I say it that way because, while I’m pretty sure I know what they are, I’m also still a little confused. In the process of trying to figure it out, exactly, I realized that I had encountered something in the past semester that I felt was rather loopy. Therefore, I’m presenting that writing (in this case, Book Delta, Chapter 6 in Aristotle’s Metaphysics) to see 1)if anyone else sees it and 2)there is agreement as to whether or not it really is a strange loop. The translation comes from The Classical Library.
'One' means (1) that which is one by accident, (2) that which is one by its own nature. (1) Instances of the accidentally one are 'Coriscus and what is musical', and 'musical Coriscus' (for it is the same thing to say 'Coriscus and what is musical', and 'musical Coriscus'), and 'what is musical and what is just', and 'musical Coriscus and just Coriscus'. For all of these are called one by virtue of an accident, 'what is just and what is musical' because they are accidents of one substance, 'what is musical and Coriscus' because the one is an accident of the other; and similarly in a sense 'musical Coriscus' is one with 'Coriscus' because one of the parts of the phrase is an accident of the other, i.e. 'musical' is an accident of Coriscus; and 'musical Coriscus' is one with 'just Coriscus' because one part of each is an accident of one and the same subject. The case is similar if the accident is predicated of a genus or of any universal name, e.g. if one says that man is the same as 'musical man'; for this is either because 'musical' is an accident of man, which is one substance, or because both are accidents of some individual, e.g. Coriscus. Both, however, do not belong to him in the same way, but one presumably as genus and included in his substance, the other as a state or affection of the substance.
The things, then, that are called one in virtue of an accident, are called so in this way. (2) Of things that are called one in virtue of their own nature some (a) are so called because they are continuous, e.g. a bundle is made one by a band, and pieces of wood are made one by glue; and a line, even if it is bent, is called one if it is continuous, as each part of the body is, e.g. the leg or the arm. Of these themselves, the continuous by nature are more one than the continuous by art. A thing is called continuous which has by its own nature one movement and cannot have any other; and the movement is one when it is indivisible, and it is indivisible in respect of time. Those things are continuous by their own nature which are one not merely by contact; for if you put pieces of wood touching one another, you will not say these are one piece of wood or one body or one continuum of any other sort. Things, then, that are continuous in any way called one, even if they admit of being bent, and still more those which cannot be bent; e.g. the shin or the thigh is more one than the leg, because the movement of the leg need not be one. And the straight line is more one than the bent; but that which is bent and has an angle we call both one and not one, because its movement may be either simultaneous or not simultaneous; but that of the straight line is always simultaneous, and no part of it which has magnitude rests while another moves, as in the bent line.
(b, i) Things are called one in another sense because their substratum does not differ in kind; it does not differ in the case of things whose kind is indivisible to sense. The substratum meant is either the nearest to, or the farthest from, the final state. For, one the one hand, wine is said to be one and water is said to be one, qua indivisible in kind; and, on the other hand, all juices, e.g. oil and wine, are said to be one, and so are all things that can be melted, because the ultimate substratum of all is the same; for all of these are water or air.
(ii) Those things also are called one whose genus is one though distinguished by opposite differentiae—these too are all called one because the genus which underlies the differentiae is one (e.g. horse, man, and dog form a unity, because all are animals), and indeed in a way similar to that in which the matter is one. These are sometimes called one in this way, but sometimes it is the higher genus that is said to be the same (if they are infimae species of their genus)—the genus above the proximate genera; e.g. the isosceles and the equilateral are one and the same figure because both are triangles; but they are not the same triangles.
