syntax and semantics. Meaning? The birth of the meaning WILARD parte 8: sintassi e semantica.
Vedremo come sintassi e semantica sono tra loro intrecciate anche in matematica. E cercheremo di capire cosa vuol dire valore intrinseco ed estrinseco dei simboli. What is meant by the meaning of a symbol
in terms of algebraic
? (See
The birth of the meaning). Does anyone remember the
boring math classes in high school including "notorious" considerable product-algebra, in particular the one that follows.
(a + b) ^ 2 = a ^ 2 + 2ab + b ^ 2 (1.)
In algebra
long chains of symbols can be transformed so mechanical . The right side of Eq .1 may be replaced by the left side, and vice versa. If we had a denominator of an expression - eg. -A similar term, we could simplify
a^2 + 2ab + b^2
--------------------- =
( a + b )^2
(a + b)^2
---------------- = 1 (2.)
(a+b)^2
Ricorderete i minuziosi passaggi, necessari per ridurre una espressione lunga e complessa, ad una più semplice formata da pochi segni. Questo è quel che si intende con forma
sintattica della matematica. O algebra.
In algebra un simbolo non ha un significato in sè e
per sè , viene trasformato seguendo
regole e
prassi estrinseche, i.e. pratiche che prescindono il contenuto del simbolo . Cosa rappresentino "a" e "b" o "a^2" "b^2", se siano alberi, pears, even numbers, Istat, person, or mini-colored pony, not a question.
The expression (1.) Is not a numerical equality, but a logical equivalence: the chain of symbols on the left is a perfectly and logically equivalent to the right. What do you mean the component parts and why they are equivalent, it is not so important from the standpoint of algebraic.
The expression (1.)
in fact was known to surveyors (
mathematical ) at least 2000 or perhaps 3000 years, with methods and practices that are not appealed the magical power of algebraic rules, but the geometry. The geometry, unlike algebra, based its meaning on the intuition that we have real space: for instance the concept of continuity, which goes beyond a formal definition, and that permeates the analysis.
What is
the expression (1.) Geometry in
? Sometimes it is a complex numerical problem back to geometry, but in this case just draw a segment
s which is the sum of two segments in
and
b , and "make the square". That is, build a square shape
with
base and height
s = a + b . The area is numerically speaking
s ^ 2 , and thus
(a + b) ^ 2. Algebraically speaking,
s ^ 2 = (a + b) ^ 2 
But what is s ^ 2 from the geometric point of view, in this case? S and draw the square by joining the points that divide s in both segments to and b, from side to side, we note that is formed by four symmetrical geometric figures: two squares and two rectangles . If we calculate the area of \u200b\u200bthese four figures, as is evident their bases and heights (are rectangles and squares, their area is given by multiplying the base by the height) we see easily that have value a ^ 2, b ^ 2 , ab, ab .
Since these four areas cover the whole area of \u200b\u200bthe square, by construction, is easy to see that the area of \u200b\u200b s ^ 2 is equal to the sum of its parts a ^ 2, b ^ 2, ab, ab, namely:
s ^ 2 = (a + b) ^ 2 = a ^ 2 + b ^ 2 + ab + ab (3.1) ie (a + b) ^ 2 = a ^ 2 + 2ab + b ^ 2 (3.2)
Just as indicated in the expression ( 1.). Ultimately, the
syntax is not a game purely mechanical and abstract syntax hides a semantics. The semantics above, and in turn
explains the syntax. QED.
But if the two expressions, and the two worlds are completely equivalent, so why waste time learning the meticulous and tedious algebra when the geometry is so easy and intuitive to understand. Probabilmente se lo sono chiesti in molti questo, per primi gli algebristi italiani del '500, e la risposta è molto semplice.
Se è facile riportare (a+b)^2 ad un modello geometrico
bidimensionale , e in modo relativamente facile è possibile riportare l'espressione (a+b)^3 ad un modello
tridimensionale intuitivo, non è possibile creare un modello intuitivo
n-dimensionale dell'espressione (a+b)^n, ad esempio non è possibile costruire un modello geometrico
quadridimensionale intuitivo dell'espressione (a+b)^4, nè soprattutto è possibile costruire un metodo generale valido per un
n qualsiasi fissato a piacere.
The algebra with its mechanisms repeatable and extensible disrupts the barriers of intuition to solve problems that otherwise would not be approached.
Believe it or not, information is based on the same principle, is born from the same philosophy: logic and reasoning in the sense understood not as Intute and content, but as mere syntactic manipulation of symbols: computers manipulating symbols, exemplified by binary numbers, the popular expression is that "computer programs are run." These symbols carry a content, which has been mechanized, and then ignores their semantic aspect, but without wasting it: hides the syntax semantics, and in some ways surpasses, the crosses. To view of many, and many calculations, the syntax is stopped, the calculations are close, the program ends, the syntax and semantics is inspired. Or output data or representational form. Image, for example. O word.
Believe it or not, the first computer was conceptually formalized by A. Turing in 1936, defined in a purely abstract and conceptual as pure mathematical tool, and some of the most important problems of information technology including the first of 7 problems of the millennium, have been formulated eight years before the first real computer was built. The
computer from this point of view is purely abstract science. Perhaps the most abstract and most difficult of the sciences. Of course, the most giovane e la più moderna.
