Notes From The Mathematical Underground

I. All I have ever written has been, one way or another, concerned with “developmental” mathematics. I have written and still write mathematics from a point of view that I like to think is the real developmental one. I have presented and discussed developmental issues with colleagues on various venues, in my own department, at joint AMS-MAA meetings, at meetings of the American Mathematical Association of Two Year Colleges (AMATYC), in the AMATYC Review, on Mathedcc and Mathspin, etc. But, for all that and for just as long, even though I have kept musing about my own position in relation to developmental mathematics and while I have never had any doubt about said position, I have always been uneasy about formulating and stating it.

On this site, the closest I have come to mentioning my motivation was in the preface to Reasonable Basic Algebra in which I quoted from one of my Notes From the Mathematical Underground in the AMATYC Review, Spring 1996 issue:

Also directly relevant to the issue is an article by Colin McGinn, Homage to Education, in the August 16, 1990 issue of the London Review of Books [. . . ]. The article is a review of a book of, and of a book about, R. G. Collingwood. The relevant part is where McGinn “spell[s] in [his] own way what [he] thinks Collingwood is getting at here.” “Democratic States are constitutively committed to ensuring and furthering the intellectual health of the citizens who compose them: indeed, they are only possible at all if people reach a certain cognitive level . . . . [. . . ]. Democracy and education (in the widest sense) are thus as conceptually inseparable as individual rational action and knowledge of the world.” [. . . ]. But what is education? “Plainly, it involves the transmission of knowledge from teacher to taught. But what exactly is knowledge? ” […]. [It] is true justified belief that has been arrived at by rational means.” […]. Thus the norms governing political action incorporate or embed norms appropriate to rational belief formation. […]. The educational system of schools and universities is one central element in this cognitive health service […].

The quasi-mathematical language in which this is stated should have a special resonance for mathematicians. “It would be a mistake to suppose that the educational duties of democratic state extended only to political education, leaving other kinds to their own devices. […]. How do we bring about the cognitive health required by democratic government? A basic requirement is to cultivate in the populace a respect for intellectual values, an intolerance of intellectual vices or shortcomings. […]. The forces of cretinisation are, and have always been, the biggest threat to the success of democracy as a way of allocating political power: this is the fundamental conceptual truth, as well as a lamentable fact of history.”

However, “people do not really like the truth; they feel coerced by reason, bullied by fact. In a certain sense, this is not irrational, since a commitment to believe only what is true implies a willingness to detach your beliefs from your desires. […]. Truth limits your freedom, in a way, because it reduces your belief-options; it is quite capable of forcing your mind to go against its natural inclination. This, I suspect, is the root psychological cause of the relativistic view of truth, for that view gives me license to believe whatever it pleases me to believe. […]. One of the central aims of education, as a preparation for political democracy, should be to enable people to get on better terms with reason—to learn to live with the truth.”

Indeed,

A political act “to enable people to get on better terms with reason—to learn to live with the truth.”

is what I used when trying to explain what this site and its contents are all about. And, in a way, that does say it all. However it certainly doesn’t spell it out and I have long felt the need to discuss this a bit further but, somehow, have always had trouble with it.

II. It was a recent article, again in the London Review of Books, 6 March 2008, Is It Glamorous by David Simpson, a review of Absent Minds: Intellectuals in Britain by Stefan Collini, Oxford 2007, that, although it had of course nothing to do with developmental mathematics, finally gave me, for whatever reason, what I needed to discuss how I see developmental mathematics and why.

The part of the article that resonated with my own feelings and motivations is where Simpson discusses Collini’s attitude regarding Edward Said in general and Said’s Representations of the Intellectual [The 1993 Reith Lectures] in particular. Following are a few excerpts of Simpson’s article directly relevant to what I will say below about developmental mathematics. However, I should say immediately that the whole article is well worth reading independently of the issues that concern me here.

[I]t seems symptomatic that the figure [Collini] finds most wanting is Edward Said.

