16 Apr 2014

Katz and Sherry’s [Pt.4.4] “Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.4 ‘Status Transitus,’ summary


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[The following is summary. My own comments and citations are placed in double brackets. All boldface and underlying are mine.]



 

Mikhail G. Katz  and David Sherry
“Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

4. Cum Prodiisset

 
4.4 Status Transitus

Brief Summary:
A state of transition (status transitus) is an infinitely small variation that explains how things can change from one state to a contrary one. For example, parallel lines might converge, so long as the angle of that convergence is infinitely small. That angle would then be the ‘between’ transitional state between its states of being parallel and non-parallel. Such an angle is infinitesimally more than completely superposing the other line, but infinitesimally less than diverging sharply from it, at the vertex of that angle. Looking away from the intersection, we would say it is not yet parallel. Looking toward it, we would say it is not yet superposed. Transition can be explained by means of a law of continuity which says that continuous changes occur by infinitely small bridging variations, allowing us to go from opposing states, so long as that change is so small that we cannot even assign it a magnitude (and thus make an ‘actual’ distinction between the states).

 

Summary
 
Previously we noted three of Leibniz’ examples for the Principle of Continuity: (1) lines can both be parallel yet also converge at an infinitely small angle [[because there is so little difference between an infinitely small angle and none at all]] (2) two lines can be equal even if they differ in length by an infinitely small amount, and (3) a parabola is an ellipse with one side extended to infinity. Yet we find him also stating that:
Leibniz introduces his next observation by the clause ‘‘of course it is really true that’’, and notes that ‘‘straight lines which are parallel never meet’’ (Child 1920, p. 148); that ‘‘things which are absolutely equal have a difference which is absolutely nothing’’ (Child 1920, p. 148); and that ‘‘a parabola is not an ellipse at all’’ (Child 1920, p. 149).
[[KS 580]]
To explain then the original three examples we observed, he proposes his notion of status transitus, or ‘state of transition’. [[This concept will introduce the idea of event, change, motion, time, process, and the like, because]] in one such a state of transition, ‘there has not yet arisen exact eqauality’.
a state of transition may be imagined, or one of evanescence, in which indeed there has not yet arisen exact equality … or parallelism, but in which it is passing into such a state, that the difference is less than any assignable quantity; also that in this state there will still remain some difference, … some angle, but in each case one that is infinitely small; and the distance of the point of intersection, or the variable focus, from the fixed focus will be infinitely great, and the parabola may be included under the heading of an ellipse (Child 1920, p. 149).
[[KS 580]]

Recall also Leibniz’s notion of terminus: in a continuous transition, the final ending (the terminus) of the transition may be included with that transition. [[Consider something in motion slowing to a stop. That final ending, rest, can be included with the motion preceding it, as a part of that motion, even if it lies at the end of the motion. The speed while it is moving could be assigned a value, and its speed at rest can be assigned the value of 0. But because of continuity, perhaps we might say that between its states of motion and rest, it is going an inassignable, infinitely slow speed. By ‘between’ we do not mean during some duration of time it is moving some extent of distance. Say we come to the end of the object’s motion. It first comes to complete rest at time point 2 (t2) at location point 2 (p2). Immediately prior it was at p1, t1. There is no p or t between them, if we are dealing with an actual infinity (one that is already divided infinitely and not perpetually divided potentially and thus interminably). But, at p1,t1, its state of motion was immediately in relation with its state of rest at p2,t2. And at p2,t2, it is still in relation to its prior state of motion at p1,t1. This is because motion is always a difference between times and locations. If we divide motion, we do not divide it ultimately into p’s and t’s. Rather, we divide it into the smallest possible differences between p’s and t’s. There is a difference between p1,t1 and p2,t2, because one lies at a state of motion, and the other lies at a state of rest. When we concern ourselves with that tiny moment when the object goes from p1,t1 to p2,t2, then we have the terminus (rest state at p2,t2) included in the motion (as it is part of the interval with the motion state at p1,t1). We also have this state of transition, status transitus.]] “Thus, status transitus is subsumed under terminus, passing into an assignable entity, but is as yet inassignable.” [KS 580] We would not say that the status transitus is a ‘limit’ (as some translators have rendered it), because a limit is an assignable entity, where that status transitus is not. [580]
Yet for Leibniz, the metaphysical reality of the infinitesimal is open to question.
whether such a state of instantaneous transition from inequality to equality, … from convergence [i.e., lines meeting—the authors] to parallelism, or anything of the sort, can be sustained in a rigorous or metaphysical sense, or whether infinite extensions successively greater and greater, or infinitely small ones successively less and less, are legitimate considerations, is a matter that I own to be possibly open to question (Child 1920, p. 149).
[KS 580]
Yet this uncertain to the ontological reality of infinitesimals should not stop mathematicians from using them effectively in their calculations. Leibniz asserts
the possibility of the mathematical infinite: ‘‘it can be done’’, without ontological commitments as to the reality of infinite and infinitesimal objects.
[KS 581]


 
Bibliography:
Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174
The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

 

Katz and Sherry’s [Pt.4.3] “Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.3 ‘Souverain Principe,’ summary


summary by Corry Shores
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[The following is summary. My own comments and citations are placed in double brackets. All boldface and underlying are mine.]




