Saturday, December 31, 2016

Welcome

I study economics as a hobby. My interests lie in Post Keynesianism, (Old) Institutionalism, and related paradigms. These seem to me to be approaches for understanding actually existing economies.

The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.

In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.

I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.

Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.

Thursday, April 16, 2015

A Plague On Both Your Houses

In a Bloomberg News piece, Noah Smith makes some false claims. I think his mistakes - what Eatwell and Milgate call an imperfectionist view - are widely shared among many macroeconomists. My belief that these mistakes are widely shared is not overthrown, I think, by the confusions put forth in these later posts by Stephen Williamson and Noah Smith, respectively.

First, we have the mistaken belief that in a perfect world, capitalist economies would move quickly towards equilibrium. Smith starts his column with an anecdote:

"One time, at a dinner, I asked a famous macroeconomist: 'So, what really causes recessions?'

His reply came immediately: 'Unexplained shocks to investment.'"

I take this to be an expression of the freshwater view, as embodied in models of Real Business Cycles. Cycles are to be understood as equilibrium paths responding to exogeneous stochastic shocks. Risk exists, but uncertainty does not. Recessions and depressions occur when workers voluntarily decide to take long vacations.

Second, we have mistaken understandings of price theory and how equilibrium is established:

"The market adjusts by the price mechanism. If the cost of something goes up, the price goes up to match. If demand falls, the price drops until the market clears."

I take this to be a claim that equilibrium prices are indices of relative scarcity, a belief shown to be without logical foundation about half a century ago. Ever since Robert Lucas put forth his critique in the 1970s, mainstream macroeconomists have claimed to be developing models with rigorous microfoundations. And those foundations are supposed to be provided by General Equilibrium Theory, in which agents optimize under constraints.

But many macroeconomists seem to be just ignorant of price theory, as experts in GET, such as Frank Hahn explained long ago. In the most rigorous neoclassical theory, with many commodities and many agents, the assumptions do not lead to the conclusion that prices behave that way. Nor do the theorists have a good story about how equilibrium is established. The mathematics used in mainstream macroeconomists does not allow one to find clear statements of assumptions. At least, I am unable to understand what assumptions mainstream economists think they are making on tastes, technology, and endowments in multicommodity models to justify their macroeconomic modeling. I would rather that economists turn to non-equilibrium modeling, a position that I think Robert Lucas still finds incoherent.

Third, suppose you hold that observed fluctuations in employment and output in capitalist economies can hardly be an equilibrium response. If you held the mistaken ideas about price theory that Noah Smith does, you would think that the empirical behavior of economies could only be explained by introducing some imperfection, some failure of competition, some information asymmetry, or some stickiness or slow adjustment into your theory. And given your empirical beliefs, you would think the development of theory in such a direction is a triumph of science:

"But despite these scattered denunciations and grumbles, sticky prices are enjoying a hard-fought place in the sun. The moral of the story is that if you just keep pounding away with theory and evidence, even the toughest orthodoxy in a mean, confrontational field like macroeconomics will eventually have to give you some respect."

But it is not the case that markets, including the labor market, would rapidly clear if only imperfections did not exist in a market economy. For economists to have reached this as a consensus position is a failure of their profession, not an achievement. Business cycles neither need to be explained as an equilibrium phenomenon, nor need sticky prices be invoked to explain the failure of markets to clear.

Is the topic of the above post orthogonal to a debate Paul Krugman overviews? I am of two minds on Krugman's post. I cannot be too hostile to a blog post illustrated with a homoclinic bifurcation. Maybe a solid appreciation of nonlinearity in macroeconomics is associated these days with heterodox, but not necessarily non-mainstream economics.

References
  • John Eatwell and Murray Milgate (2011). The Fall and Rise of Keynesian Economics, Oxford University Press.
  • Richard M. Goodwin (1990). Chaotic Economic Dynamics, Oxford University Press.
  • Murray Milgate (1982). Capital and Employment: A Study of Keynes's Economics, Academic Press.

