The symbols we use in mathematics to form equations are just an aid in clearly forming an argument and communicationg it to others. We are clearly restricted when we use this formal language. If we could only cast out any mention of this language and symbols when doing mathematics, then we would be on the right track in truly understanding reality's ways.
The notion of quantity, form, change, space, shape, order, etc. are all independent of their symbolic representation. The language can easily change trough time, but these notions will not.
Computation as we know it, is merely a formal manipulation or transformation of symbols. It can be done by hand or by a computer. Either way, there is always a notion of a conciever and an executor present, when talking about computation. These two are usually one and the same, but I like to think about them as separate entities. The executor, follows a fixed set of rules to transform given string of symbols, that a conciever has conceived having some end goal in mind. The executor blindly follows these rules and eventually, (if he's in luck and didn't get stuck somewhere blindly following the rules),he will get a transformed string of symbols representing the final result.
And the conciever is the one that anticipates this result, again as a string of symbols.
So, when doing computation, the main assumption is that, when we manipulate symbols, we manipulate the notions that they represent. Just like in the primitive times, when people practiced magic, they believed that the symbols they use in their spells represent objects from the real world.
They believed that drawing these symbols in some special sequence will result in a spell being cast, and as a result something in the real world will change according to the spell's intention. So, in an amusing way, doing mathematics can be regarded as "doing magic", not in the real world, but in the world of ideas.
Computers process strings of symbols by following a fixed set of rules that we call a program. The conciever is the programmer, and the executor is of course the computer. The processing by a computer is usually done in a one-by-one
fashion, but is much faster that doing it by hand. Computers can be seen as manipulators of symbols, or executors of programs, but the acctual thing we are after is the "manipulated" idea after the computer has done millions and millions of manipulations on it (that would be too tedious to do by hand).
So "ideas" are the ones that we are after when doing computation, because we hope that this mechanical grinding away of symbols will tell us something new and interesting about reality and nature, although this point of view was refuted a hundred years ago by Godel's famous incompleteness theorems. These theorems show that there is definately something more to mathematics and computation than just "symbol grinding". Remarkably, Godel showed this using only using some basic facts from NUMBER THEORY, nothing fancy.
And what about nature and reality ?
What are nature's rules, and what "language" is used to set these rules ? Nature is the executor, but who is the conciever ? And what is the final result ? Is it LIFE maybe ?
The answers to these questions are certainly beyond human comprehension, but there is, as always a lot if speculation about it! But, when we finally find this out, only then we can make a significant progress in truly understanding this "manipulation of ideas" notion and and "reality's ways" in general that mathematicians are still desperately and vaguely trying to capture by the notion of "computation".
Zafar, thank you for your insightful comment, I actually never saw this as a geometrical "embedding problem", nice!
About making progress, well computers know "what" to do but not "why" they do it., and in that sense, computation is "blind", so human insight is needed. On the other hand, because computers really don't "question their motives" (at least not yet) this makes them useful for doing long and tedious tasks.
Although I don't fully understand all this, I agree with the statement that doing math is like magic, not reality, but a world of ideas. Very well said.
Dear Alana, thank you for your comment :) it's always nice to see someone from high school reading and commenting posts on Research Gate, the youth is our future :) Cheers
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