Every area of mathematics has its own set of basic axioms. If you start with different axioms, you will get a different kind of mathematics, but the logical arguments will be the same. Mathematics is not about choosing the right set of axioms, but about developing a framework from these starting points. You also can’t have axioms contradicting each other. If there are too many axioms, you can prove almost anything, and mathematics would also not be interesting. If there are too few axioms, you can prove very little and mathematics would not be very interesting. For example, an axiom could be that a + b = b + a for any two numbers a and b.Īxioms are important to get right, because all of mathematics rests on them. However this is not as problematic as it may seem, because axioms are either definitions or clearly obvious, and there are only very few axioms. Mathematicians assume that axioms are true without being able to prove them. You need at least a few building blocks to start with, and these are called Axioms. How do you prove the first theorem, if you don’t know anything yet? Unfortunately you can’t prove something using nothing. One interesting question is where to start from. Raphael’s School of Athens: the ancient Greek mathematicians were the first to approach mathematics using a logical and axiomatic framework. Once we have proven a theorem, we can use it to prove other, more complicated results – thus building up a growing network of mathematical theorems. Once we have proven it, we call it a Theorem. Traditionally, the end of a proof is indicated using a ■ or □, or by writing QED or “quod erat demonstrandum”, which is Latin for “what had to be shown”.Ī result or observation that we think is true is called a Hypothesis or Conjecture. We could now try to prove it for every value of x using “induction”, a technique explained below. This equation works in all the cases above. Thinking carefully about the relationship between the number of intersections, lines and regions will eventually lead us to a different equation for the number of regions when there are x = V.Axi points on the circle: In the above example, we could count the number of intersections in the inside of the circle. It is not just a theory that fits our observations and may be replaced by a better theory in the future. Proofs are what make mathematics different from all other sciences, because once we have proven something we are absolutely certain that it is and will always be true. Instead you have to come up with a rigorous logical argument that leads from results you already know, to something new which you want to show to be true. This example illustrates why, in mathematics, you can’t just say that an observation is always true just because it works in a few cases you have tested. The sequence continues 99, 163, 256, …, very different from what we would get when doubling the previous number. Unfortunately something went wrong: 31 might look like a counting mistake, but 57 is much less than 64. Or we might decide that we should check a few more, just to be safe: The number of regions is always twice the previous one – after all this worked for the first five cases. We might decide that we are happy with this result. We can immediately see a pattern: the number of regions is always twice the previous one, so that we get the sequence 1, 2, 4, 8, 16, … This means that with 6 points on the circumference there would be 32 regions, and with 7 points there would be 64 regions. We have to make sure that only two lines meet at every intersection inside the circle, not three or more. The diagrams below show how many regions there are for several different numbers of points on the circumference. This divides the circle into many different regions, and we can count the number of regions in each case. Imagine that we place several points on the circumference of a circle and connect every point with each other.
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