Mathematics |

There is little doubt that
mathematics has influenced our understanding of the natural world. It has been
said that the world is mathematical at its deepest level. Whether mathematical
knowledge comes to us as a result of some connection to natural phenomena is
another matter. Many have likened mathematics to a logical game invented by
humans. Others consider mathematics to be a unique aesthetic experience, while
still others consider it a 'special language tool'. The following questions
offer an opportunity to reflect on the nature of mathematical knowledge, which
Diploma programme students encounter in their Group 5 subject(s).

- What does calling
mathematics a 'language' mean? Does mathematics function in the same way as
our daily written and spoken language?
- Do mathematical symbols
have meaning, in the same sense as words have meaning?
- Why is it that some
claim that mathematics is no more than a 'logical game', such as chess, for
example, devoid of particular meaning? If this were the case, how do we
account for the fact that it seems to apply so well to the world around us?
- What could Carl Sandburg
have meant by the following?

'Arithmetic
is where the answer is right and everything is nice and you can look out of the
window and see the blue sky – or the answer is wrong and you have to start all
over and try again and see how it comes out this time.'

- Is it reasonable to
claim that mathematics is effective in accounting for the workings of the
physical world?
- Could it be argued that
mathematics is simply the application of logic to questions of quantity and
space?
- What did Einstein mean
by asking: 'How can it be that mathematics, being after all a product of
human thought which is independent of experience, is so admirably
appropriate to the objects of reality?'
- What are the differences
between the
**formal**school of thought which regards mathematics as similar to an activity governed by rules, limited only by the rules of logic and the creativity of the mathematician, and the**realist**school of thought which regards mathematics as referring to the way the world actually works? - What is the foundation
on which mathematical knowledge rests? Is it discovered or invented? What is
meant by this distinction? Can it be applied usefully in other areas?
- What is the origin of
the axioms of mathematics? Are axioms necessarily self-evident to all
people? How is an axiomatic system of knowledge different from, or similar
to, other systems of knowledge?
- Do different geometries
(Euclidean and non-Euclidean) refer to or describe different worlds?

- What is the significance
of proof in mathematical thought? Is a mathematical statement true only if
it has been proved? Is the meaning of a mathematical statement dependent on
its proof? Are there such things as true but unprovable statements in
mathematics?
- Mathematics has been
described as a form of knowledge which requires internal validity or
coherence. Does this make it self-correcting? What would this mean?
- How is a mathematical
proof or demonstration different from, or similar to, justifications
accepted in other Areas of Knowledge?
- Is mathematical
knowledge certain knowledge? Can we claim that '1 + 1 = 2' is true in
mathematics? Does '1 + 1 = 2' hold true in the natural world?
- Does truth exist in
mathematical knowledge? Could one argue that mathematical truth corresponds
to phenomena that we perceive in nature or that it coheres, that is,
logically connects, to a designed structure of definitions and axioms?
- Fermat's 'Last Theorem'
remained unproved for 358 years, until 1995. Is mathematical knowledge
progressive? Has mathematical knowledge always grown? In this respect, how
does mathematics compare with other Areas of Knowledge (for example,
history, the natural sciences, ethics and the arts)? Could there ever be an
'end' to mathematics? In other words, could we reach a point where
everything important in a mathematical sense is known? If so, what might be
the consequences of this?
- Has technology, for
example, powerful computers and electronic calculators, influenced the
knowledge claims made in mathematics? Is any technological influence simply
a matter of speed and the quantity of data which can be processed?
- What impact have major
mathematical discoveries and inventions had on conceptions of the world?

- Why do many
mathematicians consider their work to be an art form? Does mathematics
exhibit an aesthetic quality?
- What could be meant by G
H Hardy's claim that: 'The mathematician's patterns, like the painter's or
poet's, must be beautiful; the ideas, like the colours or the words, must
fit together in a harmonious way. Beauty is the first test. There is no
permanent place in the world for ugly mathematics'?
- What relationships, if
any, exist between mathematics and various types of art (for example, music,
painting, and dance)? How can concepts such as proportion, pattern,
iteration, rhythm, harmony and coherence apply both in the arts and in
mathematics?
- Is the formation of
mathematical knowledge independent of cultural influence? Is it independent
of the influence of politics, religion or gender?
- What is meant by S
Ramanujan's comment that 'Every time you write your student number you are
writing Arabic'?
- If mathematics did not
exist, what difference would it make?