Introduction Calculus without Limits: the Theory - C. K....

Calculus withoutLimits

C. K. Raju

Introduction

Set theory andsupertasks

Paradoxes of settheory

Why R?

How to define thederivative?

Conclusions

Calculus without Limits:the Theory

A Critique of Formal MathematicsPart 1: Axioms and Definitions

C. K. Raju

Inmantec, Ghaziabadand

Centre for Studies in Civilizations, New Delhi

C. K. Raju

Introduction

Why R?

Conclusions

Calculus without Limits:the Theory

A Critique of Formal MathematicsPart 1: Axioms and Definitions

C. K. Raju

Inmantec, Ghaziabadand

Centre for Studies in Civilizations, New Delhi

C. K. Raju

Introduction

Why R?

Conclusions

Outline

Introduction

Set theory and supertasks

Paradoxes of set theory

Why R?

How to define the derivative?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

Why are limits important?

I We saw that it is impossible to teach limits?

I So, why are limits important?I Common answer: rigor.I Belief is that the use of limits makes calculus

rigorous.I Calculus is taught for its practical value in physics

and engineering, whileI limits are taught for rigor.

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Why R?

Conclusions

I We saw that it is impossible to teach limits?I So, why are limits important?

I Common answer: rigor.I Belief is that the use of limits makes calculus

C. K. Raju

Introduction

Why R?

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I We saw that it is impossible to teach limits?I So, why are limits important?I Common answer: rigor.

I Belief is that the use of limits makes calculusrigorous.

I Calculus is taught for its practical value in physicsand engineering, while

I limits are taught for rigor.

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Introduction

Why R?

Conclusions

I We saw that it is impossible to teach limits?I So, why are limits important?I Common answer: rigor.I Belief is that the use of limits makes calculus

rigorous.

I Calculus is taught for its practical value in physicsand engineering, while

C. K. Raju

Introduction

Why R?

Conclusions

and engineering, while

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C. K. Raju

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Why limits?contd.

I We can easily form the difference quotient ∆f∆x ,

I but as we take smaller and smaller values of ∆x thelimit might fail to exist, or it might fail to be unique.

I The rigorous approach to calculus—also calledmathematical analysis—allows us to prove theexistence and uniqueness of limits.

I The mathematician believes this answer, and otherpersons in the community of mathematicians mayshare this belief.

I But how far is it true?

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Introduction

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Why limits?contd.

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Why limits?contd.

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Why limits?contd.

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Why limits?contd.

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What is rigor actually?

I I will argue that rigor = reliance on the arbitrarydecisions of those in mathematical authority.

I What the calculus student learns—ritualisticmanipulation of symbols, and obedience toauthority—is inherent to formal mathematics.

I Tomorrow I will look at arbitrariness in the notion ofproof.

I Today I look at arbitrariness in axioms anddefinitions,

I to demonstrate the arbitrariness in calculus fromwithin formal mathematics.

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I Historically, the construction of R by Dedekind cutsinvolved Cantor’s set theory.

I Let us try to understand why set theory is needed forcalculus.

I After the calculus came to Europe (in the 16th c.)there were epistemic doubts about its validity.

I Mathematicians thought of summing an infiniteseries by actually carrying out the sum,

I and this seemed a supertask (an infinite series oftasks).

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Set theory and supertaskscontd.

I Even the simplest set theoretic statement “let x ∈ R”involves a supertask.

I This involves the claim that it is possible to selectand specify a real number, in a way that singles it outuniquely from an infinity of adjacent real numbers.

I This is a supertask.

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An example

I Consider a real number such as π which has adecimal expansion 3.14159. . . which neitherterminates nor recurs.

I This decimal expansion represents the number π asthe limit of an infinite series

∑an.

I Summing this series term by term, or calculatinga1 + a2, a1 + a2 + a3, . . . , is a supertask, for itrequires us to perform an infinity of additions.

I The fastest computers today can manage teraflops,or around 1012 floating point additions per second.

I If we use this computer exclusively to addcontinuously for a year: we can only go up to 1020

additions—still a long way from infinity.

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An example

∑an.

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An example

∑an.

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An example

∑an.

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An example

∑an.

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I To specify just one real number involves a supertask.

I this is not a task which is physically every going to bepossible.

I But set theory allows us to do it metaphsyically.I In fact, set theory allows us to specify an

uncountable infinity of real numbers!

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I To specify just one real number involves a supertask.I this is not a task which is physically every going to be

possible.

I But set theory allows us to do it metaphsyically.I In fact, set theory allows us to specify an

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possible.I But set theory allows us to do it metaphsyically.

I In fact, set theory allows us to specify anuncountable infinity of real numbers!

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possible.I But set theory allows us to do it metaphsyically.I In fact, set theory allows us to specify an

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A note

I Note that we must discriminate formal reals from thetraditional use of real numbers, such as 3.14 as anapproximation to π,

I which has a very old history, dating back to timeswhen European culture had not even begun.

I Such approximations are readily possibleI the question of a supertask arises only when we

speak of being able to specify the value of π exactlyor uniquely.

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A note

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A note

I Such approximations are readily possible

I the question of a supertask arises only when wespeak of being able to specify the value of π exactlyor uniquely.

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A note

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Dedekind’s real achievement

I The use of R for calculus means that doubts aboutsupertasks,

I which were earlier attached to the calculus,I got pushed into doubts about set theory.I From this perspective, Dedekind’s real achievement

was that he pushed doubts about supertasks andinfinity away from nubers and into the domain of settheory.

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I which were earlier attached to the calculus,

I got pushed into doubts about set theory.I From this perspective, Dedekind’s real achievement

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I which were earlier attached to the calculus,I got pushed into doubts about set theory.

I From this perspective, Dedekind’s real achievementwas that he pushed doubts about supertasks andinfinity away from nubers and into the domain of settheory.

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I which were earlier attached to the calculus,I got pushed into doubts about set theory.I From this perspective, Dedekind’s real achievement

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Irrational proofs

I From a practical perspective, this is an excellentsolution.

