Standard Proof of Fundamental Theorem of Calculus as (Backward) Magics

Here is a copy of the standard proof of the Fundamental Theorem of Calculus from the standard test book Calculus: A Complete Course, by Adams and Essex, which represents Backward Magics and not Forward Reason:

We see that Adams starts with the primitive function $F(x)=\int_a^xf(t)\, dt$ and proves that it satisfies the differential equation $F^{\,\prime} (x) = f(x)$, which is Backward Magic.

BodyandSoul vs Standard Calculus 0

BodyandSoul is Constructive or Computational Calculus while Standard Calculus as presented in the standard text book Calculus: A Complete Course by Adams and Essex, can be described as Analytic or Symbolic Calculus. I gave a basic example of the difference between a constructive and symbolic approach in the post

• Mathematics: Backward Magic or Forward Reason

about the Fundamental Theorem of Calculus. Let me here supplement by exhibiting the difference in the approaches as concerns the basic concepts of continuous function and derivate. 

In Analytical/Symbolic Calculus, a real-valued function $s\rightarrow f(s)$ with $s$ a real variable, is said to be continuous at $t$ if

• $f(t) = \lim_{\Delta t\rightarrow 0}f(t+\Delta t)$, 

  and to have derivative $Df(t)$ at $t$ if

• $Df(t) = \lim_{\Delta t\rightarrow 0}\frac{f(t+\Delta t) -f(t)}{\Delta t}$. 

In Constructive/Computational Calculus a real-valued function $s\rightarrow f(s)$ with $s$ a real variable, is said to be Lipschitz continuous with Lipschitz constant $L$ if for all $t$ and $\Delta t$

• $\vert f(t+\Delta t) - f(t)\vert\le L\vert \Delta t\vert$

and differentiable with derivative $Df(t)$ if for a positive constant $K$, for all $t$ and $\Delta t$

• $\vert f(t+\Delta t) - f(t) - Df(t)\Delta t\vert\le K\vert \Delta t^2\vert$.

We see that Analytical/Symbolic Calculus uses the concept of limit which is a difficult concept involving the mysterious process of $\Delta t$ tending to zero or becoming infinitessimally small, but holy God, not zero!

We see that Constructive/Computational Calculus does not use the difficult concept of limit, only the more basic and easy to grasp concept of local change, with a function being Lipschitz continuous if it is locally constant and differentiable if is locally linear with specified deviations.   

Why Calculus Reform Is So Difficult (But So Needed)

              Education of Calculus teachers:  Reading Calculus: A Complete Course.

The computer has given Calculus new meaning, tools and power, but Calculus education is today basically the same as when I studied at Chalmers 50 years ago, which was the same as the Calculus education 100 years ago untouched by the computer, and so on...

Real life experience shows that Calculus education resists all forces of reform. The situation is well described in the Preface to the standard text book Calculus: A Complete Course by Adams and Essex:

• Calculus is in many respects a classical subject.
• It is much older than the memories of anyone alive.
• It is taught in every institution of higher learning in the world.
• If it is so universal why do we not just reprint old text books from, say, the nineteenth century?
• The mathematics is the same isn't it?
• Of course , it is.
• If your great great grandparents had studied Calcukus in their time, there would be much that they would recognize in modern texts.
• The text books have grown larger, with many more examples, applications and exercises. Colorful ink and elaborate diagrams burst from the pages.
• The mathematics is the same but the audience is not.
• One unprecedented change began to take place more than a decade ago - a mere wink in the of the eye in the history of Calculus.
• Computer code began to appear in text books to respond to the growing awareness, access, and dependence of the audience on computers. 

But isn't it possible to question the standard dogma that Calculus is the same today as in the 17th-18th century, when it was developed by the great mathematicians of that time? 

No, the Calculus canon represented by Adams cannot be questioned because teachers of Calculus have been thoroughly trained during their studies to believe and confess to the Calculus canon and therefore are unable to go outside the cage and question the canon. And the rest of the world has nothing to say, because the only people who understand Calculus are teachers of Calculus. 

MST/BodyandSoul questions the Calculus canon, not just superficially, but on the fundamental level concerning the basic concepts of real number, continuity, derivative, the Fundamental Theorem of Calculus and more generally about the very meaning and use of Calculus. 