Friday, April 03, 2015

How To And How Not To Attack Marx's Economics

1.0 Introduction

I am currently reading John Roemer's Free to Lose. I thought I would outline some areas where Marx can be criticized on economic theory, as well as some areas where I do not think he is not so vulnerable. (I do not think I had previously absorbed Roemer's theory of the emergence of classes from an analysis of reproducible equilibrium. But then the Roemer work I know the best is Analytical Foundations of Marxian Economic Theory, which may predate this explanation.) Another motivation is irritation with a series of post here.

2.0 Labor Theory of Prices

For purposes of this post, I put aside the question of whether prices tend to be proportional to labor values. I think Marx rejected this theory, including in the first volume of Capital. He says so, for example, in this passage:

"From the foregoing investigation, the reader will see that this statement only means that the formation of capital must be possible even though the price and value of a commodity be the same; for its formation cannot be attributed to any deviation of the one from the other. If prices actually differ from values, we must, first of all, reduce the former to the latter, in other words, treat the difference as accidental in order that the phenomena may be observed in their purity, and our observations not interfered with by disturbing circumstances that have nothing to do with the process in question. We know, moreover, that this reduction is no mere scientific process. The continual oscillations in prices, their rising and falling, compensate each other, and reduce themselves to an average price, which is their hidden regulator. It forms the guiding star of the merchant or the manufacturer in every undertaking that requires time. He knows that when a long period of time is taken, commodities are sold neither over nor under, but at their average price. If therefore he thought about the matter at all, he would formulate the problem of the formation of capital as follows: How can we account for the origin of capital on the supposition that prices are regulated by the average price, i. e., ultimately by the value of the commodities? I say 'ultimately,' because average prices do not directly coincide with the values of commodities, as Adam Smith, Ricardo, and others believe." -- Karl Marx, Capital, V. 1 (last footnote in Chapter V.)

I take "average price" in the above passage to be referring to what has also been called "such classical terms as 'necessary price', 'natural price', or 'price of production'" (Piero Sraffa, PCMC: p. 9). And Marx is saying that prices of production do not correspond to labor values, even though he is abstracting from this distinction in the first volume of Capital. Others have also asserted that a contradiction in Marx cannot be found here:

"Writers ... like E. Bohm-Bawerk have asserted that there is a contradiction between the analyses of Volumes I and III which is certainly not to be found there unless one reads into them an interpretation different from that which Marx repeatedly emphasized." -- William J. Baumol, "The Transformation of Values: What Marx 'Really' Meant (An Interpretation)" (, V. 12, N. 1 (Mar. 1974): pp. 51-62,
3.0 Heterogeneous Labor Activities

Employees perform many distinct activities in laboring under the direction of capital. I do not think this observation is sufficient, in itself, to hinder the development of a theory organized around labor values. Consider jobs provided by supposedly unskilled labor, such as stocking shelves in a supermarket or working behind the counter in a fast food restaurant. These sort of jobs are often treated as homogenous, both by workers and employers. Workers in one or other such job can transition among them easily enough in times of high employment.

What are jobs that require vastly different levels or types of skills? I do not think this is a problem for Marx as long as relative wages can be treated as stable:

"We suppose labor to be uniform in quality or, what amounts to the same thing, we assume any differences in quality to have been previously reduced to equivalent differences in quantity so that each unit of labor receives the same wage." -- Piero Sraffa, (1960: p. 10).

As far as I can tell, this is a common position among the classical economists, with Adam Smith providing an early explanation of wage differentials.

A problem can arise here, however. Suppose some skills are acquired through an investment, such as paying for higher education. Perhaps there is a tendency for skilled workers to make decisions based on anticipated rates of return. Then, just as Wicksell effects express the dependence of the price of capital goods on distribution, so relative wages would vary with distribution. And labor values would be dependent on prices. One could then express labor value as a vector of different quantities of different types of non-competing workers. But would the assumption that the economy hangs together - e.g., all commodities are basic - work in this case? Or one could make the claim that even skilled labor is heavily produced in the household and outside of firms run for profits. And, thus, calculations of rates of return for acquisition of many skills for the worker are empirically unimportant. (I think I take this objection, as well as the first response, from Ian Steedman.)

4.0 Labor Values Dependent on Choice of Technique

I take labor values as being found from the processes used in production, as expressed in a Leontief input-output matrix and labor coefficients. The components of such matrices and vectors are given in physical units. The analysis of the choice of technique shows that the cost-minimizing technique varies with distribution. So, here too, labor values depend on prices, instead of vice-versa.

