How do we know if a statement is true? (Screencast 1.1.2)

How Do We Know If a Statement Is True?

Defining Mathematical Statements

• The screencast introduces the concept of mathematical statements as declarative sentences with definite truth values, distinguishing between obviously true (e.g., "two is even") and obviously false statements (e.g., "three is even").

Understanding Prime Numbers

• A positive whole number P is defined as a prime number if it is greater than or equal to 2 and has exactly two divisors: 1 and itself.

• Examples of prime numbers include 2, 3, 5, 7, and others. Non-prime numbers like 4 and 12 have more than two divisors.

Analyzing the Statement About Primes

• The statement under consideration claims that for any prime number P , the expression 2^P - 1 is also a prime number.

• This statement's truth value does not depend on opinion; it must be determined through mathematical proof.

Experimenting with Prime Numbers

• To explore the truth of the statement, examples are computed using known prime numbers.

• For P = 2 : 2^2 - 1 = 3 , which is prime.

• For P = 3 : 2^3 - 1 = 7 , also prime.

Continuing Calculations

• For P = 5 : 2^5 - 1 = 31, which remains prime.

• For P = 7 : 2^7 -1 =127, confirming another instance where the statement holds true.

Evaluating Results

• After testing primes up to seven, a question arises about whether these results confirm the statement's validity universally or just suggest it might be true.

Discovering Counterexamples

• The next test involves checking for P =11. Here, calculations show that 2^11 -1 =2047, which is not a prime (it factors into smaller integers).

Conclusion on Truth Value

• The initial claim that whenever P is prime then so too is 2^P -1, turns out to be false due to this counterexample.

• One counterexample suffices to disprove universal claims; however, no finite list of successful examples can prove such statements true.

Key Takeaways on Proof in Mathematics

Understanding Problem-Solving Approaches

The Importance of Engaging with Problems

• The main way to gauge the truth value of a problem is through exploration rather than immediate resolution. This approach has two significant benefits:

• It enhances comprehension of the problem.

• It may lead to discovering insights that directly address the questions posed.

• Engaging with examples can provide a sense of a statement's truth value, but it's important to note that examples do not constitute proof unless they contradict a general statement.

• Understanding that working through problems can yield unexpected findings reinforces the idea that exploration is crucial in mathematical reasoning.

• The discussion emphasizes being on solid ground when approaching complex problems, regardless of one's initial belief about their truthfulness.