Quite a few years ago now, I spent some time as a college philosophy major. I realized pretty quickly that I had seriously misunderstood what that meant. I was a young man looking for answers, trying to understand the core of what life was about so that I could live accordingly. And while the great philosophers – some of them, anyway – offered those answers and attempted to explain that core, the academic approach was to regard all their ideas as if they were merely moves in some endless intellectual game: Plato says this, but Aristotle denies it, Spinoza offers this, but look at what Kant says instead – my rook to your queen’s pawn, my club to your heart.
Worse, as I later learned, the academic teachers of philosophy had for generations misrepresented the teachings of ancient philosophy. Ironically, this misrepresentation arose from the writings of scholars of the 17th and 18th centuries who truly admired Plato and Aristotle, but in an idiosyncratic and conditional way: extolling the philosophers as the originators of rationalism, but condemning them for failing to maintain the kind of hyper-rationalism they themselves wanted to practice and spread.
Another element not to be disregarded was the tendency – and not just among academics – to believe that “newer” automatically means “better.” For teachers of philosophy, this translates into the belief that the speculations of Hegel or Wittgenstein or Heidegger or Foucault must be ever more complete, more scientific, more true, than those of Plato, Aristotle or Epicurus, because we have built upon, we have surpassed, their groping attempts to explain reality. In a word, philosophy, like everything else, has “evolved.”
Finally, and most damagingly, we have the triumph of the belief that “learning” is a noun, not a verb; that knowledge is a sort of commodity to be acquired and traded in measurable chunks. It’s likely that this view was inevitable once the bureaucratization of education began within industrialized society, because it enables the creation of standardized curricula and lesson plans and all the rest of the apparatus required to turn schools into factories (sorry, “manufacturing plants”). What this meant for philosophy departments, as for all others, is that the professors taught the curriculum – in other words, the entrenched misunderstandings, misreadings, biases, tendencies – and not the subject.
The subject of philosophy is, of course, wisdom; or rather, the seeking of wisdom. Not surprisingly, professors of philosophy have for several centuries shied away from attempting to teach such things, perhaps in largest part because they are so open-ended. What they teach is not philosophy, how to “do philosophy,” how to be a philosopher, but what different philosophers have said and how to quibble with it. The measure of how far the professors are removed from the actual doing of philosophy is the fact that while every one of the teachers and textbooks I encountered in my time as a philosophy major happily defined the word “philosophy” as “love of wisdom,” not one ever tried to explain what “wisdom” might be.
What I took away from my experience as a philosophy major was the belief that Western philosophy had absolutely nothing to offer to a seeker of the kind of core understanding of life I mentioned earlier. It took me a lot of years and a long roundabout trip through Eastern religion and Western occultism and mysticism to realize that I had been completely misled. When I finally returned to ancient Western philosophy, to the philosophers themselves and not the professors, I discovered that what I had been looking for in the first place was always there.
Saturday, July 3, 2010
Wednesday, June 23, 2010
A Dog's Life
One summer evening a few years ago, I was sitting in my backyard unwinding with a bottle of Warsteiner after a day’s work when something struck me that has stuck with me ever since. Our backyard at the time was a rectangle surrounded by a chain-link fence, and as I sat there I could see through to the other backyards, which were also rectangles surrounded by chain-link fences.
What struck me was how much the houses and yards along our block resembled the kennel where my wife and I had boarded our two dogs not long before. It was one of those nice kennels, where each dog had a nice big cage inside and an opening to a nice individual fenced-in “run” outside. We felt good about leaving our dogs there for a week because they were free to go in and out, they weren’t as confined as they would have been in one of those old nasty kennels where they had to sit in a cage all day waiting to be walked.
What we had done, of course, was judge the kennel according to our human standards. Without realizing it, we had boarded the dogs at a kennel that essentially was modeled on our own living space: a box in which we felt safe and sequestered when that was what we wanted, and an attached open area where we could go out and be “in” nature when that was what we wanted, but safely marked off from our neighbors’ parcels of ground. We were imputing to our dogs the same kind of need for a well-defined freedom that we felt for ourselves.
Now, it may seem invidious that I’m comparing an average American suburban home to a dog kennel, but I don’t mean it that way. On the contrary, the comparison really depends on the fact that we love our “companion animals” and want only the best for them. The point is simply that we conceive the “best” for them in the same terms we conceive it for ourselves: as having a certain kind of private, personal space in which we are free to do what we want, when we want.
If there is anything invidious in this, it’s the contrast between this rather limited – one might almost say compromised – version of freedom and the “Freedom” with a capital F that people make such a fuss over in the sphere of public discussion and action. It’s perhaps a little hard to reconcile the Freedom that people have fought and died for with the freedom to have a barbecue and burn tiki torches.
Still, the two kinds really aren’t totally unrelated. Where they are related is in the understanding that you and I have a right to do whatever we want to do in our personal spaces. (Within reason, of course: If my neighbor is committing sex crimes or torturing puppies in the house next door, I need to interfere with him doing that.) This is precisely why the ownership of a home is the core of the American Dream: because my home is a space where I can exercise my sovereignty as an individual, and of course individual sovereignty is what America is all about.
Freedom also involves, of course, the freedom to work at the job one chooses so as to be able to afford a home. And for some fortunate few, their work itself provides the kind of fulfillment we all seek, while for others work is just a means to obtain the kind of personal space we need to practice whatever else gives us that fulfillment (“I work to live, I don’t live to work”).
I happen to live in a kind of middle space in this regard: As a journalist, I sometimes am lucky enough to wander into a story that actually does some kind of good for others, and that’s about as rewarding as it gets. But I also have an inner life that I pursue in the privacy of my home that gives me some satisfaction even on those days when my job totally sucks.
I imagine a lot of us are in somewhat the same situation, doing what we can in our careers to give something to the world, and/or seeking in our “leisure” hours to cover whatever we feel as a lack in our spiritual or psychological lives. This sort of thing is, I believe, exactly what Thomas Jefferson had in mind when he wrote that we have an “unalienable right” to “the pursuit of happiness.”
What I find regrettable in our society in regard to these things is the widespread tendency to confuse means with ends. It appears that many of us expect to find fulfillment in the acquisition of the personal space and its accoutrements, rather than the use. There’s a bumper sticker that sums up the attitude: “He who dies with the most toys wins.” Many of us seem to believe that it’s the mere having of a home, or the size of the home, not the life lived inside it, that matters most. Once they have it, what are they supposed to do with it?
It appears that for many, the “pursuit of happiness” within one’s private space or in public means eating as much, drinking as much, owning as much, playing as much as one can, with no thought for the consequences to oneself or the world at large. Such an attitude is truly tragic, because it focuses on the most ephemeral things the world has to offer and leads people away from the sources of real, lasting happiness.
