Tell me something that is true, for sure, always. Tell me something that I can rely on.
New terms in this lesson:
Letís evaluate the statements below. Remember that evaluate means to examine something and decide its value.
Coming up with something that is true, for sure, always is not always easy. If you start analyzing the truth of things, youíll find that you can imagine a scenario for almost anything where it might change. If you do find something that is true it can be very comforting because you can really rely on it. Letís look at some examples.
Given the statement A:B is true, write something about B that is true, for sure. What can you say about B that is true? We know the relationship between A and B, but we donít know whether A is true or false. We need to make an assumption to get us started.
There are two possibilities: If we assume A is true, then B is true, or we can assume A is false, then B will be false. B is either true or false.
Remember that the given information is very important. We have to assume that the given statement ďA� is true without proof. Otherwise we cannot draw any conclusions other than ďB is either true or false,� which isnít usually a very helpful conclusion. The given expression is so important because it is the thing that we assume to be true to get us started. If the given changes, everything changes.
Because the given is the cornerstone on which your entire argument rests it is very important to always write the given down. You may think that you will remember what the given was, but what if someone else reads your work? Or what if you read it weeks from now? Will you still remember?
The goal of using consistent systems and consistent symbols in logic is to make everything universally understandable. Anyone who understands logic should be able to read and follow your work. You should always write your work as if you expect someone else to try to read it later; that is the best policy to avoid making mistakes. Remember that a single mistake, anywhere in the chain of logic, invalidates the conclusion.
!!!!!!!!!!!!!!!!!!!!!!!!!!!Hereís an example from algebra of evaluating the given: !!!!!!!!!!!!!
You should now be understanding why mathematicians emphasize the given so much. If you want to evaluate a set of statements for validity, you must begin by assuming something is true. And if you change this basic assumption, then the whole argument changes as well.
In mathematics, we often rely on basic yet universal laws or axioms. An axiom is a special kind of assumption. Unlike the assumptions that weíve been making so far, like ďA is true� an axiom is a widely established, accepted law or set of laws that are unproven, but assumed to be true. 1 + 1 = 2 is an example of an axiom. It is an assumption that everyone knows and uses.
We use axioms so much that it can be easy to forget that they are really just assumptions. Most people think that �1 + 1 = 2� is a fact, for sure, always, but as youíll learn later this isnít the case. You could say that axioms are really, really good assumptions, because the longer they are used without a paradox developing the more likely it is that they are correct, however, that doesnít prove that they are true, always, for sure.
Think about a busy bridge. Hundreds of people drive over it in their cars and ride over it on their bikes every day and all of them make the assumption that the bridge is well-built, well-maintained. They use that assumption to support a larger argument: the bridge is safe to drive on. Like an axiom, this is an assumption that is use an accepted by many, many people. Over time millions of people will cross the bridge, and the more people who cross it safely the more likely it is that their assumption that the bridge is safe is the correct one. However, one day, after having been true for years and years, this might suddenly change. An earthquake, rust, or simply the passage of time might weaken the bridge to the point that it collapses, possibly taking some cars and people down with it.
Similarly, scientists and mathematicians rely on axioms as the basis for theories. The Pythagorean Theorem and Einsteinís Theory of Relativity rely on unproven axioms and allow scientists to make much more complicated calculations, theories, and predictions. Occasionally, an axiom will indeed prove untrue, sometimes with devastating consequences. Just look up some of the bridge disasters that have occurred in history.
Here are some examples of axioms:
1 + 1 = 2 (Unproven but assumed to be true)
A + B = B + A (Unproven, but assumed to be true)
The United States Constitution (Unproven, but assumed to be true)
U.S. citizens and the Supreme Court assume the Constitution to be the basis for legal arguments. But there is no proof that the constitution is true. In fact, the constitution gives politicians the power to alter or amend the constitution when necessary. The power to amend the constitution is also one of its most hailed features.
There will never be an earthquake larger than magnitude 9.9 (Unproven, but assumed to be true)
The previous section discussed the importance of axioms as the basis for reasoning. We must assume that the ďgiven statement� is true to continue evaluating the argument, but where do we go from there? Generally it isnít a good idea to build an argument completely on assumptions. Remember that if any assumption is wrong it invalidates the entire argument, and the more assumptions there are the more likely it is that one of them will be wrong.
Axioms, however, are an exception. Axioms can be applied to the given to take an argument further, even though they are themselves assumptions. Because axioms are universally accepted to be true they are okay to use in your chain of logic. The following diagram illustrates the concept of using axioms as an intermediate step to evaluate other statements: