Mathematics 1
(Courant Robbins)

22. Sum of First n Squares

Courant Robbins

What is Mathematics ?

1996, Oxford University Press
ISBN13: 9780195105193
ISBN10: 0195105192

Originally Published - 1941

Richard Courant's qualifications may best be derived from this interview: http://www.aip.org/history/ohilist/4562.html - so far as I can tell he was highly influential in establishing at least some of the standard calculus curriculum.

For me, the immediate objective is a calculus animation sufficient to explain guitar harmonics to a somewhat less than mathematically inclined musician yet rigorous enough to satisfy a Calculus I teacher; and to align the project with the following statement from Courant Robbins:

page v. - "The goal is genuine comprehension of mathematics as an organic whole and as a basis for scientific thinking and acting."

I don't need the whole of mathematics, just what can be clearly related to the guitar

The notes on the first 400 pages are to help me deal with some background material and inadequacies in my math. Those marked in red are thought to be most important for me.

Sum of the First n Squares animation

It is used in the introduction to integration on page 408.

p 44 - Algorithm

Euclidean Algorithm

see p. 236 (margin note - prob. looked up "algorithm" online)

re: Mathematical Rigor - amenability to algorithm checking

(Wikipedia - ...completely rigorous proofs which tend to be (long and unwieldly) may obscure what is being demonstrated.)

...guided by the need for a suitable instrument for handling measurements... The rules for operating with new numbers (rationals, irrationals,transcendentals etc..) ... are created by us ...to preserve the fundamental laws of arithmetic.

p89...extended number domain...freely created by definition...but useless (unless the prevailing rules and properties of the original domain are preserved in the larger domain)

(Monochord string might serve as a number axis - idea of infinite division of a continuum in a limited segment)

see numberLine.html

(Given

b = Pi(a)
Pi = (approx) 22/7(a)

p 63 Introduction to limit concept

"...it sometimes happens that a certain rational number s is approximated by a sequence of other rational numbers s sub n, where the index n assumes consecutively all the values 1, 2, 3..."

p 68-69 - ... nested intervals - fundamental postulate of geometry.

p 68 "...the continuum of numbers or real number system...is the totality of infinite decimals. Finite decimals may be considered a special case where all digits from a certain point on are zero..."

nested_intervals.html
preliminary design - possible animation based on Figure 11, p 69

http://everything2.com
/e2node/nested%2520interval
%2520property

...one of several different ways of saying that the real numbers are a complete ordered field...their intersection is not empty; that is, there is at least one real number that is in all of the nested intervals...

---

A Radical Approach to Real Analysis
By David M. Bressoud
Published by MAA, 2007
ISBN 0883857472, 9780883857472
323 pages

Ch 2.4 pp 32-33 (nested interval '...defined the nature of the real number line...the crucial property that distinguishes the set of all real numbers from the set of rational numbers...guarantees...that the real number line has no holes in it. If two sequences are approaching each other from different directions, then where they ``meet'' there is always some number...'

starting with:
1. a real number may be regarded as in infinite decimal
2. infinite decimals are limits of finite decimal fractions

Cantor;
...any sequence (a sub1 ), (a sub2 ), (a sub 3 ...) of rational numbers defines a real number if it converges.

Convergence is understood to mean that the difference (a sub m - a sub n) between any two numbers of the sequence tends to zero when (a sub m) and (a sub n) are sufficiently far out in the sequence, as m and n tend to infinity.

September 30, 2008 - Algebra (in general, not treated as a specific topic in courantRobbins)

http://en.wikipedia.org/wiki/Algebra

Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.

***

http://abstractmath.org
/MM/MMMathStructure.htm

Mathematical Structure
Basic idea
A mathematical structure is a set (or sometimes several sets) with various associated mathematical objects such as subsets, sets of subsets, operations and relations, all of which must satisfy various requirements (axioms). The collection of associated mathematical objects is called the structure and the set is called the underlying set. The axiomatic method essentially always produces a mathematical structure.

