Now available on website and in bookstore

Foundations of Mathematics. A Neoclassical Approach to Infinity (FMNAI) is for (1) students interested in methodology and the foundations of mathematics – e.g. studying physics, engineering, economics, psychology, thus a broad group who use mathematics – and (2) teachers of mathematics who are sympathetic to the idea of bringing set theory and number theory into mathematics education.

The book presents:
(A) Constructivism with Abstraction, as a methodology of science.
(B) Particulars about infinity and number theory, within foundations and set theory.
(C) Correction of errors within mathematics on (B), caused by neglect of (A).

Other readers are (3) research mathematicians, who would benefit from last correction, but who must mend for that they are not in the prime target groups.

Set theory and number theory would be important for a better educational programme:
(i) They greatly enhance competence and confidence.
(ii) They open up the mind to logical structure and calculation also in other subjects.
(iii) They are fundamental for learning and teaching themselves.

The axiomatic system for set theory ZFC is shown to be inconsistent. Mathematics has been in error since Cantor 1874 because of neglecting above methodology of science.

Warning: This book proves radical new ideas and must be read with care. If you spend more than a cursory glance on this summary then it is advisable to get the hardcopy and continue reading from paper. I tend to focus my research on misconceptions that lead society away from common sense, and then I select pivots that cause crucially different points of view depending on how the argument is resolved. Such a pivot only works well if the argumentation gets proper attention.


Foundations of Mathematics. A Neoclassical Approach to Infinity

Listening to Mikroutsikos – Milva – Volpe d”amore
and Live 1995

Part surprise, part dark fear that this was always a possibility: ZFC is inconsistent.

You can read the paper here.

For who doesn’t get the letter soup: ZFC stands for the Zermelo & Fraenkel & Axiom of Choice system of axioms for set theory. ZFC is supposed to be the foundation for modern mathematics. Alas, mathematics appears to be founded in the minds of mathematicians and not on those axioms.

Around 1900, Gottlob Frege (1848-1925) started the business of axiomising set theory. When he had one of his books ready for print, Bertrand Russell (1872-1970) wrote him about what became known as Russell’s paradox. Frege still wanted his book printed, but hurriedly included a line that Russell’s paradox made him doubt.

“Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion.” (Gottlob Frege, 1903)

Logicomix tells the saga in vivid comics. The problem with Logicomix is that they haven’t read my book ALOE (1981, 2007, 2011), while there now is this new result on ZFC, that puts quite a bit of flavour on the connection between Russell and Cantor and ZFC.

Indeed, there is a new dramatic story to tell, involving the need to boycott Holland.

Also, the authors of Logicomix are all Greek, except for Annie Di Donna: which not only shows the importance of exceptions, but also indicates the surprising links in Europe, say between Holland and Greece – or my admiration for Milva in 1995 in Syracuse (Magna Graecia). But, of course, math is an issue for the whole world, so don’t feel excluded.

Logicomix, cover (Source: wikimedia commons)

Logicomix, cover (Source: wikimedia commons)

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”.


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


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 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).

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.

Philippe Legrain (1973) is a British economist who worked as an advisor for former communist and later EU commission president José Manuel Durão Barroso in 2011-2014. Legrain has now written a book on a European Spring, something that Manny tried hard to achieve, as readers of this weblog know, as I tried to advise Manny too.

The real news from Brussels is that Manny’s successor Jean-Claude Juncker doesn’t want a Spring but a Hot Summer. A showdown with Putin should unite the peoples of the EU, and also force Greece to choose sides, JCJ thinks.  Philippe Legrain’s book is old hat. I am still waiting for approval by JCJ to spill some of the beans of our luncheons, dinner parties and fire-side chats. The big secret in Europe is that Angela, François, Jean-Claude and me were all born in 1954, and that we share an obligation to make it all work, though I am for peace-with-strength and they are hopelessly confused.

This Sunday, Philippe Legrain allowed Dutch television to interview him on his Spring eulogy. Legrain is unaware that Dutch TV journalists are hypocrites by profession, and that he is sucked dry like in those vampire movies.

The interview is in English with Dutch undertitles, and you can check how interviewer Marcia Luyten (1971) takes Legrain for a walk in the woods, guts him, and leaves his body somewhere in a ditch.

