Recent Stanford econ working paper's of note
A Theory of Endogenous Institutional Change
ABSTRACT: This paper asks (a) why and how institutions change; (b) how does an institution persist in a changing environment and (c) how do processes that it unleashes lead to its own demise. The paper shows that the game theoretic notion of self-enforcing equilibrium and the historical institutionalist focus on process are both inadequate to answer these questions. Building on a game theoretic foundation, but responding to the critique of it by historical institutionalists, the paper introduces the concepts of quasi-parameters and self--reinforcement. With these concepts, and building on repeated game theory, a dynamic approach to institutions is offered, one that can account for endogenous change (and stability) of institutions. Contextual accounts of formal governing institutions in early modern Europe and the informal institution of cleavage structure in the contemporary world provide illustrations of the approach.
Deciding Between Competition and Collusion
ABSTRACT: In many studies in empirical industrial organization, the economist needs to decide between several non-nested models of industry equilibrium. In this paper, we develop a new approach to the model selection problem that can be used when the economist must decide between models with bid-rigging and models without bid-rigging. We elicit from industry experts a prior distribution over markups across auctions. This induces a prior distribution over structural cost parameters. We then use Bayes Theorem to compute posterior probabilities for several non-nested models of industry equilibrium. In many settings, we believe that it is useful to formally incorporate the a prior beliefs of industry experts into estimation, especially in small samples where asymptotic approximations may be unreliable. We apply our methodology to a data set of bidding by construction firms in the Midwest. The techniques we propose are not computationally demanding, use flexible functional forms and can be programmed using most standard statistical packages.
Monte Carlo Simulation of Macroeconomic Risk with a Continuum of Agents: The Symmetric Case
Suppose a large economy with individual risk is modeled by a continuum of pairwise exchangeable random variables (i.i.d., in particular). Then the relevant stochastic process is jointly measurable only in degenerate cases. Yet in Monte Carlo simulation, the average of a large finite draw of the random variables converges almost surely. Several necessary and sufficient conditions for such "Monte Carlo convergence" are given. Also, conditioned on the associated Monte Carlo sigma-algebra, which represents macroeconomic risk, individual agents' random shocks are independent. Furthermore, a converse to one version of the classical law of large numbers is proved.
The Economics of Civil Rights
Order Without Law? Property Rights During the California Gold Rush
Blogospere: market for ideas
Jon Udell, impersonating the webs unwashed, asks "Is that all? So what?" with regards to weblogs.
Most of our reactions to the simplicity (crudeness?) of markets is summed up in the asking of the same questions.
How can we impart something as important as the distribution of water or the protection of the environment to something that doesn't have a central authority? Who will make sure the right thing will happen? How can we be sure that all the details will be taken care of and that all risks will be properly accounted for?
Answer (in the form of a question): how do we trust the markets to provide food, clothing, housing and other human necessities?
Scarcity and value
I've been thinking about the exchange with my brother
. I said, "natural resources are NOT constrained... basically, we humans are only constrained by our creative use of resources."
This seems right, but it does ignore the fact that prices for a good are set in markets where the players are weighing the relative scarcity of the good. For me to sell my widget to you for $10 dollars implies that for widgets to become more scarce to me costs less than $10 and its worth more than $10 for you for widgets to become less scarce to you.
My point is that natural resources are often thought as given, i.e. thought of in terms of absolute scarcity, but are in fact better thought of in terms relative to their use by individuals. Markets are a great mechanism for individuals to communicate their respective outlooks on a resource's scarcity. The market for land is a perfect example. It's obvious that there is a fixed amount of land, but the real estate market reflects the fact that different people in different circumstances have different assessments of its scarcity. If I own a plot of land in a prime spot on the bank of a major river that I use for low grade farming (let's say raising sheep) and a company can use that land to build a high production factory, they're going to see the land as more valuable (i.e. more scarce) than me... I can raise sheep anywhere, but they need the river to ship their goods. The market allows me to interact with the company and to set a price.
