Wildberger (111.1-111.8) Math Foundations, 111.1-111.8: “Defining Cauchy Sequences]”, summary

 

by Corry Shores

 

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[The following is summary of Wildberger’s video lecture. You will find that he is a supremely talented teacher. Any mistakes are my own, as I am not a mathematician. Bracketed comments are my own and are not to be trusted.]

 

 

 

 

Norman J. Wildberger

 

Course Series

 

Math Foundations

 

Math Foundations B (80-149)

 

Real Numbers and Limits

 

111

“Real numbers and Cauchy sequences of rationals (I)”

[youtube page]

 

111.1-111.8

(00.08-10.07)

“Defining Cauchy Sequences]”

 

 

 

 

 

Brief summary:

(111.1) We will examine and criticize the thinking behind Cauchy sequences. (111.2) Cauchy was a brilliant and prolific mathematician who founded complex analysis, among other things. (111.3) We look now at a Cauchy sequence. We will examine the formal definition given in analysis texts. (Wildberger claims that this definition does not quite logically work.) [The basic idea seems to be the following. What is a Cauchy sequence? It is a series of increasing rational numbers (that  progressively tend toward some determinate value), such that no matter how small of a value we choose, there will always be a place in that series after which the differences between any two such values will be less than that arbitrarily small, chosen number. In other words, they converge upon a limit.]

‘A sequence S₁, S₂, S₃ of say, rational numbers is a Cauchy sequence precisely when for all ε greater than zero there is an N, a natural number, with the property [N] or such that if little n or little m are bigger than or equal to N, then the difference between S_n minus S_m is less than ε. In words, what it means is that a Cauchy sequence is a sequence that has the property that after a certain point, all the elements in the sequence are close to each other. And a little bit more precisely, it means that no matter what level of tolerance ε you choose, as long as it is a positive level of tolerance, that there is some point that you can get to, denoted by this capital N, so that for all elements of the sequence past that point, the difference between any two of them is within ε.’

(01.42)

This is a classical definition with serious problems; nonetheless, it is the foundation of the idea that real numbers are actually Cauchy sequences of rationals. (111.4) The reason that the notion of a Cauchy sequence is logically flawed is a similar reason as that for the limit of an arbitrary sequence being a number A. What is problematic so far is that we have not yet defined what a sequence is. (111.5) We wonder, why is this definition important? What role does it play in constructing the real numbers, supposedly? There is a important key fact or theorem. [It says, basically, that a Cauchy sequence is one where a series of rational numbers tend toward a limit.]

‘If S₁, S₂, S₃ is a sequence of rational numbers, with the limit of S_n = A a rational number, then this sequence is a Cauchy sequence. So if we have a sequence of rational numbers which actually does have a limit, in the classical sense of the limit that we defined in the last few videos, then that sequence is a Cauchy sequence. So sequences with limits are Cauchy sequences.’

(04.08)

(111.6) There is a standard proof of this theorem (although Wildberger does not think it is adequate.) [The basic point here seems to be that we can take that ε value (for which the differences between successive values after some point remain smaller than ε) and divide that ε in two, placing one half above the limit value and one half below it, creating a band range of values surrounding equally the limit value. The proof shows that once the terms enter into that band, they stay within it.

‘The idea is that once the terms of the sequence, whatever a sequence is, are within ε of 2 of this limit A, then they are within ε of each other.’

(4.52)

This can be explained with a diagram.

