Monthly Archives: September 2015

Listening to Izaline Calister “Mi Pais
“Atardi Korsou ta Bunita”, or Willem Hendrikse,
or Rudy Plaate “Dushi Korsou” or IC & CR “Mi ta stimabu“,
and Frank might also have liked Las Unicas “Ban Gradici Senjor” from Aruba


We need two weblogs to discuss Gerald Goldin (2003), Developing complex understandings: On the relation of mathematics education research to mathematics.

We namely start with Goldin (1992), Towards an assessment framework for school mathematics (pages 63-88 in Lesh & Lamon, Assessment of authentic performance in school mathematics).

The Framework gives a baseline and also allows us to see the long standing involvement of Goldin in mathematics education and its research. We all know about the New Math disaster in the 1960s and about Morris Kline (1908-1992) who in 1973 wrote Why Johnny Can’t Add: the Failure of the New Math. I wasn’t aware however of the heavy involvement in the USA in the 1970s with behaviourism. This US phenomenon wasn’t copied to such extent in Europe. Perhaps the European attention for the New Math partly came from the European (Bourbaki) origins. With these two assaults on mathematics education in the USA, both New Math and behaviourism, we may better understand the flower power reaction movement, partly captured by Hans Freudenthal’s scam of “realistic mathematics education” (RME), in which students are left free to discover mathematics for themselves, with perhaps at most “guided reinvention”.

Recovering from New Math and behaviourism

Thus in 1992, after two decades of struggling to recover from New Math and behaviourism, Goldin writes the following Introduction to this Framework.

Gerald Goldin, "Framework", p63-64, Introduction

Gerald Goldin (1992), “Framework”, p63-64, Introduction

The role of the general public in understanding processes and content is that Goldin wants parents to understand what Johnny is doing in his proposed approach.

Test questions, purposes and methods

Goldin (1992) discusses various possible test questions, and then he questions what they are supposed to test. This is a nice example of a problem that Johnny might be asked to handle mathematically.

Goldin (1992), p68

Goldin (1992), p68

While you are solving this problem, you must also wonder whether this fits the RME philosophy, and what would count as a successful answer. Are you allowed to guess yes/no/maybe or would you have to give a proof ?

Goldin shows how the question can be used for various didactic goals with also various possible interventions, ranging from exploration (my suggestion now too: perhaps just give the egg timers and not pose any questions) to standardized solution technique (that flower power calls boring and behaviourism exciting).

A cognitive model to prevent disasters from new fashions

Goldin cannot avoid Freudenthal’s RME (applied mathematics), and gives an answer that fits Van Hiele’s view that mathematics is directed at higher levels of insight (abstraction).

Goldin (1992), p70-71

Goldin (1992), p70-71

Goldin gives the obvious warning that when we don’t know what we are doing, then we may make all kinds of investments in training and computer programs, that later appear to be useless.

A traditional question

Subsequently, Goldin discusses a traditional question.

Goldin (1992), p72

Goldin (1992), p72

RME would allow students to find their own solution strategies, and students perhaps run afoul when they don’t find any (or arrive at the right answer in mysterious ways). Traditional methods would provide a single algorithm that always generates an outcome, but perhaps students run afoul when the question is slightly changed. I refer to Goldin’s paper for these and more angles.


This discussion is only a rough introduction to Goldin (1992) “Framework”. The idea of this text (1) is to set the stage for weblog text (2). It helps to be aware of what Goldin was already proposing in 1992 when we are going to look at his view in 2003.

My problem with Goldin (1992) is that he does not distinguish the Van Hiele levels of insight. For example, it seems somewhat obvious that traditional Problem 2 requires a different approach for novices than for experts. But it helps to be also aware that the same words may have a different meanings depending upon the level of insight.

For novices Problem 2 may require much creativity and thus can also give a lot of fun. It is only when students become experts that problem 2 becomes boring. The discussion about assessment should not be burdened by the situation that mathematics education can create the paradox that students find Problem 2 boring and still cannot answer it properly. Like in Item-Response Theory, a test does not only say something about students, but students also say something about the test (environment).