Here one could object that the choice of technique is a highly artificial problem, of interest primarily for an internal critique of neoclassical economics. In actuality, firms do not have a choice at any time of processes from a pre-existing menu. Rather technology evolves as a non-reversible process in historical time.

5.0 Volume III Invariants Cannot All Hold

In the above, I have been concentrating mostly on objections to the premises of Marx's economic theory. Let me consider a conclusion. According to Marx, accounting in labor values allows one to identify certain invariants that hold for the economy as a whole. For example, the sum of labor values for gross outputs of industry is equal to the sum of gross outputs, evaluated at prices of production. And the sum of surplus value across industry is equal to the sum of profits. According to Marx, the competition under which prices of production are formed redistributes total surplus values into aliquot quantities distributed to each industry.

Under the traditional analyses of prices of production, Marx was just wrong. For an arbitrary numéraire, not all invariants can simultaneously hold.

Four answers have been given to this issue. I do not think highly of traditional Marxists who argue that one or the other invariant should be given preference. Typically, such arguments are presented with a lot of Hegelian terminology. I find intriguing the argument that all invariants can hold if one adopts Sraffa's standard commodity as the numéraire. Duncan Foley and Gerard Duménil have proposed the new interpretation, organized around the concept of the Monetary Expression of Labor Value (MELT). As I understand it, the new interpretation makes Marx's claims too much a matter of an accounting tautology for my taste. Finally, there is the Temporal Single System Interpretation (TSSI), which I associate mainly with Alan Freeman and Andrew Kliman, although, I guess, they work with many more scholars. Of course, more invariants can be made to hold if you interpret the theory to have many more degrees of freedom.

6.0 Exploitation of Corn

A theorem in the analysis of prices of production states that the rate of profit is positive if and only if labor is exploited. Exploitation here has a technical definition; it is not an ethical concept. From John Roemer, I learn that one can argue that Marx had both ideas in mind.

Anyways, from the same analysis, one can show that same theorem holds for any commodity (that is basic or in the workers' consumption basket?). So why focus on labor? Answers have been given that deal with matters not in the math at this level of abstraction. Workers, unlike owners of commodities sold as means of production, must be brought under the direction of the capitalists when they hire them. Furthermore, the agreements laborers strike are, at best, incomplete contracts. Not all activities that the workers will be expected to perform in given situations can be prespecified. Furthermore, often some will be unpleasant, and a tug-of-war can arise between the worker and the capitalist's representative in the workplace.

Whatever you think of these rationales for focusing on the exploitation of labor, the issue of working conditions seems like a perennial concern.

7.0 Falling Rate of Profit

I do not have much to say about the theory of the falling rate of profit. I think Marx was mistaken here, but recall this is a volume 3 theory, never published in Marx's lifetime. I am aware of Marx's account of countervailing tendencies. (How is this a theory, if no explanation is given why one tendency should predominate?) And, as usual, theorists in the TSSI tradition disagree.

9.0 Outside the Theory of Value and Distribution

Such a brief overview, compared to the thousands of pages Marx wrote, and the many ways scholars and followers have read (parts of?) this work, obviously cannot cover all issues. I have said nothing about historical materialism, for instance. If this theory is read as mandating economic determinism, with no possibility of the superstructure shaping the evolution of the economic base, I, like many others, think the theory is wrong.

Nor have I said anything much about many of Marx's analyses that can be developed independently of the theory of value and distribution. For example, I like to set out Volume 2 models of simple and expanded reproduction in terms of prices of production. Whether or not Richard Goodwin's theory of the business cycle is Marxist or is descriptive of some capitalist economies at some time seems to be independent of Marx's theory of value. And Marx had many other analyses of concrete situations that might or not be worthwhile. For example, in Volume 1, he presented the introduction in Great Britain of laws regulating maximum hours of work as addressing what we would now call a prisoner's dilemma. Each mill owner would like to work their employees until their health breaks, fire them, and then hire refreshed workers. But if all mill owners are doing this for wokers from a young age, no large population of such refreshed workers will exist in the locality. So the owners need such laws after a certain level of development.