‘Here we have a sequence, S₁, S₂, S₃, S₄, S₅, S₆, etc.’ [pointing to the the first six blocks in series.] ‘So the values of the sequence are here on the y-axis. So S₁ is whatever this value is,’ [pointing to the corresponding place on the y-axis for the first block] ‘S₂ is this value, S₃, is this value. And here is the value A,’ [pointing to the green line], ‘which we are assuming this limit is. So this sequence goes to A. Now, because it goes to A, we know that if we say choose some band, and let us choose the band to be ε over 2,’ [pointing to the band between dashed red lines], ‘where ε is that number given to us in the Cauchy sequence definition. So if we are given ε, then first of all we calculate ε over 2, and then we find an N’ [points to the N], ‘so that past that point, the sequence will be within ε over 2 of A, so between A plus ε over 2 and A minus ε over 2. So in this case here, the sequence is bouncing around, but let’s say that N = 5 and beyond, the sequence then manages to stay within this band around the value A. Well in that case, the difference between the values of the sequence of any two values beyond this N will necessarily be less than ε, because the total width of this band from top to the bottom is ε over 2 plus ε over 2, which is epsilon. So the argument is that any two of these sequence elements past this point N will be within at most ε of each other.’

(5.15-07.02)

(111.7) This can be said in a more mathematically precise way. [It seems to say that no matter how small the ε interval, there will be a place along the sequence after which the values will fall in the ε region either above or below the limit value toward which the sequence is tending.]

‘Given an ε greater than 0, find N, a natural number, so that if n is bigger than or equal to N, then S_n minus A, in absolute value, is less than ε over 2. We can do that, this is possible, since we are assuming that the limit of the sequence S_n is some value A.’

(07.06)

(111.8) This theorem can be formulated in terms of a triangle inequality. [It seems to say that for any two places past the natural number (selected because after it the numbers stay within the arbitrarily small selected ε value that surrounds the limit value), the difference between those values will be less then their total distances away from the limit, and thus they will remain within that ε value, which is equal to the top half of the band added to the bottom half of the band. This is the standard proof that Cauchy sequences are ones with a limit, and this implies that the sequence of values, pairwise, get closer and closer to that limit value.]

‘So if we believe this, then if n and m are bigger than or equal to N, in other words, beyond this value 5 in this example, then if we look at the difference between S_n and S_m, so S_n minus S_m in absolute value, this is less than or equal to the absolute value of S_n minus A plus the absolute value of A minus S_m. This is a triangle inequality, so a basic fact about inequalities. So that if you have two numbers S_n and S_m, say on the number line, then that separation is less than or equal to, well if you pick any number A whatsoever anywhere and you can look at the separation between S_n and A and between A and S_m, the sum of those two has to be bigger than or equal to the separation between S_n and S_m. So that’s the triangle inequality, basic fact about inequalities. And now we are assuming that n and m are past this point N. So from what we have assumed up here, we know that S_n minus A is going to be less ε over 2, this is less than ε over 2. And similarly A minus S_m, which in absolute value is the same as S_m minus A, is also less than ε over 2. And so the sum of these 2 is less than ε over 2 plus ε over 2, which is ε, showing that, yes, once little n and little m are bigger than this capital N that we have found, then the difference between any of two of these elements is less than ε. So that is the standard proof that shows that sequences with a limit are Cauchy sequences. That is an important fact to remember. If a sequence does have a limit, then it is Cauchy sequence. So if a sequence has a limit, it implies something about the sequence itself independent of the limit. The existence of that limit implies that the sequence elements themselves will be, pairwise, getting closer and closer to each other.

(07.46-10.07)

 

 

 

 

 

 

Contents

 

111.1

[Introduction to the Topic]

 

111.2

[Brief Bio of Cauchy]

 

111.3

[The Formal Definition of Cauchy Sequences]

 

111.4

[A Problem with the Definition]

 

111.5

[A Theorem about Cauchy Sequences Tending Toward a Limit]

 

111.6

[The Standard Proof of That Theorem]

 

111.7

[A More Mathematically Precise Formulation of the Theorem]

 

111.8

[The Theorem in Terms of a Triangle Inequality]

 

Bibliography

 

 

 

 

 

 

Summary

[Mostly identical to the brief summary above.]

 

111.1

[Introduction to the Topic]

 

[We will examine and criticize the thinking behind Cauchy sequences.]

 

(00.08-00.34)

 

[Wildberger says that

In today’s video we are going to look at Cauchy sequences, and we are going to start investigating why real numbers as “equivalence classes” of Cauchy sequences is really a very flawed idea.

