war mir diese Zeitschrift bisher bekannt, noch habe ich alle Texte ganz durchgelesen, es scheint aber generell interessant zu sein. Außerdem finde ich es gut, dass das Thema inzwischen in der Gesellschaft angekommen ist.

Inwieweit allerdings solche Diskussionen auf halbwegs hohem Niveau sinnvoll sind, ist natürlich fraglich, denn letztlich werden die moralischen Entscheidungen die der Gesellschaft am Ende per Gesetz aufgezwungen werden auf Stammtischen entschieden.

Ich las vor Allem einen eher Tierrechtskritischen Artikel, "Das Bein in meiner Küche". In ihm wird die Position vertreten, dass eine Gleichbehandlung von Menschen und Tieren menschenfeindlich sei, und dass der Zusammenhalt von Menschen als solcher nicht durch deren biologische Verwandtschaft, sondern durch deren gesellschaftliche Bindungen gerechtfertigt ist, weshalb auch Haustiere wie Hunde in dieser Gesellschaft eine gewisse Sonderstellung haben, trotz des Fehlens einer biologischen Verwandtschaft, weil sie am gesellschaftlichen Leben teilnehmen.

Ich teile beide Meinungen natürlich nicht, aber mir ist ungefähr klar, wie der Autor zu seiner Gesinnung kommt - daran sind auch inhaltliche Fehler und Denkfehler beteiligt. Der Autor kann sich zum Beispiel nicht vorstellen, dass man aus einem brennenden Haus zuerst ein Tier rettet, und dann erst einen Menschen - ich hingegen hoffe einfach, dass ich niemals in die Situation kommen werde, mich zwischen zwei Leben entscheiden zu müssen. Ich wüsste nicht, ob ich einen Serienmörder eher retten würde, als einen Minensuchhund.

Letztendlich ist eine Begründung der Bevorzugung des Menschen durch gesellschaftliche Bindungen auch wieder ein Problem, denn es gibt auch genügend Menschen, die nicht an der menschlichen Gesellschaft teilnehmen, vor Allem Schwerbehinderte, oder schlichtweg Außenseiter.

Wir haben hier die Schwierigkeit, dass es sich um menschliche Moralvorstellungen handelt, die durch die Umstände nicht unbedingt begünstigt werden. Wir müssen töten um zu leben, ob es nun Tiere oder Pflanzen sind. Die ganze belebte Natur ist eigentlich ein einziges fressen und gefressen werden. Wir haben das Bedürfnis, uns darüber hinwegzusetzen, doch die Natur macht es uns nicht leicht, dafür einen sinnvollen Rahmen zu finden. Man würde gerne irgendwo die Grenze ziehen, und hätte gerne eine intuitive Begründung dafür, das Finden einer solchen Begründung zu einer gezogegen Grenze hält aber selten einer Hinterfragung stand. Wir wollen nicht einfach "Mensch sein" als moralisch sinnvolles Maß akzeptieren, aber außer der für uns nicht befriedigenden Ähnlichkeit der Erbinformationen gibt es nicht viel was alle Menschen gemeinsam haben - nicht den Verstand, nicht das gesellschaftliche Leben, noch nicht einmal das anthropomorphe Aussehen. Es gibt nahezu beliebig starke Ausreißer (die man meistens im Spektrum schwerer Krankheiten findet), denen wir aber ebenfalls Menschenrechte zuweisen wollen. Wir möchten "Leidensfähigkeit" als Maß nehmen, aber was ist Leid?

Wir werden uns in Zukunft voraussichtlich noch größeren Problemen zu stellen haben. Noch vor Jahrzehnten war es unvorstellbar, dass man Erbinformationen zwischen Lebewesen übertragen kann, oder gar künstlich herstellen kann. Wer weiß ob man irgendwann das reine "Retortenbaby" zumindest theoretisch herstellen kann. Oder zumindest den nicht von einem Menschen zu unterscheidenden Androiden. Dann wird selbst unser Begriff von "Leben" relativiert, wie einst unser Begriff von Menschlichkeit.
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The Internet Based TOEFL
Sat, 18 Feb 2012 21:01:03 GMT

At school, I often wondered why my English is judged by tasks I would not even be able to solve in German. Like listening and repeating historic facts, or conversations I just heard, or expressing my opinion on a topic I do not care about. I hated that.