Accordingly, MST/BodyandSoul is banned by KTH, and to get a link put up on a Calculus course web page at Chalmers,  as supplementary material questioning Adams, has shown to be very difficult if not impossible.  

But the "unprecedented change" of the appearance of the computer will eventually push reform...because of "growing dependence of the audience (not teachers) on computers"...    

Mathematics: Backward Magics or Forward Reason

                A primitive function magically pulled out of a hat as an area under a function graph.


There are two approaches to mathematics:

1. Symbolic mathematics: magics: objects pulled out of hats. 
2. Constructive mathematics: reason: objects constructed in stepwise computation.  

Let me give two examples:

The Fundamental Theorem of Calculus

The presentation of the Fundamental Theorem of Calculus in standard text books of Calculus, is the following: Consider the integral

• $u(t) =\int_0^t f(s)\, ds$  for $t > 0$,

defined as the area under the curve determined by the function $s\rightarrow f(s)$ for $s\in [0,t]$.

Compute the derivate $\dot u=\frac{du}{dt}$ of the function $t\rightarrow u(t)$ with respect to $t$, to find that, assuming some suitable continuity property of $s\rightarrow f(s)$:

• $\dot u (t) = \lim_{\Delta t\rightarrow 0}\frac{u(t+\Delta t) - u(t)}{\Delta t}= \lim_{\Delta t\rightarrow 0}\frac{1}{\Delta t}\int_t^{t+\Delta t} f(s)\, ds = f(t)$ for $t >0$. 

In short, the key argument is to show that the integral $u(t)$, defined as an area, satisfies a differential equation

• $\dot u(t) = f(t)$ for $t > 0$

or solves an initial value problem

• $\dot u(t) = f(t)$ for $t > 0$ with $u(0)=0$.        (*)

We thus start with a given function, the integral $u(t)$, which is shown to be the solution of a certain initial value problem. The process leads from solution to equation satisified by the solution. The equation appears as magics without reason, since the reason is put into the specification of the solution or integral $u(t)$, with appeal to a concept of area which has to be defined, and not into the equation.

But this is backwards: The more reasonable forward procedure is to start with the initial value problem (*) expressing that the rate of change $\dot u$ of $u$ is equal to $f$ as balance equation expressing some basic physics, and then proceed to the integral $u(t)$ as the solution to the balance equation constructed by time stepping. This is the approach followed in BodyandSoul. We sum up as follows

• To proceed from solution to equation is backwards magical. 
• To proceed from equation to solution by forward time-stepping is reasonable and not magical. 

There are many specific examples of this form including trigonometric and exponential functions and more generally elementary functions all better constructed by time stepping basic differential equations than magically being picked out of hats.

For example, the trigonometric functions $\sin(t)$ and $\cos(t)$ are better defined as solutions to $\ddot u + u =0$, which can be constructed by time stepping, rather than geometrically as in standard calculus as ratios of the lengths of sides of a right-angled triangle, which is not computationally constructive.

Quantum Mechanics

The same situation is met in quantum mechanics:

The backward magical process is to start from a wave function solution and discover an equation satisfied by the solution, a magical Schrödinger equation without physical basis which is a mystery to all physicists.

The more natural procedure is to start from the Schrödinger equation, which can be formulated as a rational balance equation of smoothed particle dynamics, and then construct the solution (the wave function) by forward time stepping.

Concluding Remark: In the discussion of the mathematics program at Chalmers, the standard text book by Adams represents backwards magics, while BodyandSoul represents forward reason. Pick what you think is best. But after all, who cares?

Computational vs Analytical Calculus (at Chalmers)

In connection with the apparent return to Chalmers of (a bit of) the BodyandSoul mathematics education reform project, which was run at Chalmers 1998 - 2006 but was disrupted in 2007 when I moved to KTH, it may be of some interest to compare BodyandSoul as Computational or Constructive Calculus with the current standard of Analytical Calculus as presented by e.g. the standard text book Calculus: A Complete Course by Adams and Essex, a standard which found its form 100 years ago.

Standard Analytical Calculus:

• symbolic analytical solution of a limited variety of specific algebraic and differential allowing analytical solution
• as many analytical formulas and tricks as possible = thick book with many examples
• limited generality as large collection of special cases
• elementary functions picked from a hat
• solution work done by symbolic manipulation on paper
• maximal number of formulas and tricks to convince student of generality, which is illusion,
• mastery of the many analytical tricks difficult for the ordinary student
• the general problem cannot be tackled, not even by teacher
• conceptual difficulty: tricky symbolic manipulations, existence unclear and properties magic
• in short: many difficult special cases = powerless mathematics.  

