Abstraction & numerical succession versus ‘mathematical induction’

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 1^H, 2^H, 3^H, ….. PM. We denote n^H = 1 / n, to be pronounced as per-n, see the earlier discussion on n^H.

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 n^H = 1 / n, see the earlier discussion on n^H.

Theorem: For all n ∈ N:

1 + 2 + 3 + … + n = n (n  + 1) 2^H

Proof: By numerical succession:

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

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

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

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’.

Abstraction vs Eugene Wigner & Edward Frenkel

Thinking depends upon abstraction. Let Isaac Newton observe an apple falling from a tree. The apple and the tree are concrete objects. The observation consists of processes in Newton’s mind. The processes differ from the concrete objects and leave out a wide range of aspects. This is the definition of abstraction: to leave out aspects. Perhaps nature “thinks” by means of the concrete objects, but a mind necessarily must omit details and can only deal with such abstractions. For example, when Newton suddenly is hit by the idea of the universal law of gravity, then this still is an idea in his mind, and not the real gravity that the apple – and he himself – are subjected to.

Newton discovers the universal law of gravity, (c) marcelgotlib.com

Edward Frenkel’s reference to Eugene Wigner

There is this quote:

“The concepts that Yang and Mills used to describe forces of nature appeared in mathematics earlier because they were natural also within the paradigm of geometry that mathematicians were developing following the inner logic of the subject. This is a great example of what another Nobel Prize-winner, physicist Eugene Wigner, called the “unreasonable effectiveness of mathematics in the natural sciences.” [ref] Though scientists have been exploiting this “effectiveness” for centuries, its roots are still poorly understood. Mathematical truths seem to exist objectively and independently of both the physical world and the human brain. There is no doubt that the links between the world of mathematical ideas, physical reality, and consciousness are profound and need to be further explored. (We will talk more about this in Chapter 18.)” (Edward Frenkel, “Love & Math”, 2013, p 202, my emphasis)

Hopefully you spot the confusion. Frenkel is an abstract thinking mathematician with some experience in science – e.g. with a patent – but apparently without having understood the philosophy of science. This weblog has already discussed some of his views, see here, especially his confusion about mathematics education while he hasn’t studied the empirical science of didactics. It is a chilling horror to hear him lecture about how math should be taught and then see the audience listening in rapture because they think that his mathematical brilliance will certainly also generate truth in this domain.

Eugene Wigner’s error – see the paper below – is to forget that abstraction still is based upon reality. When reality consists of {A, B, C, …, Z} and you abstract from this reality by looking only at A and leaving out {B, C, …., Z} then it should not surprise you that A still applies to reality since it has been taken from there.

Mathematical ideas have a perfection that doesn’t seem to exist in concrete form in reality. A circle is perfectly round in a manner that a machine likely cannot reproduce – and how would we check ? If the universe has limited size then it cannot contain a line, which is infinite in both directions. Both examples however are or depend upon abstractions from reality.

Since mathematics consists of abstractions, we should not be surprised when its concepts don’t fully apply to reality, and neither should we be surprised when some applications do. That is, there is no surprise in terms of philosophy. In practice we can be surprised, but this is only because we are mere human.

Paper on abstraction

This issue on the definition and role of abstraction is developed in more detail in this paper, also in its relevance for mathematics education and our study of mind and brain: An explanation for Wigner’s “Unreasonable effectiveness of mathematics in the natural sciences”, January 9 2015.

A correspondent commented:

“It seems to me that the question that Wigner is asking is “Why is mathematics so much more effective in physics (which is what he means by ‘natural sciences’) than in most other studies?”  Physics textbooks are full of formulas; these comprise a large fraction of what the field is, and have great predictive power. Textbooks on invertebrate biology have few mathematical formulas, and they comprise only a small part of the field. Textbooks on comparative literature mostly have no formulas. So an answer to Wigner’s question would have to say something about what it is about physics _specifically_ that lends itself to mathematimization; merely appealing to the human desire for abstraction doesn’t explain why physics is different from these other fields.
I have no idea what an answer to Wigner’s question could possibly look like. My feeling is that it is better viewed as an expression of wonderment than as an actual question that expects an answer.” (Comment made anonymous, January 9 2015)

I don’t agree with this comment. In my reading, Wigner really poses the fundamental philosophical question, and not a question about a difference in degree between physics and literature. The philosophical question is about the relation between abstraction and reality. And that question is answered by reminding about the definition of abstraction.