Mathematical Analysis of Infinity

p 77 - ... the concept of the infinite pervades all of mathematics since mathematical objects are usually studied not as individuals but as members of classes or aggregates containing infinitely many objects of the same type.

p 78 - (RE: Cantor's general concept of a set or aggregate)

...any collection of objects defined by some rule which specifies exactly which objects belong to the given collection...

***
... the basic notion is that of 'equivalence'...

...if the elements in two sets A & B may be paired with each other...so that:

"to each element of A there corresponds one and only only one element of B"

and

"to each element of B corresponds one and only one element of A"

then

the correspondence is said to be "biunique" and A is said to be equivalent.

***

This is in fact the very idea of counting, for when we count a finite set of objects, we simply establish a biunique correspondence between these objects and a set of number number symbols 1,2,3,...

[ror-keeping in mind some ultimate purpose that - (p 1 - Mathematical statements should be reducible to statements about the natural numbers)

Sets and Number Line (first entered September 30, 2008 - October 6, 2008 - An animation Axiomatics has been started.)

see also p 108

some external references:

http://en.wikibooks.org/
wiki/Algebra/Sets_and
_the_Number_Line

algebraic elements are often numbers
geometric elements are often points, which are infinitesimally small locations in space

The number of elements in a set could be countable or could be infinite, as long as the elements are quantifiable, definable, or determinable in some way either now or in the future

Infinity is not a number, it is a concept

(...contains various suggestions for the Number Line diagram should be used or at least referenced in the animation - numberLine.html)

***

http://learningnerd.wordpress.com/
2006/10/23/discussing-numbers-the
-number-line-types-of-numbers-sets
-inequalities-and-intervals/

Discussing Numbers: The Number Line, Types of Numbers, Sets, Inequalities, and Intervals October 23, 2006

Introductory paced, 15-20 min videos included, very good, inclusive - I'd like to find a faster paced and shorter set of videos on the same topic for comparison and different levels of viewers...I have difficulty maintaining concentration for this length of time at this pace but I can still learn a great deal if the subject is new - possibly try some sort of quick review at end...- use same questions at beginning of next topic...?

***

http://www.webmath.com/k8numlinecomp.html

Compare Two Numbers Using a Number-Line

***

http://en.wikipedia.org/wiki/Set_theory

September 29, 2008
Set theory, formalized using first-order logic, is the most common foundational system for mathematics. The language of set theory is used in the definitions of nearly all mathematical objects, such as functions, and concepts of set theory are integrated throughout the mathematics curriculum. Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects. Elementary operations such as set union and intersection can be studied in this context. More advanced concepts such as cardinality are a standard part of the undergraduate mathematics curriculum....

Set theory is also a promising foundational system for much of mathematics.... For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.

***

http://en.wikipedia.org/
wiki/Zermelo-Fraenkel_set_theory

(September 30, 2008) - Outlines a form of most commonly used Axiomatization of Set Theory using First Order logic symbols (which I don't understand) and English analogies to them (which I can at least read)

8 axioms + 9 is "Well Ordering Theorem"

***

October 30, 2008 - The following seems useful:

Mathematics: The Science of Patterns
By Keith Devlin
Published by Macmillan, 1996
ISBN 0805073442, 9780805073447
224 pages

p57 Cantor's Set Theory

...When mathematicians define some abstract mathematical object or system as a 'set of objects' satisfying certain properties, it usually does not matter what the members of the set are; rather, what counts are the operations that can be performed on those members. in fact, even that is not quite right. The real interest is in the properties of those operations... (i.e. if studying natural numbers: Their interest is in the various properties of numbers and of the operations of arithmetic, properties such as the commutativeity of addition, whether one number divides evenly into another, whether a number is prime,and so forth.

http://www.stanford.edu/~kdevlin/Math_in_2100.pdf

What will count as mathematics in 2100?
Keith Devlin

p2

After Newton and Leibniz, mathematics became the study of number, shape, motion, and change.... In the middle of the 19thcentury, however, a revolution took place. Generally regarded as having its epicenter in the small university town of Göttingen in Germany, the revolution’s leaders were the mathematicians Lejeune Dirichlet, Richard Dedekind, ...and Bernhard Riemann.

