BodyandSoul vs Standard Calculus 2

Einstein presenting his equation of general relativity $R_{ik}=0$ supposedly(?) describing the world. Simple and general. 

A Standard Calculus text book, like Calculus: A Complete Course by Adams and Essex, is filled with symbolic formulas covering more than 1000 small print pages and is difficult for the student to digest and heavy to carry along.

The objective of a Standard Calculus text book appears to be to convince the student of the usefulness of Calculus through mass demonstration by presenting so many specific problems, which can be solved with pen and paper using Symbolic Calculus, that there can be only a few left which cannot be solved this way. In short, the objective is to show that Symbolic Calculus works by presenting very many specific examples. But the massiveness is misleading since in fact very few problems can be solved symbolically with pen and paper.

The essence of the BodyandSoul approach as Constructive Calculus is the opposite: Instead of many specific problems solved by symbolic mathematics with pen and paper, one general problem containing all the specific problems of Standard Calculus and many more, is considered. The essence of the theory is then to show how and why any given instance of the general problem can be solved by the computer, as expressed in a Fundamental Theorem of Calculus.

The one general problem of Constructive Calculus, in one variable to start with, is the Initial Value Problem (IVP): Construct a function $u(t)$ of time $t$ such that

• $\frac{du}{dt} = f$ for $t > 0$ with $u(0)=u_0$,         (*) 

where $f=f(u,t)$ is a given function of $u$ and $t$ and $u_0$ a given initial value, by successive time stepping according to

• $u(t+\Delta t)$ = $u(t) + f(u(t),t)\Delta t$ with $u(0)=u_0$, 

with $\Delta t >0$ small. This is formally a finite time step version of $du = fdt$ or $\frac{du}{dt}=f$ with vanishingly small time step $dt$, and $du \approx u(t+\Delta t) - u(t) =f\Delta t\approx fdt$.

If $f$ depends only on $t$, the solution $u(t)$ is the integral

• $\int_0^t f(s)\, ds + x_0$.

If $f=u$ and $u_0=1$, the solution is $u(t)=\exp(t)$. 

More generally, with simple dependencies of $u$ and $t$ all elementary functions (exponential, trigonometric, Bessel, ...) are constructed this way and their properties follow from the specifics of the IPV they solve.Calculus in one variable can thus be reduced to a study of the IVP (*). 

Similarly, Calculus in several variables can essentially be reduced to an IVP of a generalization of (*) with $u=u(t,x)$ and $x$ a multi-dimensional space coordinate and $f$ depending on partial derivatives of $u$ with respect to space coordinates, which is solved by time stepping after finite element discretization in space. 

Constructive Calculus can thus be summarized as $\frac{du}{dt} = f$ solved by time stepping $du=fdt$. Constructive Calculus combines simplicity with generality, which is a prime goal of (computer) science and mathematics, to be compared with the difficulty of all the specific cases of Symbolic Calculus.

PS We may summarize as follows:

• Constructive Caculus is simple and general.
• Symbolic Calculus is difficult and special.

                                     Mass demonstration for (or against?) Symbolic Calculus

10 Reasons Why Standard Calculus is Wrong Today

In recent posts I have argued that Calculus in its standard symbolic mathematics form,  in the computer era of our time should be given a new form as constructive mathematics, where symbols are supported by numerical algorithms. Let me here list 10 reasons why the standard text Calculus: A Complete Course by Adams and Essex, is no good today:

1. Real numbers are introduced with geometric reference as points on a number line.
2. Completeness of the set of real numbers is identified as a geometric property of the number line to "have no holes", which as a pretentious triviality can only be mystifying to the student.   
3. The derivative is introduced through a limit process with division by infinitely small quantities, which is numerically ill-posed.
4. The integral is introduced with geometric reference to area defined by equality of sup of lower sums and inf of upper sums, which can only be mystifying to the student. The Fundamental Theorem is expressed as differentiation of area, which misses the esssence of the Fundamental Theorem with the integral arising as the solution to a basic initial value problem.
5. The exponential function is introduced as the inverse of the logarithm with the basic differential equation satisfied by the exponential arising as a strange mystifying surprise at the end.
6. Trigonometric functions are defined geometrically and not constructively as solutions to certain basic initial value problems computable by time stepping. This is wrong from mathematical point of view.
7. The central Fixed Point Theorem is not proved in the text, but left as an exercise for students to skip.
8. The proof of existence of solutions to the basic initial value problem is not given, just vaguely hinted at.
9. The presentation of calculus of several variables uses an awkward notation, and is severely limited with very little about the basic field of partial differential equations. Implicit function theorem not proved. Inverse function theorem not even stated.
10. The scope of Calculus today as constructive mathematics designed by human minds performed by computers, is missing.      