Well, today I had my "Internet Based TOEFL". Before it started, I had to sign a form, and copy (!!!) an agreement, explicitly not in printed form (I hope they can read Sütterlin). If I understood correctly, this agreement forbids to give information on the contents of that test, which is why I cannot share any examples. Because you know, if I did so, the prices for training materials could decrease - heaven forbid!

So there I was, finally, at a time as early as 9.30, after I had to take the train from Munich to Nuremberg, because there were no free places for a test in Munich at a sufficiently early date. Having payed 250 Euros to be allowed to talk to a computer for about 4 hours.

A few instructions on where to put my baggage, a quick check of my passport and a bad webcamshot of my face without glasses later, I was sitting in front of a screen on which my tired face smiled back at me - what an amazing technology! All that was missing now was the supervisor logging me (and the other students who have gathered there) into that system, and a quick calibration of the headset and the microphone.

The "reading section" began. And it was similar to the questions I saw in practice exams before - except that the text was shitty, and I could not have cared less about its topic. Same for the second text. The third one was better, though. Also, no luck with the topics in the "listening section". Some stupid questions about details that I did not consider meaningful while listening, but in general, I think it went good for me.

Then a break for 10 minutes - I actually do not see any reason why it is not possible to pause the test at another time. I mean, sometimes people need to go to the toilet or something. Whatever.

As all the people needed about the same time, the blabbering began - all students started with the "speaking section". I was a few seconds slower than my neighbour, which is why I could hear her start talking while I still had to prepare my answer, which was very distracting! You are asked about two "familiar topics" - which were both topics I have never thought about. So I came up with some shit I could make up during the twenty seconds of preparation time and tried to make the best out of it.

I mean, come on, what the fuck? I already gave talks about complicated scientific topics in English. You always find time to think about your answer for a minute. And this is an exam situation!

However, after the "writing section" was finished, I could finally leave. Some other students also left. All of us agreed on these tests being a rip-off. For 250 Euros, you could afford a real teacher testing you. At an appointment that fits you. Under fairer testing conditions.
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Randomly Found Software: cnibbles
Sun, 12 Feb 2012 18:29:29 GMT

Nibbles was a Snake-like game, which was part of MS-Dos, as an example file for QBasic. The name "Nibbles" (probably ?) came from the half-bytes - "nibbles" - since on the terminal, it uses characters that split every terminal character into two parts.

Well, there are a lot of newer Snake-like games out there. However, the special thing about cNibbles is that it is, as the original Nibbles, a pure terminal application.
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Steam on Coffee
Sat, 04 Feb 2012 20:41:10 GMT

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On Vegetarism (again)
Fri, 03 Feb 2012 18:29:23 GMT

I am a vegetarian. I blogged about that in former blog posts, especially on my old blog. In the past, I used to explain the people about my feelings and opinions about meat. Meanwhile I am more careful with whom I tell that, not because I respect the decision of a person to eat meat in any way (I would probably sign a law that forbids eating meat immediately), but because I am tired of discussing, and I did not hear a single point which I did not know before, for years. Also, I am not vegan (yet).

However, of course, when there is dinner somewhere, I sometimes have to ask whether something contains meat. And more than one time I had to endure stupid comments from carnivores around me. To me, some people seemed to feel attacked when they see a vegetarian. Some psychological study now claims to prove this (however, I do not have access to the actual study).

I can understand that somehow. Vegetarism is morally superior, and people do not feel good about that. I can remember times when I thought that veganism was exaggerated - or at least I thought that I thought this. I actually always knew that veganism is morally superior to vegetarism, and I tried to be vegan, but I could not achieve that goal so far. I also think that frutarianism is morally superior, but I think that the world is not ready for that radical form of nutrition yet - which does not lower the honor I assign to people doing this.
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On Infinity 5 - Digression: Orderings
Tue, 31 Jan 2012 15:05:00 GMT

My original intention was to write a post about ordinal numbers, but it turned out that it is a lot harder to give a clear and simple explanation than I expected. So I decided to do a little digression on ordering relations. It is not directly about infinity, but related and needed for the theory of ordinal numbers.

To understand this part 5 of my series, be sure to have read the previous parts 1, 2, 3 and 4. Especially, you should have an idea of what a bijection is, how infinity is defined, what a powerset is, and what countable and uncountable sets are.

And, to make that clear: The following part is hard. You have to say good bye to some of your "intuitive" thoughts on mathematical objects, and have to be open to new concepts that are abstract in a way that non-mathematicians usually decline. We are now entering a world, in which finally there is no place for "calculating" and visual conceptions. I will try to make it as simple as possible, omitting a lot of complicated parts, but still, the objects I will write about are not as intuitive as the objects discussed before were.