]

[contents]

 

 

 

 

 

 

111.2

[Brief Bio of Cauchy]

 

[Cauchy was a brilliant and prolific mathematician who founded complex analysis, among other things.

]

 

(00.35-01.07)

 

[ditto]

[contents]

 

 

 

 

 

 

111.3

[The Formal Definition of Cauchy Sequences]

 

[We look now at a Cauchy sequence. We will examine the formal definition given in analysis texts. (Wildberger claims that this definition does not quite logically work.) [The basic idea seems to be the following. What is a Cauchy sequence? It is a series of increasing rational numbers (that  progressively tend toward some determinate value), such that no matter how small of a value we choose, there will always be a place in that series after which the differences between any two such values will be less than that arbitrarily small, chosen number. In other words, they converge upon a limit.]

‘A sequence S₁, S₂, S₃ of say, rational numbers is a Cauchy sequence precisely when for all ε greater than zero there is an N, a natural number, with the property [N] or such that if little n or little m are bigger than or equal to N, then the difference between S_n minus S_m is less than ε. In words, what it means is that a Cauchy sequence is a sequence that has the property that after a certain point, all the elements in the sequence are close to each other. And a little bit more precisely, it means that no matter what level of tolerance ε you choose, as long as it is a positive level of tolerance, that there is some point that you can get to, denoted by this capital N, so that for all elements of the sequence past that point, the difference between any two of them is within ε.’

(01.42)

This is a classical definition with serious problems; nonetheless, it is the foundation of the idea that real numbers are actually Cauchy sequences of rationals.]

 

(01.08-03.08)

 

[ditto]

[contents]

 

 

 

 

 

 

111.4

[A Problem with the Definition]

 

[The reason that the notion of a Cauchy sequence is logically flawed is a similar reason as that for the limit of an arbitrary sequence being a number A. What is problematic so far is that we have not yet defined what a sequence is.

]

 

(03.09-03.52)

 

[ditto]

[contents]

 

 

 

 

 

 

111.5

[A Theorem about Cauchy Sequences Tending Toward a Limit]

 

[We wonder, why is this definition important? What role does it play in constructing the real numbers, supposedly? There is a important key fact or theorem. [It says, basically, that a Cauchy sequence is one where a series of rational numbers tend toward a limit.]

‘If S₁, S₂, S₃ is a sequence of rational numbers, with the limit of S_n = A a rational number, then this sequence is a Cauchy sequence. So if we have a sequence of rational numbers which actually does have a limit, in the classical sense of the limit that we defined in the last few videos, then that sequence is a Cauchy sequence. So sequences with limits are Cauchy sequences.’

(04.08)]

 

(03.53-04.40)

 

[ditto]

[contents]

 

 

 

 

 

 

111.6

[The Standard Proof of That Theorem]

 

[There is a standard proof of this theorem (although Wildberger does not think it is adequate.) [The basic point here seems to be that we can take that ε value (for which the differences between successive values after some point remain smaller than ε) and divide that ε in two, placing one half above the limit value and one half below it, creating a band range of values surrounding equally the limit value. The proof shows that once the terms enter into that band, they stay within it.]

‘The idea is that once the terms of the sequence, whatever a sequence is, are within ε of 2 of this limit A, then they are within ε of each other.’

(4.52)

This can be explained with a diagram.

‘Here we have a sequence, S₁, S₂, S₃, S₄, S₅, S₆, etc.’ [pointing to the the first six blocks in series.] ‘So the values of the sequence are here on the y-axis. So S₁ is whatever this value is,’ [pointing to the corresponding place on the y-axis for the first block] ‘S₂ is this value, S₃, is this value. And here is the value A,’ [pointing to the green line], ‘which we are assuming this limit is. So this sequence goes to A. Now, because it goes to A, we know that if we say choose some band, and let us choose the band to be ε over 2,’ [pointing to the band between dashed red lines], ‘where ε is that number given to us in the Cauchy sequence definition. So if we are given ε, then first of all we calculate ε over 2, and then we find an N’ [points to the N], ‘so that past that point, the sequence will be within ε over 2 of A, so between A plus ε over 2 and A minus ε over 2. So in this case here, the sequence is bouncing around, but let’s say that N = 5 and beyond, the sequence then manages to stay within this band around the value A. Well in that case, the difference between the values of the sequence of any two values beyond this N will necessarily be less than ε, because the total width of this band from top to the bottom is ε over 2 plus ε over 2, which is epsilon. So the argument is that any two of these sequence elements past this point N will be within at most ε of each other.’

