Tok Mathematics
Theory of Knowledge
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
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