--------------------------------------------------------------------------------
p3

...In their new conception of the subject, the primary focus was not performing a calculation or computing an answer, but formulating and understanding abstract concepts and relationships...Mathematical objects, which had been thought of as given primarily by formulas, came to be viewed rather as carriers of conceptual properties. Proving was no longer a matter of transforming terms in accordance with rules, but a process of logical deduction from concepts.

In the 1980s, however, a definition of mathematics emerged... mathematics is the science of patterns. Those patterns can be either real or imagined, visual or mental, static or dynamic,qualitative or quantitative, purely utilitarian or of little more than recreational interest. They can arise from the world around us, from the depths of space and time, or from the inner workings of the human mind.

(ror- caveat, people will create patterns out of noise, ie constellations, )

p88 - (Foundations of Mathematics)

...(continuum hypothesis - reduces to questions of what is meant by mathematical existence. "Luckily, the existence of mathematics does not depend on a satisfactory answer...intuition always remains the vital element...")

http://en.wikipedia.org/
wiki/Foundations_of_mathematics

Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. The search for foundations of mathematics is also a central question of the philosophy of mathematics: On what ultimate basis can mathematical statements be called true?

p 88-93 - Imaginary Numbers

August 4, 2008
This may not be as immediately important as I had first thought. An animation complexNumbers.html has been created but it shows little and I do not expect to develop it for some time. January 2010- revised to same effect.

February 20, 2009 - p.92 Art 2 - Geometrical Interpretation of Complex Numbers - browsing, reread and noted "utmost" - last sentence...(the geometrical approach is unnecessary to mathamatics based on the formal definitions of addition and multiplication [more or less Axiomatic Set Theory] - but "of the utmost importance" (in applications and physics)).

August 15 - might want to add following from p 89... the imaginary unit has nothing to do with counting...it is merely a symbol subject to the fundamental rule i^2 = -1...its value will depend entirely on whether by this introduction a really useful and workable extension of the number system can be effected.

p 103 - The concept of algebraic number is a natural generalization of rational number, which constitutes the special case when n =1.

Not every real number is algebraic. This may be shown by a proof due to Cantor, that the totality of all algebraic numbers is denumerable. Since the set of all real numbers is non-denumerable, there must exist real numbers which are not algebraic.

Set theory is a distinctly weak part of my math background. The articles here refer to "multiplication is not repeated addition" have not led me to any conclusions, perhaps some cautions:

...September 18,2008 - just completed a review of Devlin's second article:
http://www.maa.org/devlin
/devlin_0708_08.html

Apparently he wants the principles of axiomatic set theory and arithmetic operations not contradicted (by statements like ""multiplication is repeated addition)- and he wants the teachers to decide how to do it themselves. - it was all very contentious and I have never made much out of it except for the idea of multiplication as proportion noted in several articals below.

"As I said earlier, I don't think it would be a sensible thing to teach arithmetic by starting with the real number system; indeed, I find it hard to imagine how that could possibly succeed. But since that is the culmination of the arithmetic learning journey, it would be wise to avoid doing anything that runs counter to that final goal system."

Courant Robbins makes this point at page 70; "Some modern textbooks on mathematics repel many students by starting with a pedantically complete analysis of the real number system." (More or less what actually happened during the New Math episode.)

***

For a sympathetic argument: a detailed discussion of "The New Math" and plea for saving what was good.

http://lsc-net.terc.edu/
do.cfm/conference_material/
6857/show/use_set-oth_pres

***

As to multiplication not being repeate addition:

http://www.themathpage.com/ARITH/
multiply-fractions-parts-of-fractions.htm#def

Here is the most general definition of multiplication:

Whatever ratio the multiplier has to 1
the product shall have to the multiplicand.

