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

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
. 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.


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.


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

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

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

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.

From: Abdassamad Clarke

See also:


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