lördag 30 november 2013

Konrad Zuse on The World as Clock with Finite Precision

                                      Konrad Zuse pondering if physics is digital? If so, time has a direction.

I have recently discovered that the idea which I have pursued in different pieces of work, the idea to view physics as a Clock with Finite Precision, was expressed in 1969 by the German computer pioneer (and civil engineer) Konrad Zuse (who constructed the first working computer named Z3 in 1941) in the remarkable article Calculating Space on Digital Physics, starting out with the following experience which I share with Zuse:
  • The work which follows stands somewhat outside the presently accepted method of approach, and it was for this reason rather difficult to find a publisher ready to undertake publication of such a work. 
Here are some highlights from Calculating Space, connecting in particular to my book The Clock and the Arrow: A Brief Theory of Time about the 2nd law of thermodynamics and the direction of time:
  • It is obvious to us today that numerical calculations can be successfully employed in order to illuminate physical relationships. 
  • To what extent are the realizations gained from study of calculable solutions useful when applied directly to the physical models? Is nature digital, analog or hybrid? And is there essentially any justification for asking such a question? 
  • The examples of digitalization of fields and particles which have been preented are in their present unfinished form still far removed from being able to serve in the formulation of physical rules. Nevertheless, they give a rough impression of the possibilities for using the tools of the automaton theory to answer physical questions.
  • The question to what extent it is possible to consider the entire universe as a finite automaton depends on the assumption which we make in relation to its dimensions. 
  • An infinite information content is required for an unlimited spacetime element. It is practically impossible to simulate such a model with computers because of the necessity of infinite number of places required. 
  • The sources of error are correspondingly great in the extremely large number of collisions between gas molecules, and these errors quickly lead to deviations from theoretical processes.
  • This means that the better the causality rule is approximated in the reverse time direction, the more calculations we must be prepared to carry out in our model. This leads to the result that simulations of universal systems with causality functioning in both time directions belong to the category of “unsolvable” problems.
  • Of course, it can be said that this is true only for calculating simulative models. But this result should encourage us to reconsider the matter. Are we justified in assuming a model of nature for which no calculable simulation is possible? 
  • From this point of view, it appears that the frequently advanced argument of determination in both time directions should be fundamentally reexamined.
  • But if Zuse didn’t hit upon the concept of universal computation (as Turing did), he was interested in another very deep question, the question of the nature of nature: “Is nature digital?” He tended toward an affirmative answer. (from Afterword)



Traditional TextBook Is Dead: Long Live New WebBook


The traditional university course, in mathematics and physics in particular, is based on a traditional printed textbook, which defines the course, its subject, questions and answers.

The traditional printed textbook is cut in stone and defines knowledge by exclusion: What is not in the textbook is not of concern. The world is defined by what is inside the book and that is what defines the exam questions. The student is supposed to go inside the closed world of the textbook and not look out and get distracted. The textbook answers specific questions posed in the book, and other questions should not be asked.

The previous post reported about an education leader at Chalmers University of Technology who complained that today students no longer read textbooks; instead they check out lecture notes based on the book if any,  ask a study mates, search the web, look at a web-lecture from MIT or Khan Academy. Since they do not use the text book, they do not need to buy it either and thus can save some money.

The traditional textbook will be replaced by something on the web, some form of WebBook and the question is what that may be?

One thing seems clear and that is that the WebBook will be open to the exterior, in contrast to the traditional text book forming its own closed world without window.

The WebBook will be formed by inclusion through links to the exterior. The traditional textbook is based on exclusion and does not have links to the exterior.

The new web student will take a new active role and independently search and collect material to bring into the course, instead of passively relying on the words of the book transmitted through the teacher.

The web teacher will have to give up the omnipotent role of understanding everything by understanding everything in the book defining the world, but not necessarily much more, and join the student in search and collection of stuff to bring into the course.

These new roles are now forming and it will be very interesting to follow this process.        

fredag 29 november 2013

Den Traditionella Lärobokens Död



Mikael Enelund, programansvarig för utbildningen i Maskinteknik vid Chalmers, levererar i en brevväxling angående BodyandSoul på Chalmers följande skakande observation från vad som prisats vara den bästa teknikutbildningen i landet:
  • Min erfarenhet är att studenterna läser litteraturen i allt mindre omfattning. De praktiserar modellen "trial and error" dvs att försöka lösa problemet först och om det behövs titta på föreläsningsanteckningarna, fråga en kompis eller en lärare, söka på nätet, titta på en föreläsning från MIT eller Khan academy etc och i sista hand titta i läroboken.
Detta är, mina läsare, inget annat än en dödförklaring av den traditionella läroboken, den tjocka tryckta amerikanska kursboken på glättat papper mellan styva pärmar inhandlad för dyra pengar, eller kurskompendiet som läraren själv snickrat ihop och sålt till studenterna år efter år. 

