Computer Science, Logic, and Religion (Islam) Amr Sabry

Computer Science, Logic, and Religion (Islam)

Amr Sabry

February 3, 2002 Science and Spirit Association 2

Amr: the computer scientist

Associate professor at Indiana University

Assistant professor at the University of Oregon

Postdoctoral researcher at Chalmers University, Göteborg

Ph.D. from Rice University M.Sc., B.Sc. from Cairo University


Amr: the Muslim


Outline

Computer Science

Computability Theory; Logic

Logic and Revelation

Islam


What are computers good for?


Computers …

automate tasks and processes that humans can do (generally a lot faster)


Easy tasks

Sorting. For example, 3,6,1,10,22,0,2,15,18,9 becomes 0,1,2,3,6,9,10,15,18,22

The record for sorting is 1 terabyte in 50 minutes

1 terabyte = 1,000,000,000,000 bytes (characters)

1 terabyte: about the information contained in a million unabridged dictionaries


Harder tasks Find p and q such that p*q = n

It took the international community 17 years to solve a challenge problem where n =

14381625757888867669235779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879543541

The RSA public key encryption relies on even larger numbers: the whole thing is based on the assumption that factoring large numbers is a hard task for the computer!


More hard problems


Largest solved TSP instance

15,112 cities

110 processors

22 years of computation time


Status

There are lots of very important and very interesting computational problems that are hard to do with current technology


Is there a limit …

or is it just a matter of buying and

building more powerful computers with faster and faster processors, and larger and larger memories?


Undecidable problems

Theory of computation asserts that some problems are undecidable (unsolvable in principle, no matter how much computing power you have!!!)

Example: There is a program that you want to download and execute. Before downloading it, you want to check if it is a virus. Is it possible to be absolutely certain that the program is not a virus?


Perfect virus detection. Not. Assume that we have an

incredibly clever way to check that programs are safe…

// Let P be a program // Assume that check(P,I) can// tell us whether it is safe to// execute program P with

input I

weird (p) = if check(p,p) then delete_all_files else return_normally

What happens if we execute weird(weird)?

If our clever program says that the execution is safe, the execution is actually not safe.

If our clever program says the execution is not safe, the execution is actually safe.

So our clever technique can’t possibly exist!


Computers …

… automate tasks and processes that humans can do (generally a lot faster).

There exist tasks that cannot be done using computers.


Fine. So what?

The fact that computers cannot be used to automate certain tasks implies that rational thought (logic) is inherently an unsound or incomplete way of understanding the universe.


Logic

Aristotle, the “first teacher”

Al-Farabi (الفارابى), the “second teacher”

Gödel


Aristotle…

Start with axioms which denote “true” facts

Use rules of inference to deduce more “true” facts

If everything the logic proves is “true” then the logic is called sound

If everything that is “true” can be proved then the logic is called complete


Al-Farabi

Applied logic to the study of language (Arabic)

Developed a philosophical method to reconcile Islam and Aristotle’s logic

Still believed that human reason was superior to revelation.


Muslim Theologians Both reason and revelation necessary. Debated for 300 years, which is superior? The 3 brothers: the good brother is in Paradise, the

wicked brother is in Hell, and the one who died as a child is in a place less than Paradise. The third brother complains that by dying as a child he was not given the opportunity to worship God and enter Paradise. To this, God responds that dying early was better for him because God foresaw that he would have grown up to be wicked. To this the second brother asks why he was not made to die at a young age?

Consensus emerged that revelation is superior.


Gödel By the beginning of the 1900s, the consensus of

the mathematical community was that all of mathematical facts could be derived using logic.

Gödel in an astonishing 1931 article showed that this was impossible.

His argument is very similar to our argument that the perfect virus detector cannot exist: a simple yet devious mathematical fact which refers to itself in a way that asserts that it can’t be proved.


Some quotes … In the nineteenth and early twentieth centuries, one of the big

mathematical goals was to reduce all of number theory to a formal axiomatic system. Like Euclid's Geometry, such a system would start off with a few simple axioms that are almost indisputable, and would provide a mechanical way of deriving theorems from those axioms. It was a very lofty goal. The idea was that this system would represent every statement you could possibly make about natural numbers. So if you made the statement "every even number greater than two is the sum of two primes," you would be able to prove strictly and mechanically, from the axioms, that it is either true or false. For real, die-hard mathematicians, the words "true" and "false" would become shorthand for "provable" or "disprovable" within the system. Russell and Whitehead's Principia Mathematica was the most famous attempt to find such a system, and seemed for a while to be the pinnacle of mathematical rigor.


Continued…

Gödel's theorem dashed this hope completely. It didn't just find a hole in Russell and Whitehead, which would presumably have been patched: it showed that the entire goal is unachievable. More specifically, Gödel showed that for any formal axiomatic system, there is always a statement about natural numbers which is true, but which cannot be proven in the system. In other words, mathematics will always have a little fuzziness around the edges: it will never be the rigorous unshakable system that mathematicians dreamed of for millennia.

Kenny Felder


More quotes…

Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that rational thought can never penetrate to the final ultimate truth ... But, paradoxically, to understand Gödel's proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel's name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it.

Rucker, Infinity and the Mind


More quotes… Gödel showed that provability is a weaker notion than truth,

no matter what axiom system is involved ...

The other metaphorical analogue to Gödel's Theorem which I find provocative suggests that ultimately, we cannot understand our own mind/brains ... Just as we cannot see our faces with our own eyes, is it not inconceivable to expect that we cannot mirror our complete mental structures in the symbols which carry them out? All the limitative theorems of mathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally.