Listening to Izaline Calister “Mi Pais
“Atardi Korsou ta Bunita”, or Willem Hendrikse,
or Rudy Plaate “Dushi Korsou” or IC & CR “Mi ta stimabu“,
and Frank might also have liked Las Unicas “Ban Gradici Senjor” from Aruba


Frank Martinus Arion passed away yesterday in Curaçao. The English wikipedia site is a bit short, with his 1973 literary debut Double play. His important scientific work is his thesis: “The kiss of a slave”, that traces Papiamentu to Africa.

Kiss of a Slave, by Martinus Arion, Thesis Univ. of Amsterdam 1996

The Kiss of a Slave, by Efraim Frank Martinus (Arion), Thesis at the Univ. of Amsterdam 1996

Masha Danki !

Frank wouldn’t have wanted us to be sad. The best way to to thank him is to have the biggest party of all.


Carneval 2013 (Source: Screenshot)

I met Frank in the bar of the then hotel Mira Punda in Scharloo. These are old pictures taken by its then-owner Jose Rosales in 2005. Nowadays it is refurbished, and you should check out Hotel Scharloo or see pictures, or see


Hotel Mira Punda 2005 before the refurbishment to Hotel Scharloo (Source: Jose Rosales)

A second time in 2005-2006 Frank came by to discuss the future of the Caribbean, and we sat there on the terras of Mira Punda. I was just getting my driver’s licence so it was impossible to drive up to his place.

Just a year later, in 2006, when I had returned to Holland, his book Double Play was presented as the Dutch liberaries book of the year, and I met him again in The Hague.

Here is my view on the future of the Caribbean, no doubt influenced by these brief but powerful meetings about national independence. Perhaps the Caribbean could develop a sense of nationhood ?


Listening to Roefie Huetng with Jamie’s Blues


Roefie Hueting (1929) is an economist and jazz piano player, or a jazz piano player and an economist, who cannot decide which of the two is most important to him. See this earlier report on his double talent.

Hueting’s first public performance was on stage on liberation day May 5 1945 at the end of World War 2, when he was dragged out of his home to play for the people dancing in the streets. He still performs and thus he has been 55+15=70 years on stage.

With the Down Town Jazzband (DTJB) Hueting recorded 250 songs, played on all major Dutch stages, five times at the North Sea Jazzfestival, while the 50th DTJB anniversity of 1999 was together with the Residence Orchestra in a sold-out The Hague Philips Hall.

Hueting was one of the founders of the Dutch Jazzclub from which sprouted The Hague Jazz Club. This HJC has its current performances at the Crowne Plaza Hotel, formerly known as the “Promenade”. This hotel is at the Scheveningseweg, the first modern road in Holland, created by Constantijn Huygens in 1653, connecting the area of the Peace Palace – the area where also Grand Duchess Anna Paulowna of Russia (1795-1865) had her Summer palace – to the sea. See also these pictures of the German Atlantik Wall – to stay with the WW 2 theme.

At the celebration last Sunday September 27 other performers were Joy Misa (youtube), Machteld Cambridge, Erik Doelman (youtube) and Enno Spaanderman.

The Hague Alderman Joris Wijsmuller (urban development, housing, sustainability and culture) came to present Roefie Hueting with a book containing a picture of Mondriaan‘s Victory Boogie-Woogie – also celebrating the end of WW 2. Wijsmuller observed the erosion of “sustainability” that in the opinion of Hueting rather should be “environmental sustainability”.

Roefie Hueting and alderman Joris Wijsmuller at Crowne Plaza Hotel 2015-09-27

Roefie Hueting and alderman Joris Wijsmuller at Crowne Plaza Hotel 2015-09-27

Roefie Hueting solo at the piano, 2015-09-27

Roefie Hueting solo at the piano, 2015-09-27

Hueting introducing a jam session 2015-09-27

Hueting introducing a jam session 2015-09-27

"Victory Boogie-Woogie" by Piet Mondriaan (Source: Wikimedia Commons)

“Victory Boogie-Woogie” by Piet Mondriaan (Source: Wikimedia Commons)

Listening to Beauty in red


The Scottish MacTutor history of mathematics archive contains a webpage on Hans Freudenthal (1905-1990). It is always useful to have views from outsiders.

