Are we, as far as our thinking processes go, nothing but 'Computers made of Meat'? Is it possible that Scientific Laws will be discovered that totally explain human behaviour? Lucas' Theorem, due to the great Oxford Philosopher John Lucas FBA, explains why both these hypotheses are impossible.

Note for the rigorous/pedantic: if by computers you mean '*any entity
that can do a computation*' then of course we are, in that rather vacuous
sense, computers. (My mother was actually employed as a "computer" in the
Admiralty Research Lab in the 50s) What Lucas' Theorem shows is that,
unlike artificial computers (or anything equivalent to a Turing Machine),
we are not automata.

But whilst equations like 3*x* + 5 = 7 could be solved by rational
numbers (*x* = 2/3) other proved more intractable. Consider *x*^{2}
= 2. *x* = 7/5 is fairly close: 49/25 is only slightly less than 2
- 141/100 is even better. Clearly, by using larger and larger denominators
you can get closer and closer to the answer.

Then a pesky mathematician showed that, however closely you could approximate
the answer by rational numbers, you can never get there. For suppose that
there was a rational number *a*/*b* such that (*a*/*b*)^{2}
= 2. Then there must be a way of expressing the fraction in its lowest
terms, as c/d say, where c and d have no common factor. But if (*c*/*d*)^{2}
= 2, then *c*^{2} = 2*d*^{2}, and thus
*c*^{2}
is an even number. But the square of any odd number is an odd number, so
c must be an even number: 2*f* say. Hence (2*f*)^{2}
= 2*d*^{2} so 4*f*^{2} = 2*d*^{2}
so *d*^{2} = 2*f*^{2} so *d*^{2}
is even, and hence
*d* is even: 2*g* say. But this is impossible
because
*c*/*d* was a fraction in its lowest terms. Hence there
is no rational number which solves the equation *x*^{2} =
2.

**Note 1** it is not enough to say "ah
well, let's take *f*/*g* as the new solution" because the proof
can also be applied to the 'new fraction'
*f*/*g*. Because
this proof applies to *any* possible fraction it shows that *no*
fraction has the required property. Changing the fraction half way through
to 'avoid the contradiction' does not work.

It is said that the first mathematician to prove this was put to death for his un-orthodoxy, and it was only many years later that other mathematicians brought the result to the attention of the educated public, so that it became an undoubted result.

Now *all *the 'arguments' for scientific determinism and "the brain
is a computer" go along the lines of "of course we don't have anything
like a complete theory yet, but we are continuing to make advances and
so eventually we will have the full picture". You might have thought that
the advocates of this line, who are all (presumably) aware of the above
proof, must realise that this argument is completely fallacious, even if
the gullible public to whom they offer the pronouncements does not. But
in fact, Lucas's Theorem shows that, not only are the 'arguments' for Scientific
Determinism and The Brain is a Computer fallacious, but the hypotheses
are logically impossible. (In addition, it is now known that even if the
components of the brain were completely deterministic, which is far from
certain, the brain as a whole would not be. See here
for details).

John Lucas FBA was an Oxford Philosopher until his retirement in the
1990s. In the early days of Artificial Intelligence he proved a general
result that has fundamental importance to any discussion of the distinction
between computers and human beings, which was announced first in a paper
and then in his book *The Freedom of the Will* (OUP).. Investigators
of computers have been making great strides in their ability to do calculations
which mimic the results of the human intellect. Decision making, calculation,
symbolic manipulation, speech synthesis, expert systems, neural networks
and the like provide closer and closer approximations to given aspects
of human behaviour. Similarly, much trumpeted advances in neuroscience
provide increasingly deeper understanding of the mechanisms of the brain.
Surely it is "obvious" that eventually computers will be able to mimic
human behaviour totally, and that scientists will be able to explain all
aspects of the brain, and hence the mind. Far from being uniquely made
in God's image, humans are 'nothing but' computers made of meat?

Define a Deterministic Logical System as a set of (at least two) states S, a set of inputs I and a logical system L which uniquely defines, given a state s in S and some inputs i in I, what the next state s' of the system will be.

There is a non-empty set of human beings HFree such that, for any member
**h**
of HFree, there does not exist a Deterministic Logical System which would
accurately predict all **h**'s actions in all circumstances.

