Euclid's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checki… - George Pólya

The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. … Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information.

We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building.

<math>\frac {dy}{dx} = \frac {\omega^2x}{g}</math>...The first derivative, the result of the differentiation of <math>y</math> with respect to <math>x</math>, was written by Leibniz in the form
<math>\frac {dy}{dx}</math>...Leibniz's notation ...is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation... need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative <math>\frac {dy}{dx}</math> was considered as the ratio of two "infinitely small quanitites", of the infinitesimals <math>dy</math> and <math>dx</math>. ...it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure... it brought mathematics into disrepute... some of the best minds... such as... Berkeley, complained that calculus is incomprehensible. ...<math>\frac {dy}{dx}</math> is the limit of a ratio of <math>dy</math> to <math>dx</math>... Once we have realized this sufficiently clearly, we may, under certain circumstances, treat <math>\frac {dy}{dx}</math> so as if it were a ratio... and multiply by <math>dx</math> to achieve the separation of variables. We get
<math>{dy} = \frac {\omega^2x}{g}xdx</math>