There is a difference between the pre-scientific notion of uncertainty and the mathematical notion of probabilistic model,

There is a difference between science and that what lies before science, and a reduction of the world to what scientists have to say about the things around us, obscures the fact that the world is not a collection of things to which we, as subjects, have, miraculously , access. And of course this is not a "fact" but a worldview, that is highly debatable.

I am a mathematician , I know about modern physics, and I think that the works of Martin Heidegger should be read by scientists .

His works at first make the impression of being obscure, but this first impression is debatable as well .

Kind regards. Johannes Willem Nieuwenhuis.

I'd say that uncertainty and probability are just words to describe useful perspectives on natural situations where determinism either doesn't apply (though there may not be any such situations, strictly speaking..) or where determinism is simply useless in practice.

"Determinism" is useful in predicting things like planetary orbits or tides, where nonlinear interactions are weak enough to ignore. It's not that useful, however, in many nonlinear dynamical systems (e.g., weather, turbulence) which may be deterministic, but where the extreme sensitivity to initial conditions (that we can never know precisely enough) makes them "chaotic". Remember the title of Lorenz's original paper on chaos was titled "Deterministic nonperiodic flow". So meteorologists have had to resort to probabilistic analysis of ensembles of numerical predictions to make their forecasts, which they will readily admit have a degree of "uncertainty".

As useful tools for dealing with the world, neither AI nor quantum mechanics are all that new - quantum mechanics has been around for a century. Other probabilistic models (like Mendelian inheritance) have been around even longer.

Between them, Heisenberg's uncertainty principle and chaos theory have provided science with a good appreciation for the limits of "exact answers and precise predictions".

Yes, it's true :)

عند وضع عتبات تقارب صارمة جدًا، نقلل من الخطأ العددي وعدم اليقين الناتج عن الحساب.

لكن دائمًا يبقى هناك احتمال لعدم الوصول للتقارب المثالي بسبب حدود الطرق العددية أو جودة الدالة المستخدمة (DFT functional أو basis set).