most recent 30 from http://mathoverflow.net 2013-06-19T00:15:40Z http://mathoverflow.net/feeds/user/15644 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66377/why-is-differentiating-mechanics-and-integration-art/67170#67170 Todd Rowland 2011-06-07T18:09:03Z 2011-06-07T18:09:03Z

<p>Why is integration harder for people than differentiation? This is as much a question about people and perception than it is about mathematics. Methods are ad hoc for the integrations that inspired the question, while software, for example Mathematica, uses rules to integrate that are just as mechanical as its rules for differentiation, as <a href="http://www.wolframscience.com/nksonline/page-1177c-text" rel="nofollow">Stephen Wolfram explains</a> in his book A New Kind of Science.</p> <p>But performing the integration is only half the issue. The other half is making the answer look right, and for that the computer has to perform algebraic simplifications. This need for algebra is somehow intrinsic to these problems, e.g., even to verify an answer that looks different. So in a sense its difficulties are not different from the calculus student who needs to do algebra.</p> <p>Why then is algebra hard, in particular algebraic simplification? Without getting too theoretical, just consider a typical adhoc procedure to integrate x^2*Exp[x]. We know that taking the derivative of x^n*Exp[x] gives two terms, and we can use that knowledge to predict the answer will be a combination of x^2*Exp[x], x*Exp[x], and Exp[x]. The behavior there allows us to solve for an answer.</p> <p>Let's abstract from this example, where the terms are just linearly independent. The presence of terms is a binary condition, and one can think of taking the derivative as an operation like a <a href="http://www.wolframscience.com/nksonline/page-53" rel="nofollow">cellular automaton</a>. So an abstract model of the derivative operator is a cellular automaton. </p> <p>Now that one has a model it is possible to investigate questions like the original question. Some cellular automata are reducible, in the same way that integrating x^n*Exp[x] is reducible, while others are irreducible and undecidable questions (i.e. there is no decision procedure) about them are common. So in the end it is basically the NKS phenomenon that Wolfram discovered. </p>

http://mathoverflow.net/questions/66377/why-is-differentiating-mechanics-and-integration-art/67170#67170 Todd Rowland 2011-06-10T17:10:44Z 2011-06-10T17:10:44Z

VonJD: It is conceivable that another computational system would find integration easier. The second half of my post meant to address the human question. The CA analogy, where differentiation is like a step in a CA, shifts the problem to something easier to visualize. That reversing a CA is harder than computing its evolution is something one can see in the wild evolution of rule 30. One of the adhoc methods of integration is knowing what terms to include, and doing this is similar to reversing a CA, both in the cases where it is easy and where it is hard.