What Is Mathematics and What Should It Be? [pdf]
Dr. Zeilberger writes a rather pleasant rant. He appears frustrated with the lack of imagination in mathematicians which occurs when they focus too strongly on the axiomatic nature of mathematics, and seems to wish that we take a step back from the powerful axiomatic tools we have developed, so that we may search for what is &#x27;really true about the world&#x27; rather than &#x27;playing a mathematical game&#x2F;believing in a mathematical religion&#x27;.<p>Three specific complaints he has involve a belief in infinity (and limits), which he asserts do not exist in the real world; a delay in the publication of a pair of papers on which he worked, due to a variety of circumstances; and a &quot;pernicious&quot; influence of axiomatic mathematics which leads to &quot;stupid&quot; questions such as Hilbert&#x27;s Second Problem.<p>The failed publication of one paper is <i>particularly</i> notable, as it claimed a counterexample of Fermat&#x27;s Last Theorem (according to Dr. Zeilberger&#x27;s &quot;Opinion 123&quot;, on his Rutgers website), and (ibid) was recognized by Andrew Wiles himself as one of three <i>possible</i> counterexamples: the reason given for this oversight was in fact the acceptance of the decidedly non-rigourous statement &quot;it is easily seen ...&quot; (ibid). This particular instance seems to contradict the main thrust of Dr. Zeilberger&#x27;s rant against a mathematics overburdened by rules.<p>Unfortunately, little example of what mathematics SHOULD look like is offered - beyond a statement that &#x27;obvious things should be treated as such&#x27; and an assertion that infinities and continuities do not occur in real life&#x2F;nature&#x2F;the universe. While such a statement may be understandable coming from a respected (and clearly accomplished) combinatorist such as Dr. Zeilberger, physics has yet to demonstrate conclusively that space and time are discrete - undisproven interpretations of quantum mechanics exist which allow for continuities, and &#x27;the size of the universe&#x27; is defined as that which we can see (ie, within ~13.8 billion light-years of Earth&#x2F;Sol) so that actual infinities are ruled out only because of our inability to perceive them.<p>Perhaps Dr. Zeilberger needs to think outside his own discrete box.