Paul Seabright in The Company of Strangers
sees water as a commodity that should be bought and sold in markets:
There is no serious alternative to treating water as an economic commodity. It is scarce in many locales even if it is not globally scarce, and its local scarcity may eventually prove globally threatening [i.e. "water wars"]. But calling it an economic commodity begins rather than ends the argument. What would it mean to treat it as such? First, water's scarcity requires users to be given incentives to use it efficiently. These need not always be price incentives, but we know that price incentives often have desirable features that other kinds do not. in particular, they make possible the decentralization of decisions, when we lack the detailed knowledge and mutual trust required for direct regulation or moral persuasion. The great merit of charging a price for water is that we no longer need to argue who deserves it more: if people are poor they may deserve our help, but if water can be priced to reflect its scarcity relative to other goods, we no longer need to argue the case for helping them separately when we consider food, housing, water, clothes, and all the other aspects of their lives. Proper pricing strengthens rather than weakens the case for helping the poor...
Likewise, markets in emissions of pollutants would have a similar effect. Price incentives would give polluters an incentive to be more efficient.
The Inner Child is for Children
Freud introduced to the world the dichotomy of conscious and unconscious. He surmised that the unconscious was responsible for most behavior. To understand a person's overt, conscious behavior, he developed tools (psychoanalysis) to try to get a better understanding of the person's unconscious. Dream analysis, as outlined in The Interpretation of Dreams
, was one of these tools. For example, if I dream of being sad after the dream-death of my father, Freud would see me to have an unconscious desire to see my father die (or go away forever).
Many unconscious desires developed in early childhood. Thus dream interpretation includes sessions on early childhood experience. A dream-death, may not reflect a current desire by my unconscious to see my father dead, it may be a latent wish from my childhood. When my grandfather died when I was young, I noticed that he went away forever and I unconsciously associated going away with death. Subsequently, I was jealous of my Father's relationship with my mother and I wished him to go away forever. Thus I wished him dead and thus I dream about his death today. In this way, childhood experiences shape my unconscious.
As a healthy adult, I work to come in close contact with my unconsciousness. This is a journey to find my true self because, as all the authors we read imply, only my unconscious represents my true self. In that sense, Hesse's Damian
is a journey of character to get more in touch with his unconscious, his true self. The bulk of the book is the story of a boy, Emil, growing from 10 years old to his early 20's. Because of his family situation (they have money), he has the opportunity to meet a fascinating cast of characters that help him to come closer and closer to himself as he realizes, and then breaks through, the constraints society puts on the individual.
Although, I was by no means destitute growing up, I felt jealous of Emil. Through luck and opportunity not available to most people, the characters he finds himself surrounded by help him to find who he truly is. I certainly didn't meet such interesting people growing up and I feel like I've only recently started the journey that Emil started when he was twelve.
Of course, Emil's journey didn't have to exist in reality. Those characters could be the manifestation of his psyche during an inner journey that Emil might have taken. This journey would have been very similar to the experience of psychoanalysis. However, whether or not the journey existed in reality, Emil (and the author) has been given an opportunity that most people don't get. For most people, reality intrudes; you have to work; the kid's have to be fed.
If finding one's self is seen as a gift, it becomes a responsibility to follow through for those given the opportunity. I've been given this opportunity and I intend to strive beyond the predetermined definitions of me that society has given me to find my true self. However, there's more to journey than discovering what you're not (e.g. whatever society tells you to be). It involves a process of discovering who you want to be. So there's a reactive and a proactive aspect to the journey.
Our readings in this course are filled with characters that react to society and determine that they are going to define themselves in their own terms and of characters who fail because they don't make this discovery. In Undine Goes
, Bachmann describes a character that observes her place in society, but takes it as a given and fails to break out of the cycle of dependency she finds herself in. Again and again, she finds hope in a new relationship with a man, Hans. Again and again, she discovers the same bare truths about Hans. Again and again, she ascends the mountain to be with her Hans, her Sisyphusian rock, who then treasonously turns his back on her and rolls back down the hill.