I can agree that physics seems to be more mathematical in degree than literature, i.e. when we adopt the common notions about mathematics. This obviously has to do with measurement. Use a lower arm’s length, call this an “ell“, and proceed from there. Physics only has taken the lead – and thus has also the drawbacks of having a lead (Jan Romein’s law). Literature however also exists in the mind, and thus also depends upon abstractions. Over time these abstractions might be used for a new area of mathematics. Mathematics is the study of patterns. Patterns in literature would only be more complex than those in physics – and still so inaccessible that we call them ‘subjective’.

For example, the patterns in Gotlib’s comic literature about Newton & his apple might be more complex than the patterns in the physics of Newton & his apple, as described by his universal law of gravity. All these remain abstract and differ from the concrete Newton & his apple.

There is no “unreasonable effectiveness” in that Gotlib’s comic makes us smile.

Colour coding conventions on Europe

Listening to Theodorakis, The struggles of the Greek people

 

Last weblog referred to Pseudo Erasmus who referred to Graig Willy who referred to Thierry Medynski who referred to Emmanuel Todd.

Medynski uses a colour scheme for Todd’s categories that I find hard to remember. It also appears that Willy has given a colour to Russia while this is not available from Medynski. Thus, let me return to Medynski’s map and propose a colour coding that seems easier to remember (updated May 18).

My suggstion is: Green will be the authoritarian stem family structure that can live with inequality.  Gray blue will be the authoritarian family structure that wishes to see equality except for the patriarch. Red allows for inequality but because of liberal tendencies. Blue combines liberalism and equality. The blue-ish area identifies the region in which equality dominates.

“Todd identifies four premodern European family types according to two major criteria: Is an individual free upon adulthood or does he continue to live with, and under the authority of, his parents? Are brothers equal, notably in terms of inheritance, or are they unequal.” (Craig Willy’s summary of Todd)

My colour proposal	Authoritarian	Liberal (free from parents)
Unequal	Stem (green)	Nuclear (red)
Equal (inheritance)	Communitarian (gray blue)	Nuclear egalitarian (photon) (blue)

This gives the following map – in which the legend is also sorted from blue to red.

Traditional family systems of Europe (1500-1900) (Source: Todd – Medynski)

There is more cohesion between Germany and Norway and Sweden than commonly perceived.

Relation to the USA

My suggestion is based upon the USA Red and Blue, for the Republican versus Democratic states.

USA Red and Blue States, for Republican and Democratic Party outcomes at Presidential elections. Purple: mixtures over elections (Source: Wikipedia)

The differences between red and blue states may not be quite comparable to Todd’s scheme, but it helps to develop the idea and identification. Still, the clue is that the USA apparently has been shaped predominantly because of the nuclear family structure.

“Les États-Unis et l’Europe n’ont pas le même projet de société du fait de leurs structures familiales. Structurés sur la famille nucléaire absolue, les États-Unis expriment une dérive du fondamentalisme protestant avec cette vision messianique et civilisatrice pour diriger le monde selon leurs propres intérêts. Du fait de sa mosaïque de structures familiales, l’Europe devrait favoriser l’émergence d’un monde polycentrique. Cependant, depuis l’Acte Unique, tout se passe comme si l’identité européenne était réduite aux valeurs véhiculées par la famille nucléaire absolue, à savoir la pensée unique du néo-libéralisme. D’où l’échec de cette conception de l’Europe.” (Medynski, my emphasis)

The differences between Republicans and Democrats thus may be linked to the differences between England and the Ile de France.

Consequences for Europe and the euro

Check out Todd’s 2013 Harper’s video on the euro – with thanks to Pseudo Erasmus for alerting us to this. See also Jamie Galbraith and perhaps also not so strong John Gray. And then see my paper Money as gold versus money as water.