(c) Two things are called one, when the definition which states the essence of one is indivisible from another definition which shows us the other (though in itself every definition is divisible). Thus even that which has increased or is diminishing is one, because its definition is one, as, in the case of plane figures, is the definition of their form. In general those things the thought of whose essence is indivisible, and cannot separate them either in time or in place or in definition, are most of all one, and of these especially those which are substances. For in general those things that do not admit of division are called one in so far as they do not admit of it; e.g. if two things are indistinguishable qua man, they are one kind of man; if qua animal, one kind of animal; if qua magnitude, one kind of magnitude.—Now most things are called one because they either do or have or suffer or are related to something else that is one, but the things that are primarily called one are those whose substance is one,—and one either in continuity or in form or in definition; for we count as more than one either things that are not continuous, or those whose form is not one, or those whose definition is not one.
While in a sense we call anything one if it is a quantity and continuous, in a sense we do not unless it is a whole, i.e. unless it has unity of form; e.g. if we saw the parts of a shoe put together anyhow we should not call them one all the same (unless because of their continuity); we do this only if they are put together so as to be a shoe and to have already a certain single form. This is why the circle is of all lines most truly one, because it is whole and complete.
(3) The essence of what is one is to be some kind of beginning of number; for the first measure is the beginning, since that by which we first know each class is the first measure of the class; the one, then, is the beginning of the knowable regarding each class. But the one is not the same in all classes. For here it is a quarter-tone, and there it is the vowel or the consonant; and there is another unit of weight and another of movement. But everywhere the one is indivisible either in quantity or in kind. Now that which is indivisible in quantity is called a unit if it is not divisible in any dimension and is without position, a point if it is not divisible in any dimension and has position, a line if it is divisible in one dimension, a plane if in two, a body if divisible in quantity in all—i.e. in three—dimensions. And, reversing the order, that which is divisible in two dimensions is a plane, that which is divisible in one a line, that which is in no way divisible in quantity is a point or a unit,—that which has not position a unit, that which has position a point.
Again, some things are one in number, others in species, others in genus, others by analogy; in number those whose matter is one, in species those whose definition is one, in genus those to which the same figure of predication applies, by analogy those which are related as a third thing is to a fourth. The latter kinds of unity are always found when the former are; e.g. things that are one in number are also one in species, while things that are one in species are not all one in number; but things that are one in species are all one in genus, while things that are so in genus are not all one in species but are all one by analogy; while things that are one by analogy are not all one in genus.
Evidently ‘many’ will have meanings opposite to those of ‘one’; some things are many because they are not continuous, others because their matter—either the proximate matter or the ultimate—is divisible in kind, others because the definitions which state their essence are more than one.
My name is Bailey (@baileyfuriosa), and I’m a sophomore at Arizona State University studying Spanish, linguistics, economics, and Romanian. @companionablesniffles is my best friend, and I visited her at Goucher last spring. I loved it and now follow a lot of Goucher students on Tumblr.
I’ve enjoyed math and science for a long time but once I got to college, I fell in love with humanities and found a fantastic combination of the two in linguistics. I haven’t done anything with science/math in about a year, so I’m looking forward to getting back into it. Also, I have a very basic foundation in philosophy. I’m anticipating having a lot of questions, so the book group is very appealing to me.
Hey, guys! I’m Nony and I’ve always been interested in Math and studied Gödel’s Incompleteness Theorems years ago. I would consider myself to be an autodidact, avoiding school to study things that I was interested in on my own. This hasn’t been very conducive to being successful in our society but I have persevere!
Anyway, I’m very excited to read this book with you guys and look forward to our weekly discussions! Luckily the Goucher folks will be back soon enough and we’ll be able to discuss it in person! Be jealous!
When I was in 4th grade, the teacher had us read Stone Soup, and then we made stone soup.
The story goes like this: A poor beggar bets this man that he can make a delicious soup using only a stone. Then the beggar says something like, “You know, this soup would taste just a bit better with some potatoes.” so the man rushes off to get potatoes to put into the boiling water with the stone. And the beggar keeps asking for one more vegetable, and another, and another, until finally the guy is making vegetable soup with a stone in it.
My point, and I do have one, is that the stone isn’t the important bit. It’s the things that are added on to it after which are important, even though the hapless man is conned into thinking the stone is the important bit, the add-ons are just to make it work a little bit better.
