Cosmological variations in the fine structure constant, α, can be probed through precise velocity measurements of metallic absorption lines from intervening gas clouds seen in spectra of distant quasars. Data from the Keck/HIRES instrument support a variation in α of 6 parts per million. Such a variation would have profound implications, possibly providing a window into the extra spatial dimensions required by unified theories such as string/M-theory. However, recent results from VLT/UVES suggest no variation in α. The COsmic Dynamics EXperiment (CODEX) spectrograph currently being designed for the ESO OWL telescope (Pasquini et al. 2005) with a resolution high enough to properly resolve even the narrowest of metallic absorption lines, R > 150 000, will achieve a 2-to-3 order-of-magnitude precision increase in Δα/α. This will rival the precision available from the Oklo natural fission reactor and upcoming satellite-borne atomic clock experiments. Given the vital constraints on fundamental physics possible, the ELT community must consider such a high-resolution optical spectrograph like CODEX.