University Of Cambridge Alumni

"There has been a great deal of nonsense written... about the mysterious fourth dimension. ...4-dimensional space just consists of points with four coordinates instead of three (...similarly for any number of dimensions). ...[I]magine a telegraph ...over which numbers are ...sent in sets of four. Each set... is a point in 4-d... space."

"The general... problem... packing... in n-dimensional space. ...[T]here is nothing mysterious about n-dimensional space. A point in real n-dimensional space \R^n is... a string of real numbersx = (x_1,x_2,x_3, ...,x_n).A sphere in \R^n with center u = (u_1,u_2,u_3, ...,u_n) and radius \rho consists of all points x... satisfying (x_1-u_1)^2 + (x_2-u_2)^2+ ... +(x_n-u_n)^2 = \rho^2. We can describe a sphere packing in \R^n... by specifying the centers u and the radius."

"The classical... problem... asks: is this the greatest density..? an unsolved problem, one of the most famous..."

"The classical... problem is... how densely a large number of identical spheres ([e.g.,] ball bearings...) can be packed together. ...[C]onsider an aircraft hangar... [A]bout one quarter of the space will not be used... One... arrangement... the face-centered cubic (or fcc) lattice... spheres occupy \pi / \sqrt{18} = .7405... of the total space.... the lattice packing has density .7405... . [H]pwever, there are partial packings that are denser than the face-centered cubic... over larger regions..."

"In this chapter we discuss the problem of packing spheres in and of packing points on the surface of a sphere. The problem is an important special case of the latter, and asks how many spheres can just touch another sphere of the same size."

"We are planning a sequel... The Geometry of Low-Dimensional Groups and Lattices which will contain two earlier papers..."

"[I]n two dimensions the... [19 point] hexagonal lattice solves the packing, kissing, covering and quantizing problems. ...[T]his ...book is ...a search for similar nice patterns in higher dimensions."

"... I have said for twenty-five or thirty years that the one thing I would really like to know before I die is why the monster group exists.""

"When I was on the train from Liverpool to Cambridge to become a student, it occurred to me that no one at Cambridge knew I was painfully shy, so I could become an extrovert instead of an introvert."

"It is the privilege of a trader in a free country, in all matters not contrary to law, to regulate his own mode of carrying it on according to his own discretion and choice."

"An active imagination may find a bad tendency arising out of every transaction between imperfect morals."

"The Courts of law are not provided at the public expense, and were not intended by those who so provided them, for the settlement of any but differences which do arise in the ordinary course of business."

"My duty is, as a Judge, to be governed by fixed rules and former precedents."

"Human nature is imperfect."

"I apprehend that you may put a leading question to an unwilling witness on the examination in chief at the discretion of the Judge; but you may always put a leading question in cross-examination, whether a witness be unwilling or not."

"This railway is the most absurd scheme that ever entered into the head of a man to conceive. Mr. Stephenson never had a plan — I do not believe he is capable of making one. He is either ignorant or something else which I will not mention. His is a mind perpetually fluctuating between opposite difficulties; he neither knows whether he is to make bridges over roads or rivers, or of one size or another; or to make embankments, or cuttings, or inclined planes, or in what way the thing is to be carried into effect. When you put a question to him upon a difficult point, he resorts to two or three hypothesis, and never comes to a decided conclusion. Is Mr. Stephenson to be the person upon whose faith this Committee is to pass this Bill involving property to the extent of £400,000/£500,000 when he is so ignorant of his profession as to propose to build a bridge not sufficient to carry off the flood water of the river or to permit any of the vessels to pass which of necessity must pass under it."

"THE ETERNAL TRUTHS OF MATHEMATICS -- https://www.whyarewehere.tv/people/george-ellis/"

"WHAT IS FINE-TUNING? -- https://www.whyarewehere.tv/people/george-ellis/"

"TOP-DOWN CAUSATION -- https://www.whyarewehere.tv/people/george-ellis/"

"MORAL REALITY AND MORAL FACTS -- https://www.whyarewehere.tv/people/george-ellis/"

"Actually philosophical speculations have led to a great deal of good science. Einstein’s musings on Mach’s principle played a key role in developing general relativity. Einstein’s debate with Bohr and the EPR paper have led to a great of deal of good physics testing the foundations of quantum physics. My own examination of the Copernican principle in cosmology has led to exploration of some great observational tests of spatial homogeneity that have turned an untested philosophical assumption into a testable – and indeed tested - scientific hypothesis. That’ s good science."

"Attempts to explain values in terms of neuroscience or evolutionary theory in fact have nothing whatever to say about what is good or bad. That is a philosophical or religious question (scientists trying to explain ethics from these kinds of approaches always surreptitiously introduce some unexamined concept of what is a good life by the back door)."

"You cannot do physics or cosmology without an assumed philosophical basis. You can choose not to think about that basis: it will still be there as an unexamined foundation of what you do."

"Above all he believes that these mathematically based speculations solve thousand year old philosophical conundrums, without seriously engaging those philosophical issues. The belief that all of reality can be fully comprehended in terms of physics and the equations of physics is a fantasy."

""On the limits of quantum theory: Contextuality and the quantum–classical cut", Annals of Physics 327 (2012) 1890–1932"

"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?"

"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."

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

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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

"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."

"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."

"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."

"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."

"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."

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

"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."

"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."

"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."

"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.""

"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."