Rigorous mathematics and Deductive Reasoning

Rigorous mathematics and Deductive Reasoning

by Abdassamad Clarke

Mathematics uses five main elements: definition, axiom, hypothesis, proof and theorem.

Definitions are created to ensure that the terms and concepts being used in the arguments are clear.

The axiom is a concept which is so obviously true one does not have to prove it; it is ’self-evident’. Sometimes axioms are also called postulates. Based on the axioms, mathematicians make hypotheses or propositions.

A hypothesis is an unproven idea, a jump in the dark. Having done that, the mathematician tries to prove it.

If he furnishes a proof for it by deductive reasoning, it is called a theorem. A theorem is not the same as the saying, “O, but it’s only a theory,” by which is meant ‘a hypothesis’.

Using this method, European mathematicians set about making a logical system out of the mathematics that already existed. They were doing the same thing that Euclid had done with geometry. By the end of the eighteenth and the beginning of the nineteenth century, they had built a very impressive building of pure mathematics. The mathematics of the preceding epochs was inextricably wed to physics and astronomy. Gradually the discipline called Pure Mathematics, which had no apparent practical uses, began to appear.

Remember that they had originally set out to know the mind of God. It is like many human endeavours. Along the way they built a wonderful house of rigorously proved mathematics and they forgot God. They forgot the truth that Allah “created both you and what you do?.” (Surat as-Saffat: 95-96) They said, “We made this building.” Allah says, “If only you had said when you entered your jannah (garden), ‘It is what Allah wills! There is no strength but by Allah’.” Imam Malik said that this is a dhikr to be said upon entering one’s house, because in Arabic a man’s jannah is his house. This is the ayah about the man who owned a wonderful garden, but who did not see the hand of Allah in his own work. Scientists didn’t think of themselves as discovering the order that is in Allah’s creation, but they thought they themselves had built an amazing science.

One day, the French mathematician Laplace presented his newest, most extraordinary work, Celestial Mechanics, to the emperor Napoleon. The emperor said, “Monsieur Laplace, they tell me you have written this large book on the system of the universe and have never even mentioned its Creator.” Laplace is said to have answered, “I have no need of this hypothesis.”

It was almost precisely at this moment in the history of mathematics that the cracks in the building began to show.

Allah says, “The building they built will not cease to be a source of doubt in their hearts unless their hearts are cut to shreds. Allah is All-Knowing, All-Wise.” They thought they had built a house which was truth, i.e. completely sure and certain knowledge. Then they discovered geometries other than Euclid’s which are equally correct mathematically. They cannot both describe reality, i.e. be true. That means mathematical theorems can be mathematically correct but not necessarily true. This was a tremendous blow to the emerging religion of mathematical science. Worse was to come.

Mathematicians found that the simplest things were not really proved clearly and without doubt. Euclid’s geometry was not as sure as they had at first thought. Some of the basic axioms he used were not so clear, and he used others without saying that he was doing so. Subsequent work based on Euclid or on his methods was also not so sure.

This was a great catastrophe. Mathematicians had to go back to the beginning and try to prove a lot of what they had done again. It was as if, having built a really wonderful skyscraper, the builder discovered that there were very serious flaws in the foundations. No one would want to demolish the building and start again, and neither did the mathematicians.

Yet more serious mistakes were found. In the twentieth century, Bertrand Russell and Alfred North Whitehead wrote a book called Principia Mathematica. Russell, a philosopher, logician and mathematician, was trying to arrive “at a perfected mathematics which should leave no room for doubts.” Sceptics said that there is no absolute truth. Russell replied, “Of such scepticism mathematics is a perpetual reproof; for its edifice of truths stands unshakable and inexpungable to all the weapons of doubting cynicism.” This book is in three volumes and even for a mathematician is an almost completely unreadable attempt to prove all of mathematics logically from sure foundations.

Russell said later, “I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere… But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.” The work had failed. It was one of many blows to mathematics as a body of sure knowledge beyond doubt.

You might ask why this should matter. Most people react to pure mathematics with a commonsense, “Let’s get on with the real world.” However, science is increasingly mathematical. If maths has holes, then science has holes – big holes. However, the effect of mathematics is much further-reaching than one would have imagined.

This axiomatic approach had already pervaded all of the sciences and created new ones, although many of the new ones, such as economics, were regarded as pseudo-sciences when they first appeared, as they are in reality. An example of how far it has gone is the idea of constitutional government one of whose foundation documents is the Declaration of Independence of the United States.

Constitutionalism

The Declaration of Independence begins, “We hold these truths to be self-evident…” i.e. these are the axioms. This approach is mathematical without involving numbers or calculation. However, one does need to scrutinise each ‘truth’ which, even though it seems on the surface very wonderful and idealistic, contains a great number of contradictions.

