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I'm thinking, what a variable actually is? It seems like x is the same thing as identity(x) in any context. Can we just change any statement that incudes variables to have identity functions of those variables (instead of variables).

For example, 4x + 3y is the same as 4*identity(x) + 3*identity(y).

I have no idea why I need this, just trying to understan are there any flaws in my logic?

Thank you!

azif00
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Viacheslav Prokopev
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    This lecture should answer most your questions : https://www.youtube.com/watch?v=V49i_LM8B0E – tryst with freedom Mar 26 '22 at 12:57
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    I think your confusion stems from a more profound misunderstanding than what is explicit in the question. Do you know what a function is? – Git Gud Mar 26 '22 at 12:59
  • Buraian, thank you for the lecture! But do you have a timestemp? I'm feeling a bit lazy this saturday, but I need an answer or I can't relax lol. – Viacheslav Prokopev Mar 26 '22 at 13:02
  • Is identity a function in some programming language? – ViktorStein Mar 26 '22 at 13:02
  • Git Gud, I think that I do know what a function is, but now that you asked I am not sure again and heading to google for a good definition – Viacheslav Prokopev Mar 26 '22 at 13:03
  • @Ramanujan, no it's just how I decided to annotate it here, it's a mathematical identity function. https://en.wikipedia.org/wiki/Identity_function – Viacheslav Prokopev Mar 26 '22 at 13:04
  • I guess $x^2-4$ is the same as identity(x^2-4), which is not a "variable"? I propose that the answer to the question in the title is "no". – GEdgar Mar 26 '22 at 13:38
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    Sure, you can write $4 \cdot id(x) + 3 \cdot id(y)$ instead of $4 \cdot x+3 \cdot y$. The question is, what would be the benefit of this? You still have the variables $x$ and $y$ there in your expression, you just wrapped them inside a further function $id$. Note that you cannot write something like $f = 4 \cdot id + 3 \cdot id$: this would lose the information that the first $id$ function is applied to $x$ and the second one to $y$. – Pilcrow Mar 26 '22 at 13:53
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    A variable is a symbol, a syntactic object. – Mauro ALLEGRANZA Mar 26 '22 at 14:07
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    Notice you can also write $3=id(3)$ in every context. Do you think this will help explaining "what the number $3$ is"? – Torsten Schoeneberg Mar 26 '22 at 14:41
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    Also, to make something in @Pilcrow's comment more explicit: There is no such thing as "the identity function". There is one distinct identity function for each distinct set. And when you try to write out what that means, you will encounter variables and quantifiers already ... – Torsten Schoeneberg Mar 26 '22 at 20:01
  • @Pilcrow I really like your explanatin, can I please ask you then, what is your definition of a variable? I define it as a placeholder for an object in a sequence of operations (function). But that does not seem like a broad enough definition. I don't like the definition of a variable as "just a symbol in an expression", because that one is way too general and really tells you nothing. – Viacheslav Prokopev Mar 27 '22 at 00:03
  • @TorstenSchoeneberg Thank you! Then what is a variable? How can we define it other then "a variable is just a symbol"? – Viacheslav Prokopev Mar 27 '22 at 00:06
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    Can you provide some info about your math background? The answer I'd want to give for someone who knows multivariable calculus and is trying to think about it more formally is very different than the answer I'd want to give if you only just recently encountered expressions like $4x+3y$ for the first time. – Mark S. Mar 27 '22 at 00:34
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    @MarkS. Hi, I know my questions might sound a bit silly but I guess I need the answer for someone who (used to) know multivariable calculus. The area of math I am most comfotable with is probability, but I did all undergraduate level math about 6-7 years ago. Now I just decided to start over and picked a descrete math book, encountered a definition for a variable and really did not liked it. – Viacheslav Prokopev Mar 27 '22 at 00:49
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    I typically avoid the term “variable” entirely. I’ll say things like “if $x$ is a number and $3x+1=10$ then $x=3$.” Here $x$ isn’t a “variable”, it’s just a number. – littleO Mar 27 '22 at 03:11
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    Related: "Is there a detailed definition of the concept of a 'variable', and why do we use them as such?" – Blue Mar 27 '22 at 03:30
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    A variable, to a first approximation, is something you can assign values to (from some domain which can be specified explicitly but which is often left implicit). For example, it makes sense to say: "let $x := 3$". It does not make sense to say "let $5 := 3$" because $5$ is a constant with a fixed meaning. Similarly, it does not make sense to say "let $(x-1)^2 := 4$", although this is a consequence of assigning $x := 3$. The truth value of a proposition such as $2 x-3 < 5 +y$ then depends only on the values assigned to $x$ and $y$, whereas the meaning of $2$, $3$, $5$, $<$, $+$ is fixed. – Pilcrow Mar 27 '22 at 12:47

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