Three Hard Questions and Quest for Their Solution

Mr. Gill

As a dabbler interested in Mathematics I am really grateful to you to bring this subject up.

I find the documentary about the solution to Fermats` Theoram. I have watched this several times and always in a mesmerised awe. In my opinion there is no drama produced whcih has such rivetting drama element to it. I sometime discuss this with a friend who has a phD in Mathematics.

THe other day someone had mentioned about a hindu scientist/mathematician who has perhaps found something/something ( I am relustant to write formula because I am cannot recall) about prime numbers. I have bee looking for that name ever since. Could you please elucidate a bit on that.

Rest of your artcle seems very interesting as well and I sincerely hope it draws the attention from the right quarters especially DRUMS, Urstruly, Sameer, rsridhar, and perhaps a few others. I will read the whole article at a litle bit of leisure.

Thanks again.

While it is true that one cannot speak on the ineffable, it is just as true that we have many inneffable concepts we use to induce precision. In mathematics, the CONCEPTS of zero and infinity are also ineffable, as is true symmetry (when we compare human produced geometry with the ``perfect circle``). Thus if god is ineffable (which it is, aside from referring to it in paradox), then we have little right to speak on numbers.

We should keep in mind which PEOPLE are pursuing the ``theory of Everything`` and from where they get their foundation. Current science sees the world through the filter of the LEFT brain. This ASSUMES an innecessary premise (Proof of God`s existence must come from scientific means). Do the math, Invalid Premise = _______ conclusion. We are not told why God cannot be accessed by other means (right brained, inspiration, introspection, paranormal activities and so on), its only implied that such methods are not factual as they cannot be repeated in a lab. Well neither can a lot of physics (we still have no proof on the properties of black holes).

Left brained arguments defeat themselves simply because one is deducing laws applicable to one field (finite) and applying them to another (infinite). It is logically flawed. The raises the question of what constitutes ``proof.``

What`s provocative is that the notion of God is central to many religions, yet most religious people have likely not even thought about this stuff.... Buddha was right, it really doesnt matter.

[ ... that is, four dimensions of space and one dimension of time ... ]

I thought space has three dimensions and not four.

Gill Sahib: I read the three questions with interest. Here are some comments:
On Question 1: I enjoyed the discussion on Fermat`s last theorem where you stated clearly and simply what the theorem was and confirmed the fact that it had been solved. It would be interesting if it is possible to describe Wiles logic whereby he solved the theorem in terms that even a chowk-ignoramus like myself can understand.
On Question 2 (the God question): this is a hoary debate by now I think, and trying to put logic here is I think a chase after a wild goose that no one has ever seen. I was surprised by your omission of Kant`s famous works in this regard where he is said to have first proved, then disproved, the existence of God. I think the question will always remain one of faith, or at least until the day humanity is so far advanced scientifically that it can travel back and forth in time at will, criss-cross multiple universes, and indeed be able to shake hands with God. I dont think that day is anywhere near. So, to me this question is as relevant as trying to determine how many angels can balance on the head of a pin (as the mullahs of middle age europe used to try and do). The more relevant question, to my mind is: can we as humans live up to the values described by the 99 characteristics of God (being fair, just, merciful etc.) in our everyday life?
On Question 3: This is essentially the same question as question 2, put differently: Can man ever know everything there is to know? History shows that it is dogmatic people have repeatedly become arrogant and believed they have found some ``final`` laws of nature. And then their neat little world is shattered by new findings. Copernicus shook church dogma when he removed earth (and thereby man, and thereby the church mullahs of his day) from being the center of the universe. Einstein and Planck did this when they showed how the Newtonian edifice (which physicists of the late 19th century thought represented the final laws of nature) straddled time-space only in the same sense that an ant-hill is an entire universe (i.e. it is an entire universe only to the ants that live in that ant hill). And as we enter into the 21st century, there are rumblings of changes to come in our understanding of the workings of nature that will dwarf even the incredible achievements of Einstein.

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Gill Sahib,

The three questions you`ve considered are of quite distinct characters.