Consider this multiplication of whole numbers:

3 × 8 = 24

The multiplier 3 is three times 1; therefore the product will be three times the multiplicand; it will be three times 8.

(ror 3:1::24:8)

***

October 6-7, 2008 - An animation Axiomatics has been started to help me work out some of this.

December 11, 2008 - A set theoretic derivation of multiplication seems unlikely (for me - very) in the near future. For the time being I am keeping a proportional understanding of the operation, as follows.

http://cehd.umn.edu
/rationalnumberproject/95_2.html

TEACHERS' SOLUTIONS FOR MULTIPLICATIVE PROBLEMS
Guershon HAREL, Purdue University
Merlyn BEHR, Louisiana State University
(Received July 19, 1994)

...The main finding is that teachers who solved the problems correctly and relationally reasoned in terms of ratio and proportion concepts...

...the ratio and proportion solutions are based on relational, genuine understanding of the problem situation

http://www.wtamu.edu/ academic/anns/mps/math/
mathlab/beg_algebra/ beg_alg_tut7_mult.htm
Beginning Algebra Tutorial 7:
Multiplying and Dividing Real Numbers

http://www.wtamu.edu/academic/anns/mps/math/
mathlab/beg_algebra/beg_alg_tut8_property.htmBeginning Algebra Tutorial 8:
Properties of Real Numbers

West Texas A&M University (WTAMU). Kim Seward with the assistance of Jennifer Puckett.... created as a service to anyone who needs help in these areas of math. Last revised on Jan. 8, 2002

Also:

http://www.ltcconline.net/
greenl/courses/187/a/
WholeNumberMultiplication.htm

and:

http://www2.dsu.nodak.edu/users/
edkluk/public_html/MulTab/didactic.html

and:

http://mathforum.org/library
/drmath/view/58316.html

RE: Real Number Properties

To talk about these properties, you need to think not only about the numbers themselves, but the operations you perform on them...

Closure means that if you add any two real numbers you'll get a real number. Same if you multiply two numbers. Subtraction is closed, but division is not. You cannot divide by zero.

p112 - ...the laws 1-26 form the basis of the algebra of sets...

(this needs googling to verify if it needs modernization and/or relation to axiomitization of setTheory?)

September 22, 2008 - The following puts some perspective on Sets for me:

http://en.wikipedia.org/
wiki/Set_theory

The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the most well known.

Set theory, formalized using first-order logic, is the most common foundational system for mathematics. The language of set theory is used in the definitions of nearly all mathematical objects..."

...properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.

This site provided some terminology

http://en.wikibooks.org/
wiki/Algebra/Sets_and
_the_Number_Line

and this quote from it seems helpful:

"The objective of algebra is often to take the information available in a situation or problem, and to define or determine a set of elements such as numbers as simply and precisely as possible."

p 112 - ...use the laws 1-26 as the basis for an "algebra of logic"

...properties or attributes of objects...reduced to a formal algebraic system...

The rules for translating the usual logical terminology into the language of sets ...

[...written I believe, before any real computers existed...needs googling, verification & updating...symbols probably, if not the logic...]

***

From an even older source, a logician and mathematical historian:

Jourdaine, Philip E. B., The Nature of Mathematics, in: Newman, The World of Mathematics Vol I p 4, Simon and Schuster, New York, 1956

p 44 ...mathematical...function ...is to imitate by logical consequences of the properties assigned to it by definition, certatin processes of nature as closely as possible...

That this model of reality is constructed solely out of logical conceptions will result from our conclusion that mathematics is based on logic, and on logic alone...

p70...when the notions of variable and function are introduced into logic...all mathematical methods...and conceptions can be defined in purely logical terms...

p 120 - 121 (...understanding lies in translating the geometrical problems into the language of algebra...

...first we find a relationship (equation)...next we find the unknown quantity ...finally, ...determine whether this solution can be obtained by algebraic process that correspond to ruler and compass constructions...)

It is the principle of analytic geometry, the quantitative characterization of geometrical objects by real numbers, based on the introduction of the real number continuum, that provides the foundation for the whole theory.

