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February 2, 2004

1. Why do we need proofs? Proofs allow us to determine whether something is true or false. Proofs allow us to determine whether a piece of information leads to a true or false conclusion. Proofs allow us to determine whether the change in the truth value of a piece of information will lead to a change in the truth value of the conclusion. Proofs allow us to examine the validity of a reasoning process, of an experiment, of a diagnosis, of an analysis, and so on. Proofs allow us to discuss topics with others in a consistent manner that can be subjected to examination. Proofs allow us to embrace rationality.

2. As discussed in the class, there are direct and indirect proofs. In direct proofs, there are hypotheses, and we want to derive for the conclusion. Often, this direct proof method is used to find conclusions. For example, given a set of hypotheses, what is the set of conclusions that one can derive? Also, we use direct proofs when we can see a path from the hypotheses to the conclusion—from the initial state to the goal state. This is probably the most common approach to proofs.

3. There are several approaches to indirect proofs: Proof of the contrapositive, Proof by contradiction, and Proof by cases. In Proof of the contrapositive, we assume the negation of the conclusion, and see whether we can obtain the negation of the hypotheses. In Proof by contradiction, we also assume the negation of the conclusion. However, this time around, we use it as a hypothesis with other hypotheses and see whether we can obtain a contradiction. If we do, then we know the conclusion is true. In Proof by cases, we try to prove that each hypothesis leads to the conclusion. This decomposition allows us to focus on individual hypothesis, which might be easier to do in some situations.

4. With the proof of the contrapositive, we assume the negation of the conclusion and see whether we can obtain the negation of the hypotheses. This means that now the contrapositive becomes the hypotheses, and the negation of the hypotheses becomes the conclusion. This also means that the contrapositive can trigger the negation of the hypotheses. So, whenever we want to use this type of proof, we must be sure that the negation of the goal state can trigger the negation of the initial states. For example, if you add heat to water, water becomes vapor. So, the hypothesis is “adding heat to water”, and the conclusion is “water becoming vapor”. Using the contrapositive of the conclusion is “vapor becoming water”, and the negation of the hypothesis is “subtracting heat from water”. So, when we do the experiment to convert vapor into water (such as collecting vapor using a cold lid), we can trigger “subtracting heat from water” (by measuring the temperature).

5. With the proof by contradiction, we look at the negation of the conclusion (the goal state) by combining it directly into the set of hypotheses. This is a very useful maneuver. Why? For some proofs, it is difficult to visualize the path from the initial state to the goal state and we may find it hard to direct the proving process. But now, by assuming and incorporating the negation of the conclusion, we have the goal state in the mix of the initial state(s). This becomes much simpler and more efficient. It clears up the picture. Think about this. Let’s say you have a set of hypotheses and a conclusion. Hmm … what would happen if I assume the negation of the conclusion and combine it with the set of hypotheses? If I can derive a contradiction with one of the hypotheses, then the conclusion must be true.

6. Proof by cases, under a closed world setup, is valid. However, in real world, it may not be applicable when the hypotheses and the conclusion are not modeled correctly, considering the interdependencies among the hypotheses and the conclusion. For example, say the set of hypotheses is { “If I add salt to water, the drink is tasty.”, “ If I add sugar to water, the drink is tasty.”, “I add salt to water.”, “I add sugar to water.” } And suppose the conclusion is “The drink is tasty.” If we prove it by cases, we can combine “If I add salt to water, the drink is tasty.” and “I add salt to water.”, then I can derive “The drink is tasty.” Similarly, I can derive “The drink is tasty.” By considering “If I add sugar to water, the drink is tasty.” And “I add sugar to water.” Thus, given the above set of hypotheses, can we really say that the drink is tasty after adding both salt and sugar? Maybe not! This is the danger of using proof by cases. To be sure, we may need to add a hypothesis that says “If add both sugar and salt to water, the drink is not tasty.”

7. Many proofs in Chemistry are done through proof of the contrapositive, and many proofs in Physics are done through proof by contradiction. In astronomy, for example, it is impossible to detect the existence of a black hole because it is so dense that light cannot escape and thus it cannot be observed. How do scientists prove the existence of a black hole? Through theory, physicists were able to postulate the existence of such matters. And then, through powerful telescopes, scientists are now able to find blackholes. How? They know that when such a dense object exists, it exerts gravitational pull on its surrounding objects. So, to prove the existence of a blackhole, scientist needs to show there exists objects that wobble. If there is a dense object, then nearby objects wobble. If the dense object cannot be observed, then it is a blackhole. If an object wobbles, it is near a dense object. Thus, by detecting a wobbly object, scientists can conclude that it is near a dense object. And since the dense object is not observed, it must a blackhole. In Chemistry, chemicals have different characteristics under different conditions. Scientists apply tests (such as if chemical is A, then it exhibit property P when heated) and if the properties are observed, then they can conclude that the chemical is A. This is done often in forensics and medicine.

Student finds largest known prime number

Thursday, December 11, 2003 Posted: 10:24 AM EST (1524 GMT)

DETROIT, Michigan (AP) -- More than 200,000 computers spent years looking for the largest known prime number. It turned up on Michigan State University graduate student Michael Shafer's off-the-shelf PC.

"It was just a matter of time," Shafer said.

The number is 6,320,430 digits long and would need 1,400 to 1,500 pages to write out. It is more than 2 million digits larger than the previous largest known prime number.

Shafer, 26, helped find the number as a volunteer on an eight-year-old project called the Great Internet Mersenne Prime Search.

Tens of thousands of people volunteered the use of their PCs in a worldwide project that harnessed the power of 211,000 computers, in effect creating a supercomputer capable of performing 9 trillion calculations per second. Participants could run the mathematical analysis program on their computers in the background, as they worked on other tasks.

Shafer ran an ordinary Dell computer in his office for 19 days until November 17, when he glanced at the screen and saw "New Mersenne prime found."

A prime number is a positive number divisible only by itself and one: 2, 3, 5, 7 and so on.

Mersenne primes are a special category, expressed as 2 to the "p" power minus 1, where "p" also is a prime number.

In the case of Shafer's discovery, it was 2 to the 20,996,011th power minus 1. The find was independently verified by other participants in the project.

Mersenne primes are rare but are critical to the branch of mathematics called number theory. That said, what is the practical significance of Shafer's number?

"People are going to make posters of it to hang up on the wall," said Shafer, who is pursuing a doctorate in chemical engineering. "It's a neat accomplishment, but it really doesn't have any applicability."

As for his own standing in the world of mathematics, "I don't think I'm going to be recognized as I go down the street or anything like that."

He said the method by which the number was found -- harnessing many computers together -- is more important than the number itself.

"Somebody else could have found the number," he said. "You install the program on the computer and it takes care of itself." But "I get the credit, along with the people that developed the software."