Our consumerist economic structure of course encourages this sort of belief and behavior, and the recent shakiness of that structure is a warning about its unsustainability – as if further warning were needed on top of our repeated energy crises, our “obesity epidemic,” our high crime rates and all the other social ills that are so obviously traceable to our society’s tendency to want more, more, more.
As much as I would like to see increased regulation of businesses, I would be the last person to suggest that we impose further restrictions on people’s private behavior. “An ye harm none, do what thou wilt” strikes me as a pretty good ethical principle. The challenge is getting people to understand the “harm none” part, especially in a world in which we seem to have moved from the idea that “all men are created equal” to a belief that “individual sovereignty” means every man is entitled to be a king. Regrettably, it appears that the king everyone wants to be is this one:
What struck me was how much the houses and yards along our block resembled the kennel where my wife and I had boarded our two dogs not long before. It was one of those nice kennels, where each dog had a nice big cage inside and an opening to a nice individual fenced-in “run” outside. We felt good about leaving our dogs there for a week because they were free to go in and out, they weren’t as confined as they would have been in one of those old nasty kennels where they had to sit in a cage all day waiting to be walked.
What we had done, of course, was judge the kennel according to our human standards. Without realizing it, we had boarded the dogs at a kennel that essentially was modeled on our own living space: a box in which we felt safe and sequestered when that was what we wanted, and an attached open area where we could go out and be “in” nature when that was what we wanted, but safely marked off from our neighbors’ parcels of ground. We were imputing to our dogs the same kind of need for a well-defined freedom that we felt for ourselves.
Now, it may seem invidious that I’m comparing an average American suburban home to a dog kennel, but I don’t mean it that way. On the contrary, the comparison really depends on the fact that we love our “companion animals” and want only the best for them. The point is simply that we conceive the “best” for them in the same terms we conceive it for ourselves: as having a certain kind of private, personal space in which we are free to do what we want, when we want.
If there is anything invidious in this, it’s the contrast between this rather limited – one might almost say compromised – version of freedom and the “Freedom” with a capital F that people make such a fuss over in the sphere of public discussion and action. It’s perhaps a little hard to reconcile the Freedom that people have fought and died for with the freedom to have a barbecue and burn tiki torches.
Still, the two kinds really aren’t totally unrelated. Where they are related is in the understanding that you and I have a right to do whatever we want to do in our personal spaces. (Within reason, of course: If my neighbor is committing sex crimes or torturing puppies in the house next door, I need to interfere with him doing that.) This is precisely why the ownership of a home is the core of the American Dream: because my home is a space where I can exercise my sovereignty as an individual, and of course individual sovereignty is what America is all about.
Freedom also involves, of course, the freedom to work at the job one chooses so as to be able to afford a home. And for some fortunate few, their work itself provides the kind of fulfillment we all seek, while for others work is just a means to obtain the kind of personal space we need to practice whatever else gives us that fulfillment (“I work to live, I don’t live to work”).
I happen to live in a kind of middle space in this regard: As a journalist, I sometimes am lucky enough to wander into a story that actually does some kind of good for others, and that’s about as rewarding as it gets. But I also have an inner life that I pursue in the privacy of my home that gives me some satisfaction even on those days when my job totally sucks.
I imagine a lot of us are in somewhat the same situation, doing what we can in our careers to give something to the world, and/or seeking in our “leisure” hours to cover whatever we feel as a lack in our spiritual or psychological lives. This sort of thing is, I believe, exactly what Thomas Jefferson had in mind when he wrote that we have an “unalienable right” to “the pursuit of happiness.”
What I find regrettable in our society in regard to these things is the widespread tendency to confuse means with ends. It appears that many of us expect to find fulfillment in the acquisition of the personal space and its accoutrements, rather than the use. There’s a bumper sticker that sums up the attitude: “He who dies with the most toys wins.” Many of us seem to believe that it’s the mere having of a home, or the size of the home, not the life lived inside it, that matters most. Once they have it, what are they supposed to do with it?
It appears that for many, the “pursuit of happiness” within one’s private space or in public means eating as much, drinking as much, owning as much, playing as much as one can, with no thought for the consequences to oneself or the world at large. Such an attitude is truly tragic, because it focuses on the most ephemeral things the world has to offer and leads people away from the sources of real, lasting happiness.
Our consumerist economic structure of course encourages this sort of belief and behavior, and the recent shakiness of that structure is a warning about its unsustainability – as if further warning were needed on top of our repeated energy crises, our “obesity epidemic,” our high crime rates and all the other social ills that are so obviously traceable to our society’s tendency to want more, more, more.
As much as I would like to see increased regulation of businesses, I would be the last person to suggest that we impose further restrictions on people’s private behavior. “An ye harm none, do what thou wilt” strikes me as a pretty good ethical principle. The challenge is getting people to understand the “harm none” part, especially in a world in which we seem to have moved from the idea that “all men are created equal” to a belief that “individual sovereignty” means every man is entitled to be a king. Regrettably, it appears that the king everyone wants to be is this one:
Monday, June 21, 2010
Where There's a Will There's an Excuse
Looking back at the stuff I’ve written since I reactivated this blog a few weeks ago, it struck me that a casual reader might get the impression that my thought processes are pretty chaotic. I could claim that I’ve deliberately been picking random topics as a way to enable “emergent order” to work its magic on my muddled thoughts in the same way it’s supposed to account for the existence of order in physical processes that are alleged to be random in their underlying dynamics. But just as I believe that the order in our cosmos is there from the beginning, I also want to claim that there has been method in my madness all along.
One of the nagging questions about human beings, one that gets asked over and over again under all kinds of circumstances, is this: How could anyone do that? We hear about some awful, horrible thing that has happened, something that seems to violate every rule as we understand the rules, and we wonder how or why another human being could behave in such a grossly and grotesquely wrong way: committing serial murders, genocide, child-rape, conning old people out of their life savings, condemning miners to unmarked graves in unsafe coal pits, feeding children toxic chemicals with their formula, aiding and abetting dictators just to get at the minerals buried under their subjects’ homes, etc. etc. etc.
Frankly, I don’t think the answer is as difficult or mystifying as people seem to believe. Let’s start here: Socrates said (according to Plato) that no one does evil willingly. And Aristotle said, famously, “All beings by nature desire the good.” People do what they do because they believe, rightly or wrongly, that what they’re doing is good – if not for the world at large, at least for themselves.