I suppose I should say something about the theory of monopoly. I do not see why prices of production cannot be developed with different markups in different industries. I may not be familiar enough with the literature, but it is my impression that many accounts of markup pricing do not take into account constraints arising from the inter-industry flows emphasized in Sraffian theory and empirical work in Leontief input-output analysis. Furthermore, markups cannot be so high in a viable economy that demands total more than the net output of a viable economy. (A theory of cost-push inflation can arise here.) This is not to say that I do not think those exploring administered, full-cost, or markup pricing are not looking at something empirically important.

And Marx had many detailed empirical observations, including claims about how feudalism evolved into capitalism. I cannot address such matters of history. Finally, I have said nothing above about the sociology of economics. I think the above is quite enough for one post.

Saturday, March 21, 2015

On Mainstream Economists' Ignorance Of Real Analysis

"Logic sometimes makes monsters. Since half a century we have seen a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose. No longer continuity, or perhaps continuity, but no derivatives, etc. Nay more, from the logical point of view, it is these strange functions which are the most general, those one meets without seeking no longer appear except as particular cases. There remains for them only a small corner.

Heretofore when a new function was invented, it was for some practical end; to-day they are invented expressly to put at fault the reasonings of our fathers, and one never will get from them anything more than that." -- Henri Poincaré (1908, as quoted in Lakatos 1976, pp. 22-23).

Mainstream economists these days seem unwilling to accept claims about economics that are not backed up by mathematical models. (I think that views on mathematical formalism are pluralistic among non-mainstream economists. Mathematical models are just one of several approaches to acceptable claims about economics, and some non-mainstream economists are quite good at producing mathematical models.) Generally speaking, mainstream economists seem to me to reject norms common among mathematicians.

Anybody taking a standard undergraduate sequence in mathematics at a reasonably good university has an opportunity to be introduced to real analysis. Often, such a class is where the mathematician is introduced to a certain style of definitions and proofs, particularly epsilon-delta proofs. Besides this style, these classes teach a certain content, that is, the theory of limits, the differential calculus, and the integral calculus, from a rigorous standpoint. (I also draw on measure theory below, which, for me, was not taught at the undergraduate level.) In such a class, one should see various examples and purported counter-examples. The examples help the student to understand the range of behavior consistent with certain axioms. The supposed counter-examples help the student understand why theorems contain certain assumptions and why certain concepts useful for stating these assumptions were introduced into mathematics. Given an example inconsistent with the conclusion of a theorem, the student should identify a clause in the assumptions of the theorem that rules out the example.

To make my point, I'll list some examples. For my amusement, I'm not (initially) looking up anything for this post. Just as when someone criticizes somebody else's grammar, the probability approaches unity that they will make a typographic error, so I'll almost certainly be mistaken somewhere below. Does anybody have suggestions for additions to the following list of examples from real analysis?

  1. Define a function that is discontinuous at some point.
  2. Define a function that is continuous everywhere, but differentiable nowhere.
  3. Define a sequence of functions that converges pointwise, but is not uniformly convergent. (Or is it the other way 'round?)
  4. Define a function that is Lebesque integrable, but not Riemann integrable.
  5. Provide an example of a non-(Lebesque) measurable set.

The style of reasoning introduced in courses on real analysis has been important in economics since, at least, Debreu (1959). And economics provides many examples analogous to the answers to the above problems. Lexicographic preferences can provide an example of a complete order on a commodity space - that is, rational preferences - that cannot be represented by an utility function. Such preferences highlight the need for an assumption on the continuity of preferences, given that the commodity space is a continuum; "rationality" is not sufficient. Menu-dependent preferences suggest the possibility of specifying deeper structures that do and do not allow the construction of binary preference relation providing an order for a commodity space. I suppose the concept of hemi-continuity is proof generated in economics.

Sraffians have also provided many examples not consistent with outdated mainstream teaching. Ian Steedman's work, over the last quarter century, is particularly good on examples illustrating that the Cambridge critique is not exhausted by the possibilities highlighted by reswitching and capital-reversing. As of yet, economists have not specified any general assumption on production processes that rules out these sort of Sraffian examples and yields neoclassical conclusions. Yet many economists - who, I guess, treat their training in mathematics as a hazing ceremony for induction into the brotherhood of economists - proceed as if they have some such theorem.