Collini finds [Representations of the Intellectual] a ‘poor book’ marked by ‘simplistic binary alternatives’, a ‘kind of political free association’ […]. Above all it succumbs to an ‘insidious kind of glamour, that of being the champion of the wretched of the earth’.

[Said was] throughout his career […] a defender of those who couldn’t speak for themselves or get a fair hearing when they did.

[Intellectuals] should call our attention to ‘all those […] issues that are routinely forgotten or swept under the rug’ and should universalise every crisis so as to bring it into line with as many other crises as possible, to the point of being ‘embarrassing . . . even unpleasant’.

Collini does not like Said’s ‘existential drama’ and its ‘inescapable logic of choice’

and, last but not least,

If Collini is right that with a few variations and exceptions the view of intellectuals has been much the same across the West in the 20th century, that there is a ‘larger international pattern’ at work, what is the common influence or structure that would explain it? He says at the end of the book that there is such a structure, but somehow the matter of ‘structural rather than merely local explanations’ dwindles down to a matter of ‘alarmist cultural pessimism’ which he has ‘taken issue with on other occasions’. There might be interesting reasons why capitalist economies in tandem with representative democracies are felt to have the power to impose despair or desperation on their intellectuals. But do they do so on all of them? Are there not some who maintain an optimism of the will, and must it be ‘culpably romantic’ to do so?

Well, of course, I am not Said and since I have long admired Said I feel that directly invoking the above would be at least presumptuous, in fact impertinent, and certainly quite ridiculous. So, I will now proceed with my stance regarding developmental mathematics and leave any connection with the above entirely to the reader.

III. The first issue is why would anyone at all want to “learn” mathematics. I can see three possible answers which however involve three different meanings of the word “learn”.

1. One can see mathematics as a chore, as something necessary to be able to do other, specific things such as being able to register for another course or being able to cut rafters for a roof. But the apprentice carpenter neither needs nor wants to go through Euclid’s books in order to be able to cut rafters and the English major neither needs nor wants to factor quadratics in order to be able to discourse on Shakespeare. Developmental mathematics is therefore the prerequisite for Precalculus which, as the name indicates, is the prerequisite for Calculus which is the prerequisite for Physics and other “advanced” courses, etc.

   Of course, the dual of the question is why would most curriculums require mathematics in the first place. The answer is usually that curriculum designers include some mathematics so as to give weight to, or simply pad, their curriculum … and, by a fortunate coincidence, give jobs to mathematicians who are then expected to, and duly do, return the favor.

   The carpentry curriculum thus claims “geometry” to be a necessary background for learning how to use a carpenter’s square and the English departments thus claims that set theory and abstract algebra are essential to understand the linguistics of Harris or Chomsky, the structures of Lévi-Strauss, the Borromean knots of Lacan, as well as Catastrophe Theory and now Chaos Theory for whatever literary theory is currently fashionable. But of course, none of these claims ever held any water. Harris didn’t know any mathematics beyond the definitions of equivalence relation and semi-group and never did anything with either. Being honest, Chomsky never claimed to do anything mathematical in the first place, Lévi-Strauss didn’t know what a structure was—he just liked the word, Lacan never exhibited the least interest in rational discourse, etc.

   When all else fails, as it invariably does, recourse is then made to needs like having to be able to compute unit prices, discounts and markups at the store, etc. When you point out that very few people feel these supposed needs and, anyhow, that most everybody nowadays has a calculating cell phone, your opponent, very likely to be into teaching with calculators, gets vaguer and vaguer until s/he declares her/himself outraged by something or the other and walks away in a huff and with a snort of disgust.

   What I find amazing under these conditions is how we are able to convince most everybody that they need to know a certain amount of arithmetic and a certain amount of algebra and, when we catch them in time, how we succeed in corralling them into developmental mathematics programs. The bitter irony here is that success in these programs is largely determined by those running these very programs. There might be a good reason here as when, occasionally, the success of a developmental program is measured by the success of its graduates in ulterior courses, the results turn out to be horrendous. See, for instance, LongitudinalStudy

2. Another view is that mathematics, in some way, is somehow formative: being able to factor a few quadratics is good for you. Period. A variant is the belief that a—small—amount of mathematics is a necessary ingredient of “general knowledge”, the panoply of the well-rounded, cultivated gentleman: Gauss as well as Michelangelo, Shakespeare and Einstein. See whatever “liberal arts mathematics” book you happen to have at hand.