Which is exactly how the tortoise cons Achilles. The stone is (A) (B) and (Z).
When Hofsteder states that the implications of Godal’s thrm tore mathematics apart he brushes over the horrifing time mathematicisans had with Geometry, and it brings to mind one of my fav stories in math history.
Once upon a time (1800s) there was an old mathematician named Farkas Bolyai who had a son named Janos Bolyai. Farkas’ main focus in his mathematical carreer was geometry, and in particular Euclid’s Fifth postulate.
Euclid, a Platonic, wrote his most famous text, Elements, in 13 volumes in 300 BC. “His aproach to geometry has dominated the teaching of the subject for over two thousand years. Moverover, the axiomatic method used by Euclid is the prototype of all of what we now call ‘pure mathematics’.”* When i growup and have a rare book collection you can be sure Elements will be in it.
Euclid’s Elements has five nice and neat postulates (or axiom) to build from, i won’t list them but i’m sure you can ask wikipedia. Now the first 4 are dandy, but it is the last one that drove people to madness. We can start with a definition “Two lines, l and m, are parallel if they do not intersect, i.e., if no point lies on both of them.” (notice the assumption that all lines are in the same plane, also note all parallel lines don’t need to be equidistant)
HERE IS THE DOOSEY: THE EUCLIDEAN PARALLEL POSTULATE: For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l.
I mean this seeems clearly true right? but even thinking about it as a pen and paper project we only ever draw lineSEGEMNTS. Who knows if the actual lines will ever meet or not.
So back to the 1800s and the Bolyais. Farkas spent all his life vainly trying to prove this puzzling postulate, and when he saw his son heading into the same path he wrote him a letter
You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone…I thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geomety and return it purified to mankind. I accomplished monstrous, enourmous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction…I turned back when I saw that no man can reach the bottom of the night. I turned back unconsoled, pitying myself and all mankind
I admit that I expect little from the deviation of your lines. It seems to me that I have been in these regions; that I ahve traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happieness
Janos was not deterred by his father’s failures though. Instead he found himself in a completly new geometry, AWESOME. in 1823he writes to his father
It is now my definite plan to pubblish a work on parallels as soon as I can complete and arrage the material and an oppertunity presents itslf; at the moment I still do not clearly see my way through, but the path which I have folloed gives positive evidence that the goal will be reached, if it is at all possible; I have not quite reached it, but I have discovered such wonderful things that I was amazed, and it would be an everlasting piece of bad fortune if they were lost. When you, my dear Father, see them, you will understand; at present I can say nothing except this: that out of nothing I have created a strange new universe. All that I have sent you previously is like a house of cards in comparison with a tower. I am no less convinced that these discoveries will bring me honor than I would be if they were completed.
His father got excited about this thing his son had made (although he never really understood it). So Farkas had it published as a 26page appendix to his book, it was titled “The Science of Absolute Space with a Demonstration of the Independence of the Truth of Falsity of Euclid’s Parallel Postualte (Which Cannot Be Decided a Priori) and, in Addition, theQuadrature of the Circle in Cas of Its Falsity” and sent a copy to his friend from university Carl Friedrich Gauss, “the formost mathematician of his time”, hoping Gauss would help it to be published in its own right.
To end our majestic yet tragic tale, Gauss was a huge dick to the Bolyais. He wrote them stating that the appendix was pretty sweet but he had already been working on the same conclusion and hadn’t intended it to be published in his life time but wanted to write it all down eventually. Janos was really hurt and never tried to publish his findings and Farkas still didn’t get it and published another flawed proof of Euclid’s Fifth.
*All my quotes are from my favorite text book of all time, Euclidean and Non-Euclidean Geometries, Developemnt and History, by Marvin Jay Greenberg. This book srsly made me love the maths.
“[Godel’s Incompleteness] Theorem can be likened to a pearl, and the method of proof to an oyster. The pearl is prized for its luster and simplicity; the oyster is a complex living beast whose innards give rise to this mysteriously simple gem.”—
G.E.B., pg 17.