**Proof**: Let HLog be the non-empty set of mathematical logicians
capable of understanding Godel's theorem (see Defn 7
for a formal definition), and let **h** be a member of HLog. Suppose
there exists a *specific* Deterministic Logical System (S1,I1,L1)
which would accurately predict all **h**'s actions in all circumstances.
Since **h** can do elementary arithmetic, L1 must be rich enough to
contain elementary arithmetic and thus by Godel's theorem there exists
a proposition G_{L1} in L1 whose interpretation is "G_{L1}
cannot be decided by L1". Consider now the circumstance where
**h**
(while rational) is asked: "is G_{L1} true?". **h** will rationally
answer "yes", by reasoning along the following lines: "*L1 cannot be
contradictory, because if L1 were contradictory then (since in a contradictory
logical system, if you can prove P you and prove not-P) the next state
of the system would not be uniquely defined. But if G _{L1}
were false, then it would mean that "G_{L1} cannot be decided by
L1" can be decided by L1, which is a contradiction. Hence G_{L1}
is true*." However, by definition, L1 cannot decide G

**Note 2**: Some people try to avoid this proof by arguing that "*You've
just given me a procedure that allows me to program a computer to answer
appropriately*" and so all that would be necessary would be to extend
L1 with this procedure (making L2 say). This is exactly analagous
to the logical mistake explained in Note 1. Because
this proof applies to *any* possible Deterministic Logical System
it shows that *no* Deterministic Logical System has the required property.
Changing the Deterministic Logical System half way through to avoid the
contradiction does not work.

**Note 3**: Clearly the term "capable of understanding Godel's theorem"
is a trifle informal. The specific property we need is that **h**
is capable, for any Deterministic Logical System (S,I,L) of recognising
(and verifying, if necessary with the aid of a computer) that G_{L}
is a Godel proposition in L, and reasoning rationally about it along the
lines indicated. If you want a formal proof, one is given here.

**Corrollary 1**: All rational human beings are members of
HFree. **Proof**: it is absurd to suppose that there is anything so
epistomolgically unique about mathematical logicians, and in principle
all rational human beings are capable of being taught to understand Godel's
theorem, .

**Corrollary 2**: No computer/automaton can competely model the mind
of a human being. **Proof**: a computer/automaton is a Deterministic
Logical System, and even if there are random factors in the program, these
are ultimately either 'pseudo-random' or depend on random inputs, which
can be made inputs to the Deterministic Logical System.

**Corrollary 3**: No scientific theory can ever be discovered that
*completely
*accounts
for human behaviour. **Proof:** Any scientific theory that can be discovered
which completely accounted for anything must be a Deterministic Logical
System. But we have shown that no Deterministic Logical System can
completely account for human behavior. Hence **QED**.

**Note 4**: none of these corrollaries deny that you can make *increasingly
accurate* models or theories: they only show that the process will never
finish. Similarly, you can make increasingly accurate approximations to
the solution of the equation x^{2} = 2 in rational numbers, but
you will never get there completely.

**Note 5**: It seems that the connection with freewill is fundamentally
that a human, unlike an automaton, is able to *choose* in which logical
system to work. The human logician, faced with a contradiction if
they were limited to the logical system L, chooses to step outside it to
decide the question. An automaton cannot do this: even adding a "rule"
which says " *when faced with a contradicion in a logical system then
step outside it"* does not rescue the automaton from the proof, because
it simply extends the automaton's Logical System by this rule, making a
new logical system L' - and recall that the proof applies to *any*
logical system.

Amazingly *The Freedom of the Will* is out of print. But
similar arguments were made by the outstanding mathematician Sir
Roger Penrose FRS in The
Emperor's New Mind and Shadows
of the Mind. There is no doubt that the proof is correct, however
uncomfortable and counter-cultural it may be to many in the Western intelligensia.

(for reasons of character sets we write **E** for "there exists",
=/= for "not equal to" and x for the cross product)

**Defn 1** A relation R:X->Y defined on (possibly infinite)
sets X and Y is **automatic **if there exists a logical system LR with a
finite number (n(LR)) of axioms and terms and a term T_{LR}(x,y) such
that R(x,y) iff T_{LR}(x,y) is a theorem in LR.

**Obs 1** The join of two automatic
relations is automatic.

**Proof** Let R and S be automatic relations defined on X, Y and Y, Z resp.
The logical system L_{RS} formed by uniting LR and LS, re-labelling
terms as necessary to avoid conflict of axioms or terms, and the term T_{LR}S
(x,z) = (**E **y)(T_{LR}(x,y) and T_{LS}(y,z)) has at most
n(LR) + n(LS) +1 axioms and terms, and (RoS)(x,y) will be true iff (**E
**y)(T_{LR}(x,y) is a theorem in LR and T_{LS}(y,z) is a theorem
in LS), in which case, since all theorems about TLR and TLS are theorems in
T_{LRS} (with appropriate re-labelling) T_{LRS}(x,z) will be
a theorem in LRS. Suppose now that T_{LRS} is a theorem in LRS.
Then (**E **y)(T_{LR}(x,y) is a theorem in LR and T_{LS}(y,z)
is a theorem in LS) and hence R(x,y) and S(y,z). Hence **QED**.

**Note 6 **Not all relations are automatic, since the number of finite sets
of axioms + terms is countable whereas the number of relations defined on even
N x N is un-countable.

**Defn 2** An (abstract) Actor is a relation R defined
on S x C x A where:

S is a set of States

C is a set of Circumstances

A is a set of Actions

If R(s,c,a) is true we say that R in state s and circumstance c would take action
a.

