Éanna OBoyle

ToK Mathematics

“... what the ordinary person in the street regards as mathematics is usually nothing more than the operations of counting with perhaps a little geometry thrown in for good measure. This is why banking or accountancy or architecture is regarded as a suitable profession for someone who is ‘good at figures’. Indeed, this popular view of what mathematics is, and what is required to be good at it, is extremely prevalent; yet it would be laughed at by most professional mathematicians, some of whom rather like to boast of their ineptitude when it comes to totalling a column of numbers....Yet ... it is not the mathematics of the accountant that is of most interest. Rather, it is ... abstract structures and everyday intuition and experience” (p.173, Barrow). 2.1 Mathematical Propositions

2.1.1 Mathematics consist of A Priori Propositions (theorems) We know mathematical propositions (or theorems) to be true independently of any particular experiences. No one ever checks empirically that, for example, 364.112 + 112.364 = 476.476 by counting objects of those numbers separately, adding them together, and then counting the result. The techical term to describe this independence of experiences is to say that the propositions are a priori. Therefore we say that mathematical propositions are a priori propositions. 2.1.2 Universality

When mathematical propositions are made, they are assumed to be true for ever. It is assumed that a constant (we call it !) = Circumference / Diameter for a circle, and that it will be true forever and true everywhere in the universe. 2.1.3 The contradictory of a mathematical statement is necessarily false We can say that “2 + 2 = 3” is not only false, it is necessarily false because “2 + 2 = 4” . Of course, if “3” is used to denote the number “4”, then we have in essence 2 + 2 = 4. 2.2 Mathematical Systems

2.2.1 Mathematical reasoning “seems’’ uniquely strong

A system describes how new propositions (theorems) are deduced from axioms by a defined set of rules of deduction. The most striking feature of mathematics is the method of reasoning it employs. Mathematical proofs must consists of inferences from propositions assumed to be true without themselves being proved. Such original “foundation” propositions are called axioms, which are basically unproven assumptions or as the Greeks put it, ‘truths so evident that no one could doubt them’. Just as many of the concepts with which mathematics deals with are invented by human minds, so the axioms about these concepts are invented to suit what the concepts are intended to reveal about reality. School mathematics is based on Euclidian geometry. One of Euclid’s assumptions (the famous ‘fifth postulate’) is that given a line L and a point x not on L, there is only one line M in the plane of x and L that passes through x and does not meet L. Another of Euclid’s axioms is that ‘things that are equal to the same thing are equal to each other’. The method of proof frequently used in mathematics is deductive reasoning in which axioms or mathematical propositions which have previously been proved are used to make new propositions on the basis of logic. Ultimately mathematical proofs must consist of inferences from axioms. The certainty apparantly found in mathematics made it, for many early philosophers, an ideal paradigm (way of thinking about something) for obtaining knowledge. There is a story that over the entrance to his academy, Plato placed the words “Let none that is ignorant of geometry enter here”. After Euclidian geometry came Aristotelian Logic (from Aristotle!) which clearly demonstrated how new theorems or statements can be proved from previous ones. For example, Nicky is a husky.

All huskies like to roll in snow. Therefore: Nikki likes to roll in snow. Here, as in any deductively valid argument, it is impossible that the premises all be true and the conclusion be false. The conclusion follows from or is...