Nu finns nätet, och ingen köper längre en dyr lärobok för sedan inte titta i den, och läraren finner ingen mening i att skriva ett kompendium när allt redan finns på nätet.

Vad skall då läraren göra utan kursbok eller kompendium? Vad skall föreläsningarna bestå av om det inte finns någon kursbok eller kompendium att kopiera på svarta tavlan? Vad skall tentan bestå av om det inte finns en kursbok som anger vad som inte ingår i kursen och istället nätet tränger in från alla håll?

Se där några frågor som dagens lärare, studenter, läroboksförfattare och utbildningsledare står inför. Låt oss söka svar i några kommande poster...Vi börjar med begrunda The text book is dead uttryckt av Facebook Co-founder Chris Hughes:
  • In the next five to seven years, the textbook is no longer going to be the basic building block of education.
On a similar note, Nobel Laureate Harry Kroto said he now regards Wikipedia as more accurate than printed textbooks and far less costly for both students and educators. In preparing his GEOSET lectures, Kroto said, he never uses a single textbook.

PS Man kan också begrunda:

torsdag 28 november 2013

BodyandSoul Sågas Definitivt på Chalmers


Nedan finns korrespondens mellan mig och Stig Larsson rörande BodyandSoul (BS), som visar att ingen form av BS med mitt namn är tänkbart på Chalmers, åtminstone inte enligt Stig, vad jag förstår. 

Däremot kommer BS utan mitt namn att fortsätta att framhållas av Chalmers och särskilt M som siktar på att leverera den bästa utbildningen i landet, eller t o m världen, och då prisas av näringslivet som ett lysande exempel på den innovativa reform av ingenjörsutbildningen som alla avnämare frågar efter:
Eftersom inget material från BS med mitt namn kan användas, så måste Chalmers nu återskapa detta i egen regi utan att kopiera och då till stora kostnader. Detta verkar vara lika irrationellt som roten ur två. Att kalla kopiering för  "inspiration" som Stig gör, är inte korrekt. Det krävs engagemang och tid för att skriva något nytt, och det behövs om man inte bara skall kopiera. 

Stig säger att  han "inte är nöjd" med boken av Adams som används som grundtext,  men att den ändå "fungerar". Om BS säger nog Stig att han "är nöjd", eftersom han kopierar den, men att den "inte fungerar". Så kan det vara i matematikens logiska värld.



Brev till Stig Larsson 26/11:
Hej Stig

Skulle Du kunna beskriva lite närmare vilka överväganden som låg till grund för att BS avfördes
som kurslitteratur på Chalmers när jag flyttade till KTH, efter att ha använts 2000 - 2006 på K?

Var det Du som fattade detta beslut eller någon annan? Vem i så fall? Vilka var skälen att ersätta BS med en traditionell bok som Adams?

Du säger att Adams skall ersättas "på sikt" eftersom den som traditionell bok inte egentligen passar det reformprogram Du driver på M, antar jag. Hur ställer Du Dig till en ny version av BS utformad enligt önskemål från Chalmers? Skulle detta kunna lösa problemet att finna adekvat kurslitteratur till Ditt reformprogram på M?

Vänligen, Claes


Svar från Stig 28/11: 

Hej Claes.

Det var jag som fattade beslutet om att byta kurslitteratur. Skälet var
att BS inte fungerade bra som lärobok. Jag hade länge försvarat den mot
K-studenternas (rättmätiga) klagomål med argumentet att det finns ingen
annan bok med detta viktiga innehåll. Jag gav upp när jag insåg att vi
skulle kunna klara det med standardbok plus kompletteringar.

Adams fungerar men jag är inte nöjd med den. En ny bok behöver skrivas,
men den måste växa fram ur våra föreläsningsanteckningar. Därför avböjer
jag ditt erbjudande om en ny version av BS.


Vänligen, Stig

Brev till Stig 281/11:

Tack för detta svar Stig.

BS fungerande inte sämre än någon annan mattebok under den tid den användes 2000 - 2006, det vet Du mycket väl. Huruvida en bok funkar eller ej beror hur den presenteras av lärarna. Så länge det fanns engagerade lärare funkade BS bra, men när flera av dessa lämnade Chalmers runt 2006-7 så föll reformen ihop. Inte svårt att förstå. Förmodligen var Du utsatt för (en del av) den press jag utsatts för och som tvingade mig att lämna Chalmers.