BodyandSoul contains corrections of these errors of Standard Calculus, and much more...

PS1 The mission of Calculus: A Complete Course is stated in its Preface to be:

• Of course it is not true that computers can contain all of mathematics; so one challenge (of a new edition) is to ensure that no one imagines that they can.

PS2 The authors of Calculus: A Complete Course want to impress the reader by using a "mathematical style" with DEFINITION, THEOREM, PROOF and EXAMPLE displayed in very big letters, but then in reality offer little, since proofs of important results are generally omitted and the proofs actually offered are mostly pretentious trivialities. BodyandSoul (BS) was written in order to replace the emptiness of a standard calculus text. Seeing BS by teachers at Chalmers with experience of BS, being replaced by Calculus: A Complete Course, makes me sad or angry, or both.

BodyandSoul vs Standard Calculus 1

                                                  Symbols as signs on a blackboard.

The basic principle of the BodyandSoul mathematics education reform program is unification of symbolic and computational mathematic where symbols are given meaning by computational algorithms. The symbol (or sign) $\sqrt{2}$ thus is assigned to the real number with decimal expansion generated by Newton's method for computing a (positive) root of the equation $x^2 = 2$. Further, the symbol 

• $\int_0^t v(s)\, ds$,   

denotes the function $x(t)$ for $t > 0$ satisfying $Dx(t)=v(t)$ for $t > 0$ and $x(0)=0$, where $Dx=\frac{dx}{dt}$ is the derivative of $x(t)$ with respect to $t$, which is constructed by time stepping according to $dx = vdt$. 

Each symbol as a form of soul, thus is carried by a body in the form of a constructive algorithm defining the meaning or numerical value of the symbol.

On the other hand, in a standard presentation of Calculus, symbols are not defined by constructive processes. For example, the symbol $\sqrt{2}$ denotes an object (some form of number) with the property that $\sqrt{2}\times\sqrt{2}=2$, without however carrying any information about its numerical value. Neither does the symbol $\int_0^t v(s)\, ds$ itself carry information about its value.

Standard Calculus mainly consists of symbolic mathematics as symbol manipulation according to given rules, which is what makes Calculus books so thick, while constructive mathematics and numerical algorithms are used only in (the "few exceptional") cases when symbolics fails.

Unification of soul and body can be seen as an expression science, in contrast to religion separating holy soul (symbol) from sinful body (numerical value).

Discussing these aspects with a Standard Calculus teacher is as easy as discussing realities of immaculate conception with a catholic priest. An example is given by the fruitless discussion with teachers at Chalmers claiming to be "inspired by BodyandSoul" while delivering Standard Calculus, as if the central message of BodyandSoul was missed.

Svar från Mikael Enelund Programansvarig Matematik - Maskinteknik

Här är svar från Mikael Enelund programansvarig för den nya utbildningen i Beräkningsorienterad Matematik för Maskintekniker vid Chalmers, på min fråga om han har några kommentarer till de senaste posterna på min blogg:

Hej Claes.

Som ansvarig för maskinteknikprogrammet litar jag helt och hållet på att Stig och Anders mfl utvecklar och genomför en genomtänkt och anpassad matematikutbildning för maskinteknikstudenterna. Kurserna fungerar bra, de får fina omdömen av studenterna, andelen studenter som klarar kurserna är bra (i jämförelse med övriga kurser i programmet) och kunskaperna verkar vara tillräckliga för kurser i programmet som tillämpar matematik. Lärare på masternivå och industrin har också påpekat att studenterna har blivit mycket bättre på att använda beräkningsmatematik (såväl industriella programvaror som egenutvecklade program) för att lösa öppna, komplexa och ofta olinjära problem.

Det sker också en ständig kursutveckling och anpassning till programmets behov. Speciellt trevligt för mig som också är lärare i hållfasthetslära är att grunderna till finita-elementmetoden undervisas i grundkursen Matematisk analys i flera variabler, detta är faktiskt unikt för Chalmers maskinteknik. Så vitt jag förstår är matematikkurserna i varierande omfattning inspirerade av BodyandSoul och dina tidigare gärningar på Chalmers. Utan ditt pionjärarbete skulle det vara mycket mindre beräkningsmatematik i både matematiken och de mer tillämpade kurserna.