   Yet, there is something to that view but it is impossible to delineate and, in any case, most people don’t have the leisure or the financial means and that type of course is very much on the way out.

3. My own view is that mathematics is the simplest universe in which to learn how to make a case for one’s conjecture, in which to distinguish what we can show is true from what we can show is false and from what we don’t know to be true or false, etc. In short, mathematics is the simplest place in which to learn how to operate rationally. In that, mathematics is a lot closer to law than to what is currently peddled as mathematical proof in geometry textbooks. See The Uses of Argument by S. Toulmin.

IV. The second issue is what mathematics ought to be learned when. As pointed out above, most students are not really free to choose what mathematics they are to learn. And when it us who decide for them, as we usually do, we invariably choose developmental mathematics and/or precalculus mathematics and for the—very few—survivors, calculus. All of which according to the gospel of texts carefully packaged by an industry driven by greed bordering on the pathologically insane which, though, give us great opportunities to deploy our teaching skills, that is essentially our ability to sugarcoat the pill and grease the plank. The students be damned.

Many alternatives would of course be possible.

• One could be the arithmetic and algebra of collections-of-items and unit-prices with co-multiplication because this can quickly be generalized to “baskets” of collections-of-items and “lists” of unit-prices, that is, in other words, LinearAlgebra.
• Another could be Discrete Mathematics but it seems to lack any story line and I have nothing to suggest. All I can say is that the texts I have seen appeared to be collections of topics: a bit of sentential logic here, a bit of set theory there, some graph theory possibly somewhere in-between, etc.
• Another could be Geometry and/or Group Theory starting, say from the notion of tiling. But, while I have played a bit with the idea, I am not sure how to let it go beyond the obvious. It is a bit as with Incidence Geometry which is initially enticing since there are so few axioms but which quickly degenerates into counting arguments.
• The alternative which I have chosen to develop materials for is a strongly integrated Arithmetic-Basic Algebra-Differential Calculus three-semester sequence and this for a variety of reasons. The main one is that the mathematics of change is a well defined goal well within reach of a lot more students than learn it in the traditional sequence. Another one is that there is a simple, very strong story line which I will discuss later. Yet another is that this alternative can be fitted without too much upheaval in the current college framework. Last but not least is that I happen to like the subject of polynomial approximations and that I see this sequence as the ideal developmental mathematics. (Of course, how well it will work in practice will have to be ascertained by others than myself.)

In the next installment of this blog, I will thus discuss developmental mathematics as embodied in the Arithmetic-Basic Algebra-Differential Calculus sequence and from the point of view that “mathematics is the simplest universe in which to learn how to make a case for one’s conjecture, …”

As ever, any criticism, critique, feedback, etc is of course welcome, the more detailed, the more welcome.

A. Schremmer

This is more of a “progress report” than about issues concerning learning mathematics although, as it happens, the two are deeply linked.

The progress report. As usual, I am behind: estimated time of arrival for Reasonable Algebraic Functions (RAF) is now sometimes in Fall 2008 and for Reasonable Decimal Arithmetic (RDA) sometimes in Spring 2009.

The reason or, if you prefer, the excuse, for the delay. I spent an inordinate amount of time, even for me, trying again and again to redesign the first of the three parts that comprise RAF. The issue was that while Polynomial Functions (Part Two of RAF) and Rational Functions (Part Three of RAF) had long seemed stable enough in terms of their mathematical treatment, there had always been two problems: (1) How to present Power Functions and (2) several issues associated with the methodology used in the treatment of Polynomial Functions and Rational Functions.