 

Mikhail G. Katz  and David Sherry


“Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”


4. Cum Prodiisset


 

4.3 Souverain Principe



Brief Summary:

Another of Leibniz’s formulations of his law of continuity is: “the rules of the finite succeed in the infinite, and conversely.”



Summary

 

As a basic principle, all things are governed by reason [[the world is not irrational but operates in accordance with rational law-like regularities]]. For this reason, the “the rules of the finite succeed in the infinite, and conversely.” (KS 579). This is another formulation of the law of continuity.
[[Perhaps Leibniz is noting that both the infinitely small and the finite coincide in the world, but it cannot be that the some parts of the world are governed by some laws, and other parts by other laws, especially when considering that the infinitely small parts reside compositionally within finite bodies.]]



 

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174


The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Katz and Sherry’s [Pt.4.2] “Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.2 ‘Law of Continuity, with Examples’, summary


summary by Corry Shores
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[The following is summary. My own comments and citations are placed in double brackets. All boldface and underlying are mine.]




 

Mikhail G. Katz  and David Sherry


“Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”


4. Cum Prodiisset


 

4.2 Law of Continuity, with Examples



Brief Summary:

In Cum Prodiisset Leibniz discusses his Law of Continuity. According to one formulation, in a continuous transition, the final ending (the terminus) of the transition may be included with that transition.


Summary

 

The basis of the calculus that Leibniz formulates in Cum prodiisset is his Law of Continuity (LC). It takes a variety of forms. Here is one formulation:

In any supposed continuous transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included.9 [[ft 9: Boyer claims that Leibniz used this formulation of LC in ‘‘a letter to [Pierre] Bayle in 1687’’ (Boyer 1959, p. 217). Boyer’s claim contains two errors. First, the work in question is not a letter to Bayle but | rather the Letter of Mr. Leibniz on a general principle useful in explaining the laws of nature, etc. (Leibniz 1687). Second, while this letter does deal with Leibniz’ continuity principle, it does not contain the formulation In any supposed continuous transition, ending in any terminus, etc.; instead, it postulates that an infinitesimal change of input should result in an infinitesimal change in the output (this principle was popularized by Cauchy in 1821 as the definition of continuity in Cauchy 1821, p. 34). Boyer’s erroneous claims have been reproduced by numerous authors, including Kline (1972, p. 385).]]
[[KS 577. footnote, 577-578]]

The final terminus in this explanation is an ending of a transition. KS then give five reasons that the terminus “encompasses inassignable quantities” [[infinitesimal quantities, see KS ‘Leibniz’s Laws of Continuity and Homogeneity” http://arxiv.org/pdf/1211.7188.pdf. See Page 578 of KS ‘Leibniz’s Infinitesimal’ for the five reasons.]]


In Cum Prodiisset, Leibniz offers a number of examples for how the Law of Continuity can be applied. KS will focus on three of them [[quoting]]:

(1) In the context of a discussion of parallel lines, he writes: when the straight line BP ultimately becomes parallel to the straight line VA, even then it converges toward it or makes an angle with it, only that the angle is then infinitely small (Child 1920, p. 148).

(2) Invoking the idea that the term equality may refer to equality up to an infinitesimal error, Leibniz writes: when one straight line is equal to another, it is said to be unequal to it, but that the difference is infinitely small (Child 1920, p. 148).

(3) Finally, a conception of a parabola expressed by means of an ellipse with an infinitely removed focal point is articulated in the following terms: a parabola is the ultimate form of an ellipse, in which the second focus is at an infinite distance from the given focus nearest to the given vertex (Child 1920, p. 148).

[[KS 579, for more on the ellipse example and on the law of continuity, see this entry on Leibniz’ letter to Malebranche.]




Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174


The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Euclid, Elements [B1, D4] Book 1, Definition 4


by Corry Shores
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[The following is quotation, unless given in brackets]



Euclid


The Elements
Στοιχεῖα
 


Book 1
Βιβλίον I


Definitions
Ὅροι


Definition 4



Definition 4:
D4. A straight line is that which lies evenly between its extremities.

(Byrne, p.xviii)


4. A straight-line is (any) one which lies evenly with points on itself.

(Fitzpatrick 6)


4. A straight line is whatever lies equally with the points on it,

(Mendell)



δʹ. Εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ᾿ ἑαυτῆς σημείοις κεῖται.

