Because it is evident. Something minus the same thing leave you with nothing, zero, zilch, nada. Something is not nothing, just like nothing is not something. Just ask a five year old, they will tell you they’d rather have five marshmallows than no marshmallow at all (American example, I never liked marshmallows, but you get the general idea).
I don’t think we need to invoke “proof by contradiction” at all here. Just break it down to basic properties.
For any real number x,
- 5x – 5x = 5x + -5x (definition of subtraction)
- 5x + -5x = (5 + -5)x (distributive property)
- (5 + -5)x = (0)x (5 and -5 are additive inverses; their sum is 0)
- 0x = 0 (multiplicative identity)
We have shown that, no matter what value x has, the left side of the equation must be 0. Since the right side is not 0, the equation cannot be true.
I don’t think there’s any “proof by contradiction” there. I never assumed anything to be true, only to have it lead to a contradiction.
Even shorter. Replace Step 2 with 5x + -5x = 0 (Definition of additive identity)
I will point out that if we are working in the modulo 5 group there is a solution. x = 0
True, but then, if we’re working in the modulo 5 group, then all numbers are solutions.
Very extremely true.
This discussion reminds me of Lewis Carroll’s What the Tortoise Said to Achilles.
To get anywhere in a mathematical argument, we have to agree on what our axioms are. I’m surprised to see the OP questioning whether to accept the Excluded Middle. That’s one that mathematically naive people tend to accept without question. That every statement must be either true or false is not an axiom about the real numbers, it’s an axiom of logic, which is more fundamental.
If I were explaining this to a 5 year old child, I would not invoke proof by contradiction, which even some moderately sophisticated adults find questionable. However, most five year olds don’t know anything about algebra, and little even about arithmetic, so it’s a little unclear to me how I would even describe what the question is. I guess I would use an argument something like this:
- We want to find a number, call it x.
- We will multiply it by 5.
- We will then take that result and subtract it from itself.
- We want the result to be 5.
But if we subtract anything from itself, we get zero (hopefully it wouldn’t be too hard to convince them of this fact). So no matter what the answer is after step 2, the answer after step 3 will be zero, not 5. So no matter what number we pick in step 1, we will not get 5 after step 3.
The OP asks about use of this explicit step of reasoning.
This is mostly repeating the above, but maybe slightly more explicit.
We assume a whole raft of stuff about the nature of the space we are casting the question in. Numbers, arithmetic operations. We are also assuming that we are reasoning in Aristotelian logic, no excluded middle. If we can’t assume at least this, we can’t have a conversation.
The first step of the question goes:
consider the equation 5x – 5x = 5 where 5 is a real number and x is a variable for any real number.
So far so good. Maybe.
5,x \in \mathbb{R}, 5x - 5x = 5
asked to solve the equation for x
The problem is not that we can’t solve for x. The problem is that the original statement is not true in a fundamental way. So even asking for a solution for x isn’t valid. It isn’t that algebraic manipulation in order to solve for x leads to a problem, the expression is intrinsically unsound. Algebraic manipulation cannot proceed as the expression is false from the outset. False means something very specific. It means it isn’t a valid expression in the universe we define our usual mathematics in. The rules for doing algebraic manipulation essentially require that the expression we perform manipulation on is well formed within the space our axioms define. If 5 = 0 we violate one of the axioms from which we generate the numbers. (the numbers are injective, for a start.) So our entire space within which reasoning is to be performed vanishes.
5,0 \in \mathbb{R}, 5 = 0 is not true,
they cannot both be members of the reals. So asking for some property (like solve) is already invalid as there is no system to ask the question of. It isn’t just the lack of a solution. There is nothing at all you an ask.
There are other questions where we might reasonably have an expression and ask for a solution, yet there isn’t one. That gets us to the fundamental theorem of algebra basically that any odd order function has at least one real root. So you can define lots of even order polynomials that have no real roots, but do not break the axioms of the system.
Division by zero is another case. It isn’t that we might invoke infinity. We don’t. An expression with a division by zero has no definition within our mathematical space. Which is one way of interpreting “undefined” as the answer to divide by zero. If we allowed it we just get the same arbitrary contradictions of the fundamental axioms. Another way of interpreting “undefined” is that you may now choose anything you like as a solution, as the solution has no definition. That is less sound. Of course we have lots of tools for wrangling things that behave badly, but singularities themselves, by definition, are not wrangleable.
