Any sense of abstraction associated with a
physical reality will qualify as a model. Note that this is not a definition of
a model or even coming close to encompass all the different things meant whenever
the term ‘model’ comes up (Certainly not the Standard Model in physics).
A simple example would be to think of a city
which has some islands within it which are connected by bridges. This diagram itself suggests an abstraction
of reality which is why we would call it a model. Abstraction plays a very
important role is sciences namely it help think about specific aspects of the
given problem without referring to un-meanigful details irrelevant in the given
context. For the given system the problem is “Starting from any landmass (A,B,C
or D) can we commute in such a way so as to cross each bridge only once and
come back to the starting point (zero displacement).”
Now
before moving on,the reader must herself try to do so. A better diagram for the
given problem would be another level of abstraction so we make a new diagram.
Will it simplify things. Maybe.
Starting from A ,we can choose 2 paths,one
to C or to D, from C we can go to B or D and from D we can go to B or C. A to C
to D to A will leave B untraversed.hence we must go to B.From B we must go to C
and the only choice left is only other bridge we untraversed from C to A which
we choose and come to A. we missed bridge B to A. After trying several such
paths, we discover an important thing that if we leave from one landmass(vertex
henceforth), we need at least two bridges(edge henceforth) one for going out
and one for coming back. this criterion is obiously satsifed by every vertex.
But we notice a new difference here,that starting from any vertex if we
randomly choose edges(previously untraversed ones) and keep on travelling,from
a vertex having odd number of edges we can always move out from it and never
come back.Let’s take some random paths A->
C->A->D,A->D->B->C->A->C, we can never come back to A now,
since A has odd number of edges.
So at least in the given problem,we can say
there does not exist any such path as required by the problem.
This is also known as Konigsberg Bridge
Problem and was solved by Euler. Hence we demonstration the need and use of
model to solve physical problems abstractly.
A model incorporates some fundamental
phenomena underlying a given problem and tries to answer questions based on it
or simply check if it matches the observations. Not all the times, especially
in biological systems we know the underlying phenomena leading to the current
observation. Hence it becomes important to choose to up to what level of
details must be ingrained in the model and what kind of questions can it answer).
Modelling becomes useful when large numbers
of rules are required to play a game and a human would mess up very soon in
remembering all the details and how she should play.
“Mathematics is the sharpest tool of
rational thought. While there is not always a mathematical solution to every
problem in biology, the discipline of mathematics provides practice in clear
thinking: expressing one's assumptions, testing them out, and predicting their
consequences. Biological problems seem sooner or later to yield to mathematical
formulations.”
-
Richard Gordon.
References:
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