A benefit of publication can be informed criticism which in this case has entailed a revision. I had confused Gödelian propositions with axioms. The revision makes extensive use of Roger Penrose – a major mathematical physicist and philosopher, and to a lesser extent, Stanley Jaki, also a physicist and philosopher – so there is a certain amount of a legitimate appeal to authority. If it is a subject that seems particularly interesting, perhaps you wanted to know why Gödel’s Theorem is regarded as such a landmark, I hope the reader might read this version too. My apologies for leading anyone astray. I can only hope that this revision is evidence of intellectual integrity and not just boneheadedness on my part.
In The Master and His Emissary, Iain McGilchrist writes that a creature like a bird needs two types of consciousness simultaneously. It needs to be able to focus on something specific, such as pecking at food, while it also needs to keep an eye out for predators which requires a more general awareness of environment.
These are quite different activities. The Left Hemisphere (LH) is adapted for a narrow focus. The Right Hemisphere (RH) for the broad. The brains of human beings have the same division of function.
The LH governs the right side of the body, the RH, the left side. With birds, the left eye (RH) looks for predators, the right eye (LH) focuses on food and specifics. Since danger can take many forms and is unpredictable, the RH has to be very open-minded. Continue reading
Kurt Gödel was a Platonist, logician and mathematician who developed the intention of making a profound and lasting impact on philosophical mathematics. His next task was to think of something! Amazingly, at the age of twenty five, he achieved his goal, publishing his incompleteness theorem.
A good friend of Albert Einstein’s, Einstein once said that late in life when his own work was not amounting to much, the only reason he bothered going to his office at the Institute for Advanced Study at Princeton was for the pleasure of walking home with Gödel.
John von Neumann wrote: “Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement.”
While at university, Gödel attended a seminar run by David Hilbert who posed the problem of completeness: Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system? Continue reading
Kurt Gödel’s Incompleteness Theorem was inspired by David Hilbert’s question “Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?” Hilbert played the same role regarding Alan Turing’s proof of the halting problem. Hilbert had asked: “Is there some mechanical procedure [an algorithm] for answering all mathematical problems, belonging to some broad, but well-defined class?” In German this is called Entscheidungsproblem – the decision problem.
Turing found that he could answer this question by framing it in terms of a Turing machine – could there be a program that could determine whether any other arbitrary computer program and input would eventually stop or just loop forever? This was called the halting problem.
“Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist.”
Plato’s allegory of the cave appears in Book VII of Plato’s most famous and longest dialog, The Republic. Plato’s dialogs frequently star Plato’s teacher Socrates as a character. The dialogs involved discussions and philosophical arguments between various characters, some of whom were based on real people. Plato particularly disliked the sophists who were professional rhetoricians and who seemed to care more about money and social success than truth. In fact, Plato accused them of teaching their students how to make the worse argument appear better – enabling their students to convict the innocent and set free the guilty.
Thoughts inspired by teaching epistemology for the first time and listening to the podcasts of Jordan Peterson
Epistemology became a major topic for analytic philosophers because they trace their intellectual origins to Descartes and the British empiricists. Descartes dismantles the foundations of his beliefs and then tries to rebuild them on certain grounds. Having used the method of doubt to tear everything down, including even mathematics, he finds irrefutable evidence of the existence of his own mind and then tries to prove that the “external world” exists.
The British empiricists take their inspiration from Descartes, accept his distinction between mind and body and plump for body as the truly real. Following Galileo and Montaigne’s distinction between primary and secondary qualities, they attempt to dispose of mind-related secondary qualities as merely “subjective” and thus nonexistent.
Strangely enough, analytic philosophers complacently flip between claiming that physical reality is the paradigm of the real and also thinking that the existence of the external world is questionable and in need of proof. It is the latter that drives the analytic philosopher’s interest in epistemology. Continue reading
Having an emotional and intellectual appreciation for the sacred is necessary to live well. Without an appreciation for the sacred a person’s attunement to life is severely damaged.
The sacred can be thought of as the appearance of the transcendent in the midst of the immanent; of a slight rip in the curtain separating the two.
A human being, Nature and Beauty can all be counted as instances of the sacred. Mystics seem to suggest that in fact all reality is divine and describe the sacred as shining through the most mundane of objects. Since mystics face the problem of communicating their rare experiences to the rest of us, they frequently make use of poetry. This has the advantage of potentially engaging the reader emotionally, intellectually and imaginatively. The aesthetic experience can be an instance of when people are most alive and a poem, as an instance of the beautiful, can point beyond itself to the divine realm. A realm for which we have an affinity, claims Plotinus, as being our true home.
Gifts are universal. Every culture on Earth has and will always exchange gifts. The effect of gifts is to tie people together; to connect them. This is their ultimate meaning and significance. Many features of gifts are immune from changes in cultural context and time. They stay the same in all circumstances. They are traditional everywhere.
Marcel Mauss’ The Gift is an anthropological study of gifts. He hoped to show that gift-giving precedes mere economic transactions in chronology and significance. Successful businesses often combine gifts with the more prosaic monetary exchanges.
Some people exhibit an amazing lack of interest in reality, content to imagine living in a wholly invented world. The notion that much of subjective experience is illusory is strongly connected with the beginnings of “modern” philosophy.
Galileo and Locke claimed that only things which are physical and measurable really exist. Galileo argued that primary qualities; solidity, motion, figure, extension and number were really real – being the objective properties of objects and that secondary qualities; color, sight, sound, small, taste and touch did not actually exist per se. They are merely artifacts; products of the sense organs that really have nothing to do with the objects being perceived. They are merely what our brains do when confronted with sensory input and primary qualities.
With modern egalitarianism, the existence of the rich is regarded as an offense to the poor, the smart to the dumb, and the good looking to the plain. Pure resentment drives this phenomenon – resentment being a combination of admiration, envy and hatred. Wanting to be rich, handsome and smart, and failing to be, these things are then hated.
Many high schools are now apparently doing away with prize-giving ceremonies and the notion of a valedictorian to spare the feelings of other students.
Moral subjectivism, or relativism, reduces morality to feelings and personal opinion. This renders moral knowledge and disputes meaningless. Aesthetic subjectivism likewise insists that beauty is in the eye of the beholder and nothing more. I encountered raised voices and outrage in a class when I recently suggested otherwise. The reaction was stronger than anything I had experienced before and seemed out of proportion to the claim. Far more contentious-seeming moral issues had not inspired any such protests. My essay Aesthetic Knowledge published at the Sydney Traditionalist Forum is my argument for aesthetic objectivism.