And people are able to convince themselves that very bad things are actually very good things. Even a psycho- or sociopath may have some inkling that society in general disapproves of the bad, terrible, awful things he or she wants to do, but there’s always a way to claim that “I am right, you (all) are wrong.”
Because we all live in a constructed reality, each of us in his or her own constructed reality: an intellectual or psychological bubble built with the materials at hand, personal, social, political, intellectual, what have you.
As I pointed out here, it’s impossible for a human being to have a complete picture of the universe as it really exists at any moment. As a result, we're forced to go through life with an understanding of the universe and our place in it that is, and will always remain, largely hypothetical. The nature of reality forces us to fill in a lot of blanks with our best guesses, which often are supplied to us by those around us.
That gives us wide latitude to indulge whatever predispositions we bring to the table, whether from personal or social conditioning or out of the fundaments of our souls. In essence, we learn to construct arguments in support of whatever it is we want to believe, whatever we want to do.
We can make anything fit, if we just put our minds to the task: skimping on safety equipment in mines and on oil platforms so as to keep our costs low and our profits high, for instance; selling drugs (“prescription medications”) that ravage people’s bodies or minds, because we can whip out a “clinical study” that shows that 51 percent of the test subjects felt slightly better after swallowing our pill, and only 10 percent had “adverse reactions;” forcing the migration of indigenous people or just chewing through the ground beneath their feet because they didn’t understand the value of what was down there and weren’t exploiting it like we can; or “she said no but I could see she really meant yes.”
There does remain some fairly widespread agreement, even in our fragmented world, about what’s right and what’s wrong. Unfortunately, it seems more and more as though the people who share that agreement are the least able to do anything about it. The social, political and economic predators not only have clawed their way to the top, they’ve embedded their self-justifications at the heart of our society, to the point where demanding that a (foreign) corporation compensate people for the catastrophic damage it has caused through its utterly unconscionable activities can be characterized by a “people’s representative” as a form of extortion.
This is exactly what I mean about living in a “bubble”: Anyone who could see British Petroleum as the victim in the current catastrophe is living in his imagination, not reality. Man may be, as Aristotle said, a rational animal, but he’s very talented at putting his rationality to work in the service of what pleases him most, no matter how destructive or downright disgusting that may be.
One of the nagging questions about human beings, one that gets asked over and over again under all kinds of circumstances, is this: How could anyone do that? We hear about some awful, horrible thing that has happened, something that seems to violate every rule as we understand the rules, and we wonder how or why another human being could behave in such a grossly and grotesquely wrong way: committing serial murders, genocide, child-rape, conning old people out of their life savings, condemning miners to unmarked graves in unsafe coal pits, feeding children toxic chemicals with their formula, aiding and abetting dictators just to get at the minerals buried under their subjects’ homes, etc. etc. etc.
Frankly, I don’t think the answer is as difficult or mystifying as people seem to believe. Let’s start here: Socrates said (according to Plato) that no one does evil willingly. And Aristotle said, famously, “All beings by nature desire the good.” People do what they do because they believe, rightly or wrongly, that what they’re doing is good – if not for the world at large, at least for themselves.
And people are able to convince themselves that very bad things are actually very good things. Even a psycho- or sociopath may have some inkling that society in general disapproves of the bad, terrible, awful things he or she wants to do, but there’s always a way to claim that “I am right, you (all) are wrong.”
Because we all live in a constructed reality, each of us in his or her own constructed reality: an intellectual or psychological bubble built with the materials at hand, personal, social, political, intellectual, what have you.
As I pointed out here, it’s impossible for a human being to have a complete picture of the universe as it really exists at any moment. As a result, we're forced to go through life with an understanding of the universe and our place in it that is, and will always remain, largely hypothetical. The nature of reality forces us to fill in a lot of blanks with our best guesses, which often are supplied to us by those around us.
That gives us wide latitude to indulge whatever predispositions we bring to the table, whether from personal or social conditioning or out of the fundaments of our souls. In essence, we learn to construct arguments in support of whatever it is we want to believe, whatever we want to do.
We can make anything fit, if we just put our minds to the task: skimping on safety equipment in mines and on oil platforms so as to keep our costs low and our profits high, for instance; selling drugs (“prescription medications”) that ravage people’s bodies or minds, because we can whip out a “clinical study” that shows that 51 percent of the test subjects felt slightly better after swallowing our pill, and only 10 percent had “adverse reactions;” forcing the migration of indigenous people or just chewing through the ground beneath their feet because they didn’t understand the value of what was down there and weren’t exploiting it like we can; or “she said no but I could see she really meant yes.”
There does remain some fairly widespread agreement, even in our fragmented world, about what’s right and what’s wrong. Unfortunately, it seems more and more as though the people who share that agreement are the least able to do anything about it. The social, political and economic predators not only have clawed their way to the top, they’ve embedded their self-justifications at the heart of our society, to the point where demanding that a (foreign) corporation compensate people for the catastrophic damage it has caused through its utterly unconscionable activities can be characterized by a “people’s representative” as a form of extortion.
This is exactly what I mean about living in a “bubble”: Anyone who could see British Petroleum as the victim in the current catastrophe is living in his imagination, not reality. Man may be, as Aristotle said, a rational animal, but he’s very talented at putting his rationality to work in the service of what pleases him most, no matter how destructive or downright disgusting that may be.
Monday, June 14, 2010
Results May Vary
I’ve written previously about some of what I consider the shortcomings of orthodox financial economics. In general, it appears to me that economic theory is largely detached from reality in much the same way astronomy was before Copernicus and Galileo.
One instance of this detachment from reality is the “random walk” theory of financial markets. This model of market behavior was developed in the 1960s and still holds sway in the retail research departments of most U.S. brokerage firms. It’s based mainly on this:
What the chart shows is the Dow Jones Industrial Average from its creation in 1896 to the present (monthly closing prices; my apologies for the unreadability). The greenish line is a “linear regression,” that is, the straight line that is the best fit to the actual data points. Linear regression was a cutting-edge tool back in the ’60s, when computing power was rather limited. But there’s an unargued underlying assumption in applying it to a market, which makes the “random walk” theory an exercise in circular logic.
The assumption is that the graph of a stock index like the Dow in a way “wants” to be a straight line but can’t manage it. What economists actually say is that the market “seeks equilibrium,” by which they mean that it wants to go up continuously and at a consistent rate. But instead, there are “perturbations” that cause “fluctuations” above or below the idealized rate of gain. Those fluctuations are by nature random (hence “random walk”) and therefore are unpredictable.
Thus, investors shouldn’t try to guess when the market is going to have one of those “perturbations” – in other words, they shouldn’t try to practice “market timing” – but instead should simply buy stocks and hold them for the long term, because the underlying trend always goes higher. And here the logical circle is closed.