December 11. 2008

Abraham Kaplan cited in Newman, The World of Mathematics, Vol 2, p1297 Simon and Schuster, 1956

(Algebraic symbolism expresses the relation between quantities without the need to specify their magnitude.)

"...we proceed, not from postulates of uncertain truth, but from arithmetical propositions whose truth is a matter of logic following necessarily from the definitions of the symbols occurring in them."

Courant Robbins continues with some illustrations and explanations pretty much directly from Descartes, La Geometrie and I have already related some of this to music (cf w/ Zarlino's mesolabium, Descarte's Proportional Compass - (Origins 2, 3 animations)

p 181 - ( Logically [a geometrical object] ) is completely described by the totality of statements by which it is related to other [geometrical] statements...)

[...ordinary axioms of geometry(Euclid's) objects are abstractions from the physical world of ...chalk, stretched strings, light rays...etc...

"Geometrical [objects] have essentially simpler properties than do any physical objects, and this simplification provides the essential condition for the development of geometry as a deductive science."

p 214 - Axiomatic Method

October 6-7, 2008 - Animation Axiomatics started.

***

http://en.wikipedia.org/wiki/Axiom

...an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived.

...Axioms ... cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems).

Logical axioms are usually statements that are taken to be universally true...

non-logical axioms (e.g, a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic)

Arithmetic
The Peano axioms are the most widely used axiomatization of first-order arithmetic.

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. (ror - I presume this stuff will have to be learned to at least some minimal level due to computer programming.)

Second-order logic is more expressive than first-order logic...(ror - and what I know about it so far can best be expressed as null)

 

***

http://www.libraryofmath.com/
about-foundations-of-geometry.html
re: geometry
Axiomatic Method. The problem is to erect the entire structure of Euclidean geometry upon the simplest foundation possible; i.e. to choose a minimum number of undefined elements and relations and a set of axioms concerning them, with the property that all of Euclidean geometry can be logically deduced form these without further appeal to intuition.

Good treatment of conics:
http://www.mathacademy.com
/pr/prime/articles/conics/index.asp

Also, some time ago I had some historical connections of Descartes > Papus > Conics > Locus of Three or Four Lines > Calculus - now lost, here is a historical treatment of conics:
http://www.math.rutgers.edu
/~cherlin/History/Papers1999
/schmarge.html
Conic Sections in Ancient Greece - Ken Schmarge

p 214 - The Axiomatic Method - ...each theorem of the system is proved by showing it is a necessary logical consequence of some previously proved proposition...

(As a starting point), there must be some statements, called postulates or axioms which are accepted as true, and for which proof is not required...

p 215 - (to be of value, the axioms must be):

1. simple
2. few in number
3. consistent - (no mutually contradictory theorems can be derived from them),
4.complete - (every theorem of the system can be deduced from them)

and it is also desirable that the axioms are

5. independent (no one of them is a logical consequence of the others.)

p 215 - Differing philosophies on the "Foundations of Mathematics"

Intuitionists (Kantian) - mathematical entities are considered to exist in a realm of "pure intuition", independent of definitions...does not accept the number continuum, admits the denumerably infinite...(clarify???)

Formalists - concern is only with the formal logical proceedure of reasoning on the basis of postulates...(accepts the number continuum)...grants all the freedom necessary for theory and applications...but...imposes the necessity of proving that these axioms, accepted as arbitrary creations of the human mind, cannot possibly lead to contradictions...(per Godel etc...it appears that proofs for consistency and completeness are not possible within strictly closed systems of concepts.)

Rigorous adherance to a philosophical point of view sharpens the debate and narrows the issues however, citing and paraphrasing John von Neumann, (this controversy) "...constitutes the best caution against taking the immovable rigor of mathematics too much for granted... and I know myself how humiliatingly easily my own views regarding the absolute mathematical truth changed during this episode, and how they changed three times in succession." Newman, World of Mathematics, Vol 4, p 2059

For myself, I find philospophy fun but am more concerned with the bottleneck of a "Pons Assinorum" too narrow for the times and the numbers that must cross if there is to be any safety factor to existence on so small a lifeboat as this planet now affords so large a population.