I It provides an easy escape route for mostmathematicians who rarely go beyond naive settheory.

I They can say that it is not their job (“proof by territorylimitation”).

I they can say (as Paul Erdos nearly said), “so manypeople believe it, they can’t all be wrong can they?”(proof by numbers”), etc.

I (For more details about such proofs, see theappendix to my book The Eleven Pictures of Time,Sage, 2003.)

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Irrational proofs

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Irrational proofs

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Irrational proofs

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Irrational proofs

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Sets and supertasksSummary

I Thus, with R, doubts about supertasks in thecalculus were pushed out of what mathematiciansregard as their normal area of activity,

I and into set theory.I But were they resolved?

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I and into set theory.

I But were they resolved?

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I and into set theory.I But were they resolved?

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Traditional paradoxes of infinity

I It has long been known that infinity brings in variousparadoxes.

I A classic example is the Sanskrit sloka, the firstverse of the Isa Upanisad, “: p� Z md, p� Z Emdm p� ZA tp� Z m� dQyt� / p� Z -y p� Z mAdAy p� Z m�vAvEfyt�”.

I The second line says “if you remove the whole fromthe whole, what remains is the whole”.

I It was such paradoxes which made Descartes andGalileo suspect the calculus when it first arrived inEurope (as we will see in more detail, later on).

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Paradoxes of infinity in the Christian tradition

I A similar paradox was encountered in Christiantradition a thousand years before Descartes.

I Proclus (a commentator on the Elements) hadargued that the truths of mathematics were eternal,hence the world itself must be eternal.

I John Philoponus, in his Apology Against Proclus,defended the idea that the world was created,

I He argued that adding a day to eternity would notchange eternity. Hence, the world was not eternal.

I Curiously, he had a different attitude towards theeternal torture in hell which he thought awaitednon-Christians, a torture which he thought theywould experience for an eternity of time.

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The double standard

I A similar double-standard is found today in settheory,

I but this is much harder to spot.I Let us try.

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The double standard

I but this is much harder to spot.

I Let us try.

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The double standard

I but this is much harder to spot.I Let us try.

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Russell’s paradox

I Recall Russell’s paradox.

I Let R = {x |x 6∈ x}.I Now, if R 6∈ R, then, by definition, we must have

R ∈ R.I On the other hand, if R ∈ R then, again, by definition,

we must have R 6∈ R.I So, either way, we have a contradiction.

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Russell’s paradox

I Recall Russell’s paradox.I Let R = {x |x 6∈ x}.

I Now, if R 6∈ R, then, by definition, we must haveR ∈ R.

I On the other hand, if R ∈ R then, again, by definition,we must have R 6∈ R.

I So, either way, we have a contradiction.

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Russell’s paradox

I Recall Russell’s paradox.I Let R = {x |x 6∈ x}.I Now, if R 6∈ R, then, by definition, we must have

R ∈ R.

I On the other hand, if R ∈ R then, again, by definition,we must have R 6∈ R.

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Russell’s paradox

we must have R 6∈ R.

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Russell’s paradox

we must have R 6∈ R.I So, either way, we have a contradiction.

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The definition of a set

I The way these contradictions are resolved inaxiomatic set theory is peculiar.

I Take, for example, the von-Neumann-Bernays-Gödel(NBG) set theory.

I Here, a well-formed formula (of the sort used inRussell’s paradox) in general only defines a class.

I A set is defined as a class A for which ∃ a class B,such that A ∈ B.

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Resolution of Russell’s paradox in NBG

I Russell’s paradox is resolved in NBG by saying thatthe Russell class is a class, not a set, for we cannotfind a class S such that R ∈ S.

I The paradoxes of set theory apply to classes, notsets.

I Mathematicians can stick to sets and thus avoid theparadoxes which are now (believed to be) confined toclasses.

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So are all paradoxes resolved?

I How can we be sure that all paradoxes of set theoryare resolved.

I NBG includes classes which are paradoxical.I How can we be sure this does not make NBG

inconsistent?

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I NBG includes classes which are paradoxical.

I How can we be sure this does not make NBGinconsistent?

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I NBG includes classes which are paradoxical.I How can we be sure this does not make NBG

inconsistent?

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The consistency of NBG

I The consistency of NBG is not proven

I it is only widely believed among mathematicians.I So, basing the calculus on R and NBG does not

guarantee the surety of the results.I That’s only a belief.

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I The consistency of NBG is not provenI it is only widely believed among mathematicians.

I So, basing the calculus on R and NBG does notguarantee the surety of the results.

I That’s only a belief.

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I The consistency of NBG is not provenI it is only widely believed among mathematicians.I So, basing the calculus on R and NBG does not

guarantee the surety of the results.

I That’s only a belief.

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I The consistency of NBG is not provenI it is only widely believed among mathematicians.I So, basing the calculus on R and NBG does not

guarantee the surety of the results.I That’s only a belief.

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Metamathematics

I It is interesting to see how the belief in theconsistency of NBG is maintained.

I By Gödel’s second incompleteness theorem, theconsistency of a consistent theory cannot be provenwithin the theory.

I Therefore, to decide the consistency of set theory werequire metamathematics.

I The question is: what kind of metamathematics?I Before answering this question, let us recall some

socially accepted results of metamathematics.

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Metamathematics

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Metamathematics

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Metamathematics

I The question is: what kind of metamathematics?

I Before answering this question, let us recall somesocially accepted results of metamathematics.

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Metamathematics

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Cantor’s Continuum Hypothesis

I For a set X denote its cardinality by #(X ).

I It may be proved (by contradiction) that#(X ) < #P(X ).

I If the set X is finite, #(X ) = n, then the binomialexpansion may be used to show that #(P(X )) = 2n.

I Not clear what happens when X is infinite.I Recall that Cantor’s continuum hypothesis states that

if ℵ0 is the cardinality of the infinite set N of naturalnumbers, and c is the cardinality of R then 2ℵ0 = c.

I The metamathematical theorems of Gödel andCohen showed that the continuum hypothesis (CH)implies (but is not implied by) the axiom of choice.