Som sagt, när det gäller val av läromedel och pedagogik så lämnar jag det åt kursernas lärare.

Vänliga hälsningar

Mikael

Mitt svar var följande:

Tack för svar Mikael.

Som programansvarig tycker jag Du borde tänka själv och inte bara blint lita på leverantörer. Du skulle kunna få ett mycket bättre program för M om Du bara efterfrågade detta. Det första vore att begära att den särdeles traditionella amerikanska kurslitteraturen ersattes av något som passar vår IT värld av idag och imorgon. Varför inte pröva att ställa detta krav?

Vänliga hälsningar

Claes

More on the Fundamental Theorem of Calculus: Standard vs BodyandSoul

In recent posts we have compared the Standard Calculus approach with the BodyandSoul approach to the Fundamental Theorem as expressed in the corresponding proofs of the theorem, which concerns the relation between primitive function/integral $x(t)$, derivative $Dx = \frac{dx}{dt}$ and integrand $v(t)$ connected by the equations:

1. $Dx(t) = v(t)$
2. $x(t)=\int_0^t v(s)\, ds$.

We have noticed that in Standard Calculus as presented in e.g. the standard text book Calculus: A Complete Course by Adams and Essex, the integral $x(t)$ as an area under the graph $t\rightarrow v(t)$ is the primary given object (possibly somehow constructed as a limit of piecewise rectangular approximating areas), and the proof of the Fundamental Theorem consists of showing that $x(t)$ satisfies the differential equation $Dx = v$. 

In the BodyandSoul approach the primary given object is the differential equation $Dx=v$ with $v$ given data and $x$ unknown solution to determine, and the proof of the Fundamental theorem consists of showing that this equation can be solved by constructing approximate solutions by time-stepping

according to 

• $x(t+\Delta t)=x(t)+v(t)\Delta t$ with $\Delta t$ a time step tending to zero. 

In this approach $x(t)$ satisfies the differential equation $Dx(t)=v(t)$ by construction, because 

• $Dx(t)\approx \frac{x(t+\Delta t) - x(t)}{\Delta t}=v(t)$.

In the Standard approach the construction is hidden and the proof thus consists in verifying that the integral $x(t)$ satisfies the equation $Dx(t)=v(t)$. Now, this is cumbersome because computation of a derivative in principle is an ill-posed unstable process, which has to be regularized to be computationally meaningful in finite precision with the presence of perturbations of $x(t)$. In the Standard approach this is circumvented by performing symbolic exact differentiation. 

For example, it is verified by symbolic differentiation in infinite precision that e.g. $D t^3 = 3t^2$, which allows computation of the area under the function $3t^2$ from $t=0$ to $t=1$, as a form of magic seemingly without summation:

• $\int_0^13t^2 = 1^3 - 0^3 = 1$.   

The full picture is hidden to the student of Standard Calculus, presenting primarily the non-constructive magic of exact symbolic differentiation and less the constructive non-magic of time-stepping. BodyandSoul presents both aspects and opens to a deeper and more useful constructive understanding.

PS1 To define as in Standard Calculus the integral as an area follows the same non-constructive geometric approach as defining trigonometric functions through the lengths of the sides of rectilinear triangles (used in e.g. Adams). In BodyandSoul the integral and elementary functions such as trigonometric functions are constructed as solutions to elementary differential equations. Properties of elementary functions then come out as consequences of defining elementary differential equations and not from possibly far-fetched geometry.

The step from the (difficult) symbolic geometry of Euclide to the (easy) constructive analytic geometry of Descartes, marked the beginning of the scientific and industrial revolution and was thus not a small step for humanity. A modern Calculus course should reflect this step.

PS2 In BodyandSoul the trigonometric functions $x(t)=\sin(t)$ and $y(t)=\cos(t)$ are defined as solutions of the system describing a harmonic oscillator:

• $Dx(t) = y(t)  and Dy(t) = - x(t)$ for $t > 0$ with initial values $x(0)=0$ and $y(0)=1$,

and solutions are computed by time-stepping. 

The familiar relations $D\sin(t) = \cos(t)$ and $D\cos(t)= - \sin(t)$, thus result from the constructive time-stepping of the system where these relations are encoded. This is readily understood by the general student.

