University Of Cambridge Faculty

"Life, like science and art, is a theory about the world: a theory that in our case takes bodily form. By a succession of adaptations, most of which are favourable and none of which are lethal, living things have invested in particular expectations about the future course of their environments. If those theories are good enough, then life will prosper and multiply; but if they are outmoded by changing conditions, their embodiments will dwindle and perish."

"Where there is life there is a pattern, and where there is a pattern there is mathematics. Once that germ of rationality and order exists to turn a chaos into a cosmos, then so does mathematics. There could not be a non-mathematical Universe containing living observers."

"We say that the string is 'random' if there is no other representation of the string which is shorter than itself. But we will say that it is 'non-random' if there does exist such an abbreviated representation. ...In general, the shorter the possible representation... the less random... On this view we recognize science to be the search for algorithmic compressions."

"Each of the most basic physical laws that we know corresponds to some invariance, which in turn is equivalent to a collection of changes which form a symmetry group. ...whilst leaving some underlying theme unchanged. ...for example, the conservation of energy is equivalent to the invariance of the laws of motion with respect to translations backwards or forwards in time... the conservation of linear momentum is equivalent to the invariance of the laws of motion with respect to the position of your laboratory in space, and the conservation of angular momentum to an invariance with respect to directional orientation... discovery of conservation laws indicated that Nature possessed built-in sustaining principles which prevented the world from just ceasing to be. There were fewer roles for the Deity to play..."

"Aristotle believed that the world did not come into being at some time in the past; it had always existed and it would always exist, unchanged in essence for ever. He placed a high premium on symmetry and believed that the sphere was the most perfect of all shapes. Hence the universe must be spherical. ...An important feature of the spherical shape... was the fact that when a sphere rotates it does not cut into empty space where there is no matter and it leaves no empty space behind. ...A vacuum was impossible. It could no more exist than an infinite physical quantity. ...Circular motion was the most perfect and natural movement of all."

"Continual miniaturisation allows resources to be conserved, efficiency to be increased, pollution to be reduced, and the remarkable flexibilities of the quantum world to be tapped. Very advanced civilizations elsewhere in the universe may have been forced to follow the same technological path. Their nano-scale space probes, their atomic-scale machines and nano-computers, would be imperceptible to our coarse-grained surveys of the universe. ...This may be the low-impact evolutionary path you need to follow in order to survive into the far, far future."

"In Theories of Everything... John Barrow argued that Gödel's incompleteness theorem undermines the very notion of a complete theory of nature. Gödel established that any moderately complex system of axioms inevitably raises questions that cannot be answered about the axioms. The implication is that any theory will always have loose ends. Barrow also pointed out that a unified theory of particle physics would not really be a theory of everything, but only a theory of all particles and forces. The theory would have little or nothing to say about phenomena that make our lives meaningful, such as love or beauty."

"While we have no reason to expect that our position in the universe is special in every way, we would be equally misled were we to assume that it could not be special in any way."

"Location is not, as the estate agents say, everything. We must also consider our place in history."

"Are these good reasons for worrying about the apparent contradiction? Should either make us feel the need of an explanation?"

"Plato not only permitted live philosophical enquiry to take place in the course of every reader’s every reading of the dialogue, by putting tempting and plausible views on trial, in a situation as near as possible to the open-minded exploratory give and take of dialectical debate with a real interlocutor. He also created a most fitting memorial to the real Socrates – the man himself, who lived and died for the idea that philosophy is best done in open-ended dialogue, and with your whole way of life at stake should you be refuted."

"In the Protagoras Socrates persuades Protagoras that goodness is identical with pleasure. He advocates a form of hedonism. In the Gorgias, Callicles espouses hedonism and Socrates refutes him. Socrates gets Callicles to admit that, after all, some pleasures are not good. [...] So Socrates holds contradictory views on pleasure in the Protagoras and the Gorgias."

"Perhaps Heraclitus lived before Parmenides, perhaps he lived after, perhaps he lived at the same time. Whichever way, his sayings cry out to be read in their own right, as a radically anti-materialist project unlike anything previously known. They bitterly resist the attempt to package them along with the pre-Parmenidean thinkers; they flourish in a situation in which we are able to juxtapose them with alternatives, such as Parmenides’s world view, for which they may indeed have been a foil."