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I For a set X denote its cardinality by #(X ).I It may be proved (by contradiction) that

#(X ) < #P(X ).

I If the set X is finite, #(X ) = n, then the binomialexpansion may be used to show that #(P(X )) = 2n.

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#(X ) < #P(X ).I If the set X is finite, #(X ) = n, then the binomial

expansion may be used to show that #(P(X )) = 2n.

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expansion may be used to show that #(P(X )) = 2n.I Not clear what happens when X is infinite.

I Recall that Cantor’s continuum hypothesis states thatif ℵ0 is the cardinality of the infinite set N of naturalnumbers, and c is the cardinality of R then 2ℵ0 = c.

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expansion may be used to show that #(P(X )) = 2n.I Not clear what happens when X is infinite.I Recall that Cantor’s continuum hypothesis states that

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expansion may be used to show that #(P(X )) = 2n.I Not clear what happens when X is infinite.I Recall that Cantor’s continuum hypothesis states that

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Axiom of Choice

I The axiom of choice (AC): every set has a choicefunction.

I That is, if X is a set the elements of which arenonempty sets, then there exists a function f withdomain X such that ∀A ∈ X , f (A) ∈ A.

I A choice function f for a set X allows us to pick anindividual element f (A) ∈ A for each A ∈ X .

I Equivalent is Zorn’s Lemma: in a partially orderedset if every chain is bounded above, then there mustbe at least one maximal element,

I or Hausdorff maximality principle: in a partiallyordered set every chain is contained in a maximalchain etc.

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Axiom of Choice

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Axiom of Choice

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Axiom of Choice

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Axiom of Choice

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Axiom of choicecontd.

I These are today part of the everyday equipment ofmathematical reasoning.

I The AC is needed to prove what are regarded aseveryday results today:

I the existence of a Lebesgue non-measurable set orTychonoff’s theorem (that the product of compactsets is compact) etc.

I Zorn’s lemma is used to prove the Hahn-Banachtheorem etc.

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Banach-Tarski paradox

I However, the AC (and the existence of Lebesguenon-measurable sets) also leads to theBanach-Tarski paradox.

I Namely, let A, B ⊂ Rn, with n ≥ 3.I Further, let A, B be bounded and have non-empty

interior.I Then, there exist finite partitions of A, B, such that

A =⋃k

i=1 Ai , B =⋃k

i=1 Bi , and each Ai is congruent(under Euclidean motions) to Bi .

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I Namely, let A, B ⊂ Rn, with n ≥ 3.

I Further, let A, B be bounded and have non-emptyinterior.

I Then, there exist finite partitions of A, B, such thatA =

⋃ki=1 Ai , B =

⋃ki=1 Bi , and each Ai is congruent

(under Euclidean motions) to Bi .

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interior.

I Then, there exist finite partitions of A, B, such thatA =

⋃ki=1 Ai , B =

⋃ki=1 Bi , and each Ai is congruent

(under Euclidean motions) to Bi .

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interior.I Then, there exist finite partitions of A, B, such that

A =⋃k

i=1 Ai , B =⋃k

i=1 Bi , and each Ai is congruent(under Euclidean motions) to Bi .

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Banach-Tarski Paradoxcontd

I This paradox conflicts violently with geometricintuition, for it means that a ball in 3-dimensionalspace may be broken into a finite number ofnon-overlapping pieces, which may be reassembledby rotation and translation (without stretching) intotwo balls of the same volume as the original.

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The theorems of Gödel and Cohen

I such paradoxes created fears that AC may lead toinconsistency of NBG.

I However, the metamathematical theorems of Gödeland Cohen showed that both the continuumhypothesis (CH) and AC are independent of theremaining axioms of NBG.

I Usually taken as reassurance about CH and AC.I We look at the formal contrapositive:I if set theory is inconsistent with AC, then it must be

inconsistent without AC.

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I Usually taken as reassurance about CH and AC.

I We look at the formal contrapositive:I if set theory is inconsistent with AC, then it must be

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I Usually taken as reassurance about CH and AC.I We look at the formal contrapositive:

I if set theory is inconsistent with AC, then it must beinconsistent without AC.

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What kind of metamathematics

I To return to the original question.

I Metamathematics needed to prove consistency ofNBG,

I But what kind of metamathematics?I Specifically, can principles like AC and CH be

admitted in metamathematics?

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I To return to the original question.I Metamathematics needed to prove consistency of

I But what kind of metamathematics?I Specifically, can principles like AC and CH be

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NBG,I But what kind of metamathematics?

I Specifically, can principles like AC and CH beadmitted in metamathematics?

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NBG,I But what kind of metamathematics?I Specifically, can principles like AC and CH be

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Deciding decidability

I By Gödel’s first incompleteness theorem, any formaltheory large enough to contain natural numberscontains a proposition asserting its own negation

I which cannot hence be either proved or disprovedwithin the theory (if the theory is consistent; if it isinconsistent, every statement is provable).

I However, if such a theory is decidable, then thestatement can be either proved or disproved withinthe theory.

I That is, if set theory is decidable it must beinconsistent.

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Is set theory consistent?

I Decidability of a formal theory is usually understoodin the sense of recursive decidability.

I But, why should we limit metamathematics to finiterecursion?

I Conjecture: Transfinite recursion (an easyconsequence of AC), makes set theory decidable(hence inconsistent).

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Is set theory consistent?contd.

I Usually AC etc. are excluded from metamathematicson the grounds that metamathematics should onlyuse conservative techniques of proof.

I But if we distrust transfinite induction, why allow it inset theory?

I And if we find it trustworthy, why not allow it also inmetamathematics?

I So, standard of proof in metamathematics 6=standard of proof in mathematics. Why?

I The only answers is from mathematical authority. Soformal mathematics ultimately depends uponauthority, not reason.

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Interim summary

I Use of limits in calculus does not guarantee anysurety in the results.

I All it does is to push the doubts about supertasksinto the domain of set theory.

I The consistency of set theory is not proven: it isbelieved.

I This belief is maintained by using two standards ofproof.

