"help"
Hi all, I'm a bit confused here. Is algebra a formal language? They both seem to be rules that allow transformation from one sequence of symbols to another, yet wikipedia seems to consider them entirely different things. I would appreciate it if someone could explain the difference.
Well, no, algebra is not a formal language. Algebra is written in human language.
I guess a "formal language" is a set of rules operating on sequences of letters. In other words, in a formal language you make operations on letters without necessarily making any sense of it; that can also be programmed to a computer. At any point you are completely deterministic about what you can and can't do with your letters.
On the other hand, algebra is a completely human activity: we make sense of what algebra says. Using formal transformations of letters is simply a means of helping our understanding of the world.
Note that these formal transformations by no means cover all things that can be done in algebra. For example, if we say "Take an element x of the set S", we do not define the meaning of the word "element" and "set" - this is one of many manifestations of the fact that algebra is not a formal language.
I guess a "formal language" is a set of rules operating on sequences of letters. In other words, in a formal language you make operations on letters without necessarily making any sense of it; that can also be programmed to a computer. At any point you are completely deterministic about what you can and can't do with your letters.
On the other hand, algebra is a completely human activity: we make sense of what algebra says. Using formal transformations of letters is simply a means of helping our understanding of the world.
Note that these formal transformations by no means cover all things that can be done in algebra. For example, if we say "Take an element x of the set S", we do not define the meaning of the word "element" and "set" - this is one of many manifestations of the fact that algebra is not a formal language.
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From what I understood, Algebra can be expressed with a formal language but it is not a formal language. Don't quote me on this though.
| From what I understood, Algebra can be expressed with a formal language but it is not a formal language. Don't quote me on this though. |
However, you define a formal language in the usual human mathematical language.
Furthermore, a formal language is a sequence of letters following formal rules; however checking that a given sequence of letters is in fact following the predefined rules is again a human activity.
True, it can be programmed to a computer, however, the program itself is written by human too. Our trust in computers is again a completely human activity.
| however checking that a given sequence of letters is in fact following the predefined rules is again a human activity. True, it can be programmed to a computer, however, the program itself is written by human too. Our trust in computers is again a completely human activity. |
I am sorry, but what does this have to do with anything?
Well, the question was (after a minor interpretation) what is the difference between algebra and a formal language model of algebra.
A summarized version of my answer is: a formal model of algebra is a sequence of letters (written on a piece of paper or, say, on a computer screen). Algebra itself a mathematical discipline exercised by humans.
A summarized version of my answer is: a formal model of algebra is a sequence of letters (written on a piece of paper or, say, on a computer screen). Algebra itself a mathematical discipline exercised by humans.
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How is an equation non deterministic ? I only know (and currently am concerned with) elementary algebra.
But then, I guess I can see the differences between a formal grammar and algebraic transformations.
Though my real interest is whether algebra is comparable to natural language. Both have rules of structure and both express meaning, right? At first I figured that a formal language would be but now it seems that it isn't.
But then, I guess I can see the differences between a formal grammar and algebraic transformations.
Though my real interest is whether algebra is comparable to natural language. Both have rules of structure and both express meaning, right? At first I figured that a formal language would be but now it seems that it isn't.
I think that algebra is unseparable from natural language (it is one part of human expression).
If your question is whether you can model natural language as a formal language, we first gotta resolve the trouble of what exactly is natural language (note that it is different even between different people from the same country and same culture, or even when comparing a single person in different points in time).
To me it seems obvious that the natural language can be, to the extent that it can be defined, formally modelled.
If your question is whether you can model natural language as a formal language, we first gotta resolve the trouble of what exactly is natural language (note that it is different even between different people from the same country and same culture, or even when comparing a single person in different points in time).
To me it seems obvious that the natural language can be, to the extent that it can be defined, formally modelled.
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What exactly makes algebra like a formal language?
The definitions of formal languages/formal grammars/L-systems are given in algebraic terms but I don't really see how the other way round would work.
I think formal languages are a subset of algebra ( LOL )
The definitions of formal languages/formal grammars/L-systems are given in algebraic terms but I don't really see how the other way round would work.
I think formal languages are a subset of algebra ( LOL )
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You can convert a theorem such as "To every number p there exists a number coprime with p" to a formal string, something like
(\forall p \exists q ) (d | p & d | q -> (d==1)),
where the sequence d | p (read as "d divides p") should in turn be substituted by
\exists e (e*d==p)
In turn, a proof to the above formal statement can similarly be written with a sequence of cryptic letters. A proof can be verified by a computer by carrying out a predefined set of substitutions/simplifications.
A formal model of algebra can be defined as the set of all sequences of letters representing all provable theorems, together one sequence of letters representing one formal proof for each provable theorem.
(\forall p \exists q ) (d | p & d | q -> (d==1)),
where the sequence d | p (read as "d divides p") should in turn be substituted by
\exists e (e*d==p)
In turn, a proof to the above formal statement can similarly be written with a sequence of cryptic letters. A proof can be verified by a computer by carrying out a predefined set of substitutions/simplifications.
A formal model of algebra can be defined as the set of all sequences of letters representing all provable theorems, together one sequence of letters representing one formal proof for each provable theorem.
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| algebra is inseparable from natural language |
| What exactly makes algebra like a formal language? |
Algebra: a * (b + c) becomes a * b + a * c; Formal language: (given S->ab) Sc becomes abc.
I see how algebra is much more powerful, this I like your though about formal language being a subset.
@tition
Not all algebra is easily computable.
@hamsterman
In algebra the equality
Neither
( + and * may be defined differently for different sets )
Not all algebra is easily computable.
@hamsterman
In algebra the equality
a * (b + c) = a * b + a * c determines a property ( distributivity ), not a transformation.Neither
a * (b + c) nor a * b + a * c define the meaning of + or *, given the definitions of + and * you can say that the property a * (b + c) = a * b + a * c is true.( + and * may be defined differently for different sets )
@Bazzy
... and that is why I am not jobless... ;) noone here says it's easy... ;)
Btw, without having read any articles on the subject, I know there are complexity issues with formalizing mathematics (i.e. you can't do it effectively with a simple algorithm).
In other words, a mathematician's job can be thought of "improving the world's algorithms for producing and verifying mathematical theorems".
@ hamsterman
in your examples of rules, you missed some of the most common mathematical operations: "there exists x" and "for all x". Those can also be written with formal symbols (\forall and \exists in LaTeX, the commonly used mathematical typesetting environment).
| Not all algebra is easily computable. |
... and that is why I am not jobless... ;) noone here says it's easy... ;)
Btw, without having read any articles on the subject, I know there are complexity issues with formalizing mathematics (i.e. you can't do it effectively with a simple algorithm).
In other words, a mathematician's job can be thought of "improving the world's algorithms for producing and verifying mathematical theorems".
@ hamsterman
in your examples of rules, you missed some of the most common mathematical operations: "there exists x" and "for all x". Those can also be written with formal symbols (\forall and \exists in LaTeX, the commonly used mathematical typesetting environment).
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