PM

PM 1. For completeness and comparison, this is the colour scheme of Medynski’s image. We changed only red and yellow but it still makes a difference in reading.

Thierry Medynski	Authoritarian	Liberal
Unequal	Stem (green)	Nuclear (yellow)
Equal	Communitarian (red)	Nuclear egalitarian (blue)

PM 2. Never forget about the Heineken Eurotopia map.

PM 3. Check whether there is a relation with the other French intellectual, Thomas Piketty.

PM 4. Russia would have the gray blue too, which confirms Willy’s adaptation of Medynski’s image.

“Cette mosaïque de systèmes familiaux distingue l’Europe des Etats-Unis (structurés sur la famille nucléaire absolue) et de la Russie (structurée sur la famille communautaire exogame) où seul un des termes, l’individualisme ou le système communautaire, est privilégié.” (Medynski, my emphasis)

The myth on Germany’s decentralised bargaining

Dustmann et al. reported on Germany’s transformation from the Sick Man of Europe around 2000 into the Economic Superstar after 2010, in 2014a VoxEU and 2014b the Journal of Economic Perspectives.

Their VoxEU summary is:

“In a slow-growth, high-unemployment continent, Germany’s performance stands out. The success is often ascribed to the politically difficult Hartz labour-market reforms. This column discusses evidence to the contrary. The Hartz reforms played no essential role. Rather, the key was the threat of offshoring to central Europe together with the pre-Hartz structure and autonomy of the German labour-market institutions. This structure allowed trade unions to make wage concessions necessary to adapt to the new realities. Other nations should decentralise bargaining to the firm level while keeping workers’ representatives.” (Dustmann, Fitzenberger, Schönberg, Spitz-Oener, VoxEU, Feb 3 2014, my emphasis)

The latter statement in bold is a New Myth.

Response to the New Myth

In 2014 former IMF managing director Johannes Witteveen (1921) gave his valedictorian lecture on the Dutch policy of wage restraint and the surplus on the external account. In comment C I already responded to the Dustmann et al. analysis:

“The Dutch surplus actually started in 1981, see my 2009 paper “A macro-economic lesson from Holland“.
Gerhard Schröder (BRD Kanzler 1998-2005) started to copy the Dutch model of wage restraint. The consequence was that Germany and Holland out-competed the rest of Europe, creating the imbalances of the eurozone.
See The Economist “Model Makers” May 2 2002.
In VoxEU Feb 3 2014, Dustmann et al. look at wage-bargaining structures and argue that it wasn’t Hartz 2002-2005 that caused the German low wage policy, but rather the German Reunification in 1989. Okay, but: (1) The Dutch example helped Schröder to target lower rather than higher wages, (2) It remains important to maintain macro-economic co-ordination. Herein lies the main policy objective rather than in such wage-bargaining structures. The analysis by Dustmann et al. might be interpreted as suggesting the abolition of national bargaining but that would be false. My advice is also an Economic Supreme Court per country.“ (Colignatus, comment C, May 21 2014 now bold)

The Dutch Royal Association on Political Economy (KVS) had a Preadvies on collective bargaining in December 2013.

(Former) KVS chairman Arnoud Boot (1960) – also a member of the Dutch Social Economic Council (SER) that monitors collective bargaining in Holland – liked the discussion so much that I felt encouraged to inform him in Februari 2014 about the drawbacks of the analysis by Dustmann et al.

Notably:

• A factual analysis does not quite refute a counterfactual. The scenario of wage restraint might also have been adopted by central bargaining. When Germans grew aware decentrally that they needed restraint then this might also have surfaced centrally. The Dutch model of wage restraint was well known in Germany at that time.
• Macro-economics remains more relevant than industrial relations. Papers on industrial relations say more about national conventions rather than about outcomes. Outcomes are rather determined by market conditions and policy decisions.
• One should rather criticise the lack of co-ordination in Germany. Wage restraint (in West Germany) is no solution for the lack of investments (in East Germany). Certainly not when this causes problems in the rest of Europe.
• If the advice by Dustmann et al. is followed then co-ordination in Southern Europe would be abolished, perhaps with ever lower wages without investments, which is a recipe for continued Depression. Beware of looking at the world through the glasses of industrial relations.
• My comments on the KVS Preadviezen 2013 still stand: they are deficient on the Dutch policy of wage restraint and the lack of investments. Political Economy causes the suggestion of an Economic Supreme Court.
• For the fall of the Berlin wall, see this memo.

