The rules that were written, and the rules that were intended, are well-established by other answers. The answer is definitely “no.”
But why is it “no”? This answer is an attempt to capture the narrative reasons, the lore behind preparing spells. Note that this necessarily requires delving into pre-5e descriptions, because 5e doesn’t talk about it terribly much. The general assumption here is that D&D is consistent unless something says it isn’t. Unfortunately, one of the things that is explicitly inconsistent in 5e is spell preparation, so this is necessarily going to get a bit subjective.¹
The first thing to understand is that D&D’s magic system is based off the works of Jack Vance, in his Dying Earth series. There, all who used magic were forced to work as D&D wizards do, with a spellbook and with pain-staking memorization. In fact, preparing spells was a great deal harder in those works than they ever have been in D&D. Gary Gygax explicitly noted that D&D’s “Vancian magic system” comes from the series:
Just what portions of these works, the subsequent AD&D game, stemmed from
inspiration related to the writing of Jack Vance? Several elements, the unquestioned foremost being the magic system used in these games. To my way of thinking, the concept of a spell itself being magical, that its written form carried energy, seemed a perfect way to balance the mage against other types of characters in the game. The memorization of the spell required time and concentration so as to impart not merely the written content but also its magical energies. When subsequently cast—by speaking or some other means—the words or gestures, or whatever triggered the magical force of the spell, leaving a blank place in the brain where the previously memorized spell had been held. Because I explained this often, attributing its inspiration to Jack Vance, the D&D magic system of memorized then forgotten spells was dubbed by gamers “the Vancian magic system”.
(“Jack Vance & the D&D Game,” Gygax 2001, pg. 2)
Several key notes here:
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a spell itself [was] magical, […] its written form carried energy
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The memorization of the spell required time and concentration so as to impart not merely the written content but also its magical energies.
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When subsequently cast […] leaving a blank place in the brain where the previously memorized spell had been held.
Here we see that spell preparation isn’t merely memorization—it’s also imbuing oneself with the magical energy of the spell itself. The prepared spell is, itself, “a thing”—and a thing that could go away. Casting the spell removed it from your memory—and also meant the energy required for it went away.
Now, there is a key note here: prior to 4e, wizards had to prepare each individual spell in a specific slot. You want to be able to cast fireball three times today? Prepare fireball three times, in three separate slots. You want to cast fireball from a 5th-level slot for more damage?² You have to decide that when you’re preparing your spells, and then you have to decide how many 5th-level spell slots you’re using that way. When you cast a prepared spell, you lost all of it—the energy, the knowledge, everything. You might have another copy, but that was in a separate spell slot. And prior to 3e,³ all spellcasters worked this way—wizards prepared from a spellbook, clerics prepared by praying, etc., but all prepared spells into individual slots.
In 5e, things were changed to make spell preparation a little easier: you no longer have to prepare each spell into a specific slot, and thus decide how many slots are being devoted to each spell. Effectively, they divorced “the power to cast the spell” (spell slot) from “knowledge of the spell” (prepared spell), where they had previously been one and the same. This was almost-certainly done for gamist reasons,⁴ because playing a wizard in earlier editions was notoriously complex—and notoriously time-consuming. From a gameplay perspective, it’s a great change. But it does make it a little harder to connect 5e spellcasting to the Vancian vision.
But this change actually started in 3e—not with the wizard, but the (new at the time) sorcerer. The mechanics of the sorcerer’s spellcasting didn’t change much between 3e and 5e. It was presented as having a certain number of spells “permanently memorized,” but only enough energy—spell slots—to cast a limited number of them each day. A given slot wasn’t dedicated to a particular spell until the moment you decided to cast it—and once you did, you could always do so again so long as you had slots remaining that were high enough in level. Already we have a case where knowledge of a spell wasn’t part-and-parcel with the ability to cast it. D&D 3e referred to casters who didn’t prepare spells as “spontaneous.”
And in 3e,⁵ there was also a spellthief class from the Complete Adventurer supplement. The spellthief did what it says on the tin, it stole spells. For a wizard (or another class that prepared spells), this was very clear: you stole the spell they’d prepared, knowledge and energy both. For sorcerers and others like them, however, it was less clear how this worked narratively. Complete Adventurer has this to say:
The target of a steal spell attack loses one 0-level or 1st-level spell from memory if she prepares spells ahead of time, or one 0-level or 1st-level spell slot if she is a spontaneous caster. A spontaneous caster also loses the ability to cast the stolen spell for 1 minute. […]
For example, a 1st-level spellthief who uses this ability against a 1st-level sorcerer could choose to steal magic missile. Assuming the sorcerer knew that spell, a successful steal spell attack would eliminate one 1st-level spell slot and temporarily prevent her from casting magic missile. If the same spellthief stole magic missile from a wizard who had it prepared, the wizard would lose one prepared magic missile spell (but wouldn’t lose any other magic missile spells she might also have prepared).
(Complete Adventurer, D&D 3.5e, pg. 16)
This demonstrates that in some sense, even when we separate knowledge of the spell from spell slots, “the spell” is still “a thing,” that can be stolen. Both knowledge and magic come together when the spellthief successfully takes one.
So it is still in 5e, now applying to wizards too. A prepared spell has magic of its own, separate from the spell slot that will eventually be dedicated to casting it. That applies equally whether the spell was prepared that morning from a spellbook, or “prepared” years and years ago and written in one’s blood, as with a sorcerer. “Just” knowing the details of the spell—even knowing all the details of the spell—isn’t enough to prepare it; you need the magic too.
And that is why Keen Mind does not affect how many spells a wizard may prepare.
Spell preparation remained pretty consistent through the TSR era. Wizards of the Coast added some new variations in 3e—e.g. the sorcerer, who didn’t prepare spells at all—but the 3e wizard was much the same as the wizards and magic-users before it. 4e changed things—as in everything, not just spell preparation—massively, but 5e undid most of those changes. Nonetheless, 5e spell preparation is still different from 3e and earlier spell preparation in some key ways that do affect this answer.
Prior to 5e, casting a spell from a higher-than-necessary spell slot had no inherent benefit. In 3e, however, metamagic wasn’t a sorcerer-specific thing; instead, each was a feat. Instead of costing sorcery points, metamagic feats required you to cast the spell from a higher spell level—and some of them could, indeed, cause the spell to deal more damage in exchange. For instance, a maximized fireball counted as a 6th-level spell, and instead of doing Xd6 fire damage, it would simply deal 6×X fire damage, as if you’d rolled a 6 on every die.
Read: all those that Gygax worked on, since 3e was published by Wizards of the Coast well after Gygax was out of the game.
I’m sure someone from the D&D development team discussed this change somewhere, but I haven’t had any luck finding any such comments.
Actually “the v.3.5 revised edition.”
