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Philosophical questions (or even just a matter of taste) regarding some mathematical constructions can give rise to reformulations of whole theories, for example, we can develop (Non-standard) Analysis from the beginning based on a rigorous definition of infinitesimal number. We have example of this outside of Mathematics as well, for example, Lagrangian Mechanics is a reformulation of Newtonian Mechanics based on the principle of stationary action.

Now to my $\textbf{1st question}$: What else change in a reformulation of a theory besides the new viewpoint ? Take Non-standard Analysis, is there any theorem that is valid in "Standard" Analysis that is not valid in "Non-standard" Analysis, or the other way around ?

$\textbf{2nd question}$: What about more serious change in the approach to the whole of Mathematics, for example, a finitistic approach to all of Mathematics ?

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    $\begingroup$ Perhaps you misunderstand what non-standard analysis is about. It did not come about through a new theory to make infinitesimals rigorous, but instead is guaranteed by technical machinery of model theory. In a sense, we can find a structure that has "infinitesimals" in it and it satisfies the same first order properties as the real numbers. We are not changing the theory of the reals at all, we are just finding a new structure that satisfies the same things as the "standard" structure. And as for 2, you may want to be a bit more precise. For example, (in general) proofs are finite... $\endgroup$
    – Ryan Sullivant
    Feb 26, 2014 at 5:04

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Many mathematical theories are incomplete, and so admit multiple non-equivalent models. In many cases, one model is selected as the standard model. In such a case, other models are nonstandard models. Models of nonstandard analysis are examples of this scenario.

In the case of nonstandard analysis, the model satisfies the same first order formulas as standard analysis; to define the differences between the models, you must use a second (or higher) order language.

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