(5.15-07.02)]

 

(04.41-07.02)

 

[ditto]

[contents]

 

 

 

 

 

 

111.7

[A More Mathematically Precise Formulation of the Theorem]

 

[This can be said in a more mathematically precise way. [It seems to say that no matter how small the ε interval, there will be a place along the sequence after which the values will fall in the ε region either above or below the limit value toward which the sequence is tending.]

‘Given an ε greater than 0, find N, a natural number, so that if n is bigger than or equal to N, then S_n minus A, in absolute value, is less than ε over 2. We can do that, this is possible, since we are assuming that the limit of the sequence S_n is some value A.’

(07.06)]

 

(07.03-07.45)

 

[ditto]

[contents]

 

 

 

 

 

 

111.8

[The Theorem in Terms of a Triangle Inequality]

 

[This theorem can be formulated in terms of a triangle inequality. [It seems to say that for any two places past the natural number (selected because after it the numbers stay within the arbitrarily small selected ε value that surrounds the limit value), the difference between those values will be less then their total distances away from the limit, and thus they will remain within that ε value, which is equal to the top half of the band added to the bottom half of the band. This is the standard proof that Cauchy sequences are ones with a limit, and this implies that the sequence of values, pairwise, get closer and closer to that limit value.]

‘So if we believe this, then if n and m are bigger than or equal to N, in other words, beyond this value 5 in this example, then if we look at the difference between S_n and S_m, so S_n minus S_m in absolute value, this is less than or equal to the absolute value of S_n minus A plus the absolute value of A minus S_m. This is a triangle inequality, so a basic fact about inequalities. So that if you have two numbers S_n and S_m, say on the number line, then that separation is less than or equal to, well if you pick any number A whatsoever anywhere and you can look at the separation between S_n and A and between A and S_m, the sum of those two has to be bigger than or equal to the separation between S_n and S_m. So that’s the triangle inequality, basic fact about inequalities. And now we are assuming that n and m are past this point N. So from what we have assumed up here, we know that S_n minus A is going to be less ε over 2, this is less than ε over 2. And similarly A minus S_m, which in absolute value is the same as S_m minus A, is also less than ε over 2. And so the sum of these 2 is less than ε over 2 plus ε over 2, which is ε, showing that, yes, once little n and little m are bigger than this capital N that we have found, then the difference between any of two of these elements is less than ε. So that is the standard proof that shows that sequences with a limit are Cauchy sequences. That is an important fact to remember. If a sequence does have a limit, then it is Cauchy sequence. So if a sequence has a limit, it implies something about the sequence itself independent of the limit. The existence of that limit implies that the sequence elements themselves will be, pairwise, getting closer and closer to each other.

(07.46-10.07)]

 

(07.46-10.07)

 

[ditto]

[contents]

 

 

 

 

 

 

Bibliography:

 

Wildberger, Norman J. (2014). “Real numbers and Cauchy sequences of rationals (I) | Real numbers and limits Math Foundations 111.”  Part 111 of the course series:  Math Foundations. Video.

