On 05 Apr 2017, at 06:28, Jason Resch wrote:
In my view, Penrose's theory that computation could not explain
human thought was based on the flawed idea that there exist problems
that humans could solve which no computer could. I prepared the
following to offer my explanation for why this is an unsupported
supposition:
-------------
In general, modern theories of consciousness react to the
consequences of the Church-Turing thesis in one of three ways, which
has yielded the following three philosophical camps:
Non-computable physicists
Weak AI proponents
Computationalists
Believe human thought involves physical processes that are non-
computable, and therefore conclude that it’s impossible to replicate
the behavior of a human brain using a computer.
Believe the behavior of the human brain can be replicated by
computer, but assume such a reproduction, no matter how good, would
not possess a mind or consciousness.
Believe the behavior of the human brain can be replicated by a
computer, and assume that when the reproduction is sufficiently
faithful, it possesses a mind and conscious.
Descartes reason for rejecting mechanism
I agree with you post (above and below). I just add that my work
(helped by Löb and Solovay's result and others of course) show that
the machine already make the Penrose-Lucas "mistake" when they use the
first person view, until they bet on "non-solipism", going from "I
know that my soul is not a machine" to "if I am a machine, then only
God can know that my soul is supported by (infinitely many) machines
(and then I know I am confronted with a hierarchy of non-computable
phenomena, which computer scientist have already discovered: the
degrees of unsolvability.
Then I would not say that Descartes rejected Mechanism (indeed the
mainstream would say he (re)discovered it), but he imposed on it a
human dualism, ... but I think it was just to avoid being burned by
the french clergy. Eventually he flew out of France.
Bruno
was that he believed no machine could be engineered with enough
complexity to account for human language and thought. Yet today, it
is nearly universally accepted (among biologists, physicists,
cognitive scientists and philosophers) that such machines are
possible, for unless human beings violate the laws of physics, we
ourselves are examples of such machines: machines that can think.
Yet, not everyone agrees that the behaviors exhibited by we “human
machines” can be replicated by the kind of machines Turing
described. It may be that some element of human reasoning relies on
a physical process that is not computable. For example, something
that is infinite in complexity or fundamentally nondeterministic.
The English mathematician and physicist Roger Penrose falls into
this camp. He has argued that human thought cannot be emulated by
any algorithm or computer. This argument, known as the Lucas-Penrose
argument, is based on the observation that there are certain
problems which no computer is able to solve. A famous example of
which is the halting problem, first described by Turing in his 1936
paper.
The halting problem is: given an arbitrary computer program,
determine whether that program will, if started, ever complete
(halt). In the same paper, Turing proved that regardless of the
approach (algorithm) employed by the computer, there would always
exist some programs for which it was impossible to answer the
halting question.
The British philosopher John Lucas and Roger Penrose both considered
this limitation of computers to be clear evidence of the superiority
of man over (Turing) machines. It further inspired Penrose to
propose, in his 1989 book The Emperor’s New Mind, that non-
algorithmic (not Turing emulable) processes must be involved in the
brain. This would have to be in order to explain why humans could
solve these problems while computers could not.
Yet, all currently known laws of physics (with the notable exception
of wave-function collapse) are deterministic, and can be modeled by
Turing machines. This led Penrose to propose that the currently
known laws of physics are inadequate to explain the workings of
human thought, and that some undiscovered physics holds the key to
understanding the workings of the mind.
Penrose, together with the anesthesiologist and consciousness
researcher Stuart Hameroff, speculated that the brain may harness
quantum mechanical effects, such as wave-function collapse, or
somehow makes use of (the presently not well understood) quantum
gravity to achieve its unique reasoning abilities.
However, Lucas’s and Penrose’s argument contains an overlooked
assumption: that humans were capable of solving the halting problem.
Certainly, a human programmer or team of programmers can examine a
program and figure out whether or not it will ever complete. Even if
it is a very complex program, given enough time, or a skilled enough
team, a determination can eventually be made. Or so they believed.
As it turns out, it is quite easy to write a computer program such
that whether or not it ever completes depends on some, as of yet
unknown, result in mathematics. For example, in number theory the
Goldbach conjecture hypothesizes that every even number greater than
2 can be represented as the sum of two prime numbers. Despite
countless efforts by mathematicians since 1742, and even an offered
prize of $1,000,000 to anyone who could prove or disprove it, the
truth or falsehood of the conjecture remains unknown.
A program based on the Goldbach conjecture might look something like
this:
Step 1: Set X = 4
Step 2: Set R = 0
Step 3: For each Y from 1 to X, if both Y and (X – Y) are prime, set
R = 1 Step 4: If R = 1, Set X = X + 2 and go to Step 2
Step 5: If R = 0, print X and halt
This simple program searches for a counterexample to the Goldbach
conjecture. If the Goldbach conjecture is false, this program (given
enough time) will disprove it by printing an even number that can’t
be represented as the sum of two primes and entitle its discoverer
to everlasting fame. As of 2014, every even number up to
4,000,000,000,000,000,000 has been checked, and so far no exceptions
to the rule have been found.
But the fact that an exception hasn’t been found doesn’t mean there
isn’t one. There remain quite a few numbers left to try. Yet if no
exceptions exist, this program will go on forever, stuck in an
infinite loop in which the computer searches for a number that
doesn’t exist (as fruitlessly as trying to find an even number that
ends in 5).