We hold these truths to be self-evident: that all men are created
equal, …

Of course, they are largely equal in having two eyes each, two legs, hearts, livers, love and hate, and in other matters, but are they equal in wealth, intellect, talent, beauty, social standing, strength, wisdom or any other thing? If the equality does consist in having two eyes and other physical attributes does this mean that invalids, crippled people and physically impaired people are less equal, an idea abhorrent to modern people?

…that they are endowed by their Creator with certain unalienable
Rights, …

How did their Creator endow them with these rights? Where is this written? In what revelation? Has the Creator revealed that in Christianity, Judaism, Islam or in any other historical revelation? Or is the writer of this declaration a new prophet with a new revelation?

…that among these are Life,…

Life is a fact rather than a right. Disease, natural disaster and accident may terminate it. People who haven’t read the Declaration of Independence or who do not agree with it may put an end to it.

…Liberty, …

Where does liberty end? Am I at liberty to take my neighbour’s life? Obviously not, because he has the “inalienable right” to life. But am I at liberty to sleep with his wife or his daughter if I so wish and if we all think that we are not going to hurt anybody? Perhaps my neighbour even agrees to that. If I am not so at liberty, why not?

…and the pursuit of Happiness.

If my pursuit of happiness makes someone else miserable, then what? What happens if I do not want to be happy? Perhaps I would like to be miserable. For example, perhaps I want to accumulate a great deal of money and be resented, feared and disliked by large numbers of people, like the late Howard Hughes. Did Genghis Khan want happiness? Did he want to be liked? Had he the inalienable right to do what he did and to seek his fulfilment? Did he care whether he had or not?

That to secure these rights, Governments are instituted among Men,
…

Here the passive verb “are instituted” cleverly avoids confronting the question “who institutes them and how?”

…deriving their just powers from the consent of the governed.
…

How do the governed show their consent? This is not a small problem. Is it the consent of all of the governed, or most of them. How do ‘most of them’ get defined – we have not even broached the problematic nature of statistics in this work. What questions do you ask them to find their consent?

…That whenever any Form of Government becomes destructive of
these ends, it is the Right of the People to alter or to abolish it,
…

What defines its being “destructive of these ends” and who is to decide that it is so? Who are the people? If I disagree with the majority am I then not one of “the people”? If the majority are ignorant and one person is knowledgeable, must he bow to their will? If he knows that some activity is suicidal or destructive and the majority do not care and indeed rather like it, must he be silent?

…and to institute new Government, laying its foundation on such
principles and organizing its powers in such form, as to them shall
seem most likely to effect their Safety and Happiness.”

“Seem most likely” was what the Bolsheviks thought when they formed the Soviet State, and that resulted in almost eighty to a hundred million dead people in Russia, China, Cambodia, etc., and an incalculable amount of human misery.

As you see, whatever seems axiomatic to one person is not always going to be so to another. In fact the people who drew up the above document were as aware of all of the above objections as we are, but they thought that it was only a matter of filling in the details, e.g. by defining the people as “the majority of the people” as shown in elections, etc. This leads to the great complexity of detail involved in constitutional government, to the plethora of paragraphs and sub-clauses, amendments, plebiscites, referendums, etc. Orientalism

Another example of the spread of this technique into other areas than mathematics, and perhaps the most anti-scientific and dishonest example of which we can think is the contentious subject of orientalism. We use this term here to refer to western studies of Islam. Orientalism too contends that it is a scientific discipline. It is clearly a weapon of the enemies of Allah. However, the Muslims have largely failed in dealing with it because they deal piece-meal with its multitudinous propositions, whereas what should be dealt with is its dishonestly unstated axiomatic base. Euclid and subsequent mathematicians stated all their definitions of terms and their axioms; orientalists state none of these things.

For example, what is perhaps the bible of orientalism is the Leiden-published
Encyclopaedia of Islam. This is a distressing and ugly, but apparently erudite, set of tomes. It is the clearest evidence of the entirely unscientific axiomatic base of orientalism – we seek refuge with Allah from the evil of it and His forgiveness for mentioning it – that is that the Messenger of Allah was a perfectly ordinary human being who did not receive revelation but compounded the entire edifice of Islam from fragments of poorly comprehended jewish and christian materials, and that where it disagrees with the Qur’an, the Biblical literature is always decisive. They insist that subsequent Islamic thought elaborated Islam on this basis, and added into it stolen pieces of Buddhism and neo-Platonism, etc., etc. The proof that these axioms are false is that they have never been clearly stated as being the axioms and the premises of orientalism. An axiom, to be a correct basis for a scientific study, must be so self-evidently true that it needs no proof. This ‘axiom’ is a mere prejudice, and at best a proposition which is impossible to substantiate and which, if stated as a proposition, would be easily refutable.

