Bad Math In Good Math

1.0 Introduction and Overview of the BookMark 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 MathBesides 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 ProofAnyways, 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 ProofI 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 SuggestionsI 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 x ≤ r}

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 ofand 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 D_a(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?

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

The Supreme Court and the Democrats’ Preemptive Surrender on Obamacare

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  By Dean Baker, Truthout | Op-Ed

  As the Supreme Court listened to arguments over subsidies in the state exchanges Democrats were making their plans for preemptive surrender. Many were warning that an adverse ruling would be the death of Obamacare. There is no doubt that a ruling for the plaintiffs would be bad news. It would deny millions of people subsidies in the states without their own exchanges.

  Read more…

The Economics of Media Bias

Timothy Taylor
conversableeconomist@gmail.com

Here are four basic questions about media bias:

“First, is media news reporting actually slanted? …

Second, if reporting is biased, what is the reason? Is such bias driven by the
supply-side, as when reporting reflects the prejudices of an outlet’s owners or journalists? …

Third, what is the effect of media competition on accuracy and bias?  …

Finally, does media reporting actually matter for individual understanding and
action? Does it affect knowledge? Does it influence participation in the political
process? Does it influence how people vote?”

The questions are posed by Andrei Shleifer in his paper on “Matthew Gentzkow, Winner of the 2014
Clark Medal,” in the Winter 2015 issue of the Journal of Economic Perspectives. As background, the Clark medal is given by the American Economic Association each year “to that American economist under the age of forty who is judged to have made the most significant contribution to economic thought and knowledge.” Shleifer is describing the academic work for which Gentzkow won the award last year. Shleifer argues: “In a very short decade, economic research has obtained fairly clear answers to at least some of these questions.”

(Full disclosure: My paid job has been Managing Editor of the JEP since the first issue in 1987. All papers in JEP from the first issue to the most recent are freely available on-line courtesy of the American Economic Association. Shleifer was Editor of JEP, and thus my boss, from 2003-2008.)

On the first question of the existence of media bias, how does one go beyond anecdotes about how different newspapers or TV channels covered certain stories to come up with a defensible quantitative way of detecting media bias? The modern approach has been to use text analysis. For example, have a computer search a dataset of all speeches given in Congress during the year 2005. Have the computer search for phrases that are much more commonly used by Republicans or by Democrats. For example, in 2005 Democrats were much more likely to refer to the “war in Iraq” while Republicans were more likely to refer to the “war on terror.” Now do a search on the text used by media outlets, and see if they are more likely to be using Republican phrases, Democratic phrases, or an even mixture of the two.

Gentzkow didn’t invent this approach to meausuring media bias. For earlier work on the subject in the research literature, a starting point would be the article by Tim Groseclose and Jeffrey Milyo, “A Measure of Media Bias,” in theQuarterly Journal of Economics in 2005 (120:4, pp. 1191–1237). But in work with co-author Jesse Shapiro, Gentzkow applied the approach to newspapers across the US and was thus able to provide hard evidence that many newspapers indeed exhibit partisan bias in how they report the news.

Does the bias of newspapers reflect their owners, or their customers? Here’s how Shleifer describes it:

“Gentzkow and Shapiro then collected data on the use of these highly diagnostic phrases in US daily newspapers and used these data to place news outlets on the ideological spectrum comparable to members of Congress. In addition to this large methodological advance in how to measure partisan newspaper slant, the paper used detailed information on newspaper circulation and voting patterns across space to estimate a model of the demand for slant and to show that—consistent with the theory—consumers gravitate to like-minded sources, giving the newspapers an incentive to tailor their content to their readers. They also show that newspapers respond to that incentive and that variation in reader ideology explains a large portion of the variation in slant across US daily newspapers. … [A]fter controlling for a newspaper’s audience, the identity of its owner does not affect its slant. Two newspapers with the same owner look no more similar in their slant than newspapers with different owners. Ownership regulation in the US and elsewhere is based on the premise a news outlet’s owner determines how it spins the news. Gentzkow and Shapiro produced the first large-scale test of this hypothesis, which showed that, contrary to the conventional wisdom and regulatory stance, demand is much more influential in shaping content than supply as proxied by ownership.

Does more competition in the media tend to increase or diminish this bias? This question is tough  to answer, but in a different paper by Gentzkow and Shapiro, they look at a closely related topic of how people of different political beliefs use the Internet. Specifically, do people tend to cluster at the websites that that match their ideology, or do they surf around? Shleifer describes the result:

One might worry that the increase in choice among news suppliers as a result of the Internet would allow news consumers to self-segregate, reading only news that confirms their preconceptions. Gentzkow and Shapiro test this claim using data from a panel of Internet users for which they have a survey-based measure of political ideology and tracking data on online news consumption. They find that ideological segregation is surprisingly low online. The average conservative’s news outlet on the Internet is about as conservative as usatoday.com; the average liberal’s is as liberal as cnn.com. Strikingly, the Internet is less ideologically segregated than US residential geography: two people using the same news website are less likely to have an ideology in common than two people living in the same zip code.

Finally, is it the case that people just choose the media outlets that reflect their bias, in which case the media bias doesn’t affect their opinions or their voting patterns? Or is there reason to believe that the extent of media bias does affect opinions and voting patterns?

In one study, Gentzkow looked at historical data on how television coverage spread across the United States, and what changes in voting patterns followed. As Shleifer writes; “He estimates a huge negative effect: the availability of television accounts for between one-quarter and one-half of the total decline in voter turnout since the 1950s. Matt argues that a principal reason for this is substitution in media consumption away from newspapers, which provide more political
coverage and thus stimulate more interest in voting.”