"Philosophy has come to include, for us, a wide range of theoretical questions that typically look beyond what we can answer by experimental enquiries. While science asks how matter behaves, and tests its theories with observation, philosophy asks what matter is, or how observation can teach us anything. While mathematics asks what the sum of 2 and 7 is, philosophy asks what the number 2 is, and whether 2 plus 7 could ever make anything but 9."

"For Heraclitus, the logos is something that we need to learn to notice if we are to understand the true significance of the world. It manifests itself all around us but, Heraclitus suggests, only a few intelligent people ever realize what is going on."

"Much of the content of so-called Pythagorean teaching appears to be a mix of mystical gobble-de-gook and adulatory veneration of the genius of the founder."

"Both Democritus and Anaxagoras try to explain the puzzling behaviour of ordinary reality by appeal to a microscopic replica of reality, in which another set of tiny bodies or minute scraps of stuff move around and cause things to happen. As a way to overcome the difficulty of explaining changes in the world, this ultimately emerges as unsatisfying: if there were problems with explaining chemical and physical events as they appear to us, there will be the same problems with explaining the reactions between smaller and yet smaller bodies."

"Plato’s metaphysics grew out of that of Parmenides, together with a strong feel for Heraclitus’s account of the physical world as a world of incessant change. His ethics were deeply inspired by Socrates, but his views on the soul also pick up on motifs that emerge in Pythagoras."

"Even if Melissus’s analysis of the concept of existence is faulty, his procedure is very interesting. He challenges the data of sense experience by appealing to conceptual truths, facts about what a certain predicate (here ‘true’) must entail. These facts seem to escape the need to appeal to sense experience. We check up what is true about being true by examining our notion of being true, not by checking any things in the external world. So the argument seems to find a way of challenging the value of sense experience without begging the question. Melissus casts doubt on the senses by privileging the logical grammar of the word ‘true’. But, we might ask, did we learn how to use the word ‘true’ without relying on the senses?"

"Sophistry is one of the methods by which politicians dress up their policies in alien clothing, to pass them off as more desirable than they really are. Spin doctors thrive best where ‘democracy’ is the slogan."

"As the Presocratic philosophers bow out and Plato arrives to direct the next drama in the series, the Sophists make an astounding final act. All singing, all dancing, they ask society to question its raison d’être, its political beliefs, its moral values, its religious beliefs, its educational system, its legal codes, and its codes of etiquette. They draw attention to the power of the media and ask us to consider whether, without the media, there would be any truth at all. The antagonism that they generate, as portrayed by the Socrates imagined in Plato’s dialogues, starts the ball rolling for some of the most exacting philosophical endeavours the world has ever seen."

"Whether or not Zeno was merely trying to defend Parmenides from the ridicule of others, there is no doubt that he has pushed the analysis of reality onto a new plane. He makes us think not just about objects in space, but about space as a structure within which they exist; about motion not just as the behaviour of physical bodies, but as a theoretical concept involving conceptual divisions in space and time; about number not just as a way of counting finite bodies but as a rational system potentially (or actually) continuing ad infinitum, with the problematic consequences that that might entail; about the notions of ‘before’ and ‘after’ in time, and how long the duration of the present is."

"By taking us on a cumulative sequence from our own familiar gods, through those of other ethnic groups, to those of animals, Xenophanes shows that our own images have no more authority than those of animals."

"Parmenides did for science what Plato would later do for morality and aesthetics as well: he alerts us to the fact that opinions are just opinions, and they may differ widely. There may yet be a single truth, which need not be as anyone thought. To search for knowledge is to search for access to the truth, not to collect other people’s opinions, and philosophy conducts its unrelenting search for truth in the steps of Parmenides, by respecting sound and rigorous logical argument rather than the variegated tapestry of unexamined opinions."

"Xenophanes might be saying that we have only superficial understanding, and we never get to knowledge of the clear truth."