Since the 1980s, the random walk theory has come under increasing criticism, and many economists outside the Wall Street houses acknowledge that it’s not a realistic model. Many still want to cling to some variation of randomness, however, and have developed hybrid models that include some degree of self-recursiveness along with the randomness, giving us things like the exotically named GARCH model: “generalized auto-regression with conditional heteroskedacity.” However, scientific studies have shown that these models have zero value in predicting market movements.
Another argument in favor of “buy and hold” investing is the claim that the stock market consistently over the years has provided an average annual percentage return that is higher than other kinds of investments. But this depends very much on how you calculate the annual return. The usual figure is 8 percent, compared with half that return or less from interest-bearing investments such as bonds.
It’s true that if you take the closing price of an index like the Dow on a given day and calculate the percent change from the same day the year before, and you average that calculation over the past 100 years, you’ll get a figure something like that 8 percent number. But the calculation doesn’t bear any resemblance to how people actually invest: You can’t “buy the Dow” every day and sell it a year later.
I’ve constructed a model that I think represents more accurately how people really invest. I had to make some simplifying assumptions, but I believe the result is still more indicative of the kind of average returns investors can expect.
Here’s the idea: Let’s suppose that a worker sets up a program in which he or she invests a set percentage of his or her income each month. This program continues for 30 years, at which point the worker retires and cashes out. I’ve also assumed that the worker gets a cost-of-living raise at the beginning of each year, based on the nominal inflation rate (based on the Consumer Price Index) for the previous year. (That’s something few workers are actually seeing today, so the model may actually overstate the investor’s returns somewhat.)
The following chart shows the average annual return a worker would have received by following this investment strategy:
The time scale (if you can read it; right-click to open it bigger in a new window) indicates the date upon which the worker began the monthly investment program, and the vertical scale shows the percentage return at the end of 30 years for a worker who started investing on the date shown. For example, a worker who started an investment program in the very first month that Charles Dow calculated his industrial average in 1896 (i.e., the very beginning of the blue line) would have earned an average annual return of 2.86 percent over the next 30 years.
The very highest average return, 18.11 percent, would have been earned by a worker who began a monthly investment program in December 1969 and cashed out in December 1999. Obviously, the average annual return for someone who started 10 years later and cashed out last year would have been quite a bit less, just 3.92 percent at the low point in February 2009.
Worse yet, people who started investing in late 1901 to mid-1903 would have lost money, as would almost everyone who started in 1912. Perhaps most surprisingly, anyone who embarked on this kind of program in the late 1940s to mid-1950s -- which we're used to thinking of as boom times -- would have earned a fairly paltry annual return of about 2 percent or less when they cashed out in the late 1970s-early 1980s -- which were not so booming.
Overall according to this scenario, investors who have cashed out to date have earned an average annual return of 5.05 percent or a median annual return of 4.44 percent. Those figures aren’t all that much above the long-term average or median returns on interest-bearing investments. As I said earlier, the results may overstate the average returns because of my assumption about annual salary increases. In addition, the numbers don’t include any taxes or transaction costs such as brokerage commissions.
However, the real lesson of this exercise isn’t about long-term average returns, it’s about the wide variability of the real-time returns. What it boils down to is that even for a long-term investor, your results still depend entirely on when you start investing and when you cash out. If we relate that to our entry into the career world, it shows how much the decision is out of our hands: We can’t choose what year we’re born. Whether we like it or not, we’re all market-timers.
One instance of this detachment from reality is the “random walk” theory of financial markets. This model of market behavior was developed in the 1960s and still holds sway in the retail research departments of most U.S. brokerage firms. It’s based mainly on this:
What the chart shows is the Dow Jones Industrial Average from its creation in 1896 to the present (monthly closing prices; my apologies for the unreadability). The greenish line is a “linear regression,” that is, the straight line that is the best fit to the actual data points. Linear regression was a cutting-edge tool back in the ’60s, when computing power was rather limited. But there’s an unargued underlying assumption in applying it to a market, which makes the “random walk” theory an exercise in circular logic.
The assumption is that the graph of a stock index like the Dow in a way “wants” to be a straight line but can’t manage it. What economists actually say is that the market “seeks equilibrium,” by which they mean that it wants to go up continuously and at a consistent rate. But instead, there are “perturbations” that cause “fluctuations” above or below the idealized rate of gain. Those fluctuations are by nature random (hence “random walk”) and therefore are unpredictable.
Thus, investors shouldn’t try to guess when the market is going to have one of those “perturbations” – in other words, they shouldn’t try to practice “market timing” – but instead should simply buy stocks and hold them for the long term, because the underlying trend always goes higher. And here the logical circle is closed.
Since the 1980s, the random walk theory has come under increasing criticism, and many economists outside the Wall Street houses acknowledge that it’s not a realistic model. Many still want to cling to some variation of randomness, however, and have developed hybrid models that include some degree of self-recursiveness along with the randomness, giving us things like the exotically named GARCH model: “generalized auto-regression with conditional heteroskedacity.” However, scientific studies have shown that these models have zero value in predicting market movements.
Another argument in favor of “buy and hold” investing is the claim that the stock market consistently over the years has provided an average annual percentage return that is higher than other kinds of investments. But this depends very much on how you calculate the annual return. The usual figure is 8 percent, compared with half that return or less from interest-bearing investments such as bonds.
It’s true that if you take the closing price of an index like the Dow on a given day and calculate the percent change from the same day the year before, and you average that calculation over the past 100 years, you’ll get a figure something like that 8 percent number. But the calculation doesn’t bear any resemblance to how people actually invest: You can’t “buy the Dow” every day and sell it a year later.
I’ve constructed a model that I think represents more accurately how people really invest. I had to make some simplifying assumptions, but I believe the result is still more indicative of the kind of average returns investors can expect.
Here’s the idea: Let’s suppose that a worker sets up a program in which he or she invests a set percentage of his or her income each month. This program continues for 30 years, at which point the worker retires and cashes out. I’ve also assumed that the worker gets a cost-of-living raise at the beginning of each year, based on the nominal inflation rate (based on the Consumer Price Index) for the previous year. (That’s something few workers are actually seeing today, so the model may actually overstate the investor’s returns somewhat.)
The following chart shows the average annual return a worker would have received by following this investment strategy:
The time scale (if you can read it; right-click to open it bigger in a new window) indicates the date upon which the worker began the monthly investment program, and the vertical scale shows the percentage return at the end of 30 years for a worker who started investing on the date shown. For example, a worker who started an investment program in the very first month that Charles Dow calculated his industrial average in 1896 (i.e., the very beginning of the blue line) would have earned an average annual return of 2.86 percent over the next 30 years.
