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).

Definition of Mathematics

• 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.'

Mathematics and Reality

• 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?

Mathematics and Knowledge Claims

• 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?

Mathematics and Values

• 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?