They don’t have a webpage on Pierre van Hiele (1909-2010) yet.

I have found that Freudenthal committed fraud w.r.t. the work by Van Hiele.

Being erased from history is not so bad. What is bad is being misrepresented.

Recently, the math war in Holland reached a new low point, when a psychologist who rejects Freudenthal’s “realistic mathematics education” also started attacking Van Hiele, rather than saving him. See my letter to Jan van de Craats.

In other words, Freudenthal so massively abused Van Hiele’s work, that people may see neither Van Hiele’s real contribution nor the abuse: and then some people bunch his work together with the errors by Freudenthal.

David Tall in the UK thinks that he himself now invented what Van Hiele already had invented, see here. What will the history books later say ?

I wondered whether the MacTutor history website only concerned mathematicians with results in mathematics, or also those looking at mathematics education. It appears that they also do a bit of the latter, e.g. by discussing Emma Castelnuovo (1913-2014).

Van Hiele isn’t mentioned on Castelnuovo’s MacTutor page. A google didn’t show yet whether Castelnuovo refers to work by him. This google did recover the Karp & Schubring (ed) Handbook on the History of Mathematics Education (2014) in which they both are mentioned of course.

Freudenthal however is mentioned on her MacTutor page. Van Hiele has declared that Freudenthal misinformed others about his work and what it was really about. Thus if Castelnuovo depended upon Freudenthal for her interpretation of Van Hiele’s work, then there would be a problem.

For example, the page on Castelnuovo contains a confusion between the distinction of mathematics versus applied mathematics (Freudenthal’s “realism”) and the distinction between concrete versus abstract (Van Hiele). See here. See also Research Italy’s interview with Nicoletta Lanciano.

A major reason why Van Hiele is important for mathematics itself is that you need the Van Hiele theory on levels of insight (abstraction) to understand what mathematics is about, actually. See this discussion on epistemology.

Indeed, you can read a novel without actually knowing what a novel is. (wikipedia) Similarly, mathematicians may do mathematics without quite knowing what it is. But it helps to be aware of what you are doing.

For historians it also helps to be aware what history writing is.

PM 1. Check that Amir Alexander doesn’t know what history writing is. PM 2. For those who like irony: Freudenthal wrote on history too. PM 3. The following is not a simpleton’s reaction but the result of seven years of patience that reaches its endpoint: Jan van de Craats refused to properly answer to that letter, and now is in breach of scientific integrity himself, see here. Check how Van de Craats supports mathematics education that tortures kids with fractions.

Screenshot of MacTutor History of Mathematics Archive

Screenshot of MacTutor History of Mathematics Archive

Listening to Just like river water in the spring


Professor Jan van de Craats (University of Amsterdam, now emeritus) is in breach of integrity of science. In an email to me in 2008 he confirms some of my criticisms on mathematics education, but since then he has been effectively neglecting this and refusing to discuss matters. He founded and now advises a foundation SGR for better education in arithmetic, and they employ dubious methods, including neglect and refusal to discuss and refer to criticism. Their criterion on “good” must also contain “keep a closed mind”.

SGR was founded in 2008 and has a Committee of Recommendation. Perhaps that list requires a date, or must be updated, since SGR now supports a particular commercial product, the education method Reken Zeker at a particular publisher, and at least two persons on the list have joined the national council on education that is supposed to be impartial (Maassen van den Brink en Van der Werf at Onderwijsraad).

Let me given an indication how Van de Craats’ breach of scientific integrity also causes bad mathematics education. Let me take two screenshots from two instruction videos from this SGR website.

Two screenshots of videos at SGR

The first video discusses a division of mixed numbers, and the second video discusses the conversion of a square meter into square decimeters. The screenshots are such that you don’t need to understand Dutch. The issues are clear enough. The didactic problem lies in the presentation. An invitation to you is:

Assignment: Spot the problem in didactics of mathematics.