In contrast, Bachmann's Charlotte in A Step Towards Gomorrah
realizes her plight and schemes to release herself from it. She doesn't like being defined by her marriage and her husband. Scheming, she constructs her own alternative reality outside the bounds of society. She creates her own kingdom. Her failing is that she doesn't realize her kingdom. It only stays in her head and her husband comes home the next morning; the alarm clock is set.
In general, the readings have not dealt as well with the proactive aspect of the journey to self. This is why the end of Damian is so unappealing. After having discovered himself and begun to define himself outside the constraints of society, Emil joins a cult. What!? He gives himself over to the covenant with Damian and his mother. Similarly, Charlotte can't realize her newly found kingdom.
The problem these characters run into is simply this: your own personal kingdom is a lonely place. It is meaningless to create an alternative (social) reality unless you can convince other people to live in that reality. For example, I have imagined an alternative form of marriage that involves 3 consenting adults, public nudity and cinnamon. The details aren't important, but the fact the other people are involved requires that I get those other adults to consent to my alternative reality.
This means that there is an unexpected twist to the proactive leg of the journey to the self. Once the bonds of society are broken and an alternative vision of reality is created, to truly realize myself, I have to build consensus in the community to see my reality realized. This problem is what Nietzsche tackles in Thus Spoke Zarathustra
. In his various journeys to and from the mountain, Zarathustra attempts to create his own ministry. He wants to tell the people about his idea of the Ubermensch. He soon discovers not everyone is willing to hear his story, that some think he's talking about something else and that without apostles most people that can hear his message, won't hear it.
Communication, a process of consensus building, is the biggest challenge to those that want to sell their version of reality. For example, the ideas in this paper occurred to me very easily. The writing of the paper is much more difficult.
But is it even possible to communicate? Was Zarathustra's journeys up and down the mountain, like Undine's, like Sisyphus', in vain? The answer from Kafka in The Metamorphosis
appears to be "no" to the first and "yes" to the second question. Gregor's situation screams of the inevitability and inescapability our ascribed social definition. Just as Gregor can't break free of his buggy existence, we are stuck too, like insects in a hive. What's terrible about Gregor's story is that he's able to have an internal dialog; he is aware of his predicament and can do nothing about it. Kafka's message is that Nietzsche was deluded. He admits that we're free to realize the bounds society puts on us; we're even free to imagine some alternative reality for ourselves. Its in the critical final leg in the journey to self, the reaching out to the community, the consensus building, that is impossible. Zarathustra can speak, but he won't be heard.
I'm an optimist and I don't believe the journey to my self is delusional. Also, I don't think that the above analysis reflects Kafka's true intentions. I think he's trying to communicate something entirely different. It is curious that the main character, Gregor, dies (and is forgotten!) three fourths through the story. Main characters don't die if there's no significance in it. Perhaps this strange turn of events is Kafka telling us to avert our eyes. He wants us to ignore the tragedy of the Gregors in the world, those that see their fates as predetermined by society, and to see opportunities, like the one given Gregor's family is given after his death. After all, even if all of our lives are preordained, isn't it much more fun to think that they're not.
Aesthetically, I'm turned off by Freud. He's right that the unconscious is an important part of who we are. He's also right that the modern sense of self can be seen as a conflict between external forces and the unconscious. I'd also agree that childhood experiences determine the shape of the unconscious and that we need to get in touch with the unconscious. I do not like how the conversation seems to end there with Freud, as if expressing the unconscious, discovering the true self, is enough. I don't like how adults, encouraged by Freud and others of his ilk, conduct their lives as a search for their inner child, as if being a child is the ideal state of being.
I relish being an adult because to me, to be an adult is to be on a journey to self, but to have the courage to reach out and to create the world in your own image.
Is nature a 'force' ?
My brother read my statement of purpose
"Economic life is dominated by two forces..."
It seems to me there are at least three forces. You describe two social forces, but I feel that nature (the environment) is as dominant a force as the other two. I have to imagine resources play an important role in an economy. Re: pricing...isn't supply (and i'm referring to real world supply/scarcity not that created through monopoly, etc.) an important factor?
