René Descartes' philosophy gave rise to the Cartesian model of knowledge, which suggests that arguments which are mathematically certain are preferable to others. Many rhetorical theorists, espoused the model of rhetorical probability.
Which of the neoclassicists and epistemologists made the best case for his view? And why would it be considered best?
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Many contemporary philosophers would have problems with Descartes' dualism. But they would also probably agree that a mathematical proof is more certain than say a theory of social behavior. So, stemming from this idea that mathematically certain theories are preferable, subsequent philosophers would try to provide more certainty to fields of knowledge including and in addition to mathematics.
Spinoza was influenced by Descartes, although he did depart from Descartes' mind/body dualism. Spinoza argued that mind and body were one, that God and Nature were one (pantheism), and that all things were made of the same substance and subject to the same laws. This led Spinoza to a view that everything was determined. Things happen out of necessity, being based on physical laws. Although Descartes' work on mathematics is sound, he never provided an equally mathematical or physical explanation for how the mind interacts with the body (how an abstract can affect material). Spinoza's monism (body and mind are of the same substance) provides a more sound argument for how minds interact with bodies.
In the neoclassical/Enlightenment period, perhaps the two most important philosopher/scientists were Newton and Liebniz. Both developed calculus. Newton developed his three laws of motion based on math and physics. These laws would describe motion including the paths of the planets (until Einstein's theories of relativity and gravity would surpass them). Newton's third law, for example, is mathematically driven:
When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.
You push on a wall with x newtons of force and the wall "pushes" back with the same amount. Equal and opposite reactions: similar to the balance of a math equation.
Newton also contributed to the development of mechanical philosophy. This is the view that all things function like machines. The whole (machine, let's say a living thing) is more complex and complicated than its individual parts. Hence, the idea that the whole is different than the sum of its parts. This is a qualitative notion of the whole based on the quantitative (numbers) function of its individual parts. Subsequent thinkers would refine mechanical philosophy (including later materialists). One of Newton's problems was that he could show the effects of gravity with his laws but he couldn't really explain how gravity works. Among many other thinkers following Newton, Laplace, who would come about a century later, expanded on the idea of mathematical certainty. He is known for supporting the view of scientific determinism wherein everything can be attributed to some cause. Thus, there should be a formula or equation for any event.
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