The hypothesis goes:
I want humans to be casually interstellar and expand across the galaxy. I want this to occur in less than N years, N << 10^10.
If humans are casually interstellar and expansionist, then we would colonize the galaxy in M years, M << 10^10.
Humans are not special, therefore if another alien species existed in the past at current human tech levels, then they would do the same as humans would and colonize the galaxy within N + M << 10^10 years after achieving current human tech level.
We believe we can detect such a civilization. Therefore, if we are more than N + M years after a another civilization with current human tech level, then we would see them.
We do not detect any such civilizations.
Therefore one of the assumptions must be wrong.
1. We can not detect a galactic civilization.
2. No alien civilization has reached current human tech level within N + M years of now.
3. Humans are special. Only humans would colonize the galaxy if they had casual interstellar travel.
4. Humans are not special, but we will never invent casual interstellar travel/spread to the stars.
So, if you think humans are not special (3 is false) and you hope humans will spread to the stars in the future (hope 4 is false), then you hope the answer is 1 (humans are bad at detection) or 2 (humans are early).
If you rule out 1 and 2, then 4 can only be true if 3 is true. If you rule out 3 as well, then you must conclude humanity will never spread to the stars for unknown reasons.
I want humans to be casually interstellar and expand across the galaxy. I want this to occur in less than N years, N << 10^10.
If humans are casually interstellar and expansionist, then we would colonize the galaxy in M years, M << 10^10.
Humans are not special, therefore if another alien species existed in the past at current human tech levels, then they would do the same as humans would and colonize the galaxy within N + M << 10^10 years after achieving current human tech level.
We believe we can detect such a civilization. Therefore, if we are more than N + M years after a another civilization with current human tech level, then we would see them.
We do not detect any such civilizations.
Therefore one of the assumptions must be wrong.
1. We can not detect a galactic civilization.
2. No alien civilization has reached current human tech level within N + M years of now.
3. Humans are special. Only humans would colonize the galaxy if they had casual interstellar travel.
4. Humans are not special, but we will never invent casual interstellar travel/spread to the stars.
So, if you think humans are not special (3 is false) and you hope humans will spread to the stars in the future (hope 4 is false), then you hope the answer is 1 (humans are bad at detection) or 2 (humans are early).
If you rule out 1 and 2, then 4 can only be true if 3 is true. If you rule out 3 as well, then you must conclude humanity will never spread to the stars for unknown reasons.