The Turing machine was invented by a human mind. Presumably, nothing less powerful than a Turing machine can invent a Turing machine.
However, a Turing machine has infinite tape, whereas the mind is situated in a finite universe, and thus can only be a TM with finite tape. A TM with finite tape can be simulated by a finite automata, which is strictly less powerful than a Turing Machine. This seems to imply that a Turing machine was invented by an automata that is less powerful than a Turing machine, contradicting the initial premise.
Is this a problem? Or, is the initial premise false? Is it possible for a finite automata to somehow "invent" a machine that is more powerful, a kind of bootstrapping, if you will?
While it is difficult to formally define "invent," it does not seem a finite automata can even represent finite TMs effectively. For example, on the wikipedia page, it states a DFA will require quadrillions of states to represent a TM with a few hundred states. So, to even just represent the useful subset of halting TMs, my impression is that a DFA representation will probably exceed any computational capacity we have. There appears a big disconnect between DFAs and TMs, such that it is hard to imagine a plausible bootstrapping to go from one to the other.
There is also the related issue of how a finite automata can prove the halting problem for TMs.