We see that if the Goldbach conjecture is false, the program finds
an exception and halts, but if it’s true then the program loops
forever and never reaches the halted state.
With this example we see that despite the simplicity of the program,
determining whether or not this program halts has yet to be solved
by any person. And since proving that it halts or doesn’t is
equivalent to solving the Goldbach conjecture, there were a million
dollars available to anyone who could have proved whether or not the
above program halts. Yet, despite being offered for two years
(between 2000 and 2002) the prize went unclaimed.
Lucas and Penrose assumed the impossibility of the halting problem
didn’t apply to humans, but as it happens, it’s impossibility
applies equally to humans as it does to computers. In 1931, the
Austrian logician Kurt Gödel proved that within any given
mathematical framework some mathematical statements cannot be proved
or disproved.
When a program’s continued operation depends on statements that are
unprovable under some mathematical framework, then solving the
halting problem is equally impossible for humans as it is for any
computer when working within that framework. So the halting
problem’s impossibility does apply to humans, and there never was
any evidence to suggest we possess special, inherently non-
computable, mental abilities or that there remains some undiscovered
non-computable physics that gets put to use by our brains.
Penrose’s idea that human thought requires an undiscovered physics
was based on an invalid assumption. Accordingly, this theory has
remained a fringe view, and has attracted criticism from computer
scientists and physicists.
The artificial intelligence researcher Marvin Minsky said,
“[Penrose] tries to show, in chapter after chapter, that human
thought cannot be based on any known scientific principle.”, and
elaborated, “one can carry that quest too far by only seeking new
basic principles instead of attacking the real detail. This is what
I see in Penrose’s quest for a new basic principle of physics that
will account for consciousness.”
Max Tegmark also tested the feasibility Penrose’s and Hameroff’s
idea that the brain might harnesses wave-function collapse as part
of its operation. Tegmark determined that particles in the brain can
remain in a state of superposition for at most one 10 trillionth of
a second. This time period is billions of times shorter than the
fastest neurons can fire and thus the effect could have no
meaningful impact on the behavior of neurons in the brain.
Ongoing work by IBM's Blue Brain Project, which has the stated goal
of reverse engineering the mammalian brain, appears to further
refute the idea that what the brain does is not computable. The
director of the Blue Brain Project, Henry Markram believes that he
and his team have created the first biologically accurate model of a
small section of a rat’s brain.
Applying simulated electrical impulses to their neural model, the
simulated neurons began to respond just like a real neural circuit:
clusters of connected neurons fired together, the cells wired
themselves together and large scale patterns of activity flashed
across the entire model. This model did not include any quantum
mechanical effects, (nor any unknown physical laws), yet its
behavior matched the observations of groups of real neurons.
The positive results of the Blue Brain Project suggest that the
brain operates according to presently known physical laws, all of
which are computable. Given this, together with the realization that
the halting problem is generally insoluble (not just insoluble for
computers), we can, with some confidence, reject the hypothesis that
human thought requires non-computable physics. In doing so, two
possibilities remain: weak AI and computationalism.
Jason
On Tue, Apr 4, 2017 at 9:47 AM, David Nyman <da...@davidnyman.com>
wrote:
I've been thinking about the Lucas/Penrose view of the purported
limitations of computation as the basis for human thought. I know
that Bruno has given a technical refutation of this position, but
I'm insufficiently competent in the relevant areas for this to be
intuitively convincing for me. So I've been musing on a more
personally intuitive explication, perhaps along the following lines.
The mis-step on the part of L/P, ISTM, is that they fail to
distinguish between categorically distinct 3p and 1p logics which,
properly understood, should in fact be seen as the stock-in-trade of
computationalism. The limitation they point to is inherent in
incompleteness - i.e. the fact that there are more (implied) truths
than proofs within the scope of any consistent (1p) formal system of
sufficient power. L/P point out that despite this we humans can
'see' the missing truths, despite the lack of a formal proof, and
hence it must follow that we have access to some non-algorithmic
method inaccessible to computation. What I think they're missing
here - because they're considering the *extrinsic or external* (3p)
logic to be exclusively definitive of what they mean by computation
- is the significance in this regard of the *intrinsic or internal*
(1p) logic. This is what Bruno summarises as Bp and p, or true,
justified belief, in terms of which perceptual objects are indeed
directly 'seen' or apprehended. Hence a computational subject will
have access not only to formal proof (3p) but also to direct
perceptual apprehension (1p). It is this latter which then
constitutes the 'seeing' of the truth that (literally) transcends
the capabilities of the 3p system considered in isolation.
If the foregoing makes sense, it may also give a useful clue in the
debate over intuitionism versus Platonism in mathematics. Indeed,
perceptual mathematics (as we might term it) - i.e. the mathematics
we derive from the study of the relations obtaining between objects
in our perceptual reality - may well be "considered to be purely the
result of the constructive mental activity of humans" (Wikipedia).
However, under computationalism, this very 'perceptual mathematics'
can itself be shown to be the consequence of a deeper, underlying
Platonist mathematics (if we may so term the bare assumption of the
sufficiency of arithmetic for computation and its implications).
Is this intelligible?
David
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