I Infinite procedures (even AC) allowed for proofs inmathematics, but disallowed in metamathematics.

I This is a hypocritical social consensus amongauthoritative Western mathematicians. Ideally, thereshould be one standard of proof for bothmathematics and metamathematics.

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Interim summary

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Interim summary

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Interim summary

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Interim summary

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Interim summary

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Completeness of R

I Why is R needed for calculus?

I Conventional answer: because R is complete (as ametric space).

I The field of rational numbers Q is not.I The usual algorithm for square-root extraction (first

stated by Aryabhat.a) gives for√

2 a sequence ofrational numbers 1.4, 1.41, 1.414, 1.4142, . . . .

I This is a Cauchy sequence: for successive termsdiffer only in the next decimal place,

I so the difference between the mth and nth term canbe made less than 10−q where q = min{m, n}.

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Completeness of R

I Why is R needed for calculus?I Conventional answer: because R is complete (as a

metric space).

I The field of rational numbers Q is not.I The usual algorithm for square-root extraction (first

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Conclusions

Completeness of R

metric space).I The field of rational numbers Q is not.

I The usual algorithm for square-root extraction (firststated by Aryabhat.a) gives for

√2 a sequence of

rational numbers 1.4, 1.41, 1.414, 1.4142, . . . .I This is a Cauchy sequence: for successive terms

differ only in the next decimal place,I so the difference between the mth and nth term can

be made less than 10−q where q = min{m, n}.

C. K. Raju

Introduction

Why R?

Conclusions

Completeness of R

metric space).I The field of rational numbers Q is not.I The usual algorithm for square-root extraction (first

C. K. Raju

Introduction

Why R?

Conclusions

Completeness of R

C. K. Raju

Introduction

Why R?

Conclusions

Completeness of R

C. K. Raju

Introduction

Why R?

Conclusions

Completeness of R

I However, this Cauchy sequence does not convergein Q since Q is not complete.

I The limit would be√

2, but easy to prove that there isno rational number p such that p2 = 2.

I From the construction of R as the set of equivalenceclasses of Cauchy sequences in Q, this does nothappen in R which is complete.

I What happens in a field larger than R?

C. K. Raju

Introduction

Why R?

Conclusions

Completeness of R

C. K. Raju

Introduction

Why R?

Conclusions

Completeness of R

C. K. Raju

Introduction

Why R?

Conclusions

Completeness of R

C. K. Raju

Introduction

Why R?

Conclusions

Archimedean Property

I R has the Archimedean property (AP).

I Namely, given x ∈ R, x ≥ 0, ∃ n ∈ N, such that x < n.I Here, n = 1 + 1 + 1 · · ·+ 1 (n times), is defined in

any ordered field (so AP makes sense in any orderedfield).

I AP characterizes R. That is, R is the largest orderedfield with AP.

I Consequently, if we have an ordered field S ⊃ R,then the AP must fail in S.

C. K. Raju

Introduction

Why R?

Conclusions

I R has the Archimedean property (AP).I Namely, given x ∈ R, x ≥ 0, ∃ n ∈ N, such that x < n.

I Here, n = 1 + 1 + 1 · · ·+ 1 (n times), is defined inany ordered field (so AP makes sense in any orderedfield).

C. K. Raju

Introduction

Why R?

Conclusions

I R has the Archimedean property (AP).I Namely, given x ∈ R, x ≥ 0, ∃ n ∈ N, such that x < n.I Here, n = 1 + 1 + 1 · · ·+ 1 (n times), is defined in

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

Infinities and infinitesimals in an ordered field

I Such a field S in which the AP fails, must have bothinfinities and infinitesimals.

I Thus, since the AP fails, we must have an x ∈ S suchthat x > n for all n ∈ N.

I Such an x is what we intuitively understand as aninfinitely large number.

I Further, since S is an ordered field, this x must havea multiplicative inverse 1

x . This must satisfy0 < 1

x < 1n for all n ∈ N.

I Thus, 1x corresponds to what we intuitively

understand as an infinitesimally small number.

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

Limits in a field without AP

I What would happen to limits in such a field?

I Still possible to say that

limn→∞

but the limit would not be unique,I for the infinitesimal 1

x is another limit on the ε–δdefinition of limit,

I since

|1n− 1

x| < 1

n≤ |1

n− 0| < ε.

I Note: we are here not talking about non-standardanalysis: the infinities and infinitesimals in the field Sdo not arise merely at an intermediate stage: theyare “permanent”, so to say.

C. K. Raju

Introduction

Why R?

Conclusions

I What would happen to limits in such a field?I Still possible to say that

limn→∞

but the limit would not be unique,

I for the infinitesimal 1x is another limit on the ε–δ

definition of limit,I since

|1n− 1

x| < 1

n≤ |1

n− 0| < ε.

C. K. Raju

Introduction

Why R?

Conclusions

limn→∞

I since

|1n− 1

x| < 1

n≤ |1

n− 0| < ε.

C. K. Raju

Introduction

Why R?

Conclusions

limn→∞

I since

|1n− 1

x| < 1

n≤ |1

n− 0| < ε.

C. K. Raju

Introduction

Why R?

Conclusions

limn→∞

I since

|1n− 1

x| < 1

n≤ |1

n− 0| < ε.

C. K. Raju

Introduction

Why R?

Conclusions

Example of an ordered field without AP

I This example also required for later philosophy ofzeroism.

I Consider the set P of all polynomials with realcoefficients, in one indeterminate,

P = {f (x) =n∑0

aixi |ai ∈ R, an 6= 0}.

I Define f (x) > 0 if f (x) > 0 for all sufficiently large x .I Likewise, define f > g if f − g > 0.I Since R is a field, it is well known P must be an

integral domain.

C. K. Raju

Introduction

Why R?

Conclusions

P = {f (x) =n∑0

aixi |ai ∈ R, an 6= 0}.

integral domain.

C. K. Raju

Introduction

Why R?

Conclusions

P = {f (x) =n∑0

aixi |ai ∈ R, an 6= 0}.