These sentences alone made me stop reading and stare off to visualize every oyster/pearl pairing I’ve met. What a beautiful description of that feeling you get when you’re working through a nasty problem with a hodgepodge of data, but finally something clicks, it all makes sense…and you’ve found the pearl.
Is it so simple to "talk about language in language?"
On page 17 (of my edition at least), Hofstadter writes "While it is very simple to talk about language in language, it is not at all easy to see how a statement about numbers can talk about itself." I understand the point that he is making here, trying to illustrate the fact that it’s hard to understand self-reference within number theory, but it seems strange to me that, within a chapter on the issues that arise because of self reference and “strange loops,” the author would assert the simplicity of self-referential language.
Maybe it’s just the fact that I spent last semester grappling with readings in Derrida and Heidegger that claimed that language can’t talk about itself that made that sentence set off alarm bells when I read it. Maybe I’m confusing two completely different ideas here and the problems of self-referential language are irrelevant to his comparison to number theory. But I feel like his example (on page 21) of a strange loop in language:
The following sentence is false.
The preceding sentence is true.
indicates that it is not exactly simple to “talk about language in language,” unless we interpret that claim to mean that it’s only simple to talk about language in language; not that it’s simple to achieve accuracy, relevancy or meaning by doing so. But then what is so simple - just to make words that seem to refer to language? In any case, I didn’t really know what to do with that sentence when I came across it. Thoughts?
I’m Nathan, aka nnailling, a 27 year-old accountant from Arkansas. I try to look for creative avenues to think beyond the cubicle life (hence my Tumblog). I’m hoping GEB will do this, as well as help fulfill a resolution to read more books. It looks like a real challenge. I’ve struggled with reader’s block the last couple of years (and my whole life, really), so I’m excited about having this group to discuss things that are beyond my expertise.
A bit of an introduction. I’m Amanda Udoff and I live in Baltimore, MD where I’m heading back to school for my M.A.T. to teach high school social studies. I graduated from Goucher in 12/2008 with a degree in Political Science and Theatre. I’m pretty right brained, so the reading of this will be an exercise out of my comfort zone, but I’m really looking forward to it.
I’ve been thinking a lot recently about patterns and behavior as they relate to people and governance and history, so I’m hoping that this book will tangentially inspire some revelations about that. Otherwise, my expectations of this are minimal… it’s more an effort to broaden my horizons than anything else.
If ya’ll are interested in what Beecher is talking about in regards to how un-rigorous* math was in the beginning of the century, the book The Artist and the Mathematician is a very approachable short history of brining rigor back into the mathematicx. It also goes on to discuss the movements in this time period with word play, and languagge play. I wish i had a copy to quote from but i don’t.
*rigor is the whole idea that Hoffstetler is talking about where you need to be exact in your steps, you can’t “handwave” (another math term maening you say something just clearly follows) becuase then how do you know that you’ve actually proven anything.
It’s a comic, and a very quick read. If you’re unfmaliar with the state of mathmatics prior to Godel’s theorems this is an excellent way to understand the context around Godel’s work. In the first few chapters you’re going to hear about Bertrand Russell and his quest for absolute truth. This story is about him, and why he did his work attempting to codify the foundations of math.
Hey, I’m Ben Beecher. I’m living and working in nyc as a computer programmer, free software advocate/linux nerd, and amature go player. I graduated from goucher back in 2007, and I still make it down there from time to time. I’ve got a pretty left-brain centric mind, so I’m going to be coming at this from a mathematical and logical perspective, and I’m happy to have people with more artistic leanings to correct my misconceptions about Bach and Escher. This will be the third time I’ve read GEB, so I’m a little ahead on the concepts, especially the math related ones. GEB might be my favorite book, and I’m a huge fan of Hofstadter - after reading it I went on to read I Am a Strange Loop , Metamagical Themas and The Mind’s I. Recently I’ve been reading about cellular automaton and I’m interested to see how close an isomorphism exists so I plan to read it with an eye for that. I tend to be rather strongly opinionated and hot headed - sorry in advance if I say something that offends you!
