Review
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Logic and the Mathematical
بسم الله الرحمن الرحيم وصلى الله على سيدنا محمد وعلى ءاله وصحبه أجمعين وسلّم
Title: Logic and the Mathematical
Author: Abdassamad Clarke
Publication date: 23/2/2013
Assalamu alaykum. Welcome to the Civilisation and Society Programme of the MFAS. This is the fourth of 12 sessions which make up the Technique and Science module. The lecture will last approximately 40 minutes during which time you should make a written note of any questions that may occur to you for clarification after the lecture.
Old fashioned story-telling begins at the beginning, and it would certainly be a foolish man who would dismiss the old-fashioned out of hand. But for our purposes, we will begin with the new, and approach history in flashback. So this is a review of some few contemporary sciences in order to set the scene, however partially. The significance of all of the following hypotheses and theories is that they are highly mathematical.
We turn first to the most conspicuous of the sciences, cosmology. Because of their discomfort with the idea of the Big Bang – it looked too much like Genesis’s concept of Creation – cosmologists came up with several competing models. Some have multiple parallel universes of different dimensionalities, in some of which the conditions for conscious life exist, as is the case with the universe we happen to be in. The modelling of this multiverse is highly mathematical. Other models have sequences of universes each beginning with a big bang that is the ‘big crunch’ of the previous universe. As we saw, the issue with such hypotheses is that their propounders have some work to do to find ways to prove them wrong, let alone to prove them right. In the sense of Karl Popper’s criterion of falsifiability they may never be theories.
However, this concept, or one which looks to all intents and purposes identical with it, appears in the works of a number of Muslim writers as a theoretical given and, moreover, often in the accounts of some of the Sufis of worlds they have visited in vision or by other means. Whereas the Judaeo-Christian worldview might be threatened by it, the Muslims do not necessarily have to be.
The anthropic principle is based on the observation that there are many physical constants which, if they were even minutely different, then conscious life could not exist. Discussion of it revolves around, not the issue of intelligent design, since both parties accept that, but whether to interpret it teleologically, i.e. that creatures are designed purposefully, by the Creator and for a purpose, or eutaxiologically, i.e. that there is a cause for their design such as natural selection. The teleological argument is as much about the purpose for which conscious beings are made as it is about their Maker. The eutaxiological argument does not presuppose atheism, but rather leaves that issue aside to deal with the apparent workings out of causality in the physical world.
Simply expressed, the teleological statement of the anthropic principle is that the universe is designed from the very first moment and at every stage of its growth to bring about conscious beings who would have knowledge of it. It is only a very small step to the famous statement of the Divine which the Sufis often cite: “I was a hidden treasure and wished to be known, and so I created the creation in order to be known.”
Leaving that aside, the existing cosmological model of a single universe is said to be pretty complete, except for the missing dark matter which is estimated to constitute 84% of the matter in the universe, and dark energy, which currently accounts for 73% of the total mass–energy of the universe. Dark matter and dark energy are essentially hypotheses to account for rather large holes in the mathematical model. But as yet, no one knows what dark matter or dark energy could actually be.
“The holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region…the theory suggests that the entire universe can be seen as a two-dimensional information structure "painted" on the cosmological horizon, such that the three dimensions we observe are only an effective description at macroscopic scales and at low energies.”
The echoes of the Qur’ānic understanding of the lau maḥfūẓ are quite striking.
Searching for some more fundamental building block of matter, now that the a-tom has proved quite divisible, and its sub-atomic particles bewilderingly paradoxical including the quarks that were hypothesised to solve the whole issue, mathematicians have suggested a way out: string theory. Here the ultimate indivisible constituents of matter are considered to be small strings vibrating in a large number of dimensions. This is an attempt to resolve one of the great rifts of the 20th century: the mutual exclusivity of quantum mechanics, which works on the sub-atomic scale, and general relativity, which works on the cosmic scale, and both of which depend on highly abstruse mathematics.
Chaos Theory is an overarching theory that groups quite a few separate articulations of the theory beneath a single umbrella. In essence they explain phenomena, such as weather patterns, which when modelled by computers then very minute initial differences produce massively different outcomes. This subject was opened up by the ability of computers to do many computations to high degrees of accuracy. Until that time, such phenomena were considered ‘chaotic’ but the new maths revealed an exquisite degree of order. Thus the name is a misnomer, and the term Exquisite Order Theory would be more appropriate.
