The mechanistic paradigm, also known as the Newtonian paradigm, assumes that things in the environment around humans are more like machines than like life.  It was more common in the 19th century.  This is a set of loosely related beliefs that affects all sciences:

In physics it presents atoms as made of particles in preference to say the wave theory of light.  In fact, both the particle and wave view are required to explain everything light does, but are rarely presented as equals.  Many physics students graduate fully understanding the hydrogen bomb but having no clear idea of how musical instruments work.
In astrophysics it assumes that the universe is like clockwork and works on its own without our choice making any difference - this fits relativity but is not easy to combine with quantum mechanics.
In cosmology it accepts models of our universe more easily if reject the continuous creation of matter, energy or any new "other universes".
In chemistry it assumes that molecules are like building blocks, and have no unpredictable or strange and unique effects when combined.  This is the basic assumption of mechanosynthesis which some scientists think will create a molecular assembler.
In biology it assumes that everything about life - all of biochemistry - is predictable from chemistry and physics.
Believers tend to regard ecology and psychology as "less scientific" than physics, chemistry and biology.
Believers may reject ideas like psychoneuroimmunology, Gaia philosophy or Fecund universes for no scientific reason, simply because these theories assume "higher level phenomena" sometimes drive lower levels.
Greedy Reductionism may be more likely among believers in mechanistic ideas.  For instance, they may want to always see things as made of smaller, even invisible, parts that no one can see or prove is real - like in string theory.
For this reason, when they work in psychology, they may prefer theories like behaviorism that deny free will and try to explain human behaviour as a function of biology.
The philosophy of mathematics of most believers is a form of Platonism.  This assumes there is a perfect or ideal form that theories only approximate.  This lets them explain inaccuracies in mechanistic theories as being due to an imperfection in human ability to reason, instead of imperfections in math itself.

Believers in this paradigm sometimes say that those who do not believe in it are following a cognitive paradigm - but almost no one uses this term, since it is redundant - cognitive science is already accepted as the most basic idea in the philosophy of science.  But mechanists reject some of the ideas of cognitive scientists, like cognitive science of mathematics.

Mechanistic thinking also assumes that philosophy of perception is much less important than cognitive scientists say it is - that humans and their beliefs and equipment do not generally add a lot of bias to a scientific theory.  Thomas Kuhn said otherwise, that these things matter, and that the major assumptions of science, can shift drastically.  This he called a paradigm shift.  The shift from mechanistic to cognitive paradigm is an example of this.
Later he used other words to describe the assumptions and beliefs, like mind-set, but the word "paradigm" is still used.  Some say it is much over-used.

Economics is often said to "suffer from" assumptions of the mechanistic paradigm.  Sometimes those who believe in neoclassical economics and also in the mechanistic paradigm say they "seek to unify physics and economics," as if people and particles behaved as two examples of the same kind of thing.

Technology is often easier to make if people accept a mechanistic paradigm - but it may be harder to say why it does not work, if one believes in these ideas.  For instance, creating diagnostic trees might be easier if one works from experience, not from an idea of how a technology should or must work.

A controversial idea is that mechanistic ideas are just an older idea called scholasticism, with more mathematics.  Both tried to work from what should or must be, instead of what experiment seemed to show.

Another controversial idea is that scientism, belief in science as if it were a religion or ethical tradition, comes from this paradigm.  Most scientists who are mechanistic do not say they see science as a guide to ethics, but try to keep them separate.

Physics
Chemistry
Biology