Whether using optical or non-optical telescopes, spectroscopy or interferometry, light is intrinsically involved in both the message (the astronomical data observed) and the various instruments used to detect, explore, and interpret that message. Regarding each of these functions, mathematics necessarily enters as an “instrument” without which the aesthetic purpose embedded within such “technological instruments” cannot be realized, nor the observations attain precision of meaning. As the French mathematician Henri Poincaré explained at the first International Congress of Mathematicians in 1897, mathematics must not only aim to provide a “mathematical instrument for the study of nature” (such as his New Methods of Celestial Mechanics); mathematics must also serve the “aesthetic purpose” of aiding philosophers and physicists “to fathom the notions of number, of space, of time,” for only mathematics can “express relations so delicate, so rich, and so precise.” When theoretical and technological advances compelled 20th-century physicists to interrogate a dynamic universe expanding from an initial singularity, cosmologists embraced immense conceptual and material challenges to make light a precise instrument for illuminating its cosmic messages. This talk will analyze several episodes in modern astronomy, relativistic physics, and observational CMB cosmology in which practitioners grappled with unresolved tensions within modern cosmology. We will see that an underexamined and often misunderstood issue is how to overcome the limits of geometric empiricism — an issue first posited in its modern form 130 years ago by Poincaré.