First proposed by John von Neumann and later developed by John Nash, Game Theory, a highly sophisticated analysis of what happens in games such as poker, was marred, as John Nash himself came to admit, by an assumption of psychopathic selfishness on the part of human beings.1 Significantly, this assumption and its later elaboration in Game Theory lay behind both cold-war geo-political analysis that may have brought the world close to the brink of nuclear catastrophe and predictive economic studies which actually pushed the world over the financial brink. And here the nub of the issue looms large. Game theory is an utterly sophisticated mathematical analysis that overlooked something extremely basic: its own unscientific assumptions about the psychopathic selfishness of human nature, and yet those assumptions, that were demonstrably wrong, have produced precisely the world they had posited as an assumption: one of psychopathic selfishness.
The mathematician Georg Cantor, doggedly subjected the idea of infinity to rigorous mathematical treatment, something mathematicians had avoided for centuries. However, the outpouring of puzzling results was staggering and outraged many mathematicians and philosophers: for example, some infinities are bigger than others, and they can be added and subtracted from each other. Cantor himself suffered severe depressions until the end of his life, which some attributed to his discoveries but others to the opposition and ridicule he experienced.
So that brief and very incomplete picture of the contemporary scientific picture and that from the recent past, concluded, we may now proceed to our history.
Mathematic’s legendary almost mythical founder in the Western tradition is Pythagorus (b. about 570 – d. about 495 BC), just as Logic’s is Aristotle (384 BC – 322 BC). Notably, Pythagorus is clearly a religious figure in whose worldview mathematics had an almost mystical significance. Apart from that, logic and mathematics share a certain character; they are rule bound. When we see the word ‘rules’ we perhaps rightly suspect the presence of a synonym of ‘method’ and ‘technique’. Logic is as it were the mathematics of language, and mathematics is the logic of numbers, algorithms, figures and more. They will reach a kind of synthesis later in our story.
Logic and mathematics have significance early on for their presence in the trivium and quadrivium which are the basis of the Greeks’ ‘liberal’ paideia which will assume importance later for the mediaeval world. Augustine of Hippo (354-430) was instrumental in their marriage to Christianity. Denis Lawton and Peter Gordon, in their A History of Western Educational Ideas, write:
Augustine …concentrated on bringing classical learning into the service of Christianity. Part of pagan culture treated in this way was the curriculum based on the seven Liberal Arts: arithmetic, geometry, astronomy and music (the quadrivium); and grammar, rhetoric and logic (the trivium). Augustine worked hard to harmonise the two traditions and largely succeeded.2
The classical trivium comprises grammar, logic (or dialectic) and rhetoric, which are in Dorothy Sayers’ phrase “The Lost Tools of Learning”. Grammar governs the rules of language, logic the rules of thought and rhetoric the rules of expression. When students had mastered these tools (not subjects) they could advance with them to the quadrivium, the four ‘subjects’: arithmetic, geometry, music and astronomy. All of this was only preparatory to the study of philosophy and theology. However, careful consideration of this ‘scholastic’ tradition leaves one with little doubt that rather than the mediaeval scholastics being overturned by the Renaissance, they themselves had laid the foundations for the extraordinary bursting out of mathematical scientific knowledge that was to take place. These seven liberal arts cultivate an orderly and logical form of thought and expression, and an outlook on the world that is clearly leading to the new scientific thinking. The foundations for widespread dissemination of this way of thinking were laid in the 12th century when education outside of the monastery began to take shape. Lawton and Gordon write:
The Church often wanted to retain control of education and reacted by saying that only those teachers licensed by bishops could be permitted to teach. The Lateran Council (1179) made this an official requirement for all Christendom. This ostensibly gave some credibility to the developing non-monastic schools and perhaps improved the quality of some of them. The schools adopted a 'liberal' curriculum of the classical trivium (grammar, rhetoric and logic) and the quadrivium (geometry, arithmetic, astronomy and music) which became the accepted advanced curriculum for the later Middle Ages.3
We have seen the rise of this extraordinary mathematical science and its development during the Reformation. We have noted also that disquieting doubt: is it a human creation or a discovery? Two arts that had been a kind of training for the intellect and a preparation for philosophy and theology had become something in themselves. The servant had become the master. Heidegger writes:
“Logic originated in the ambit of the administration of the Platonic-Aristotelian schools. Logic is an invention of schoolteachers, not of philosophers.”4
In the Enlightenment, the mathematical and logical approach was applied to all existing sciences and was in at the invention of quite new ones. This had gone as far as the philosophes’ application of science to politics and the drawing up of governments based on constitutions.5 As we have pointed out, it led to the pseudo-science of economics. I say ‘pseudo-science’ because, although economics elaborates its schema quite rigorously, at its root lie basic assumptions that are simply untested. They come in disguised as axioms, self-evident truths that need no proof, but are in fact little better than prejudices, as I demonstrated in “From the State to the Market”, which was lecture No. 10 in our Politics of Power course, with respect to the “Law of Supply and Demand”.