The very highest average return, 18.11 percent, would have been earned by a worker who began a monthly investment program in December 1969 and cashed out in December 1999. Obviously, the average annual return for someone who started 10 years later and cashed out last year would have been quite a bit less, just 3.92 percent at the low point in February 2009.
Worse yet, people who started investing in late 1901 to mid-1903 would have lost money, as would almost everyone who started in 1912. Perhaps most surprisingly, anyone who embarked on this kind of program in the late 1940s to mid-1950s -- which we're used to thinking of as boom times -- would have earned a fairly paltry annual return of about 2 percent or less when they cashed out in the late 1970s-early 1980s -- which were not so booming.
Overall according to this scenario, investors who have cashed out to date have earned an average annual return of 5.05 percent or a median annual return of 4.44 percent. Those figures aren’t all that much above the long-term average or median returns on interest-bearing investments. As I said earlier, the results may overstate the average returns because of my assumption about annual salary increases. In addition, the numbers don’t include any taxes or transaction costs such as brokerage commissions.
However, the real lesson of this exercise isn’t about long-term average returns, it’s about the wide variability of the real-time returns. What it boils down to is that even for a long-term investor, your results still depend entirely on when you start investing and when you cash out. If we relate that to our entry into the career world, it shows how much the decision is out of our hands: We can’t choose what year we’re born. Whether we like it or not, we’re all market-timers.
Saturday, June 12, 2010
The Human Factor
Before I moved back to Petersburg in the fall of 2008, I had been working for five years for the Post and Courier in Charleston, S.C., as assistant business editor. In that capacity, I was asked to write a business blog, which I did for about six months before I accepted a buyout and left.
There was a sort of cognitive dissonance between me and the bosses about that blog: what I was writing turned out not to be what they had expected me to write (and I wasn't even doing any of the cosmic stuff then). So all of my posts there were taken offline pretty quickly after I left.
That's a shame, because personally I think some of them were pretty good, although that might just be my memory playing tricks on me. But one of them in particular I want to try to reconstruct, because the point it made was one that needs to be remembered.
I'm sure at some point or other, all of us have seen a sign at a business that says, "Our people are our most important asset." It's a nice sentiment, but if there's any truth at all in what it says, it's purely symbolic. Under "generally accepted accounting principles," people are not an asset.
If you look at actual corporate financial statements, you won't find "people" listed anywhere. But if you know where to look, you can find where they're hidden. It's not on the statement of assets. On the contrary, people show up on the income statement as a cost of doing business. And if the company owes money to its employee pension fund, that shows up on the statement of liabilities.
As a result, when business slows down (or goes down the toilet), it's a no-brainer for the MBAs and CPAs and other bean-counters to look at the financial statements and think, "Hey, here's a quick and easy way to make the numbers look better: Fire some workers and dump the pension plan."
If people really were treated as an asset in some way - if companies were required to account for the potential cost of training replacements, for example - then it wouldn't be such an easy decision to fire them en masse. That's because anytime a company has to write down the value of an asset, the writedown has to be reflected on the income statement as an expense. So laying people off wouldn't automatically make "the bottom line" look better.
It's ironic, or something, that businesses do account for their "property, plant and equipment" as assets, but not the employees who actually make those things work to put out products or provide services, and in general to create profits. In the real world - the world beyond the spreadsheets and trial balance ledgers and forecasting models - people do actually have value.
There was a sort of cognitive dissonance between me and the bosses about that blog: what I was writing turned out not to be what they had expected me to write (and I wasn't even doing any of the cosmic stuff then). So all of my posts there were taken offline pretty quickly after I left.
That's a shame, because personally I think some of them were pretty good, although that might just be my memory playing tricks on me. But one of them in particular I want to try to reconstruct, because the point it made was one that needs to be remembered.
I'm sure at some point or other, all of us have seen a sign at a business that says, "Our people are our most important asset." It's a nice sentiment, but if there's any truth at all in what it says, it's purely symbolic. Under "generally accepted accounting principles," people are not an asset.
If you look at actual corporate financial statements, you won't find "people" listed anywhere. But if you know where to look, you can find where they're hidden. It's not on the statement of assets. On the contrary, people show up on the income statement as a cost of doing business. And if the company owes money to its employee pension fund, that shows up on the statement of liabilities.
As a result, when business slows down (or goes down the toilet), it's a no-brainer for the MBAs and CPAs and other bean-counters to look at the financial statements and think, "Hey, here's a quick and easy way to make the numbers look better: Fire some workers and dump the pension plan."
If people really were treated as an asset in some way - if companies were required to account for the potential cost of training replacements, for example - then it wouldn't be such an easy decision to fire them en masse. That's because anytime a company has to write down the value of an asset, the writedown has to be reflected on the income statement as an expense. So laying people off wouldn't automatically make "the bottom line" look better.
It's ironic, or something, that businesses do account for their "property, plant and equipment" as assets, but not the employees who actually make those things work to put out products or provide services, and in general to create profits. In the real world - the world beyond the spreadsheets and trial balance ledgers and forecasting models - people do actually have value.
Friday, June 11, 2010
On Edge
The twisted-universe model I wrote about last time clearly must involve curved space. The idea that space can be curved is pretty familiar by now, mostly because of Einstein’s idea that gravity results from a bending or curving of space by a mass of matter. That seems to be the consensus these days about how gravity works; the most popular alternative, that gravity is somehow transmitted from one mass to another, suffers somewhat from the failure so far to detect any of the “gravity waves” this theory would require.
I always had a hard time figuring out how gravity could have an effect on space, which essentially has no properties of its own to be affected. (Under relativity and quantum mechanics, empty space seems to be occupied by a “quantum vacuum” that does have some properties, but that’s not the same thing as space.) However, when I started thinking about the properties of my twisted space, I realized that any space does have one property: dimension. Which means that the only kind of change you can make to a space is a change of dimension.
As I explained last time, giving a spherical universe a half-twist through the fourth dimension raises the dimension of the universe as a whole to 4. The dimension at any point within the universe would appear to a normal physical observer to be 3, but in fact it would be slightly more than 3; we’ll say it’s D=3+(n<1). And if you add up that n<1 for all the points (at some arbitrarily selected but uniform scale, such as a light year or a parsec) in an orbit of the universe, the sum will be 1.
Now, even though this cosmos is finite, it’s still very large. So the n<1 – which I’m going to declare a fundamental universal constant, and call Ü, mostly because I like umlauts – is going to be very small, almost ininitesmal. The 3+Ü that exists at any point would then be the natural dimension of space in this bent universe of mine.