However, it should be understandable, but you might need to read a few parts of it at least twice.

So, this post is about the theory of ordering relations. Many parts of applied mathematics are, in the end, about orderings. Sorting algorithms, for example. Whenever you look for a name in an alphabetical list, you apply basic properties of ordering relations.

Every ordering relation has a domain D, which is the set of elements which it compares. Orderings are usually denoted by a relation symbol R in infix-notation, which means that you denote it by aRb for two elements a,b\in D. For example, the default ordering on \mathbb{N} is denoted by a\le b, for a,b\in\mathbb{N}. We denote this ordering as pair (R,D), for example, the default order on \mathbb{N} we denote by (\le,\mathbb{N}).

There are several axioms for several purposes which can be postulated for orderings, but the one thing that all orderings have in common is transitivity: If aRb and bRc, then aRc. This should be pretty clear when looking at the given example (\le,\mathbb{N})! And if you look at other ordering relations, like the alphabetical (lexicographical) ordering of names, you will notice that you implicitly assume this property: When searching in a phone book, you look up one item, then decide whether you have to browse further, and find a second item, and then decide whether the item you search for lies between those two items.

While transitivity is an important property of orderings, it is not sufficient. For example, the relation (=,\mathbb{N}) is also transitive, but it is not an ordering relation (it is a so-called equivalence relation, another class of relations we will not discuss here). We need to give account to the "direction" of the ordering somehow. This is done by the axiom of antisymmetry: If aRb and bRa, then a=b. Again, for (\le,\mathbb{N}), you should immediately accept that.

A third axiom, the axiom of reflexivity, is less intuitive: aRa for all a. It holds for (\le,\mathbb{N}). The problem here is that there are two major concepts of ordering relations, the strict ordering relations and the partial ordering relations. An example for a strict order would be (<,\mathbb{N}). In general, partial orders are orders of the type "less-than-or-equal", while strict orders are of the type "less-than". We will only deal with partial ordering relations, for which reflexivity holds.

So the axioms for (R,D) to an ordering relation are:
  • Reflexivity: For all a\in D we have aRa.
  • Antisymmetry: For all a,b\in D, from aRb and bRa follows a=b.
  • Transitivity: For all a,b,c\in D, from aRb and bRc follows aRc.
Make sure you fully understand that. Use the example (\le,\mathbb{N}) if necessary. We give some further examples.

Of course, there are the canonical examples (\le,\mathbb{Z}), (\le,\mathbb{Q}), (\le,\mathbb{R}).

Another common example is the divisibility relation between non-negative integers: For two integers a,b\in\mathbb{N}_0 we say a|b (a divides b), if there is an x\in\mathbb{N}_0 such that a\cdot x=b, or equivalently, if \frac{b}{a}\in\mathbb{N}_0 (where in this case we set \frac{0}{0}=1). For example, 1|2, 2|4, but 2\not |3. Clearly, a|0 for a\in\mathbb{N}_0, but 0\not |a for a\in\mathbb{N}_0\backslash\{0\}.
Trivially, a|a.
If a|b and b|a, then both \frac{a}{b}\in\mathbb{N}_0 and \frac{b}{a}\in\mathbb{N}_0, so clearly, a=b.
If a|b and b|c, then \frac{b}{a}\in\mathbb{N}_0 and \frac{c}{b}\in\mathbb{N}_0, so \frac{c}{a}=\frac{c}{b}\cdot\frac{b}{a}\in\mathbb{N}_0, therefore a|c.
Thus, (|,\mathbb{N}_0) is an ordering relation.

People familiar with arithmetics will know that we can extend the divisibility relation to negative integers, in the same way. However, this would not be an ordering in our sense anymore (exercise: why?).

Somewhat related to the divisibility relation is the subset relation \subseteq on powersets, which turns out to also be an ordering relation: Say you consider \mathfrak{P}(A), the powerset of A, then of course, by definition, B\subseteq B for B\in\mathfrak{P}(A). Let B\subseteq C, and C\subseteq B, then x\in B implies x\in C and vice versa, so B=C. Let B\subseteq C and C\subseteq D, then x\in B implies x\in C, and x\in C implies x\in D, so B\subseteq D. Therefore, for every set A, (\subseteq,\mathfrak{P}(A)) is an ordering relation.

