Topic: Physical Insight, Mathematical Formalism, and the Limits on our Knowledge of the Universe
Abstract: Mathematics might best be described as the science and the language of  patterns. Natural science, on the other hand, might best be described as the search for patterns in the midst of the complexity of nature. Indeed, science sits squarely between the practical world of observation and experiment and the more abstract world of mathematics.  Mathematics often leads science to new insights, while science often demands of  mathematics new tools.  Physical intuition guides mathematics, while mathematics moulds physical intuition.  As noted by Eugene Wigner, the "unreasonable" success of this dance between mathematics and science is  itself an  extraordinary statement about both the nature of mathematics and the nature of the Universe.

             Our evolving understanding of the Cosmos offers a wonderful illustration of this interplay.  From the simple geometrical cosmology of the Greeks, to Galileo's telescope, to the elegance of Newton's Principia, to the modern vistas of the Hubble and Chandra Observatories, to the mathematics of complexity and beyond, the history of our evolving understanding of the Universe has been a history of the interplay between mathematics and science.  But what are the limitations on this remarkable collaboration? Karl Popper's "The Logic of Scientific Discovery" lays out the ground rules for answering that question.  Insights into the world obtained through mathematics are never better than the falsifiable predictions they allow us to make about the outcomes of further observation and experiment. The loop must be closed.  But what do we do when the mathematics of successful and testable theories of the Universe require us to accept extraordinary statements about realities that can never be observed, even in principle?  For modern astronomers and cosmologists, this question is anything but academic.  At the end of the day, what does it mean "to know," and what are the limits on that knowledge?