1. Essentially, the mathematical approach, due to Lagrange, is to use local polynomial approximation and I discussed in some detail how I used it in a Precalculus-Calculus One sequence. By itself, its implementation does not raise any problem, at least certainly nothing like what is involved when using limits:
   • For Affine Functions, all that is needed is replacing x by x₀+h.
   • For Quadratic Functions, all that is needed is an addition formula for the expansion of (x₀+h)².
   • For Cubic Functions, all that is needed is an addition formula for the expansion of (x₀+h)³.
   • For Quartic Functions, etc …
   • For Polynomial Functions of degree n, we need the Binomial Theorem.
   • For Rational Functions, all that is needed is polynomial division albeit both in ascending powers to localize near bounded inputs including the poles, if any, and in descending powers to localize near infinity.
   • For Radical Functions, we need to solve a—fortunately sparse—system of equations to find the coefficients of the expansion.

   While it might not entirely satisfy mathematicians, this approach is, I think, very much in the spirit of mathematics in that cases can be made—and readily turned into proofs—for most of the theorems being used. At the very least and if nothing else, there is none of that “After plotting these points, you can see that they appear to lie on a line, as shown on Figure 1.1. The graph of the equation is the line that passes through the five plotted points.” (Larson-Hostetler, Precalculus 6th edition. For the effect this kind of “philosophy” is likely to have on serious students, see Sim Wobpa which was written à propos the 4th edition of the very same opus.) In the rare occasions when, for whatever reason, a case is not made, this is duly noted and explained.

2. There is, however, an important difference between RAF and Lagrange DC which is that: a) in order to be usable in Precalculus One, the concept of derivative had to go and, b) in order to still make sense as a “mathematical theory”, the treatment in RAF had to be, in some sense, “minimum and closed”. As it turned out, though, these two requirements were easy to satisfy. Concerning a), the analysis in RAF could be done without mentioning derivatives and by just using the coefficient of hⁿ. Concerning b), first RAF does not include Radical Functions on the grounds that the way the localization is obtained brings them closer to Transcendental Functions defined as solutions of differential equations and then, second, since Affine Functions have no turning point and Quadratic Functions have no inflection point with Cubic Functions being able to exhibit both, RAF did not have to deal with any other Polynomial Functions to illustrate these behaviors and the list of contents for RAF is:
   • Power Functions
   • Constant Functions
   • Affine Functions
   • Quadratic Functions
   • Cubic Functions
   • Rational Functions

3. A very important unifying fact is that, in some ways, both Polynomial Functions and Rational Functions are “essentially” little more than Power Functions with some number of fluctuations, that is of minimum-maximum pairs, thrown in. To be a bit more precise, though, a big difference is that the fluctuations are bounded in the case of Polynomial Functions whereas they can be infinite in the case of Rational Functions. Specifically, the essential bounded graph is defined as the simplest bounded graph compatible with the local graph near infinity and the local graph near the pole(s). Beyond that, of course, the essence of their nature is that Polynomial Functions are locally approximately polynomial while Rational Functions are locally approximately Laurent-polynomial.

   Still, an important issue is that of locating these fluctuations. Fortunately, in the above list of contents, things are naturally taken care of: after expanding near x₀, we locate the turning point by killing the coefficient of h in the localization which, in the case of a quadratic function, entails solving an affine equation and in the case of a cubic function entails solving a quadratic equation. Similarly, we locate the inflection point of a cubic function by killing the coefficient of h² which entails solving an affine equation. So, in a way, there is a “closure” of sorts.

4. However, in the case of students who need to learn that mathematics has a logical development, as opposed to “math” being a set of “skills”, an extremely important issue is that of the order in which to introduce the necessary language and the necessary concepts so as to create an impression of logical flow. The “progression” from Polynomial Functions to Rational Functions thus seemed quite “logical”, at least to me. But I realized eventually that it still resulted in a certain “ad hoc” feeling with the students. And another place where the flow was far from being “obvious” occurred when moving from Part Two - Polynomial Functions to Part Three - Rational Functions: there was this sudden, new concern as to whether there might be bounded inputs with infinite outputs. Again, at first, this didn’t seem to be a big deal as, after all that is why they were dealt with in two different Parts, but, still, it sure didn’t contribute to continuity in the story line. And, of course, there was the issue that the flow in the progression
   • Positive-power Functions
   • Negative-power Functions
   • Polynomial Functions (of degree ≤ 3)
   • Rational Functions

   which, if it certainly works, was somewhat less than felicitous. But then, the flow in the progression

