Computer Science, Logic, and
Religion (Islam)
Amr Sabry
Amr: the computer scientist
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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
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Amr: the Muslim
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Outline
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Computer Science
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Computability Theory; Logic
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Logic and Revelation
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Islam
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What are computers good for?
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Computers …
automate tasks and processes that
humans can do (generally a lot faster)
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Easy tasks
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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
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Harder tasks
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Find p and q such that p*q = n
It took the international community 17
years to solve a challenge problem where n
=
143816257578888676692357799761466120102182967212423625625618429357069
35245733897830597123563958705058989075147599290026879543541
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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!
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More hard problems
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Largest solved TSP instance
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15,112 cities
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110 processors
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22 years of
computation time
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Status
There are lots of very important and
very interesting computational
problems that are hard to do with
current technology
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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?
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Undecidable problems
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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?
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Perfect virus detection. Not.
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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
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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!
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Computers …
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… automate tasks and processes that
humans can do (generally a lot faster).
There exist tasks that cannot be done
using computers.
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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.
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Logic
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Aristotle, the “first
teacher”
Al-Farabi (‫)الفارابى‬,
the “second
teacher”
Gödel
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Aristotle…
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Start with axioms
which denote “true”
facts
Use rules of
inference to deduce
more “true” facts
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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
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Al-Farabi
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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.
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Muslim Theologians
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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.
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Gödel
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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.
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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.
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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
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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
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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
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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
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Our Universe
Falsehood
Fact
Fact
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.
Fact Fact
Fact
Falsehood
Fact
Fact
Falsehood
Falsehood
Fact
Fact
Falsehood
Fact
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Falsehood
Fact
Fact
Fact
If you don’t believe in logic you can’t
even distinguish facts from
falsehoods in this region.
Fact
Falsehood
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Falsehood
Fact
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The Ultimate Truth
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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
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Searching for the truth
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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…
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Revelation: mercy and guidance
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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.
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Main message of revelation
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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.
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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.
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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.
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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:
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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?
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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!
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Part of belief system: Money
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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.
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Proving your Belief: Money matters
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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 AllEmbracing, All-Knowing.
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Are you going to put your money in a savings account
or give it to the poor?
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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!
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How about charity without Belief?
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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.
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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
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So…
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Believe in the Unseen
Reason speculatively
Seek guidance
Do your best
Hope for the best
Repeat…
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