Linguistic study of math/physics equation "language"
January 8, 2010 6:26 PM Subscribe
I was thinking the other day about "all Greek to me!" as I was reading a physics book w/equations (using the Greek symbols)
And equations are a sort of language, of course.
So I wondered if there's some sort of linguist who's ever looked at the grammar or syntax of math/physics equations and tried to derive, whatever the hell it is linguists derive!
Does this sound like something anyone has heard of? If so, have any links?
It's not clear what you're trying to ask here.
Math--I suppose you mean algebraic expressions, or equations--can be considered a "language" in a technical sense, and you could specify a grammar for it, or a set of rules that say which strings of symbols are valid expressions/equations and which are not. But it's not a very complicated one, even compared to other formal grammars (for example, those of programming languages).
You could try to construct the same kind of grammar for a natural language like English, but it's very hard, because natural languages are much more complicated than formal ones.
posted by k. at 6:39 PM on January 8, 2010
Math--I suppose you mean algebraic expressions, or equations--can be considered a "language" in a technical sense, and you could specify a grammar for it, or a set of rules that say which strings of symbols are valid expressions/equations and which are not. But it's not a very complicated one, even compared to other formal grammars (for example, those of programming languages).
You could try to construct the same kind of grammar for a natural language like English, but it's very hard, because natural languages are much more complicated than formal ones.
posted by k. at 6:39 PM on January 8, 2010
I was just thinking about this the other day. Like, an RPN calculator is kind of SOV (subject object verb) while infix arithmetical notation is SVO. Most imperative programming languages are SVO but the subject is always understood to be the computer.
10 (COMPUTER) PRINT "HELLO WORLD"
20 (COMPUTER) GOTO 10
In a regular language, you can make a new word by cramming a bunch of roots together (or prefixes/postfixes, depending on the language). In functions you can make a new function by something like h(x) = f(g(x)) and so forth.
posted by DU at 8:42 AM on January 9, 2010
10 (COMPUTER) PRINT "HELLO WORLD"
20 (COMPUTER) GOTO 10
In a regular language, you can make a new word by cramming a bunch of roots together (or prefixes/postfixes, depending on the language). In functions you can make a new function by something like h(x) = f(g(x)) and so forth.
posted by DU at 8:42 AM on January 9, 2010
I'm not sure exactly what you're question is, but hopefully at least some of the following points you in the right direction.
Some of the big questions in the philosophy of mathematics (see the SEP entry) are in some sense linguistic questions.
What do mathematical sentences mean? One common approach to this is to look at the meanings of the parts of sentence. There's a fair amount that goes into this.
We need to figure out what the names refer to ("chndrcks" refers to me: chndrcks).
We need to figure out the predicates ("is a Mefite" : those of us with metafilter accounts fall under this predicate and no one else does).
"chndrcks is a Mefite" means chndrcks (the thing picked out by the word "chndrcks") falls under that predicate (for a basic understanding, think of it as being a member of a set).
Now we need to apply this to mathematics. What do "0" and "1" refer to? Do they exist independently of any mathematician? Do they exist but depend on minds? Do they not really exist at all? If they exist, what sort of thing are they? If they don't exist, how do we make sense of mathematical statements that we take to be true? What's the reference of mathematical predicates? etc.
If this is an area that interests you, Shapiro's Thinking about mathematics is fairly easy to read and quite comprehensive.
Anothere area you might want to investigate is Frege's reduction of math to logic: Die Grundgesetze der Arithmetik (Basic Laws of Arithmetic). Such attempts are known as Logicism (or sometimes neo-logicism). There are lots of connections between logic and language.
You might also look around the Foundations of Mathematics.
You can actually do a lot with a fairly basic mathematical language. For example, look at Robinson's Arithmetic:
Things (the Domain): Natural Numbers
Primitives:
  s: the successor function
  +: addition function
  .: multiplication
  <>, ~, E, A, =* - these are the standard logical conncetives
Now we can say the following axioms which make up Q Robinson's Arithmetic:
1) Successor function is 1 to 1
  AxAy (s(x) = s(y) -> x = y)
2) 0 is not the successor of any number:
  Ax (0 =/ s(x))
3) every number except 0 is the successor of another number
  Ax (x =/ 0 -> Ey x = s(y))
4) Any number plus 0 is itself:
  Ax (x + 0 = x)
5) For any two numbers, x and y, x plus the successor of y equals the successor of x+y:
  AxAy(x + s(y) = s(x+y)
4-5 give you addition
6) Any number times 0 is 0
  Ax(x . 0 = 0)
7) x times the successor of y equals (x times y) + x
  AxAy (x . s(y) = (x . y) + x)
6-7 give us multiplication
So for example:
2+2 = 4
s(s(0)) + s(s(0)) = s(s(s(s(0)))
1. ss0 + ss0 = s(ss0 + s0) (rule 5)
2. ss0 + s0 = s(ss0 + 0) (rule 5)
3. ss0 + 0 = ss0 (rule 4)
4. ss0 + s0 = sss0 substitution of identicals lines 2, 3
5. ss0 + ss0 = ssss0 substitution of identicals lines 1, 4
We can also define some further notions:
say y divides x if there's a number (z) such that y times z = x
  x | y =def Ez(y . z = x)
Then we can say when x is Prime: x is prime if every number that divides x (as defined above) is either x itself or 1 (and x must be greated than 1)
Pr(x) =def Ay (x | y -> [y = x v y=1]) ^ 1 < x
*Controversial, but acceptable
posted by chndrcks at 9:24 AM on January 9, 2010
Some of the big questions in the philosophy of mathematics (see the SEP entry) are in some sense linguistic questions.
