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If I have an expression with a free variable and I am looking at a particular assignment is it accceptable to see that string of symbols as a perfectly valid name for a number?

For example if my expression is simple like $x+1$ and $x=1$ I can see $x$ as a perfectly valid new way to refer to $1$ locally and $x+1$ as a string a perfectly valid new way to refer to the number $2$?

This way in order for $x+1=2$ we need only assign $x=1$ and not 'subsitute' into it? I can then see something like $x+1$ as being perfectly equivalent to $1+1$ in form simply that $x+1$ names a variable number and $1+1$ a specific number? It is just that in many text books the value of an expression is referred to by 'subsitution'.

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    Whatever answer you get... does it really matter? – JMoravitz Nov 09 '22 at 17:27
  • I'm not really understanding the question. The value is just the result of a calculation. So if $x=2$, then the value of $x+2$ is $4.$ – Accelerator Nov 09 '22 at 18:11
  • @Accelerator so the value of the expression is a number it represents, so $x+2$ is another way of writing $4$ if $x=2$ and we need to say '$x=...$ before we can describe the value of $x+2$, it describes the operations but also represents the value of the result of those operations. –  Nov 09 '22 at 18:19
  • I don't think it's that deep. – Accelerator Nov 09 '22 at 18:36
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    My father's name was "John", but he always went by "Jack". You can call me "Jack's son", "John's son", or just "Paul". It doesn't matter which you use. It still refers to me in all cases (in contexts that preclude other Johns/Jacks/Pauls or my brothers). When you say $x = 1$, you are giving the alternative name "$x$" to the number $1$. Then the expressions $x + 1$, $1 + 1$, and $2$ all refer to the same number. "Substitution" just means replacing one name for a number with a different name. Why do you think it means something different than what you are doing? – Paul Sinclair Nov 10 '22 at 13:02
  • @PaulSinclair fair enough sometimes text books make it sound as if $x+1$ is not a valid name for a number but $(1)+1$ is and we need to subsitute first, but obviously this is not the case, in which case the expressions take the denotion and aren't simply just some kind of 'forms'. –  Nov 10 '22 at 13:43
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    Note that qualification I had to give: "in contexts that preclude other Johns/Jacks/Pauls or my brothers". Proclaiming that "Jack's son" refers to me in all contexts would make for some really uncomfortable family gatherings. Usually we use variables without a set value. It could refer to any of a range of numbers. When you then say "$x = 1$", you have in a sense replaced this unset variable "$x$" with a more specific one that can only take on one value. That in itself is a substitution. When the books talk about substituting, they mean replace an unset variable by a specific value. – Paul Sinclair Nov 10 '22 at 14:18
  • Does this answer your question? Variables as symbols or objects and interpretation – Mauro ALLEGRANZA Nov 14 '22 at 12:26

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