"Mathematics is concerned only with the
enumeration and comparison of relations."
 Carl
Friedrich Gauss

Sections
Introduction
1.
Doing
mathematics on the computer in Sage
About
ME/CFS
Introduction:
I keep having trouble with
my eyes  keratoconjunctivitis
sicca  and therefore this is
again only a brief Nederlog. And as you may have noticed, I kept
changing the subjects the last days, and I am doing so again, today.
This time my subject is mathematics, by
computer, and if you are not interested in mathematics or programming, this
is not for you, and there probably will be another subject tomorrow.
1.
Doing mathematics on the
computer in Sage
I like mathematics and I
can program and therefore  one might think  doing mathematics should
be a whole lot easier on computers. The answer is yes and no, and
mathematicians seem to be divided about it too. Here is an interesting
reference  if mathematics and programming interest you  namely
by Natalie Wolchover, who
writes
Anyone who relies on calculators or spreadsheets might be
surprised to learn that mathematicians have not universally embraced
computers.
Part of the problem is
mostly this, as she writes:
Computers are now used extensively to discover new
conjectures by finding patterns in data or equations, but they cannot
conceptualize them within a larger theory, the way humans do. Computers
also tend to bypass the theorybuilding process when proving theorems,
said Constantin Teleman, a professor at the University of California at
Berkeley who does not use computers in his work.
Others differ, and indeed
it is difficult to see why one cannot, in
principle, put a lot of theory in software. Then again, software may be
bugged in all manner of subtle ways, and the problem with results
generated by a computer that one cannot verify mentally or by hand,
with paper and pencil, is that one has to take it on faith  though
this might be overcome or lessened when different programs working on
the same probloem arrive at the same result.
In fact, there are some
theorems that depend on computers. The most
famous is the proof of the
that you find well
explained on Wikipedia by way of the last link. There are some others,
but it seems by and large computers are mostly used  so far  in pure
mathematics to generate data and find patterns.
It turns out that there is
interesting and useful set of free and open
source programs
collectively called
Apart from its being free
and open source, part of its strengths are
that it comprises a lot of other free and open source mathematical
software; that it is (mostly) based on Python,
which is a clear, elegant and easy to learn programming language (that
is not needed for using Sage, but helpful if one uses it); that it can
do a whole lot of mathematics and graphing  algebra, geometry,
differentiating, integrating, matrices, groups and more  and quite
elegantly too; and that one can use it in a web browser, preferably
Firefox (maybe others, but Firefox is my main browser anyway, so I did
not look further).
In case you are interested,
here are some helpful links  with one
proviso: I only know this from Linux.Ubuntu, and not from
Windows or Mac on which it also can run.
First, there is the the
Sage project's site
with a lot of
documentation, videos and also
There are various tutorials
and videos explaining it, but I found the
following one, by a user of it, quite helpful  and I provide a link to
the first of a series of 11 files, that explain what it is, how to get
it, and how to work with it:
This is all well done, as
is the second in the series, that explain how
to get it
I followed the Installation
instructions that the last link provides
for a version available in 2009. This works, but in 2012 there are a
few gotchas:
 the download of the
latest version is almost 500 MB
 one needs at least 3 GB
free to install it
 I downloaded from a
British site that added an lmza compression
 it was explained on the
site how to unpack this (on Linux) and indeed that worked, but
 the above Installation and Testing are formulated for the Linux terminal but on
Ubuntu one can use the file manager and the archive manager as well,
and faster, although
 I do start Sage from a
terminal (and the above link gives explanations how to make a startup
file for one's desktop)
It all worked as promised,
though one should not forget to enable
JavaScript for the Notebook version that runs in Firefox, which is
quite neat. I think this is the best way to run it, but it is pleasant
one can also run it in a terminal aka
virtual console  which thereby becomes an extremely powerful
calculator, at the very least.
I find it all quite
impressive and useful, and an excellent example of
what free
and open source can achieve  in which context I will end with
three remarks.
First, there are commercial
programs  not free and not open source 
such as Magma,
Maple
and Mathematica
that are  said to be  as powerful as Sage. The price is several thousands of dollars
for one such program, with the practical certainties that one cannot
see their source, and that they are developed by far fewer persons.
(Besides, this is not helpful for those who do not grow money on their
backs, such as students, whether in universities or high schools.)
Second, I have played a
little with Mathematica and Maple, long ago,
and could get less done with them then than I could with Sage now,
though that probably is due to the passage of time or to my personal
tastes and limitations.
Third, while I have used
spreadsheets some, I generally have done
mathematical things in some programming language I know, mostly in
Smalltalk, JavaScript or Python, because that tended to be easier and
more versatile for me, but my guess is that such mathematics as I do
will be easier in Sage, because it was built for it, and one can use
Python in it.
So... if you are interested
in mathematics, you should check out Sage. And in case you are learning mathematics: I
am sure I would have loved this as a teenager, and would have
been helped a lot by it, because it is fast, powerful and has excellent
graphics, and also it seems a lot smarter, kinder and more capable than
my teachers of mathematics. [1]

[1]
None of whom was a mathematician: They were former merchant marines,
who were allowed to teach mathematics, which was a profession that only
served their interests and not the interests of their pupils. (But in
the early sixties pupils were supposed to obey, also if, as in my case
with mathematics, the teaching was utterly incompetent and horrible.)