När Du nu tackar nej till varje form av samarbete med BS och mig, så kan jag inte tolka det på annat sätt än att Du fortfarande är utsatt för press och inte är fri att handla rationellt. När jag läser vad Du skriver så ser jag att det mesta är hämtat från BS i form av vad Du kallar "inspiration", men som lika gärna skulle kunna kallas kopiering. I denna anda har Du har sedan 2007 skrivit ett litet kompendium i Beräkningsteknik och jag antar att resten av BS på samma sätt skall "växa fram ur föreläsningsanteckningar", men vad är realismen i detta? Är det verkligen meningsfullt att ägna en massa tid åt att göra det som redan är gjort, eller med BS i bagaget lätt skulle kunna göras? Vore det inte bättre att ägna tillgängliga resurser åt att göra något nytt och unikt som kan läggas till BS? För att på detta sätt få ett riktigt starkt program på Chalmers som Du skulle kunna ha glädjen av att ha skapat. Skulle inte det vara bra? Den enda haken skulle vara att mitt namn skulle finnas på någon del av texten.

Är det verkligen fortfarande så att BS utan mitt namn är lysande (Du har ju prisats för detta),
medan BS med mitt namn är så dåligt att det måste sågas? Ja, kanske är det så. Men vore det
inte mer rationellt att säga om nu tankarna i BS är så bra, så kanske personen som framfött tankarna också skulle kunna accepteras, mer eller mindre, om man nu tänker på att huvudmålet är att ge studenterna en modern adekvat matteutbildning?

Skulle Du inte, för M-studenternas skull och M-ledningen och hela M-utbildningen, kunna tänka en gång till och se om det inte trots allt vore rationellt att åtminstone sätta ihop en kravspecifikation på en ny version av BS, utan någon som helst förbindelse att köpa om jag skulle kunna leverera? Skulle inte M-ledningen, som kanske inte är lika allergisk mot mitt namn som Matematik, kunna tycka att denna tanke ändå kunde prövas, med tanke på studenterna som nog är ännu mindre allergiska?

Vad säger Du Stig? Vi har ju väsentligen samma vision vad gäller matteutbildning och det vore väl korkat om vi då inte samarbetar till båtnad för studenterna och M och Chalmers?

Vänligen, Claes

Svar från Stig: = 0 

Response från Mikael Enelund 28/11:

Hej Claes.

Som jag sagt tidigare lämnar jag valet av kurslitteratur till kursens lärare. Mitt fokus är kursernas lärandemål, att kurserna passar in i M-programmet, kursernas resultat och i viss mån pedagogiken samt att kurserna ständigt utvecklas.

Stig och Anders har meddelat att de inte är nöjda med Adams och jag har förstått att de planerar att utveckla egna texter baserat på sina föreläsningsanteckningar. Detta är jag nöjd med. Jag har som sagt förstått att de använder dina texter och idéer som inspiration och även det tycker jag är bra. Huruvida de refererar till dina texter korrekt och i tillräcklig omfattning har jag ingen åsikt om. Det får ni klara upp er sinsemellan.

Jag är mycket nöjd att ha Stig och Anders som lärare på M-programmet. Jag är övertygade om att de är Chalmers bästa mattelärare och de som är mest lämpade att undervisa blivande M-ingenjörer. Kursen Inledande matematik för M1 är precis utvärderad och studenterna är mycket nöjda. I kursenkäten blev medelvärdet av studenternas sammanfattande betyg på kursen 4,3 (5). Det är ytterst få kurser i programmet som når detta betyg och unikt högt för en första kurs i matematik.

Jag tycker att fokuseringen på kurslitteraturen är olycklig. Min erfarenhet är att studenterna läser litteraturen i allt mindre omfattning. De praktiserar modellen "trial and error" dvs att försöka lösa problemet först och om det behövs titta på föreläsningsanteckningarna, fråga en kompis eller en lärare, söka på nätet, titta på en föreläsning från MIT eller Khan academy etc och i sista hand titta i läroboken. Jag tror att denna modell kan var effektiv åtminstone i de kurser jag undervisar i verkar det fungera. I det här sammanhanget blir det viktigt att vi tillhandahåller "rätt" problem och tillåter studenterna att formulera egna problem.

Detta, matematikinnehåll och pedagogik diskuterar jag gärna (men inte vilken kurslitteratur lärarna skall använda). Jag vill också påpeka att min inställning till pedagogiskt utvecklingsarbete är att hela tiden göra små förbättringar och kontinuerligt utvärdera. Jag kan förstå att det är frustrerande för dig att det synes gå långsamt men det har hänt saker och jämfört med annan ingenjörsmatematik i Sverige vill jag påstå att vi har kommit längst mot en "simuleringsbaserad matematik", åtminstone på grundnivå. Jag kan inte heller gå in och detaljstyra innehåll i kurser, jag är ansvarig för 140 kurser. Däremot har jag tydliga ambitioner och riktlinjer för hela utbildningen och utvecklingen mot simuleringsbaserad matematik är prioriterat.