I am a big fan of safety factors since loading a tank on the deck of a Victory ship with a rather neglected jumbo boom back in 1968 or so. Those who had done the calculations did not do the loading. It was a success, perhaps the tank is still being usefully employed by those it was intended to shoot, and luck is good, sometimes it is all that is available, likely more of it will be available if it is not the only thing relied upon.

p 216 - (regardless of the philosophy)...The axiomatic approach to a mathematical subject is the natural way to unravel the network of connections between the various facts and to exhibit the essential logical skeleton of the structure.

....but a significant discovery or insight is rarely obtained by an exclusively axiomatic proceedure...The constructive intuition of the mathematician brings to mathematics a non-deductive and irrational element which makes it comparable to music and art

p 217-The totality of axioms of geometry provides the implicit definition of all "undefined" geometrical terms...

if the formal axioms did not agree more or less with the properties of physical objects, then geometry would be of little interest...(this) decides the direction of mathematical thought.

number continuum

see numberLine.html

p 222 - ...the base chosen for the logarithm is of no importance, since change of base merely changes the unit of measurement. [section dealing with non-euclidean distance]
[ p 444 - The principle property of the logarithm is expressed by the formula:

log a + log b = log (ab)

(Intuitively, this formula could be obtained by looking at the areas defining the three quantities log a, log b, and log (ab)]

[p 442 ...the basic concepts of the calculus furnish a much more adequate theory of the logarithms and the exponential function than (the usual school instruction)...]

see p 44

An important point for me - When something is placed on the world wide web a range of viwers from beginner to expert might profitably be taken into account...

http://www.maths.uwa.edu.au/~schultz
/3M3/L28Weierstr,Dede,Cantor.html

Richard Dedekind
Continuity and Irrational Numbers, 1872

"Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny."

***

(ror - another idea belongs here, source missing, paraphrased to the effect 'a rigorous treatment, by reason of its considerable detail, tends to obscure rather than demonstrate the conceptual treatment.'')

p 272 - Functions and Limits

The main body of modern mathematics centers around the concepts of functions and limits..

p 272 - The concept of function enters whenever quantities are connected by a definite physical relationship...The whole domain of periodic phenomena - the motion of tides, the vibrations of a plucked string, the emission of light waves from an incandescent filament - is governed by the simple trigonometric functions sin x and cos x.

(check the above, somewhere in the book a distinction between a function in physics and one in mathematics is noted)

p 276 - A mathematical function is simply a law governing the interdependence of varying quantities.

It does not imply the existence of any relationship of "cause and effect" between them...

It is only the form of the connection between the...variables which is relevant to the mathematician.

(?- Fourier analysis IS the function u = f(x, t)?)

plucked string

p 289 - ...the description of the continuity of a function is based on the limit concept.

p291...epsilonDelta concept of limit (geometric series)

(...limit of a sequence here is just an introduction to the concept of limit ?] see p 63 see p 303

November 14, 2008 There is now an animation, Limit 1; today it is just links to the youTube videos I find useful but It may be expanded a bit later.

The non-video pages being looked at:
http://www.mc.maricopa.edu/
~dschultz/LimitStudent.html
#MapleAutoBookmark1

http://www.themathpage.com/
acalc/limits.htm

There are some epsilon delta applets - Limit Concept

http://www.slu.edu
/classes/maymk/Applets
/EpsilonDelta.html

http://math.hws.edu
/javamath/config_applets
/EpsilonDelta.html

(July 21,2008 - currently thinking of a variation of Equal Temperament 1 and 2 animations, but single animation, both number line and sequence sideBySide - vague at this moment - will have to be much shorter and very clear to be worthwhile...

December 11, 2008 - I have been reading this page and will reread, so far there appears little danger of comprehension.