Ok - so today is the first day of our first week! This week we’re going to power through the Introduction, A Musico-Logical Offering, the first dialogue, Three Part Invention, and the first chapter, The MU-Puzzle.
The introduction is a broad overview of Hofstadter’s project. You’ll be introduced to Bach through an amazing story about the creation of the Musical Offering, and we’ll meet M.C. Escher and Kurt Gödel and get a history of the last hundred years of mathematics in 2 pages as well. The first dialogue and chapter are pretty straightforward - they will lay the ground rules for future discussions of mathematical theory and give us our first puzzle.
Don’t be discouraged if you come across a part you struggle with or don’t understand - that’s what the group is here for! Post your question, leave it as a comment, or send it to me and I’ll post it up.
Hi, I’m Andy Cohen, A.K.A. Inadvisable on Tumblr. I work at Newsweek magazine in New York and co-host a weekly call-in radio show on WFMU called SHUT UP, WEIRDO (which has it’s own Tumblog).
I’ve been trying to read GEB since it first came out in 1979, because as a kid I was really into Escher. The book sounded exciting, but the math and music was way over my head. I tried again 20 years ago, but didn’t make it to the start of Chapter 1. Since then, I’ve become a fan of Bach and read a biography of Godel, so I’m ready for a third attempt. This is my first “book club” experience.
In recent years I’ve become interested in the nature of truth and reality. Some specific areas of interest:
What makes a news story “accurate”?
Why do smart people make big financial mistakes?
How do magic tricks work? Why are people fooled by them?
What makes jokes funny?
Can you ever really trust your brain and/or perceptions?
Hey guys, my name is Jon and I’m pretty pumped to have a group to read this book with. I’m reading my Dad’s old copy, and if I get past chapter two it will have beaten the record my 14 year old self set 8 years ago.
I’ve always been a closet philosophy nerd, but don’t expect formal language or ‘real’ knowledge. I’m also a closet artist, and I think that’s what makes this book appeal to me so much, finally getting a chance to look at all these mathematical and logical ideas in a visual way. It will make the math go down easier.
Also, thanks for posting the first Bach piece we needed to hear, and I hope someone in this group gets (like really gets) music theory. Its not me.
"The ‘Strange Loop’ phenomenon occurs whenever, by moving upwards (or downwards) through the levels of hierarchical system, we unexpectedly find ourselves right back where we started."
Chris had mentioned he didn’t know as much about Bach’s musical pieces as he would have liked, and I assume it’s a bit similar for other people. I found a good youtube channel for the Kuijken Ensemble, of which I have a lot of Bach’s other pieces on my itunes, and thought I’d share it with you all. It’s 11 parts and the first half of the youtube channel, but it’s pretty great quality, especially on catching the notes of the harpsichord.
“In a way, this book is a statement of my religion. I hope that this will come through to my readers, and that my enthusiasm and reverence for certain ideas will infiltrate the hearts and minds of a few people. That is the best I could ask for.”— Douglas Hofstadter
…but, I decided to pregame our journey’s beginning tomorrow by reading the Preface to the Twentieth-Anniversary edition of GEB. Just having finished it, I can tell you that the prospect of attacking the book’s main text is both exciting and terrifying.