This whole idea of a non-integer dimension is exactly what Benoit Mandelbrot means by the term “fractal” that he coined to describe objects with a “fractional dimension.” And he has demonstrated that a very wide variety of objects are fractals, which means that many (perhaps most) of the objects we think of as, say, three-dimensional are in fact three-plus dimensional. The more complex the object, the higher the fractional excess; so a big, many-branched tree would be “more fractal,” if you will, than, say, a bowling ball.
But it seems likely that the fractional dimension of even a fairly smooth 3D+ object, like a planet, would have to be greater than the near-infinitesmal value of Ü. And as a result, the planet would “stretch” the dimension of the space around it, causing the kind of contour that Einstein’s conjecture associates with gravity. This would be true even with very small objects, which would account for the kind of “clumping” that scientists believe took place in the early universe, leading eventually to the formation of galaxies and so on.
There’s something else that the structure of the twisted-universe model might help account for that has puzzled me for a long time. I think some visual aids may help here.
We often hear airplane pilots talking about flying “straight and level.” But they’re actually doing anything but. What they’re really doing is flying at a constant altitude above the Earth’s surface. Because the Earth’s surface is curved, however, the plane’s actual path is also curved, as shown above. What would happen if an airplane really flew “straight and level” looks like this:
Now, the thing I’ve wondered about for years is this: We all know that the speed of light is a sort of universal speed limit, that nothing can go faster without violating all sorts of natural laws. But I’ve always wondered why it’s precisely the speed it is, 299,792.5 kilometers per second, or about 186,000 miles per second.
We’re used to the idea, again thanks to Einstein, of light traveling a curved path around massive, high-gravity objects. But since I’m supposing here that all light must travel a curved path in a curved, twisted universe, the “normal” path of light would look something like this:
Obviously, the curvature of this “universe” is highly exaggerated, but it illustrates how the rays of light in a sense “flow” along the contour of the space. What I’m going to suggest is that the speed at which light (and of course other forms of radiation) travels is actually determined by that contour or curvature, because if it travelled at a higher speed, this would happen:
What this would actually mean is hard to say. It might mean that the energy disappears into the fourth dimension, or it could even mean that it exits the universe, whatever that might entail.
Mention of the fourth dimension brings up one final point I want to make before closing this largely pointless expostulation. You’ll remember the Möbius strip from last time:
In looking at this illustration, I want you to imagine that the strip is actually transparent, because what we’re talking about here is empty space, not paper. So there’s really nothing separating point A from point B, or C from D, except space; or rather, except the twisted structure of this space. But the separation is nevertheless complete and inviolable: The only way to get from point A to point B is to go around the strip; you can’t go through it.
Why not? Well, if you travel around the strip from A to B, you’re in effect adding up Ü units, or in a sense travelling uphill dimensionally. By the time you reach point C, you’re in a dimension that’s 0.5 higher in relation to A, and when you reach B, the space you’re in is a full 1 dimension away from A. It’s still 2+ÜD from a local point of view (in the illustration; in the twisted universe, it would be 3+ÜD), but A is 3D from the perspective of B (4D in the real universe), and vice versa. So naturally, there’s no way to perceive one space from the other, much less to go there directly.
This is precisely what constitutes the boundary or “edge” of the universe, this dimensional barrier. And what that means is that every point in the universe is on the edge of the universe.
Somehow, that reminds me of the famous Hermetic saying quoted by Giordano Bruno and Pascal, among others: “God is an intelligible sphere whose center is everywhere and whose circumference is nowhere.” In the twisted universe, the circumference is everywhere, but I’m not sure whether there’s a center anywhere.
I always had a hard time figuring out how gravity could have an effect on space, which essentially has no properties of its own to be affected. (Under relativity and quantum mechanics, empty space seems to be occupied by a “quantum vacuum” that does have some properties, but that’s not the same thing as space.) However, when I started thinking about the properties of my twisted space, I realized that any space does have one property: dimension. Which means that the only kind of change you can make to a space is a change of dimension.
As I explained last time, giving a spherical universe a half-twist through the fourth dimension raises the dimension of the universe as a whole to 4. The dimension at any point within the universe would appear to a normal physical observer to be 3, but in fact it would be slightly more than 3; we’ll say it’s D=3+(n<1). And if you add up that n<1 for all the points (at some arbitrarily selected but uniform scale, such as a light year or a parsec) in an orbit of the universe, the sum will be 1.
Now, even though this cosmos is finite, it’s still very large. So the n<1 – which I’m going to declare a fundamental universal constant, and call Ü, mostly because I like umlauts – is going to be very small, almost ininitesmal. The 3+Ü that exists at any point would then be the natural dimension of space in this bent universe of mine.
This whole idea of a non-integer dimension is exactly what Benoit Mandelbrot means by the term “fractal” that he coined to describe objects with a “fractional dimension.” And he has demonstrated that a very wide variety of objects are fractals, which means that many (perhaps most) of the objects we think of as, say, three-dimensional are in fact three-plus dimensional. The more complex the object, the higher the fractional excess; so a big, many-branched tree would be “more fractal,” if you will, than, say, a bowling ball.
But it seems likely that the fractional dimension of even a fairly smooth 3D+ object, like a planet, would have to be greater than the near-infinitesmal value of Ü. And as a result, the planet would “stretch” the dimension of the space around it, causing the kind of contour that Einstein’s conjecture associates with gravity. This would be true even with very small objects, which would account for the kind of “clumping” that scientists believe took place in the early universe, leading eventually to the formation of galaxies and so on.
There’s something else that the structure of the twisted-universe model might help account for that has puzzled me for a long time. I think some visual aids may help here.
We often hear airplane pilots talking about flying “straight and level.” But they’re actually doing anything but. What they’re really doing is flying at a constant altitude above the Earth’s surface. Because the Earth’s surface is curved, however, the plane’s actual path is also curved, as shown above. What would happen if an airplane really flew “straight and level” looks like this:
Now, the thing I’ve wondered about for years is this: We all know that the speed of light is a sort of universal speed limit, that nothing can go faster without violating all sorts of natural laws. But I’ve always wondered why it’s precisely the speed it is, 299,792.5 kilometers per second, or about 186,000 miles per second.
We’re used to the idea, again thanks to Einstein, of light traveling a curved path around massive, high-gravity objects. But since I’m supposing here that all light must travel a curved path in a curved, twisted universe, the “normal” path of light would look something like this:
Obviously, the curvature of this “universe” is highly exaggerated, but it illustrates how the rays of light in a sense “flow” along the contour of the space. What I’m going to suggest is that the speed at which light (and of course other forms of radiation) travels is actually determined by that contour or curvature, because if it travelled at a higher speed, this would happen:
What this would actually mean is hard to say. It might mean that the energy disappears into the fourth dimension, or it could even mean that it exits the universe, whatever that might entail.
