An analogous statement has been used to show that humans are subject to the same limits as machines: “Lucas cannot consistently assert this formula”.
[4] Still, this criticism only works if we assume that we can replace Lucas' reasoning with a formal system whose theorems can be listed by an algorithm that has a Gödel sentence, but the Penrose-Lucas argument tries to prove otherwise: our ability to understand this level of arithmetic is not an effective procedure that can be simulated in a Turing machine.
Penrose argued that while a formal proof system cannot prove its own consistency, Gödel-unprovable results are provable by human mathematicians.
[5] He takes this disparity to mean that human mathematicians are not describable as formal proof systems (which theorems can be proved using an abstract object such as a computer), and are therefore running a non-computable algorithm.
[7] The inescapable conclusion seems to be: Mathematicians are not using a knowably sound calculation procedure in order to ascertain mathematical truth.
Penrose proposed that isolated systems may still undergo a new form of wave function collapse, which he called objective reduction (OR).
The rough threshold for OR is given by Penrose's indeterminacy principle: where: Thus, the greater the mass-energy of the object, the faster it will undergo OR and vice versa.
Penrose claimed that such information is Platonic, representing pure mathematical truth, aesthetic and ethical values at the Planck scale.
He argued that mathematicians do not progress by mechanistic search through proofs, but by trial-and-error reasoning, insight and inspiration, and that machines do not share this approach with humans.
Also, Artificial Intelligence based on Reinforcement Learning can work by taking actions in an environment in order to maximize the notion of cumulative reward, acting like trial-and-error procedures.
[33] Mistakes do not directly imply that the human mind is inconsistent per se: biological organisms are subject to cognitive turmoils, reduced long-term memory and attention shifts; these reduce our reasoning capabilities and make humans act unconsciously without taking into consideration all the possible variables of a system.