Something you should basically know from phone books and dictionaries is the lexicographic order. It bases on an already given order (R,O), for example, the set of letters with alphabetical order, but every other ordered set is sufficient, too. You then look at the set O^{<\omega} of finite sequences, and define the ordering (R_{lex},O^{<\omega}) by

  • () R_{lex} a for all sequences a, that is, the sequence of length 0 is smaller than everything else (in dictionaries, however, this is usually not needed, as there is no such word)
  • (a_1,\ldots,a_n) R_{lex} (b_1,\ldots,b_m) if and only if either a_i=b_i for 1\le i \le n and m>n, or for j=\min\{i|a_i\neq b_i\} we have a_i R b_i - in words, either (b_1,\ldots,b_m) starts with (a_1,\ldots,a_n), or for the smallest j in which they differ, we have a_j R b_j
You should convince yourself that this is actually the intuitive lexicographical order.

As mathematicians have to deal with a lot of structures, they try to find similarities between them, and use techniques found for known structures on unknown ones. That is why they rather talk about a general "ordering relation" than about a concrete given order - there are many principles that simply hold for every such structure. From this desire comes the technique of finding morphisms between structures. Morphisms are (usually) structure-preserving functions, that is, functions that you can apply on elements without losing special relations between them. This is a highly abstract concept and we will not discuss it here in detail, but knowing this may be motivating. So, we are going to see certain kinds of morphisms between ordering relations. Keep in mind that this is abstract - if you do not understand its purpose, write it down with concrete given relations for a better chance of understanding.

Now, let two ordering relations (R,D) and (Q,C) be given. A function f:D\rightarrow C between the domains is called (order-)homomorphism, if aRb implies f(a) Q f(b). If additionally f is bijective, it is called (order-)isomorphism. If such an f exists, then the ordering relations are called homomorphic/isomorphic.

An intuitive understanding of this should be best given when thinking about homomorphisms and isomorphisms as some sort of "renamings" of the elements, which is permitted without losing relevant properties. That is, if you have some true proposition about (R,D), and rename every occurence of a\in D by f(a)\in C, and every occurence of R by Q, then you will (mostly) get a true proposition again.

Let us look at the above examples. Obviously, from (\le,\mathbb{N}), there is a homomorphism, namely f(x)=x, into (\le,\mathbb{R}) - as \mathbb{R} is just a superset of \mathbb{N}. But also, g(x)=2\cdot x+9 is a homomorphism, since a\le b is equivalent to 2\cdot a+9\le 2\cdot b+9. However, as there cannot be a bijection between \mathbb{N} and \mathbb{R}, both orders are clearly not isomorphic.

Of course, f and g are also homomorphisms between (\le,\mathbb{Z}) and (\le,\mathbb{Q}), and the question arises, whether these orderings are isomorphic. On the other hand, between two distinct rational numbers there are always infinitely many other rational numbers, and this is not the case for integers, so both orderings cannot be isomorphic.

If a|b, then a\le b for a,b\in\mathbb{N}_1. Thus, f is also a homomorphism from (|,\mathbb{N}_1) into (\le,\mathbb{N}_1) (note that the same does not hold for \mathbb{N}_0). To see that (|,\mathbb{N}_1) and (\le,\mathbb{N}_1) are not isomorphic, notice that there is exactly one element a=2 which has exactly two smaller elements (1 and 2), while there are infinitely many a with exactly two divisors, namely all the primes.

You may have noticed that we did not yet give an example for isomorphic orderings. That is due to the fact that it is rather hard to find such orderings which are mathematically interesting, not overkill, and not somehow "artificial" - mathematically, isomorphic structures need not be distinguished, and most theorems over structures are only "up to isomorphism". However, there are some examples, one is (\le,\mathbb{Q}) and (\le,\mathbb{Q}^{>0}). That is right, they are the same ordering, just that the one's domain is bounded. One isomorphism can be given by

h(x) =\left\{\begin{array}{cl}\frac{1}{1-x} & \mbox{for } x\le 0 \\ x+1 & \mbox{for } x>0 \end{array} \right.

We leave the proof as an exercise (it should be provable by school math).

It turns out that an even more special kind of ordering relations, the well-orderings, are useful to describe infinitary objects, and get a way of "enumerating" them. However, we will discuss this in the next part.


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Interestingly Structured Fat
Tue, 31 Jan 2012 15:00:00 GMT

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