I'm attempting to find a latin translation for the contrasting meaning of ignoramus et ignorabimus - We do not know and will not know i.e, We must know, we will know
Ignoramus et ignorabimus is a Latin maxim coined by the German physiologist Emil Du Bois-Reymond (1818–1896) in his work Über die Grenzen des Naturerkennens (1872).
His sentence supports the apparent impossibility for the human being to know and give reason to all concepts of reality.
In this context, a translation diametrically opposed to the above maxim could be
Cognoscimus et cognoscemus
which remarks the cognition and awareness. This translation, however, highlights mainly the present moment in which we "reach the cognition", literally means "we get to know and we will".
Otherwise, as @cnread has suggested, you can use the logic perfect
Cognovimus et cognoverimus
meaning "we have reached the cognition [and now we possess the knowledge] and we will have reached it [so we will have it]."
There's a nuance of meaning between the two, the choice is yours.
P.s. As a side note, I confess I agree more with Du Bois-Reymond's sentence, appearing more realistic and true on a phylosophical and human perspective.
The opposite of ignoramus et ignorabimus is scimus vel sciemus. "We know or we shall know". (DeMorgan's law requires changing "and" to non-exclusive "or", which is vel in Latin. The other Latin word for "or", aut, would be incorrect.) There is not a good way to express "must" in this instance in Latin without rephrasing that I would judge awkward in this case. Assuming this is a situation where forgetting is impossible, I'd suggest sciemus as a good single word opposite.
Note that the question originally began by asking for the opposite of ignoramus et ignorabimus.
For the Hilbert quote, you could say "Oportet nos scire. Sciemus." Or "Oportet scire. Sciemus.", which is probably clear. ("It is necessary [for us] to know. We will know.) Or put it all in the passive and omit the first-person angle entirely: "Sciendum est. Scietur.", which you could probably shorten to "Sciendum. Scietur." if you wanted to sound pithy. ("It must be known. It shall be known ") Passive voice isn't exactly favored in Latin but it is used more in Latin than in English, and it is often the best way to express necessity.
By the way, in case you are wondering why I prefer scire to cognoscere for the verb is that scire is part of the etymology of scientia, which is what Hilbert would taken himself to be talking about. See also, for exmaple the motto of the University of Chicago "crescat scientia; vita excolatur"..."Let knowledge grow and so life be enriched".