Cosmological variations in the fine structure constant, $\alpha$, 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 $\alpha$ 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 $\alpha$. 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, $R \gt 150000$, will achieve a 2-to-3 order-of-magnitude precision increase in $\Delta\alpha/\alpha$. 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.