In Standard Calculus, $\sin(t)$ and $\cos(t)$ are defined geometrically and the relations $D\sin(t) = \cos(t)$ and $D\cos(t)= - \sin(t)$ have to be discovered and proved by tricky trigonometry, which the teacher may love, but the general student finds difficult.

Again, presenting this argument to a Standard Calculus teacher will lead nowhere.

Beräkningsinriktad Matematikutbildning för Maskinteknikprogrammet på Chalmers vs BodyandSoul

Efter att ha först ha deltagit i BodyandSoul (BS) programmet 2000 - 2005  för kemistudenter och sedan lagt ner detsamma 2006 i samband med att jag flyttade till KTH, flyttade Stig Larsson över sina reformerfarenheter till maskinteknik och lanserade där en ny Beräkningsinriktad Matematikutbildning presenterad i en artikel författad tillsammans med programansvarige Mikael Enelund vid Utvecklingskonferensen KTH 2008 med informationen (som forfarande är aktuell):

• Kurslitteraturen är två traditionella läroböcker, kompletterade med ett kompendium i  Beräkningsmatematik.  

Den nya Beräkningsinriktade Matematikutbildningen beskrivs i artikeln precis som om den vore BS, dock utan minsta referens till BS. Kompendiet är kopia av några inledande kapitel i BS och innehåller en referens till BS som "inspiration" och artikeln avslutas med

• Ansvarig matematiklärare, professor Stig Larsson, tilldelades Maskinteknikprogrammets pedagogiska pris för sina insatser för utvecklandet och genomförandet av matematikkurserna. Priset baseras på en omrröstning bland studenterna. Stig erhöll även Chalmers pedagogiska pris 2008 för sitt arbete med att integrera matematikämnet i ingenjörsämnet”. Pristagarna utses av en jury. Studenternas sammanfattande betyg på matematikkurserna har för alla kurser legat över fyra på en femgradig skala och slutsatsen är att studenterna är mycket positiva till kurserna och genomförandet.

Idag 5 år senare ser allt ser likadant ut och reformaktiviteten verkar ha avstannat: samma "traditionella läroböcker" (Adams och Lay) och samma lilla kompendium i Beräkningmatematik, samma programansvarig, samma revolutionerande reformmatematik värdig pedagogiska priser, men naturligtvis fortfarande utan synbar närvaro av BS. Verkligheten ter sig ibland konstig, men allt torde ha en rationell förklaring som så småningom kan uppenbaras.

PS1 För den fortsatta lanseringen av maskinteknikprogrammet, som beskrivs som Sveriges bästa och prisats av Teknikföretagen som Årets Teknikutbildning Högskola 2012, se:

• Integration of Computational Mathematics Education in the Mechnical Engineering  Curriculum at Chalmers. (2011) Samma på svenska (2011)
• MSC Programme in Mechanical Engineering (2013)

Webbaserad läobok/undervisningsmaterial i matematik listas som önskvärd framtida utveckling. När jag meddelar Enelund och Larsson att sådan finns i form av webversionen MST av BS, möts jag av kalla handen. Planen verkar vara att expandera kompendiet i Beräkningsmekanik att fylla detta syfte, möjligtvis med BS/MST som "inspiration" enligt tidigare mall, men att denna utveckling får anstå eftersom "orken inte räcker".

Larsson säger att den traditionella läroboken av Adams avses avvecklas "på sikt" men att BS/MST inte kommer att användas som ersättning, eftersom "BS/MST inte går att använda som kurslitteratur" (även om det gick bra under 6 år med BS för kemistudenter), utan bara som "inspiration", för läraren och kursboksförfattaren, men inte för studenten eller priskommitten.

PS2 När Larsson 2006 presenterade sin plan för den nya Beräkningsorienterad Matematikutbildningen utformad helt enligt BS inför CDIO, fanns en vag antydan av BS i texten i följande ordalag:

• The Mechanical Engineering program at Chalmers University of Technology has taken part in the development of the CDIO model of engineering education since 2000. 
• At the same time, new mathematics courses for the engineering education have been developed at Chalmers and implemented in the Chemical Engineering and Bioengineering programs since 1999. These courses emphasize mathematical modeling, simulation, the use of modern computational tools, and interaction with courses in chemistry and chemical engineering. This is achieved by taking a computational (constructive) approach to the teaching of mathematics. 

