Islam: The Religion of… Logic? (3/3)

Not often described as the Religion of Logic, Islam had a golden age spanning from the 8th-12th century (CE). This is the last in a three-part post on logic and rational thought in Islam. The previous post looked at the Mu’tazilites, who believed in reason and rational thought above all else. Here we will look at applications and types of logic in Islamic law making.

Islamic Jurisprudence (fiqh)
In the modern day there are many schools of fiqh (madhhab) which can be seen in this world map. The four accepted Sunni madhhabs are Hanafi, Shafi’i, Maliki and Hanbali. The Shia’s have the Ja’fari, Isma’ili and Zaidi madhhabs. Mu’tazilism, being a system of theological interpretations, doesn’t exactly have a madhhab. This gives a rather confusing situation where you can have “Sunni Mu’tazalites” and “Shia Mu’tazalites”. This would be Mu’tazilites roughly following, for example, a Hanafi or Zaidi madhhab. The movement was predominantly of Sunnis, notably the founder Wasil ibn Ata (a good friend of Zayd ibn Ali) and scholar Abd al-Jabbar, but there were also Shi’ite Mu’tazilah like Ayatollah Hilli and the poet Ibn al-Rumi. The introduction of Mu’tazilism on the kalam of Judaism even brought about “Jewish Mu’tazilites” such as David ibn Merwan al-Mukkamas. Though again we see another area of Islam wherein Mu’tazilites are shown to be very different from standard Islamic views.

The Sources of God’s Law (Sharia)
Fiqh is the process used for creating, understanding and applying religious laws, for the main part madhhabs can be seen to follow a process of stages involving different religious sources: The main ones being the Qur’an, Sunnah, Ijma, Qiyas (Sunni) and ‘aql (Shia). The madhhabs give different weight to different sources. Hanbali, for example, give credence to the first two stages and ignore the last ones.

1 2 3 4
Sunni Qur’an Sunnah Ijma Qiyas
Shia Qur’an Sunnah Ijma ‘Aql
Mutazilah ‘Aql Qur’an Sunnah Qiyas*

ijma – meaning “consensus” either by religious authorities (Sunni), the Imam (Shia) or the Muslim community (Ibadi). Sunni’s often use the Companions of Mohammad (Sahabah) as the religious authorities, looking for their consensus in Hadiths. Using ijma is invalid for Mu’tazilah because of a rational scepticism towards peoples’ ability to make mistakes. For this same reason Hadiths are treated with caution and discarded if they contradict the Quran. Both the Shia and Mu’tazilah held critical views about the first generation of Muslims, with Ibadi also viewing Uthman and Ali as less than righteous.

Qiyas – meaning “deductive analogy” in reference to what is written in The Qur’an and Hadiths with what is being assessed. Because most Mu’tazilites follow Hanafi teachings Qiyas were often accepted, though not always. Notably Ibrahim an-Nazzam who denied Qiyas, Ijma and even Sunnah as sources for Sharia stating that only The Qur’an and ‘aql were acceptable.

‘Aql – meaning “reason” is intellect in terms of the rational faculty of the soul, deep understanding of God’s words, Imams have ‘aql. The term is slightly different when seen from Mu’tazilah doctrine as much closer to rational logic than religious understanding. Where Qiyas are analogical reason, ‘aql is pure logical reason.

A Note on Ibadi’ism
I found two completely contradictory Ibadi views on Qiyas and ijma, the first was Ibadi madhhab rejects the 3rd and 4th stages as a form of innovation (bid‘ah). This follows from the generally conservative nature of Ibadi Islam, but then in a state-published book on Ibadi’ism from Oman it says they follow 5 stages of fiqh: Qur’an, Sunnah, ijma, Qiyas & Induction (istidlal). The use of ijma can be seen to follow directly from the democratic nature of the Ibadi caliph. So I’m not sure which is the case as they both make sense for different reasons.

Sources: Initially Wikipedia then al-islam.org (Shia), livingislam.org (Sunni) then videos of scholars, academics and documentaries on Youtube (Sunni & Shia & Ibadi). The information was sporadic and sometimes contradictory so please feel free to correct me if I made mistakes.

