Penrose’s first Gödelian argument

Penrose’s first Gödelian argument

The application of Gödel’s

theorems

to outside

metamathematics

,
notably the philosophy of mind, was initiated by Gödel himself. He
had a strong philosophical bent towards Platonism, which also
motivated his mathematical discoveries (Saint-Andre, 1998). Gödel first thought that his
theorems established the superiority of mind over

machine

(Wang, 1996a; 1996b). Later,
he came to a less decisive,

conditional

view: if machine can equal
mind, the fact that it does cannot be proved (Bojadziev, 1997).

”either…the human mind ( within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable

diophantine problems

.” – Godel, 1951*, p. 310

The importance of Gödel’s theorem for understanding of human mind was
revived by Lucas (1961) and was brought in the scope of brain scientists
recently by Roger Penrose (1989, 1994).

In 1989 Roger Penrose suggested that human mind is nonalgorithmic (noncomputable) and more powerful than any

formal system

and this follows from Gödel’s first theorem. Indeed his

argument

aimed to show that we (humans) can unmistakenly find a true statement

G

⁢

(

g

ˇ

)

for given formal system

F

, which the formal system

F

cannot prove, but which we (humans) know is true. Thus stated the statement for mind noncomputability follows from statement which is

stronger

than Gödel’s first theorem, and due to numerous errors and ambiguities in the exposition it is not clear whether Roger Penrose in his book

”The Emperor’s New Mind”

claims to have proved mind noncomputability from result stronger than Gödel’s first theorem, or proves it directly from Gödel’s first theorem. That is why below we provide concise but not

equivalent

formulations of Penrose’s main thesis. Indeed it seems that Penrose himself makes no

difference

between these two.

Penrose’s first Gödelian argument (formulation 1, mind noncomputability is derived from statement stronger than Gödel’s first theorem): Human mathematician sees the truthness of the Gödelian

sentence

G

⁢

(

g

ˇ

)

of given formal system

F

, therefore human mind is nonalgorithmic (noncomputable).

Penrose’s first Gödelian argument (formulation 2, mind noncomputability is derived directly from Gödel’s first theorem): Human mathematician understands (proves) Gödel’s first theorem, therefore human mind is nonalgorithmic (noncomputable).

Both of these arguments are provably erroneous, and as stated are false. The correct full text of Gödel’s first theorem that can be proven by every sufficiently strong formal system is the following.

Gödel’s first theorem: for every sufficiently strong to capture basic

arithmetic

formal system

F

which is recursively axiomatizable (i.e. there is an

algorithm

for determining for each string of the underlying

language

whether it is an axiom or not) in
which all valid sentences are

representable

by finite strings of symbols
there exists true sentence

G

⁢

(

g

ˇ

)

, which is

undecidable

within

F

if

F

is

consistent

. (Note:

g

ˇ

is the numeral of the Gödel number g of the

formula

G

⁢

(

g

ˇ

)

)

Here we should note that if the formal system is inconsistent it will be able to prove any statement including the Gödel’s first theorem. If the formal system is consistent and sufficiently strong it will also be able to prove Gödel’s first theorem, and this was shown by Gödel’s himself in 1930.

Gödel’s first theorem is both human understandable and
it is a theorem in any sufficiently strong finite size formal system

F

that captures basic arithmetic. Thus

computers

also prove Gödel’s
first theorem.

Penrose’s argument (formulation 1) insisting that we
directly see the truthness of

G

⁢

(

g

ˇ

)

in

F

is false because we
have to be able to directly decide the

consistency

of arbitrary finite
size formal system

F

, which is clearly not the case.
Therefore what Penrose has overlooked is that human when given a formal system

F

must be able to decide nonalgoritmically and unmistakenly whether

F

is consistent or not, in order to determine whether

G

⁢

(

g

ˇ

)

is true or not. Clearly

G

⁢

(

g

ˇ

)

is false for inconsistent

F

! Since no human can decide nonalgoritmically and unmistakenly whether arbitrary

F

is consistent or not then Penrose’s first Gödelian argument (formulation 1) is false, and the human mathematician cannot see directly that

G

⁢

(

g

ˇ

)

is true, as

G

⁢

(

g

ˇ

)

might be false for inconsistent

F

.

