Lesson Nine - Proofs

Let’s start today with the statement: There is no almighty God. We want to prove that there is no almighty God. How will we do that?

Here is a proof:

  1. If there is an almighty god, that god has the power to change anything, everything and anything.
  2. Now let’s look at the expression 1 + 1 = 2. This is true, isn’t it? We all know that.
  3. Even a god cannot change that.
  4. Therefore there is no almighty God, and the proof is finished.

That was simple. Let’s move on now.

Wait! Do you see a fallacy in the proof? If you read this statement, do you see anything that looks like a fallacy, any reasoning errors? Check your notes or look back to a previous section. There is a fallacy. Try and find the name of it.

Hint: The statement “We all know that” is the fallacy. Can you get it now?

The fallacy here is that of appealing to popular sentiment. The only justification for the argument is popular sentiment. Since this is a fallacy, this is not a logical proof. What do you do when you see a fallacy in an argument? You start to ask questions. Thus, the argument is not as clear-cut as it looks, and perhaps the expression 1 + 1 = 2 is not always true. Imagine two people walking into an empty house later 3 people coming out. Could that ever happen? What is the probability that 1 + 1 = 3?

Since the previous “proof” contains a fallacy, it is not a valid argument. So this raises some questions – what is “1”? What does “1” stand for? What is the definition of “1” or “1.0”

For instance if we evaluate the expression x = 0.5 + 999/1000 the precise answer is x = 1.5001001… However, the answer x = 1.50 would be considered correct as well because it represents the interval 1.495 < = x < 1.55 and the precise answer is within that interval. In most algebra problems “1.50” represents all numbers between 1.495 and 1.55.

Similarly, “1” can stand for the set of real numbers between 0.5 and 1.4999… if we consider rounding, which in many cases we do.

Example 1

1.49 represents any real number between 1.475  x  1.485. This means if you write “x = 1.48”, this means the true answer is somewhere within the range 1.475 - 1.485 because we almost always round. And this is correct!

Example 2

Find the value for x in the equation 3x + 1 = 2
Subtract 1
3x + 1 – 1 = 2 – 1

Divide both sides by 3 3x/3 = 1 / 3 x = 1/3 or .3333…..

The true answer for this is x = any real number between 0.325 and 0.335. So when you say the answer is 3, that means the answer is within the defined interval. Similarly, when you say the answer is 1, that means the answer is somewhere within the interval of 0.5 and 1.4999…

Since “1” stands for any number between 0.5 and 1.49999…., then “1” represents a range of numbers on a number line.

Number Line:

Thus if you have 1 + 1 = ?, then you have a range of numbers between 0.5 and 1.5. Now choose a number between 0.5 and 1.5 for x and a number between 0.5 and 1.5 for y. If you choose 1.4 for both numbers, the solution is 2.8 rounded to three. Thus there is a certain probability that 1 + 1 = 3.

You might be thinking, wait a minute. We learned that 1 + 1 = 2 is true always, for sure. In everyday English this is true. However 1 + 1 = 2 is an axiom, an improvable assumption, and there is a much broader, mathematical definition of one. Once we define “1” to represent real numbers between 0.5 and 1.4999…, then we cannot simply use the axiom 1 + 1 = 2 any more. Things get more complicated and more interesting!

Let’s use a step by step process to understand how to evaluate the statement 1 + 1 = 3.

  1. First rewrite the equation 1 + 1 = ? as x + y = ?
  2. Pick a number between 0.5 and 1.4999… for x. Let’s say 1.3.
  3. Pick a second number between 0.5 and 1.4999… for y. Let’s say 1.4
  4. Add the first number (1.3) to the second number (1.4).
  5. 1.4 + 1.3 = 2.7
  6. Then round
  7. 2.7 ≈ 3

Thus there is a certain probability that 1 + 1 = 3 after rounding the solution. Note that you can only round the solution, and not the numbers such as 1.49999.

This range of solutions can be represented by a graph.

So our focus now turns to answering the question, what is the probability that 1 + 1 = 3?

We know it is possible for 1 + 1 = 3, but we want to figure out, how often to expect this answer. Similarly we want to know the probability of 1 + 1 = 2 and 1 + 1 = 1.

What is the probability that 1 + 1 = 3?

  1. Pick a number, y, where 0.5 C y  1.5
  2. Pick a number, x, where 0.5 C x  1.5
  3. Plot sample space
  4. Calculate event space: x + y  2.5. This is what we know about the event space. Everything above 2.5 is rounded to three, so we do not want any values that would round up to three. There is a certain probability that x and y add up to three because 1 represents any number between 0.5 and 1.5.
  5. Plot event space
    Next, to plot the event space, we want Y all by itself.
  6. Compute the probability that 1 + 1 = 2 and 1 + 1 = 3. The probability is equal to the size of the space above the line.

If you pick a value for x and a value from y from this corner, it means that both x and y are large. Then if you round the sum, you get something close to three. If the first number is big and the second number is big, then, if you round it. You get something close to three.

Now we have to compute the probability. The probability is equal to the size of the event space over the size of the sample space, which is a percentage of total area. We need to calculate the area of the event space in comparison with the whole square.

Begin by dividing the square up into smaller and smaller quadrants. How many times does the small triangle fit into the small square? The small square space above the line is 1/8 (0.125) of the total square, or 12.5% of the area.

It is not very big, but it is something. Thus, the probability that 1 + 1 = 3, is 12.5%. There is more than a 10% chance that 1 + 1 = 3!

Can you use the same technique to calculate the probability that 1 + 1 = 1?

Start with a similar formula and adjust the value to be less than 1.5, because any sum less than 1.5 is rounded to 1.
x + y > 1.5
x – x + y = 1 – x
y = -x + 1.5

Similarly there is a 12.5% chance that 1 + 1 = 1. This means that 1 + 1 = 2 is correct exactly 75% of the time, which is why we usually always assume it to be true and take it for granted.

Here is the distribution function for 1 + 1 = ?.

To review: We began this lesson proving that there is no almighty God. Based on the axiom 1 + 1 = 2, the proof looked valid until we discovered a fallacy of popular sentiment. What do you do when you discover a fallacy? Ask questions. What kind of questions did we ask? What is the definition of one? We asked for the definitions. Then we learned that the definition of one is not as clear-cut as we thought because one actually means a lot. We realized that although everyone thought they knew that 1 + 1 = 2, our collective knowledge was limited. We then computed the probability that 1 + 1 = 1, 1 + 1 = 2, and 1 + 1 = 3. We found that 75% of the time, 1 + 1 = 2, and therefore most people believe that 1 + 1 = 2. But that is not always the case.