Mention of the fourth dimension brings up one final point I want to make before closing this largely pointless expostulation. You’ll remember the Möbius strip from last time:
In looking at this illustration, I want you to imagine that the strip is actually transparent, because what we’re talking about here is empty space, not paper. So there’s really nothing separating point A from point B, or C from D, except space; or rather, except the twisted structure of this space. But the separation is nevertheless complete and inviolable: The only way to get from point A to point B is to go around the strip; you can’t go through it.
Why not? Well, if you travel around the strip from A to B, you’re in effect adding up Ü units, or in a sense travelling uphill dimensionally. By the time you reach point C, you’re in a dimension that’s 0.5 higher in relation to A, and when you reach B, the space you’re in is a full 1 dimension away from A. It’s still 2+ÜD from a local point of view (in the illustration; in the twisted universe, it would be 3+ÜD), but A is 3D from the perspective of B (4D in the real universe), and vice versa. So naturally, there’s no way to perceive one space from the other, much less to go there directly.
This is precisely what constitutes the boundary or “edge” of the universe, this dimensional barrier. And what that means is that every point in the universe is on the edge of the universe.
Somehow, that reminds me of the famous Hermetic saying quoted by Giordano Bruno and Pascal, among others: “God is an intelligible sphere whose center is everywhere and whose circumference is nowhere.” In the twisted universe, the circumference is everywhere, but I’m not sure whether there’s a center anywhere.
Wednesday, June 9, 2010
A Twisted Universe
Last time, I argued that the overall shape or structure of the universe is unknowable, an argument that might (in light of Goedel’s incompleteness theorems) be unprovable. But despite perfectly good reasons to abandon the goal, I’m now going to present an argument in favor of what I believe may be a novel model of the cosmos.
First, what do we mean by the words “universe” or “cosmos?” The start of the Wikipedia entry for “universe” seems to me to sum it up nicely, especially because it says pretty much what I hoped it would say: “The Universe comprises everything perceived to exist physically, the entirety of space and time, and all forms of matter and energy.”
That second clause, “the entirety of space and time,” raises a point that I think a lot of people overlook: The universe is not in space, rather, all space is in the universe. This means the universe cannot have a spatial boundary: You can’t travel to the edge of the universe and find that the universe ends while space continues. As a result, the shape or structure of the universe must be such that if you travel through it in a straight line, you never reach a point where the universe is on one side and something else is on the other.
It seems to me that there are only two ways this can be possible. The first is if the universe is infinitely large. This would certainly allow for a never-ending trajectory, but it also creates serious difficulties.
First, given the equivalence between space and time, it seems necessary that an infinitely large universe also be infinitely old. Even if there were no inherent reasons for rejecting an infinite universe (and I think there are), this requirement that it be infinitely old would directly contradict the prevailing scientific cosmological model, the Big Bang theory, which holds that the universe is “only” about 13 billion years old.
Also contradicting the Big Bang theory would be any requirement that matter/energy be consistently distributed throughout an infinite space. Such consistency is required if we aren’t to allow different regions of space to have drastically different properties; in other words, if we do want to guarantee that universal laws really are universal. But if we distribute matter/energy similarly throughout an infinite universe, it’s clear that we must have an infinite amount of matter/energy to distribute.
Since matter/energy can be neither created nor destroyed, an infinite universe must have contained the same amount – i.e., an infinite amount – of matter/energy from its beginning. In other words, a less-than-infinite volume of space must have held an infinite quantity of matter/energy. This strikes me as very unlikely. In short, I don’t see any possibility that an infinite universe could have had a beginning in time (without divine agency), because it must already have been infinite at its beginning, something that contradicts the entire basis of the Big Bang theory.
But if an infinitely large universe must also be infinitely old, then one would expect certain physical phenomena to have advanced long ago to their extremes. Most obviously, the second law of thermodynamics requires that the total entropy of the universe increase over time. If it has already been increasing for an infinite time, then one might reasonably expect that entropy would long ago have reached its maximum, known as “heat death,” in which no free energy is available to cause motion or sustain order. Clearly, that is not the case.
So I’m convinced that infinity is just not on, which means we need to find another structure “such that if you travel through it in a straight line, you never reach a point where the universe is on one side and something else is on the other,” as I said earlier.
To explain what I think is the best alternative to infinity, I’ll start with an object that may be familiar to many readers:
This is the famous Möbius strip, discovered by German mathematician August Ferdinand Möbius. It’s just a strip of paper with the ends glued or taped together, with the important requirement that the paper be given a half-twist before gluing or taping, but it has a number of interesting properties, which many of you are undoubtedly familiar with.
For one, if you take a pen and draw a line along the length of the strip, the line will return to its starting point, having covered both “sides” of the paper, without lifting the pen off the paper at any time.
The standard explanation for this is that mathematically (topologically), despite all appearances to the contrary, the strip has only one surface. But there’s another way of explaining it:
If you took a strip of paper and stuck the ends together without the half-twist, the only way to draw a line on both sides of the paper would be to lift the pen from the paper and move it around to the other side. In other words, the pen would have to leave the two-dimensional surface of the paper and travel through three-dimensional space to the other side.
But the half-twist turns the Möbius strip as a whole into a three-dimensional object, and so the tip of your pen does in fact make that trip through 3D space as it makes its “orbit” of the entire strip, without ever leaving the (apparently) 2D plane.
If you consider the twist as spread evenly over the whole of the strip, then in effect the dimension of the surface at any one point is slightly more than 2D; it has a dimension of 2.00…n, with the magnitude of the fractional part at any point depending on how long the strip is. When you add the fractional parts over the whole length, the sum will be 1.0, which, when added to the nominal 2.0 dimension of the surface, makes the dimension of the object as a whole 3.0.
Now, suppose we take a three-dimensional object – a sphere, for example – and give it a half-twist and join its ends together. Obviously, this isn’t something we can actually do in three-dimensional space – a sphere has no “ends” in 3D space. (By the same token, a 2D being couldn’t make a Möbius strip.) But I hope that by bootstrapping up from the example of the Möbius strip, we can conceptualize the result of this procedure.
As was the case with the Möbius strip, the half-twist raises the dimension of the object as a whole by 1.0, in this case to 4.0. And as with the Möbius strip, the dimension at any point within the sphere will appear to be unchanged. But in fact, it will be 3.00…n, with the fractional part again adding up over a full orbit of the space to 1.0, making the total dimension 4.0.