Jag försäkrar dig att ambitionen är att ständigt utvecklas och det gäller såklart inte bara matematiken. För att ge dig en vidare bild, utvecklingsarbete som står högt på agendan är verktyg för och träning i optimering av konstruktioner och produkter, CAE-verktyg tidigt i utbildning och framställning av modeller (fysikaliska och digitala) för att testa och utvärdera. Vi var bland de första utbildningar att använda 3D-skrivare och studenterna har i nuläget tillgång till två 3D-skrivare, nästa år kommer vi att minst 6 stycken. Simuleringsbaserad matematik är som du inser en grundpelare i allt detta som syftar till att utbilda ingenjörer som är väl förberedda för att utveckla bättre produkter och konstruktioner och mer effektivt än dagens ingenjörer.

Vänliga hälsningar

Mikael


Mitt svar till Mikael 29/11: 

Tack för svar Mikael:

Du tar upp en viktig punkt när Du säger att Du

"tycker att fokuseringen på kurslitteraturen är olycklig. Min erfarenhet är att studenterna läser litteraturen i allt mindre omfattning. De praktiserar modellen "trial and error" dvs att försöka lösa problemet först och om det behövs titta på föreläsningsanteckningarna, fråga en kompis eller en lärare, söka på nätet, titta på en föreläsning från MIT eller Khan academy etc och i sista hand titta i läroboken"

Att studenterna inte längre läser kurslitteraturen beror på att den inte upplevs som så meningsfull i dagens IT-värld, där informationen finns på nätet. Man kan ju säga att om ingen läser kursboken, och i många fall inte ens köper den eftersom den är dyr, så kan kursboken vara vilken som helst. Men läraren måste ju ha något att komma med i form av nätbaserad kurs som definierar kursens kärna och som leder ut till det vidare nätet.

Jag arbetar nu med att utforma en sådan web-version av BS under namnet Mathematical Simulation Technology.

Skulle inte Du kunna titta på detta, kanske tillsammans med några intresserade studenter? Tanken är inte att pracka detta på M, utan bara en trevare till en eventuell diskussion om hur kurslitteraturen idag och imorgon kan se ut, när nu den traditionella kursboken har förlorat sin mening och ingen längre läser den.

Vad säger Du om detta?

Vänliga hälsningar, Claes


PS Mikael skriver:
  • Jag har som sagt förstått att de använder dina texter och idéer som inspiration och även det tycker jag är bra. Huruvida de refererar till dina texter korrekt och i tillräcklig omfattning har jag ingen åsikt om. Det får ni klara upp er sinsemellan.
Var går då gränsen mellan inspiration ock kopiering? Det är naturligtvis roligt om mina texter och ideer kommer till användning, i samarbete. När Stig avvisar varje form av samarbete är det inte lika roligt längre.

tisdag 26 november 2013

A History of BodyandSoul at Chalmers

Some of the history of the BodyandSoul mathematics education reform project at Chalmers 2000-2007 is documented in the following conference presentations by Stig Larsson, who participated in the project:
  1. Stig Larsson,
    A reformed mathematics education at Chalmers,
    Högskoleverkets kvalitetskonferens, Norrköping, September 25-27, 2001.
    (abstractamslatexdvipdf)
  2. K. Eriksson, N. Ericsson and S. Larsson,
    Integration of chemistry in math courses,
    Chalmers Strategic Effort on Learning and Teaching C-SELT,
    Conference at Lingatan, August 13-15, 2002.
    (amslatexpdf)
  3. C. Niklasson, M. Christie, S. Larsson, L. Öhrström, and J. Bowden,
    Integration of Mathematics/Numerical Analysis with Chemistry/Chemical Engineering,
    preprint 2003.
    (pdf)
  4. L. Öhrström, G. Svensson, S. Larsson, M. Christie, and C. Niklasson,
    The pedagogical implications of using Matlab in integrated chemistry and mathematics courses,
    Int. J. Engrg. Education 21 (2005), no. 4, 683-691. (abstractpdf)
  5. M. Enelund and S. Larsson,
    Development of a new computational mathematics education for the mechanical engineering program at Chalmers University of Technology,
    Second International CDIO Conference, Linköping University, June 13-14, 2006.
    (pdf)
  6. M. Enelund and S. Larsson,
    A computational mathematics education for students of mechanical engineering,
    World Transactions on Engineering and Technology Education 5 (2006), no. 2, 329-332.
    (wordpdf)
  7. M. Enelund, H. Johansson och S. Larsson,
    Beräkningsinriktad matematikutbildning för maskinteknikprogrammet på Chalmers,
    Utvecklingskonferensen för ingenjörsutbildning 2008, KTH, Stockholm, 26-27 november 2008.
    (amslatexpdf)
  8. M. Enelund, S. Larsson, and J. Malmqvist,
    Integration of a computational mathematics education in the mechanical engineering curriculum,
    Proceedings of the 7th International CDIO Conference, Technical University of Denmark, Copenhagen, June 20 - 23, 2011.
    (pdf)
Here are some key quotations from 1. - 8. describing in particular the presence and absence of the BodyandSoul book:

