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.
   (abstract, amslatex, dvi, pdf)
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.
   (amslatex, pdf)
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. (abstract, pdf)
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.
   (word, pdf)
7. M. Enelund, H. Johansson och S. Larsson,
   Beräkningsinriktad matematikutbildning för maskinteknikprogrammet på Chalmers,
   Utvecklingskonferensen för ingenjörsutbildning 2008, KTH, Stockholm, 26-27 november 2008.
   (amslatex, pdf)
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.   

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.  

Royal Swedish Academy of Sciences Echoes IPCC CO2 Alarmism

Echo and Narcissus.

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

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.  

What is $\pi$ and $e$?

There are two numbers with a special stature in mathematics:

• $\pi$ usually defined geometrically as the ratio of a circle's circumference to its diameter,
• $e=\exp(1)$ is the base of the natural logarithm, 

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

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.