Islam: The Religion of… Logic? (2/3)

Often described as the Religion of Violence, Islam had a golden age spanning from the 8th-12th century (CE). The previous post brushed over the Islamic Golden Age and Kalam. This post introduces a prominent theological school that lived and died in that time.

The Mu’ tazila

Logical-Koran-2

Paradoxical statements in the Qur’an had to be logically qualified.

Their theology was and is incredibly different to mainstream Islam for many reasons, the big ones being belief in Free Will, Atomism, rationalising discrepancies in the Qur’an and that the Qur’an itself was created. Now although the Qur’an is a primary source for knowing God’s laws – pure analytical reason gets the final say! Listed below are the five fundamental beliefs in Mu’tazilism, although monotheism and divine justice are standard for virtually all forms of Islam the interpretation makes them very important. The First Principle, as stated in the Mu’tazila text Kitab Al-Usul Al-Khamsa, is that good and evil can be known solely through human reason (without revelation) and that it is a Muslim’s duty to try and know God in this way. This principle, the autonomy of intellect, underpins the five fundamentals below:

  • MonotheismTawhid,  better expressed as the oneness of God, has had slightly different meanings over the history of Islam. Usually the Qur’an is seen to be an essence of God (His word) and thus co-eternal with Him. Mu’tazilites strict interpretation of tawhid says the Qur’an cannot both be part of Him and apart from Him, so the Qur’an cannot be eternal and thus is created. This became one of the most contested positions in Islamic thought. A view shared by the Ibadi Muslims of present day Oman.
  • Divine JusticeAl-‘Adl, there are many divine attributes and although virtually all schools of thought believe God to have justice as one – the strict analysis of it by Mu’tazilites brings about a controversial view. In answer to the Problem of Evil, like in Zoroastrianism, they respond by introducing Free Will – something completely opposed to the determinism of mainstream Islam. This is because if God is divinely just then he cannot create someone, command them to do evil then punish them for doing so.
  • The Promise and the Threat – At-wa’d wa al-wa’id, this is divine retribution. For this reason one must try to know God through rationality in order not to inadvertently disobey Him (and burn in Hell).
  • The Intermediate PositionAl-Manzilah Bayna al-Manzilatayn, when a Muslim sins they do not become a disbeliever (kafir) but neither do they stay a true believer (mu’min). If they die in this state they will be judged by God separately from a mu’min and a kafir. This view sits between the Kharijite position that sinning is disbelief (a view shared by modern day Islamic Extremists) and the Murjite position that a sinner is still a believer until Judgement Day where God will decide.
  • Commanding Good, Prohibiting EvilAl-‘amr bil ma’ruf wa al-nahy ‘an al-munkar, an obligation for all mu’min. This is the maxim that led to political intervention in the Abbasid Caliphate.

Mu’tazila: Sunni, Shia, Ibadi?
There are quite a few sects/schools/movements in Islam, this infograph shows the types that are around today. It doesn’t mention Mu’tazila and in my reading I have seen them referred to as Sunni sub-set, their own sect and even not Muslims at all. I would say there is enough difference in views to call them an independent sect. If we look at views on who can be a caliph we see that Sunni say the caliph must be from the tribe of Mohammed, Shia say the caliph must be from the family of Ali then the Ibadi say it can be anyone of strong faith. The Ibadi view was one adopted from their predecessor the Kharijites, the Mu’tazila share the Ibadi and Kharijite view on who can be caliph. But their rational dispute with the texts and onus on self-reasoning is contrary to conservative Ibadi/Kharijite views. Although the lines do blur as we will see in the next post, Mu’tazilites were quite the contrarians of their time – and even of our time.