Penrose’s argument (formulation 2) is modification that derives mind noncomputability directly from Gödel’s first theorem in a fashion

”the human mathematician can ”see” that

G

⁢

(

g

ˇ

)

is true for consistent

F

however the consistent

F

cannot prove

G

⁢

(

g

ˇ

)

”

. It should be noted however that this latter claim is not strong enough for one to establish superiority of mind over formal systems (algorithms) and does not prove noncomputability of mind, because the human does not ”see” the truthness of

G

⁢

(

g

ˇ

)

, the human derives (proves) it given the

premise

for consistent

F

. Thus Penrose puts the human and the formal system in not equivalent positions: the human knows explicitly that

F

is consistent, while the

F

is given no such access on grounds that consistent formal system cannot know its own consistency as implied by Gödel’s second theorem. This inequivalence of the initial ”given” (not ”seen”!) knowledge with which the human and the formal system start leads to erroneous final

conclusions

in the book

”The Emperor’s New Mind”

, and makes it impossible for the reader to understand whether Penrose makes distinction between the presented above formulation 1 and formulation 2 of his argument. The text of Gödel’s first theorem is provable within the formal system

F

and it says exactly that

”

G

⁢

(

g

ˇ

)

is true for consistent

F

”

. Therefore no matter what the

ontology

of our mind is, there is nothing novel about the Gödel’s first theorem and about the Gödel’s statement

G

⁢

(

g

ˇ

)

that the human knows but the artificial intellect does not prove.

References

1. Bojadziev D (1997)

http://nl.ijs.si/ damjan/g-m-c.html

Mind versus Gödel. In: M. Gams, M. Paprzycki and X. Wu (eds.),

Mind Versus Computer

, IOS Press, pp. 202-210.

2. Crossley JN, Brickhill C, Ash C, Stillwell J, Williams
N (1972)

What is mathematical logic?

Oxford University Press.

3. Detlovs V, Podnieks K (2006)

http://www.ltn.lv/ podnieks/mlog/ml.htm

Introduction

to
Mathematical Logic

. Hypertextbook for students in mathematical logic

4. Gödel K (1931) Über formal unentscheidbare Sätze der
Principia und verwandter Systeme I.

Monatshefte
für Mathematik und Physik

38

: 173-198.

5. Gödel K [1951*]

Collected Works, Vol. III. Unpublished Essays and Lectures

. Oxford University Press, 1995.

6. Lucas JR (1961) Minds, Machines and Gödel.

Philosophy

36

: 112-127.

7. Mendelson E (1997)

Introduction to Mathematical
Logic

, 4th ed. London: Chapman & Hall.

8. Penrose R (1989)

The Emperor’s New Mind: Concerning
Computers, Minds, and The Laws of Physics

. Oxford University Press.

9. Penrose R (1994)

Shadows of the Mind: A Search
for the Missing Science of Consciousness

. Oxford University Press.

10. Podnieks K (2006)

http://www.ltn.lv/ podnieks/gt.html

What is Mathematics? Gödel’s
Theorem and Around

. Hypertextbook for students in mathematical logic

11. Saint-Andre P (1998)

http://www.saint-andre.com/thoughts/wang-godel.html

Objectivism Without Platonism: Hao Wang on Kurt Gödel

12. Wang H (1996a) A Logical Journey: From Gödel to Philosophy. Cambridge, Massachusetts, The MIT Press.

13. Wang H (1996b)

http://www.fortunecity.com/emachines/e11/86/moremind.html

Can Minds do more than Brains? In

How Things Are: A Science Tool-Kit for the Mind

, John Brockman and Katinka Matson (eds.)

Title

Penrose’s first Gödelian argument

Canonical

name

PenrosesFirstGodelianArgument

Date of creation

2013-03-22 17:05:45

Last modified on

2013-03-22 17:05:45

Owner

dankomed (17058)

Last modified by

dankomed (17058)

Numerical id

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Author

dankomed (17058)

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