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I don't know if the word I'm thinking of is Anthropocentric, since I don't mean to say the universe exists only for us, but only that we are the result of more than just simple natural processes.
From the Merriam-Webster dictionary:"ANTHROPOCENTRIC"adjective
an·thro·po·cen·tric | \ ˌan(t)-thrə-pə-ˈsen-trikhttps://www.merriam-webster.com/dictionary/anthropocentric?pronunciation&lang=en_us&dir=a&file=anthro03Definition of anthropocentric:
considering human beings as the most significant entity of the universe.
interpreting or regarding the world in terms of human* values and experiences.
*As a species in this context.
Yes, you are using the word correctly, at least in context with the second definition. But it is easily one of the weakest Fermi Paradox arguments; that our circumstances are so special that a) we are alone and unique in that regard or b) we as well be because the conditions are so rare and the distances are so vast. It mistakenly assumes that the processes that resulted in our existence apply to the universe in its entirety, and suffers from single-sample bias. Our existence is not evidence of anything aside from the possibility that these things are possible in the universe. This is really assuming that Earth is the best possible host for life just because we are on Earth and we think we are so special. I also dislike the term 'cosmic lottery'. It entails that even in a random system (which an ecosystem is not) intelligence in life-forms is a 'goal' for evolution in some way and is a very arrogant stance. Evolution has no goals, and this ignores the examples of 'intelligence' in other Earth organisms that could have a shot at technological societies - like certain species of avian. We also do not know if intelligent life existed at some point in Earth's prehistoric past. Obviously not high technology, but at least stone age. Evidence of that would not fossilize very well if older then some millions of years. We have not discovered any animal lineages that resulted in intelligence, but the possibility remains.
Therefore, we can consider ourselves a statistically AVERAGE
sample of the universe's life pool. To state the opposite is essentially an argument from ignorance, even if it is based on scientific observations suggesting that the conditions
suitable for technological intelligent life-forms are rare. Those observations do not say that technological life *is* rare or non-existent, they only mediate the factors that lead to intelligent life. That is it. We don't have enough of these observations to say with even 50% confidence that humanity on Earth is extremely fortunate or rare. Many people would say "Glass half-empty, glass half-full" if confronted with this rough probability game, but if we consider the vast wilderness of the Milky Way alone, taking into consideration galactic habitability zones and star type, a brute-force application of probability skews the likelihood of life being present at some evolutionary stage in a plethora of environments.
That is my counter to your resistance (on principle alone), and may contain points you've probably heard of before, and will inevitably dismiss.