   • Positive-power Functions (Needed to localize Polynomial
     Functions.)
   • Polynomial Functions (of degree ≤3)
   • Negative-power Functions (Needed to localize Rational
     Functions.)
   • Rational Functions

   while it was on an “as needed basis” and thus seemed a bit more logical, didn’t deal with the concern about the sudden concern about whether there might be bounded inputs with infinite output.

5. There was also another problem in that I was also introducing the terminology on an “as needed” basis. For instance, the term “turning point” was introduced in the context of Quadratic Functions and the term “inflection point” in the context of Cubic Functions. While it felt right not to start with a massive set of definitions at the beginning of the course, later on, it created problems of reference and ended up exacting a heavier price than I had thought.

   And then, given that RAF is a standalone, there was the issue of where to introduce the various necessary concepts, those usually thrown-in into a catch all “review chapter. What took me four months to arrive at was the idea of introducing all that was needed on functions defined by graphs. In particular, given the local graph near infinity of a function, a natural question is under what conditions does joining smoothly the local graph near –∞ to the local graph near +∞ result in the bounded graph. The condition of course was the answer to the Essential Question—do all bounded inputs have bounded outputs or is there a bounded input with an infinite output?—which until now had come up only with Rational Functions—and Chapter 3 was entirely devoted to the issue of smooth interpolation. But then, finally, I came to realize that it was the notion of outlying graph, that is of the local graph near infinity together with the local graph near the poles, if any, that provided the means to make the contents consistent in that for “all” functions it was the outlying graph which controlled the (essential) bounded graph.

All of this to say that, as of this Bastille Day, the list of contents for the first five chapters of RAF is:

1 - Introduction
1.1 Relations
1.2 Functions
1.3 Functions Specified by an Input-Outpur Rule
1.4 Signed-numbers Graphically
1.5 Signed-numbers Qualitatively
1.6 Large and Small Numbers
1.7 Qualitative Rulers
1.8 Screens
1.9 Functions Defined By A Graph
1.10 The Fundamental Problem*

2 - Graphic Local Analysis
2.1 Local Graphs
2.2 Local Language
2.3 Place of a Local Graph
2.4 ∞-Height Inputs and 0-Height Inputs
2.5 Shape of a Local Graph
2.6 Local Behavior
2.7 Feature-Sign Change Inputs
2.8 0-Slope and 0-Concavity Inputs
2.9 Extremum Inputs

3 - From I-O Rule To Global Graph
3.1 Smoothness
3.2 Interpolation
3.3 The Essential Question
3.4 The Essential Bounded Graph
3.5 Essential Notable Inputs

Part I - Power Functions

4 - Positive-Power Functions
4.1 Input-Output Pairs
4.2 Normalized Input-Output Rule
4.3 Local Graph Near ∞
4.4 Types of Local Graphs Near ∞
4.5 The Essential Question
4.6 Essential Bounded Graph
4.7 Notable Inputs
4.8 Local Graph near 0
4.9 Types of Local Graphs Near 0
4.10 Essential Global Graph
4.11 Types of Global Graphs

5 - Negative-Power Functions
5.1 Input-Output Pairs
5.2 Normalized Input-Output Rule
5.3 Local Graph Near ∞
5.4 Types of Local Graphs Near ∞
5.5 The Essential Question
5.6 Types of Local Graphs Near 0
5.7 Essential Bounded Graph
5.8 Local Graph near 0
5.9 Notable Inputs
5.10 Essential Global Graph
5.11 Types of Global Graphs

*The Fundamental Problem is the problem of deriving a graph from an input-output rule. It provides the story line of RFA. Larson-Hostetler and all the others notwithstanding, this cannot be done by “joining five points smoothly”.