Obviously, despite my generalization, some economists, both mainstream and non-mainstream understand and accept mathematical analysis. Maybe more mainstream economists understand than my generalization would suggest. The refusal I have seen of economists to accept their own logic may be the manifestation of anti-intellectualism and boundary-patrolling that I think is so common among properly socialized economists. The general public must not come to understand how vulnerable the conclusions of mainstream economists are to slight perturbations in model assumptions. Demonstrations of the failure of the logic in the teaching and public pronouncements of economists must be distracted in blather about credentials or (false?) irrelevancies about empirical results. What economists say in public and what they say in professional seminars need not be consistent. (This is not quite the right link from Dani Rodrik making his point.) I can easily be led to believe that explanation for some behavior I have seen is more a matter of the sociology of economics and less a lack of understanding of mathematics. So, in general, are economists still exhibiting a century-outdated attitude to mathematics?

Answers
  1. This is an easy question. For amusement, I'll name a function that exhibits a discontinuity of the second kind, if I correctly remember the terminology. Consider the limit of the following function of the reals as x approaches zero: f(x) = sin(1/x), if x ≠ 0; 0, if x = 0.
  2. Various space filling curves provide examples. I think both Hilbert and Sierpinski provide examples.
  3. I'm vague on this one, but consider the Fourier series for a square wave, where the value of the square wave at points of discontinuity is the midpoint of the left-hand and right-hand limits. I think mathematicians greeted Fourier's work on functions that were only piecewise continuous with some degree of incredulity.
  4. f(x) = 0, for x rational; 1 for x irrational.
  5. Consider a decomposition of the real numbers between zero and unity, inclusive, into equivalence classes. For this example, two real numbers in the range are considered equivalent if the difference between them, modulo one, more or less, is a rational number. The axiom of choice allows one to select a real number in each equivalence class. Take the union, with the index set for the union formed by the choice from each equivalence class. The index set contains an infinite number of elements, and the union is the desired closed interval. Furthermore, each equivalence class can be put into a one-to-one correspondence with any other equivalence class. Thus, the measure of each equivalence class must be the same. And these measures must add up to one, since that is the Lebesque measure of the closed interval. But assigning a measure of zero to each equivalence class will not do, and the sum over equivalence claess for any finite measure would be positive infinity. So any equivalence class formed in this way in non-measurable.
References
  • Gerard Debreu (1959). Theory of Value: An Axiomatic Analysis of Economic Equilibrium. John Wiley & Sons.
  • Imre Lakatos (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.
  • Walter Rudin (1976). Principles of Mathematical Analysis, Third edition. McGraw-Hill.

Thursday, March 12, 2015

Purge of Heterodox Economists Underway at Manitoba?

I stumbled across an article published yesterday in "The students' newspaper of the University of Manitoba". Apparently, the Canadian Association of University Teachers (CAUT) published a report, Report of the Ad Hoc Investigatory Committee into the Department of Economics at the University of Manitoba. They are concerned with the violation, in the economics department, of the academic freedom of professors of economics.

Monday, March 09, 2015

Newton Method, Re-Iterated

Figure 1: Cube Roots Of Unity, Rotated, Newton's Method

I have been re-visiting my program for drawing fractals with Newton's method. Newton's method is an iterative method for finding the roots of non-linear systems of equations. That is, it is used to find zeros of functions. For my purposes, Newton's method can be used to draw fractals, although I was pleased to learn a bit more about methods in numerical analysis. I made various improvements to my program, including the the implementation of:

  • More polynomial functions whose zeros are desired.
  • Rotations and reflections.
  • Two additional iterative methods for root finding.

I was pleased that I had thought to define a Java interface for functions whose zeros were sought. (When one looks at one's own code from a couple years ago, one might as well as be looking at code by somebody else.) Each new function could be added by defining a class implementing this interface. Besides specific functions, I defined a general polynomial, with complex coefficients, that maps complex numbers into complex numbers. I defined rotations and reflections by the transformations to the zeros of this general polynomial. A different strategy would need to be specified if one wanted to create a program for drawing fractals for functions that are not limited to being polynomials.

Halley's method is derived from a second-order Taylor approximation. (Newton's method is derived from a first order approximation.) As nearly, as I can see, Halley's method does not produce as interesting fractals. In implementing the method, I had to review a bit about tensors, since the second derivative of a function mapping the real plane into the real plane is a tensor.

