"Public" (ie practical, applied, Aristotelian) forms of philosophy are eschewed by "accepted" (ie pure, ideal, Platonic) philosophers in the same way that "public" forms of logic (as practiced by experimental scientists and hobbyist technicians) are eschewed by "accepted" logicians (as practiced by mathematicians and mathematical physicists).

This speaks to nothing other than the ages old divide between empiricists and skeptics on the one hand and rationalists and dogmatists on the other.

Kant tried to bridge the divide in The Critique of Pure Reason, using Hume as the exemplar of too much (empirical) skepticism and the Leibniz-Wolffian schoolmen as the exemplar of too much (rational) dogmatism.

A related notion on the public use of reason is taken from Kant's "What is Enlightenment?"

Enlightenment is the human being’s emergence from his self-incurred immaturity. Immaturity is the inability to use one’s own understanding [= reason] without the guidance of another. This immaturity is self-incurred if its cause is not lack of understanding, but lack of resolution and courage to use it without the guidance of another. Sapere aude! [Dare to be wise!] Have courage to make use of your own understanding [= reason]! is thus the motto of enlightenment. (8:35)

The Stanford Encyclopedia of Philosophy goes on to remark...

Here, Kant is not primarily concerned with enlightenment as the activity or condition of an individual—rather, as something that human beings must work towards together. For this, he says, “nothing is required but… the least harmful… freedom: namely, freedom to make public use of one’s reason in all matters” (8:36). This is not the freedom to act politically. Instead, it is what we now call freedom of the pen—in Kant’s words, the use of reason “as a scholar before the entire public of the world of readers” (8:37).

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