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The earlier discussion on Stellan Ohlsson brought up the issue of abstraction. It appears useful to say a bit more on terminology.

An unfortunate confusion at wikipedia

Wikipedia – no source but a portal – on abstraction creates a confusion:

  1. Correct is: “Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, selecting only the aspects which are relevant for a particular purpose.” Thus there is a distinction between abstract and concrete.
  2. Confused is: “For example, abstracting a leather soccer ball to the more general idea of a ball selects only the information on general ball attributes and behavior, eliminating the other characteristics of that particular ball.” However, the distinction between abstract and concrete is something else than the distinction between general and particular.
  3. Hopelessly confused is: “Abstraction involves induction of ideas or the synthesis of particular facts into one general theory about something. (…) Bacon used and promoted induction as an abstraction tool, and it countered the ancient deductive-thinking approach that had dominated the intellectual world since the times of Greek philosophers like Thales, Anaximander, and Aristotle.” This is hopelessly confused since abstraction and generalisation (with possible induction) are quite different. (And please correct for what Bacon suggested.)

A way to resolve such confusion is to put the categories in a table and look for examples for the separate cells. This is done in the table below.

In the last row, the football itself would be a particular object, but the first statement refers to the abstract notion of roundness. Mathematically only an abstract circle can be abstractly round, but the statement is not fully mathematical. To make the statement concrete, we can refer to statistical measurements, like the FIFA standards.

The general statement All people are mortal comes with the particular Socrates is mortal. One can make the issue more concrete by referring to say the people currently alive. When Larry Page would succeed in transferring his mind onto the Google supercomputer network, we may start a philosophical or legal discussion whether he still lives. Mutatis mutandis for Vladimir Putin, who seems to hope that his collaboration with China will give him access to the Chinese supercomputers.

Category (mistake) Abstract Concrete
General The general theory of relativity All people living on Earth in 2015 are mortal
Particular The football that I hold is round The football satisfies FIFA standards
The complex relation between abstract and general

The former table obscures that the relation between abstract and general still causes some questions. Science (Σ) and philosophy (Φ) strive to find universal theories – indeed, a new word in this discussion. Science also strives to get the facts right, which means focusing on details. However, such details basically relate to those universals.

The following table looks at theories (Θ) only. The labels in the cells are used in the subsequent discussion.

The suggestion is that general theories tend to move into the abstract direction, so that they become universal by (abstract) definition. Thus universal is another word for abstract definition.

A definition can be nonsensical, but Σ strives to eliminate the nonsense, and officially Φ has the same objective. A sensible definition can be relevant or not, depending upon your modeling target.

(Θ) Aspects of scientific theories (Σ) Science (Φ) Philosophy
(A) Abstract definition (developed mathematically or not) (AΣ) Empirical theory. For example law of conservation of energy, economics Y = C + S, Van Hiele levels of insight (AΦ) Metaphysics
(G) General (GΣ) Statistics (GΦ) Problem of induction
(R) Relation between (A) and (G) (RΣ) (a) Standards per field,
(b) Statistical testing of GΣ,
(c) Definition & Reality practice
(RΦ) (a) Traditional epistemology,
(b) Popper,
(c) Definition & Reality theory

Let us redo some of the definitions that we hoped to see at wikipedia but didn’t find there.

Abstraction is to leave out elements. Abstractions may be developed as models for the relevant branch of science. The Van Hiele levels of insight show how understanding can grow.

A general theory applies to more cases, and intends to enumerate them. Albert Einstein distinguished the special and the general theory of relativity. Inspired by this approach, John Maynard Keynes‘s General Theory provides an umbrella for classical equilibrium (theory of clearing markets) and expectational equilibrium (confirmation of expectations doesn’t generate information for change, causing the question of dynamic stability). This General Theory does not integrate the two cases, but merely distinguishes statics and its comparative statics from dynamics as different approaches to discuss economic developments.

Abstraction (A) is clearly different from enumeration (G). It is not impossible that the enumeration concerns items that are abstract themselves again. But it suffices to assume that this need not be the case. A general theory may concern the enumeration of many particular cases. It would be statistics (GΣ) to collect all these cases, and there arises the problem of induction (GΦ) whether all swans indeed will be white.

Having both A and G causes the question how they relate to each other. This question is studied by R.

This used to be discussed by traditional epistemology (RΦ(a)). An example is Aristotle. If I understand Aristotle correctly, he used the term physics for the issues of observations (GΣ) and metaphysics for theory (AΦ & GΦ). I presume that Aristotle was not quite unaware of the special status of AΣ, but I don’t know whether he said anything on this.

Some RΦ(a) neglect Σ and only look at the relation between GΦ and AΦ. It is the price of specialisation.

Specialisation in focus is also by statistical testing (RΣ(b)) that only looks at statistical formulations of general theories (GΣ).

The falsification theory by Karl Popper may be seen as a philosophical translation (RΦ(b)) of this statistical approach (RΣ(b)). Only those theories can receive Popper’s label “scientific” that are formulated in such manner that they can be falsified. A black swan will negate the theory that all swans are white. (1) One of Popper’s problems is the issue of measurement error, encountered in RΣ(b), with the question how one is to determine sample size and level of confidence. Philosophy may only be relevant if it becomes statistics again. (2) A second problem for Popper is that AΣ is commonly seen as scientific, and that only their relevance can be falsified. Conservation of energy might be relevant for Keynes’s theory, but not necessarily conversely.