If you cannot spot the problem, try to draw the inference: that you need to brush up on your awareness of didactics, and that you ought to read my book Elegance with Substance, (EWS) 2009, 2nd edition 2015 (with pdf online since 2009, so that you don’t have the excuse of a paywall either).

Thus, if you hate to read EWS, and hate to drag professor Van de Craats to the courts of justice and have him hanged or drawn & quartered, to remain with the subject of fractions, then you will be encouraged to really think and spot the didactic problem that arises from comparing these two images. Clicking on the screenshots will bring you to the videos in Dutch, but only these screenshots are relevant now. Please scroll the computer window in such a way that you don’t see the discussion of the solution below till you have formulated your solution or give up.

Division by two mixed numbers at SGR (Source: website SGR)

Division by two mixed numbers at SGR (Source: website SGR)

Conversion of a square meter to decimeters at SGR (Source: SGR website)

Conversion of a square meter to square decimeters at SGR (Source: SGR website)


The didactic problem with these two screenshots

In the second screenshot 1m or 1 m represents multiplication, or 1 × m, without writing the multiplication sign. In the first screenshot 2 + ⅓ is written as 2⅓ = 2 × ⅓ = ⅔.

One might hold that it is “1 m” with a space and “2⅓” without a space, so that the notations are well defined. This is difficult to maintain in handwriting, especially for kids. It still is needlessly confusing, and thus didactically wrong.

One might also hold that the form a b/c can be recognised as a “number next to a fraction” so that kids should be able to spot the fraction b/c, and then understand that the whole expression would mean a + b/c. This is dubious. If you agree that 10 dm = m so that dm = m / 10, then above example gives a m / 10, so that kids would need to understand this as a + m / 10. Is that really your reasoning ?

If your response now would be that dimensions like m and dm must be treated differently, so that dm = m / 10 is wrong and must be dm = 1/10 m, then you are changing mathematics and introducing a second arbritrary rule just for the reason that you don’t want to admit that you were wrong. It means that you already tortured kids and don’t mind to torture more if it helps to maintain your ego and investments in textbooks full of errors.

The notation for mixed numbers was invented at some time deep in the past, but without proper didactic considerations, and the only reason to maintain it is that mathematicians don’t mind torturing kids.

See Elegance with Substance (EWS) (2009, 2015). I discuss this in 2008, Van de Craats refers to it in his email of 2008, and it could have been solved in 2009, so that it could have been in all methods that were put on the market in 2010, not only Reken Zeker.

In his other own “remedial book” Van de Craats prefers 5/2 over 2½ with the stated reason “because 5/2 is easier to calculate with”, which is a misrepresentation of the real didactic issue.

PM. The first video stops at 49/66, which might be justified since it cannot be simplified anymore or written in mixed number format. The small supplementary problem is that this should be checked and mentioned, which is’t done. The algorithm thus isn’t fully discussed. This is not the key issue here. It just surprises me since SGR puts such an emphasis on algorithms.

Van de Craats and Wilbrink on Pierre van Hiele

Van de Craats also refuses to look into and to refer to criticism w.r.t. the manner how psychologist Ben Wilbrink abuses the work by Pierre van Hiele, even though he has an extensive section with links to the site of Wilbrink. See my discussion of Van de Craats’ breach again.

One of Van Hiele’s suggestion was that fractions can be abolished. See the discussion here. Thus, SGR spends a lot of time on teaching kids fractions that can actually be abolished. Perhaps kids at some stage, when they understand the inverse of multiplication, must be instructed that old-fashioned people write mixed numbers in another fashion. But this is a short explanation. This would not obstruct the whole learning process of mastering arithmetic.


We spotted another case of the elementary sick Dutch mindset that requires a decent boycott.

In this case it is mathematics again. The key issue is that mathematicians are trained for abstract thought and not for empirical science. This is world problem.

The combination of this Dutch mindset with mathematics is especially disastrous.

