Sergei Yakovenko's blog: on Math and Teaching

Monday, November 25, 2013

Lecture notes for the first part of the course

Here is the syllabus of the first part of the course, together with the updated lecture notes.

Lecture 1 (2013-Analysis-01)

  • First encounters with infinity
  • Formal language of mathematics: sets, operations, functions, logical constructions, quantifiers…
  • Infinite operations (union, intersection, …)
  • Enumeration. Comparison of sets.
  • Sequences: stabilization, accumulation.

Lecture 2 (2013-Analysis-02)

  • Numbers: how we discover them and how we “build” them.
  • Arithmetic operations. Equations.
  • Adjunction missing solutions of equations that we want to be solvable.
  • Rational numbers: the structure of a field.
  • Why can’t one divide by zero and what is the price to pay if we allow such division
  • Irrational algebraic numbers: how to operate with them

Lecture 3 (2013-Analysis-03)

  • Order on rational numbers.
  • Density.
  • Holes between rational numbers
  • Infinite systems of two-sided inequalities
  • Dedekind cuts and operations on them.
  • Completeness of the new “number system” \mathbb R
  • Uncountability of the set of real numbers \mathbb R.

See you tomorrow to talk about convergence and limits.


Sunday, November 17, 2013

Applied Mathematics: How-Not-To

Filed under: opinion — Sergei Yakovenko @ 2:04
Tags: , ,

The November 2013 issue of the Notices of the American Mathematical Society, the journal which purports to appeal to the most broad community of mathematicians of all walks and denominations, published a paper by T. L. Saaty and H. J. Zoffer, “Principles for Implementing a Potential Solution to the Middle East Conflict”. This publication attracted opinion of many readers of the Notices, who expressed their concern about politicizing the journal and lending its pages for a biased and controversial text.

Addressing these concerns, the President of AMS, Prof. David Vogan, responded that in his view there is nothing wrong with publication of papers which arise public interest, even if their conclusion is questionable.

“It is my belief that the Notices should publish articles describing reasonable mathematical modeling of interesting problems, even if the conclusions reached are controversial or disagreeable. The real world is a complicated, controversial, and sometimes unpleasant place.”

Let me express my strongest disagreement. I believe that, rather than being a reasonable mathematical modeling of interesting problems, the paper in question is a travesty of applied mathematics.

I will briefly recall the main points of the authors’ model for achieving an agreed solution to the conflict. They believe that both sides possess a number of “bargaining chips”, concessions that they can make towards the other side, but the value of these bargaining chips is different for each side (a symbolic gesture with zero cost for one side may be of immense practical value for the other and vice versa). The authors see the main problem of negotiations in the fact that possible “bundles of concessions” which can be traded for each other, are too numerous and too difficult to the sides to determine themselves, and hence a highbrow science should be called for the rescue.

The area of conflict resolution is not new: the Game Theory is one of the best developed fields between Applied Mathematics, Mathematical Economics and Operations Research. Its success in developing a system of key notions changed the way mathematicians and strategists think about competition and cooperation; several Nobel prizes in Economics were awarded for works in this area. However, the authors of the discussed paper ignore some basic principles which are necessary to make such analysis valid.

First, mathematical modeling is all about eliminating ambiguity of notions and terms or at least reducing it to the minimum. In this sense the list of possible “concessions” (Table 1a) is surprisingly obfuscating. Its terms are at best vague, and at worst intentionally misleading.

What exactly concede Israelis by “abandoning the idea of a Jewish state”? Is it the same what the Palestinians expect when reading these words? Attorneys could fill tens of pages by the legal fine print, defining the precise meaning of this “concession”, and achieving an agreement on terms of reference may itself be a subject of protracted negotiations. Simply throwing it “as is” would only plant the seeds of contention at the stage of implementation (if ever reached).

Another “concession”, this time on the Palestinian side, is to “acknowledge Israel’s existence as an independent state”. Well, you can hardly concede something which is beyond your control, exactly like approving the next day’s sunrise cannot be considered as a “concession”, and you can’t expect much in reward for such a “concession”. In a similar way, another Israel’s possible “concession” is to “comply with all applicable UN resolutions”. These resolutions themselves were subject to differing interpretations, most famous is the absent definite article “the” in the Resolution 242, requiring Israel to withdraw from territories (not “the territories”) acquired in the defensive 1967 war. Formally Israel already accepted “all applicable UN resolutions” to the extent the other side fulfills its obligations. What remains to concede then? The term “Eastern Jerusalem”, appearing several times in “concessions” of both sides, is also ill-defined: is the Jewish Quarter of the Old City and the Western Wall part of the Eastern Jerusalem to be “conceded”?