The Definition & Reality methodology consists of theory (RΦ(c)) and practice (RΣ(c)). The practice is that scientists strive to move from the particular to AΣ. The theory is why and how. A possible intermediate stage is G but at times direct abstraction from concreteness might work too. See the discussion on Stellan Ohlsson again.

Conclusions

Apparently there exist some confusing notions about abstraction. These can however be clarified, see the above.

The Van Hiele theory of levels of insight is a major way to understand how abstraction works.

Paradoxically, his theory is maltreated by some researchers who don’t understand how abstraction works. It might be that they first must appreciate the theory before they can appreciate it.

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The discussion of Putin’s proof gave me an email from Alexis Tsipras, who just resigned as prime minister of Greece and is busy with the general elections of September 20 soon. Rather than reporting on it, I might as well fully quote it.

To: Thomas
From: Alexis@formerprimeministerofgreece.org
Subject: My proof of Fermat’s Last Theorem
Date: Fri, 28 Aug 2015 11:58:03 +0100
Google Unique Message Identifier: 23DFGA@671

Dear Thomas,

Thank you very much for your discussion of President Putin’s proof when he was a youngster of Fermat’s Last Theorem. I know his mother Vera Putina very well. The Putin family has a vacation home here in Greece, and she can stay there on the condition that she immediately leaves when Putin himself comes down. She has shown me his proof too. I can only agree with your conclusion that it shows how smart President Putin was when he was young.

Putin’s proof inspired me to find a proof too. I am sometimes exhausted by the tough negotiations with the European Heads of State and Government, if not with members of my own party. Thus I resort often to a sanatorium for recuperation. Thinking about such issues like Fermat’s Last Theorem helps to clear my mind from mundane thoughts. I was very happy last Spring to indeed find a much shorter and more elegant proof. 

For the theorem and notation I refer to your weblog. My proof goes as follows.

Theorem. No positive integers n, a, b and can satisfy an + bn = cn for n > 2.

Proof. (Alexis Tsipras, April 31 2015)

Let us assume that an + bn = cn holds, and derive a contradiction.

There are two possibilities: (1) n is even, or (2) n is uneven.

(1) If n is even, then we can write A = an/2 and B = bn/2 and C = cn/2 such that A, B and C are still integers. Then we get the following equation:

A2 + B2 = C2

This equation satisfies the condition that n = 2, and thus it doesn’t satisfy the condition n > 2.

(2) If n is uneven, then we can write A = a(n-1)/2 and B = b(n-1)/2 and C = c(n-1)/2 such that A, B and C are still integers. Then we get the following equation:

a A2 + b B2 = c C2

This equation does not satisfy the form of an + bn = cn so that it falls outside of Fermat’s Last Theorem.

In both cases the conditions of the theorem are no longer satisfied. We thus reject the hypothesis that an + bn = cn holds. Q.E.D.

This is much shorter that President Putin’s proof. And, I prove it while he only came close. I have been hesitating to tell him, fearing that he might become jealous, and be no longer willing to support Greece as he does in these difficult times for my country. Now that you have confirmed how wonderful his proof at only age 12 was, I feel more assured. Will you please publish this proof of mine too, like you did with President Putin’s proof ? I have put my best efforts in this proof, just like at the negotiations with the European Heads of State and Government. Thus I hope that it will be equally convincing, if not more.

After the next elections I will probably be exhausted again. I would like to work on another problem then. Do you have any suggestions ?

Sincerely yours,

Alexis

Fermat and Tsipras (source: wikimedia commons)

Fermat and Tsipras (source: wikimedia commons)

The door rang. I was surprised to see Vera Putina. It appeared that Putin’s mother was visiting her granddaughter’s fiancé’s family in Holland. “It is not safe for me to go to Moscow,” she explained, expressing the sentiment of many.

When she was settled in the safety in my living room with a good cup of Darjeeling, it also appeared that there was more.

VP: “I am upset. The Western media depict my son only as a sportsman. They show him doing judo, riding horses, fighting bears, and the last week they featured him as a diver in a submarine. Of course he is very athletic, but he is also a smart man. I want you to look at his intellectual side too.”

Me: “It is fine of you to ask, because there indeed is a general lack of awareness about that.”

VP: “Let me tell you ! When my father had his love affair with Grand Duchess Kira Kirillovna of Russia [see the Putin family tree here], one of their secret meeting places was in the royal archives of the Romanovs. Sometimes my father, because he was unemployed, took some of the old documents to sell on the market. You know, my mother Kira had an expensive taste.”

VP said this without blinking an eye. The unperturbedness when taking other people’s possessions and territories must have a family origin.

VP: “My father once found the application letter by Pierre de Fermat for membership of the St. Petersburg mathematical society. It detailed his proof of his Last Theorem.”

Me: “Ah, that might explain why he never published it ! He used it for his application, and this got lost in the archives ?!”

VP: “I don’t know about that. My father sent it in for the Wolfskehl Prize of 100,000 gold marks, but it was rejected for it didn’t satisfy the criterion of having been published in a peer-reviewed journal.”

Me: “This is historically very interesting. If you still have that letter by Fermat, no doubt a historical journal will gladly publish it. It doesn’t matter on content since Andrew Wiles now proved it.”

VP: “No, no, no !” She gestured with passion as an Russian woman can do. “Fermat is not important ! It is what Vladimir did ! When he was twelve, he also looked at Fermat’s letter, and he found an omission ! Moreover, he worked on the problem himself, and almost solved it. Here, I brought along the papers to prove it to you.”