I Define f (x) > 0 if f (x) > 0 for all sufficiently large x .

I Likewise, define f > g if f − g > 0.I Since R is a field, it is well known P must be an

integral domain.

C. K. Raju

Introduction

Why R?

Conclusions

P = {f (x) =n∑0

aixi |ai ∈ R, an 6= 0}.

I Define f (x) > 0 if f (x) > 0 for all sufficiently large x .I Likewise, define f > g if f − g > 0.

I Since R is a field, it is well known P must be anintegral domain.

C. K. Raju

Introduction

Why R?

Conclusions

P = {f (x) =n∑0

aixi |ai ∈ R, an 6= 0}.

integral domain.

C. K. Raju

Introduction

Why R?

Conclusions

example of an ordered field without APcontd

I Note that the AP fails in P.

I Thus, the unit element is the polynomial f (x) = 1,and if g(x) = x , we see that g(x) > n no matter whatn we choose.

I The integral domain P can be extended naturally toits field of quotients S, consisting of all rationalfunctions.

I The formal quotient, such as x−2x−3 is defined

whenever the denominator is a non-zero polynomial,even though, as a function, it may be infinite (or fail tobe defined) at a finite set of points (the roots of thedenominator).

I To avoid quibbles concerning the form 00 , we can

define two rational functions to be equivalent if theydiffer only on a finite set of points. (This can happenalso with equivalent formal quotients, e.g. x(x−1)

x−1 andx(x−2)

x−2 .)

C. K. Raju

Introduction

Why R?

Conclusions

I Note that the AP fails in P.I Thus, the unit element is the polynomial f (x) = 1,

and if g(x) = x , we see that g(x) > n no matter whatn we choose.

x−1 andx(x−2)

x−2 .)

C. K. Raju

Introduction

Why R?

Conclusions

x−1 andx(x−2)

x−2 .)

C. K. Raju

Introduction

Why R?

Conclusions

x−1 andx(x−2)

x−2 .)

C. K. Raju

Introduction

Why R?

Conclusions

x−1 andx(x−2)

x−2 .)

C. K. Raju

Introduction

Why R?

Conclusions

Interim summary

I As we will see in more detail later on, this is how thecalculus originally developed in India.

I Order counting (with rational functions) was used inplace of limits.

I and it was acceptable that limits are not unique.I Right now the question is only this: why do calculus

in R? why not use such an S which makes calculuseasier and more intuitive?

I The only answer is that conventional calculusteaching uncritically imitates the European historicalexperience of the calculus.

C. K. Raju

Introduction

Why R?

Conclusions

Interim summary

C. K. Raju

Introduction

Why R?

Conclusions

Interim summary

I and it was acceptable that limits are not unique.

I Right now the question is only this: why do calculusin R? why not use such an S which makes calculuseasier and more intuitive?

C. K. Raju

Introduction

Why R?

Conclusions

Interim summary

C. K. Raju

Introduction

Why R?

Conclusions

Interim summary

C. K. Raju

Introduction

Why R?

Conclusions

Inadequacy of the classical definition

I There are other practical reasons why it is necessaryto involve infinities and infinitesimals.

I Classical (ε–δ) definition soon proved inadequate forapplications to physics.

I With this definition a differentiable function must becontinuous.

I So, a discontinuous function may not bedifferentiated.

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

The Dirac δ

I But, in physics, there regularly arose the need todifferentiate discontinuous functions.

I The classical example of a discontinuous function isthe Heaviside function:

H(x) =

{0 if x < 0

1 if x > 0

I Its derivative of this is the Dirac δ function.I The Dirac δ had a sad childhood:I physicists denied that it was physical, and used it as

purely a mathematical artifice.I Mathematicians, on the other hand, considered it as

something non-mathematical and non-rigorous—amere construct used by physicists.

C. K. Raju

Introduction

Why R?

Conclusions

The Dirac δ

H(x) =

{0 if x < 0

1 if x > 0

C. K. Raju

Introduction

Why R?

Conclusions

The Dirac δ

H(x) =

{0 if x < 0

1 if x > 0

I Its derivative of this is the Dirac δ function.

I The Dirac δ had a sad childhood:I physicists denied that it was physical, and used it as

C. K. Raju

Introduction

Why R?

Conclusions

The Dirac δ

H(x) =

{0 if x < 0

1 if x > 0

I Its derivative of this is the Dirac δ function.I The Dirac δ had a sad childhood:

I physicists denied that it was physical, and used it aspurely a mathematical artifice.

I Mathematicians, on the other hand, considered it assomething non-mathematical and non-rigorous—amere construct used by physicists.

C. K. Raju

Introduction

Why R?

Conclusions

The Dirac δ

H(x) =

{0 if x < 0

1 if x > 0

purely a mathematical artifice.

I Mathematicians, on the other hand, considered it assomething non-mathematical and non-rigorous—amere construct used by physicists.

C. K. Raju

Introduction

Why R?

Conclusions

The Dirac δ

H(x) =

{0 if x < 0

1 if x > 0

C. K. Raju

Introduction

Why R?

Conclusions

Formalizations

I Heaviside, however, used it for electrical engineering.

I The resulting physical intuition was, however, soonformalised by

I Sobolev,I Schwartz (theory of distributions),I by Gel’fand and Shilov in the theory of generalised

functions, andI by Mikusinski in the operational calculus.

C. K. Raju

Introduction

Why R?

Conclusions

Formalizations

I Heaviside, however, used it for electrical engineering.I The resulting physical intuition was, however, soon

formalised by

I Sobolev,I Schwartz (theory of distributions),I by Gel’fand and Shilov in the theory of generalised

C. K. Raju

Introduction

Why R?

Conclusions

Formalizations

formalised byI Sobolev,

I Schwartz (theory of distributions),I by Gel’fand and Shilov in the theory of generalised

C. K. Raju

Introduction

Why R?

Conclusions

Formalizations

formalised byI Sobolev,I Schwartz (theory of distributions),

I by Gel’fand and Shilov in the theory of generalisedfunctions, and

I by Mikusinski in the operational calculus.

