has a similar sense of exhilaration at understanding as the original author of the proof, much as, he argues, the viewer of a masterpiece has a sense of exhilaration similar to the original painter or sculptor. Indeed, one can study mathematical and scientific writings as literature. Philip J. Davis and Reuben Hersh have commented that the sense of mathematical beauty is universal amongst practicing mathematicians. By way of example, they provide two proofs of the irrationality of . The first is the traditional proof by contradiction, ascribed to Euclid; the second is a more direct proof involving the fundamental theorem of arithmetic that, they argue, gets to the heart of the issue. Davis and Hersh argue that mathematicians find the second proof more aesthetically appealing because it gets closer to the nature of the problem. Paul Erdős was well known for his notion of a hypothetical "Book" containing the most elegant or beautiful mathematical proofs. There is not universal agreement that a result has one "most elegant" proof; Gregory Chaitin has argued against this idea. Philosophers have sometimes criticized mathematicians' sense of beauty or elegance as being, at best, vaguely stated. By the same token, however, philosophers of mathematics have sought to characterize what makes one proof more desirable than another when both are logically sound. Another aspect of aesthetics concerning mathematics is mathematicians' views towards the possible uses of mathematics for purposes deemed unethical or inappropriate. The best-known exposition of this view occurs in G. H. Hardy's book A Mathematician's Apology, in which Hardy argues that pure mathematics is superior in beauty to applied mathematics precisely because it cannot be used for war and similar ends. Journals Philosophia Mathematica journal The Philosophy of Mathematics Education Journal homepage See also Definitions of mathematics Formal language Foundations of mathematics Golden ratio Model theory Non-standard analysis Philosophy of language Philosophy of logic Philosophy of science Philosophy of physics Philosophy of probability Rule of inference Science studies Scientific method Related works The Analyst Euclid's Elements "On Formally Undecidable Propositions of Principia Mathematica and Related Systems" "On Computable Numbers, with an Application to the Entscheidungsproblem" Introduction to Mathematical Philosophy "New Foundations for Mathematical Logic" Principia Mathematica The Simplest Mathematics Historical topics History and philosophy of science History of mathematics History of philosophy Notes Further reading Aristotle, "Prior Analytics", Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK,