One of the main features that distinguishes modern and early medieval physics is the mathematical structure of modern physics. The idea that physics ought to be described in geometrical and mathematical terms dates back to the ancient Greeks. Whether he was the first to do so is less clear, but the Greeks strongly associated the idea with Pythagoras. Plato was particularly influenced by the idea, and his academy were probably the first group of people to have a serious attempt at developing a mathematical physics.

They failed.

On the other side of Athens, another man was developing his own system of philosophy. Aristotle, though a student of Plato, and similar enough in most aspects of his thought that the Romans classed him as part of the same school, taught that physics should be about structure and causality. At the heart of his thought was the idea that to explain the phenomena of change, a being could exist in numerous different states (or potentia as he called them). Change was the excitation of matter from one state to another (one of the potentia was actualised). Physics becomes, in part, a description of which transitions are possible, under what circumstances they are likely to occur, and which transitions occur naturally on account of the inherent tendencies of matter towards particular goals. This list of allowed transitions and interactions allows us to understand the properties of the beings. Whether something is hard or soft would depend on its natural tendencies under compression; whether it is red or blue would depend on its natural interactions with the colours of light. In particular, each system of beings would have a natural tendency towards its natural state (or what we would call its equilibrium state). Bodies made of earth fell because they had a natural tendency towards a resting point at the centre of the earth. Aristotle's physics and astronomy, as refined by the later Greeks, was immensely successful. It accounted for almost everything. Therefore it was accepted as being almost certainly true. The mathematical approach was laid aside.

But not forgotten.

In fourteenth century Oxford and Paris, a group of scholars wondered whether Pythagoras and Aristotle could be reconciled. Whether it was possible to develop a mathematical representation of Aristotle's physics. Physical space, they believed, could be represented by geometrical space. Events in space could be mapped to geometrical points. Trajectories of matter could be represented by geometrical lines, following fixed rules according to the will of God, regularly directing matter through the secondary causes, its natural tendencies.

They almost succeeded.

And where they failed was not in their mathematical approach, but their application of it to Aristotle. Anomalies were found in Aristotle, places where the physics failed to match up with observation. Pendulums, the details of projectile motion, the movements of the planets: it became clear that Aristotle's description of physics was lacking. So the medieval natural philosophers improved on it. They introduced the idea of impetus, whose magnitude was a body's mass times its velocity, and said that it was the impetus of matter that gradually changed when a body was thrown into the air. Thus they were able to gradually improve physics, while still remaining Aristotelian in other areas of their philosophy.

But there was a tension.

It began to look increasingly as though the universe was a great machine, with matter governed not by internal tendencies but external laws. The only change that the mathematical approach could capture was movement from place to place, so that became the only type of motion accepted. Aristotle's vision of potentia became redundant, as internal change within substances was seen as impossible. The fundamental building blocks of matter, corpuscles, were indestructible, constant, and different arrangements of them were seen as being sufficient to account for the vast array of objects around us. God was reduced to being no more than a lawmaker, and then an idle observer of matter. Thus Aristotle's philosophy was being challenged by the mathematical mechanical world view. But the two lived side by side. Mechanism was just the surface picture: Aristotle still provided the foundation.

The high medieval philosophers almost made the key breakthrough into modern physics.

But, then, disaster. The black death and wars decimated the universities. The next generation of scholars did not understand the achievements of their predecessors. They scorned both the Aristotelian tradition, and the mathematical approach. Instead they contented themselves with alchemy and astrology, being content to make observations, plot them on a graph, and join them up without looking to gain a deeper understanding. The renaissance was a backwards step in philosophy and the theoretical understanding of physics. It almost cost European scholarship everything. It abandoned the possibility of knowledge of theory.

But it did emphasise and improve the empirical method.

It was an Italian, Galileo Galilei, one of the few of his generation well trained in the medieval mathematics, who fully realised that mathematical physics and empirical physics were not enemies, but allies. He combined them, to devastating effect. Alongside Descartes' advances in geometry, and Tycho's and Kepler's astronomical observations, he destroyed the Aristotelian physics once and for all.

And Galileo and Descartes were avowed mechanists, accepting no compromise with Aristotle.

From Galileo and Kepler, the scientific revolution spread back to protestant England. Barrow, Boyle, Hooke, and others lead the way; Newton provided the finishing flourish, both in developing the mathematics, and applying it. The laws of motion were established; the mechanical principle governed everything. Gradually more and more more fields of physics were added to the purview of mechanical physics. By the end of the nineteenth century, with mechanical theories of thermal physics, electricity, magnetism and optics complete, it seemed as though nothing could stand in its way. With rapid progress in chemistry, and Darwin's theory explaining the development of life in terms of mechanical principles, nowhere in science was left untouched by mechanism.

The task of making philosophical sense of the new science was left to the philosophers.

The philosophers were divided. Descartes was there at the start, but though he believed that the physical world was governed by mechanical principles, he realised that this could not explain the mental world, where we give meaning to words and thoughts, meaning which cannot exist in a purely mechanical system. There were those who were committed to the empiricist science, as though Galileo had never occurred, such as Berkeley, Locke and Hume, who recognised that without Aristotelian natural purposes, there was no good foundation of ethical thought: no way of getting from a fact to a purpose and consequently a value in a mechanical system. Kant tried to bridge the gap between the two, and found himself questioning how our knowledge related to reality. Marx applied these theories to social and political sciences, but was blinded by the idea of class conflict. The social sciences, the theologians, the educational theorists, they all tried to jump onto the bandwagon of the success of the mechanical science. The Aristotelian concept of natural purposes and functions was banished from all corners of the academy.

