I'm a french speaker and normally the negation should be "ne" or "ni" but the problem was that this sounds very low in comparison with the "not" in english for example.
To let clearly known to your interlocutor that is a negation, in an early stage of the langage, a word has been added to describe exactly what you do not. For example, with the verb to walk (marcher), the negation was constructed "Je ne marche pas" (in english "I walk no step"). In french a "pas" is a single footstep (For example "a stepper motor" is "un moteur pas-à-pas").
In the same way in old french you can said "Je ne mange mie" ("I eat no bread"). La "mie de pain" is the bread crust.
In the same way "Je ne vois point" ("I see not a point", a "point" in french is a dot) is still used today , and "Je ne bois goutte" ("I drink no drop").
In my understanding, it is quite strong because "Je ne mange mie" would say "I don't eat anything, not even bread crumbs". Maybe must I say the smallest part possible just above nothing (Have a look at the added words : a drop, a dot, a tiny portion of bread, a single footstep) and even that you can't walk, drink, eat or see.
Note : "Je ne mange mie" and "je ne bois goutte" aren't used anymore by native speakers (and probably unknown for 99%+ of them). Even "Je ne mange mie" sounds strange in french. "Je ne bois goutte" also.
From all of this "added" words ("pas", "point", "mie" & "goutte", and probably others) only the two firsts remains in the language and it is now a part of the normal negation. "Je ne mange pas", "Je n'étudie pas", "Je n'ai pas d'argent" ou "De l'argent ? je n'en ai point !" ("Money ? I haven't !").
When you said, "je ne sais", "je ne peux", "je ne puis" you have difficult to hear that is a hard negation. Indeed the three (and in my opinion it is true for all verbs) are correct but only used today in the write form (or between "educated" person) and as we said in french "dans un langage soutenu" (elevated language ?) like "Je ne vois point"*. "Point" is more "elevated" that "pas". For exemple the latin expression "Extra Ecclesiam nulla sales" ("outside the Church there is no salvation") is usually traduced in "Hors de l'Eglise, point de salut". Source : https://en.wikipedia.org/wiki/Extra_Ecclesiam_nulla_salus
So for example "L'oublier, je ne peux." (Thinking of an old love, "Forget him/her, I can't") is correct but is near "d'une figure de style" due to the missing part "pas". The missing "pas" in place of a weakening effect has an stronger one. Of course, if you spoke about something less important you said / write : "L'oublier, je ne peux pas."
Sorry for my poor english ;)