C. K. Raju

Introduction

Why R?

Conclusions

Formalizations

formalised byI Sobolev,I Schwartz (theory of distributions),I by Gel’fand and Shilov in the theory of generalised

functions, and

I by Mikusinski in the operational calculus.

C. K. Raju

Introduction

Why R?

Conclusions

Formalizations

formalised byI Sobolev,I Schwartz (theory of distributions),I by Gel’fand and Shilov in the theory of generalised

C. K. Raju

Introduction

Why R?

Conclusions

Schwartz theory

I In the Schwartz theory, one averages a function andthen differentiates it.

I Formally, this corresponds to the formula forintegration by parts:∫ ∞

−∞f ′g = −

∫ ∞

−∞fg′.

I Here, f is the function (possibly discontinuous) whichone seeks to differentiate,

I and the derivative f ′ is now being defined by the righthand side, where the derivative is transferred to

I the test function g which is assumed to be infinitelydifferentiable: g ∈ C∞.

C. K. Raju

Introduction

Why R?

Conclusions

Schwartz theory

−∞f ′g = −

∫ ∞

−∞fg′.

C. K. Raju

Introduction

Why R?

Conclusions

Schwartz theory

−∞f ′g = −

∫ ∞

−∞fg′.

C. K. Raju

Introduction

Why R?

Conclusions

Schwartz theory

−∞f ′g = −

∫ ∞

−∞fg′.

C. K. Raju

Introduction

Why R?

Conclusions

Schwartz theory

−∞f ′g = −

∫ ∞

−∞fg′.

C. K. Raju

Introduction

Why R?

Conclusions

Test functions

I The test function g is usually assumed to becompactly supported

I or to vanish rapidly at infinity etc.,I so that the term fg vanishes at infinity,I and the above formula corresponds to the formula for

integration by parts.I This works equally well for functions of several

variables, and we can write∫Rn

f ′g = −∫

Rnfg′,

for g ∈ D(Rn) where D(Rn) is the space of compactlysupported and infinitely differentiable functions.

C. K. Raju

Introduction

Why R?

Conclusions

Test functions

I or to vanish rapidly at infinity etc.,

I so that the term fg vanishes at infinity,I and the above formula corresponds to the formula for

f ′g = −∫

Rnfg′,

C. K. Raju

Introduction

Why R?

Conclusions

Test functions

I or to vanish rapidly at infinity etc.,I so that the term fg vanishes at infinity,

I and the above formula corresponds to the formula forintegration by parts.

I This works equally well for functions of severalvariables, and we can write∫

Rnf ′g = −

fg′,

C. K. Raju

Introduction

Why R?

Conclusions

Test functions

integration by parts.

I This works equally well for functions of severalvariables, and we can write∫

Rnf ′g = −

fg′,

C. K. Raju

Introduction

Why R?

Conclusions

Test functions

f ′g = −∫

Rnfg′,

C. K. Raju

Introduction

Why R?

Conclusions

The space of test functions

I Formally, D(Rn) is a topological vector space with thetopology of uniform convergence on compacta to allorders.

I Technically, this topology is obtained as follows.

I Take a sequence of compact sets Ki such that Ki iscontained in the interior of Ki+1 and

⋃∞i=1 Ki = Rn.

I On C∞(Rn) define the seminormspN(f ) = max{|Dαf (x)| x ∈ KN , |α| ≤ N}.

I Here α = (α1, α2, . . . , αn) is a multi-index, and

Dα =(

∂∂x1

∂∂x2

. . .(

∂∂xn

.I These seminorms pN generate a vector topology on

C∞(Rn), in which the space of compactly supportedtest functions D is a closed subspace.

C. K. Raju

Introduction

Why R?

Conclusions

I Technically, this topology is obtained as follows.

I Take a sequence of compact sets Ki such that Ki iscontained in the interior of Ki+1 and

⋃∞i=1 Ki = Rn.

Dα =(

∂∂x1

∂∂x2

. . .(

∂∂xn

C. K. Raju

Introduction

Why R?

Conclusions

I Technically, this topology is obtained as follows.I Take a sequence of compact sets Ki such that Ki is

contained in the interior of Ki+1 and⋃∞

i=1 Ki = Rn.

Dα =(

∂∂x1

∂∂x2

. . .(

∂∂xn

C. K. Raju

Introduction

Why R?

Conclusions

i=1 Ki = Rn.I On C∞(Rn) define the seminorms

pN(f ) = max{|Dαf (x)| x ∈ KN , |α| ≤ N}.

Dα =(

∂∂x1

∂∂x2

. . .(

∂∂xn

C. K. Raju

Introduction

Why R?

Conclusions

pN(f ) = max{|Dαf (x)| x ∈ KN , |α| ≤ N}.I Here α = (α1, α2, . . . , αn) is a multi-index, and

Dα =(

∂∂x1

∂∂x2

. . .(

∂∂xn

I These seminorms pN generate a vector topology onC∞(Rn), in which the space of compactly supportedtest functions D is a closed subspace.

C. K. Raju

Introduction

Why R?

Conclusions

pN(f ) = max{|Dαf (x)| x ∈ KN , |α| ≤ N}.I Here α = (α1, α2, . . . , αn) is a multi-index, and

Dα =(

∂∂x1

∂∂x2

. . .(

∂∂xn

C. K. Raju

Introduction

Why R?

Conclusions

Which derivative?

I The Schwartz theory requires that the integral be theLebesgue integral and not the Riemann integral.

I with the Schwartz theory every integrable functionsis differentiable.

I ε–δ definition of the limit and the correspondingderivative was not “natural”.

I That was just a consensus among mathematicians,which has changed, because the earlier definitionwas not adequate for physics.

C. K. Raju

Introduction

Why R?

Conclusions

Which derivative?

C. K. Raju

Introduction

Why R?

Conclusions

Which derivative?

C. K. Raju

Introduction

Why R?

Conclusions

Which derivative?

C. K. Raju

Introduction

Why R?

