No matter how dubious

The following quote stuck with me when I read it years ago. Looking back I appreciate it even more.

Now, when solving differential equations, or indeed solving any problem, it is permissible to use any methods at all, no matter how dubious, provided that once the solution has been found it can be proved to satisfy all the conditions of the problem.

You could make a bell curve meme out of this. A novice would say “Sure, if it works it works.” An expert would agree. But someone in between who has recently been introduced to rigorous mathematics would object. They might say, for example, “You can’t just treat dy/dx like a fraction!” even though they did a few weeks ago.

Mathematics is discovered inductively but taught deductively. This creates the false impression that math advances deductively. It does not. That is a rationalist fantasy. To use Iain McGilchrist’s metaphor from “The Master and His Emissary,” the intuitive master solves problems, then assigns his analytical emissary to check the work.

I’ve looked for the source of the quote above several times without success. I was convinced it was a footnote in Boyce and DiPrima, but could never find it there. I recently ran across the line when I was looking for something else. The quote come in fact it comes from Applied Functional Analysis by D. H. Griffel, 1985.

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