Dear Professor Baez,
My name is Paolo Bizzarri and I am following your invitation to
provide some feedback on the topic “Why is mathematics boring?”
Let me introduce myself: I am 37 years old, got a Degree in Computer
Science in 1994. In the last fourfive years I have started studying
mathematics for fun and passion.
As my job is different, I have a limited time to dedicate to my
passion. Even if I find mathematics extremely interesting, I have
often found studying maths boring, difficult and hard to pursue. So, I
have done some simple reflection on why I find difficult and boring
something for which I have anyway passion.
My conclusions are as follows:
 mathematics is taught in an unnatural way;

mathematics is a practical discipline, but it is taught as an abstract
one;

there is lots of implicit knowledge that is not made available through
books.
I will try to expand each of these sentences in the following.
Mathematics is taught in an unnatural way.
My point is here referred mainly to textbooks, and their typical
structure of definition/lemma/theorem/definition.
Where is the problem? The problem is that this approach is unnatural.
It is not in that way that mathematicians reason and produce their
work.
If you see a demonstration, it is as terse and essential as it has to
be. Each passage is perfectly connected with the previous passages.
Each hypothesis is done exactly when it is was needed, and it is
absolutely minimal.
The question (my question) was: yes, everything works perfectly, but
this is not science. This seems more an Hollywood film, where
everything happens for a precise reason.
But mathematicians do not work in that way (or, at least, this is my
understanding). The real problem in mathematics is often not to
demonstrate a theorem: it is to find a good object to study. The
definition comes AFTER a theorem has been demonstrated. The
demonstration itself is refined numerous times, in order to obtain the
“perfect”, textbook demonstration. Which is the only demonstration you
see, and I found them quite unnatural exactly because they were
perfect.
The real point should be that mathematics text book should provide the
context, the reason WHY they are studying something, and what they are
trying to study. Which bring us to my second point.
Mathematics is a practical discipline, but it is taught as an abstract one.
Again, this is based on my limited experience, and can be pretty
typically Italian. But, anyway, it is the only experience I can
provide.
One of the main points about mathematics is that it is “abstract”,
“pure”, not tied to any practical problem.
Which is false, from an hystorical point of view. But it is false also
on a more concrete, daybyday, practical matter.
Maths is boring for non fulltime mathematicians because they don’t
know the “tools of the trade”. They are not used to manipulate mental
objects like grups or vector spaces. When I have read for the first
time about groups, I have found difficult to understand lots of things
about them and their importance. When I have started to see them used,
they became much more clearer to me.
Perhaps mathematicians do not feel this problem strongly, because they
are used to work with abstract objects. It is the same problem that a
nonIT professional has when he has to use a computer program done for
an IT professional.
This separation is strong also because you are not supposed to use the
tools you learn by yourself: the demonstrations are already provided,
and you have not to improve them (you would not be able anyway…).
You have to use them in some cooked up situations, but again there is
little understanding that this is done for a specific reason. Which
bring us to the third, and possible final argument.
There is lots of implicit knowledge that is not made available through books.
It is one of the most striking things I have seen: there is lots of
thing about mathematics that are not explicitly expressed in
mathematics textbooks.
One is what all mathematics call “elegance”. It is a fuzzy concept,
but it is fundamental. I have not seen a single, explicit reference to
it (except, perhaps, in Topics in Algebra). But this is a fundamental
criterion to create and judge mathematical theories, and a strong guide
about which structure you expect.
The second is about the “styles” of demonstration you adopt. Given a
certain domain, it is quite common to see demonstrations that use a
limited set of methods to be carried out, but these methods are never
expressely nominated or indicated.
But, without a name, it is difficult to effectively teach these
things. We cannot even speak properly about them.
Conclusions.
Well. If you didn’t find boring these writings of mine, I owe you a
pizza in Pisa, if you ever come to my city.
Best regards.
Paolo Bizzarri
Re: Why Mathematics Is Boring
‘Modern mathematics’ is indeed boring and devoid of meaning in most papers. That’s why I stick to reading the works of the greats like von Neumann, Wiener, Einstein, and hosts of others. What made them great? They *explain* things, and then go on to carry out tremendeously complicated calculations. I think the advent of calculators and numerical methods has hurt the advance and understanding of math to a large degree. However, on the flip side, new symbolic methods allow computations which are a great help and lead to new relations not previously seen, so maybe there is hope yet.