Vid nästa presentation 2008 efter det att det nya epokgörande programmet startat på maskinteknik, var BS fullständigt utrensat, som om det aldrig funnits, som om det inte var väl dokumenterat i form av 3 böcker hos Springer och omfattande websida med extramaterial, som om Larsson aldrig hade undervisat enligt BS i 6 år.

PS3 Se följande legendariska foto av BodyandSoul-teamet som drev matteutbildningen på kemiteknik 2000-2006. Kan personerna identifieras?

 

anderslogg‏@anderslogg

Klassiskt foto runt sekelskiftet från Prof. Leibschnitzels (3 fr v) gästföreläsning om Lipschitz-kontinuitet. #tmv225 pic.twitter.com/l1tZGaRWd5

•  Reply 
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PS4 Jag leddes till denna undersökning av sakernas tillstånd (som jag var lyckligen ovetande om) då Anders Logg, efter att ha avslutat den inledande första kursen ht 2013 i reformprogrammet för Beräkningsorienterad Matematikutbildning vid Maskinteknik Chalmers, frågade mig om det fanns någon som ville ta över websidan för BodyandSoul, som Anders haft hand om sedan 2000, eftersom Anders inte längre såg att websidan fyllde någon funktion för utbildningen på Chalmers och eftersom Anders nu tillträtt sin professur på Chalmers.

PS5 Jag har bett Mikael Enelund, programansvarig för maskinteknik, om kommentar, men det verkar inte som Mikael vill kommentera. Att tiga verkar vara guld för ansvariga vid högskolan, men håller det i längden?

PS6 Anders har på mitt förslag lagt upp en länk till MST samt några relevanta kapitel ur BS på hemsidan för den inledande kurs som Anders givit, detta efter det kursen avslutats och eftersom Anders ansett det vara korrekt. Mitt förslag till Stig att göra detsamma för andra kurser i programmet har inte tillmötesgåtts, förmodligen följdriktigt.

PS7 Den intresserade kan jämföra utdraget ur BS enligt ovan med Stig Larssons kompendium Beräkningsmatematik och kanske då förstå varför "BS inte passar som kursbok" medan den svenska översättningen av Stig passar jättebra, eller inte förstå det. Kan problemet vara att BS är på engelska?

Under tiden 2007 - 2013 är det bara detta kompendium i Beräkningsmatematik på 49 sidor som "nyskrivits" av Stig som översättning till svenska av några kapitel ur BS. Att översätta hela BS på 2048 sidor  i samma takt skulle ta 7 x 41 = 287 år! Men många priser och utmärkelser skulle det bli.

PS8 Det sorgliga i sammanhanget är inte att BS kopierats utan korrekt referens, det sker hela tiden i den akademiska världen, och det får man ju se som ett tecken på att BS är tillräckligt intressant för att attrahera kopiering,  utan att så lite av BS har kopierats under falska förespeglingar av att leverera detsamma som BS. Det är ju därför priserna har inhöstats, inte för att en traditionell amerikansk standard Calculusbok som Adams har använts. 

Who Is Denier?

Judy Curry has a post on the concepts of denier and denial:   

• When used in the context of the climate debate, particularly when scientists discuss another scientist or their arguments (e.g. Mann calling JC a ‘denier’), the use of denial is intellectual tyranny at its worst. Scientists bullying their opponents is not new; Isaac Newton provides a prime example. When a scientist uses the word about the arguments of another scientist or the scientist themselves, they are giving the public a message that they don’t need to think for themselves, but rather they only need to listen to the person that is claiming a consensus and is screeching the loudest.

stimulated by a a post on Prospect Magazine by Skidelsky:

• The extension of the “denier” tag to group after group is a development that should alarm all liberal-minded people. One of the great achievements of the Enlightenment—the liberation of historical and scientific enquiry from dogma—is quietly being reversed. 

I have also been ridiculed as "denier of the greenhouse effect" because of my study of the proof of Planck's and Stefan-Boltzmann's Laws of blackbody radiation. Or a "denier of modern physics" because of my studies of the basics of relativity theory and quantum mechanics. Or a "denier of mathematics" by questioning the standard presentation of the Fundamental Theorem of Calculus.  

My experience is that the mere questioning of ruling dogma can make people upset and upset people often react by a killer instinct.

Standard Calculus as Ill-Posed Unstable Backward Magic

        Jacques Hadamard (1865-1963) was a gentle man with strong opinions on mathematics.