Pseudo Erasmus and the European Soul

This issue becomes a bit more relevant since I came across a good summary of the New Myth by Pseudo Erasmus. This forms more entertaining reading, since this author isn’t quite interested in actual policy making, but rather searches for the elusive European Soul.

What makes Germany, France, England and so on tick, and keeps them apart ?

Reading requires these steps:

1. First savour “Der Todd des euro“ that debunks Emmanuel Todd‘s anti-German-isms.
2. Secondly savour Pseudo Erasmus’s link to Graig Willy’s summary of Emmanuel Todd’s anthropological analysis on Europe.
3. Thirdly savour Pseudo Erasmus’s discussion of the “anthropology” of the crisis, in which the New Myth also pops up. Pseudo Erasmus call’s Dustman et al. the “best” account of what happened but doesn’t think that the scheme would work in all countries. He basically agrees that the advice for decentralised bargaining in all countries is a myth indeed. His reasons differ from those above but can be supplemented with those.
4. Finally check out that democratic nations need Economic Supreme Courts.

PM 1. I am reminded of Frank Sulloway, “Born to rebel”, 1996, who suggested a link with inheritance laws, in Catholic countries all going to the eldest son, and in Protestant countries an equal split. PM 2. Pseudo Erasmus states: “adjustment-through-recession is a painful process which might be alleviated, even avoided, if everyone could “share the pain”, rather than a select group of people be unemployed. But that just does not happen, because it cannot — except in Germany.” This isn’t quite true, since the Dutch model of wage restraint (exporting unemployment) is ex ante sharing of such pain (even before a recession). PM 3. Of course Germany should not be lumped together: there are important regional differences.

Since Pseudo Erasmus has an anonymous profile the picture prize is for Emmanuel Todd.

Emmanuel Todd (1951) in 2014 (Source: wikimedia)

The End of the History Manifesto

Jo Guldi (Brown) and David Armitage (Harvard) wrote The History Manifesto (html or PDF). On May 12 professor Armitage came to Amsterdam to defend it at the Academy of Sciences KNAW.

The authors argue that historical research gets lost into short-term-ism and overspecialisation, and that there is a growing need for the longue durée (Fernand Braudel 1902-1985) and “big stories”. The Manifesto closes with a call-to-arms:

“Once called upon to offer their advice on political development and land-reform, the creation of the welfare state and post-conflict settlement, historians, along with other humanists, effectively ceded the public arena, nationally as well as globally, to the economists and occasionally lawyers and political scientists. (When was the last time a historian was seconded to Downing Street or the White House from their academic post, let alone consulted for the World Bank or advised the UN Secretary-General?) It may be little wonder, then, that we have a crisis of global governance, that we are all at the mercy of unregulated financial markets, and that anthropogenic climate change threatens our political stability and the survival of species. To put these challenges in perspective, and to combat the short-termism of our time, we urgently need the wide-angle, long-range views only historians can provide. Historians of the world, unite! There is a world to win–before it’s too late.” (The History Manifesto p125 – my emphasis)

Amsterdam, May 12 2015. Wonderful weather outside of KNAW.

Remarkably, on the three indicated areas economics has much more to offer than history:

• For climate change and survival of the species there is my book on the Tinbergen & Hueting Approach (2009, 2015).
• On unregulated financial markets and income inequality, there are my books DRGTPE (2000, 2005, 2011) from before the crisis and CSBH (2012) from after the crisis – see above About page.
• On global governance there is the analysis in DRGTPE that each nation better adopts a constitutional Economic Supreme Court (ESC) – so that the national ESCs can exchange information and thus contribute to global co-ordination and stability. For example, see this memo in the RES Newsletter of Fall 2014.

These issues can only be resolved by economics. One needs to study political economy (see DRGTPE for its definition) and have a solid background in econometrics and macro-economic modeling to understand and judge the issues. It is necessary indeed to take the long view, since it are this kind of topics. By implication the economist looking into these issues might be regarded as being a “historian” – and perhaps historians are willing to respect this even though such an economist might have no formal training in such an MA course.