Some “concessions” are actually troubling: “Evacuate settlers of Jewish settlements claimed by the Palestinians with or without compensation” is not a “concession”, but in today’s world is considered as an ethnic cleansing in everything but the name and can well bring the “conceding side” to the International Court (in the past things looked differently). Can one seriously contemplate such “concession”? If yes, then why should it be one-sided only?

All these remarks actually mean that it is not the definite options that are offered for mutual concessions, but rather empty slogans open to different interpretations. Politicians are well familiar with such things, but they are not the kind of objects admitting mathematical analysis.

Second, even stronger objection is that the model has a built-in flaw. It treats the two sides of the conflict as if they were two centrally commanded warring parties or two economic agents with well defined preferences which could be theoretically revealed by resorting to experts’ opinion. Nothing can be more distant from reality. Each of the two sides is a multi-million people, very heterogeneous. Any idea that five or six self-appointed experts can give a reliable assessment of the gain/cost of “concessions”, is ridiculous: Judea and Samaria dwellers clearly have a different attitude towards the “concession” of getting expelled from their houses than the Tel-Aviv residents, and inhabitants of Palestinian refugee camps in Lebanon most certainly have the set of priorities strongly different from the established Ramallah bureaucracy. In the democratic society the process of revealing preferences of various groups of population is implemented via elections. Instead of the anonymous experts deciding what can and what can’t be disposed off, the model should analyze the platforms of various political parties and aggregate them with the weights obtained at the ballot box. This approach will be very difficult even under the assumption that elections are honest and held in time, which is not the case with one of the sides in the conflict.

These two objections are already sufficient to discard not just the conclusions, but the model itself (or, to be more precise, the current attempt to apply the AHP as the authors call their approach to this particular conflict resolution). However, several minor instances raise the question about the possible bias of the authors. Attempting to resolve a conflict, one has at least to pretend to be impartial to its sides. Looking at the illustrations to the paper and the accompanying captions, one can easily see that even the vocabulary used by the authors (settlers vs. Palestinians, – an example of dehumanization, “occupied” vs. “disputed” lands, in disagreement with the principal UN resolutions) clearly follows the narrative of one side rather than uses a neutral language. It is far from obvious whether arbiters with such credentials would pass for jurors in a court.

The world in which we live may indeed be nasty and unpleasant. Still, certain things seem to be impossible to spell out even as a conjectural solution. If a realistic model (which could easily be built on much more solid foundations) predicts that the cheapest way to deal with the problem of the Earth overpopulation is to let starve to death 80% of the Third World population, would Prof. Vogan still consider analyzing such a model a legitimate discourse in addressing this (no doubt, much more pressing) problem? will he make a “concession” and agree to publish a paper with such proposed conclusion in the Notices of AMS?

Last but not least. Prof. Vogan sees no harm done by discussing a controversial mathematical model as soon as the proposed solution is not implemented.

I do.

Today mathematics is still considered as one of the last bastions of objective knowledge, and mathematically justified conclusions are generally accepted by the public opinion as a reliable truth modulo a possible human error, unlike many other areas, e.g., climate science, political science etc. The imprimatur given by publication in the respectable and peer-reviewed mathematical journal will carry with itself two very sad consequences. First, the parties at the real negotiation tables will be pushed in the wrong direction: indeed, how can you argue against “the optimal solution obtained by rigorous mathematical methods”? But such misconception may and most likely will distort realistic expectations of the sides and result in more protracted (or even failed) negotiations. The other implication, more sad for our professional community, will be an unavoidable conclusion that mathematics can be twisted to the whim of politically driven people, and mathematicians are no more honest in their work than scientists massaging or even fabricating their experimental data or sociologists who derive pre-designed conclusions from biased polls. Of course, mathematicians are also humans and may hold very polar views on various subjects, however, any attempt to justify their views by invoking the authority of the sterling mathematical models is plain wrong and harmful.

Post Scriptum. Several colleagues asked me why I don’t send this text to “Notices” as a response to the paper of Saaty and Zoffer. The short answer is: it would be inconsequential. One of the main points of the text is absolute inadmissibility of politicizing math and sciences. Sending this unscientific protest to the professional journal would be in itself a step towards further politicizing the atmosphere.

However, if any journal or blog would like to copy this text or disseminate its content in any form, there are no restrictions to it.

Wednesday, November 13, 2013

Homework assignment 2

Filed under: Uncategorized — innaento @ 10:42

Homework 2 can be found here:


It is due on November 28th (in 2 weeks).

Sunday, November 3, 2013

Tuesday, Nov. 5th – come as usual (9:15-12:15)

Filed under: Uncategorized — innaento @ 11:52

We will have a recitation on RSA and isometries (for Prof. Novikov’s course) instead of Analysis lecture.

Homework 1 correction

Filed under: Uncategorized — innaento @ 11:51

In Problem 1, part b, it should say aN is included in bN (otherwise it is wrong – for example, for a=6, b=3). Thanks to Menuha for pointing this out.

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