-1

When doing many mathematical problems/proofs I encounter statements like '$x$ is a real number.

These statements feel lacking in precision to me, $x$ is a variable, it 'represents' a number, or acts like a placeholder for one, but to describe something as being 'a number' implies it must be a specific thing for which the statement is true, so for example '1 is a number', '$2$ is a number' and so on, I understand referring to $x$ as 'a number' in that it does 'represent' where a number can be a part of our structure, but it seems more informal to say this.

What I do understand is that if we have $x$ placeholding for a real number then '$x$ is a number' will be true under every assignment/interpretation.

Is it formally correct to make this statement or is it simply informally allowing the reader to understand that the variables represent mathematical objects of a certain type.

I can understand the meaning in a sense, however is there any other ways to describe the nature of the objects that $x$ can be considered for?

  • Some times, such as when solving equations, $x$ (or whatever letter we use) truly is a single, specific real number, it's just described a little indirectly. Or we know very well which number it is, it's just a hassle to write down exactly, like $\pi$ or $e$ or $147$. – Arthur Oct 02 '22 at 11:15
  • See also this post. We can think of formal power series in a formal variable $x$, for example. Then $x$ is not just "placeholding". – Dietrich Burde Oct 02 '22 at 11:18
  • Not sure what you mean by "true" here. If, say, one wants to show that $\sqrt 2$ is not rational, it is perfectly common to start with something like "let $x$ be a rational number such that $x^2=2$" the idea being to derive a contradiction. In such a case, is it "true" that $x$ is a rational number? – lulu Oct 02 '22 at 11:18
  • Cool question. I think you're getting confused about the cases where $x$, as a variable, is bound or unbound; this is related to the distinction between a formula and a sentence. For instance, "$x$ is a even" is a formula, where $x$ is bound, and "$\forall x \in \mathbb{N}$, $x$ is even" is a sentence, where $x$ is bound. You may be interested in the Wikipedia pages on "first-order logic" and "free variables and bound variables". – Sambo Oct 02 '22 at 11:22
  • @DietrichBurde especially in polynomial rings where $X$ also becomes it's own object. –  Oct 02 '22 at 11:50
  • @lulu for the cases that our derivation is a contradiction for $x$, then '$x$ is rational' is true. –  Oct 02 '22 at 11:52
  • Why? Such an $x$ does not exist so in what sense is it "true" that $x$ is rational? – lulu Oct 02 '22 at 12:06
  • If you want to be fussy, you can say, "$x$ is a real variable". – Gerry Myerson Oct 02 '22 at 12:32
  • Compare "Two is a number" with " "Two" is a word". Can you see the difference? – Mauro ALLEGRANZA Oct 02 '22 at 14:40
  • Do you have the same sort of confusion with a geometrical proof starting with " P is a point..." ? – Mauro ALLEGRANZA Oct 02 '22 at 14:43
  • See this post if you want to understand what really is going on when you start talking about a real number $x$ that has never been mentioned before. – user21820 Oct 03 '22 at 11:09
  • @Sambo: Note that whether a variable involved in a ∀-reasoning is bound or not in a formal system actually depends on the formal system. In Fitch-style systems such as this one, it is common to consider variables in a ∀subcontext as being bound, and it is also common to not allow variable shadowing. I mention this because the issue here isn't just about single sentences but about where they occur in a proof. – user21820 Oct 03 '22 at 11:11
  • @lulu my mistake, I would say it's something we 'assume' true, but is not, but we 'assume' that 'x is a rational number' is true. –  Oct 03 '22 at 20:37
  • I consider "$x$ is a real number" to mean "$x$ is an element of the set of real numbers" because I read $x \in \mathbb{R}$ as "$x$ is a real number" too. You're running into the assignment versus equality problem because "is" is overloaded just like $=$ is. So if I say $y=x^2$ am I defining $y$ to be $x^2$ or am I to evaluate the claim that $y$ is equal to $x^2$? Typically you can figure it out from context though so it's not that problematic but the notation $:=$ is used for defining like we would normally mean when we write $y=x^2$. It makes a big difference in computer science though. – CyclotomicField Oct 04 '22 at 12:06

1 Answers1

0

Suppose I want to know, since $2$ is even, is $2^2?$ How about the case with $4^2?$ And $6?$ I can go on, but eventually you'll get tired of this. Here's how you can save the time.

If $x$ is an even integer, then we can write $x=2y$ for some other integer $y.$ You may then demonstrate with a few examples. Now that this is established, we have $x^2=(2y)^2=4y^2$ and with intuition, we can see that $4y^2$ comes in the form of an even integer.

So without testing an infinite amount of numbers $$2, 4, 6, 8, \ldots$$ we can simply consider a general form $$x\in\{2y|y\in\mathbb{Z}\}=2\mathbb{Z}$$ and see how $x$ performs in our analysis. The idea behind "let $x\in2\mathbb{Z}$" is simple. We are just choosing a collection of objects to test, and we run the test by applying things we know about $x$ and trying to obtain the desired result, a proof.

More than anything, the importance comes from the fact that $x$ sufficiently resembles the objects we wish to investigate within a certain problem. We assume nothing more about $x.$

Edit: Coming back to the problem, I see how badly I missed the mark in clearing the confusion. I, however, doubt there's any point to fixing my answer.

Nickolas Bogan
  • 101
  • 6