It puts the horse behind the cart when one presumes that a student of the past would, by this kind of academic study, hit upon the proper advice to deal with these issues for the future. I am afraid that Guldi & Armitage are seriously mistaken here. It is important for a political economist to delve into history, and historians can provide valuable service here, but a historian would have to become a political economist if he or she is to say something about these subjects.

Consistent links

Of course, when the survival of the species is at stake, and thus also the survival of historians, then one can imagine that some historians feel the need to say something about this. But rather than starting to re-invent the wheel themselves, they are advised to check out the designated smiths. Dutch readers may check my question on ecological survival for the Dutch research agenda to 2025.

Admittedly, I may not be the typical economist, and Guldi & Armitage merely “misunderestimated” the situation. Originally I wanted to study archeology but it was because of Biafra and the world problems that I decided to turn to econometrics. At the KNAW session I indeed met an archeologist who confirmed that he was quite comfortable with long time scales. My recent revisit of the old interest is in “The simple mathematics of Jesus” (2012). An apt reference is also to this weblog text.

I am also struck by this statement in the manifesto:

“There is no public office of the long term that you can call for answers about who, if anyone, is preparing to respond to these epochal changes.” (The History Manifesto, p 1)

Traditionally people have the right to petition the monarch, and democracy seems to reduce the need for that, but our model of democracy still fails. The creation of an Economic Supreme Court would amend that. Some analysts who see short-term-ism everywhere might fear that the ESC would also fall victim to it. However, it is the task of the ESC to check the quality of the information for policy making. Hence, it looks at both the short and the long terms.

A question during the session

The discussion monitor at KNAW invited an economist in the audience to ask a question. My question was:

“The Dutch prime minister Mark Rutte studied history. However, he shows quite a disrespect for science. How is it possible that an academic study causes this kind of disrespect, and what can be done to ensure that the study of the past maintains scientific standards ?” (Rephrased.)

Dutch readers can check that I essentially asked the same question for the Dutch National Research Agenda for 2025. A key problem is: when a historian in a public debate makes an error against science then he or she is seldomly corrected by another historian.

Mark Rutte is a counterexample to the Guldi & Armitage bracketed question above: here we have a historian with access to the corridors of power. Check on Rutte e.g. here.  Another example appears to be William Hague. One can also understand that my question is essentially critical of the History Manifesto too: Are Guldi & Armitage not similarly disrespectful of science, by claiming more for history than history can do ?

The Dutch word for science is “wetenschap” and it does not distinguish between the humanities and the “hard” sciences. My question fell a bit in the trap of the distinction made in the Anglo-Saxon world. Professor Armitage referred to C.P. Snow and the “two cultures” lecture of 1959, and indicated that the gap ought to be bridged. He explained how students of history at Harvard are encouraged to look into the sciences, including mathematics and computer science. He did not go into the issue that the scientific method should be used in history as a science too. Fortunately the History Manifesto on occasion refers to the humanities as a science too (say p 10), but then it should also apply the methodology of science – with the comparative advantage of Verstehen.

Thus his answer left me unsatisfied. For these same considerations as Snow had, the Econometric Society was founded in 1930, with an offshoot in cliometrics. The gap between the “two cultures” can only be bridged when it becomes mandatory that a sizeable section of the humanities have also a background in the sciences – i.e. when the humanities stop claiming that they would be so very special and when they concentrate their contribution on what they indeed are special in and what indeed is their comparative advantage. Indeed, this will also require a re-engineering of the education of mathematics, see “Elegance with Substance” (2009).

The panelists

Invited panelists were Mathijs Bouman (economic journalist), Rens Bod (director of Digital Humanities), Hanco Jürgens (Germany Institute Amsterdam) and Siep Stuurman (emeritus UU).

Rens Bod was most ambitious: historians should also predict the future. This indeed fits the scientific method, and prevents that historians just tell fancy stories and enhance those with the authority of age. The ambition to look at the future fits the ancient tradition in history, check e.g. Thucydides and the wish to study history to learn from it. The ambition however runs the risk of failure when historians are not up to the task. Thucydides of course wrote at a time when he could not rely on a well-developed discipline of political economy.