She delved into the bag that she had brought along and produced a stack of papers. I also saw a wire bound notebook such as children use in school.

Me: “Almost solving means not solving. Mathematics is rather strict on this, gospodina Putina. But it is historically interesting that you have Fermat’s original proof and that your son worked on it.”

VP: “For this, he had to learn Latin too !’

She gave me the stack. There was a great deal of difference between her nonchalant and triumphant handing over of the papers and my hesitant and rather reverent accepting of them.

VP: “You look it over, and inform the Western media that my son almost solved Fermat’s Last Theorem when he was only twelve ! If I hadn’t told him that he had to go to his judo lessons, he would have finished it for sure !”

She said the latter as proof that she had been a good mother, but also with a touch of regret.

Confronted with such motherly compassion I could only respond that I would oblige. Hence, below is Vladimir Putin’s proof. First I translate Fermat’s own proof from Latin (also using the Russian transcript that Putin made) and then give Putin’s correction.

Fermat (1601-1665) and Putin (1952+)

Fermat (1601 – 1665) and Putin (1952 – ∞) (Source: wikimedia)

Fermat’s Last Theorem, using middle school algebra

Theorem. No positive integers n, a, b and can satisfy the equation an + bn = cn for n > 2.

Proof. (Pierre de Fermat, April 31 1640, letter to czar Michael I of Russia)

Without loss of generality b. Take k = n – 2 > 0. We consider two cases:

(1) a2 + b2c2

(2) a2 + b2 > c2.

(1) When a2 + b2c2 then a2 + b2 + d = c2 for d 0

Then a < c and b < c. Then also ak < ck and bk < ck for k > 0.

If the theorem doesn’t hold, then there is a k > 0 such that:

ak+2 + bk+2ck+2

ak a2 + bk b2 = ck c2 = ck (a2 + b2 + d)

a2 (akck) + b2 (bkck) = d ck ≥ 0

negative + negative ≥ 0

Impossible. Thus the theorem holds for (1).

(2) If a2 + b2 > c2 then obviously (see the diagram) for higher powers too: an + bn > cn.

Fermat's drawing for his proof (right rewrites left)

Fermat’s drawing for his proof (RHS re-orders LHS)

Since (1) and (2) cover all possibilities, the theorem holds.

Q.E.D.

Putin’s correction, age 12

The comment by schoolboy Vlad on this proof is:

“While (2) is obvious, you cannot rely on diagrams, and you need to fully develop it. At least I must do so, since I find the diagram not so informative. I also have problems reading maps, and seeing where the borders of countries are.”

Hence, young Putin proceeds by developing the missing lemma for (2).

Lemma. For positive integers n, a, b and c: if a2 + b2 > c2 then an + bncn for n > 2.

Proof. (Vladimir Putin, October 7 1964)

Without loss of generality a b. Take k = n – 2 > 0.

If a2 + b2 > c2 then a + b > c. (Assume the contrary: a + b c then a2 + b2 < (a + b)2c2, which contradicts a2 + b2 > c2.)

Expression an + bn > cn is equivalent to (an + bn)1/n > c. The LHS can be written as:

f[n] = a (1 + (b / a)n)1/n  with a b.

This Lemma has the Pythagorean value f[2] = √(a2 + b2) > c. The function has limit f[n → ∞] = a. (See a deduction here.) Thus f[n] is downward sloping from f[2] >  to limit value a. We have two cases, drawn in the diagram below.

Case (A) Diagram LHS: c ≤ a, so that there will never be an intersection f[n] = c.

Case (B) Diagram RHS: a < c < f[2] = √(a2 + b2). There can be an intersection f[n] = c, but possibly not at an integer value of n. Observe that this case also provides a counterexample to Fermat’s claim that “obviously” f[n] > c, for, after the intersection f[n] < c. Young Putin already corrects the great French mathematician ! This is a magnificent result of the future President of the Russian Federation, at such a young age. His grandfather’s Marinus van der Lubbe’s submission to the Wolfskehl Prize would also have failed on this account.

a (1 + (b/a)^n)^(1/n) and parameter cases

f[n] = a (1 + (b / a)^n)^(1/n) and parameter cases

At this point, young Putin declares that Case (A) on the LHS is proven, based upon above considerations. He adds:

“I accept this proof on the LHS, even though I have difficulty understanding that limits or borders should not be transgressed.”

As so often happens with people who are not entirely sure of their case, the schoolboy then develops the following simple case, just to make certain.

Case (≤ b). Use numerical succession from a2 + b2 > c2.

Given an + bn > cn then prove an+1 + bn+1 > cn+1.

a an + b bnb an + b bn = b (an + bn) > b cnc cn

Thus the Lemma holds for this case.

To be really, really, sure, Putin adds an alternative proof that assumes the contrary:

ak a2 + bk b2ck c2

ak a2bk a2 + bk a2 + bk b2bk c2ck c2bk c2

a2  (akbk) + bk (a2 + b2c2) ≤ c2 (ck bk)

nonnegative + positive  ≤  nonpositive

Impossible. Thus the Lemma holds for (≤ b).

It would have been better when he had looked at Case (B) on the RHS, notably by proving that f[n] = c cannot hold for only integers.

At this point in his notebook, young Putin writes:

“I have to go to judo training. Perhaps I will continue tomorrow.”

