So now I like the theorem, and ten minutes ago, I didn't care. I still have no use for it. Edit: The proof, by the way, is to focus on the circle rather than the triangle. Therefore it bisects their angle, and that means that the center lies on the angle bisector between any two sides of a triangle to which it is tangent.
If that's not Euclid's proof, it certainly could be. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Create a free Team What is Teams? Learn more. How to explain that proof is important Ask Question. Asked 7 years, 11 months ago. Active 7 years, 11 months ago. Viewed 4k times. Bach Bach 3 3 silver badges 16 16 bronze badges. Proofs can present widely-applicable techniques and sharpens your thinking skills, while theorems themselves also provide tools for tackling problems and serve as building blocks for all theory.
That being said, it is nice that you are trying to get your brother to stop taking things on faith! Knowing why the statements are what they are and how they can be proved is also equally important is what I feel. You may have to calculate probabilities, but you'd better bet that it isn't as simple as looking up Permutation on wikipedia and punching the numbers in to the formula.
Understanding why things work is far more valuable than what the simplest way to calculate a textbook problem. At least in my experience. Add a comment. Active Oldest Votes. This happens all the time in higher mathematics; heaps of published proofs are wrong, and every so often a false theorem ends up being believed for many years.
However, the question was more about theorems that have been independently proved by many people, which are therefore almost certainly correct. Still, I believe the principle applies.
Developing that questioning and skeptical attitude for later, when the theorems are not so clear cut. This talk will explain why, and Mark Ronan will present a fascinating array of mathematical examples. The moral of the lecture is two-fold. We need to question our proofs, however good they look, and we must question our assumptions. If we don't, we may miss something vitally important, and non-Euclidean geometry is a glorious example.
It has become extremely useful to modern mathematics, in areas surprisingly far removed from geometry. His conquests brought the Greek language and Greek ideas to large parts of the Near East, and in Egypt he founded a new city, Alexandria that later became a great focal point for trade and new ideas. Ptolemy founded the great library of Alexandria, which became the foremost centre for scholarship in the world.
There around BC, Euclid worked. Euclid wrote The Elements, the greatest textbook of all time. Starting with 23 definitions and 5 postulates for plane geometry, he proceeded by means of lemmas, theorems and proofs. His five postulates are roughly paraphrased as:. The Elements were passed on to the Roman world, and in about translated into Arabic for the Islamic world.
In the attempt to prove the degree result mathematicians including Bolyai stumbled across another very strange surface, called the hyperbolic plane , on which the angles in a triangle add up to less than degrees. The hyperbolic plane is hard to visualise, but it is similar to a kale leaf that gets more and more crinkly as you move towards the edge see here to find out more.
Although we don't come across that strange surface in everyday life, it is very important. Einstein's special theory of relativity is formulated using hyperbolic geometry. Out of special relativity grew the general theory of relativity, without which modern satnav devices and GPS enabled phones wouldn't work. Mathematicians often pride themselves on the fact that all they need to do their work is their brain and a pencil and paper.
But over the recent decades this has begun to change: computers have entered mathematics and sparked a lot of controversy. The controversy doesn't concern making the odd calculation using a calculator or computer. Mathematicians use those devices to make their life easier, just like everyone else. It concerns whole proofs that rely on computers.
There are two ways in which this can happen. In computer assisted proofs a computer is used to perform a large number of steps that a single human couldn't possibly manage in any reasonable amount of time. See here for more on computer assisted proofs. Over recent years computer scientists have also developed automated theorem provers ATPs — computer programs that can derive a result from some basic premises using the rules of logic and thereby prove it.
So far ATPs still need a lot of human input to work properly, but it's conceivable that in the future they will become far more potent. Whether or not they will ever be able to replace humans remains to be seen, and it's a topic that's hotly debated. See The future of proof for more information. The limits of maths Does mathematics really live up to the noble claim that every statement it makes can be proven true or false beyond any doubt?
Unfortunately not entirely. At the beginning of the twentieth century people worked had to put all of mathematics, rather than just sub-areas like plane geometry, on a rigorous footing, making sure that every true statement can be derived from a few basic axioms. It wasn't an easy task. Their system also contained a flaw. They couldn't show that it didn't contain any contradictions.
Suppose you have chosen a set of axioms you think should underlie all of maths. That set of axioms would be no good if it didn't allow you to define and draw conclusions about the natural numbers and their arithmetic, so let's also assume that your set of axioms is strong enough to do that.
Let's also assume that as you build up all of mathematics from your axioms, proving one statement after the other, you don't encounter any contradictions: the system you can build with your axioms is contradiction-free. This is quite a shocking result: it means that no matter what set of axioms you choose, the mathematics you can build up from it will always be incomplete.
Mathematicians have concrete examples of statements that cannot be proved using the accepted axioms of mathematics. When you come across such an undecidable statement, you essentially have to make up your own mind as to whether you believe it's true or not. You must have engaged in some reasoning to convince yourself that it is correct. Although that reasoning is not a proof, it is a precursor to a proof.
When you write software, you engage in proof-like thinking, demonstrating that a particular method or function definition does what you think it does. You will not find just what you want proved in any mathematics book, because you have a custom claim that is only of interest to you.
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