Hanco Jürgens wondered whether there really was a crisis in the humanities and a lack of public attention for history. He referred to the commemorations of WW II this month. That Russia had taken the Crimea put an end to Francis Fukuyama’s End of History and meant a Return of History. There is rather a change in the public role of the intellectual. In Holland in 2030 likely half of the student population will have higher education (?) and that will mean a change in the kind of public debate.

Siep Stuurman emphasized professionalism. Historians should remain critical. The “protestant wind” that destroyed the Armada is an obvious bias – see “Whig history“. The method of the longue durée rather looks for turns that enlighten issues (and that were surprising to participants too). Thomas Malthus had fine data from the past but lacked insight in what technology had in store. History can puncture myths, like that protectionism hindered development and that free trade supported it: in the past protectionism was the norm, and England only imposed free trade when it had the advantage anyway. His main point was that history provides serial contexts.

The other economist: Mathijs Bouman

Bouman supported the criticism of short-term-ism by giving the example of the 2008 financial crisis. He labeled financial analysts as “historians” since they used time series data. They used only a few decades, and thus they missed systematic national risk on house prices. He agreed that a good economist is also a historian. On the other side he found the History Manifesto biased against neo-liberalism. Marx and Piketty got too much attention, even though the latter indeed presented data over a much longer period. The long term would require an unbiased view too.

Bouman supported my critical question above by stating that developing a view about the future does not only require data from the past but also model assumptions: and what theory was Armitage using ?

Of course, Bouman did not explain to professor Armitage that the directorate of the Dutch Central Planning Bureau has been censoring my work since 1990. In general, visitors who come to Holland are treated as guests and they meet with kind people and have pleasant discussions. These visitors will tend to think that Holland is an open and tolerant country.  KNAW functions as a Potemkin village for foreign scientists.

Conclusion

Given that economics already solved the three major problems mentioned by Guldi & Armitage – so that it is only a matter that the Parliaments of the world start studying on those solutions – I can only advise that the historians enter these notions in their history books. By consequence there is little use for the History Manifesto.

What we should appreciate is its opening statement of “speaking truth to power“. This can best be done by embracing the scientific method, to first find that truth, and then say it.

Addendum May 14

A statement in the manifesto about what Paul Warde would have shown made me very curious and caused me to check this on the internet: I found a discussion by Pseudo Erasmus on errors in the manifesto also on other statements. Perhaps this is a case in which historians correct other historians.

I also found the criticism by Cohen & Mandler in the AHR rather convincing. The best thing would be to retract the manifesto as an apparently insufficiently researched opinion piece that needs better contemplation.

Amsterdam, May 12 2015. Still wonderful weather after the KNAW session.

D66 con on the 2015 UK General Elections

Holland has Proportional Representation (PR) and the UK has District Representation (DR). The 2015 UK General Elections under the current DR system have a result that differs dramatically from a PR result, see the graphs here. The Conservatives with 36.9% of the vote get 50.9% of the seats – and given the turnout of 66.1% they only have 24.4% of the electorate. Their only restraint now is the prospect of losing in the next election. In Holland, parties tend to be forced to form coalitions. Compromises tend to make for less extremes.

To understand the issue, first check this short paper of mine, that appeared in Mathematics Teaching 222, January 5 2011, the journal of the UK Association of Teachers of Mathematics (ATM). The short paper summarizes this longer more involved comparison of PR vs DR.

Mathematics Teaching 222, January 5 2011, journal of the ATM

The Liberal Democrats in the UK want a change from DR to PR. They entered the 2010 coalition also to get the 2011 referendum. Curiously, Nick Clegg selected the Alternative Vote (AV) system. The AV system is needlessly complex and it was duly rejected in the referendum. Nick Clegg might have had more success by proposing a simple PR as in Holland.

“The campaign was described in retrospect by political scientist, Professor Iain McLean, as a “bad-tempered and ill-informed public debate”.” (Wikipedia article)

Now check the following curious point: the Liberal Democrats in Holland – the party D66 formerly known as Democrats 1966 – want Holland to change from PR into DR.