About a key moment in the economic crisis:

“There was a panic across the Eurozone, and people thought that governments could not pay their debts. And actually it was a panic that ultimately the ECB was able to solve. In the case of the Netherlands it wasn’t even a panic. This was just a mistake made by this government, under pressure from Brussels, and Berlin.” (Philippe Legrain, minute 35).

It is absurd to portray Holland as a victim of Brussels. It is the other way around. Europe is a victim of Holland:

  • It were rather speculators who saw an opportunity to make a killing. It wasn’t a panic but a real threat, caused by wrong legal rules.
  • The solution provided by the ECB still is an improvisation, and we still need a new treaty, see my piece on the two Mario’s.
  • It were Dutch hawks who joined with Germany and imposed austerity on the rest of Europe, also using Brussels as a front for the Dutch electorate.

Other major errors in the interview are:

  • Legrain apparently never really studied Holland. For him it is a small country that falls under his radar. For hm, Holland is not a perpetrator but a victim. For him, these are nice people, and not hypocrites. Apparently he regards Holland as a free, tolerant, open-minded country, that just happens to have made some policy errors, without kids delving into the trash as now happens in Greece. He can’t make himself see Marcia Luyten as co-responsible for making kids in Greece do so. But she is a hypocrite and co-responsible. She never properly informed the Dutch viewers.
  • Legrain just answers the questions that Marcia asks. He doesn’t expose Holland as a major perpetrator. He does not expose Marcia as belonging to the hypocrites who helped cause the problems. He joins Marcia in her “frame” that it are politicians who do not want to lose face, while a major part of the story is that it are journalists who have been misreporting and misrepresenting.
  • Legrain doesn’t expose Jeroen Dijsselbloem as a major stumbling block: an agricultural economist who got into politics too quickly and who apparently isn’t able to deal with these issues properly. (Dijsselbloem already failed on the policy w.r.t. the education on mathematics.) (See Dijsselbloem on Dutch exports.)
Marcia Luyten reads the English book title "European Spring", by Philippe Legrain

Marcia Luyten reads the English book title “European Spring”, by Philippe Legrain (minutes 22-40)

Thus, Philippe, the next time that you visit Holland, first consider my books DRGTPE and CSBH, and let us discuss those, before you face Dutch journalism.

Incidently, we appear to agree with a lot. For example, that the surplus on the Dutch or German exports account was invested abroad, and basically was squandered when investments failed, is a key feature in my analysis since 1990. See also Johannes Witteveen, a former executive director of the IMF.  But the hypocrites on Dutch television will not allow viewers to hear this from home grown economists. They will welcome people like you, Philippe, who criticize Germany and France, instead of the local Dutch incrowd hawks and perpetrators, and their messenger prime minister Mark Rutte, who has a degree in history and who does not know much about economics but still is zealot about Margaret Thatcher, and who got the Rathenau prize on freedom while he censors economic science in Holland.

But, you also state that you have a proposal how a currency union would work with decentralised decision making. I wonder how you could achieve that. My suggestion has been that each nation adopts an Economic Supreme Court. I wonder whether your ideas are the same. Apparently those are behind a pay wall. This will not work.  Even a crisis should not force people to buy into books that might be scientifically unwarranted.

PM. This hypocritical Buitenhof TV broadcast also contains a discussion with Nikos Koulousios, six minutes before Philippe Legrain. This is somewhat amusing, but not really so.

  • The suggestion is that when Greece and Germany collaborate on jokes, and the Germans admit that they don’t have a sense of humour, then we see a proper collaboration in Europe: and all is fine, and people should feel satisfied that the notion of a unified Europe might work. I am afraid that this only confirms national stereotypes and isn’t real humour.
  • Koulousios calls it typical that a letter signed by economists like Joseph Stiglitz and Thomas Piketty did appear in the Greek paper but not in other papers, except in the Financial Times in which it appeared originally. He seems to suggest some kind of conspiracy, in which the Greek readership is manipulated and not given the right information. However, it may just be that the FT expects royalties. Check why I don’t blog at the FT. Thus the proper diagnosis is that journalists can be quite hypocritical, not only in Holland but even in Greece.

There is one real conclusion from all this. I will discuss it with JCJ the next time I see him. Rather than joining Putin in a hot Summer war on the Ukraine, the EU should pay Russian journalists more money for accurate reporting about the state of the world.

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.


(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’.

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)

Newton discovers the universal law of gravity, (c)

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.

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)

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, purple mixtures (Source: Wikipedia)

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 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)


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