What do mathematical sentences mean? One common approach to this is to look at the meanings of the parts of sentence. There's a fair amount that goes into this.
We need to figure out what the names refer to ("chndrcks" refers to me: chndrcks).
We need to figure out the predicates ("is a Mefite" : those of us with metafilter accounts fall under this predicate and no one else does).
"chndrcks is a Mefite" means chndrcks (the thing picked out by the word "chndrcks") falls under that predicate (for a basic understanding, think of it as being a member of a set).
Now we need to apply this to mathematics. What do "0" and "1" refer to? Do they exist independently of any mathematician? Do they exist but depend on minds? Do they not really exist at all? If they exist, what sort of thing are they? If they don't exist, how do we make sense of mathematical statements that we take to be true? What's the reference of mathematical predicates? etc.
If this is an area that interests you, Shapiro's Thinking about mathematics is fairly easy to read and quite comprehensive.
Anothere area you might want to investigate is Frege's reduction of math to logic: Die Grundgesetze der Arithmetik (Basic Laws of Arithmetic). Such attempts are known as Logicism (or sometimes neo-logicism). There are lots of connections between logic and language.
You might also look around the Foundations of Mathematics.
You can actually do a lot with a fairly basic mathematical language. For example, look at Robinson's Arithmetic:
Things (the Domain): Natural Numbers
Primitives:
  s: the successor function
  +: addition function
  .: multiplication
  <>, ~, E, A, =* - these are the standard logical conncetives
Now we can say the following axioms which make up Q Robinson's Arithmetic:
1) Successor function is 1 to 1
  AxAy (s(x) = s(y) -> x = y)
2) 0 is not the successor of any number:
  Ax (0 =/ s(x))
3) every number except 0 is the successor of another number
  Ax (x =/ 0 -> Ey x = s(y))
4) Any number plus 0 is itself:
  Ax (x + 0 = x)
5) For any two numbers, x and y, x plus the successor of y equals the successor of x+y:
  AxAy(x + s(y) = s(x+y)
4-5 give you addition
6) Any number times 0 is 0
  Ax(x . 0 = 0)
7) x times the successor of y equals (x times y) + x
  AxAy (x . s(y) = (x . y) + x)
6-7 give us multiplication
So for example:
2+2 = 4
s(s(0)) + s(s(0)) = s(s(s(s(0)))
1. ss0 + ss0 = s(ss0 + s0) (rule 5)
2. ss0 + s0 = s(ss0 + 0) (rule 5)
3. ss0 + 0 = ss0 (rule 4)
4. ss0 + s0 = sss0 substitution of identicals lines 2, 3
5. ss0 + ss0 = ssss0 substitution of identicals lines 1, 4
We can also define some further notions:
say y divides x if there's a number (z) such that y times z = x
  x | y =def Ez(y . z = x)
Then we can say when x is Prime: x is prime if every number that divides x (as defined above) is either x itself or 1 (and x must be greated than 1)
Pr(x) =def Ay (x | y -> [y = x v y=1]) ^ 1 < x
*Controversial, but acceptable
posted by chndrcks at 9:24 AM on January 9, 2010
hrm, the   looked ok on preview, those should just be spaces. Also, sorry for the "you're" typo in the first line
posted by chndrcks at 9:26 AM on January 9, 2010
posted by chndrcks at 9:26 AM on January 9, 2010
Response by poster: Thanks for all the excellent links!
Some of the more mathy stuff is definitely above my head, but this is an excellent reference to start from if I ever do look at the topic in more depth.
Thanks again!
posted by symbioid at 10:40 AM on January 12, 2010
Some of the more mathy stuff is definitely above my head, but this is an excellent reference to start from if I ever do look at the topic in more depth.
Thanks again!
posted by symbioid at 10:40 AM on January 12, 2010
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Quite a lot's been written about the history of mathematical notation: see The earliest use of Various Mathematical Symbols page, or Stephen Wolfram's essay Mathematical Notation: Past and Future. Unfortunately, the Wikipedia entry is comparatively short. If you want a book, the standard reference seems to be Florian Cajori's History of mathematical notations.
posted by James Scott-Brown at 4:05 AM on January 9, 2010