Later you describe the environment as an effect of the two social forces ("I want to study the dynamic between these two forces....how can markets be used to protect the environment..."), when I'm saying it's both a result and a cause...a circular reference if you will ;)
I guess since you can't really change the "natural force", it may become viewed more as a result of practices put in place by the social forces. However, I still feel it is a dominant/governing force because the individual and the community need to act (maybe react is a better word) and make their economic decisions based on the limits/resources in their environment...making it a cause in every decision's result.
I guess I'm making the implicit claim that while [main line] economic thought has been dominated by the "scarcity of (natural) resources" line of thought, I believe economic life is determined by the arrangement made between individuals and community. I'd contend, over all time, natural resources are NOT constrained... basically, we humans are only constrained by our creative use of resources (i.e. we rely on individual genius to invent new, more efficient, ways to use resources and social institutions to diffuse those inventions). Think of how much more gdp we produce today per ton of coal as compared to 100 years ago. This pov makes natural resources infinite.
Another way to see it... the economy is less and less built on resource extraction, but on services. Take a look at the gdp numbers for these industries.
So, I see the study of economics to be the study of the environment that fosters new/better ideas. This is an interaction between individuals and community.
Statement of Purpose - final
My search for truth had always been synonymous with the study of science. The method, models and rigor of science attracted me from an early age. At five my grandmother was taking me to the planetarium, at 10 I began a subscription to Scientific American (I usually just read the summary paragraphs, the rest was over my head) and as a teenager I became a huge fan of the writings of Carl Sagan and other science popularizers. Sagan was able to explain profound and beautiful things in simple language. At the time, I took this aspect of his writing to be useful because it made it easier for me to understand how the world worked. In the back of my mind stuck his determination to help others to understand.
As a child, I had great contempt for subjects of human nature; English and social sciences. They seemed to me to be deeply narcissistic. Besides, I felt they had nothing to do with a greater pursuit of the truth. I remember arguing vehemently with my 8th grade literature teacher about whether or not "The bumblebee flew anyway" had a subtext. "If the author had some lesson for his readers to learn" I argued, "why didn't he just write the lesson out in plain English?!" Part of this bias can be explained by my growing up in the middle of the Redwood Forests of Northern California; conservative and largely unpopulated. Most of my experiences, playing in the creek or hunting tweety birds, were governed by the natural world rather than people.
However, when I went to school in Berkeley, I saw my world shift from being governed by nature to a near 100% saturation of human pursuits. Like the set of a play that is brought in to cover the bare rafters and structures that make up the stage, Berkeley covered all I knew about how the world worked with the constructs of modern human civilization. I was struck by the fact that each tree was placed by a human; every rule to follow was a human law rather than natural. I realized that people, especially large groups of them, had a logic of their own, perhaps distinct from nature. I began to ask myself, "What laws govern these people?" There were the obvious answers: government and politics, the police and power. These were brute force kinds of answers. Possibly true, but uninteresting. What struck me were more subtle issues that dealt with the layers of abstraction involved in everyday interactions of people. Buying apples at the store involves a tremendous amount of faith in these unsaid laws of human interaction. Why does money have value and why would the apple vendor take it in exchange for an apple (which by the away arrived to the vendor via a complex scheme of domestic and international trade)?
My years in industry have deepened this mystery. I've been involved in both the buying and selling of software. I've struggled with creating new products, setting prices, and trying to find buyers. I've created 'value propositions.' Why do people buy things at the market price or any price for that matter? I've taken finance classes where I've learned value is the NPV of net future cash flow. What happens in cases where future cash flows are uncertain or unknowable? Even without perfect information, the price is set, people still buy and people still sell. My experience in the real world, while sparking my interest in the theory of price setting, has shown me how important these issues really are.
For such an important and regular activity, determining price and trying to place value on stuff is an extremely hard thing to do. Aesthetically and innately, I'm attracted to the prospect of understanding this problem better, to the science of determining value and to the study of the theater that allows for such transactions to take place.