Previous posts on the Fundamental Theorem of Calculus have exposed two approaches to the connection between primitive function/integral $x(t)$, derivative $Dx = \frac{dx}{dt}$ and integrand $v(t)$ connected by the equations:

1. $Dx(t) = v(t)$
2. $x(t)=\int_0^t v(s)\, ds$.

In the standard approach as presented in e.g. the standard text book Calculus: A Complete Course by Adams and Essex, the integral $x(t)$ as an area under the graph $t\rightarrow v(t)$ is the primary given object and the proof of the Fundamental Theorem consists of showing that $x(t)$ satisfies the differential equation $Dx = v$. 

In the BodyandSoul approach the primary given object is the differential equation $Dx=v$ with $v$ given data and $x$ unknown to determine, and the proof of the Fundamental theorem consists of showing that this equation can be solved by time stepping producing the integral $x(t)$ as the solution. The process from input data $v(t)$ to output solution $x(t)$ by solving $Dx=v$ by time stepping, is well-posed or stable in the sense that small perturbations of data or solution process results in small perturbations of the solution $x(t)$.  

The mathematician Hadamard identified well-posedness and stability to be a necessary requirement in order for a mathematical problem to be meaningful, in the sense that a meaningful solution can be found. The process of integration from integrand $v(t)$ to integral $x(t)$ is well-posed and meaningful.

On the other hand, the process from integral/primitive function $x(t)$ to derivative $Dx(t)$, is ill-posed and unstable, in the sense that small perturbations in $x(t)$ may give rise to large perturbations in the derivative, because

• $Dx(t)=\lim_{\Delta t\rightarrow 0}\frac{x(t+\Delta t)-x(t)}{\Delta t}$

and a small perturbation in $x(t+\Delta t)$ or $x(t)$ gets divided by the quantity $\Delta t$ tending to zero and thus gets amplified by the large factor $1/\Delta t$. The standard approach to the Fundamental Theorem puts the emphasis on the ill-posed or unstable process of differentiation. 

We sum up as follows: 

1. The standard approach to the Fundamental Theorem is ill-posed, unstable and of questionable meaning. As illposed problem it rests on symbolic mathematics of infinite precision, which appears as magics.
2. The approach in BodyandSoul is well-posed, stable and clearly meaningful. As well-posed problem it can be solved by numerical mathematics in finite precision, which is reasonable and not magics.

These aspects would be possible to discuss constructively with the man on the street, but may be very difficult to present to a teacher of standard Calculus for which Adams' book is the bible.

BodyandSoul Not Back at Chalmers

The previous post BodyandSoul Back at Chalmers showed upon closer inspection to not represent reality.

The reality is that the BodyandSoul mathematics education reform program, which had been successfully run during 2001 - 2005 for chemistry students, was dismantled in the Fall 2006 and replaced by a standard Calculus program based on the standard text book Calculus: A Complete Course by Adams and Essex. This happened at the very moment I moved from Chalmers to KTH.

There is a narrative propagated by the department of Mathematics at Chalmers, that BodyandSoul has resurfaced as an important "inspiration" for the current mathematics program for mechanical engineers, for sure based on Adams standard book but still presented as an innovative reform program.

When I look at this program I see 90% standard Adams and 10% BodyandSoul "inspiration". To say that BodyandSoul is back at Chalmers thus does not seem to represent reality, only fiction. Unfortunately, in many cases, reality is more true than fiction.

50% For and 50% Against = Minimal Consensus (about AGW)

Judy Curry reports on:

• A comprehensive survey has been conducted of the American Meteorological Society membership to elicit their views on global warming: Meteorologists’ views about global warming: A survey of American Meteorological Society professional members.

The survey is collected in the following table

The rightmost column shows that 52% believe in Mostly human global warming, while 48% believe in the negation including Equally human and natural, Mostly natural, insufficient evidence, et cet. 

In short: Roughly 50% believe in Mostly human global warming and 50% in Not Mostly human global warming. This is Minimal Consensus: Any deviation from 50 - 50 would give more weight to one of the beliefs and thus would support more consensus. 

What is particularly remarkable is that 26% non-expert non-publishers ask for more evidence of human cause of warming (to believe that it is real), while only 9% expert publishers do that. 

In other words, non-expert scientists show more of a critical scientific attitude than expert scientists do. 

Does that tell us something about the state of (climate) science today?  Is the reason that experts are government funded, while non-experts are not by definition, and governments do not ask for evidence which may confuse the public.

The 50 - 50 non-consensus of the study is to be compared with the IPCC proclaimed 97 - 3 consensus of government funded experts.