`Fermat`s Last Theorem` is a straightforward and very specific statement about numbers. It is easily the most precisely-defined and unambiguous of your questions. An utterly mind-boggling amount of new mathematics has been discovered (created, if you prefer) in the nearly four centuries since Fermat made that marginal annotation in his copy of Diophantus. New mathematical fields and sub-fields have been discovered and explored in great depth, whose very existence was utterly unsuspected or at best only dimly perceived in Fermat`s day; sub-fields have in turn sometimes reunited in unexpected and fruitful ways, etc. Fermat were he to be resurrected today would find the subject matter and technical apparatus of contemporary research mathematics unrecognizable at first glance. However, the `rules of the game` of coming up with a proof of FLT have not changed (or changed very little) in the period between Fermat`s conjecture and Wiles` proof. Given a couple of years to catch up on three-and-a-half centuries worth of mathematics, Fermat would probably agree today that Wiles proof is valid and fair, though far more sophisticated than he could have conceived.

The so-called question of the `existence of God` is at the other end of the spectrum of precise definition. It is evident that different people have a multitude of quite different, though commonly vague, imprecise, and ill-defined, notions of God, when they raise this question. More often than not the very same person who ponders and obsesses over this question has multiple notions of God in his mind and blithely conflates them without being aware of his bemuddlement.

At one extreme, one might identify God with the Universe or with Reality itself, or at a few removes of abstration, with aesthetically perceived Harmony, Symmetry, Elegance, abstract Order in the laws governing natural phenomena. This notion is something that many if not most scientists recognize and believe in. At another extreme one might conceive of a sort of ultimate ineffable, an unmoved mover, an uncaused cause, all-encompassing, infinity of infinities, etc. This notion is connected with familiar antinomies. At a third extreme one might have in mind a personal and basically anthropomorphic God, in turns jealous, beneficient, vindictive, just, merciful, omnipotent, demanding fealty and obedience, requiring worship and propitiation, laying down laws by which human beings ought to live and judging individuals by their adherence to these laws, etc; a being who supposedly cares about the lives and affairs of human beings and societies. This is the familiar Judaeo-Christian-Islamic divinity, and has clear tribal origins. I have yet to see a good reason to believe that these several distinct conceptions (and there are no doubt many others) - even if they are all well-founded - refer to one and the same entity.

As to various alleged `proofs` of EoG, these are not airtight logically impeccable proofs in the mathematical sense. They are better regarded as attempts to create faith in the mind of the True Believer, and in that sense they serve the same function as supposed `miracles` and `Signs to the Faithful`. On close examination some - such as Anselm`s ontological argument - are seen to depend for their force on the confused and imprecise uses of concepts. Others - such as the cosmological and teleological arguments - boil down in their essence to `the negation of this proposition is inconceivable, therefore it must be true`. As far as I can discern, their overall effect seems to be to befuddle, bemuddle, bedazzle, becloud, bemuse, bewilder, bewuther and in short to bludgeon, befog and benumb the believer`s baffled brain into accepting religious faith, on the grounds that something confusing and convoluted but supposedly profound must be sound. So perhaps God might be fruitfully identified with the befuddled believer`s bafflement.

Some of the more defiantly honest, if logically unsustainable, excuses for religious faith have been Tertullian`s `Certum est, quia impossible` (`It is certain, because it is impossible`) and Pascal`s `The heart has its reasons of which reason knows nothing`. Many people who proffer arguments for religious belief lack this intransigient honesty. Requests that they defend or explicate their claims are typically sought to be dismissed with an airy hand-waving ``Why, it`s obvious!``. Should the tactless questioner be persistent he can expect to encounter a smokescreen of obfuscation concealing vacuity.

Finally, the sought-after `Theory of Everything` is a theory of physical reality but probably not of all conceivable phenomena. Criteria of success of a ToE seem more vague than for FLT, and with a strong aesthetic component. Nevertheless this is probably a less futile endeavour than the searchfor a proof of EoG.

GhalibZaman

``THe other day someone had mentioned about a hindu scientist/mathematician who has perhaps found something/something ( I am relustant to write formula because I am cannot recall) about prime numbers. ``

Prime numbers were always a fascination to errr... ``Hindu mind``. I think you are talking about the recent breakthrough in the field of ``Number theory`` by a team comprising of a couple of IIT Kanpur kids Neeraj Kayal, Nitin Saxena and Dr. Manindra Agrawal. Basically they have developed an algorithm to find out quickly and definitively whether a number is prime. You can download the celebrated paper from this IIT Kanpur website that contains the media reports, paper and the photographs of the team members.

http://www.cse.iitk.ac.in/news/primality.html

For a more laymannish explanation, you can visit the New York Times report, the link to which is provided by the above site.

PS Number theory also fascinated one of the greatest Indian mathematicians of the modern era, Srinivas Ramanujam. I remember from our earlier exchange that you are well aware of his achievements.

