When does circular reasoning go wrong?

Question

Consider the following erroneous usage of L'hopital's rule:

$$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{D_h(f(x+h) - f(x))}{D_h(h)} = \lim_{h \to 0} \frac{f'(x+h)}{1} = f'(x) \tag1 $$

Note that if $\lim_{h \to 0} f'(x+h)$ exists, it is always equal to $f'(x)$.

There are two remarks one can make about $(1)$. The usage of L'hopital's rule is entirely fallacious, since we're essentially using a rule which uses the concept of differentiation on the definition of the derivative itself, and that, notwithstanding this, its usage yields the correct answer. Note also that $(1)$ is essentially a generalization of the common (explicit) example of this fallacy where the limit is $\frac{\sin x}{x}$.

My question is twofold:

Is the fact that the circular reasoning above gave us the correct answer entirely a mathematical coincidence, or is there some "deeper" reason behind it? In either case, is there an example in mathematics, analagous to $(1)$, where circular reasoning gives an incorrect answer?

Answer 1

Circular reasoning always works. Logic would make for a pretty bad system of deduction if the truth of a proposition $P$ was not a consequence of the hypothesis that $P$ is true!

The notation $P \vdash Q$ means, that from the hypothesis $P$, you can logically deduce $Q$.

$P \vdash P$ is a theorem of logic.

---

Furthermore, circular reasoning is a good thing. When we learn a subject, such as calculus, starting from first principles we develop and study sophisticated ideas and advanced techniques.

But once we know sophisticated ideas and advanced techniques, they are far easier to use than the basic principles.

e.g. if $P$ a basic fact of calculus (or otherwise something easy to prove at the beginning of your calculus education), it is not uncommon for someone who has learned calculus to realize

$$\text{calculus} \vdash P$$

much more easily than having to work out

$$\text{first principles of calculus} \vdash P$$

---

The fallacious application of circular logic is when someone tries to make the following argument:

• $P \vdash P$
• Therefore, P.

This is an invalid argument form.

Answer 2

Circular reasoning is commonly defined to be an argument that assumes that which is to be proven (or supported). The conclusion may be true, false, or indeed meaningless. The invalidity of circular reasoning is independent of whether the conclusion is true or not.

In the immediate Question l'Hôpital's Rule is invoked on the limit of the difference quotient for $f(x)$ that defines $f'(x)$. However nothing explicitly was said to justify this application. While the denominator $h$ is clearly differentiable with respect to $h$ and tends to zero as $h\to 0$, no proof is given that the numerator is differentiable with respect to $h$ and tends to zero as $h\to 0$.

Whether this is an illustration of "circular reasoning" depends on if we are being asked to assume the differentiability of $f$ in order to conclude that property of $f$. The fallacy of such reasoning as presented, with its gaps in justification, is evident from its showing any function $f$ whatever has a derivative.

This of course is one illustration of drawing a false conclusion from circular reasoning, but circular reasoning is invalid even when turned to a purpose of supporting a true proposition. It is not necessary to find a false conclusion in order to disparage an argument for relying on circular reasoning.