Hofstadter, Gödel, Escher, Bach


More quotes The symbolic systems we use to describe the universe are not

separate from the universe: they are a part of the universe just as we are a part of the universe. Since we are within the system, our small understandings are 'the system modeling itself' (system meaning reality in this case). Completion of the model can never happen because of the basic self-referential paradox: the model is within the universe, so in effect the universe would have to be larger than itself. Or you can view it iteratively: the model models the universe. The universe includes the model. The model must model itself. The model must model the model of itself.. ad absurdum.

So Gödel's incompleteness is something to expect. It is even something that can be intuitively understood without a mathematical approach and proof: the incompleteness concept appears in clearly recognizable form in Zen Buddhism.

http://www.myrkul.org/recent/godel.htm


Our Universe

Fact

Fact

Fact

Fact

FactFactFact

Fact

Fact

Fact

Fact

Fact

Fact Fact

Fact

Fact

If you don’t believe in logic you can’t even distinguish facts from falsehoods in this region.

But if you do believe in logic, then logic shows that logic:-either proves everything including the falsehoods, or - does not prove certain true facts.

Falsehood

FalsehoodFalsehood

Falsehood

Falsehood

FalsehoodFalsehood

Falsehood


The Ultimate Truth

Logic (rational thought) gets into contradictions and is not appropriate beyond a certain point.

One can:

•Never venture beyond the limits of logic

•Venture beyond the limits of rational reason in search of the truth


Searching for the truth

Muslim theologians a thousand years ago wrote the “Book of Five Fundamentals” which starts as follows:

If it is asked: What is the first duty that God imposes upon you? Say to him: Speculative reasoning which leads to knowledge of God…


Revelation: mercy and guidance

Speculative reasoning may go astray. Revelation is a mercy and guidance. [39:18] Those who listen to the word,

then follow the best of it; those are they whom Allah has guided, and those it is who are the men of understanding.


Main message of revelation

Revelation repeats the same message over and over and over:

Think, reflect, speculate, reason, until you believe

Once you believe, you must believe in the whole thing

And then you must act to ascertain that the belief is real.


First chapter of the Qur’an after the opening prayer starts with…

2.1 A.L.M.

2.2 This is the Book; in it is guidance sure, without doubt, to those who fear Allah;

2.3 Who believe in the Unseen, are steadfast in prayer, and spend out of what We have provided for them;

2.4 And who believe in the Revelation sent to thee, and sent before thy time, and (in their hearts) have the assurance of the Hereafter.


Historically the first verses were…

[96.1] Read in the name of your Lord Who created.

[96.2] He created man from a clot. [96.3] Read and your Lord is Most

Honorable, [96.4] Who taught (to write) with the

pen [96.5] Taught man what he knew not.


Reflection…

[3.191] Those who remember Allah standing and sitting and lying on their sides and reflect on the creation of the heavens and the earth: Our Lord! Thou hast not created this in vain! Glory be to Thee; save us then from the chastisement of the fire:


Belief & speculative reason

75.36 Does man think that he will be left uncontrolled, (without purpose)?

75.37 Was he not a drop of sperm emitted (in lowly form)?

75.38 Then did he become a leech-like clot; then did (Allah) make and fashion (him) in due proportion.

75.39 And of him He made two sexes, male and female.

75.40 Has not He, (the same), the power to give life to the dead?


Belief is not just lip service!

2.214 Or do ye think that ye shall enter the Garden (of bliss) without such (trials) as came to those who passed away before you? they encountered suffering and adversity, and were so shaken in spirit that even the Messenger and those of faith who were with him cried: "When (will come) the help of Allah?" Ah! Verily, the help of Allah is (always) near!


Part of belief system: Money

The Earth and all its resources belong to God.

Humans are trustees on Earth. The goal of a trustee is not to

accumulate resources at the expense of others, but rather to work to multiply the resources and then to make sure they reach those who need them the most.


Proving your Belief: Money matters The bank promises you 7% interest (that is seen) God promises 700% for charity and even more

(that is unseen)

[2:261] The likeness of those who spend their wealth in Allah's way is as the likeness of a grain which groweth seven ears, in every ear a hundred grains. Allah giveth increase manifold to whom He will. Allah is All-Embracing, All-Knowing.

Are you going to put your money in a savings account or give it to the poor?


Money: a test not a sign of honor

89.15 Now, as for man, when his Lord trieth him, giving him honour and gifts, then saith he, (puffed up), "My Lord hath honoured me."

89.16 But when He trieth him, restricting his subsistence for him, then saith he (in despair), "My Lord hath humiliated me!"

89.17 Nay, nay! but ye honour not the orphans!

89.18 Nor do ye encourage one another to feed the poor!-

89.19 And ye devour inheritance - all with greed,

89.20 And ye love wealth with inordinate love!


How about charity without Belief? Not clear but:

18.103 Say: "Shall we tell you of those who lose most in respect of their deeds?-

18.104 "Those whose efforts have been wasted in this life, while they thought that they were acquiring good by their works?"

18.105 They are those who deny the Signs of their Lord and the fact of their having to meet Him (in the Hereafter): vain will be their works, nor shall We, on the Day of Judgment, give them any weight.

18.106 That is their reward, Hell, because they rejected Faith, and took My Signs and My Messengers by way of jest.


No guarantees anyway!

70.27 And those who are fearful of the chastisement of their Lord--

70.28 Surely the chastisement of their Lord is (a thing) not to be felt secure of--


So…

Believe in the Unseen Reason speculatively Seek guidance Do your best Hope for the best Repeat…

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Computer Science, Logic, and Religion (Islam) Amr Sabry