1.(2001):
  • A full program for a new reformed engineering mathematics education has been developed by K. Eriksson, C. Johnson, Chalmers University of Technology, and D. Estep, Colorado State University. The program includes the books Computational Differential Equations, Cambridge University Press 1996, and Applied Mathematics — Body and Soul, to appear at Springer-Verlag, and various pieces of supporting software. 
  • The full program covering the basic mathematics courses of 25–30 credit points has been implemented at Chalmers since 1999 for a group of 90 students in the “Bio Engineering” (Kb) and “Chemical Engineering with Engineering Physics”  (Kf) programs and has been extended to include also the “Chemical Engineering” (K) program with a total of 160 students 2001. 
  • The implementation of the reform program... is carried out in a cooperative effort of the mathematicians M. Asadzadeh, K. Eriksson, C. Johnson, S. Larsson, M. Larson, K. Samuelsson, and N. Svanstedt at Chalmers University of Technology. 
2.(2002):
  • A basic idea of the reformed math program, delivered by the Department of Computational Mathematics at Chalmers, is a full integration of the “body” and “soul” aspects of mathematics, that is of the concrete/computational/numerical parts representing the “body”, including programming, and the abstract/analytical/symbolic aspects representing the “soul”.
  • (Remark: No reference to the upcoming BodyandSoul book)
3.(2003):
  • In this paper, we examine how changes in mathematics education with integration into engineering subjects influence the different teaching and learning methods for subsequent subjects in the chemical engineering programs at Chalmers. 
  • Material for the first courses has been developed in a new textbook (BodyandSoul).
4.(2005):
  • The last years the undergraduate chemistry and mathematics courses at Chalmers University of Technology has undergone a major curriculum reform.
  • As textbook in mathematics we use a new book (BodyandSoul) written especially for this approach to teaching and learning mathematics at a technical university.
5. and 6.(2006):
  • .... 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.
  • This approach is based on the textbook Applied Mathematics BodyandSoul and has been implemented since 1999 in the Chemical Engineering and Bioengineering programs at Chalmers University [2]. However, the book (BodyandSoul) has proved to be somewhat too difficult for the students and we plan to use traditional textbooks complemented by lecture notes.
7.(2008):
  • Här presenterar vi en reformerad matematik-utbildning. I den reformerade matematikutbildningen integreras traditionell symbolisk matematik med numeriska beräkningar och datorn används som ett verktyg. 
  • Erfarenheterna är mycket positiva. 
  • (Remark: No reference to BodyandSoul)
8.(2011):
  • Here we present the integration of a computationally oriented mathematics education into the CDIO-based MSc program in mechanical engineering at Chalmers. We found that the CDIO-approach was beneficial when designing a reformed mathematics education and integrating the mathematics in the curriculum. 
  • In the reform of the mathematics education, traditional symbolic mathematics is integrated with numerical calculations and the computer is used as a tool. 
  • The experience is very positive. 
  • (Remark: No reference to BodyandSoul)
We see that the BodyandSoul book was used 2000-2006 as the basic text defining a new reformed mathematics program, which is presented to the world at several national and international conferences.  We know that BodyandSoul was replaced 2007 by a traditional book (Calculus: A Complete Course by Adams and Essex) not connected to the reform, complemented by lecture notes in the form of Beräkningsmekanik, which is a translation into Swedish of a couple of introductory chapters of BodyandSoul with reference reduced to a source of "inspiration". 

We see that the experience of the reform is very positive and that the reason BodyandSoul was replaced by a traditional book with no connection to reform, is stated to be that BodyandSoul  "proved to be somewhat too difficult for the students".

PS1  For his reform work Stig Larsson has been awarded several prizes at Chalmers:
  • Ansvarig matematiklärare, professor Stig Larsson, tilldelades Maskinteknikprogrammets pedagogiska pris för sina insatser för utvecklandet och genomförandet av matematikkurserna. Priset baseras på en omröstning 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.

I have asked Stig Larsson if a "somewhat less difficult" version of BodyandSoul according to specification by Chalmers might be of interest as a possible replacement of the traditional text book by Adams, which has little connection to reform, which I could deliver free of charge and save the students  some couple of hundred dollars. I will report the reaction from Stig when (if) it arrives.