Nobody Expects the Islamic Inquisition! (Minha)
The fifth doctrine, commanding good and prohibiting evil, invoked political action during the Abbasid Caliphate. Pro-Mu’tazila ulama (religious officials), including the caliphs al-Ma’mun and al-Mu’tasim, began interrogating scholars and ulama who did not believe in the Jahmite and Mu’tazilte view of Quranic Createdness. Imprisonment, punishment and even death would fall on those who did not concede. There was a growing ‘traditionalist’ re-serge in Sunni Islam at the time which promoted much the opposite and the fact that Mu’tazila shared views in-line with Zoroastrians and Shia muslims did not win them any favours with the populous. In the field of kalam two more systematic schools emerged in response, these were the Ashi’arites and the Maturidis. The latter was a hardline reaction that advocated Quranic literalism and threw out rational applications. The Ashi’arite school was the middle ground and found much support.

The 10th caliph of the Abbasid reign, Al-Mutawakkil, reversed the order of Minha and with that the ulama freely became less accepting of Mu’tazilites. The general community were already rather against them and Mu’tazilites quickly lost any power or influence they once held. Even after the Mu’tazilites had all gone – their practices still continued, mainly with the Zaidi Shias of Yemen, but also Ismaili Shias, Karaite Jews and certain Sufi schools had by this time all adopted different aspects of Mu’tazila doctrine. The Ashi’arite school of theology which advocated a lighter version of rationalism became moderately accepted in the Sunni mainstream.

The next post is on interpreting sharia (law).

Sources: Initially Wikipedia then mutazilah.com, asharis.comal-islam.orglivingislam.org, scholars on Youtube (Sunni & Shia, including Wahhabist Feiz Mohammad), academics and documentaries online. The information was sporadic and sometimes contradictory so I also bought the book Defenders of Reason in Islam which is a detailed analysis of the movement’s initial serge all the way up to it’s revival in our modern times.

Islam: The Religion of… Logic? (1/3)

Often described as the Religion of Peace, Islam had a golden age spanning from the 8th-12th century (CE). This is the first in a three-part post on logic and rational thought in Islam. Each post will look at a different relationship analytical thought has had with the Arab-speaking populous.

Liberalisation of Science and Philosophy
During the Golden Age great advances were made, importantly the House of Wisdom was set up in Baghdad. From here the first ever international scientific venture in history began; Wherein, large volumes of written knowledge from Persian, Greek, Latin, European and Indian origin were translated to Arabic. The great Arabic polymaths such as Al-Kindi, who had the earliest writings on encryption by frequency analysis and wrote On the Use of the Indian Numerals, worked from the House of Wisdom. The biggest star of the Golden Age was ibn Sina (Avicenna) who made so many contributions – the biggest being The Canon of Medicine which, written in 1025, was used as a medical standard from England to China for about 600 years. Like many thinkers were at the time, Avicenna was also a logician and he disliked Aristotelian logic. For example when looking at ‘if p, then q’ he believed it too presumptuous to assert an such a strong relation between p and q. His response, ‘q while p’, is the beginnings of Temporal Logic. Below are some examples of his Temporalis (While Logic) :

  • Whenever the Sun is out, then it is day
  • It is never the case that if the Sun is out, then it is night
  • It is never the case that either the Sun is out or it is day
  • If, whenever the Sun is out, it is day, then either the Sun is out, or it is not day
Considered the Founder of Optics, and with it Experimental Physics, ibn Al-Haythem appears on the Iraqi dinar. He also divided the first rigorous attempt at testing - making him the Founder of the Scientific Method.

Ibn Al-Haythem (Alhazen), a Persian Islamic thinker whom worked from the House of Wisdom, is considered the Founder of Optics and with it Experimental Physics. He also devised the first rigorous attempt at testing – making him the Founder of the Scientific Method.