Conclusions

Which derivativecontd

I Oddly enough, some people continue with bothdefinitions.

I though both definitions cannot go together: if afunction admits both a classical derivative almosteverywhere and a Schwartz derivative, it is notnecessary that the two should agree.

I E.g., the Heaviside function H(x) is differentiablealmost everywhere (i.e., except on a set of Lebesguemeasure zero), and the derivative H ′ = 0 almosteverywhere.

I However, the Dirac delta is not the zero distribution,since

∫δ(x)dx = 1.

I Thus, for purposes of physics, we need to settle onone of the two as the right definition, and clearly theSchwartz definition is better than the older ε–δdefinition.

C. K. Raju

Introduction

Why R?

Conclusions

∫δ(x)dx = 1.

C. K. Raju

Introduction

Why R?

Conclusions

∫δ(x)dx = 1.

C. K. Raju

Introduction

Why R?

Conclusions

∫δ(x)dx = 1.

C. K. Raju

Introduction

Why R?

Conclusions

∫δ(x)dx = 1.

C. K. Raju

Introduction

Why R?

Conclusions

Difficulty of point values and products

I However, using the Schwartz theory creates anotherproblem in the formulation of the basic differentialequations of physics

I the Schwartz theory reinterprets a function as afunctional on a function space.

I Hence, it is no longer possible to speak of the valuef (x) of the function f at a point x .

I This loss of point values already occurred in theLebesgue theory of integration.

I However, it has more serious consequences in theSchwartz theory.

I Pointwise products of functions are no longerdefined.

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

The Schwartz productI Pointwise product

fg(x) = f (x)g(x)

defined only in the special case where the functions fand g are smooth (C∞).

I Possible to give a natural-looking definition of thepointwise product when only one of the functions isC∞.

I Called the Schwartz product. If g is a distribution,and f ∈ C∞, define

〈fg, h〉 = 〈g, fh〉

for all test functions h, where 〈f , h〉 ≡∫

fh.I If f ∈ C∞ and h is a test function, f .h is again a test

function. Hence, the right hand side is well defined.

C. K. Raju

Introduction

Why R?

Conclusions

fg(x) = f (x)g(x)

〈fg, h〉 = 〈g, fh〉

C. K. Raju

Introduction

Why R?

Conclusions

fg(x) = f (x)g(x)

〈fg, h〉 = 〈g, fh〉

I If f ∈ C∞ and h is a test function, f .h is again a testfunction. Hence, the right hand side is well defined.

C. K. Raju

Introduction

Why R?

Conclusions

fg(x) = f (x)g(x)

〈fg, h〉 = 〈g, fh〉

C. K. Raju

Introduction

Why R?

Conclusions

Schwartz impossibility theorem

I Schwartz proved that there does not exist a productof distributions which

I (a) agrees with the Schwartz product (definedabove),

I (b) is associative (that is (fg)h = f (gh) for alldistributions f , g, h), and

I (c) satisfies the Leibniz rule (that is (fg)′ = fg′ + f ′gfor all distributions f , g).

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

Taub’s remarkI Taub1 asserted, “Fortunately, the product of such

distributions [as arise] is quite tractable”.

I Thus, for example, consider the Heaviside function θ.I

θ2 = θ,

I Apply the “Leibniz” rule (for the derivative of aproduct of two functions) to conclude that

2θ · θ′ = θ′,

I Since θ′ = δ, this can be rewritten as

2θ · δ = δ,

which immediately tells us that

θ · δ =12· δ.

1A. H. Taub, J. Math. Phys.,21 (1980) pp. 1423–31.

C. K. Raju

Introduction

Why R?

Conclusions

distributions [as arise] is quite tractable”.I Thus, for example, consider the Heaviside function θ.I

θ2 = θ,

2θ · θ′ = θ′,

2θ · δ = δ,

θ · δ =12· δ.

C. K. Raju

Introduction

Why R?

Conclusions

θ2 = θ,

2θ · θ′ = θ′,

2θ · δ = δ,

θ · δ =12· δ.

C. K. Raju

Introduction

Why R?

Conclusions

θ2 = θ,

2θ · θ′ = θ′,

2θ · δ = δ,

θ · δ =12· δ.

C. K. Raju

Introduction

Why R?

Conclusions

Taub’s remarkcontd

I This is simple enough except that we also have

θ3 = θ,

I from which, by the same logic, it would follow that

3θ2θ′ = θ′.

I Sinceθ2 = θ,

this corresponds to

θ · δ =13· δ.

I Comparing the above two leads to the interestingconclusion that 1

2 = 13 !

C. K. Raju

Introduction

Why R?

Conclusions

θ3 = θ,

3θ2θ′ = θ′.

I Sinceθ2 = θ,

this corresponds to

θ · δ =13· δ.

2 = 13 !

C. K. Raju

Introduction

Why R?

Conclusions

θ3 = θ,

3θ2θ′ = θ′.

I Sinceθ2 = θ,

this corresponds to

θ · δ =13· δ.

2 = 13 !

C. K. Raju

Introduction

Why R?

Conclusions

θ3 = θ,

3θ2θ′ = θ′.

I Sinceθ2 = θ,

this corresponds to

θ · δ =13· δ.

2 = 13 !

C. K. Raju

Introduction

Why R?

Conclusions

Infinities of quantum field theory

I However, infinities arise in quantum field theory (qft).

I The propagators of qft are fundamental solutions ofthe Klein-Gordon and Dirac equations.

I Products of these propagators arise in the S-matrixexpansion.

I These products are Fourier transformed intoconvolution integrals, which are divergent.

I If we apply this to δ2 we see that

(δ2) = δ ∗ δ = 1 ∗ 1 =

∫1 = ∞.

C. K. Raju

Introduction

Why R?

Conclusions

I However, infinities arise in quantum field theory (qft).I The propagators of qft are fundamental solutions of

the Klein-Gordon and Dirac equations.

I Products of these propagators arise in the S-matrixexpansion.

(δ2) = δ ∗ δ = 1 ∗ 1 =

∫1 = ∞.

C. K. Raju

Introduction

Why R?