If these were not the axioms of orientalism we should expect at least an equal amount of literature examining propositions based on the opposite axioms, i.e. that Muhammad, may Allah bless him and grant him peace, is the Messenger of Allah and the seal of the Prophets and Messengers, and that the Qur’an is the revealed Speech of Allah confirming that which came before it of other revelations to other prophets and messengers and clarifying the many distortions to be found in other scriptures. However, sceptical orientalism regards this as merely a proposition which has to be proved, whereas the other is self-evidently true and needs no proof, only needing subsidiary propositions merely to fill in the details in this prejudicial picture.

Thus, many naive Muslims strive mistakenly within the academic nexus thinking that orientalism is a rather well-meaning judaeo-christian affair which just needs to be guided aright. They combat bravely various sub-theses of this monstrous lie, without ever confronting the central thesis, the deceitfully unmentioned axioms. Modernist Islam

Within Islam too, this mathematical approach has found a home. Muslims who believe in deriving shari’ah from ‘Islamic principles’ are following the same method. The ‘principles’ are axioms, and are not the same as the traditional usul – ‘roots’ or ‘sources’. These latter are the sources of the din – among which are the Book, the Sunnah, and the consensus of the people of knowledge, etc. By following ‘Islamic principles’ one very often arrives directly at a result that contradicts the well-known shari’ah of Islam.

For example, Abu’l-A’la Maududi, founder of the Jama’at al-Islamiya in India and Pakistan is a classic case of someone who follows this method. We take a few examples at random from only one of his books, The Islamic Way of Life, to illustrate this approach.

In Chapter one, The Islamic Concept of Life, Maududi says, “There are certain postulates which should be understood and appreciated at the very outset.” Postulate is another term for axiom, so here Maududi has clearly set out his intention to create a new type of Islam based on this mathematical approach, rather than on the traditional usul.

Under Basic Postulates Abu’l-A’la includes in number one, “Man has also been invested with freedom of will and choice and the power to use the resources of the world in any manner he likes. In short, man has been given a sort of autonomy while being appointed God’s vicegerent on the earth.” First, although the author invokes the mathematical approach with the use of the term postulate, this postulate which he uses has none of the rigour, tight definition and exactness of mathematical postulates. There are so many elements in this one statement that it is meaningless to call it a postulate.

Without entering the fruitless and forbidden debate between advocates of free-will and advocates of predestination, Maududi has clearly and immediately given very strong indications that he is ideologically a member of the group who used to be known as the Qadariyyah – the proponents of free-will and those who deny the decree of Allah. The second part of that assertion is more evident by emphasis and omission than by any declaration. When the angel Jibril, peace be upon him, asked the Prophet, may Allah bless him and grant him peace, to describe Iman – belief – he enumerated its elements as, “That you believe in Allah, His Messengers, His Books, His angels, the Decree – the good of it and the bad of it, and the Last Day.” The last two terms deal with the great paradox of human existence that everything is decreed by Allah, exalted is He, from before the creation of the cosmos, and that the human being must face a reckoning for his deeds.

Imam Malik said, “The people who believe in the doctrine of free-will (al-Qadariyyah) are the worst of mankind. I see them as fickle people of shallow intelligence and innovations because of many ayat which there are against them, of which there is the words of Allah, mighty is He and majestic, ‘The building they have built will not cease to be a bone of contention in their hearts’ (Surat at-Tawbah, 111), and of which there is ‘And He revealed to Nuh, “No one of your people will believe except for he who has already believed”,’ (Surah Hud, ayah 36) and He said, ‘And they will not give birth to any but wicked disbelievers,’ (Surah Nuh, 27), ‘You will entice no one to them except for him who is to roast in the Blazing Fire,’ (Surat as-Saffat, 163) and He said, ‘but Allah was averse to their setting out so He held them back” (Surat at-Tawbah, 46) and in many other ayat.’”

The Prophet, may Allah bless him and grant him peace, is reported to have said a number of things about the proponents of free-will, including that they are the worst of mankind. Imam an-Nawawi narrated, “It is established as an authentic statement that he said, may Allah bless with him and grant him peace, ‘The Qadariyyah are the Magians (i.e. dualists) of this community’”. It is not our purpose here to refute them or to enter into a polemic on the matter. Rather we want to show how much a modernist like Abu’l-A’la al-Maududi has imbibed the mathematical approach as shown by his language of postulates. His first postulate contains by omission and by emphasis something, which is further repeated and elaborated throughout his book, entirely against the Sunnah.

