# [FOM] Axiomatization of sciences

José Félix Costa fgc at math.ist.utl.pt
Tue Jan 10 12:56:30 EST 2006

```TRIAL to discuss axiomatization of physics (I promised to continue after
return):

Richard Haney wrote:

«So it seems that the question of the truth of an axiom is a question of how
representative that idealization and abstraction is of some perceived
"reality" -- i.e., of some "enduring" pattern of experience.»

Let us take Landau's formulation of mechanics, without too many mathematical
details (we can add them later):

Axioms (inter alia):

A) Defining a inertial frame

1) Space is homogeneous.
2) Space is isotropic.
3) Time is homogeneous.

B) Principle of least action (a variational principle over the mathematical
Lagrangian function L(r,t,v)).

THESIS :: A free particle moves at constant speed.

PROOF:
The Lagrangean L can not depend explicitly on the position vector r by 1);
can not depend expliciply on time t by 3). Lagrangean L will then depend
solely on the velocity vector v. But L can not depend on the vector itself
by 2). Then it can only depend on the 'modulus' v^2.
Also we have that d_r  L  =  0.
Then
d_t   d_v   L   =   0
and if follows that d_v  L  =  constant. Conclusion: vector v = constant.

Remark. This is the law of inertia deduced.

Richard:

--- Is this an acceptable aximatization for you? ---

+++++++++++++++++++++++++++++++++++++++++++++++
J. Felix Costa
Departamento de Matematica
Instituto Superior Tecnico
Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL
tel:      351 - 21 - 841 71 45
fax:     351 - 21 - 841 75 98
e-mail:   fgc at math.ist.utl.pt
www:    http://fgc.math.ist.utl.pt/jfc.htm
+++++++++++++++++++++++++++++++++++++++++++++++

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