Conclusions

the Klein-Gordon and Dirac equations.I Products of these propagators arise in the S-matrix

expansion.

(δ2) = δ ∗ δ = 1 ∗ 1 =

∫1 = ∞.

C. K. Raju

Introduction

Why R?

Conclusions

expansion.I These products are Fourier transformed into

convolution integrals, which are divergent.

(δ2) = δ ∗ δ = 1 ∗ 1 =

∫1 = ∞.

C. K. Raju

Introduction

Why R?

Conclusions

expansion.I These products are Fourier transformed into

convolution integrals, which are divergent.I If we apply this to δ2 we see that

(δ2) = δ ∗ δ = 1 ∗ 1 =

∫1 = ∞.

C. K. Raju

Introduction

Why R?

Conclusions

Arbitrariness in the definition of the product

I Problem today is not that a product cannot bedefined.

I Many definitions have been given including one bythis author (1982)

I The problem is to select one definition from amongthe 40-odd definitions that have been proposed inthe literature.

I Quantum field theorists use the Hahn-Banachdefinition useless for classical physics (shock waves).

I Mathematicians use Colombeau’s product uselessfor physics (since it is both associative and satisfiesthe Leibniz rule).

I What are the principles on which the choice is to bedecided?

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

Which definition of the product?

I One possibility is to use comparison theorems.

I However, Hahn-Banach product used in qft hasδ2 = Aδ. Not comparable with Hormander’s productwhich does not define has δ2 or with my productwhich defines δ2 as an infinite distribution.

I Another possibility is to by social consensus amongauthoritative mathematicians.

I This is decided by “other considerations”.Colombeau product exactly like naive product ofnon-standard distributions.

I Since associate law and Leibniz rule holds, it has aproblem as follows.

C. K. Raju

Introduction

Why R?

Conclusions

I One possibility is to use comparison theorems.I However, Hahn-Banach product used in qft has

δ2 = Aδ. Not comparable with Hormander’s productwhich does not define has δ2 or with my productwhich defines δ2 as an infinite distribution.

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

C. K. Raju

Introduction

Why R?

Conclusions

Shock waves

I For smooth fluid flows one can use either (a)conservation of mass, momentum, and energy, or (b)conservation of mass, momentum and entropy.

I This is no longer true for non-smooth flows involvingshocks.

I (Here a shock is regarded as a surface ofdiscontinuity.)

I Historically, Riemann made the mistake of choosingform (b), and arrived at physically incorrectconditions for shocks.

I The correct conditions, using (a) were given byRankine and Hugoniot.

I With the Colombeau theory, it is not possible todiscriminate between forms (a) and (b).

C. K. Raju

Introduction

Why R?

Conclusions

Shock waves

C. K. Raju

Introduction

Why R?

Conclusions

Shock waves

C. K. Raju

Introduction

Why R?

Conclusions

Shock waves

C. K. Raju

Introduction

Why R?

Conclusions

Shock waves

C. K. Raju

Introduction

Why R?

Conclusions

Shock waves

C. K. Raju

Introduction

Why R?

Conclusions

I Calculus with limits is taught on grounds of rigor.However, this purported rigor depends upon theimposition of a variety of arbitrary choices.

I The choice of metamathematics is arbitrary .Calculus with limits requires infinite procedures(spertasks), incorporated in R which is constructedusing axiomatic set theory, such as NBG. Supertaskslead to paradoxes of set. Consistency of NBG canonly be proved or disproved in metamathematics.The consistency is maintained by an arbitrary choiceof metamathematics: refusing to allow inmetamathematics the sort of infinite procedures forproof that are admitted in NBG.

C. K. Raju

Introduction

Why R?

Conclusions

I Calculus with limits is taught on grounds of rigor.However, this purported rigor depends upon theimposition of a variety of arbitrary choices.

I The choice of metamathematics is arbitrary .Calculus with limits requires infinite procedures(spertasks), incorporated in R which is constructedusing axiomatic set theory, such as NBG. Supertaskslead to paradoxes of set. Consistency of NBG canonly be proved or disproved in metamathematics.The consistency is maintained by an arbitrary choiceof metamathematics: refusing to allow inmetamathematics the sort of infinite procedures forproof that are admitted in NBG.

C. K. Raju

Introduction

Why R?

Conclusions

Conclusionscontd

I The choice of the number system underlying thecalculus is arbitrary. It is possible to do calculusmore intuitively in non-Archimedean fields largerthan R.

I The definition of the derivative is arbitrary . Theclassical ε–δ definition of the derivative is notadequate for physics, since the derivative ofdiscontinuous functions naturally arises in physics.

C. K. Raju

Introduction

Why R?

Conclusions

Conclusionscontd

I The choice of the number system underlying thecalculus is arbitrary. It is possible to do calculusmore intuitively in non-Archimedean fields largerthan R.

I The definition of the derivative is arbitrary . Theclassical ε–δ definition of the derivative is notadequate for physics, since the derivative ofdiscontinuous functions naturally arises in physics.

C. K. Raju

Introduction

Why R?

Conclusions

Conclusionscontd

I The definition of the product of distributions isarbitrary The classical definition of derivative isusually replace by the Schwartz definition which isincomplete since it does not address the issue ofproducts of distributions. Colombeau’s simplisticdefinition is today being promoted by mathematicalauthority, although it is inadequate and inappropriatefor physics

I As seen by the fate of the classical definition ofderivative, ultimately mathematical definitions haveto be related to practical value not mathematicalauthority.

C. K. Raju

Introduction

Why R?

Conclusions

Conclusionscontd

I The definition of the product of distributions isarbitrary The classical definition of derivative isusually replace by the Schwartz definition which isincomplete since it does not address the issue ofproducts of distributions. Colombeau’s simplisticdefinition is today being promoted by mathematicalauthority, although it is inadequate and inappropriatefor physics

I As seen by the fate of the classical definition ofderivative, ultimately mathematical definitions haveto be related to practical value not mathematicalauthority.

Introduction Calculus without Limits: the Theory - C. K....

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