Dialectical Exploration of Theology (kalam)
Baghdad may have had the House but Basra had the Circle, the circle of Al-Hasan Al-Basri (Hasan) to be exact. In this gathering of minds instead of scientific discussion there was theological questions emerging. Questions asked were to do with the nature of God, His attributes, good and evil, how to understand the Qur’an. Not all Muslims supported the idea of kalam but those who did would relish the chance to debate other religions – especially ahl al-kitab (People of the Book). A practitioner of kalam is called a mutakallim, and this word was used for non-Muslims aswell. From these endeavours there later developed Jewish Kalam, schools of thought in this time influenced each other a great deal. There was even athiest mutakallimun such as Ibn al-Rawandi. The first Muslims that practiced early on were the Qaadariyah (believers in Free Will) who were mockingly compared to Zoroastrians and the Jahmites (believers in Quranic Createdness and non-literal interpretations of the Qur’an). These early terms were for people holding certain singular dogmas. Later, as kalam evolved, three distinct schools of thought emerged (the Mu’tazila, Ash’ari and Maturidi). The next post looks specifically at the Mu’tazila.

Rough Mandelbrot Sets

I’ve been reading up on Zdzisław Pawlak’s Rough Set Theory recently and wanted to play with them. They are used to address vagueness in data so fractals seem like a good subject.

Super Quick Intro to Rough Sets:
A rough set is a tuple (ordered pair) of sets R(S) = \langle R_*, R^* \rangle which is used to model some target set S. The set R_* has every element definitely in set S and set R^* has every element that is possibly in set S . It’s roughness can be measured by the accuracy function \alpha(S) = \frac{|R_*|}{|R^*|} . So when |R_*| = |R^*| then the set is known as crisp (not vague) with an accuracy of 1.

A more formal example can be found on the wiki page but we’ll move on to the Mandelbrot example because it is visually intuitive:

The tiles are 36x36 pixels, the Mandelbrot set is marked in yellow. The green and white tiles are possibly i the Mandelbrot set, but the white tiles are also definitely in the Mandelbrot set.

The tiles are 36×36 pixels, the Mandelbrot set is marked in yellow. The green and white tiles are possibly in the Mandelbrot set, but the white tiles are also definitely in it.

Here the target set S contains all the pixels inside the Mandelbrot set, but we are going to construct this set in terms of tiles. Let T_1, T_2, T_3,\dots , T_n be the tile sets that contain the pixels. R^* is the set of all tiles T_x where the set T_x contains at least one pixel that is inside the Mandelbrot set, R_* is the set of all tiles T_x that contain only Mandelbrot pixels. So in the above example there are 28 tiles possibly in the set including the 7 tiles definitely in the set. Giving R(S) an accuracy of 0.25.

Tile sizes: 90, 72, 60, 45, 40, 36, 30, 24, 20, 18, 15, 12, 10, 9, 8, 6, 5, 4.

Tile width: 90, 72, 60, 45, 40, 36, 30, 24, 20, 18, 15, 12, 10, 9, 8, 6, 5, 4. There seems to be a lack of symmetry but it’s probably from computational precision loss.

Obviously the smaller the tiles the better the approximation of the set. Here the largest tiles (90×90 pixels) are so big that there are no tiles definitely inside the target set and 10 tiles possibly in the set, making the accuracy 0. On the other hand, the 4×4 tiles give us |R_*| = 1211 and |R^*| = 1506 making a much nicer:

\alpha(S) = 0.8 \overline{04116865869853917662682602921646746347941567065073}

For much more useful applications of Rough Sets see this extensive paper by Pawlak covering the short history of Rough Sets, comparing them to Fuzzy Sets and showing uses in data analysis and Artificial Intelligence.

Fractal Binary

I have previously talked about Complex Bases but I wanted to look again at Base (-1+i). It’s a really hefty number system so the length of the bit-strings increase very quickly, I’d quite like to know if there is a way to assess Radix Economy for complex and negative bases, so if there are any mathematicians out there who know – Please tell me!

Base (-1+i) to Base 10.Visualising Numbers
Today I wrote a little C++ program to act on Base 2 arithmetic but convert to decimal as if it was Base (-1+i), this meant I could increment through the bits in an ‘ordered’ fashion. The image to the left is the text output of the program, it doesn’t have a very obvious pattern to it – infact the pattern-order we derive from it is somewhat an imposed one. This is because complex numbers do not have a linear order (or Total Order) and I’m trying to list them in a linear manner. They can, on the other hand, be Well-Ordered in correspondence with the natural numbers like we’re doing here.
If we take the real and imaginary values of each number and use them as the x and y co-ordinates (like I did for generating the Mandlebrot Set fractal) then the fractal “Twindragon” appears:

Colour maps of number length in Base (-1+i).