Logic stems from the Greek word logos. Heidegger says:
“But Aristotle is the first to give the clearer metaphysical interpretation of the logos in the sense of the propositional statement.”6
Later he says:
“However, we would still like to raise one question. What does "logic" mean? The term is an abbreviation for epistēmē logikē, the science of logos. And logos here means assertion. But logic is supposed to be the doctrine of thinking. Why is logic the science of assertion?”7
Of all the possible types of expression, logic deals with the propositional statement or the assertion. This is clearly a sub-set of human expression. Why in all the movements of thinking, reflecting, pondering, marvelling and wondering has the ‘assertion’ assumed such importance? We assume the understanding that thought itself is quite a mysterious matter and by no means obvious. Our language contains hints: “It occurred to me”. Do we think our thoughts or do they come to us? Are they thinking us? What exactly is our relationship to thought? It is quite probable that logic stems from an impatience with this very obscurity, like an explorer whose encounter with the fabulous jungle leads him, not to revere it or to find ways through it at peril of getting lost, but, to clear swathes of it and plan roads across it. Logic is the technology of thought. But never forget that a clear questioning of logic is not necessarily the same as an endorsement of ‘illogical’ thinking, just as an enquiry into science can itself be a scientific matter and is not necessarily anti-science.
Mathematics and the Mathematical
Heidegger writes:
From Kant comes the oft-quoted but still little understood sentence, "However, I maintain that in any particular doctrine of nature only so much genuine science can be found as there is mathematics to be found in it" (Preface to Metaphysical Beginning Principles of Natural Science).8
In an examination of what exactly ‘the mathematical’ is, in which Heidegger shows that what we call mathematics is only one part of it, he explores the Greek roots of the word itself and says:
“Mathēsis means learning; mathēmata, what is learnable.…The mathēmata are the things insofar as we take cognizance of them as what we already know them to be in advance, the body as the bodily, the plant-like of the plant, the animal-like of the animal, the thingness of the thing, and so on. This genuine learning is therefore an extremely peculiar taking, a taking where one who takes only takes what one basically already has.…Teaching corresponds to this learning. Teaching is a giving, an offering; but what is offered in teaching is not the learnable, for the student is merely instructed to take for himself what he already has. If the student only takes over something that is offered he does not learn. He comes to learn only when he experiences what he takes as something he himself really already has. True learning occurs only where the taking of what one already has is a self-giving and is experienced as such. Teaching therefore does not mean anything else than to let the others learn, that is, to bring one another to learning. Teaching is more difficult than learning; for only he who can truly learn—and only as long as he can do it—can truly teach. The genuine teacher differs from the pupil only in that he can learn better and that he more genuinely wants to learn. In all teaching, the teacher learns the most.