Computational or Constructive Calculus:

• constructive solution of general algebraic and differential equations (by time stepping)
• focus on a few basic analytical formulas and tricks
• focus on a few basic constructive algorithms (fixed point iteration, Newton's method)
• generality by computation
• elementary functions constructed by solving differential equations by time stepping
• solution work done by computer
• few basic analytical formulas and computational algorithms can be mastered by the ordinary student
• student can tackle the general problem
• conceptual simplicity:  construction by basic algorithms gives existence and properties without magics
• in short: a few simple general cases = powerful mathematics.    

As you can see, the two approaches give very different weights to Complete Calculus as the union of Analytical and Computational or Constructive Calculus.

The fact the basic text at Chalmers is Adams, shows that the weights still are those of a standard course, and that BS in more complete form is still waiting to return to Chalmers.  As long as the text book is Adams, the standard since 100 years will continue to control (and limit) young minds, at Chalmers and elsewhere. It is a tragedy.

PS For more comparison between analytical and computational math, see next post.

Why Mass Education Fails

                             The Swedish school system with a minimal canon for all students.


Everbody agrees that the Swedish school system does not work well, but nobody claims to understand why and to know what to do.

The principal goal of the Swedish school system is to secure that all students reach a certain minimal level as a minimal canon of Swedish and mathematics, and the big mystery why this goal shows to be so difficult to reach.

The question is why teaching simple arithmetics like 1/2 + 1/ 3 = 5/6 no longer works, when it worked so fine in the 1950s?

Let us seek an answer by first asking if knowledge and skill as the objective of schooling, has a different meaning in the IT-society of today than in the 1950s? A moment of reflection reveals:

• Knowledge and skill may have a large value on the market, if it is not shared by everybody.
• Knowldege and skill shared by everybody may have little value.

It is the uniqueness of some piece of knowledge and skill which makes it valuable. The more unique, potentially the more valuable, right? 

In the 1950s knowing how to  compute 1/2 + 1/3 represented unique competence carried by few which opened to few good jobs in the administration. But with mass education this no longer works because the competence is not unique at all (and everbody cannot be an administrator). 

We conclude that mass education giving all students the same minimal canon, may by the students be understood as an education of little value and thus of little interest. The more trivial such a mass education is made in order to get students through, the more will student motivation and interest drop and the education will end up in a vicious circle towards zero. This is what we are now observing.

We have now understood the basic problem or contradiction of the Swedish school system, and our minds are thus free to explore education offering 

• variety of unique knowledge and skill to a variety of motivated students   

instead of as in the present system

• the same minimal canon to all students. 

This means nothing but a revolution in education and it has already started powered by MOOC and web education.

The above analysis also shows the strong potential of the mathematiccs education reform program  BodyandSoul/MST as bringing unique knowledge and skill into the minimal canon of mathematics in standard engineering education.


 

BodyandSoul Tillbaka på Chalmers

BodyandSoul (BS) som reformerad matematikutbildning initierades och drevs vid Chalmers 1998 - 2006, men lades ner då jag flyttade till KTH 2007.  BS återuppväcks nu på Chalmers i form av delar rekommenderade som "fördjupningstext" inom M-programmet hämtade ur web-versionen Mathematical Simulation Technology (MST).

Chalmers teknologer välkomnas till MST, som ger Chalmers en konkurrensfördel jämfört med KTH, där MST är förbjudet.

PS1 Mina synnerligen motiverade frågor till Anders och Stig enligt nedan möts med kompakt tystnad. Kanske finns det inga svar som tål ljuset.