Yet, what excites me most is the prospect of teaching people the tools of economics so that they can better make decisions. Decisions on small, but important, things like the setting prices and decisions on large things on how to best structure society. If as a teacher of economics, my only accomplishment is to be a counterbalance to the knee jerk reaction toward socialism, to remind people of the dangers of such systems and of the various successes of capitalism, I will have felt accomplished at the end of my career. If I accomplish nothing else but an understanding of why Communism failed and I resist, both internal to myself and as a teacher, the temptation of similar ideologies, then I'll have been a success. However it is much harder to be for something than it is to be against something; thus, I strive for a deeper understanding of economic life.
Economic life is dominated by two forces; the spirit of the individual and the instinct for community. People are individuals; it is physically impossible to 'stand in someone else's shoes.' It shouldn't be a surprise to anyone that people think of themselves first and act accordingly. Because physics determines such individuality, this force has been easier to quantify and it is easier to borrow models from the physical sciences to help determine its form. Much of economics up to this point has been just this. Models built on assumptions of rationality or utility maximization are examples. The other force, community, gets less attention. It is hard to quantify the social nature of humanity and it is hard to understand the social institutions that it creates. Because it is hard to understand, some deny its validity or see it as an aberration. Some go as far as to say that people are acting irrationally when they act in ways that are in their community's best interest rather than their own (e.g. voting, fighting in wars, etc). However, this force has as much to do with economics as does greed or selfishness. As Thurow pointed out in his 'Future of Capitalism,' Capitalism depends on a strong foundation of social institutions and as Seabright argues in his 'The Company of Strangers,' we seem to be pre-built with the drive to build such institutions.
I want to study the dynamic between these two forces. What policies help these forces to not come in conflict? More specifically, how can markets be used to protect the environment, ensure a more equitable distribution of wealth and foster growth in our country and the less developed countries?
As you can see, I have a strong motivation to continue my studies in economics and to become a teacher of the subject. My background and interests have given me the focus and maturity to succeed in your program.
Computers have limited memory that is divided into words. Each number (integer) that is going to be represented in the computer needs to fit into a word. This means that the size of the number to be represented is restricted. For example, if a word is 32 bits, then the largest number that can be stored in one word would be at most 32 bits long. This works out to numbers up to about 4.5 billion (i.e. 2^32). This is complicated by the fact that you want to represent negative numbers.
The naive approach to negative numbers would to stick a bit at the end of the number to indicate positive or negative. In other words, 00001001 would be positive nine and 10001001 would be negative nine. You can see that this extra bit restricts the size of the largest positive number to 2^31 (assuming 32 bit words still). You’re going to get this problem, no matter what your representation of negative numbers is… You can think of the word in memory as your information bank. You can store 32 bits of information and the information about negativeness will always be at least one bit.
The real problem with this approach is that it results in very complicated arithmetic for the computer. How do you add -9 and +9? The algorithm, as written in primitive assembly, to do this would be very complex (i.e. look at the last digit, do this if its 1 and do that if its negative, etc, etc).
To simplify the math, the ‘ones compliment’ approach was taken to represent negative numbers. The idea is to invert all the bits for negative numbers. Negative nine would be 11110110. Again, the first digit represents the sign but the arithmetic becomes much easier. The problem with this method is that you get two zeros… positive and negative zero (i.e. 11111111 and 00000000). You’d be stuck writing special cases in all your software for each of these zeros. So then one’s compliment is inadequate.
The approach in response to the short comings of one’s compliment, the one that is used by most computers today, is two’s compliment. Basically, it’s the same as one’s compliment but you add one to the result of flipping the bits. This eliminates the extra zero (i.e. 00000001 plus 11111111 is just 00000000 so negative and positive zero have the same representation). This method also allows for simple arithmatatic.
K9's problem with 'permanent' links
Somebody tell the newbie
about permanent links
(see item 2). The links to my brother's BCS related posts are broken in the post below
are the updated and unbroken (for now) links.