 

Youtube page for this video.:

https://youtu.be/6JjPA3msnbo

 

Course Youtube Playlist:

Math Foundations A (1-79)

https://www.youtube.com/playlist?list=PL5A714C94D40392AB

Math Foundations B (80-149)

https://www.youtube.com/playlist?list=PLIljB45xT85DpiADQOPth56AVC48SrPLc

Math Foundations C (150 - )

https://www.youtube.com/playlist?list=PLIljB45xT85AYIeGfDQwHM8i6PQEDnnTI

 

Norman J. Wildberger, youtube channel:

[njwildberger]

Insights into Mathematics

https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ

 

 

 

 

Russell, Ch.35 of Principles of Mathematics, ‘Cantor’s First Definition of Continuity’, summary notes

 

by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]

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[The following is summary and quotation. All boldface, underlining, and bracketed commentary are mine. Please see the original text, as I did not follow it closely.]

 

 

Bertrand Russell

Principles of Mathematics

Part 5: Infinity and Continuity

Ch.35: Cantor’s First Definition of Continuity

Brief Summary

Cantor has an early definition of continuity [the later ‘ordinal’ definition is discussed in the next chapter.] For a series to be continuous, it must be perfect and cohesive. It is cohesive if it it is complete dense, having no finite gaps among its terms. It is perfect if “it consists of exactly the same terms as its first derivative; i.e. when all its points are limiting-points, and all its limiting-points belong to it.” [294]

 

Summary

 

§271

We will be talking about Cantor’s continuity. We previously considered a series continuous if it has a term between any two. “This definition usually satisfied Leibniz,† and would have been generally thought sufficient until the revolutionary discoveries of Cantor.” [290] [[Note, Russell obtains this reference from a line in one of Leibniz’ letters:

Eo ipso, dum puncta ita sita ponuntur, ut nulla duo sint, inter quae non detur medium, datur extensio continua. [‘Leibniz an des Bosses’ 26 Maji 1716’, p515 Philosophische Schriften]

When points are situated in such a way that there are no two points between which there is no midpoint, then, by that very fact, we have a continuous extension. [‘Leibniz to Des Bosses, 26 May 1716 [excerpts]’, p.201-202 Philosophical Essays]

Leibniz also invented infinitesimal calculus, which examines when points draw so close such that there are no points between but only an infinitesimal difference between them. That seems to contradict his claim here. However, note that in the above statement Leibniz says that the continuity of extension results from there being points between any two others. The infinitely small magnitude does not expand through extensive space. It is an intensity of variation.]] Previously Russell refered to the continuity of the rational numbers as compactness. From now on, he will never refer to this sort of continuity but rather only to the Cantorian sorts.

§272

”In order that a series should be continuous, it must have two characteristics: it must be perfect and cohesive.” [291]

(1) Cohesiveness

A series is cohesive when it contains no finite gaps. Quoting Cantor:

“We call T a cohesive collection of points, if for any two points t and t' of T, for a number ε given in advance and as small as we please, there are always, in several ways, a finite number of points t₁, t₂, . . . t_ν, belonging to T, such that the distances tt₁, t₁t₂, t₂t3, . . . t_νt' are all less than ε.”
[291]

[[So there are always points between others. This could also define the infinitesimal, but apparently it somehow does not.]]

The condition that distances in the series are to have no minimum is satisfied by real or rational numbers [...]. Hence every cohesive series must be compact, i.e. must have a term between any two. [292]

However, not every compact series is cohesive. [292]

§273

(2) Perfect series. “A series is perfect when it coincides with its first derivative.” [293] The points of a series can be of two kinds: isolated points and limiting points. “A finite series has only isolated points; an infinite series must define at least one limiting-point, though this need not belong to the series. A limiting-point of a series is defined by Cantor to be
a term such that, in any interval containing the term, there are an infinite number of terms of the series”. [293]  The limiting point does not need to be a part of the series. “The assemblage of all limiting-points is called the first derivative of the series. The first derivative of the first derivative is called the second derivative, and so on.”  [293] [It seems possibly that the Peano definition to follow is saying that the first derivative is a number very close to the lowest value of a class’s series of values. It seems it would be similar to the infinitesimal. It seems also to be the lower limit of the series.]