Figure 2: Cube Roots Of Unity, Rotated, Halley's Method

I do not have much of an understanding of the rationale for the Chun-Neta method. I can see that it takes less iterations than Newton or Halley's method, although more calculations per iteration than either of those two methods. (The visual result of less iterations is a lighter color around the roots in the image below, as compared with above.) As I understand it, the black lines in the figure are an artifact of my implementation, probably resulting from dividing by zero.

Figure 3: Cube Roots Of Unity, Rotated, Chun-Neta Method

I conclude with an example from a general polynomial, where I defined roots so that the resulting figures would have no obvious symmetries.

Figure 4: A Fourth Degree Polynomial, Halley's Method
Figure 5: A Fourth Degree Polynomial, Chun-Neta Method
References
  • Chun, C. and B. Neta (2011). A new sixth-order scheme for nonlinear equations. Applied Mathematics Letters.
  • Scott, Melvin, B. Neta, and C. Chun (2011). Basin attractors for various methods. Applied Mathematics and Computation, V. 218: pp. 2584-2599.
  • Yau, Lily and A. Ben-Israel (1998). The Newton and Halley methods for complex roots. American Mathematical Monthly, V. 105: pp. 806-818

Friday, February 27, 2015

Bad Math In Good Math

1.0 Introduction and Overview of the Book

Mark C. Chu-Carroll's blog is Good Math, Bad Math. His book is Good Math: A Geek's Guide to the Beauty of Numbers, Logic, and Computation.

A teenager recently asked me about what math he should learn if he wanted to become a computer programmer or game developer. One cannot recommend a textbook (on discrete mathematics?) to answer this, I think. If you do not mind the errors, this popular presentation will do. I like how it presents the building up of all kinds of numbers from set theory. And the order of this presentation seems right, starting with the natural numbers, but then later providing a set theoretic construction in which the Peano axioms were derived. (I suppose Chu-Carroll could also present a complementary explanation of the need for more kinds of numbers by starting out with the problem of finding roots for polynomial equations in which all coefficients are natural numbers. Eventually, you would get to the claim that an nth degree polynomial with coefficients in the complex numbers has n zeros (some possibly repeating) in the complex numbers.)

The book also has an introduction to the theory of computation, with descriptions of Finite State Machines, lambda calculus, and Turing machines. There is an outline of how the universal Turing machine cannot be improved, in terms of what functions can be computed. It doesn't help to add a second or more tapes. Nor does it help to add a two-dimensional tape. The book concludes with a presentation of a function that cannot be computed by a universal Turing machine. The halting problem, as is canonical, is used for an illustration.

2.0 Bad Math Not In Good Math

Besides being interested in popular presentations of mathematics, I was interested in seeing a book developed from blog posts. Chu-Carroll wisely leaves out a large component of his blog, namely the mocking of silly presentations of bad math. I could not do that with this blog. But there is a contrast here. The bad economics I attempt to counter is presented by supposed leaders of the field and heads of supposed good departments. The bad math Chu-Carroll usually writes about is not being to used to make the world a worse place, to obfuscate and confuse the public, to disguise critical aspects of our society. Rather, it is generally presented by people with less influence than Chu-Carroll or academic mathematicians.

2.1 Not a Proof

Anyways, I want to express some sympathy for why some might find some propositions in mathematics hard to accept. I do not want to argue such nonsense as the idea that Cantor's diagonalization argument fails, by conventional mathematical standards; that different size infinities do not exist; or that 0.999... does not equal 1. Anyways, consider the following purported proof of a theorem.

Theorem:

Proof: Define S by the following:

Then a S is:

Subtract a S from S:

Or:

Thus:

The above was what was to be shown.

Corollary: 0.999... = 1

Proof: First note the following:

Some simple manipulations allow one to apply the theorem:

Or:

That is:

2.2 Comments on the Non-Proof and a Valid Proof

I happen to think of the above supposed proof as a heuristic than I know yields the right answer, sort of. A student, when first presented with the above by an authority, say, in high school, might be inclined to accept it. It seems like symbols are being manipulated in conventional ways.

