"Wanting something to happen"

Solution 1:

Given your comment "I am trying to qualify that belief by saying perhaps I believe it to be true because I want it to be true," I might suggest optimistically or even over-optimistically in place of wishfully, depending on how much doubt you want to express.

Solution 2:

Longing / Desire / Hope

Definitions and examples extracted from the Merriam-Webster dictionary:

Longing: a strong desire especially for something unattainable

• They looked with longing toward freedom.
• She cast a look of longing at the shop window.
• She never told anyone about her secret longings.

Desire: conscious impulse toward something that promises enjoyment or satisfaction in its attainment

• Desire is a common theme is music and literature.
• The magazine tries to attend to the needs and desires of its readers.
• Both sides feel a real desire for peace.
• His decisions are guided by his desire for land.
• They expressed a desire to go with us.
• They have a desire to have children.
• a strong desire to travel around the world
• He was overcome with desire for her.

Hope: desire accompanied by expectation of or belief in fulfillment

• When they started their life together, they were young and full of hope.
• Rescuers have not yet abandoned hope that more survivors will be found.
• The drug has brought hope to thousands of sufferers.
• We allowed ourselves to entertain hopes that the crisis would end soon.
• The goal raised the hopes of the team.
• The hope is that there will be a settlement soon.
• The lawyers do not want to raise false hopes of an early settlement.
• He told them the truth with the hope that they would understand.
• He had little hope of attending college.
• The latest reports hold out hope for a possible end to this crisis.

Note: I suggested nouns only because I thought that it made sense to choose another noun to follow acceptance in your sentence.

Solution 3:

I like "wishful" very much, but "wishfully" is an adverb and "acceptance" is a noun. So I'd just change the sentence to read something like:

The first is the acceptance, though perhaps it is a wishful one, that understanding who we are is achievable in my lifetime.

Actually I don't really understand the use of the word "acceptance" here - don't you mean something like "belief"? "Acceptance" seems to imply (perhaps grudging) acquiescence that something is true.

I didn't like the second comma that Joshua added much, so I took it out. No real need to slide in a comma between a subject and its verb... it wasn't that long a sentence!

Solution 4:

As an off-topic aside: it sounds as though you are writing a scholarly paper on the subject of consciousness. If I were you, I would definitely not want to convey that I sometimes come to believe things because I want them to be true. It's a form of magical thinking, and is generally incompatible with science and academia. That being said...

I agree that the use of "acceptance" implies that it's established fact that we will understand the brain within our lifetime. Not that we won't--just that there's no consensus on the matter. Also, I changed to past tense to parallel "propelled" in the previous sentence.

Here are some possibilities:

Maybe you feel that the belief was somewhat foolhardy:

The first was coming to believe, perhaps with undue optimism, that understanding the brain is achievable within my lifetime.

Maybe you want to describe the onset of your belief:

The first was coming to believe, gradually, but with mounting certainty, that understanding the brain is achievable within my lifetime.

Or maybe the belief was almost a guilty pleasure:

The first was allowing myself to truly believe that understanding the brain is achievable within my lifetime.

You get the idea. This slight re-structuring allows lots of room for tailoring the sentence to your liking.