Peano gives the definition of the first derivative of a class of real numbers as follows: Let u be a class of real numbers, and let x be a real number (which may or may not be a u) such that the lower limit of the absolute values of the differences of | x from terms of u other than x is zero; then the class of terms x satisfying this condition is the first derivative of u.* This definition is virtually identical with that of Cantor, but it brings out more explicitly the connection of the derivative with limits. A series, then, is perfect, when it consists of exactly the same terms as its first derivative; i.e. when all its points are limiting-points, and all its limiting-points belong to it.
[293-294]

§274

 

But recall that in the series of rationals, there are sub-series with irrational limits, which means the series of rationals has limits falling outside it, and thus it does not contain all the terms of its first derivative, and hence it is not a perfect series. [294]

What we must say is, that a series is perfect when all its points are limiting-points, and when further, any series being chosen out of our first series, if this new series is of the sort which is usually regarded as defining a limit, then it actually has a limit belonging to our first series.
[294]

§275

 

In this section, Russell repeats “the arguments against assuming the existence of limits in the class of series to which the rational numbers belong.” [296-298. See these pages for the details.]

Sources [unless otherwise notes, all bracket page citations are from]:

Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].

Otherwise:

Leibniz. Philosophical Essays. Ed. and Transl. Roger Ariew and Daniel Garber. Indianapolis/Cambridge: Hackett, 1989.

Leibniz. Die philosophischen Schriften von G. W. Leibniz, zweiter band. Ed. G.I. Gerhardt. Berlin. Weidmann, 1879. Available online at:
https://archive.org/details/diephilosophisc00leibgoog

Archimedes’ [P24] ‘Quadrature of the Parabola’, Proposition 24

by Corry Shores
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Archimedes

Quadrature of the Parabola

Proposition 24 [quoting]

 

Proposition 24.

P24. Every segment bounded by a parabola and a chord Qq is equal to four-thirds of the triangle which has the same base as the segment and equal height.

Suppose K = 4/3ΔPQq,

where P is the vertex of the segment ; and we have then to prove that the area of the segment is equal to K.

For, if the segment be not equal to K, it must either be greater or less.

I. Suppose the area of the segment greater than K.

If then we inscribe in the segments cut off by PQ, Pq triangles which have the same base and equal height, i.e. triangles with the same vertices R, r as those of the segments, and if in the remaining segments we inscribe triangles in the same manner, and so on, we shall finally have segments remaining whose sum is less than the area by which the segment PQq exceeds K.

Therefore the polygon so formed must be greater than the area K; which is impossible, since [Prop. 23]

A + B + C+ ... + Z < 4/3A,

where A = ΔPQq.

Thus the area of the segment cannot be greater than K.

II. Suppose, if possible, that the area of the segment is less than K. (Heath 252)

If then ΔPQq = A, B = 1/4A, C = 1/4B, and so on, until we arrive at an area X such that X is less than the difference between K and the segment, we have

A + B + C + ... + X + 1/3X = 1/4A [Prop. 23]

= K.

Now, since K exceeds A + B + G + ... + X by an area less than X, and the area of the segment by an area greater than X, it follows that

A + B + C + ... + X > (the segment);

which is impossible, by Prop. 22 above.

Hence the segment is not less than K.

Thus, since the segment is neither greater nor less than K,

(area of segment PQq) = K = 4/3ΔPQq.

 

Archimedes. “Quadrature of the Parabola.” In The Works of Archimedes. Ed. T.L. Heath. Cambridge UP, 1897. Obtained at

https://archive.org/details/worksofarchimede00arch

 

 

Archimedes’ [P23] ‘Quadrature of the Parabola’, Proposition 23

by Corry Shores
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Archimedes

Quadrature of the Parabola

Proposition 23 [quoting]

 

Proposition 23.

P23. Given a series of areas A, B, C, D, ... Z, of which A is the greatest, and each is equal to four times the next in order, then

A + B + C + ... + Z + 1/3Z = 3/4A.

Take areas b, c, d, … such that

b = 1/3B

c = 1/3C,

d = 1/3D, and so on.