A Birthday Greeting for Zulfikar Ali Bhutto.

He left with many questions.

With Pardons.

Ras H. Siddiqui

AlephNull #7:
[As far as I can discern, their overall effect seems to be to befuddle, bemuddle, bedazzle, becloud, bemuse, bewilder, bewuther and in short to bludgeon, befog and benumb the believer`s baffled brain into accepting religious faith, on the grounds that something confusing and convoluted but supposedly profound must be sound. So perhaps God might be fruitfully identified with the befuddled believer`s bafflement. ]

Thanks AlephNull for the best, beautiful, precise and ``concise` statement. Is this your own brainchild?

I was not going to interact at this thread but your post forced me to post the first and the last post here. The reason for boycotting was simple. It was the use of supposed finesse but sneakish in reality, of sandwiching - an antique, mediocre at best, a poor theoretical religious contruct developed by mediocre fringe elements at the periphery of Pharoah and Babylonian civilizations, forced humanity into acceptance with all possible cruel means leading today to about two third of world population into believing it - existence of god in between two perfect scientific endeavors.

There are cleat principles, explained in a number of books, for empirical methods in scientific studies. It is not about hitting darts on a huge dart board. Most of all, as you also mentioned, is the scientific understanding of mathematically explainable scientific and laws of nature. That is what Newton achieved by mathematically coming up relationship between gravity and force.

The existence of god is not a hard question at all; it is a desire for some believers to convince those remaining non-believers whose forefathers were fortunate not to succumb to the might of swords, guns and cannons and onslaught of other forms of pressure tactics or agnostics.

The so-called hard question, ``does god exist`` could have been sandwiched in between science fictions such as time machine and cold fusion. That is where this `hard` question belongs with other similar `hard` questions. Sandwiching between two perfectly scientific areas is akin to cheating or playing trick or a poor attempt to elevate the intellectual level of a silly and, at best, a mediocre figment of imagination of antiquity.

Okay here is a very simple but interesting puzzle taken directly from ``The Man who knew Infinity``.

Say the house numbers in a particular mohalla start from 50 and continue as 51, 52... and so on. The numbers can only be integers. You have to find the house number of Mr. Saxena. The only thing you know is that the sum of all house numbers before Mr. Saxena`s house is equal to the sum of all the house numbers after Mr. Saxena`s house in the row. Now try finding out Mr. Saxena`s house number and the final house number in the row. The final house number cant be greater than 500.

For instance, one such solution is (6,8), i.e. Saxena`s house number is 6 while the final house number is 8. Now 1+2+3+4+5=7+8. There are infinite solutions to the above problem. If anybody succeeds in solving the above puzzle(writing a small code is the easiest way), can you obtain a more general form of the solution so that the you can tell the solution given a range of numbers.

These are unanswerable, insoluble, quests. I haven`t read this essay. Not because I feel above it, or I have a condescending attitude towards it. Far from it.

It is so, because my mind will start from ``casual chains``, and go to ``conjunctures`` -- thereby negating my own visualization of `causation`
-- and ultimately to `X` which Huxley, you would recall, called ``God``. All right, we start all over again from this wrap-up, from this terminus!

Now, all this would be fun if lives were not measurable in months -- let`s say 480 months, average, since in the first 240 you don`t even know which end of the world is up! And the logical end of such pursuit, ag`r t`lb sadiq ho, to, will be suicide, madness, lunacy or, the m`a^iraj of quest, takhta-é dar!

I said above that I haven`t read your essay or term paper. Now, that is not entirely correct. I have in a way `thumbed through it`. After all, I didn`t wish to sound utterly ignorant. Yet, both you and I are!! You see, once you and I have ``mastered`` the nature of the problem, concept, thought, conviction, schema or import, it becomes a ``yester-day`s
solved`` riddle or puzzle; a consumer of our meagre treasure of months ``, or parts thereof``. See if the following feels worth humming.

[Yeh jama-é s`d chak b`dl lainé maiN kya thha,
Moh`lat hii na dee Faiz hamaiN bakhia-g-ree naiN]

I am familiar, after a fashion, with your academic background. We share a common universe of discourse, to a degree. But where do we go from here? Where do I go? I can`t even understand what I have just written!

The presentation of an idea [and here we can, to express our `conception` of this word, use as many synonyms from the built-in thesaurus as we wish or to the point where we physically or cerebrally tire] is always -- as a rule of rational behaviour -- constrained by the ambience, the milieu of its presentation or its birth . This is a sensitive, time-space and `available-web resources` ``hole``. I believe that the square peg of the posited question is too big to even be relevant in this milieu.