The appeal to boycott Holland is targeted at the censorship of economic science since 1990 by the directorate of the Dutch Central Planning bureau. This example of the Dutch mindset confirms the analysis on the need of a boycott.

PM. For Dutch readers:

This is a petition on having a parliamentary enquiry into the censorship of economic science.

This is a petition on having a parliamentary enquiry into mathematics education.

How SGR teaches children fractions (Source: wikimedia commons on Dieric Bouts (1415-1475))

The medieval method how Van de Graats and SGR teach children fractions (Source: wikimedia commons on Dieric Bouts (1415-1475))

Listening again to Girls of Ali Mountain


I had some fun today with Google Translate. For other people this is serious research and business, but a lay translator may be excused to play a bit. Unfortunately, play causes questions, it isn’t a free lunch.

Google Translate and the pronunciation of numbers

We discussed the pronunciation of numbers in English, German, French, Dutch and Danish before. Here is a suggestion to develop a standard.

Kids of age 4-6 live and think in spoken language before they learn reading and writing. Thus proper pronunciation of numbers will help them mastering the written number system and arithmetic. A first phase of reading is reading aloud, a later phase is subvocalisation (i.e. become silent), and perhaps later the latter may disappear. Thinking would still be much in “silent spoken language”, while only later the formulas like 1 + 1 = 2 would benefit from thinking in forms (symbol sense).

Ms. Sue Shellenbarger in the Wall St. Journal September 15 2014 discussed The Best Language for Math. Confusing English Number Words Are Linked to Weaker Skills”. 

Hence I wondered how Google Translate deals with this, with their pronunciation icon, and, whether they could support the development of such a standard.

  • When you type in 11, and ask for the pronunciation, then you get eleven.
  • When you type in ten one then you get ten one.
  • Ergo, it would be feasible to create a language tab English-M so that 11 gives pronunciation ten one. (And normal English again for not-numbers.)
Speech examples

When you type in 1111  then Google speech gives eleven eleven, which is wrong. Please do not alert them on this, because I want to keep the example intact. Only 1,111  generates spoken one thousand, one hundred and eleven, which it also should be for 1111. Except that English-M  would give thousand, one hundred, ten one.

Numbers also occur in full sentences. For example translate I will give you 11 dollars into Dutch. Again eleven and elf. Now suddenly 1111 is spoken correctly, perhaps because it are dollars ?

A switch between language and language-M

It might be a single option to select mathematical pronunciation, for all languages. But the tab would need to show English-M and Dutch-M to prevent confusion. Also, at one time, one might wish for a translation from English-M to traditional Dutch. Best could be a selector icon in the row of language tabs that allows you to switch between traditional and mathematical pronunciation.

Google Translate is already prim on the distinction between UK and US English. There is only one English tab, and the translation of say Dutch strengheid gives both rigor and rigour. But this is a spelling issue. Mathematical pronunciation of numbers isn’t spelling reform but an enrichment of language. And it is neither the difference between Oxford English and Cockney. There may be more sites explaining dialects than Oxford English.

Indeed, when we try to translate Me want money from English to English, to remove grammatical or spelling errors, with the options I want money or We want money, then Google Translate doesn’t allow this. It just doesn’t permit translation from English to English. The translation to Dutch selects the Me  I option. “Mij wil geld” is a literal translation but Google corrects into proper grammer “Ik wil geld”. One would however feel that crummy English should be translated as crummy Dutch.

A bit of greater fun is that Google Translate accepts spoken 1 plus 1 = 3, but refuses the input of 1 +1 = 2, perhaps because they think that + is no accepted sign in the English language, or perhaps because they think that it doesn’t need translation.

Language research

Google Translate acknowledges use of results by numerous scientists around the world. A key source is WordNet. (In Holland Piek Vossen is involved in this.) When you look at what they are doing, it is huge and impressive.

By comparison, the pronunciation of the numbers is trivial. Let us start with the 20% of effort that generates 80% of results. It is a suggestion for WordNet and Google Translate to look into this.