"So, with due thanks to those great heroes, the ancient authorities, we can now move on with a more cheerful heart to the rest of Presocratic philosophy. Many of the Presocratics’ words are lost, but we may still catch a glimpse of their strange forgotten worlds, woven into a splendid patchwork of ancient quotations and interpretations."

"Many aspects of Parmenides’s thought remain puzzling even when we have collected all the scraps of evidence from his own writings and those of later thinkers who discussed his views. But his immense significance in philosophical terms has never been obscured by the difficulties in the nitty-gritty of interpretation. For one thing, it is obvious that Parmenides throws at us the challenge of whether we should trust our reason or our senses, in circumstances when they seem to conflict."

"Often, when thinking about Socrates (or about Plato’s depiction of Socrates), we need to remember that he is reacting to the Presocratics, but the reverse is never true."

"If we are to understand what is going on in Empedocles’s writings, we need to think about the philosophical motives that drive him, and we need to make use of the bits of text we already had before the papyrus turned up."

"This principle, which is thus made the foundation of the operations and results of Symbolical Algebra, has been called "The principle of the permanence of equivalent forms", and may be stated as follows: "Whatever algebraical forms are equivalent, when the symbols are general in form but specific in value, will be equivalent likewise when the symbols are general in value as well as in form.""

"Besides the ‘how many?’ question, Empedocles seems to be answering two other ancient questions: ‘How did the world come to be as it now is?’ and ‘How did it come to have the creatures that it now has?’. [...] His answers are subtle and intricate."

"Symbolical algebra adopts the rules of arithmetical algebra, but removes altogether their restrictions: thus symbolical subtraction differs from the same operation in arithmetical algebra in being possible for all relations of value of the symbols or expressions employed... all the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form, though particular in value, are results likewise of symbolical algebra, where they are general in value as well as in form: thus the product of a^{m} and a^{n}, which is a^{m+n} when m and n are whole numbers, and therefore general in form though particular in value, will be their product likewise when m and n are general in value as well as in form: the series for (a+b)^{n}, determined by the principles of arithmetical algebra, when n is any whole number, if it be exhibited in a general form, without reference to a final term, may be shewn, upon the same principle, to the equivalent series for (a+b)^n, when n is general both in form and value."

"I have endeavoured... to present the principles and applications of Symbolical, in immediate sequence to those of Arithmetical, Algebra, and at the same time to preserve that strict logical order and simplicity of form and statement which is essential to an elementary work. This is a task of no ordinary difficulty, more particularly when the great generality of the language of Symbolical Algebra and the wide range of its applications are considered, and this difficulty has not been a little increased, in the present instance, by the wide departure of my own views of its principles from those which have been commonly entertained."

"In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs + and - denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as a + b we must suppose a and b to be quantities of the same kind; in others, like a - b, we must suppose a greater than b and therefore homogeneous with it; in products and quotients, like ab and \frac{a}{b} we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science."

"Philosophy asks for a reason, not just a scientific fact."

"I assure you that I shall never cease to exert myself to the utmost in the cause of reform, and that I will never decline any office which may increase my power to effect it. I am nearly certain of being nominated to the office of Moderator in the year 1818-1819, and as I am an examiner in virtue of my office, for the next year I shall pursue a course even more decided than hitherto, since I shall feel that men have been prepared for the change, and will then be enabled to have acquired a better system by the publication of improved elementary books. I have considerable influence as a lecturer, and I will not neglect it. It is by silent perseverance only, that we can hope to reduce the many-headed monster of prejudice and make the University answer her character as the loving mother of good learning and science."

"This work... was designed in the first instance to be a second edition of a Treatise on Algebra, published in 1830, and which has been long out of print; but I have found it necessary, in carrying out the principles developed in that work, to present the subject in so novel a form, that I could not with propriety consider it in any other light than as an entirely new treatise."

"It is now more than twenty years since I somewhat rashly undertook to write the Life of Dr. Young. For many years, however, after making this engagement, I found myself so much occupied by the duties of a very laborious college office, that I had no leisure to commence the work; and when the possession of leisure would have enabled me to have done so, my health became so seriously deranged that I felt myself unequal to any continued and severe literary labour. The undertaking was consequently abandoned, and it was proposed to transfer it to other hands; but it was not found easy to secure the services of a person who possessed sufficient scientific knowledge to enable him to write the life of an author whose works were so various in their character and not unfrequently so difficult to understand and analyse, as those of Dr. Young."