So, my brother finally turned on his RSS feeds. No more slogging my way to his website
everyday; now I can read his nonensense from the comforts of Outlook (I use Newsgator
as my RSS feed reader).
Anyway, here's his thoughts on Cal's slight in the BCS
and the Heisman trophy
(he even made an excel spreadsheet
... I guess I'm not the only geek in the family). Also, sticking with the sports theme, he discusses the Pat Tillman situation
. Can you be a hero in an unpopular war?
The later actually reminds me of something that happened on the bus today. The guy sitting next to me, a large middle-aged man, had a baseball cap on that said "Vietnam Veteran." I was compelled to shake his hand and say "thank you", but I couldn't muster the courage. This man went half way around the world to get shot at and I couldn't get up the balls enough to say hi...
Preemptive war and decision under uncertainty
I've enjoyed John Quiggin's discussion of decision making under uncertainty
in the context of preemptive war. The whole discussion was triggered by Judge Posner's first post on his weblog.
In response to Prof. Quiggin's first post
, I commented:
I hope you’ll elaborate on the second point about war being a negative sum game. Intuitively, this seems wrong. I would think it easy to construct an example, from history or otherwise, of a war with positive gains in sum. Off the cuff, imagine a war were the attacked lay down their arms without a shot fired. The victors march in, institute democracy, grow the economy and are adored by the conquered. Wouldn’t this ‘war’ be positive sum?
Also, your arguments on this point seem to just have the effect of increasing the costs as discussed in Posner’s post. You’ve discovered some ‘hidden costs’ in the Posnerian calculation. That’s fine. Realize those costs and then redo the calculation.
Later in the comment thread, I'm chided:
Will, that’s silly. Of course we can construct an imaginary example of a ‘win-win’ war; it’s just that the preconditions very rarely obtain in reality. The neocons are rightly considered fools for kidding themselves that Iraq was an exception.
If the past teaches us anything, it is that war tends towards ‘lose-lose’ with occasional ‘win-lose’. The likelihood of ‘lose-lose’ has to enter our prior calculations. And as others point out calculations geared to ‘win-lose’ ignore the welfare losses of the losers.
Yes, the example was silly, but it was meant to illustrate that some such war exists. We're talking about imagining
future states of the world such that we can create policy (waging war being a policy). To allow my example is to see what the "neocons" where up to when they were planning this war. They, and I, believe there is such a thing as a zero-sum or positive sum war. To dismiss the possiblity out of hand, is to miss the point and to misunderstand why we're in Iraq.
He then made a second post
, where he linked to a well written paper on the precautionary principle
. I commented:
Thanks for the paper. For this budding economics student, the discussion of more and more general models of uncertainty was enlightning. I think I’ll print it out and use it as a cheat sheet.
Your analysis is VERY dependent on how the decesion maker determines which state of the world is “status quo” and what would be an “innovation”. Where you see preemptive war in Iraq as an uncertain innovation, Bush sees uncertainty in the status quo, i.e. an unknown (and unknowable?) connection between Saddam’s WMD and terrorist willing to use them. This would be analoguous to the “innovation” that you observe in the doubling of green house gas emmisions if we maintain the status quo.
Similarly, there is some fudgy-ness in the statement of the incompletness hypothesis. “Incomplete estimates will generally be over-optimistic.” Your over-optimism may be my over-pessimism.
I’m reminded of Rumsfeld’s famous line about unknown unknowables and some such. What rule of nature implies only bad stuff is more unknowable?
We'll see if anyone responds.
What are numbers?
Prof. Derden gave a talk
on math philosophy at the HSU
math colloquium. He was addressing the issue of what numbers really are and his talk was a response to a commonly held belief (in philosophy) that numbers can’t be sets. Besides being sets, he wants to argue that numbers are a particular type of sets called Russell sets. Below is an excerpt of an email I sent the professor. I’ve yet to receive a reply:
As I saw it, your talk consisted of one of two things. There was a strong argument in favor of using Russell sets as a model for the natural numbers OR there was a relatively weak claim that sets, in general, can't be in the set of "things numbers aren't."
