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• It is sometimes said that mathematical reasoning is a process of logical deduction. If this is true, and if the conclusion of a proof must always be implied by (contained in) its premises, how can there
ever be new mathematical knowledge?
• We can use mathematics successfully to model real-world processes. Is this because we create
mathematics to mirror the world or because the world is intrinsically mathematical?
• In the light of the questions above, is mathematics invented or discovered?
• Are all mathematical statements either true or false?
• It has been argued that we come to know the number 3 through examples such as three oranges or
three cups. Does this support the independent existence of the number 3 and, by extension, numbers in general? If so, what of numbers such as 0, -1, i (the square root of -1) and a trillion? If not, in what sense do numbers exist?
• Can mathematics be characterized as a universal language?