memoranta said:
Hello.
I just came to realize that in mathematics, a problem is defined as a set of strings.
But then, for example, if I state a problem as "Find an addition of 1 and 2," how are strings (e.g. find) connected? Is any form of a string that connects these strings that is a string?
Hey memoranta and welcome to the forums.
You might have to clarify your question a little more. I will give you a few different perspectives on your question which will help you do so.
In one aspect of mathematics, we have a way to treat each number as a 'string' and then we have rules for generating the other strings based on previously generated strings.
For example let's say you want to define the natural numbers. We start off with the number 1. Then we say how we can generate the rest of the numbers by creating a 'string' that represents the previous number 'plus 1' which is related to a different string.
However one should be careful about 'language' since we can get situations that 'don't make sense'.
You should take a look at this for an example:
http://en.wikipedia.org/wiki/Liar_paradox
Now in terms of connections, we could define a mathematical statement that conforms to what is known as a 'grammar'.
Basically a grammar states the rules of how a sentence or representation is constructed. The grammar makes sure that you don't create sentences that are basically 'garbage' (in other words they don't 'make sense' in the context of what it represents and how it is used).
For this check out the following:
http://en.wikipedia.org/wiki/Backus–Naur_Form
We use the above to make sure that computer code is in the right form and if it is the structural information is used to convert the complex interactions and code forms into simple instructions that a processor can understand.
But again with reference to the liars paradox, you need to be aware that many linguistic forms can create situations where you get something that is 'unresolvable': you find instances where you get things like 'looped logic' or contradictions of some kind or even things that can not correspond to something that has a solution.
This kind of problem is something that is studied in logic because a big part of logic is concerned with proving whether a statement is true or false, whether a solution exists or does not exist and so on.
There is a lot of overlap between various fields of mathematics and computer science for this kind of thing and I have only provided not even an inkling of what is out there.
If you have any other questions I'll do my best to answer them.