Then, since b = 1/3B,

and B = 1/4A,

B + b = 1/3A.

Similarly C + c = 1/3B.

Therefore

B + C + D + ... + Z + b + c + d+ ... + z = 1/3(A + B + C + ... + Y).

But b + c + d + ... +y = 1/3(B + C + D + ... + Y).


Therefore, by subtraction,

B + C + D + … + Z + z = 1/3A

A + B + C + … + Z + 1/3Z = 4/3A.

Archimedes. “Quadrature of the Parabola.” In The Works of Archimedes. Ed. T.L. Heath. Cambridge UP, 1897. Obtained at

https://archive.org/details/worksofarchimede00arch

 

Lagrange's Algebraicization of Analysis and Wronski's Critique, in Bottazzini's The Higher Calculus and in Boyer's History of the Calculus

by Corry Shores
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From Bottazzini's The Higher Calculus:

In his Théorie des fonctions analytiques (1772), Lagrange attempted to

present the principles of the calculus in a systematic manner without making any reference to infinitesimals, evanescent quantities, differentials or limits. He instead stressed the need to reduce the calculus to simple algebraic manipulations of finite quantities. (Bottazzini 48ab)

He defines a function thus:

We call a function of one or more quantities every expression of the calculus in which these quantities enter in any way whatsoever, mixed or not with other quantitites that one takes as having given and invariable values, while the quantitities of the function can have all possible values. (Lagrange, 1797, p. 1; 1813, p. 15, qtd in Bottazzini 48bc)

Langrange's next step was to show that any given function can be expanded as a series. Bottazzini quotes Lagrange as writing:

"We therefore consider a function f (x) of any variable x. If in place of x we put x + i, i being any indeterminate quantity whatever, it becomes f (x + i) and, by the theory of series [Bottazzini's emphasis] we can expand it as a series of this form:

in which the quantities p, q, r, ... , the coefficients of the powers of i, will be new functions of x, derived from the primitive function x and independent of the [indeterminate] quantity i ... (1797, p.2; 1813, pp.21-2, qtd in Bottazzini 48d).

Because Lagrange is interested in the formation and calculation of these different functions, the new calculus for him is differential or fluxional (48d).

Lagrange critiqued early renditions of the calculus, arguing instead that "'the true metaphysics' of the calculus lies in the fact that the errors resulting from neglecting infinitesimals of higher degrees were 'corrected or compensated' by the procedures of the calculus themselves, when they were limited to infinitesimals of the same degree" (49a). Euler and D'Alembert failed to realize this, although they addressed the problem by claiming that infinitely small values equal zero, but the limits of the ratios of indefinite differences have quantitative value (49b). Lagrange critiques Newton's notion of the motion of quantities so to avoid the notion of infinitesimal value by saying

on the one hand, to introduce movement into a calculus that only has algebraic quantities as its object is to introduce a foreign idea which obligates one to think of these quantities as the lines traversed by a moving body. On the other hand, it must be granted that we do not even have a really clear idea of what the velocity of a point is at every instant when this velocity is variable (Lagrange 1797, pp. 3-4; 1813, p.17; qtd in Bottazzini 49bc).

Lagrange thinks that Landen and Arbogast make steps in the right direction, because their aim according to Lagrange was to deal with the functions arising when expanding any given function, and then to apply these "derived" functions in math and science. Lagrange claims that his approach is free of "every illicit supposition" and "all metaphysics," and that it bears "the rigor of the ancient demonstrations" [see Hegel's Science of Logic section 587] by basing it on his method of primitive functions and derivatives (49d). "Nevertheless," Bottazzini writes

the crucial point in his entire construction is his assertion that it is possible to expand any function in a series of ascending powers of an indeterminate increment i. (49d)

Because he wants to offer proof, Lagrange considers the

form of the series that must represent the expansion of every function f (x) when one substutites x + 1 in place of x, and which we have supposed must contain only integral and positive power of i. (Lagrange 1797, p.7; 1813, p. 22; qtd in Bottazzini 50a).