I'm particularly fond of the stronger argument. That there would be a correspondence between all sets of pairs and the things we count as two seems to jive with my intuition of what numbers are. When I say "two", I just mean all those things that I can pair together. Two is, in a sense, a pronoun for all pairs of things (or a synonym maybe?). And when I say "two plus two equals four", I'm saying something like "two, for example, sheep, plus two, for example, rattle snakes, equals four things."
Given the title of the talk was "What Natural Numbers Must Be," its clear you intended to make the stronger claim. In any case, you at least argued the weaker claim. That there is a plausible argument at all for Russell sets as a model for numbers, means that sets COULD be such a model.
The word 'model' is tripping me up. Isn't a model just an approximation of the real thing? Models are built to mimic the behavior of what is being modeled. As long as the model behaves as the thing being modeled (at least in the area of concern), it is said to be a good model. In this sense, the flight of fixed wing airplanes is a good model for the flight of birds. Of course, physics has shown that they are far from the same thing. In the same sense, the natural numbers can be modeled by sets, but that doesn't mean that they are sets.
This is the same point Benacerraf makes in his essay. I couldn't find the Benacerraf's essay online but I found this article discussing the article. From that secondary article, "Benacerraf explains that to characterise the numbers is only to describe the structure, without any identification of the individual elements, and that this is why numbers are not objects at all." So the model of the numbers helps us describe the structure of the numbers, but it shouldn't be confused for the real deal.
Also, I can think of one more problem with trying to understand what the numbers really are. In your talk, you mentioned that the Platonists 'push' the location of ideas into the mind of god. Oh yeah, where's he?
Similarly, if we ever find out what the number really are, won't we just be pushing the problem? For example, let's say, at your talk, we decided that the numbers were Russell sets. But then an intrepid undergrad would ask, in his required email follow-up, "Thanks Prof. Derden for letting us
know what numbers are. Ok, so what are Russell sets?" To which you
spend another colloquium talk discussing and then narrowing the possibilities. Suppose, at the end of that talk we all decided that Russell Sets are really something called blarbs. Then we'd have to explain what the blarbs are, then what those things are, and so on into an infinite regress. Where would this all end?
Technology in math education
’s discussion of technology in the teaching of Multivariable Calculus
was a quick introduction to Matlab, Geometer’s sketch pad and Symbolic toolbox (in Matlab). Much of the talk was also a ‘geewiz’ exposition of some cool things you can do you in Geometer’s sketch pad. I left the discussion not having a good sense for how these technologies might be used to replace (or if they should be used to replace) current curriculum.
Overall, after reviewing Mr. Arnold’s web site
, I’m impressed by his extensive use of technology in the teaching of Calculus. He’s even done away with exams and grades exclusively based on performance in labs/quizzes and homework.
The question for me is: is this appropriate. Mr. Arnold has demonstrated that technology can replace a more analytical approach, but should it? For example, you can use the sketch pad to find the arc length of the cycloid without the need of any the analytical tools you traditionally learn in second year calculus. Do you lose anything by orienting yourself to the geometric interpretation in the technology?
One might argue that the analytical tools were invented 100’s of years ago in lieu of tools like the geometer’s sketch pad. After all, the inventor’s of these tools had specific geometric problems in mind (i.e. the orbit of planets) when inventing the Calculus. Thus, its okay to dispense with the analytics to be replaced by high powered technology.
The problem with that argument is that the analytic tools have general application. Not all problems have a geometric interpretation… for example, what picture can you draw that will help you solve calculus problems involving more than 3 variables? The tools are there to help you abstract to these higher dimensions were intuition fails you. Tangentially, some people are more comfortable “pushing symbols” than working with geometry. Personally, I have a hard time imagining shapes in 3 dimensional spaces.
In the end, I applaud Mr. Arnold’s work. My question to him: how does the technology help students get beyond their intuitions of geometry to help equip them with more general analytical tools?