Or is automated proof search impossible for humans as well?
Arguably, humans require more energy per operation. So, presumably such an argument hinges upon what types of operations are performed in conducting automated proof search?
Arguably, humans require more energy per operation. So, presumably such an argument hinges upon what types of operations are performed in conducting automated proof search?
The context here is that certain problems cannot be solved by an algorithm, which doesn't necessarily translate into "cannot be performed by a machine".
It only means that there cannot be any Turing machine capable of solving them, no matter what "operations" it represents.
Either the (unreferenced) study was actually arguing that "automated proof search" can't be done at all, or that human neural computation is categorically non-algorothmic.
Grid search of all combinations of bits that correspond to [symbolic] classical or quantum models.
Or better: evolutionary algorithms and/or neural nets.
I don't quite understand the role of neural nets in that context, though. Those just classical computations in the end and should be bound by the same limits that every other algorithms are, shouldn't they?
Neuromorphic engineering has expanded since the 1980s. https://en.wikipedia.org/wiki/Neuromorphic_engineering
Quantum computing is the best known method for simulating chemical reactions and thereby possibly also neurochemical reactions. But, Is quantum computing necessary to functionally emulate human cognition?
It may be that a different computation medium can accomplish the same tasks without emulating all of the complexity of the brain.
If the brain is only classical and some people are using their brains to perform quantum computations, there may be something there.
Quantum cognition: https://en.wikipedia.org/wiki/Quantum_cognition
From "Quantum Memristors in Frequency-Entangled Optical Fields" (2020) https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7079656/ :
> Apart from the advantages of using these devices for computation [12] (such as energy efficiency [13], compared to transistor-based computers), memristors can be also used in machine learning schemes [14,15]. The relevance of the memristor lies in its ubiquitous presence in models which describe natural processes, especially those involving biological systems. For example, memristors inherently describe voltage-dependent ion-channel conductances in the axon membrane in neurons, present in the Hodgkin–Huxley model [16,17].
> Due to the inherent linearity of quantum mechanics, it is not straightforward to describe a dissipative non-linear memory element, such as the memristor, in the quantum realm, since nonlinearities usually lead to the violation of fundamental quantum principles, such as no-cloning theorem. Nonetheless, the challenge was already constructively addressed in Ref. [18]. This consists of a harmonic oscillator coupled to a dissipative environment, where the coupling is changed based on the results of a weak measurement scheme with classical feedback. As a result of the development of quantum platforms in recent years, and their improvement in controllability and scalability, different constructions of a quantum memristor in such platforms have been presented. There is a proposal for implementing it in superconducting circuits [7], exploiting memory effects that naturally arise in Josephson junctions. The second proposal is based on integrated photonics [19]: a Mach–Zehnder interferometer can behave as a beam splitter with a tunable reflectivity by introducing a phase in one of the beams, which can be manipulated to study the system as a quantum memristor subject to different quantum state inputs.
Quantum harmonic oscillators have also found application in modeling financial markets. Quantum harmonic oscillator: https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator
But there are real-world instances that contradict this hypothesis already. Humans don't exhibit Turing-computable behaviour because they did indeed solve problems that aren't solvable by Turing Machines.
I can give a very concrete example as well. Consider the following program (in Python for simplicity's sake):
def nat(x):
yield x
yield from nat(x+1)
for a in nat(2):
for b in nat(2):
for c in nat(2):
if a*a*a + b*b*b == c*c*c:
print(f"{a}^3 * {b}^3 = {c}^3")
exit()
Yet it has has been proven centuries ago that the above program won't halt.
The mathematicians couldn't have used a Turing Machine-compliant computation, so something else must be involved.
Now this of course doesn't mean that humans are generally capable of solving every problem that's not solvable by an algorithm, but it means that they can solve problems that algorithms can't solve.
Yes, there is. There's a machine which does nothing but print that exact proof.
You seem to have conflated solving an instance of a problem with solving the entire problem. Turing's theorem states that no one Turing machine will get the answer right for every instance of the halting problem, but for any instance that one specific machine gets wrong, there's another machine that also gets that instance right (in addition to everything else the original machine gets right).