I am interested in possibly starting with some variation of the phrase near the bottom of the page:

"...we must base our definition (of limit) on what we have to do if we wish actually to check the statement a(sub n) --> a."

It seems the most readily comprehensible part of the explanation, necessary and reasonable and leads me into the discussion rather than chasing me away from it.

(Also, I would like to find some reason beyond a test score as to why anyone should go to the trouble to learn so convoluted an explanation. I have studied (and written) rather equally difficult things in the practice of law but usually for the purpose of confusing rather than clarifying an issue)

Possibly from quantum mechanics and the uncertainty principle which apparently post-date the explanation but may justify it:

Erwin Schrodinger, Causality and Wave Mechanics, cited in Newman, World of Mathematics, Vol 2, p 1057, 1056 - "Observations are to be regarded as discrete, disconnected events. Between them there are gaps which we cannot fill in."... no model shaped after our large-scale experiences can ever be true...however we think it, it is wrong."

Niels Bohr and Courant had been caballing on such ideas for many years and despite its paradoxes it is rather successful. It would provide a good reason to work through this. A good test score is inadequate save for those who can't get anything other than a good score, whatever the reason )

p 297 - The importance of the limit concept in mathematics lies in the fact that many numbers are defined only as limits...

This is why the field of rational numbers, in which such limits may not exist, is too narrow for the needs of mathematics.

p 305 - 306 Only by limiting processes (epsilonDelta) can the fundamental notions of the calculus - derivative and integral - be defined...(the intuitive concept of continuous motion has eluded all attempts at an exact mathematical formulation)...Certainly the intuitive idea of a continuum has a psychological reality in the human mind. But it cannot be called upon to resolve a mathematical impossibility; there must remain a discrpency between the intuitive idea and the mathematical language designed to describe the scientifically relevant features of our intuition in exact logical terms.

(see also pg 406)

November 14, 2008 - There is now an animation, Limit 1;

There are a few applets (put links here) what I might want to do is an explanation accompanying the animation...

apparently distinguishes continuous motion from logical/mathematical description of motion (using idea of a number continuum?)

January 22, 2010 The only thing keeping this entry in place is the reference to "nested intervals" (understood as equivalent to Dedikind Cuts and Cantor's idea of Real numbers as 'infinite decimals') - none of which is well understood by me. August 11, 2008 Preliminary animation for Bolzano's Theorem - barely started, to complete will require a closer look at epsilon-delta.

Section 5 Article 2, second sentence, "Our objective is to reduce the theorem to fundamental properties of the real number system, in particular to the Dedikind-Cantor postulate concerning nested intervals. (p 68)

The second theorem of continuity is Weierstrass' Theorem on Extreme Values. (p313)

Another (less fundamental) - Compact Sets (p315)

p 342 - As a rule in scientific thinking it is better to consider the individual features of a problem rather than to rely exclusively on general methods, although individual efforts should always be guided by a principle that clarifies the meaning of the special proceedures used.

This indeed is the role of the differential calculus in extremum problems.

I am thinking of "general methods" as referring to computational methods applicable to all, or at least most cases and problems - then this seems to say -calculus provides wide ranging perspective though there may be more accurate means for solutions. (possibly referring to "discrete math" as more accurate?)

see page 400

p 361 - One of the characteristic features of higher mathematics is the important role played by inequalities.

The solution of a maximum problem always leads, in principle, to an inequality...

...consider the important inequality between the arithmetical and geometrical means.

arithmetical and geometrical means interests me because I have already done some animations with them...

A short animation sketching out some of the ideas here...more of a placeholder for possible future development. Maxima and Minima.

(...positive square root = geometric mean

g = sqrt(xy) )

p 381 - ...natural phenomena often follow some pattern of maxima and minima...

p 382 - It was observed that stable equilibrium of a mechanical system is attained if the system is arranged in such a way that its "potential energy" is a minimum.

p 382 - Not only the laws of equilibrium, but also those of motion, are dominated by maximum and minimum principles.