Thus the WordNet research group might consider supporting the development of this standard for the pronunciation. Developing the standard might take some time, given the need for consensus to develop. Likely there will be stages: first in education, then in law.

The resources and energy of Google Translate might also make a difference for practical developments, notably by providing example implementations. Formation of English-M need not wait for French-M.

Eventually, Google Translate may develop into Google Language, with checkers on spelling and grammar, thesaurus, rhyme, and what have you. Some users might want writing support, like a warning message that a text is too abstract and that an example is required.

It shouldn’t be too difficult either to make an app how to pronounce numbers in English-M, but this weblog isn’t about commerce.

Pierre van Hiele and the levels of insight

Pierre van Hiele presented a theory of levels of insight as a general theory for all epistemology. Geometry was where he started, and what he used as his key example case. Many people didn’t listen well and assumed that he thought that the levels apply only to geometry. See the error on wikipedia that I just linked to, or the misconception by David Tall, who thinks that he was the first one to discover the generality, but who at least supports the notion.

A consequence for language

A consequence of the theory of levels is that students speak different languages.

They use the same English words but mean something else. There will generally be great confusion in the classroom and lecture hall, except for the teacher, who can mediate between students at different levels of insight, including those who are making the shift.

Thus, depending upon the particular field F ∈ {mathematics, physics, biology, economics, …} Google Translate ought to have English-F-1, …, English-F-n. Mathematics would have the highest level because of the notion of formal proof. Perhaps that the majority of fields F might work with only three levels: novice, verbally fairly competent but reproductive, and reasoning informally.

These would also be the levels required for wikipedia-1, …, wikipedia-n. Wiki-articles on math topics are dominated by MIT students who copy their textbooks, which produces gibberish for novices, which isn’t quite the purpose of an encyclopedia. (And some students think they know it better anyway, see here.)

When Google Translate could translate English-M-2 to English-M-1 (as far as possible), then Google Translate would turn into a teacher’s assistent.

Language spaghetti

It may be that current translators, say from English to Spanish, might not be aware of the Van Hiele levels. The issue might not be quite urgent.

  • When translators focus on “words only” then they might translate English words into say Spanish words, and then let others deal with what those words mean to them.
  • Speakers of English-4 might use sentences that contain a few words that users of English-3 don’t use much – e.g. the very word “proof” – so that the translation from English-4 to Spanish-4 would tend to work.

Other cases might simply be spaghetti that perhaps might be neglected.

For example, users of English-2 could use terms from English-4, that they actually don’t understand. They may translate into Spanish-4 – e.g. “I got a proof” becomes “Tengo una prueba”. They wouldn’t understand either of those – since they don’t understand the notion of proof yet – so that this might not be a great loss.

It is a wary notion that Google Translate will perhaps be mostly busy in translating what people don’t understand anyway. Perhaps an exam needs be taken before you offer something to be translated. But we live in a fast world.

It remains valuable to be aware of levels

The upshot is that it would still be a valuable idea to identify Van Hiele levels. Words that seem the same have different meanings, because of those levels.

Wikipedia already uses the disambiguation. They seem to regard it as the minimal word that isn’t ambiguous itself, and take quite some space to explain it so that misunderstandings are excluded. I still wonder about the Van Hiele levels. A novice would only be aware that the same word has different uses (A. Einstein might also be Alfred Einstein), while a more experienced wiki disambiguator would see ripe fruits everywhere.

Google Translate already knows about different communities – say, bubble originates in the soap industry but is used metaphorically (a form of abstraction) in economics (stock market bubble). The word translates nicely into Spanish burbuja, and Google already indicates that also the Spanish speaking world would be aware of the notion of living in a bubble – check here. But perhaps we are missing some higher levels of abstraction here, like 1 bubble + 1 bubble can have all kinds of outcomes, sometimes 0, 1,2, 3, … bubbles. Not only in reality, but also in economics, and perhaps some topological models, or when a man in a bubble meets a woman in a bubble. For some a bubble is just a word, for others a world.


Your level of fun may increase by maintaining a lay level of insight.

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.


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.