To wit, then: `` We share a common universe of discourse, to a degree. But where do we go from here? Where do //I// go? I can`t even understand what I have just written?``

P.S. At the time I wrote these lines only Ghalib Zaman`s post had appeared. He seems to have benifitted by it and I am happy for him.

At the time of posting this, a day later: Sameer, love, as we say in Pakistan, ``how`s ya doin`?`` At last I caught you. Greetings and good thoughts. Aap ham-d`m-é déreena haiN, aur aap ka milna kum nahiN, mulaqaté Masiha `O Khiz`r sé. Zinda r`haiN, aur khush!

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Question -- IF God created MAN -- WHO creaed God?

Answer -- Man created God

Question -- who came FIRST -- God or Man?

Answer -- MAN MAN -- MAN came first -- THEN and THEN only -- came GOD...

want to verify it? -- ask ur dog -- if he knows who GOD is --

Equazon solved -- folks

#13

Hasan sahab,

As someone who works with science, I`m sure you appreciate the incredible complexity of the human body (just the variety of calcium transport mechanisms in the renal nephron is enough to stagger the mind; I know, I have to memorise them!). Contrast this with the simple unicellular zygote, which is where we all started from. It`s bewildering how so much could have gone wrong so many many times and yet it didn`t.

God doesn`t need us to worship him. Actually, God doesn`t need us, period. But can we really imagine our own nothingness?

Aamir

Gill sahib, I really enjoyed this article of yours. Coming from a physical sciences background myself (though with applications in medicine) I find mathematics and the like fascinating. Simon Singh`s book on Fermat is excellent...

I also am interested in philosophy too and the arguments for/against God and also Sufism (being a murid of the Most High Naqshbandi Order)

You are right that logically (taking logic in its strict Aristotlian sense) God`s existence cannot be proven. Imam Ghazzali came to the same conclusion and suffered a nervous breakdown of sorts as he himself tells us in his autobiography al Munqidh min al Dalal...but his faith in God`s existence never left him and his faith was confirmed and proven through the spiritual path of the Sufis. Thus we can say that there IS a spiritual proof of God`s Existence which can confirm one`s blind faith; yet the path to this proof is long and ardous and full of difficulties--ie. the Sufi Path.
Becoming annihilated in God (fana) is the goal of this path and of Islam.
When one does reach it, in Ghazzali`s words one has as much certainty of the existence of God as one does of the existence of a physical object which one is holding in one`s hands. For then `God` is no longer an abstraction but a Being of which the Sufi has experience of spiritually.

``My period of retreat had lasted about ten years,148 during which I had innumerable inexhaustible revelations. It will be enough to say that the mystic Sufi follow to an uncommon degree the Way of God. Their behavior is perfect, their Way is straight, their character is virtuous. If to this were added the good sense of the reasonable, the wisdom of the wise, and the knowledge of the doctors of law, could one be sure that this would improve their behaviour or character? Surely not! Every action or state of theirs, their outward appearance and their inward conscience, is illuminated by the flame of prophecy sitting in its niche149 beyond which there is no other light on the face of the earth.

What can be said about such a Way? Its purification consists above all of cleansing the heart of everything which is not God, the Almighty. This begins, not with the state of sacralization which opens prayer,150 but by the fusion of the heart with God`s name, and is completed by the total annihilation of the self in God.151

Even this completion is only the first step with regard to one`s free will and all that one has learned. It is the first step on the Way itself. What went before152 was only the waiting room.

Once one has started on the way, one begins to receive inspirations and visions. The mystics keep vigils in which they even see angels and the spirits of the prophets. They hear their voices and have the benefit of their counselling. From these visions of images and symbols they ascend further to degrees of spirituality which cannot be described. Nobody can attempt to express these states of the soul without failing miserably.

In a word, the mystics achieve a nearness to God which for some can be a virtual in-dwelling,153 total union,154 or fusion155 with God. This is not true, as we have shown in our treatise al Maqsid al-Asna (The Greatest Goal). A person in such a state should say nothing but these lines.156 ``Whatever has happened, I shall not speak of it. Think well about this. Do not question me about it.``
http://www.muslimphilosophy.com/gz/md/IIA-02main.htm (from Deliverance from Error by al-Ghazali)

``Whereof one cannot speak, thereof one must be silent.``-Tractatus, Wittgenstein Online edition..