I have looked in the remainder of the notebook but did not find further deductions on Fermat. Apparently the next day young Putin continued with what was more on his mind. It appears that he had a fantasy land called Dominatia in which he played absolute master, and it took much of his time to determine what was happening there. Something of the unruly nature of the natural numbers however must have stuck in his mind. In a perfect fantasy land everything is already as wished, but in young Putin’s Dominatia land he fantasizes about unruly citizens who must be put under control.

Conclusions

The above supports the following conclusions:

  • The theorem & lemma are not yet proven for Case (B) on the RHS. We must still rely on Andrew Wiles.
  • Nevertheless, Vladimir Putin doesn’t do just sports but also has amazing intellectual powers, at least when he was at age twelve.
  • Fermat’s original own proof of his theorem seems to have had a serious error, but it is not precluded that it was only chance that it did not get published (with or without corrections).
  • Fermat’s Last Theorem has dubious value for education. It seems more important to develop the notion of limits, and in particular the notion that you should not transgress borders. When students do not understand this properly at a younger age then this may cause problems later on.
Appendix 1. Case (c > a ≥ b)

It may be nice to see how f[n] = c is sandwiched, when a + b > f[2] > c > a ≥ b.

Case (c > a ≥ b)  There is a point f[n] = c or an + bn = cn for reals but perhaps not for integers.

(i) At the intersection:

ak a2 + bk b2 = ck c2

Take ak c2 + bk c2 and substract the above on both sides:

(c2 – a2) ak + (c2 – b2) bk = (ak + bkck) c2

positive + positive = ?

The latter must be positive too, and hence: ak + bk > ck

Thus, assuming that the theorem doesn’t hold for n requires that it holds for k = n – 2.

(ii) After the intersection: Since f[n] is downward sloping we have f[n+1] < f[n] = c. Reworking gives:

an+1 + bn+1 < cn+1

Another way to show this is:

(a – can + (b cbn < 0

a  an+ b bn  < c (an + bn)

an+1 + bn+1cn+1

Comparing (i) and (ii) we see the switch from > to <.

Appendix 2. Parameter restrictions in general

Assume that an + bn = cn holds. There are restrictions for this to occur, notably by the remarkable product:

(an bn) (an + bn) = (an bn) cn

a2n b2n = (an bn) cn

an (an cn) = bn (bncn)             (*)

For example: when c = a, then an + bn = cn is only possible in (*) if c = a = b, but this is actually also impossible because it requires that cn + cn = cn. The table collects the findings, with the LHS and RHS now referring to equation (*).

an + bn = cn c <(LHS +) c =(LHS 0)
c > a  (LHS -)
c < (RHS +) (=),
but Case (c b)
impossible opposite signs
c b  (RHS 0) impossible impossible
cn + cn = cn
impossible
c > b  (RHS -) opposite signs impossible (=) the only risk

This table actually also proves Case (A) that Putin took for granted. Only Case (B) remains, and requires proof that f[n] = c cannot hold for only integers.

Listening to Ο Διγενής (Ριζίτικο) – Ν. Ξυλούρης, Γ. Μαρκόπουλος

 

Some economists signed below letter in the Financial Times, June 5, on the final hour. When I do my petitions then I try to make a point. Let us read this open letter and identify its holes.

Title: In the final hour, a plea for economic sanity and humanity

The suggestion of a final hour is scare mongering. It is not true either, because it is a safe bet that there will be another hour after that. Subsequently it is insulting to Angela Merkel: as if she has not been working hard on economic sanity and humanity. If you want her to read something then find an appealing title, something like: “How to get Putin pay attention and listen”.

Signatures

Groucho Marx didn’t want to join a club who wanted him as a member. One can look at the signatures for a long while and wonder why one would want to be on that list. If these would be 100,000 economists from Europe, well, perhaps. A single European Nobel Prize winning economist might be sufficient, but there isn’t one on the list. I need to ask Hillary Wainwright from Amsterdam what she thinks about the censorship of economic science in Holland. (After Greek Statistics we also have Dutch Economics.)

Prof Joseph Stiglitz, Columbia University; Nobel Prize winner of Economics, Prof Thomas Piketty, Paris School of Economics, Massimo D’Alema Former prime minister of Italy; president of FEPS (Foundation of European Progressive Studies) Prof Stephany Griffith-Jones IPD Columbia University Prof Mary Kaldor London School of Economics Hilary Wainwright Transnational Institute, Amsterdam Prof Marcus Miller Warwick University Prof John Grahl Middlesex University, London Michael Burke Economists Against Austerity Prof Panicos Demetriadis University of Leicester Prof Trevor Evans Berlin School of Economics and Law Prof Jamie Galbraith Dept of Government, University of Texas Prof Gustav A Horn Macroeconomic Policy Institute (IMK) Prof Andras Inotai Emeritus and former Director, Institute for World Economics, Budapest Sir Richard Jolly Honorary Professor, IDS, Sussex University Prof Inge Kaul Adjunct professor, Hertie School of Governance, Berlin Neil MacKinnon VTB Capital Prof Jacques Mazier University of Paris Dr Robin Murray London School of Economics Prof Jose Antonio Ocampo Columbia University Prof Dominique Plihon University of Paris Avinash Persaud Peterson Institute for International Economics Prof Mario Pianta University of Urbino Helmut Reisen Shifting Wealth Consultancy Dr Ernst Stetter Secretary General, FEPS (Foundation fro European Progressive Studies) Prof Simon Wren-Lewis Merton College Oxford

Copyright

The copyright of the open letter has been transferred to the Financial Times:

Copyright The Financial Times Limited 2015. You may share using our article tools.
Please don’t cut articles from FT.com and redistribute by email or post to the web.” (Financial Times website)

This is not handy. If you want to distribute your open letter to all kinds of newsmedia, you should not transfer the rights to a single newspaper. See why I don’t blog at the Financial Times.