Note S, C and A may be infinite.

An actor is non-trivial if **E**(s_{1},c_{1},a_{1},s_{2},c_{2},a_{2})
(R(s_{1},c_{1},a_{1}) and R (s_{2},c_{2},a_{2})
and s_{1} =/= s_{2}, c_{1} =/= c_{2} and a_{1}
=/= a_{2}.

**Defn 3** An **Automaton **is an actor for which
R is automatic.

**Defn 4** An actor R2 **predicts** an actor R1 iff
there exist automatic ‘prediction relations’ Scan_{12}:
S1 x C1 -> S2 x C2 and Map_{21}: A2 -> A1 such that: R1(s_{1},c_{1},a_{1})
iff **E**(s2,c2,a2) such that R2(s2,c2,a2) and Scan12(s_{1},c_{1},s2,c2)
and Map21(a2,a_{1}) (ie R1 = Scan12 o R2 o Map21)

**Obs 2** The relation predicts is
reflexive and transitive.

Pf R predicts R with Scan and Map being equality relations. And if R1
predicts R2 and R2 predicts R3 then Scan_{13} = Scan_{12} o
Scan_{23} and Map_{31} = Map_{32} o Map_{21}
show that R1 predicts R3, since by Obs 1 Scan13 and Map31
are automatic.

**Obs 3** Any Actor predicted by
an automaton is an automaton.

**Proof** If R2 predicts R1 then R1 = Scan_{12}
o R2 o Map_{21} and hence by Obs 1 R1 is automatic.

**Defn 5** An (abstract) **Mathematical Logical Question**
(MLQ) is a pair (P,L) where L is a finite logical system and P is a proposition
in L. The answer to a MLQ is known-true or not-known-true if P is **true**,
or **false**/**undecided** (resp.)

**Defn 6** An (abstract) **Mathematical Logician**
is an actor M:P x L->Answer whose set of circumstances
C is all Mathematical Logical Questions (MLQs) and whose
Actions A are giving the answers to MLQs and:

(a) M(P,L) is known-true only when P is true in L. NB it may also give
the answer not-known-true if P is true in L and

(b) If M(P,L) is known-true then M(not P, L) is not-known-true and

(c) If A is an axiom of L then M(A,L) is known-true

**Defn 7** A Mathematical Logician M is "**Capable of
understanding Godel’s Theorem**" if for any logical system of a non-trivial
Automaton L1 there exists a Mathematical Logical Question (GL1,L1) where GL1
is a Godel proposition in L1 whose interpretation is "GL1 is undecidable in
L1" and M(GL1,L1) = known-true (Note that Godel’s
theorem such a proposition exists. Note also that M(GL1,L1) = known-true is
correct, because L1 cannot be contradictory, since if L1 were contradictory
then the automaton not be non-trivial. But if GL1 were false, then it
would mean that "GL1 cannot be decided by L1" can be decided by L1, which is
a contradiction. Hence GL1 is true.).

**Thm 1 (Lucas' Theorem) ** No Mathematical
Logician Capable of Understanding Godel’s Theorem (MLCUGT) is an Automaton.

**Proof:** Suppose M =R1:P ´ L->A were a Mathematical Logician Capable
of Understanding Godel’s Theorem (MLCUT). If M is an Automaton then R is automatic,
so (by Defn 1) there exists a logical system LR1 in which R1(P,L) = known-true
iff TLR1((P,L), known-true) is a theorem in LR1. Now by the defintion
of MLCUGT there exists a Godel proposition GLR1 in LR1 whose interpretation
is "this proposition is undecidable in LR1" for which M(GLR1,LR1)
= known-true. However by defintion of "a Godel proposition"
(TLR1((GLR1,LR1), known-true) is not a theorem in LR1. Hence LR1 does
not predict M. Hence no such automatic R1 can exist. **QED**.

**Thm 2 **No Mathematical Logician Capable
of Understanding Godel’s theorem can be predicted by
an Automaton.

**Proof:** By Obs 3 it would then be an automaton. **QED**

**Defn 8 **An Actor R:S x C->Answer is said to be **A
Mathematical Logician Capable in principle of Understanding Godel’s Theorem
with the aid of a sufficiently powerful computer** **when in a suitable state**
if there exists (logically) an automaton PC: P x L -> C and a state s_{1}
in S such that the Actor PC o R|s_{1} defined as [PC o R|s_{1}]
(P,L) = R(s1,PC(P,L)) is a MLCUGT.

**Thm 3** No Mathematical Logician Capable
in principle of Understanding Godel’s Theorem with the aid of a sufficiently
powerful computer when in a suitable state is, or can be predicted
by, an automaton.

**Proof:** If R is such an actor then R predicts an
MLCUGT (using the automatic relation PC o R|s_{1}
as Scan and the identity as Map). Consequently by Thm
2 R is not an Automaton. But by Obs 3 it cannot
be predicted by an Automaton either.
**QED**

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