To some extent, all the above was done simply to help me fill in some rather serious gaps in my formal education and general understanding - especially algebra.

Any possible use of calculus beyond some minimal parroting of formulas apparently requires some familiarity with those subjects

January 22, 2010 I have now done a bit of research and work here - all to the conclusion that a very good understanding of introductory Algebra is prerequisite. For my part, the preliminary sketch for the Fourier animation will serve mostly to sharpen interest in the study of the subject at themathpage.com

p 399 - (re: Calculus) There is no longer any reason why this basic instrument of science should not be understood by every educated person.

( Preface v - The goal is genuine comprehansion of mathematics as an organic whole and as a basis for scientific thinking and acting.

Understanding of mathematics cannot be transmitted by painless entertainment.)

p 400 - Integral - The need for a more general method of computing areas arise when we ask for the area of a figure bounded not by polygons but by curves...

(...Archimedes (287-213 B.C.) computed the areas of curvilinear domains as the limit of the areas of a properly chosen sequence of inscribed polygonal domains with an increasing number of sides...a "naive" attitude that curvilinear areas are intuitively given entities to be computed...in the 17th century the areas for a number of other curvilinear geometric entities were computed with special methods for each...since Gauss (1777-1855 A.D.), the idea is that this process is not so much calculating an area, but defining the integral of a function p464)

[ror - September 30,2008 - as I understand it today, this makes sense when differentiation is understood and understood as an inverse relation to integration...p 427, 436 ]

.

http://pages.intnet.mu/
cueboy/education/maths/flash.htm
> Le camembert d'Archimède
(the third link)
flash animation with French vocals on Archimedes method of exhaustion (today - integration; area of circle as a limiting process)

p 401 - (calculus replaced such special and restricted proceedures for individual objects by a general and powerful method)

p 404...the word integral was coined to indicate that the whole or integral area A is composed of the "infinitesimal" parts f(x)dx.

...the limit concept (ror - stated elsewhere to be a "process") and nothing else is the true basis for the definition of the integral.

Courant Calculus 406 - Has been started (November 14, 2008) At the moment it just explains some difficulties I have.

In the book, some rather similar trivial functions are used to lead up to the "simplest passage to a limit" in the case of differentiation. (p 418)

CourantRobbins at least makes it appear the process can be comprehended by these simpler (trivial) cases - There might be no need to go into a greater amount of complexity.

For these reasons I am thinking of using the three functions here in my animation...Nothing is written in stone...there are almost 90 pages on the calculus I would like to study for a bit...

December 16, 2008 -The following animations address a number of related issues- they have been updated to link back here:

Integration 1, Integration 2 and Integration 3 Based on the mathtutor video noted therein. The video filled in some detail missing or at least difficult for me to understand from Courant Robbins. The mathtutor site seemed what was needed at the time and its stated purpose includes help for those with some deficiencies of math background. As usual, creating these animations helps me to analyize the processes so they can be put in the new format and I really cannot do much math unless it is focused on something immediately practical.

Preliminary sketch for the idea that the tangent to the curve of a function is the limit (of a sequence of secants.) secant.html

Secant - (L. secare - to cut)

p 417

f(x) = height of curve above point x
f ' (x) = slope of curve above point x

p 426 - 2nd derivative

rate of change of slope
acceleration

animation idea - fade function curve out and just show linear slope line in motion....

inflection?, change of direction of concavity

preliminary animation Fig 275 p 437...

fundamentalCalculus.html

This is not introduced until a thorough study of both integration and differentiation - I was thinking 'if its fundamental, maybe I should learn it first' - but that does not appear to be a good idea - more like - 'learn integration and differentiation first then learn how they work together to create fundamentally good stuff.' _ I'll googlize this a bit more.

cf William Dunham, "Journey Through Genius " Penguin (1991) Wiley (1990) p165-171 [gets roots of any order]

[I am thinking here of the quadrants in Cartesian coordinates - they were not really invented for 2 generations after this but a prior author (Bull?) indicated Newton as "thinking in those terms+-"