It actually means that I am also handicapped in deconstructing this open letter. I can quote, of course, but deconstructing everything becomes less inviting. I intended to deconstruct it all, but now, seeing that awkward copyright statement, I lose my enthusiasm.

Opening paragraph

“Sir, The future of the EU is at stake in the negotiations between Greece and its creditor institutions, now close to a climax. To avoid failure, concessions will be needed from both sides. From the EU, forbearance and finance to promote structural reform and economic recovery, and to preserve the integrity of the Eurozone. From Greece, credible commitment to show that, while it is against austerity, it is in favour of reform and wants to play a positive role in the EU.” (Letter by Stiglitz et al. FT June 5 2015)

  • Please don’t address Angela Merkel as “Sir”.
  • Why is it a concession from the EU to exercise forbearance and finance, which they have been doing all the time ?
  • Why is it a concession from Greece to present a credible commitment ? Don’t they claim that they have been doing so all the time ? Or, are Greece’s “best friends” now back-stabbing the Greeks ?
Are these impartial economists or members of Syriza ?

“Syriza is the only hope for legitimacy in Greece. Failure to reach a compromise would undermine democracy in [sic] and result in much more radical and dysfunctional challenges, fundamentally hostile to the EU.” (Letter by Stiglitz et al. FT June 5 2015)

  • Are these impartial economists ?
  • Aren’t there other political parties who have some views on reform and such ?
  • Shouldn’t economists be highly critical of incompetent Yanis Varoufakis, the Syriza minister of finance ? See my earlier criticism, or discussion of Angela riding this minotaur.
A new Marshall Plan ?

“Consider, on the other hand, a rapid move to a positive programme for recovery in Greece (and in the EU as a whole), using the massive financial strength of the Eurozone to promote investment, rescuing young Europeans from mass unemployment with measures that would increase employment today and growth in the future”  (Letter by Stiglitz et al. FT June 5 2015)

  • Yes, of course, see for example my Economic Plan for Europe, and this short text in eKathimerini 2011. Was there one single response from a Greek economist – such as Yanis Varoufakis ? No. See actually my list of papers on the crisis.
  • There is one crucial difference between the rosy view by Stiglitz & Piketty & friends and my more mundane analysis: they assume that the Eurozone will forget about the past, and for the future believe Greece again in whatever Greece promises, while I adopt the more realistic position that there has been a break-down of confidence already. 

Compare: Berlusconi did not ask for a bail-out but Tsipras does. Why would Tsipras think that the Eurozone loves him more than Berlusconi ? Earlier I indicated the link between Russia and Greece – the orthodox church – but why does Tsipras make a solo trip to Russia without taking account of sentiments in the rest of Europe on the Ukraine, or like in Holland on MH17 ?

Berlusconi did not ask for a bail-out but Tsipras does

Berlusconi did not ask for a bail-out but Tsipras does (2008, Source: wikimedia commons)

Thus, a new Marshall Plan is not going to work, unless there are firm regulations in place.

  • It boggles my mind that Greek policy makers do not understand this.
  • It boggles my mind that a political party like Syriza performs such populism, to promise relief without actually presenting a credible plan to achieve this. Surely, populism gets you elected, but does one not have a grain of responsibility ?
  • It boggles my mind that above supposedly serious economists support such populism instead of developing a plan that would actually work. James Galbraith is partly excused for presenting the “Modest Propopsal” jointly with Yanis Varoufakis and Stuart Holland. But he should know about my criticism. Obviously Angela Merkel will not be easily convinced either (e.g. on eurozone bonds).
Conclusion

Economic scientists must observe impartiality. Economic proposals should be backed-up with a minimum of a plan. Analyses should clearly indicate where parties are in error, and otherwise allow for a “core” (see the Edgeworth diagram) with a range of possible compromises. None of these are provided by Stiglitz & Piketty et al. The letter is a miserable failure.

Listening to Markopoulos & Xulouris – O Digenis

 

Abstraction has been defined in the preceding discussion. A convenient sequel concerns what is commonly called ‘mathematical induction’. This is an instance of abstraction.

Mathematical induction has a wrong name

Mathematical induction has a wrong name. It is a boy called Sue. It is czar Putin called president. There is no induction in ‘mathematical induction’. The term is used to indicate that each natural number n has a next one, n+1. Thus for number 665 the mathematician induces 666: big surprise. And then 667 again, even a bigger surprise after 666 should be the end of the world. The second confusion is that the full name of ‘proof by mathematical induction’ is often shortened to only ‘mathematical induction’: which obscures that it concerns a method of proof only.

This method applies to the natural numbers. It actually is a deduction based upon the definition of the natural numbers. Since the natural numbers are created by numerical succession, a proper name for the method is proof by numerical succession.

Let us define the natural numbers and then establish this particular method of proof. It is assumed that you are familiar with the decimal system so that we don’t have to develop such definitions. It is also assumed that zero is a cardinal number.