Lagrange wants to establish that this holds for all functions, so he shows that "when x and i remain indeterminate, the series cannot contain fractional or negative powers of i" (50b). But there is no way to prove such an expansion of a function, "there is only the fact that the series contains only positive powers of i," and it is only after the function is expanded that we learn what each terms means. So we begin with

where i is any indeterminate quantity whatever, P is a new function of x and i. We can determine this other function P in terms of x and i by moving fx to the other side of the equation, and dividing both sides by i so to obtain:

We then let i vanish so to separate P insofar as it is independent of i (which does not equal zero as it vanishes), leaving a differential value p. Lagrange then shows that

P = p + iQ.

Q can be further determined, and so on, hence:

But each time we substitute we continue to multiple i's with other i's

Lagrange says we may make i so small that any given term will be larger than those following it in the series.

He recognizes that this expansion does not hold for every possible function.

Lagrange then identifies the functions p, q, r, ... with f 'x, (f ''x/2), (f '''x/3!), ... , where f 'x, f ''x, f '''x, and so on are the successive derivatives of the function fx. [See the Taylor Series]

What is most notable about his technique is that it is purely algebraic.

Lagrange's theory was attacked a year later by Hoëne Wronski. In his Introduction à la philosophie de la mathématique et technie de l'algorithmie, he comes to conclude that all functions may take the form:

"where the

are any functions of x and the determination of the coefficients

of the series depend on the determinants that are today called 'Wronskian'" (54-55).

Wronski then vigorously critiques Lagrange in Réfutation de la théorie des fonctions analytiques de Lagrange. In it, Wronski notes that Lagrange's theory is based on two assumptions:

Wronski attacks three aspects of Lagrange's theory:

1) He wonders what the grounds are for the first formulation. There seem not to be any.

2) Lagrange claims that the second formulation is true because it is verified when i equals one, but Wronski wants to know what about when i does not equal 1?

3) Lagrange defines f '(x) as the coefficient of the second term in the expanded series in the bottom equation. But Wronski notes that the position of a function in a series does not by its place alone determine the meaning of that function; it seems Lagrange slips them together invalidly.

Bottazzini, Umberto. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Transl. Warren Van Egmond. Paris: Springer-Verlag, 1981.

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from Boyer's History of the Calculus

Under the influence of Leibniz' differential method and Kant's transcendental philosophy, Hoëné Wronski objected against Lagrange's ban on the infinite in analysis.

Wronki rightly asks where Lagrange obtained the series

Wronski believed instead that mathematics should be based on what he considered the "suprime algorithmic law:"

"where the quantities

are any functions of the variable x. Being the supreme law of mathematics, the irrecusable truth of this law he held to be not mathematically derived, but given by transcendental philosophy" (Boyer 261c.d).

Wronski was correct to criticize Lagrange's theory for being limited to only expandable functions. But he otherwise held highly unconventional and controversial views on the calculus (262a).

Whereas Lagrange had attempted to give a formal logical justification of the subject, Wronski asserted that the differential calculus constituted a primitive algorithm governing the generation of quantities, rather than the laws of quantities already formed. (262ab).

The calculus' propositions, which express absolute truths, cannot be deduced within the sphere of mathematics. Wronski objected to construing calculus in terms of limits, ultimate ratios, vanishing quantities, and functions, because he thought it was fruitless to abandon the notion of the infinite (262b).

Although mathematicians continued to hold to the limit concept,

Wronski represents an extreme example of a view which we shall find recurring throughout the nineteenth century. In regarding the calculus as a means of explaining the growth of magnitudes, followers of this school of thought were to attempt to retain the concept of the infinitely small, not as an extensive quantity but as an intensive magnitude. Mathematics has excluded the fixed infinitely small because it has failed to establish the notion logically; but transcendental philosophy has sought to preserve primitive intuition in this respect by interpreting it as having an a priori metaphysical reality associated with the generation of magnitude. (262-263).

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.