Hopefully, this architecture will support the rest of RFA in a manner satisfactory to Students interested In mathematics With only a background in polynomial algebra.

As ever, any criticism, critique, feedback, etc is of course welcome, the more detailed, the more welcome.

A. Schremmer

P.S. While the original entry was written on Bastille Day, it was slightly modified on July 23.

P.P.S. The navigation in FreeMathTexts is of course perfectly atrocious and, as soon as I understand css better, I will use some open source code I found on the web to remedy this rather unfortunate situation. In other, unrelated news, I am also considering learning about how to set up a forum/listserv.

Back in April, someone, signing as CCDN, started a new thread at http://mathforum.org/kb/thread.jspa?threadID=1733976&tstart=0 with the following post:

Marketplace and Math Class?

Does anyone have any ideas regarding todays current economic climate and an intriguing math lesson for High School students?

to which I responded with the somewhat desultory,

The Fed prints only so many dollar bills. If one percent of the population takes a disproportionate amount of these dollars, then these are dollars that we don’t have. Put more simply, the richer they get, the poorer we get. It’s called the law of conservation of money, AKA Where did the money go?

Students seem fascinated.

The ensuing exchange was rather unsatisfactory:

1. Having been reared in Mechanics, specifically in Fluid Dynamics, I had unwittingly taken for granted that “the law of conservation of money” would be seen for what it is, namely an avatar of Stokes’ Theorem, other avatars of which include the Law of Conservation of Mass, the Law of Conservation of Momentum and the Law of Conservation of Energy as well as the Fundamental Theorem of the Calculus … and the “extended accounting equation”:
   Given a business, at any time t, Assets(t) is what it owns, Liabilities(t) is what it owes and
   
   NetWorth(t) = Assets(t) - Liabilities(t)

   
   Thus, NetWorth(t) is the state that the business is in at time t. In accounting terminology, NetWorth(t) = Assets(t) - Liabilities(t) is called the “basic accounting equation” but it is really just the definition of NetWorth(t). Written as Assets(t) = NetWorth(t) - Liabilities(t), the “basic accounting equation” says how the assets were financed: with the owner’s own money (owner’s equity), NetWorth(t), or with borrowed money, Liabilities(t).
   At any time t, the business sustains actions, that is it earns Income(t), and incurs Expense(t) so that, over a period of time Δt = t₂ - t_1,

   NetWorth(t₂) - NetWorth(t₁) = ∫_t1^t2 [Income(t) - Expense(t)] dt
   

   which, in accounting terminology, is the “extended accounting equation”.

2. Essentially, the rebuttal to my own statement consisted only of the statement:
   (~Z) [The market] is not a zero sum game.

   At that point, I thought an interesting discussion had begun but I was disappointed and there wasn’t any real one. Still, I was intrigued by the fact that ~Z had been presented as, if not a self-evident truth, at least as a universally-known one. So, knowing just about nothing about game theory, I thought I would investigate a bit and started asking around in various contexts “Where did the money go?” The response was indeed universal: The market is not a zero sum game. On the other hand, nobody was able even to begin giving me an argument as to why ~Z should be true: “It’s just so, don’t you know?” I didn’t and still don’t.
   At that point, though, I thought I should see how I would support my own assertion. However, since it had been felt by some that the subject was “politics” and thus inappropriate for the mathedcc forum and since I had just started this blog, I thought I would post the result of my cogitations here.

3. But, in spite of my faith in the universality of conservation laws, and even though I still think that economics is essentially a very simple matter, I do realize that economics involves a great deal of psychological issues and is thus a very tricky business and I don’t claim to lay down the truth. Comments, corrections, etc are thus more than welcome.

In discussing this, I will also occasionally point out how prevalent these ideas are and how relevant they are to our more mundane preoccupations as “math teachers”.