Definition of the natural numbers

A finite sequence of natural numbers is N[5] = {0, 1, 2, 3, 4, 5}.  Since we can imagine such sequences for any number, there arises the following distinction given by Aristotle. He called it the difference between potential and actual infinity. 

(1) Potential infinity: N[n] = {0, 1, 2, 3, …., n}. This reflects the human ability to count. (1a) It uses the successor function (“+1”): s[n] = n + 1. For each n there is a n+1. The successor function is a primitive notion that cannot be defined. You get it or you don’t get it. As a formula we can ‘define’ it by writing ‘For each n there is a n+1′, but this is not really a definition but rather the establishment of a convention how to denote it. (1b) Numerical succession might actually be limited to a finite number, say for a window of a small calculator that allows for 6 digits: 0 ≤ n ≤ 999,999. The crux of N[n] however is that n can be chosen and re-chosen at will. For each N[n] we can choose a N[n+1].

(2) Actual infinity: N = {0, 1, 2, 3, …}. This reflects the human ability to give a name to some totality. Here the name is ‘the natural numbers’.

Another formulation uses recursion: N = {n | n = 0, or n-1 ∈ N}. Thus 1 ∈ N because 0 is. 2 ∈ N because 1 is. And so on. Thus, we now have defined the natural numbers.

The potential infinite deals with finite lists. Each list has a finite length. The distinctive property of these lists is that for each such number one can find a longer list. But they are all finite. It is an entirely different situation to shift to the actual infinite, in which there is a single list that contains all natural numbers.

There need be no doubt about the ‘existence’ of the natural numbers. The notion in our minds suffices. However, our mental image may also be a model for reality. If the universe is finite, then it will not contain an infinite line, and there cannot be a calculator with a window of infinite length. But, on every yardstick in the range [0, 1] we have all 1H, 2H, 3H, ….. PM. We denote nH = 1 / n, to be pronounced as per-n, see the earlier discussion on nH.

The relation between potential and actual infinity

The shift from N[n] to N is an instance of abstraction. N[n] is a completed whole but with a need to build it, with a process of repetition. N ‘leaves out’ that one is caught in some process of repetition, while there still is a completed whole. Let us use a separate symbol @ for the particular kind or instance of abstraction that occurs in the shift from (1) to (2).

(3) N[n] @ N. This records that (1) and (2) are related in their concepts and notations. In the potential form for each n there is a n+1. In the actual form there is a conceptual switch to some totality, caught in the label N.

Since we already defined (1) and (2) to our satisfaction, (3) is entirely derivative and does not require an additional definition. It merely puts (1) and (2) next to each other, while the symbol ‘@’ indicates the change in perspective from the potential to the actual infinite.

(There might be a link to the notion of ‘taking a limit’ but it is better to leave the word ‘limit’ to its well-defined uses and take ‘@’ as capturing above instance of abstraction.)

Proof by numerical succession

The method of proof by numerical succession follows the definition of the natural numbers.

Definition:  Let there be a property P[n] that depends upon natural number n. The property can be established – or become a theorem – for all natural numbers n ≥ m, by the following method of proof, called the method by numerical succession: (i) show that P[m] holds, (ii) show that P[n-1] ⇒ P[n]. (The validity of the proof depends upon whether these two steps have been taken well of course.)

When m = 0 then the property might hold for all natural numbers.  The second step copies the definition of N: If n-1 ∈ N and P[n-1], then n ∈ N and then it must be shown that P[n]: if it is to hold that P[n] for all n ∈ N.

PM 1 below contains an example that uses a more conventional notation of going from n to n+1.

The definition of the method of proof doesn’t state this explictly: In the background there always is (N[n] @ N) w.r.t. the fundamental distinction between the finite N[n] and the infinite N. Conceivably we could formulate a method for N[n] separately that emphasizes the finitary view but there is no need for that here.

Conclusions

(1) A prime instance of abstraction is the relation N[n] @ N, i.e. the shift from the potential to the actual infinity of natural numbers.

(2) The method of ‘proof by numerical succession’ is a deductive method based upon the definition of the natural numbers.

(3) ‘Proof by numerical succession’ is a proper name, for what confusingly is called ‘proof by mathematical induction’.

(4) Without further discussion: There is no unreasonable effectiveness’ in the creation of the infinity of the natural numbers and the method of proof by numerical succession, and thus neither in the application to the natural sciences, even when the natural sciences would only know about a finite number (say number of atoms in the universe).

PM 1. An example of a proof by numerical succession

We denote nH = 1 / n, see the earlier discussion on nH.

Theorem: For all n ∈ N:

1 + 2 + 3 + … + n = n (n  + 1) 2H

Proof: By numerical succession:

(i)  It is trivially true for n = 0. For n = 1: 1 =  1 * (1 + 1) 2H . Use that 2 2H = 1.

(ii) Assume that it is true for n. In this case the expression above holds, and we must prove that it holds for n+1. Substitution gives what must be proven:

1 + 2 + 3 + … + n + (n + 1) =?= (n  + 1)(n + 2) 2H

On the LHS we use the assumption that the theorem holds for n and we substitute:

n (n  + 1) 2H + (n + 1) =?= (n  + 1)(n + 2) 2H

Multiply by 2:

 n (n  + 1) + 2 (n + 1) =?= (n  + 1)(n + 2)

The latter equality can be established by either do all multiplications or by separation of (n+1) on the left. Q.E.D.

