Arguments by mathematical induction seem to provide an entire class of examples of the phenomenon, where computation is replaced by a higher level of reasoning.

With induction, one uses a comparatively abstract understanding of how a property propogates from smaller instances to larger instances, in order to arrive at a fuller understanding of the property in particular cases, without need for explicit calculation. Thus, one can see that a particular finite graph or group or whatever kind of structure has a property, not by calculating it in that instance, but by an abstract inductive argument, on size or degree or rank or whatever. A complex graph-theoretic calculation is avoided by understanding what happens in general when a point is deleted.

And there are, of course, extremely concrete elementary instances. We all know, for example, how to use induction to prove that $1+2+\cdots+n=n(n+1)/2$. Thus, the comparatively abstract inductive argument predicts definite values for concrete sums $1+2+\cdots+105$. Similarly, we often understand the iterates of a function $f^n(x)$ without calculating them, or the powers of a matrix $A^n$, or the successive derivatives of a function, all without calculation, by understanding the inductive relationship in effect at each step.

Surely mathematics is covered with dozens or hundreds of similar examples, of every degree of complexity and every level of abstraction.

Arguments by mathematical induction seem to provide an entire class of examples of the phenomenon, where computation is replaced by a higher level of reasoning.

With induction, one uses a comparatively abstract understanding of how a property propagates from smaller instances to larger instances, in order to arrive at a fuller understanding of the property in particular cases, without need for explicit calculation. Thus, one can see that a particular finite graph or group or whatever kind of structure has a property, not by calculating it in that instance, but by an abstract inductive argument, on size or degree or rank or whatever. A complex graph-theoretic calculation is avoided by understanding what happens in general when a point is deleted.

And there are, of course, extremely concrete elementary instances. We all know, for example, how to use induction to prove that $1+2+\cdots+n=n(n+1)/2$. Thus, the comparatively abstract inductive argument predicts definite values for concrete sums $1+2+\cdots+105$. Similarly, we often understand the iterates of a function $f^n(x)$ without calculating them, or the powers of a matrix $A^n$, or the successive derivatives of a function, all without calculation, by understanding the inductive relationship in effect at each step.

Surely mathematics is covered with dozens or hundreds of similar examples, of every degree of complexity and every level of abstraction.

Arguments by mathematical induction seem to provide an entire class of examples of the phenomenon, where computation is replaced by a higher level of reasoning.

To give a very elementary example, we all know how to use induction to prove that $1+2+\cdots+n=n(n+1)/2$. Thus, the comparatively abstract inductive argument predicts definite values for concrete sums $1+2+\cdots+105$. And surely mathematics is covered with dozens or hundreds of similar examples, of every degree of complexity. We understand the iterates of a function $f^n(x)$ without calculating them, or the powers of a matrix $A^n$, or the successive derivatives of a function, all without calculation, by understanding the inductive relationship in effect at each step.

The same phenomenon is exhibited in more abstract applications of induction. To prove a property of all finite graphs, all finite groups or whatever kind of structure you have, by induction on the size, or degree or rank or whatever you are inducting on, replaces a calculation of the property in the individual cases with an abstract understanding of how the property propogates to more complex objects.

Arguments by mathematical induction seem to provide an entire class of examples of the phenomenon, where computation is replaced by a higher level of reasoning.

With induction, one uses a comparatively abstract understanding of how a property propogates from smaller instances to larger instances, in order to arrive at a fuller understanding of the property in particular cases, without need for explicit calculation. Thus, one can see that a particular finite graph or group or whatever kind of structure has a property, not by calculating it in that instance, but by an abstract inductive argument, on size or degree or rank or whatever. A complex graph-theoretic calculation is avoided by understanding what happens in general when a point is deleted.

And there are, of course, extremely concrete elementary instances. We all know, for example, how to use induction to prove that $1+2+\cdots+n=n(n+1)/2$. Thus, the comparatively abstract inductive argument predicts definite values for concrete sums $1+2+\cdots+105$. Similarly, we often understand the iterates of a function $f^n(x)$ without calculating them, or the powers of a matrix $A^n$, or the successive derivatives of a function, all without calculation, by understanding the inductive relationship in effect at each step.

Surely mathematics is covered with dozens or hundreds of similar examples, of every degree of complexity and every level of abstraction.