"I have separated arithmetical from symbolical algebra, and I have devoted the present volume entirely to the exposition of the principles of the former science and their application to the theory of numbers and of arithmetical processes: the second volume, which is now in the press, will embrace the principles of symbolical algebra: it will be followed, if other and higher duties should allow me the leisure to complete them, by other works, embracing all the more important departments of analysis, with the view of presenting their principles in such a form, as may make them component parts of one uniform and connected system."

"I Gladly avail myself of the opportunity of inscribing to you, for a second time, a work of mine on Algebra, as a sincere tribute of my respect, affection and gratitude. I trust that I shall not be considered as derogating from the higher duties which, (in common with you), I owe to my station in the Church, if I continue to devote some portion of the leisure at my command, to the completion of an extensive Treatise, embracing the more important departments of Analysis, the execution of which I have long contemplated, and which, in its first volume I now offer to the public, under the auspices of one of my best and dearest friends."

"History is only true for the time being; each new generation of scholars rewrites the work of its predecessors. But such revisers rarely go back to the beginning and start from scratch. Instead they build uncritically on “generally accepted” foundations laid down by their predecessors. These traditional, established truths of history have a large symbolic component of which their exponents are usually unaware."

"Time, as we experience it, is continuous; it contains no discrete “events.” The events are put there by reflection on the past. As the past becomes more remote the remembered events become fewer in num­ ber and more limited in kind. It is for psychologists to say just why we remember this and forget that, but at the end of the day, the remembered past reflects our interests. It makes us what we are now."

"The history of European colonialism covers many centuries and takes diverse forms, but whereas the European explorers and conquerors of the Americas, Africa, and Oceania usually took it for granted that the local inhabitants could be enslaved or butchered or driven into the hinterland at the whim of the invaders, the literate nations of Asia were initially treated as peoples toward whom the courtesies of European diplomacy should be applied. At the end of the day these Asian civilizations were likewise mostly subdued by force of arms, but such conquest needed some kind of moral justification, a mythical charter. The Rig Veda as interpreted by Max Müller and his contemporaries provided just such a myth."

"The origin myth of British colonial imperialism helped the elite administrators in the Indian Civil Service to see themselves as bringing `pure' civilization to a country in which civilization of the most sophisticated (but `morally corrupt') kind was already nearly 6,000 years old. Here I will only remark that the hold of this myth on the British middle-class imagination is so strong that even today, 44 years after the death of Hitler and 43 years after the creation of an independent India and independent Pakistan, the Aryan invasions of the second millennium BC are still treated as if they were an established fact of history."

"Yet here is precisely where anthropologists might make a useful contribution, if only their scholarly associates would stop thinking of the Rig Veda as a garbled history book."

"Because of their commitment to a unilineal segmentary history of language development that needed to be mapped onto the ground, the philologists took it for granted that proto-Indo-Iranian was a language that had originated outside either India or Iran. Hence it followed that the text of the Rig Veda was in a language that was actually spoken by those who introduced this earliest form of Sanskrit into India. From this we derived the myth of the Aryan invasions. QED."

"The Aryan invasions never happened at all. Of course no one is going to believe that."

"Where the Indo-European philologists are concerned, the invasion argument is tied in with their assumption that if a particular language is identified as having been used in a particular locality at a particular time, no attention need be paid to what was there before; the slate is wiped clean. Obviously, the easiest way to imagine this happening in real life is to have a military conquest that obliterates the previously existing population!"

"The details of the theory fit in with this racist framework. Just as each member of the total family of Indo-European languages is lineally descended from one or another of a number of extinct “protolanguages,” so also are the speakers of these languages; hence the people who speak any particular language constitute an independent racial stock."

"Why is this sort of thing attractive? Who finds it attractive? Why has the development of early Sanskrit come to be so dogmatically associated with an Aryan invasion? In some cases the association seems to be a matter of intellectual inertia."