PM 2. Background theory

See CCPO-PCWA (2102, 2013) section 4, p16, for more on @.

PM 3. Rejection of alternative names

The name ‘mathematical succession’ can be rejected since we are dealing with numbers while mathematics is wider. The name ‘natural succession’ can be rejected since it doesn’t refer to mathematics – consider for example the natural succession to Putin. The name ‘succession for the natural numbers’ might also be considered but ‘numerical succession’ is shorter and on the mark too.

PM 4. Wikipedia acrobatics

Earlier we diagnosed that wikipedia is being terrorized by students from MIT who copy their math books without considering didactics. The wiki team seems to grow aware of the challenge and is developing a ‘simple wiki’ now. Check the standard article on mathematical induction and the simple article.  The next steps for the wiki team are: to establish the distinction between easy and their notion of simplicity, then reduce the standard wiki into an easy one, and subsequently ask the MIT students to do both their copying and their experiments on simplicity at this ‘simple wiki’.

Vladimir Putin called me this morning. He was his usual confidence but I sensed a tad of worry.

When Putin calls there must be a reason.

Vlad: “I did what you advised but it doesn’t work.”

Me: “Okay, I am listening.”

Vlad: “I didn’t kill Garry Kasparov yet, as you suggested, and I made sure that he was on Dutch television last Sunday. But I don’t see the headlines.”

Me: “Well, he complimented you by calling you “the most dangerous man the world has ever seen, potentially”. He even compared you to Hitler, but now with nuclear weapons. Many Dutch people are more afraid of you than ever. So you should agree that it works.”

Vlad: “Yes, of course, I watched the programme, shooting seventy tv sets to pieces. We agreed that I should experiment with democracy, so I let him have his say, so that everyone can see what idiot he is. But I don’t see a headline in The New York Times “Kasparov shows himself a great fool”. If this is democracy then I am glad that I am against it.”

Me: “But if you want people to understand that you are the most dangerous man the world has ever seen, then you need clowns like Kasparov who say so, since nobody else will dare this. Thus you cannot have the NYT to expose him as a clown, since then people will no longer listen to him, and people will no longer believe that you are the most dangerous man the world has ever seen.”

Putin went silent on the other side of the line.

Me: “Listen, democracy is a game in which you can never lose. You only have to understand its rules.”

Vlad: “I don’t play by rules. Why do you think that I am called dangerous ?”

Me: “Excuse me, I should have said “understand how it works”. You have to hand it to Kasparov: how he explained that you are no chess player since chess has rules while you are rather a poker player so that you can win even when your cards are lousy. Can’t you remember that chess game by you and me ?”

Vlad: “I thought that a silly comparison. When I play poker then I don’t have to bluff since I can always put in some nukes. But okay, I begin to understand why this interviewer Pieter Jan Hagens didn’t fall from his chair from laughter. He wanted his viewers to think that the idiot was given his freedom of speech.”

Me: “Exactly. Do also observe that Kasparov spoke with an interviewer and not with some top Dutch politicians. Kasparov could have asked them some embarrassing questions on MH17 and the Dutch Shell co-operation with Gazprom. The politicians on their part could have asked Kasparov for some real measures to hurt you. Neither happened. The trick of Dutch journalists is that they have wedged themselves into a position where they ask the questions and get paid a top income for that. Of course, such journalists are actually superfluous. People in top positions are quite capable to ask such questions themselves. They only need someone to announce who will be on the show: and anybody can do so and at a minimum wage. But this is how democracy works.”

Vlad: “And Pieter Jan Hagens thus made sure that there was no real political debate. I had to pay him for that too. I like the guy. I should invite him to Moscow to teach his tricks to my people. And they could teach him their tricks too.”

I could not suppress a shudder. I felt happy that this was a normal phone without views.

Vlad: “Still, Angela Merkel had this idiot Tsipras visiting her, and she got media coverage from all over the world, while my democratic experiment with Kasparov went unnoticed. I let the joker live ! Isn’t anybody grateful for that ?”

Me: “That is the price of being a dictator. This is a democratic world and you are the odd-man out. You will see that reaction again when Tsipras will visit you on April 8. I already wondered why you didn’t see the plight of the Greek people. If you receive and treat him while behaving as a dictator, then the world press will regard it as a non-event, but if you receive him as the inventor of democracy and a great inspiration for the European future, then the media will go berzerk.”

Vlad: “I don’t get you. You want Russia to take its example from Greece ?”

Me: “That would be a great headline ! You are doing fantastic ! Your small experiment with Kasparov on Dutch television is opening up your mind to possibilities that I hadn’t thought of myself ! Yes, look into that weird Greek system of democracy in which the largest party gets 50 seats extra. Check how Russian corruption can learn from Greek corruption in a democracy. Check how Tsipras has an inner circle with other clowns like Yannis Varoufakis, so that Kasparov’s discussion about your inner circle replacing you becomes even more silly. Check how a small determined country can wreak havoc on the world economic system, while you need a huge army and your nukes and still get expelled from the G8. I regard our discussion as very fruitful and promising. My compliments to you, the most dangerous man the world has ever known, potentially.”

Vlad, apparently pleased, but still with a tad of worry, as always when he was considering a democratic idea: “I don’t like that “potentially”. I am thinking that I will let Kasparov live a bit longer. I want him to see what I am potentially capable of.”