“The most difficult learning is coming to know actually and to the very foundations what we already know. Such learning, with which we are here solely concerned, demands dwelling continually on what appears to be nearest to us, for instance, on the question of what a thing is. We steadfastly ask the same question—which in terms of utility is obviously useless—of what a thing is, what tools are, what man is, what a work of art is, what the state and the world are.”9
This clearly is a very different understanding of what mathematics is than we are accustomed to. He continues:
“The mathēmata, the mathematical, is that "about" things which we really already know. Therefore we do not first get it out of things, but, in a certain way, we bring it already with us. From this we can now understand why, for instance, number is something mathematical.”10
Then later he writes:
“Therefore, Plato put over the entrance to his Academy the words: Ageometretos medeis eisito! "Let no one who has not grasped the mathematical enter here!" These words do not mean that one must be educated in only one subject—"geometry"—but that one must grasp that the fundamental condition for the proper possibility of knowing is knowledge of the fundamental presuppositions of all knowledge and the position we take based on such knowledge. A knowledge which does not build its foundation knowledgeably, and thereby notes its limits, is not knowledge but mere opinion.”11
In its elaboration, this mathematical approach begins with definition of terms, and then statement of axioms, i.e. those mathematical truths in the sense we have just explored that are ‘self-evident’ and need no proof. And we already saw in lecture No.2 something of the strange new application of Galileo, Descartes and Newton of this approach in just one aspect: the First Law – or Axiom – of Motion, “Every body continues in its state of rest, or uniform motion in a straight line, unless it is compelled to change that state by force impressed upon it" which suddenly didn’t look quite as axiomatic or self-evident as we had been brought up to believe. For an examination of the fuller elaboration of this approach, I refer you to my essay “Mathematics’ Imperious Sway”12 in which I examine in an exploratory manner, its extrapolation into areas as diverse as constitutional law via the American Declaration of Independence, Orientalism, and Islamic modernism and Islamic banking.
Inasmuch as philosophy concerns itself with matters such as the state, it concerns itself with power, even if philosophers themselves are ordinarily remote from its precincts, except for a few famous and sometimes infamous occasions.
Although confined to the realms of academia, philosophers can still exert a vital force on power by, for example, merely remaining silent over some issues and at appropriate moments, for as we saw in the 20th century, silence is sometimes eloquent. When the scholar is silent before a scandalous lie, he allows the lie to pass over into the realms of truth by merely not denying it. Power can also affect the course of philosophy, science and scholarship by the judicious use of what it manages best: funds. So a cosy arrangement can be arrived at, and it is the oldest story in the universe. Here we are dealing often with the popularisation of ideas as much as and more than the ideas themselves. And this is a part of the history of philosophy. Once an obscure study for the exceptionally talented and dedicated, it was at some point placed before the masses. In that sense, popular science is philosophy for the masses, but ill-digested, not thoroughly thought-through, and not rigorously defined. Scientists become rightly incensed when popularly misunderstood scientific ideas are ascribed to them. But in this sense, philosophy has become one of the essential supports of power in our time. More significantly, when it stops asking vital questions in line with its “what a thing is, what tools are, what man is, what a work of art is, what the state and the world are”, when it stops reiterating these questions and doesn’t ask the really obvious new questions that no one will ask such as, what is money, let alone what is a credit-swap derivative, then it has failed in a vital function and has connived with the actual power of the age, the power of the new money.
As a paid-up member of the academic élite, philosophy has also connived at the subversion of education and its transformation into training programmes for the workforce, and, in the case of philosophers themselves, into training for the most highly sought after trainee-bankers. The banks just snap up first class philosophers perhaps as much as they do mathematicians.
Therefore it must fall to those who have a love for logic, mathematics, philosophy and science, to see them ensconced within the urge to bring about the new nomos that the earth and its people are mutely crying out for, after the failure of the humanist project and its descent into its current barbarity.
Given that the entire cycle began with a paideia based on the trivium and the quadrivium, the potential in them and in a renewed philosophy must be evident, if placed within the secure parameters of a genuine nomos, rather than in the disintegrating Judaeo-Christian, civil-war ridden Reformation.
That brings us to the end of today’s lecture. I recommend you to listen to Melvyn Bragg’s In Our Time of 1st October 2010 on Logic, read Dorothy Sayers’ “The Lost Tools of Learning”, and chapter four “The Restriction of Being”, the section on “Being and Thinking” of Martin Heidegger’s Introduction to Metaphysics, and his “Modern Science, Metaphysics and Mathematics”. The subject of our next lecture is Being in the World, and recommended preparatory reading includes the Zollikon Seminars of Medard Boss and Martin Heidegger. Thank you for your attention. Assalamu alaykum.
1 Brilliantly treated by Adam Curtis in “The Trap”.
2 Denis Lawton and Peter Gordon, A History of Western Educational Ideas, p.41
4 Martin Heidegger, Introduction to Metaphysics, p.126
5 See my paper “Mathematics’ Imperious Sway” for an examination of the American Declaration of Independence as an example of a mathematical approach.
6 Martin Heidegger, Introduction to Metaphysics, p.61
8 Martin Heidegger, “Modern Science, Metaphysics and Mathematics”, from Martin Heidegger, Basic Writings, p.273