I've noticed that recently I've been struggling with some aspects of our logical ideas in Algebra and they seem to revolve around the idea of the true concept of a 'variable'. I mean this in the most simple use of the word, as a form of 'placeholder'. Something like xi has another variable in it's index, and provides other problems, if there is xi and x1 we can thing of xi as a more general form of the latter, if we have variables x and y we can call them 'separate variables' in the case of xi and x1 we could have i=1 in which case x1 and xi are not separate?
Another thing that I would be interested to know is based around the way we talk about them, for example:
We have an expression for example:
3x+1
We can talk about this in a general way, what values do we expect this quantity to take for differing values.
We might write y=3x+1, we then write something like
'if x=3'
and we'll say y=10 but y is being used as a general term and we jump between these 'contexts', this seems odd to me, I understand the idea but it seems strange all the same.
In the same way, having
3x+1=10 is a statement and we can talk about it generally where x is no particular number, and even quantify over it using quantification logic, but again we may write
if 'x=3' and we will write then 10=10 and the statement is true.
It seems strange to use this notation 'x=...' and to use 'if' or 'when' because are these 'variables' really changing or are we really just expressing the structure for where there is a specific number instead of a placeholder?
It begs the question, is it just a placeholder for 'some' value, or does it almost represent something that is somehow changing by itself (how can it do so on a piece of paper), if it is simply a place holder, saying 'when x=' or if 'x=1' seems strange to me, would we not perhaps best define some kind of 'replacement' operation instead of this kind of 'assignment'? Does x somehow have a value at all times? Or are we constantly jumping from context to context? Where each context is hypothetical?Can x somehow represent multiple values at once.
If we talk about x=1
Can we also write x+1=2 and have x as a proxy for 1
and hence have ax=a1 for all values?