And where did this all leave God?

God had moved from being the sustainer of all the universe, to the sustainer of natural laws, to the creator of natural laws, to an innocent bystander, to an unnecessary appendage, to something which didn't exist at all. Western society fell into a practical, if not yet conceptionally accepted, atheism. The religious, of course, still held onto their miracles; but the academic elite laughed at them. What evidence could be strong enough to accept observations and events which challenged their 'scientific' philosophy? After all what alternative was there to the mechanism/dualism/empiricism/Kantian duality/relativism/existentialism/philosophical anarchy (delete as applicable) which each individual philosopher and natural scientist believed to be firmly established by science?

But physics had other ideas.

In the early part of the twentieth century, experiments started showing results that were inconsistent with the mechanical philosophy. Objects carrying energy and momentum were found to be neither particles nor waves, but something with some properties of particles and others of waves. It was found that experimental results were fundamentally unpredictable, not because because there were unknown variables in an underlying classical system, but because there is no underlying classical system. Nature appears to be fundamentally indeterminate. It is no longer sufficient to specify a particle's energy and momentum to determine its state, indeed it is impossible. Instead matter can exist only in a number of discrete energy levels or states, identified by their momentum, angular momentum, spin and a few other quantum numbers, which do not include the particle's location. Only certain transitions between these states are permitted; some have a natural tendency to spontaneously occur. The properties of matter are computed from knowledge of the underlying states. It is impossible to state all the possible properties of a being simultaneously; if some are known, then others are wholly undetermined.

A disconnect starts to emerge between modern physics and modern philosophy.

The first attempt at constructing a theory describing these results was quantum mechanics. The various formulations of quantum mechanics preserved as much of the mechanical philosophy as they could. Particles were still seen as being indestructible. Instead of the particles's position being described by a deterministic differential equation, now it had a probability amplitude that evolved deterministically. But, though successful in many respects, quantum mechanics still did not fully match experiment. So, an alternative was constructed, quantum field theory, which discarded the remaining embers of mechanism. There are no differential equations describing a deterministic evolution of anything. New particles of matter are continuously created and destroyed. The probabilities of a future state of matter are computed by computing all possible ways the universe can evolve from state A to state B, with each way signified by the different particles being created and destroyed. There are no forces or potential energies as Galileo and Newton understood them. The only principle of mechanism that remains is that space and time can be mapped to a geometrical space. Quantum physics is thus fundamentally incompatible with enlightenment philosophy.

But it fits in very well with Aristotelian philosophy.

Not perfectly well; Aristotle's philosophy needs to be modified to fit in with a geometrical view of space and time. So I develop a precise mathematical definition of form, and show how Aristotelian causality can be expressed in terms of operators acting on a Hilbert space. While modern forms of causality are inconsistent with quantum physics, classical causality is mandated by it. Every aspect of Aristotle's natural philosophy has an analogue in quantum physics.

So where does this leave God?

The standard objections against God bring in presumptions from failed enlightenment philosophy, and have no force in classical philosophy. But the classical arguments for God are given new strength. Defining God as an uncreatable creator, we can show how His standard attributes, and His existence, can be carefully deduced from the underlying metaphysical principles derived from quantum field theory. In particular, the principles imply a non-mechanical understanding of physics: that it is a description of how God sustains the universe. The metaphysical principles lead to a solid grounding for objective ethical reasoning, contrary to most of the ways people in modern society tend to ponder morality.

But can we understand physics from knowledge of God?

Quantum field theory can be constructed from a small number of axioms, among them: indeterminacy, re-normalisability, locality, cluster decomposition and various local symmetries of the action. Together, these give the basic mathematical form of field theory, leaving only some parameters and the numbers of each type of particle to be determined and measured through observation. If we assume that physics is a description of God's upholding of the universe, and that God has the attributes assigned to him by classical theism, then we can deduce the fundamental axioms of quantum field theory. If we further assume that God desired to create a universe which supports complex and rational life, then the parameters which can't be otherwise computed are constrained to very narrow set of values - which agree with our measurements. Thus from accurate knowledge of physics, we can deduce the existence of God, and from accurate knowledge of God, we can deduce the form of physical law.

But is more direct evidence for God possible?

A miracle is best defined as a singular event which provides evidence that God is not indifferent to mankind. The standard arguments against miracles presuppose a mechanistic understanding of physics. In those frameworks, a miracle (defined incorrectly as a breaking of the laws of physics) cannot occur. But in theism, everything is under God's direct control, without the intermediary of physical law. He can do what He likes, including special acts of providence to benefit certain people or mankind as a whole. So the theistic perspective can account for all the experimental evidence underlying quantum physics, and also the various pieces of evidence supporting miraculous events. The mechanist has an alternative model of physics, which might explain the scientific observations, but which contradicts the evidence for miraculous events. Since theism explains all the evidence, and non-theistic models of physics at best require highly implausible excuses to account for some of it, clearly theism is to be preferred.

It's just the scientific method of hypothesis, deduction, and comparison against observation.