1. A basic idea, at the root of civilization, is that of exchange. I was given an apple for my lunch and you were given a banana but I prefer bananas and you prefer apples. So we exchange the apple for the banana.
   Exchanging is of course at the heart of one way to see systematic counting (See Reasonable Decimal Arithmetic. To appear.): We can exchange a single collection of four hundred sixty two one-dollar bills for a combination of three collections, a collection of four hundred-dollar bills, a collection of six ten-dollar bills and a collection of two one-dollar bills with the advantage that now we can represent each collection with one of TEN digits.
   Exchanging allows us to exchange a combination of two collections for a single collection. If I have three apples and two bananas and if I can exchange each apple for five carrots and each banana for two carrots, then I can exchange my combination of three apples and two bananas for a collection of nineteen carrots. Similarly, if I have three quarters and two dimes and if I can exchange each quarter for five nickels and each dime for two nickels, then I can exchange my three quarters and two dimes for nineteen nickels, i.e. 3/4 + 2/10 = 19/20. (See Reasonable Decimal Arithmetic.)
2. Exchanging, though, begs the question of what the rate of exchange is to be. I have apples and you have bananas. I would like to exchange some of my apples for some of your bananas and you would like to exchange some of your bananas for some of my apples. But the obvious, immediate question is how do we arrive at a mutually agreeable rate of exchange.
   This usually depends on past history and/or on aggregation. Depending on how we acquired them, my apples and your bananas probably do not have the same “value” for each of us. At a stock exchange, the question of the rate of exchange is solved in the aggregate: If, altogether, three thousand apples and two thousand bananas are being offered, then a rate of exchange of three apples for two bananas is deemed to be satisfactory to everybody. End-of-the-World Trade by Donald MacKenzie, in the May 8, 2008 issue of the London Review of Books, is an excellent piece on the setting of exchange rates and its role in the credit crisis.
   In the case of systematic counting, the rate of exchange, i.e. the base, does not really matter: we count in base THREE exactly in the same manner as in base TEN. In the case of fractions, since the idea of the procedure has already been dealt with, the issue reduces to the more technical one of finding common multiples.
3. At its most fundamental, money is just a particular goods more durable than most other goods so that one can delay bartering by exchanging one’s goods for money and thus not have to commit oneself. I have five apples and I know that I need only three but at this time I don’t know if I want to exchange my other two apples for bananas or for carrots. So, I exchange my extra two apples for money which I can hang on to until I know whether I want bananas or carrots at which time I will exchange the money I got for my two apples. The price of goods is thus the amount of money it can be exchanged for. Given a vector space of “baskets” and its dual space of “price lists” (rates of exchange for money), co-multiplication gives the value of a basket under a price list. If I have three apples and four bananas in my basket and apples are today at 15 cents per apple and bananas are at 10 cents per apple, then the value of my basket today is 85 cents.
   The monetary mass thus corresponds to the amount of available goods. A change in the money supply without a corresponding change in the amount of available goods causes the rate of exchange to change. I will use the terms inflation and deflation which, according to Wikipedia is a sense only used by classical economists while the usual contemporary definition involves the “general price level”—whatever that means.) Thus, if I burn a one-dollar bill (which I believe to be illegal), this results in an—infinitesimal—deflation. If I print a one-dollar bill (which I definitely know to be illegal), this results in an—infinitesimal—inflation. On the other hand, if I pick your pocket and steal a one-dollar bill, I am one-dollar richer and you are one dollar poorer. In all cases, the law of conservation of money prevails.
   An interesting example of this is the well-known way George Soros originally made his fortune, i.e. by, inter alia, “breaking the Bank of England”. However I am more familiar with the case in which he borrowed a huge amount of Liras from Italian banks which he then put up for sale thus forcing the price of the Lira down. At this point, the European Rate Mechanism kicked in and Italy was forced to devalue the Lira which allowed Soros to buy them back at a reduced price and thus pay off his loans. In other words, Soros’s gain came from the some 60 millions Italians left with devalued Liras. In the case of the Bank of England, he is said to have merely “sold short”. The role of the European Rate Mechanism in both cases was crucial but only in the sense that it saved time: Soros could probably have achieved about the same result by buying back slowly. This is where psychology comes in and it is only recently that the subject has started to be investigated by economists—as opposed to the standard blanket recourse to “human nature”.
   By the way, and rather obviously, the entity which succeeds in having a particular good accepted as money makes an enormous profit. In fact, the privilege of minting money used to be called seigniorage. According to Wikipedia, the U.S. Treasury estimates that it has earned about US$5 billion in seigniorage revenue from the 50 States series of quarters which many people collect and thus keep out of circulation. Another instance of this is the issue of petrodollars, namely the dollars we use to buy oil and which are not meant ever to come back home and cause inflation with, of course, the ensuing conflict with Europe which doesn’t see why it couldn’t issue petroeuros.