In a different study, Gentzkow and co-authors look at the patterns of newspapers being born and dying from 1869 to 2004, and compare this with voting patterns. Shleifer writes:

They find that newspapers have a large effect in raising voter turnout, especially in the period before the introduction of broadcast media. However, the political affiliation of entering newspapers does not affect the partisan composition of an area’s vote. The latter result contrasts with another important finding, by DellaVigna and Kaplan (2007), that the entry of Fox News does sway some voters toward voting Republican. An interpretation consistent with these findings is that newspapers motivate but don’t persuade, while television does the opposite.

Research on media bias and its political effects is certainly not settled, but for what it’s worth, I’d sum up the existing evidence in this way. There’s lots of political bias in the media, mainly because media outlets are trying to attract customers with similar bias. But in the world of the Internet, at least, people of all beliefs do surf readily between news websites with different kind of bias. The growth of television to some extent displaced the role of newspapers and lowered the extent of voting. For the future, a central question is whether a population that gets its news from a mixture of websites and social media becomes better-informed or more willing to vote, or whether it becomes a population that instead becomes expert at selfiesm, cat videos, World of Goo, Candy Crush, Angry Birds, and the celebrity-du-jour.

Why Aren’t Americans Feeling the United States’ Economy’s Improvements?

By Robert Reich

Shutterstock

This post originally ran on Robert Reich’s Web page.

The U.S. economy is picking up steam but most Americans aren’t feeling it. By contrast, most European economies are still in bad shape, but most Europeans are doing relatively well.

What’s behind this? Two big facts.

First, American corporations exert far more political influence in the United States than their counterparts exert in their own countries.

In fact, most Americans have no influence at all. That’s the conclusion of Professors Martin Gilens of Princeton and Benjamin Page of Northwestern University, whoanalyzed 1,799 policy issues — and found that “the preferences of the average American appear to have only a miniscule, near-zero, statistically non-significant impact upon public policy.”Instead, American lawmakers respond to the demands of wealthy individuals (typically corporate executives and Wall Street moguls) and of big corporations – those with the most lobbying prowess and deepest pockets to bankroll campaigns.

The second fact is most big American corporations have no particular allegiance to America. They don’t want Americans to have better wages. Their only allegiance and responsibility to their shareholders — which often requires lower wages to fuel larger profits and higher share prices.

When GM went public again in 2010, it boasted of making 43 percent of its cars in place where labor is less than $15 an hour, while in North America it could now pay “lower-tiered” wages and benefits for new employees.

American corporations shift their profits around the world wherever they pay the lowest taxes. Some are even morphing into foreign corporations.

As an Apple executive told The New York Times, “We don’t have an obligation to solve America’s problems.”

I’m not blaming American corporations. They’re in business to make profits and maximize their share prices, not to serve America.

But because of these two basic facts – their dominance on American politics, and their interest in share prices instead of the wellbeing of Americans – it’s folly to count on them to create good American jobs or improve American competitiveness, or represent the interests of the United States in global commerce.

By contrast, big corporations headquartered in other rich nations are more responsible for the wellbeing of the people who live in those nations.

That’s because labor unions there are typically stronger than they are here — able to exert pressure both at the company level and nationally.

VW’s labor unions, for example, have a voice in governing the company, as they do in other big German corporations. Not long ago, VW even welcomed the UAW to its auto plant in Chattanooga, Tennessee. (Tennessee’s own politicians nixed it.)

Governments in other rich nations often devise laws through tri-partite bargains involving big corporations and organized labor. This process further binds their corporations to their nations.

Meanwhile, American corporations distribute a smaller share of their earnings to their workers than do European or Canadian-based corporations.

And top U.S. corporate executives make far more money than their counterparts in other wealthy countries.

The typical American worker puts in more hours than Canadians and Europeans, and gets little or no paid vacation or paid family leave. In Europe, the norm is five weeks paid vacation per year and more than three months paid family leave.

And because of the overwhelming clout of American firms on U.S. politics, Americans don’t get nearly as good a deal from their governments as do Canadians and Europeans.

Governments there impose higher taxes on the wealthy and redistribute more of it to middle and lower income households. Most of their citizens receive essentially free health care and more generous unemployment benefits than do Americans.

So it shouldn’t be surprising that even though U.S. economy is doing better, most Americans are not.

The U.S. middle class is no longer the world’s richest. After considering taxes and transfer payments, middle-class incomes in Canada and much of Western Europe arehigher than in U.S. The poor in Western Europe earn more than do poor Americans.

Finally, when at global negotiating tables – such as the secretive process devising the “Trans Pacific Partnership” trade deal — American corporations don’t represent the interests of Americans. They represent the interests of their executives and shareholders, who are not only wealthier than most Americans but also reside all over the world.

Which is why the pending Partnership protects the intellectual property of American corporations — but not American workers’ health, safety, or wages, and not the environment.

The Obama administration is casting the Partnership as way to contain Chinese influence in the Pacific region. The agents of America’s interests in the area are assumed to be American corporations.

But that assumption is incorrect. American corporations aren’t set up to represent America’s interests in the Pacific region or anywhere else.

What’s the answer to this basic conundrum? Either we lessen the dominance of big American corporations over American politics. Or we increase their allegiance and responsibility to America.

It has to be one or the other. Americans can’t thrive within a political system run largely by big American corporations — organized to boost their share prices but not boost America.