PS2 Mikael Enelund is main responsible for the Engineering Mechanics program at Chalmers "buying" the math reform program from the Math Department and Stig Larsson. Enelund says that he relies 100% on the judgement of the Math Department selling/delivering the courses of the reform program, and then voices the widely spread conviction among non-professional mathematicians, that mathematics can only be understood by professional mathematicians, as an effect of traditional mathematics education. Enelund is thus in principle main responsible as buyer, but has translated the responsibility entirely to the math department as seller.

It is thus impossible for Enelund to ask himself how it is possible to use a fully traditional text book like Adams as basic text in a reform program with a fundamentally non-traditional scope and approach? If Enelund had only taken the reform program he would have understood that mathematics can be understood also by non-professional mathematicians and then he would have had the courage to ask the question.  But....

PS3 But why bother? Are we not discussing trivialities? Who cares about basic math eduction, really? Not professors of mathematics anyway, since they consider teaching Calculus to be a triviality better left to underpaid instructors ordered to use some traditional American text book, year after year delivering a steady income to the professors.

But basic math education at a technical university like Chalmers sets the standard for the whole education, and if the math education is antiquated so will the entire engineering education be. So math education is important but the debate is dead and the progress is zero. Or backwards, as when BodyandSoul was replaced by Adams at Chalmers in 2007.   

måndag 25 november 2013

Constructive and Symbolic Calculus

The basic idea of BodyandSoul is to support a mathematical symbol like $\sqrt{2}$ and $e=\exp(1)$ by a constructive numerical algorithm through which the decimal expansion of $\sqrt{2}=1.421356237...$ and $e=2.718281828...$ can be computed to arbitrary precision.

The symbol $\sqrt{2}$ thus is assigned to the positive root of the algebraic equation $x^2=2$ computable by e.g. Newton's method and $\exp(t)$ for $t > 0$ is assigned to the function $u(t)$ which by time stepping solves the differential equation $Du(t) = u(t)$ for $t > 0$ such that $u(0)=1$, where
  • $Du(t)=\lim_{\Delta t\rightarrow 0}\frac{u(t+\Delta t)-u(t)}{\Delta t}$,
is the derivative of $u(t)$ defined as a limit in infinite precision. 

The symbols can sometimes be used in symbolic computation as symbol manipulation according to given rules, without using the actual numerical values, only the defining algebraic/differential equations.

For example, one can compute $\sqrt{2}\times\sqrt{2}\times\sqrt{2}\times\sqrt{2}=2\times 2 = 4$ knowing that $\sqrt{2}\times\sqrt{2}=2$ without knowing that $\sqrt{2}\approx 1.414$. 

More importantly, one can compute derivatives symbolically by using defining differential equations and the chain rule, for example:
  • $D\exp(2t)= \exp(2t)\times 2$ 
using that $Dexp(s) = exp(s)$ and $D2t = 2$ with $s=2t$. Doing so we circumvent the ill-posed unstable aspect of numerical computation of derivatives. This is done in symbolic computation software as Maple.

However, there is no corresponding general symbolic method for solving differential equations as different forms of integration, which thus in general is performed by constructive numerical methods. This works because integration is a well-posed stable process which can be performed in finite precision.  

söndag 24 november 2013

Royal Swedish Academy of Sciences Echoes IPCC CO2 Alarmism


I have repeatedly asked the Royal Swedish Academy of Sciences to revise its Statement THE SCIENTIFIC BASIS FOR CLIMATE CHANGE from 2009 based on IPCC AR4  from 2007, in particular in the light of the new IPCC AR5. The reaction I get from the main author of the Statement, professor Lennart Bengtsson, is that the Statement is so wisely formulated that it fits any thinkable new development and thus does not need to be revised.

To see if this is true, let us analyze the main messages of the Statement by the Academy:

The Academy first makes it clear that the Academy considers IPCC Working Group I to represent the scientific basis of climate science/change:
  • The Intergovernmental Panel on Climate Change (IPCC) Working Group 1 (The Physical Science Basis) has given a broad, systematic summary of the scientific literature on climate change and has concluded that the anthropogenic emission of greenhouse gases has led to an increase in the surface temperature of the Earth
  • IPCC has undertaken modelling studies to estimate the effect of anthropogenic greenhouse gases and aerosols on climate during the next 100 years based on different emission scenarios. These studies indicate a global surface warming at the end of the 21st century of 1.5-3.5 °C compared to present-day conditions. 
  • With the warming projections examined by IPCC other changes may follow, for example in the hydrological cycle, that might cause more problems than the temperature changes themselves. Whilst the lower range of warming may be acceptable, at least in some regions, the upper range is likely to cause very severe problems worldwide.
With this solid scientific basis the Academy then delivers the following General Advice to the Swedish King, Government and People to serve as the basis for Swedish Climate Politics:
  • In view of the potential long-term negative effects of increasing atmospheric CO2 and other greenhouse gases on climate and ocean chemistry (for example, acidification), development of mitigation technologies should be given priority. These should seek ways of reducing CO2 emission and the other components of anthropogenic forcing (including CH4, N2O, tropospheric ozone and black carbon aerosols) as well as focusing on CO2 sequestration, involving both the biosphere and the geosphere.
We read that the Royal Swedish Academy of Sciences echoes IPCC science without independent evaluation and questioning, and then accordingly echoes IPCC politics with a direct message to Swedish politicians to reduce CO2 emission and focus on CO2 sequestration

