# It Ain t What You Prove, It s the Way That You Prove It. a play by Chris Binge

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1 It Ain t What You Prove, It s the Way That You Prove It a play by Chris Binge (From Alchin, Nicholas. Theory of Knowledge. London: John Murray, Pp ) Teacher: Good afternoon class. For homework I asked you to investigate triangles and to try and find some of their properties. Can anyone tell me what they have discovered? Alpha: Yes. I have found that the angles of a triangle always add up to 180. Teacher: Perhaps you could explain how you came to this conclusion. Alpha: Well, I drew a great many triangles of varying shapes and sizes and found that in nearly every case the angle sum was 180. Beta: Just a moment, did I hear you say 'nearly' every case? Alpha: Yes - I admit there were a few that seemed to come to 181 or even 179. Beta: So your resuk should say that 'The angles of a triangle nearly always add up to 180 : Alpha: No, the evidence was so strong that I can explain the few that didn't by inaccuracies of measurement. Beta: What you are trying to say is that you cling to your hypothesis despite evidence to the contrary. These are clearly counter-examples to your theory and it is most unmathematical to dismiss them so quickly. Alpha: There is always experimental error when measurement is involved - it must be expected, not considered as a counter-example. Beta: Teacher I protest. Alpha is using language that is more at home in a science laboratory where vague concepts such as 'strength of evidence' and 'experimental error' may be good enough, but this is a maths class. We are concerned with exactness and absolute truth. Alpha: Even if I remeasured my triangles more accurately and got 180 every time, I expect you are such a sceptic that you would always say there may be a counter-example I haven't yet found. Beta: For once you are absolutely correct. No amount of so called 'evidence' will convince me that your hypothesis, however likely, must be true. You are using an inductive argument which I cannot accept. I will only believe that when I have a vigorous deduaive proof that it is the case. Teacher: I am sure we are all agreed that such a proof would be desirable. Can anybody provide one? Gamma: Yes. I have a proof that will satisfy Beta. May I demonstrate? Teacher: Please do.

4 It Ain t What You Prove, It s the Way That You Prove It Page 4 that it would. Phi: Mmmm... yes. And you know I'm not even sure that the two lines are parallel. Can you be certain that parallel lines can be drawn on a plane? I suggest that any two lines you draw will meet somewhere, if we have a long enough piece of paper. I challenge you to provide an infinitely long piece of paper to prove me wrong. Alpha: Any lines I draw will be subject to error in measurement and inaccuracy in construction. Beta: Oh don't start that again, we have had enough science for one day. There is a better way round the problem. Alpha: Which is? Beta: Which is to state clearly all assumptions that we are going to call on, and make our definitions subject to those assumptions. I shall call the assumptions 'axioms' and from then we can deduce 'theorems'. Phi: But what if your assumptions are false? Beta: Truth or falsehood doesn't enter into it. We assume our assumptions, obviously. That's why they are called assumptions. Therefore anything that follows from them is true in any world where they hold. If you can't find a world where they hold then it doesn't invalidate the theorems or the argument used to deduce them. Phi: let us hear your axioms. Beta: Certainly. There is one and only one straight line between two points. Any finite straight line can be pnoducedindefinitely. All right angles are equal. A circle can be drawn with any point as centre to pass through a given point. Through any point one and only one line can be drawn parallel to a given line. Teacher: (an aside to audience) The axioms were first suggested by thegreek mathematician Euclid over 2000 years ago. They were accepted as the basis for geometry until the nineteenth century when new systems of axioms were considered and new geometries were explored. including that of the sphere. Gamma: So if we consider these axioms as our starting point, they define what we might call two-dimensional Euclidian space and it is not necessary or meaningful to question their truth since they are the starting point.

6 It Ain t What You Prove, It s the Way That You Prove It Page 6 the real world, since the real world, or at least ~u our view of it, will change. Beta: I agree with Delta. I also noticed Alpha's attempt to slander axiomatic systems by calling them games. He is probably so upset at being called a scientist that he wanted to throw a few insults of his owo. However, he has failed miserably as I do not consider the word 'game' an insult at all. The game of chess is a very good analogy. In chess the pieces have names and their rules for movement are the axioms. A position is allowable only if it can be reached by using the rules. But the pieces are not defined in tenns of anything outside chess. We call a bishop a bishop and a knight a knight but their rules of movement bear no relation to any bishops or knights outside the game of chess (if they did then the phrase 'queen mates with bishop on back row' would have a completely different meaning). No attempt is made to use the game as a picture of reality. The pieces are purely man-made concepts and the game is a formal logical structure. Mathematics is a formal logical structure derived from rules in the same way - and the greatest game of ajl.

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