(Heiberg)



[From Mendell’s commentary:]

The meaning of this definition is very unclear.  It is common to translate 'ex isou', here 'equally' as 'evenly', cf. def. 7.

(Mendell)



 
 

Sources:

Euclid; Oliver Byrne. The First Six Books of The Elements of Euclid, in which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners. London: William Pikering, 1847.

Available at:

https://archive.org/details/firstsixbooksofe00byrn

https://archive.org/details/firstsixbooksofe00eucl



 

Euclid; J.L. Heiberg; Richard Fitzpatrick. Euclid’s Elements of Geometry. Ed. Richard Fitzpatrick. Online edition. 2007. Available at:

http://farside.ph.utexas.edu/euclid/elements.pdf



 

Euclid; Heath.  The Thirteen Books of the Elements. Vol. 1. Books I-II. 2nd Edition. Ed. and Trans. Thomas L. Heath. Cambridge: Cambridge UP, 1968.

Euclid; Heath.  The Thirteen Books of the Elements. Vol. 1. Books I-II. 2nd Edition. Ed. and Trans. Thomas L. Heath. New York: Dover, 1956.

Available at:

https://archive.org/details/euclid_heath_2nd_ed

Euclid; Heath.  The Thirteen Books of the Elements. Vol. 1. Books I-II. 1st Edition. Ed. and Trans. Thomas L. Heath. Cambridge: Cambridge UP, 1908.

Available at:

https://archive.org/details/thirteenbookseu02heibgoog


Euclid; Heath.  The Thirteen Books of the Elements. Vol. 2. Books III-IX. 2nd Edition. Ed. and Trans. Thomas L. Heath. New York: Dover, 1956.

Available at:

https://archive.org/details/euclid_heath_2nd_ed

Euclid; Heath.  The Thirteen Books of the Elements. Vol. 2. Books III-IX. 1st Edition. Ed. and Trans. Thomas L. Heath. Cambridge: Cambridge UP, 1908.

Available at:

https://archive.org/details/thirteenbookseu00heibgoog


Euclid; Heath.  The Thirteen Books of the Elements. Vol. 3. Books X-XIII. 2nd Edition. Ed. and Trans. Thomas L. Heath. New York: Dover, 1956.

Available at:

https://archive.org/details/euclid_heath_2nd_ed

Euclid; Heath.  The Thirteen Books of the Elements. Vol. 3. Books X-XIII. 1st Edition. Ed. and Trans. Thomas L. Heath. Cambridge: Cambridge UP, 1908.

Available at:

https://archive.org/details/thirteenbookseu03heibgoog


 

Mendell, Henry. Euclid. In Vignettes of Ancient Mathematics:

http://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/VignettesAncientMath.html



Joyce, David E. Euclid Elements. With diagrams, java applets.
http://aleph0.clarku.edu/~djoyce/java/elements/toc.html


Online Greek. Dimitrios E. Mourmouras:

http://www.physics.ntua.gr/~mourmouras/euclid/index.html

15 Apr 2014

‘D nce n ked with your body on the high hill of happiness’, Clifford Duffy’s ‘'As you”


by Corry Shores
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D nce n ked with your body on the high hill of happiness


Clifford Duffy has a great poem up that I highly recommend not just for its beauty but as well for those who have heard of Zizek and Badiou and have maybe felt that something is not quite right with what they are doing. I have tried to summarize and comment, but I cannot do any better than Duffy has written. Please check it out if you can!




Democritus’ Paradox of the Cone, in Plutarch’s discussion of the Stoic doctrines of the elements, in De communibus notitiis adversus Stoicos


by Corry Shores
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Democritus’ Paradox of the Cone,

in Plutarch’s discussion of the Stoic doctrines of the elements,

in his De communibus notitiis adversus Stoicos



Summary from Sections 37-39:



According to common conceptions, two bodies cannot occupy the same location. But the Stoics have a confused understanding of the world and its parts. For example, they say that if you mix one cup of wine with two cups of water, the ‘one’ of the wine becomes ‘two’, because it extends throughout the whole of the two cups of wine. [In the Stoic theory of mixtures, there is a type of mixture called ‘total blending’, and the wine example illustrates it. For more, see
Sellars’ explanation of the Stoic theory of mixtures. One odd thing about this kind of mixture is that the water and wine on the one hand form a third thing, but on the other hand the parts maintain their qualities, and can be extracted. So insofar as the wine retains its qualities and can be extracted, it is one. But insofar as it has thoroughly integrated with the water to make a third body, it is two.] The one is also three, because all the parts now total three parts. It is also four, because by being mixed with the two other parts, the original one is now two. So we have the original wine, which is now two, and the original water, which is also two, now making four.