I do not know that I expect a student to notice how various questions are begged above. What does it mean to take an infinite sum? To multiply an infinite sum by a constant? To take the difference between two infinite sums? To define an infinitely repeating decimal number? But suppose one does ask these questions, questions whose answers are presupposed by the proof. And suppose one is vaguely aware of non-standard analysis. Besides how does inequality in the statement of the theorem arise? One might think the wool is being pulled over one's eyes.

How could one prove that 0.999... = 1? First, one might prove the following by mathematical induction:

Then, after defining what it means to take a limit, one could derive the previously given formula for the infinite geometric series as a limit of the finite sum. (Notice that the restriction in the theorem follows from the proof.) Finally, the claim follows, as a corollary, as shown above.

3.0 Errata and Suggestions

I think that this is the most useful part of this post for Chu-Carroll, especially if this book goes through additional printings or editions.

  • p. 7, last line: "(n + 1)(n + 2)/n" should be "(n + 1)(n + 2)/2"
  • p. 11, 7 lines from bottom: "our model" should be "our axioms".
  • p. 19: Associativity not listed in field axioms.
  • p. 20: Since the rational numbers are a field, continuity is not part of the axioms defining a field.
  • Sections 2.2 and 3.3: Does the exposition of these constructions already presume the existence of integers and real numbers, respectively?
  • p. 21: Shouldn't the definition of a cut be (ignoring that this definition already assumes the existence of the real number r) something like (A, B) where:
A = {x | x rational and xr}
B = {x | x rational and x > r}
  • p. 84, footnote: If one is going to note that exclusive or can be defined in terms of other operations, why not note that one of and or or can be defined in terms of the other and not? Same comment applies to if ... then.
  • p. 85, last 2 lines: the line break is confusing.
  • p. 95, proof by contradiction of the law of the excluded middle: Is this circular reasoning? Maybe thinking of the proof as being in a meta-language saves this, but maybe this is not the best example.
  • p. 97, step 1: Unmatched left parenthesis.
  • p. 106: Definition of parent is not provided, but is referenced in the text.
  • p. 114, base case: Maybe this should be "partition([], [], [], []).
  • p. 130: In definitions of union, intersection, and Cartesian product, logical equivalence is misprinted as some weird character. This misprinting seems to be the case throughout the book (e.g., see pp. 140, 141, and 157).
  • p. 133 equation: Right arrow misprinted as ">>".
  • Chapter 17: Has anybody proved ZFC consistent? I thought it was the merely the case that nobody has found an inconsistency or can see how one would come about.
  • p. 148: Might mention that the order being considered in the well-ordering principle is NOT necessarily the usual, intuitive order.
  • p. 148: Drop "larger" in the sentence ending as "...there's a single, unique value that is the smallest positive real number larger!"
  • p. 163" "powerset" should be "power set".
  • p. 164, line 6: "our choice on the continuum as an axiom" is awkward. How about, "our choice about the continuum hypothesis as an axiom"?
  • p. 168, Table 3: g + d = e should be g + d = g.
  • p. 171-172: Maybe list mirror symmetry or write, "in addition to mirror symmetry".
  • Part VI: Can we have something on the Chomsky hierarchy?
  • p. 185; p. 186, Figure 15; p. 193): Labeling state A as a final state is inconsistent with the wording on p. 185, but not the wording on p. 193. On p. 185, write "...that consist of any string containing at least one a, followed by any number of bs."
  • p. 190: Would not Da(ab*) be b*, not ab*?
  • p. 223: "second currying example" should be "currying example". No previous example has been presented.
  • p. 225, towards bottom of page: I do not understand why α does not appear in formal definition of β.
  • p. 229: Suggestion: Refer back to recursion in Section 14.2 or to chapter 18.
  • p. 244, 5 lines from bottom: Probably γ should not be used here, since γ was just defined to represent Strings, not a generic type. Same comment goes for α.
  • p. 245, last bullet: It seems here δ is being used for the boolean type. On the previous page, β was promised to be used for booleans, as in the first step of the example on the bottom of p. 247.
  • p. 249 (Not an error): The reader is supposed to understand what "Intuitionistic logic" means, with no more background than that?
  • p. 257: Are the last line of the second paragraph and the last line of the page consistent in syntax?
  • Can we have an index?