Again in Chapter three, Essential Features of Islamic Political System, Maududi begins, “The political system of Islam has been based on three principles, viz., Tawheed (Unity of God), Risalat (Prophethood) and Khilafat (Caliphate).” Here postulates are exchanged for principles, but the thinking is the same. We only note that the equation here fails in the first term “political system of Islam” because it introduces two terms which would not be recognised classically: “political” and “system”, and introducing matters into Islam is called classically bid’ah. Maudud was enamoured of all things western so much that he wanted to remake Islam entirely in its image. Yet, he did not have enough knowledge of western society to know of the immense literatures in criticism of ‘politics’ and ’systems’.

By the process of these three principles Maududi further arrives at ‘the State’ and ‘Islamic democracy’. The State was no part of early Islamic thinking and is clearly another innovation. If we examine the Arabic term dawlah, which is often translated as ’state’, it means a ‘turn of fortune in battle’. The word does not occur in the Qur’an but another word from the same root does, doolah and it means a ‘turn of
fortune in terms of wealth’. The former is ‘political’ and the latter ‘economic’. However, our term ’state’ is related to ’static’ which is precisely the direct opposite of the Arabic term. The obsessive drive to create a state is a desperate fear of the dynamic nature of history and of the turning this way and that way of fortune, i.e. Allah’s eternal decree of the winners and losers. The state in western thinking is also something which legislates, i.e. creates laws. The dawlah for the Muslims may never create laws, but it can only implement Allah’s revealed law.

With respect to Islamic democracy, another innovation, Maududi says, “Every person in an Islamic society enjoys the rights and powers of the caliphate of God and in this respect all individuals are equal.” Here we see the ‘equality’ of the French Revolution raising its not so unexpected head. These words hark back to a group called the Khawarij – literally ’seceders’ – who also affirmed a kind of radical understanding of equality, which led them to murder Sayyiduna ‘Ali, may Allah honour him, murder being the ultimate weapon of egalitarians.

Again later in the same chapter, Abu’l-A’la introduces the concept of Fundamental Human Rights for all mankind. This is clearly another innovatory introduction of something foreign into Islam.

In Chapter IV Islamic Social Order, he begins, “The foundations of the social system of Islam rest on the belief that all human beings are equal and constitute one single fraternity.” We have already questioned the concept of equality. It is doubly questionable here because Allah, exalted is He, specifically refutes it in many ayat in the Qur’an, e.g. where He says, “Say: ‘Are the blind and sighted equal? Or are darkness and light the same?’” (Surat ar-Ra’d: 17) And, “Do you make the giving of water to the pilgrims and looking after the Sacred Mosque the same as believing in Allah and the Last Day and doing jihad in the Way of Allah? They are not equal in the sight of Allah. Allah does not guide the people of the wrongdoers. Those who believe and emigrate and do jihad in the way of Allah with their wealth and themselves have a higher rank with Allah. They are the triumphant.” (Surat at-Tawbah: 19-20) In this latter ayat, Allah differentiates the people who struggle in the way of Allah from other believers. The ayat which differentiate Muslims from jews, christians and other unbelievers are too numerous and too well-known to mention. It is complete nonsense to say that humans are all equal and one great brotherhood. But of course ‘brotherhood’ is the last term from the slogan of the French Revolution. That brotherhood was to be achieved, as Marat proposed, by the removal of, “Two hundred and sixty thousand aristocrat heads.”

Al-Maududi continues in the same chapter to say, “The foremost and fundamental institution of human society is the unit of the family,” but the family is not the unit of the society, but a unit possibly of a clan or a tribe or a race. Islamic society begins when people pledge allegiance to their leader, not because of any familial relationship or tribal culture.

Most significantly in Chapter V, Economic Principles of Islam, Maududi, says, “Islam has laid down some principles and prescribed certain limits for the economic activity of man…” Note here that Islam rather than being ’submission’ and ’surrender’ has now become an active entity laying down principles. This leads on to something quite crucial, “Islam does not concern itself with time-bound methods and techniques of economic production or with the details of the organisational pattern and mechanisms.” The statement is ambiguous. It can lead easily to the interpretation that the economic patterns of the right-acting first generations are not a source for our shari’ah. That cuts us off from a clear model of a non-usurious economy and leaves us adrift in the sea of ‘Islamic Principles’. That was what actually and quite conveniently led many of Maududi’s followers into directorships of Islamic banks and other similar usurious institutions.

Perhaps, this is sufficient to show the penetration of this type of mathematical approach into the thinking of just one of the exponents of now out-dated modernist Islam. However, please note that any mathematician, philosopher or person trained in that type of thinking would faint at the weakness of thought displayed here, the falsity of its logic, and the emptiness of its conclusions.

We put these examples here, to illustrate how widespread is the basic idea which is at the core of the mathematisation of science, and how much damage it can do.