Within mathematics the concepts of true and certain are no longer synonyms, well carried out Maths without errors following all the rules or axioms that we have learnt and implemented would be 100% certain but does that make it true? You can be certain of your answer and can trace back the steps you followed to arrive at it and be absolutely certain that your answer is well certain but your answer being true is a completely separate concept.
All of Maths stems from axioms which are the ‘rules’ or ‘laws’ of Maths and all of our axioms make sense in the real world or at least on the plane they are supposed to serve their purpose; we can all go and test and see for ourselves that the rule ab=c means c/b=a we can be certain that this rule is accurate but does that mean it is true? Another concept we are all familiar with is the triangle; 3 straight sides, meeting at 3 vertices with all its interior angles adding up to 180 degrees, we now know that is wrong or at least wrong when you draw the triangle onto a sphere then the angles can add up to 270 degrees and other numbers that are a far cry from the 180 we grew up hearing. Yet the idea of 180 degrees is one we still use and apply and it has still brought us to many new formulae and concepts that we can prove our accurate and certain, which leaves the question how can something false yield provable accurate formulas for our reality, and if the base axiom is false isn’t everything derived from it also false?
Well no, for all our purposes as long as the triangle remains on a 2D plane it’s angles will add up to approximately 180 degrees and we can use other axioms to explain why for example the parallel lines which by following the axioms of corresponding angles and how angles in a straight line add up to 180 degrees we can easily prove this and it serves our purposes. This relates back to the concepts of truth and certainty within Maths we can be certain of our answers but whether they are true or not rests on the truth of the axioms we used to derive the answer. This sounds simple enough as long as we make sure our axioms are true then all our Maths will be true, but proving the truth of an axiom is incredibly difficult. For years and years, the belief of angles in a triangle adding up to 180 was absolute until Euclides (?) thought to put them on a sphere thus disproving the theory. Perhaps all of our axioms can be proven false in ways we simply haven’t devised yet but for now, they serve our purposes and have given us certain answers and concepts.
Thanks Saniya. You are correct that truth and certainty are different things in maths. However your claim that it all depends on the truth of your axioms is not quite right – the point is that in maths any particular set of axioms can be accepted – the *truth* is not really the issue. So think of the axioms as the ‘laws of the game’; you can change the rules if you like, but then you are playing a different game. The rules are not really true or false in themselves.
So when you say “For years and years, the belief of angles in a triangle adding up to 180 was absolute until Euclides (?) thought to put them on a sphere thus disproving the theory. Perhaps all of our axioms can be proven false in ways we simply haven’t devised yet but for now, they serve our purposes and have given us certain answers and concepts.” it’s not quite right – what you mean is that alternative axioms might lead to alternatives to 180 degrees in a triangle. It’s not that anything was disproved as such – on a flat plane the angles there DO add p to 180. But he shows that there are other possibilities, like a sphere. (it was not Euclid by the way – he was the guy about the planes, not teh spheres)
makes sense?