G-M argues "that mathematics is not really a science at all, if a science is understood to be a discipline devoted to the description of nature and its laws: "Mathematics is more concerned with proving the logical consequences of certain sets of assumptions." We obtain from the above quote definitions and context for describing what he considered the fundamentals. The quotes also suggest methods of ranking.
We need a definition of 'fundamental' and to answer the question: 'Fundamental to what?' before making and ranking a list. Context can alter what is considered important. That which is fundamental will be relative to the individual and circumstances.
Most fundamentals are described by models, or other types of analogy. It is prudent to think of any model as a sort of bead that can slide along a wire. At one end is 'certainty = 1' and the other is 'impossibility = 0'. Almost all useful models describe conditions that are somewhere between 0 and 1. The other useful idea is that we tend to view situations from somewhere between the beginning and the future, i.e. we rarely start at the absolute beginning and it is not possible to proceed beyond the present. If we use historical information as a basis we are somewhere between the present and the beginning.
Each living individual engages in activities. Fundamental to living is control of activity, especially the 'next activity'. The prior activities are prologue. Time irreversibility decrees that the past is unchangeable. The relative importance of knowledge should therefore be judged by relation to control of future activities.
Personal well being and physical survival are of major importance. In well developed socio-economic circumstances these are generally available to many of the younger members of the society. However with advancing age there is personal deterioration so that more resources are required to sustain high levels of survival and well being. In many situations significant portions of the population experience low levels of survival and well being. Peace is supposed to be a desirable state for improving conditions..
In the most highly developed nations, and for the wealthier segments of most societies, basic survival and well being is largely supplanted by competitive goals such as keeping up with or exceeding levels enjoyed by contemporaries. Some individuals work toward goals of personal power and apprehended leadership.
The answer to: 'what should be the next action' seems to be be the place where most mechanisms begin control of activity.
Physics is a good place to start thinking about fundamentals in the modern world where physical things play such a large role. Physics attempts to describe the smallest particles and largest systems and their interactions. Good descriptions (Laws) allow predictions of what will happen next. Physics is just a name for a collection of activities. Which activities should be labeled 'physics' and what they encompass are evolving and depend on contemporary status and arbitrary categorization.
Some of the key notions included in contemporary physics have changed in the preceding century. Since the adoption of quantum and relativity theories the ideas that it should be possible to predict what will, or has happened with certainty have been abandoned in favor of predictions with some level of probability . The other important idea contained in quantum mechanics is that there are levels below which things are not divisible. The continuous models models of mathematics are only approximations that smooth out the discrete effects. Before quantum mechanics were accepted it was believed that predictions could be made with certainty if the correct law, i.e. model was known and used. Similarly laws stated as continuous models tend to leave notions that things may get very large or small.
Quantum mechanics suggests that there are limits to smallness and level of certainty. The idea that if you know the start point that you can predict any intermediate stage with certainty has to be abandoned in favor of prediction with some level of probability. This change probably also means that predictions of past states are also uncertain. i.e. filling in the gaps in histories may be only predictable with some uncertainty. Similarly what may happen in the future can only be predicted with some level of probability, not with certainty. Darwin and others have articulated the mechanism of selection and survival.