But, since the Academy evidently is an echo of IPCC, the Academy now must revise its Statement in accordance with the new IPCC AR5 Summary for Policymakers:
  • It is extremely likely (95%) that human influence has been the dominant cause of the observed zero warming during the last 17 years.
Isn't this correct, Lennart Bengtsson and the Royal Academy? Then go ahead and revise the Statement!

If the Academy simply echoes IPCC passively, then its Statement should be revised to reflect AR5 instead of AR4. If after all, the Academy is a real scientific academy of real independently thinking scientists and is not simply a passive echo of IPCC, then the Statement should be completely rewritten.

PS Here is a copy of a another letter sent today asking the Academy to revise its Statement (in Swedish):

Staffan Normark KVA

Hej Staffan:

Jag upprepar min begäran att KVA måtte revidera sitt Statement The Scientific Basis for Climate Change i ljuset av IPCC AR5: http://claesjohnson.blogspot.se/2013/11/royal-swedish-academy-of-sciences.html

Jag har framställt denna begäran flera gånger men inte fått någon som helst respons från KVA. Jag skulle uppskatta någon form av svar på min framställan och inte bara tystnad som om den inte nått KVA.

KVAs Statement utgör grunden för svensk klimatpolitk med fokus på minskade koldioxidutsläpp. Om KVA skulle ändra sitt Statement bort från koldioxidalarmism, så skulle också svensk klimatpolitik ändras. KVA bär således ett huvudansvar för svensk klimatpolitik och måste agera därefter.

Vänliga hälsningar, Claes

Claes Johnson, prof i tillämpad matematik KTH

lördag 23 november 2013

Constructive Calculus in Finite Precision


The Constructive Calculus of BodyandSoul essentially consists of constructing solutions to algebraic and differential equations by computational numerical methods and thus must take into account the finite precision representation as single, double or multiple precision used by computers when performing computations with real numbers using round-off.

The basic concepts of (Lipschitz) continuity and differentiability are introduced in BodyandSoul without reference to the difficult concept of limit and therefore naturally generalize to include finite precision, while the heavy use of limits in Standard Calculus does not.

It is thus natural to define a real-valued function $f(t)$ of a real variable $t$ to be Lipschitz continuous with Lipschitz constant $L$ in finite precision $\epsilon > 0$, if for all $t$ and $\Delta t$
  • $\vert f(t+\Delta t) - f(t)\vert\le L(\vert\Delta t\vert + \epsilon)$.  
We see that here $\Delta t$ will effectively be bounded below by $\epsilon$.

Further, it is natural to define a real-valued function $t\rightarrow f(t)$ of a real variable $t$ to be differentiable with derivative $Df(t)$ in finite precision $\epsilon$, if there is a constant $K$ such that for all $t$ and $\Delta t$
  • $\vert f(t+\Delta t) - f(t) - Df(t)\Delta t\vert\le K(\vert\Delta t\vert^2 + \epsilon )$. 
In this case $\Delta t$ will effectively be bounded below by $\sqrt{\epsilon}$.

We compare with the limit definition of continuity:
  •  f(t) = $\lim_{\vert\Delta t\vert\rightarrow 0}f(t+\Delta t)$,
which requires infinite precision to make sense.

Even more importantly, we compare with the limit definition of the derivative:
  •  Df(t) = $\lim_{\vert\Delta t\vert\rightarrow 0}\frac{f(t+\Delta t) - f(t)}{\Delta t}$,
with division by the infinitely small (but non-zero) quantity $\Delta t$, which requires infinite precision and even so is difficult to grasp, in particular for the student.

The limit definitions can be (and are) used in Symbolic Calculus with derivatives determined by symbolic and not numerical computation, but present severe difficulties in Constructive Calculus in finite precision. The definitions without limits of Constructive Calculus can also be used in Symbolic Calculus by setting $\epsilon =0$, and thus are more versatile.

The essence of Constructive Calculus is to compute solutions to differential equations involving derivatives, thus essentially to compute integrals numerically in finite precision by time stepping as a well-posed numerically stable process, while derivatives may be computed symbolically on restricted classes of functions such as the piecewise polynomials of finite element methods thus circumventing the ill-posedness of numerical differentiation.  

fredag 22 november 2013

What is $\pi$ and $e$?


