The way I see math in the most basic level is, relations between numbers. The only numbers we really can use are the natural ones, they are always the same but the relations we make between natural numbers are the ones that let us get to the other types of numbers. Then we have two parts, natural numbers and "relations". "Relations" are the most important and revolutional things in our lifes as we create more and deduce its properties.
Is there any place were i can find a calculator that doesnt gives numbers but relations, one that has a table of variables were i can introduce a list of numbers for each variable and it finds the relations that are between the numbers?
Math "relations"
What if there is no simple relation? What should the algorithm return? Sounds like an NP problem to me for anything other than trivial inputs, but interesting question.
EDIT: it occurs to me there are in fact an infinite number of different relations between any two datasets - which one of them should the algorithm return?
EDIT: it occurs to me there are in fact an infinite number of different relations between any two datasets - which one of them should the algorithm return?
The only numbers we really can use are the natural ones
What do you mean? Last time I checked rationals/reals/complex worked quite well.
Is there any place were i can find a calculator that doesnt gives numbers but relations, one that has a table of variables were i can introduce a list of numbers for each variable and it finds the relations that are between the numbers?
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Do you mean something like data fitting?
EDIT: it occurs to me there are in fact an infinite number of different relations between any two datasets - which one of them should the algorithm return?This. Otherwise, we could program a calculator to find every answer to every possible problem.
The only numbers we really can use are the natural ones
I am a little confused by this. Are there no circles? Are there no triangles?
I think your assumption that mathematics is about numbers is an incorrect one. A Boolean algebra, for example, has no numbers and you would be hard pressed to argue that it is not mathematics. I can constructs valid algebraic fields using only the abstract notion of an identity, a relation, and an operation, none of which are really required to be any sort of number. I can spend hours doing deductive geometric proofs in which the essence of cardinality only sullies the purity and beauty of the form.
If you're limiting yourself to a formal system of numerical arithmetic you will still find there is going to be an infinite number of linear transformations from N to N. In fact, there is an interesting result that states that if you do manage to enumerate all the relations R:N->N, there are an infinite number more that exist that you can't enumerate within your formal system and have it remain consistent and closed. If someone sells your a calculator that gives you and answer, you have been duped.
I am confused by the whole thread. I know what a binary relation is, but I doubt that's what the OP meant by "relation".
The way I see math in the most basic level is, relations between numbers.
I'm not sure I buy this. There are many branches of mathematics, many of which have very little to do with numbers. For example, combinatorics and graph theory don't rely on numbers. Source code control systems work using graph theory (creating directed acyclic graphs), and they deal with text.
And like Bacterius mentions, there are an infinite ways you could relate any set of numbers...
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