PS2 Jämför med de senare posterna

• Computational vs Analytical Calculus
• Mathematics: Backward Magics or Forward Reason
• Why Calculus Reform Is So Difficult (But So Needed)
• Standard Proof of Fundamental Theorem as (Backward) Magics 
• More on Standard Calculus as Backward Magic
• ....
• Beräkningsinriktad Matematikutbildning på Maskinteknik Chalmers vs BodyandSoul
• More on the Fundamental Theorem of Calculus: Standard vs BodyandSoul

Högsta Förvaltningsdomstolen Ger Prövningstillstånd för KTH-Gate

Högsta Förvaltningsdomstolen meddelar 2013-10-31 prövningstillstånd avseende Rätt att ta del av allmän handling och PuL:

• Fråga om e-postmeddelanden, som inte har anknytning till något ärende och som enbart skickats mellan befattningshavare inom samma myndighet, kan anses vara färdigställda och därmed upprättade i tryckfrihetsförordningens mening. (Mål nr 5339--5340-13, Kammarrätten i Stockholms mål nr 3640-13 och 3141-13).

Prövningstillståndet gäller mitt överklagande av Kammarrättens i Stockholm beslut att avslå mina överklaganden av KTHs beslut att avslå min begäran att ta del av epost mellan rektor-dekan-avdelningsföreståndare-studierektor avseende ärendena Gomorron Sverige och MST/BodyandSoul. 

Mitt fall tas nu alltså upp av Högsta Förvaltningsförvaltningsdomstolen med avsikt att skapa ett prejudikat för Tryckfrihetsförordningen, som tolkats väsentligen olika av Kammarrätterna i Göteborg och Stockholm, med vida konsekvenser vad gäller möjligheterna till insyn i den nya högskolans linjeorganisation och beslutsapparat formad efter privat näringslivsmodell.

Frågan är om det finns någon skillnad mellan den statliga högskolan KTH (eller Stiftelsen Chalmers) finansierade med skattemedel med forskning/utbildning för Sveriges allmänna bästa som uppgift, och AB ASEA med uppgift att ge ägaren Wallenberg utdelning?

Fallet bevakas av Academic Rights Watch:

• Här finns en djup konflikt med den svenska offentlighetsprincipen som demokratifundament.
• Det vore ett svårt slag för den svenska demokratin om domen i Stockholm står sig. 
• Men vi är tillräckligt luttrade för att inte ta någonting för givet när det gäller svenska rättsinstansers tillförlitlighet i ärenden där myndighetshierarkins bevarande står mot den enskilde individens rättigheter. 

Ny Teknik skrev om censur av mitt verk på KTH 2010. Ny Teknik borde nu fortsätta sin rapportering.

PS. Notera att Högsta Förvaltningsdomstolen i sin beskrivning av mitt fall går på min linje:

• Fråga om e-postmeddelanden....kan anses vara färdigställda och därmed upprättade...    

Knäckfrågan är alltså vad som menas med begreppet "färdigställd":

• Kammarrätten i Göteborg säger att handling är "färdigställd" om den inte är avsedd för vidare bearbetning (och har skickats). 
• Kammarrätten i Stockholm påstår att det krävs "ytterligare någon åtgärd" (utöver avsändande), som dock ej specificeras, för att en handling skall vara "färdigställd".  

Om Kammarrätten i Stockholm skulle ges rätt, så skulle Tryckfrihetsförordningen (avsedd att värna medborgarens rätt gentemot Staten och Myndigheten) förlora sin mening. 

Half a Million Page Views

This blog started in 2009 has just passed half a million page views with the following top posts:

• Photoelectric Effect, Einstein, Arrhenius and Global Warming: 5888 page views
• Radiative Heat Transfer: 2448
• KTH-Gate: Climate (Mathematics) "Stopped": 2395
• IPCC Censorship: 2330
• Power of Language: "Refrigerator Effect" vs "Greenhouse Effect": 1832
• Is Crazy-Physics = Pseudo-Science?: 1198

With a total number of posts of about 1000, the average number of page views per post is about 500.

Not so bad, compared to a scientific article which may typically attract 5 readers.  

MOOC as Disturbance to the Traditional University.

The exploding MOOC market is now rapidly changing the role of the teacher, in school and at university, in particular in science and engineering education. The traditional teacher role includes as key elements

• control of subject: scientific paradigm,
• control of teaching material: text book, lecture notes,
• control of the mind of the student.  

The control is exercised through questions formulated by the teacher in

• homework,
• quizzes,
• exams.

In short, the traditional university/teacher

• controls the standard,
• controls the examination,
• owns the student,
• is omnipotent. 

All of this is now challenged by MOOC. 

The gut reaction to this challenge from the traditional university/teacher is to view MOOC as a disturbance which will soon fade away, if it is ignored.