And again as in the case of the Möbius strip, a straight-line trajectory in any direction will eventually return to its starting point, having traversed both “sides” of the sphere. But mathematically, just as the Möbius strip has only one surface, this “Möbius sphere” has only one volume, though it would appear to a four-dimensional observer to have separate “inside” and “outside” volumes, just as a three-dimensional observer (you or me) sees an “inside” and an “outside” surface on the Möbius strip.
A space of this kind would satisfy the requirement that “if you travel through it in a straight line, you never reach a point where the universe is on one side and something else is on the other.” In fact, in this twisted universe, no matter which direction you travel, there will always be as much of the universe in front of you as behind, and as much above as below. In other words, for an ordinary (physically constituted) traveler, such a universe is likely inescapable.
There are a number of other implications of this form as a cosmological model, some of which I find rather odd. But this is already a rather lengthy post, so I’ll have to take up the ramifications next time.
First, what do we mean by the words “universe” or “cosmos?” The start of the Wikipedia entry for “universe” seems to me to sum it up nicely, especially because it says pretty much what I hoped it would say: “The Universe comprises everything perceived to exist physically, the entirety of space and time, and all forms of matter and energy.”
That second clause, “the entirety of space and time,” raises a point that I think a lot of people overlook: The universe is not in space, rather, all space is in the universe. This means the universe cannot have a spatial boundary: You can’t travel to the edge of the universe and find that the universe ends while space continues. As a result, the shape or structure of the universe must be such that if you travel through it in a straight line, you never reach a point where the universe is on one side and something else is on the other.
It seems to me that there are only two ways this can be possible. The first is if the universe is infinitely large. This would certainly allow for a never-ending trajectory, but it also creates serious difficulties.
First, given the equivalence between space and time, it seems necessary that an infinitely large universe also be infinitely old. Even if there were no inherent reasons for rejecting an infinite universe (and I think there are), this requirement that it be infinitely old would directly contradict the prevailing scientific cosmological model, the Big Bang theory, which holds that the universe is “only” about 13 billion years old.
Also contradicting the Big Bang theory would be any requirement that matter/energy be consistently distributed throughout an infinite space. Such consistency is required if we aren’t to allow different regions of space to have drastically different properties; in other words, if we do want to guarantee that universal laws really are universal. But if we distribute matter/energy similarly throughout an infinite universe, it’s clear that we must have an infinite amount of matter/energy to distribute.
Since matter/energy can be neither created nor destroyed, an infinite universe must have contained the same amount – i.e., an infinite amount – of matter/energy from its beginning. In other words, a less-than-infinite volume of space must have held an infinite quantity of matter/energy. This strikes me as very unlikely. In short, I don’t see any possibility that an infinite universe could have had a beginning in time (without divine agency), because it must already have been infinite at its beginning, something that contradicts the entire basis of the Big Bang theory.
But if an infinitely large universe must also be infinitely old, then one would expect certain physical phenomena to have advanced long ago to their extremes. Most obviously, the second law of thermodynamics requires that the total entropy of the universe increase over time. If it has already been increasing for an infinite time, then one might reasonably expect that entropy would long ago have reached its maximum, known as “heat death,” in which no free energy is available to cause motion or sustain order. Clearly, that is not the case.
So I’m convinced that infinity is just not on, which means we need to find another structure “such that if you travel through it in a straight line, you never reach a point where the universe is on one side and something else is on the other,” as I said earlier.
To explain what I think is the best alternative to infinity, I’ll start with an object that may be familiar to many readers:
Möbius strip (Adapted from Wikipedia)
This is the famous Möbius strip, discovered by German mathematician August Ferdinand Möbius. It’s just a strip of paper with the ends glued or taped together, with the important requirement that the paper be given a half-twist before gluing or taping, but it has a number of interesting properties, which many of you are undoubtedly familiar with.
For one, if you take a pen and draw a line along the length of the strip, the line will return to its starting point, having covered both “sides” of the paper, without lifting the pen off the paper at any time.
The standard explanation for this is that mathematically (topologically), despite all appearances to the contrary, the strip has only one surface. But there’s another way of explaining it:
If you took a strip of paper and stuck the ends together without the half-twist, the only way to draw a line on both sides of the paper would be to lift the pen from the paper and move it around to the other side. In other words, the pen would have to leave the two-dimensional surface of the paper and travel through three-dimensional space to the other side.
But the half-twist turns the Möbius strip as a whole into a three-dimensional object, and so the tip of your pen does in fact make that trip through 3D space as it makes its “orbit” of the entire strip, without ever leaving the (apparently) 2D plane.
If you consider the twist as spread evenly over the whole of the strip, then in effect the dimension of the surface at any one point is slightly more than 2D; it has a dimension of 2.00…n, with the magnitude of the fractional part at any point depending on how long the strip is. When you add the fractional parts over the whole length, the sum will be 1.0, which, when added to the nominal 2.0 dimension of the surface, makes the dimension of the object as a whole 3.0.
Now, suppose we take a three-dimensional object – a sphere, for example – and give it a half-twist and join its ends together. Obviously, this isn’t something we can actually do in three-dimensional space – a sphere has no “ends” in 3D space. (By the same token, a 2D being couldn’t make a Möbius strip.) But I hope that by bootstrapping up from the example of the Möbius strip, we can conceptualize the result of this procedure.
As was the case with the Möbius strip, the half-twist raises the dimension of the object as a whole by 1.0, in this case to 4.0. And as with the Möbius strip, the dimension at any point within the sphere will appear to be unchanged. But in fact, it will be 3.00…n, with the fractional part again adding up over a full orbit of the space to 1.0, making the total dimension 4.0.
And again as in the case of the Möbius strip, a straight-line trajectory in any direction will eventually return to its starting point, having traversed both “sides” of the sphere. But mathematically, just as the Möbius strip has only one surface, this “Möbius sphere” has only one volume, though it would appear to a four-dimensional observer to have separate “inside” and “outside” volumes, just as a three-dimensional observer (you or me) sees an “inside” and an “outside” surface on the Möbius strip.
A space of this kind would satisfy the requirement that “if you travel through it in a straight line, you never reach a point where the universe is on one side and something else is on the other.” In fact, in this twisted universe, no matter which direction you travel, there will always be as much of the universe in front of you as behind, and as much above as below. In other words, for an ordinary (physically constituted) traveler, such a universe is likely inescapable.
There are a number of other implications of this form as a cosmological model, some of which I find rather odd. But this is already a rather lengthy post, so I’ll have to take up the ramifications next time.
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