Garry Kasparov on Dutch tv, 2015-03-22 (Source: screenshot Buitenhof tv)

Garry Kasparov on Dutch tv, 2015-03-22 (Source: screenshot Buitenhof tv)

My lunches with EU Commission President Manuel Barroso are generally rather boring but occasionally Manny springs a surprise on me. “Both the Russians and the Americans want this transcript to be leaked but they are afraid of being traced. You have a widely read weblog so you are perfectly placed for the job. You still owe me one,” he whispered yesterday. I hadn’t been aware that I owed him a favour but I can’t quite refuse when he feels like that. So here is the transcript. I include some notes within brackets for those who are new to diplomacy.

BEGIN OF THE TRANSCRIPT

March 12, 2014. Moscow 22:00 hours. Washington 14:00 hours.

“Hello Mr President. /  Hello Mr President” (Both have been instructed to say this at the same time so that there is no discussion who is calling whom.)

Obama: “Dober dan.” (He has been told that this is Russian.)

Putin: “Dober dan.” (He has been told that this isn’t English.)

Obama: “I like to thank you for the crisis on the Ukraine. It is saving my Presidency. I seemed to be lost in frustration on national debt and health care but foreign policy allows me to be Commander in Chief again. I feel reborn.”

Putin: “Glad to be of service. It is part of Russian history to always help the West. And if we Presidents didn’t care for each other, nobody would. It is lonely at the top, isn’t it ? Perhaps you could give me an excuse to crack down on dissenting journalists again ?”

Obama: “I especially appreciate your patience now that Angela Merkel is calling you all the time. Michelle wonders whether she doesn’t have a husband she should be looking after.”

Putin (laughing): “I have an actor here, who does my voice, and who knows “Danke schön” in German. I also have a highly trained team to provide him with the empty non-committing phrases she appears to enjoy. I had to promise my actor that he would not have to meet her, for he would strangle her.”

Obama (laughing): “Well, I would  be an actor too, except for your Ambassador who is watching across the Oval Office. Would you like to say hello to each other ?”

Putin: “No, I hear his whistle now. You may block Apple and Google technology from coming to Russia but we have some elementary tricks that you seem to have forgotten about.”

Obama: “Like taking the Crimea as you have done. My compliments, Vladimir ! This will go down in history as a classic. We Americans sense that it is something like the Battle of the Alamo, and we can recognise how important that is to Russia. You have earned the warm respect and appreciation of the American people.”

Putin: “I now understand what the Battle of the Alamo means to the American people. So how would you feel if I would take the entire Ukraine ?”

Obama: “Ah, yes. I was hoping to talk about that too. Did someone ever tell you about my problems with health care in America ?”

Putin: “Why are you changing the subject ? Of course I have been told about your Health Plan. You want my advice, on how I take the Ukraine, that you might copy that for your Health Plan ?”

Obama: “I meant to say: if you would take the Ukraine then your problems would be similar like those that I have. Are you aware of the quagmire, the horror … ? It is Hell on Earth ! You have one actor now to deal with Angela Merkel, a bit like I have Joe Biden to go around and pat people on their shoulders. If you would take the Ukraine, you would need at least a hundred actors to deal with the fall-out. You know how obnoxious actors can be, even if they aren’t gay.”

Putin: “Russian actors aren’t gay.”

Obama: “Well, if you have a hundred Russian actors and you pack them in the Kremlin to control the crisis then some may well become gay. Do you really want this to happen ?”

Putin: “дooрьмо́. (“Shoot”.) Pray Mother Mary to save us !”

Obama: “Say, I have been thinking. Anyone can understand that Russia needs its naval base and cannot be dependent upon whatever government in the Ukraine. As I said, you earned the respect and the appreciation of the American People by taking the Crimea as you did. Why don’t we keep it like that ? And what if I lift most of the bans on Apple and Google technology ? My Ambassador in Moscow has been wanting to give you a smart-phone for ages.”

Putin: “Mатерщи́на. (Something about mothers.) I don’t even have a BlackBerry. You in the West always depict me as a huge dictator but the common person in the West has more power under his fingertips than I do.”

Obama whistles himself.

(67 seconds silence.)

Putin: “Okay, I want to acknowledge for diplomatic purposes that I have received a smart-phone from the US Ambassador, who has returned to his seat across the room, blowing his nose and wiping his tears. Apparently the phone comes with a prepaid annual subscription registrered at the US Embassy in Moscow to my name. I hate to say this, but do so for diplomatic purposes, that I am deeply touched. Does this mean that Americans understand that Russians are cowboys too ?”

Obama: “I presume that we will be seeing each other more often now, either at Camp David or at you dacha. Can I ask Michelle to send you the details of our diets ?”

Putin: “Ah, yes, the details.”

Obama: “And can I beg you not to tell your actor about this ? For some reason she thinks that the world is interested in her opinion, and for now I would like to keep it like that. Nothing worse than a scorned woman.”

Putin: “Of course. Well, I am holding our traditional red phone in one hand and my new smart-phone in the other. I feel torn apart. Shall we use the Apple next time ?”

Obama: “Ahem. Well, this is a bit embarrassing … We have a secure direct link now. I hesitate using my smart-phone because of all the people listening in. You understand what I mean ?”

Putin: “Ah ! Don’t forget that I was at the KGB ! You gave me a bugged phone ?”

Obama: “No, no, not at all ! A secure version, actually. But let me tell you what I understand of this NSA stuff. (…)”

END OF THE TRANSCRIPT