Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
From: Ben Bacarisse <ben@bsb.me.uk>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Fri, 30 May 2025 01:05:28 +0100
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WM <wolfgang.mueckenheim@tha.de> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte
des Unendlichen" and "Kleine Geschichte der Mathematik" at Technische
Hochschule Augsburg.)

> On 29.05.2025 02:46, Ben Bacarisse wrote:
>> WM <wolfgang.mueckenheim@tha.de> writes:
>> 
>>> On 28.05.2025 01:06, Ben Bacarisse wrote:
>>>> Mikko <mikko.levanto@iki.fi> writes:
>>>
>>>>> Maybe the idea is that an exposure to nonsense helps students to learn
>>>>> to identify nonsense when they see it.
>>>> They have to regurgitate the nonsense to get the marks.  I once asked WM
>>>> what would happen if a student presented real mathematics in the exam
>>>> and he said they would not get the marks.
>>>
>>> No, you are lying.
>> That's harsh.
>
> Your statement.
>
>>  I may simply have misremembered what you said about this
>> before.  If so I apologise.  But I see you /don't/ in fact say they that
>> would get the marks.  You only say that they would need to be convinced
>> they were wrong.  What if they were not convinced and stuck by the
>> answer they had written in the exam?
>
> That depends. What answer do you have in mind?

See below.

>>> I would have asked him to explain his position
>> In the UK (at least at the universities I am familiar with), exam papers
>> must be marked according to a pre-written mark scheme.  There is no
>> option to interview the student after they submit their paper.
>
> Above I assumed a personal discussion.

>> Does
>> this really happen in Germany?  And if so, does the interview have only
>> one outcome -- agree or else?  Do you not have to write marking schemes
>> for your exams?  And if in fact you do, what do yours say about

Ah, so you can't do what you suggested and talk to the student, after
the exam, to try to persuade them that you are right!  That did sound a
bit crazy.

> In the exam there are questions like these:
>
> - Beschreiben Sie, was man unter der Abzählbarkeit aller positiven Brüche
>  versteht und erörtern Sie ein Gegenargument.
> - Beschreiben Sie, was man unter der Überabzählbarkeit der reellen Zahlen
>  versteht, und erörtern Sie ein Gegenargument.
> - Beschreiben Sie das Spiel "Wir erobern den Binären Baum" und die damit
>   verknüpfte Aussage.
> - Sehen Sie eine Parallele zwischen MacDuck und der Nummerierung aller
>   Brüche? Wenn ja, welche?
> - Was halten Sie vom wissenschaftlichen Wert der Mengenlehre unter
>   Berücksichtigung von Banach-Tarski-Paradoxon und Verteilung der Brüche in
>  (0, 1) und (1, oo).

From this, it is not obvious that you want students to say anything I'd
consider to be wrong in an exam.  So maybe someone could indeed get full
marks without having to deny mathematics.  Are there any claims in your
lectures that someone at the university down the road would object to?

> Try to answer. Then I will give you marks.

I'll try a couple of questions...

> - Beschreiben Sie, was man unter der Abzählbarkeit aller positiven Brüche
>  versteht und erörtern Sie ein Gegenargument.

The positive fractions are said to be countable because the function

  b(0) = 1
  b(n+1) = s(b(n))
  where s(q) = 1 / (2*floor(q) - q + 1)

is a bijection between the natural numbers and the positive fractions
according to the definition in Prof. Mückenheim's textbook.

I am not aware of a valid counter argument since this is simply a
definition of what the term "countable" means.  There may be weaker
systems in which b can not be proved to be bijective but the course did
not include any such formal system.

> - Beschreiben Sie, was man unter der Überabzählbarkeit der reellen Zahlen
>  versteht, und erörtern Sie ein Gegenargument.

The real numbers are said to be uncountable because no bijective
function exists between N and R.

I am not aware of any valid counterargument because this theorem is
well-established.  Of course, some alternative axiomatisation might
render this theorem unprovable, but no such set of axioms has been
presented to me.

The way you word the questions does seem to allow for correct answers.
What does your mark scheme say for these questions?  Would you accept
any answers that I would consider to be wrong?

>> Do you not have to write marking schemes for your exams?  And if in
>> fact you do, what do yours say about alternative answers?

Anything to say about this?

-- 
Ben.