4. Finally, a couple more remarks:
   • If I lead you to believe that something has more value than it has and exchange it for something whose value is more stable, then, in essence, I stole the difference in value from you.
   • Goods are created and destroyed. But the Laws of Conservation account for this with sources and sinks.

Altogether then, I still don’t see what the flaw(s) in my original statement might be, that is, I was not able to imagine any situation that would not be a zero-sum game. Again, comments, rebuttals, corrections are welcome.

A. Schremmer

In the Fall 1994 issue of the AMATYC Review, I started a new column, Notes From the Mathematical Underground, with the following:

Since this is a new column, a caveat might be in order. As we all know, there currently is a “crisis” in mathematics education. (As has been the case, roughly, as far back as I can remember, which is, alas, quite far enough.) Since the latter in­volves mathematics, faculty and students, not necessarily in that order, this is where its origin must be sought and this is what I would like to do here. By defi­nition how­ever, faculty are above suspicion or, at least, will be in this column. Students being the given of the situa­tion and, in any case, presumably not part of the readership, any discussion of how they ought to be would be futile and I will leave them mostly alone and not discuss pedagogy. This leaves mathematics and this is what I intend to discuss: No “how to do it”, no sugar-coating by way “ap­plications” or “math his­tory” or whatever, no “high tech” religion. But mathemat­ics does not exist in a vacuum and my distin­guished colleagues will, of necessity, have to be considered as parame­tersin the equation.

It might have occurred to the reader that I left textbooks out of this equation and it is in­deed fashionable in academic circles to deplore the state of the textbook art (Except among authors of course. See, for instance, Anton (1991)) and blame it for much of the educational fiasco. As publishers, though, are fond to point out, it is fac­ulty who design the courses, who order the textbooks and who write them. Moreover, when they wish to be nasty, they are quite prone to listing all the non-con­formist texts on which they say they lost their corporate shirt.

Which brings me, precisely, to the main issue that I intend to pursue here, that of the alternatives to the math­ematical underpinnings of our teaching. One way then in which I would like to do this is to discuss textbooks that didn’t make it as mainstream texts because, presum­ably, they were “too different”. Here, I mainly think of calculus texts such as those by Levi, Keisler, Strang, Flanigan-Kazdan, Freed, etc. I would also like to discuss ideas that briefly appeared in texts but were dropped in subsequent editions—when there was one, such as Munroe’s defini­tion of variables or Gillman-McDowell’s definition of the integral. There are also very simple ideas, such as Lang’s treatment of the transcendental functions or that of Finney-Ostbey, that appear in more advanced texts but which, somehow, never made it to “elementary” textbooks .

Fast forward to this day, April 24, 2008.

I don’t think I have to change a word of the above. I should, though, add a cautionary explanation.

In the Spring 2003 issue of the Notes From the Mathematical Underground, I wrote:

This is Part I of the last of my Notes From The Mathematical Underground[1] which I would like to devote to another major subject of enquiry in mathematics education, one as important as content analysis and just as ignored, namely language analysis.

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[1] I would hope though that, even in these severely conservative times, some sort of Mathematical Underground will survive.

And, indeed, the column came to an end with the Fall issue. This was because I had embarked on a new project (See: About the author) of which this very site is the logical extension.

But, I am a polemicist at heart. And while I am having fun with the “inflammatory” footnotes in the texts on, or to be on, this site, I was missing the format of the column. Hence the logical extension of the site to include this blog which I saw no reason not to call … Notes From the Mathematical Underground.

A. Schremmer