Digital electronics is a technological cornerstone in our modern society which has covered the increasing demand in computing power during the last decades thanks to a periodic doubling of transistor density and power efficiency in integrated circuits. Currently, such scaling laws are reaching their fundamental limits, leading to the emergence of a large gamut of applications that cannot be supported by digital electronics, specifically, those that involve real-time analog multi-data processing, e.g., medical diagnostic imaging, robotic control and remote sensing, among others. In this scenario, an analog computing approach implemented in a reconfigurable non-electronic hardware such as programmable integrated photonics (PIP) can be more efficient than digital electronics to perform these emerging applications. However, actual analog computing models such as quantum and neuromorphic computation were not conceived to extract the benefits of PIP technology (and integrated photonics in general). In this work, we present the foundations of a new computation theory, termed Analog Programmable-Photonic Computation (APC), explicitly designed to unleash the full potential of PIP. Interestingly, APC enables overcoming some of the basic theoretical and technological limitations of existing computational models, can be implemented in other technologies (e.g. in electronics, acoustics or using metamaterials) and, consequently, exhibits the potential to spark a ground-breaking impact on our information society.