Perhaps you would not need to explain to a five-year-old why 5\neq0. You might be forced to explain it to an older kid rigorously if they start asking questions like what is 5, what are the real numbers, what’s subtraction…
More precisely, if we can’t assume at least a common framework, then we can’t have a conversation, but it’s possible to have a conversation in a framework other than Aristotelian logic and standard arithmetic. But those frameworks are both so widely-used that, if one wants to use a different framework, it really needs to be explicitly specified.
A few issues, here. First, the fundamental theorem of algebra deals only with polynomials, not with functions in general, and it’s not hard to find an odd function that doesn’t have any roots. Second, why exclude imaginary numbers? The OP didn’t, and his equation still has no solutions even when we include imaginaries.
That’s what a mathematician would call “indeterminate”, not “undefined”.
Yes. There are other logic systems, but Aristotelian logic is most commonly used because it works the best and most generally in the universe we observe.
IMHO you’re making things more complicated than they have to be.
As a basic algebra textbook will tell you, to solve an equation means to find the solution set: the set of all numbers that satisfy the equation when you replace the variable with them.* That set may have zero members, or one, or two, or infinitely many. We certainly can solve the equation 5x - 5x = 5 in the sense of finding its solution set—it just turns out to be the empty set.
(* In the interest of simplicity I’m assuming an equation in one variable and avoiding consideration of what kind of numbers (real, complex, etc.) we consider as possible solutions.)
But the OP’s question was not “solve the equation”. It was “why do we assume only true and false are possible?”
Can you give us an example? It certainly cannot be a continuous one.
In fact there are no solutions in any field.
I think that is incorrect. I was taught in college that an indeterminate is a number that has an unknown value (or maybe no determinable value) as opposed to a variable that can take on any value in the domain/range. 3x + 2y = 19 has both x and y as variables whereas 7n - 4 = 0 has n as an indeterminate since it has a value, we just don’t know it until we solve for it.
First of all, I’m not saying I disagree with the Law of the Excluded Middle but it is possible to call it into question. When does a color shift from red to not-red? Can we definitively say not-not-red is red? This can clearly be resolved by defining red as specific wavelengths of light to which I counter with white light as it has elements of red and not-red light.
f(x) = 1/x. Which is, as you note, not continuous.
That’s not how I interpret the OP’s question, but I’m not entirely sure I know what he was looking for. I kind of wish @Ynnad would come back and let us know if we’ve given an answer that satisfied him.
I think LSLGuy paraphrased my question quite well. I am just not satisfied with the answer, “Aristotle said so and that’s how we’ve always done it.” I want to know why.
If by an odd function you mean a function f that is defined everywhere and satisfies f(-x) = -f(x), then every odd function has a root, because you must have f(0) = 0.
[aside]
I tend to frown on “explain it to me like I’m 5 years old” because from observation I know customarily the explanation to a 5 year old who insists on knowing “why” tends to go along the lines of “you’ll understand when you’re older”. Followed after a few rounds by “shut up, kid”. 
[/aside]
This has been mentioned already, but it’s called the Law of the Exclude Middle (LEM). If you assume the LEM, then every logical statement is either true or false, hence showing that something is not true is the same as showing that it’s false.
Likewise, showing that it can’t be false is the same as showing that it must be true, which is what feels strange to many people. For this reason (and others), some mathematicians do study systems of logic in which the LEM doesn’t hold. That generally goes under the heading “constructive mathematics.” The idea behind the name is that, for example, to show that something exists, it’s not enough to show that it’s impossible for it not to exist (a proof that would use the LEM), but you need to show how to actually construct or find the thing.
This is viewed as a valid if fringe area of math. Most mathematicians are happy to continue using the LEM because it’s just so darned convenient. On the other hand, constructivists have had enough influence that, if you have a choice between using a proof by contradiction or a direct proof, the direct proof is preferred just to avoid it being a “nonconstructive” result.
But, in the end, it comes down to what you want to assume in your system of logic. All mathematics depends on underlying assumptions, and the LEM, being convenient and having a long history, is commonly one of those assumptions.