There are three things, coming out of the Preface, that I want share before we get going:
1. Hofstadter talks a lot about “strange loops” in these opening pages and how they are related to an idea that he thinks arises out of Godel’s work- namely, that the “I” arises out of meaning made by finding patterns in a system of otherwise meaningless symbols (incidentelly, please feel free to correct me if this point is crude or flat out incorrect- I’m having a great deal of trouble wrapping my head around it). Turning this thought over in my mind, I began to think of it in terms of one of my great passions, both academically and recreationally- the superhero comic book. I’ll develop this thought more thoroughly later but, essentially, the mainstream superhero comic book is simply a series (in fact, several different sorts of series) of inherently meaningless symbols out of which arise patterns which create meaning- the only reason that, for instance, the Bat Signal is effective is because when Commisioner Gordon points at the sky, Batman shows up. This becomes particularly interesting when talking about characters whose symbols are taken from elsewhere and are meaningful not only through their association with that character but also with whatever pattern from whence their meaning originally comes. In other words, Captain America (whose spandex is literally referred to as “the flag” in the comics) might have something really interesting to say about GEB.
2. Something else I saw as I was reading (in a manner not dissilimair from the way Hofstadter talks about M.C. Escher entering the work) were ideas that I had been dealing with in a class I just took on Aristotle’s Physics and Metaphysics. Again, this is a thought that needs further developing, but I have a feeling that concepts like Aristotelian motion are going to be a useful lens through which to view some things that we come across. Interestingly, Hofstadter himself seems to implicitly reject the notion of being-at-work-staying-itself in the preface’s final section.
3. The following dialogue comes from the Coen Brothers’ film A Serious Man, which I saw earlier today:
Clive Park: Uh, Dr. Gopnik, I believe the results of physics mid-term were unjust. Larry Gopnik: Uh-huh, how so? Clive Park: I received an unsatisfactory grade. In fact: F, the failing grade. Larry Gopnik: Uh, yes. You failed the mid-term. That’s accurate. Clive Park: Yes, but this is not just. I was unaware to be examined on the mathematics. Larry Gopnik: Well, you can’t do physics without mathematics, really, can you? Clive Park: If I receive failing grade I lose my scholarship, and feel shame. I understand the physics. I understand the dead cat. Larry Gopnik: You understand the dead cat? But… you… you can’t really understand the physics without understanding the math. The math tells how it really works. That’s the real thing; the stories I give you in class are just illustrative; they’re like, fables, say, to help give you a picture. An imperfect model. I mean - even I don’t understand the dead cat. The math is how it really works. (via IMDB)
Yes, I’m Christie, though you may know me as christielouwho. I don’t know if I was even given the option of reading GEB, I was simply told to read it as well. Regardless, the decision was made and I immersed myself into the preface and knew it’d be easily interesting. I’m rather glad this group has started up. With so many different minds, expertises, and backgrounds to feed from, it’ll be interesting to see the intricacies of the book through different minds.
I completely and whole-heartedly admit my disdain and general ignorance for mathematical theorems and data. I’m a ‘questioner’ at heart, not a decider, so the black/white qualities of math have always frustrated me. Yet, the writing style Hofstadter uses and his passion for explaining Gödel’s theorem to the layman make even the most ignorant (*cough*, me) able to comprehend in context with Escher and Bach’s work.
I will admit that I do have a rather extensive background in art and classical music. I trained to be an artist and worked as a master’s apprentice for three years, was accepted into an art school, and even had two very artistically-minded parents raise me. I’ve always loved Escher’s art and even did a study of it in middle school off of his Metamorphosis. As for Bach… I’ve listened to Bach and his respective peers since I was a child, and swear I have some of his pieces in my subconscious along with Bizet’s Carmen. I have quite a plethora of Bach’s works and will do my best to post the pieces on the site for your listening pleasure off of my personal tumblr, and the tumblr I help run fyeahclassical.
I have a tumblr called thethirdoption (links to my other tumblrs and the name explanation there). My main areas of expertise are metaphysics and meta- just about everything, which may explain why MC Escher has been my favorite artist (and MC) (that was a joke) (just the MC part was a joke) (yes I overuse parentheses) since I was a little kid.
For the period of time in middle school when I was really into classical music, Bach was my favorite composer, because of the intricacies and complexities as well as the speed. So you might say this has been a long time coming.
I have recently taken up a thoroughly right-brained point of view for just about everything, including typically left-brained pursuits.
You might say I’m the red-headed stepchild of this bunch.