There are two numbers with a special stature in mathematics:
where $\exp(t)$ is the exponential function usually defined by 
  • $\exp(t)=\lim_{n\rightarrow\infty}(1+\frac{t}{n})^n.$     (*)
The geometric definition of $\pi$ does not give direct information about its numerical value and the definition of $e$ may appear to be ad hoc without connection to either geometry or physics.

In BodyandSoul as Constructive Calculus both $\pi$ and $e$ are constructively defined through solutions of basic initial value problems solved by time stepping. More precisely, $\pi$ is defined as the smallest positive root of the equation $\sin(t)=0$, where $u(t)=\sin(t)$ and $v(t)=cos(t)$ is the solution to the initial value problem modeling a harmonic oscillator: 
  • $\frac{du}{dt} -  v =0$ and $\frac{dv}{dt} + u=0$ for $t > 0$, $u(0)=0$ and $v(0)=1$.  (**) 
Further $u(t)=\exp(t)$ is the solution to the basic initial value problem (expressing "exponential growth" with the growth rate $\frac{du}{dt}$ equal to $u$ itself):
  • $\frac{du}{dt}=u$ for $t > 0$ and $u(0)=1$.   (***)
In particular, 
  • $(1+\frac{t}{n})^n$ 
is the result of solving (***) by time stepping with time step $\frac{t}{n}$ with
  • $u(t+\frac{t}{n})=u(t)+\frac{t}{n}u(t) = (1+\frac{t}{n})u(t)$, 
which converges to $\exp(t)$ as the time step $\frac{t}{n}$ goes to zero with $n$ tending to infinity.

Introducing and defining the numbers $\pi$ and $e$ this way, gives both an understanding why they are so fundamental (by expressing basic properties of solutions to basic mathematical equations connecting to basic physics) and also shows how the (decimal expansions of the) numbers can effectively be computed.

PS1 Notice that once $\sin(t)$ and $\cos(t)$ have been defined as solutions of (**), it follows that $(\cos(t),\sin(t))$ can be geometrically interpreted as the coordinates of a point moving along a unit circle with unit speed and thus $t$ a measure of angle as arc length. The geometric interpretation of $\pi$ thus follows from numerical algebra and not the other way around as in Standard Calculus.

PS2 In Standard Calculus $\sin(t)$ and $\cos(t)$ are defined geometrically as quotients of the lengths of the sides of right-angled triangles again without access to the numerical values except for a few special values of the angle $t$.

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


onsdag 20 november 2013

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.

måndag 18 november 2013

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.


onsdag 13 november 2013

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



tisdag 12 november 2013

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.

måndag 11 november 2013

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:
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?

Klassiskt foto runt sekelskiftet från Prof. Leibschnitzels (3 fr v) gästföreläsning om Lipschitz-kontinuitet.



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.
  • 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.   

söndag 10 november 2013

More on Standard Calculus as Backward Magic

                                                       Riemann sum from MST.

The Backward Magic aspect of Standard Calculus is expressed by the fact that a main role of the Fundamental Theorem is to compute the area $A(a,b)$ under the graph of a real-valued function $v:[a,b]\rightarrow R$ defined on an interval $[a,b]$, by finding a primitive function $x(t)$ of $v(t)$ satisfying $\frac{dx}{dt} = v$, and then computing
  • $A(a,b) = x(b) - x(a)$.
Here $v(t)$ can be momentary velocity and $x(t)$ traveled distance from some position. The laborious work of computing the area $A(a,b)$ by summing the contributions over a partition of $[a,b]$ into many small intervals, that is computing a Riemann sum, is thus avoided and magically replaced by simply evaluating the difference $x(b) - x(a)$. This was the magic which could be performed by Leibniz and Newton in front of a stunned audience at the end of the 17th century, and this is the trick each standard Calculus teacher performs today in front of a mystified class of students.

But the magic was based on somehow analytically finding a primitive function, that is by solving the differential equation $\frac{dx}{dt} =v$ analytically. This could and can be done for certain functions $v$, but the analytical machinery may be very involved and often simply impossible. The natural generalization to $v$ depending on $x$ is even more difficult analytically.

Today the computer can solve the equation $\frac{dx}{dt}=v$ by time-stepping corresponding to computing a Riemann sum as an approximation of the area, and there is no need to resort to the magics of finding a primitive function analytically. The generalization to $v$ depending on $x$ is direct and easy by time-stepping with computer. What was difficult to Leibniz and Newton, and largely motivated Calculus, is easy for the student today. This gives Calculus a different meaning as Forward Rational time-stepping, which is not the Backward Magic offered students of standard Calculus.