The program I wrote runs through binary numbers starting at 0 colouring the pixel (x=r, y=i) discretely depending on number length. The result shows all Gaussian integers representable by all possible 16,12 and 8 bit complex binary strings in base (-1+i). The colour mapping relates to the position of the Most Significant Bit (essentially the bitstring length). 0 and 1 are both of length one and are the dark blue in the center of the fractal. The 12-bit and 8-bit fractal maps have been zoomed in on to emphases  the self-similarity of the shape.

Colouring the fractals like this is a nice way of showing the distribution of numbers in the complex system but, going back to the math, a number system isn’t useful without arithmetic. Luckily the (-1+i) system is closed under addition, subtraction and multiplication. For addition and multiplication it is the same as normal binary with the difference being in the carry. Below is a table of all possible carry situations:

1+1 = 1100
1+1+1 = 1101
1+1+1+1 = 111010000
1+1+1+1+1 = 111010001
1+1+1+1+1+1 = 111011100
1+1+1+1+1+1+1 = 111011101
1+1+1+1+1+1+1+1 = 111000000

Division in the systems is rather complex, an explination of that and examples of addition/subtraction/multiplication can be found in a short paper called “Arithmetic in Complex Basis” by William Gilbert. The paper also talks about an equivalent to decimal which is base (-3+i) using the digits [0,1,2,3,4,5,6,7,8,9].

Randolph Diagrams

This is what a genius looks like.

There is something aesthetic and elegant about Randolph diagrams, unfortunately they aren’t commonly used. I found out about them when reading Embodiments of Mind by the glorious and bearded Warren McCulloch.

The Original Proposal:
In the book he refers to them as “Venn functions” and they are briefly explained as being derived from Venn’s diagrams for sets but in McCulloch’s case they were used to express logical statements. If you draw a Venn diagram of two circles intersecting you are left with four spaces ( a/b, a&b, b/a, U ), adding a jot into a space to denote truth or leaving it blank for false gives you the 16 possible logic combinations. They are great examples of the isomorphism between logic and set theory:

He used these as tools to help teach logic to neurologists, psychiatrists and psychologists. Later he developed them into a probablistic logic which he applied to John vonn Neumann‘s logical neuron nets. Which I will discuss in the next post.

Randolph’s Diagrams:

The truth values for three statements.

McCulloch does mention that they could be used to apply more than two statements but doesn’t show how, later John F. Randolph developes the system as an alternative visualisation of set relations neatly coping with more than two sets (something Venn diagrams begin to struggle with after five). For each additional statement/set a new line is introduced in each quadrant. Four statements would be a large cross with four smaller crosses, one in each quadrant.

Wikipedia has an example of the tautological proof for the logical argument, modus ponens, which can be found here, but I thought it would be good to show how three values are handled – so we’ll use syllogism, as in “Socrates is a man, all men are mortal, therefore Socrates is mortal” being reduced in it’s logical form to tautology:

((A implies B) and (B implies C)) implies (A implies C)

Movie Idea

The Vienna Circle was a group of mathematicians, physicists, economists and philosophers alike who met to disscus epistemology and philosophy of science. This lead to a powerful movement called Logical Positivism which dispelled metaphysics, aesthetics and ethics from science as niether right nor wrong – of no value what so ever.

Summary:
Based on the Vienna Circle, but taken well out of context and set in ancient Japan. A collection of yojimbo (bodygaurd), ninja, ronin (masterless samurai) and farmers – analog to the real members, who meet in secret to talk skeptically about spirituality. Portrayed as pro-industrial natural philosophers.

Imagine these guys discussing Hilbert's Decision Problem, Samurai comming to attack them, then them being a bunch of badass logic-ninja defending science!