Let us consider the equation 5x – 5x = 5 where 5 is a real number and x is a variable for any real number. If we are asked to solve the equation for x we end up with the equation 0 = 5 which is false. Therefore, we conclude that the equation 5x – 5x = 5 has no solution. Explain this to me like I’m five years old. I do not dispute that 5x – 5x = 5 has no solution. I do not dispute that the process of transforming the equation into an obviously false statement shows that the equation has no solution. What is the logical reasoning behind this process and how do we know the reasoning is sound?
It is proof by contradiction aka an indirect proof.
If we have a true statement in mathematics then it cannot lead to a mathematically untrue statement through a valid process.
Since mathematical statements can only be true or false, if it is not true then it must be false
Thus we assume the statement is true, show it leads to an untrue statement thus proving the statement is false.
In your example, we assume x has a value that is a solution to the equation.
Our process is to find the value of x. SInce it is a linear equation, I know the process is valid.
But in carrying it out, we get a false statement, that 0 = 5.
Therefore the statement “x has a value that is a solution to the equation” is false.
All of this is based on Aristotelian Logic.
Your explanation makes sense if the above statement is true but how do we know the above statement is true?
If people instead said “either 0=5, or the equation has no solution,” would you find that acceptable?
I’m not sure if I understand the question, but the “solution of an equation” is the set of all values of the variable (or variables, if you have multiple of them) for which the equation is true. The equation “5x - 5x = 5” is never true, no matter what value x has, and so its solution is the empty set.
Why include the X variable? The equation is the same as 5 - 5 = 5. The reasoning is logic. Including the X-variable changes nothing. It just complicates it.
We know 5 - 5 = 0
What is it that you are asking?
We may start by noting that there is a map from, let’s say, polynomials in one variable over the real numbers to the real numbers given by substituting some real number for x. No matter what value you put in, though, by definition evaluating a constant just yields that constant. In other words, the left-hand side always evaluates to zero which is not equal to 5 in the real numbers.
PS or what @Chronos said
NB it is entirely possible for a non-zero polynomial to be identically zero after evaluation in finite characteristic. That will not work over the real numbers, though; a non-zero polynomial cannot have infinitely many real roots.
I guess another way of asking my question is, “How do we know that proof by contradiction is valid?”
Because math and science is based upon observable things in nature. If you have 5 things and remove 5 you are alway left with zero. You can observe that.
There are a lot of mathematical systems for which conforming to an understanding outside of the system is not useful, and very few rigorous mathematical systems allow you to end a proof with that kind of observation.
The real test is when that kind of understanding is incorporated into the basic axioms of the system. Those axioms are part of the definition of the system, so they can be used within the system as equivalent to that out-of-system knowledge while retaining the integrity of the proof.
Within the mathematical system of Arithmetic, that observation (take away n of n items and you have zero items) is a derivative of the additive identity axiom:
x+0=x
A little arithmetic, you arrive at a restatement of the axiom which says the same thing as the intuitive out-of-system understanding:
x-x=0
If your proof-by-contradiction arrives at a conclusion that contradicts an axiom, the premise must be false.
- The expression 5x – 5x is equivalent to 0.
- Therefore, the equation 5x – 5x = 5 is equivalent to the equation 0 = 5.
- The equation 0 = 5 has no solutions. (It is never true.)
- Therefore, the equation 5x – 5x = 5 has no solutions. (It is never true.)
Is there one of these steps that you’re having trouble with, that you need to have explained or justified?
But why? Remember, I am a five-year old in this scenario. I don’t recall seeing a real number axiom stating a statement must be either true or false and nothing else. If you can show me this is an axiom of the real numbers then I will shut up and go away.
Yep, and there’s infinitely many of them. Perhaps the simplest is X + 1 = X.
Bertrand Russell & Alfred North Whitehead’s Principia Mathematica 1+1=2 is a 360 page proof. I am not sure there is an ELI5 for you for this if you don’t accept the most simple logic that X - X = 0.
Under Aristotelian Logic, statements are either true or false. There are other logic systems where this Law of the Excluded Middle does not apply.
More precisely, the conclusion that 1+1=2 appears on page 360 of the book. But most of those pages were concerned with other matters.
I like your take on the task “explain to me like I was a five year old”. Now explain it to me like I was finishing my Math PhD.
The thing is: it is not. At least, not always. See non-classical logic, for instance. But that logic is probably more difficult to grasp that classical logic for a five year old.
But proof by contradiction is applicable to your equation. The equation states that something is (numerically) equal to something that is (numerically) different, and even a five year old can see that this is not the case. The five year old could use the fingers to see that. It is evident.
TBH I have not read that book. My understanding is it lays out the proofs that lets someone say 1+1=2 with confidence that it is correct (sort of the building blocks of math that get you to 1+1…the fundamental stuff).
But why? What makes 5x - 5x = 5 bivalent and amenable to proof by contradiction when other statements such as the Liar’s Paradox are not?
Because there is no paradox in 5x - 5x = 5. It’s just not true.