Originally in 1966 D66 seems to have opted for the first-pass-the-post DR as nowadays still exists in the UK and USA. Around 2003 they shifted their preference to (something like) the German electoral system with two votes: one vote for the local representative and another vote for the national level to assure PR. This system tends to obscure the true preferences and is often abused to by-pass the 5% electoral threshold, by voting locally for a small party and nationally for a bigger one.

FAQs:

1. Are LDs just being obnoxious ? Answer: The Dutch D66 for 100% and the UK LD for 50%.
2. Do they ever talk with each other ? Answer: Perhaps, but they don’t listen anyway. See below.
3. Why did Nick Clegg in the May 2010 coalition agreement propose the complex Alternative Vote system for the 2011 referendum, while he could have proposed the simpler Dutch PR system and then gained more support ? Answer: There still is that 50% of being obnoxious. Clegg (half-Dutch) didn’t diagnose that his Dutch political friends of D66 are 100% obnoxious.
4. Do the LDs ever study what science has to say on electoral systems ? Answer: Perhaps the UK LD do but the Dutch D66 apparently do not.

The point is this: For D66, the plea for a DR started as a gimmick in the roaring 1966s, it became one of their crown jewels, and they never really studied it.

Two other “democratic crown jewels” of D66 are – and of which they neglect scientific findings as well:

• Direct election of prime minister and mayors. In reality: direct elections are riddled with paradoxes, see my book “Voting Theory for Democracy“. It is better to have PR for Parliament, and then let Parliament use the more complex systems to select the prime minister (including bargaining).
• Referenda. In reality: these have similar paradoxes like direct elections, while there is great sensitivity on the exact phrasings, and while the examples of demagoguery abound. It is much better to have coalition government in a PR system, so that decisions can be taken with more information and by representation of the relevant interests.

D66 has fluctuated between 3 and 24 seats in a 150 seat Dutch Parliament. They tend to gain votes in opposition and then lose seats when in government, often by showing incompetence. D66 founder Hans van Mierlo (1931-2010) who in 1966 selected the “democratic crown jewels” out of thin air because of his infatuation with the USA, also was foreign minister during the Srebrenica massacre in July 1995 and didn’t resign.

D66 seems to have mastered the trick of “seeming reasonable”, by expressing pro’s and con’s of arguments. They put much emphasis on education whence they have some appeal to students. There is always some young face who appeals to reason and more democracy, and who gets some fancy of the public – like demagogues tend to do.

Dutch Boris van der Ham, an actor by education and profession, also D66 parliamentarian in 2002-2012 and now chairman of Humanistisch Verbond, explains how the UK LD lost so many seats in the 2015 UK General Elections: see his text in English. The LD went from 23% to 8% of the votes and from 57 to 8 seats.

1. Van der Ham holds that Nick Clegg “took his responsibility” and voters punished him for it. In reality, it is a political debate also whether Clegg promised too much, or was naive, or failed w.r.t. the 2011 referendum as explained above, or made some bad political calculations.
2. Van der Ham is fully silent about the above, i.e. his own responsibility in the communication with Nick Glegg in November 2010 about the important issue of electoral reform. He could have warned Clegg that AV is complex and that there are simpler alternatives. He could also have told Clegg that D66 did not really study the issue scientifically and entertains a bias in favour of say the German DR.
3. May 13: Since Van der Ham is silent on this, I sent some questions to him, and he replied. His reply allowed me to change this present text a bit from hypothetical to confirmed. Indeed, when D66 would get more than 50% of the vote in Holland, then they might change the current PR into the less transparent German DR system, and use the new-speak that this would be “more democratic”.

The Dutch version of Van der Ham’s article on Joop.nl proudly shows his picture with Nick Clegg from November 2010. Here is my response in Dutch, with reference to an earlier article by me on Joop.nl on the math of elections, and here is my pamphlet arguing that reasonable people in D66 should abolish their party for its anti-scientific foundations.

Boris van der Ham (D66) & Nick Glegg (LD) in November 2010 (Source: Van der Ham on Joop.nl)