Now this fine subtilty is a consequence of their putting bodies into a body, and so likewise is the unintelligibleness of the manner how one is contained in the other. For it is of necessity that, of bodies passing one into another by mixture, the one should not contain and the other be contained, nor the one receive and the other be received within; for this would not be a mixture, but a contiguity and touching of the superficies, the one entering in, and the other enclosing it without, and the rest of the parts remaining unmixed and pure, and so it would be merely many different things. But there being a necessity, according to their axiom of mixture, that the things which are mixed should be mingled one within the other, and that the same things should together be contained [p. 412] by being within, and by receiving contain the other, and that neither of them could possibly exist again as it was before, it comes to pass that both the subjects of the mixture mutually penetrate each other, and that there is not any part of either remaining separate, but that they are necessarily all filled with each other.
[Plutarch
Section 37]

Another strange result of the Stoic theory of mixture is that one drop of wine in the sea mixes throughout the sea’s entirety. “And this Chrysippus admits, saying immediately in his First Book of Natural Questions, that there is nothing to hinder one drop of wine from being mixed with the whole sea. And that we may not wonder at this, he says that this one drop will by mixtion extend through the whole world; than which I know not any thing that can appear more absurd.” [Plutarch Section 37] As we will see, this idea that the drop extends to the whole world would be a main source of problems for the Stoic theory of mixtures, in Plutarch’s assessment.


Thus a further strange conclusion is that bodies have no ends to them, and we extend infinitely. [The drop of wine unites with the water which mixes with the air which falls and is absorbed in the soil and perhaps in this way everything is mixed with everything else] So also we cannot say that one body is larger than another, if both extend to infinity. Thus inequality becomes impossible [since nothing has determinate bounds. All things permeate with all other things]. But if there is no inequality, then there is also no roughness or unevenness of bodies. “unevenness is the inequality of the same superficies with itself, and roughness is an unevenness joined with hardness.” But in the Stoic theory, bodies do not terminate in a last part [because they extend into larger and larger mixtures] [and thus for example a surface cannot be said to be rough, because it does not actually have determinate boundaries of any kind, jagged or otherwise.] [Plutarch Section 38]


Plutarch then quotes the Stoic Chryssippus. [Here it seems that Chryssipus is explaining the composition of bodies. If he says our body for example is divisible into smaller parts, the question is, do they terminate? If so, why cannot those parts not be divided? Or, if it goes on to infinity, would that not mean there is no fundamental composition to bodies, since there are no basic parts? It would be divisions all the way down with no basic substance. So Chrysippus might want to argue that the basic parts are neither finite nor infinite, even if he cannot specify what they might instead be.]

Chrysippus says: ‘If we are asked, if we have any parts, and how many, and of what and how many parts they consist, we are to use a distinction, making it a position that the whole body is compacted of the head, trunk, and legs, as if that were all which is enquired and doubted of. But if they extend their interrogation to the last parts, no such thing is to be undertaken, but we are to say that they consist not of any certain parts, nor yet of so many, nor of infinite, nor of finite.’
[
Plutarch Section 38]

Plutarch notes this is a contradiction. Chrysippus wants it both ways, and in the end undermines the possibility of his claim having any truth value.

For if there were any medium between finite and infinite, as the indifferent is between good and evil, he should, by telling us what that is, have solved the difficulty. But if—as that which is not equal is presently understood to be unequal, and that which is not mortal to be immortal—we also understand that which is not finite to be immediately infinite, to say that a body consists of parts neither finite nor infinite is, in my opinion, the same thing as to affirm that an argument is compacted of positions neither true nor false.... [Plutarch Section 38]


Plutarch says that Chryssippus elaborates on this impossible status of infinite and finite with the example of the sides of a pyramid. [The distinction he makes will be clearer in the following Democritus paradox. But for now…] A pyramid is made of triangles who sides incline toward the juncture at the top. [perhaps are to think of the triangular sides as being made of smaller and smaller triangles down to the very small. Or Perhaps the triangles are embedded by in another way. At any rate, in light of the following example, Plutarch seems to be saying that if we were to consider the pyramid as being made of stacking layers, that one layer to the next can either be equal in size to the lower or unequal in size.] Chrysippus says that [these layers] are unequal [meaning the next one up is slightly smaller] and yet no one exceeds [its neighboring layer.] But this again leads into a contradiction.

To this he with a certain youthful rashness adds, that in a pyramid consisting of triangles, the sides inclining to the juncture are unequal, and yet do not exceed one another in that they are greater. Thus does he keep the common notions. For if there is any thing greater and not exceeding, there will be also something less and not deficient, and so also something unequal which neither exceeds nor is deficient; that is, there will be an unequal thing equal, a greater not greater, and a less not less. [Plutarch Section 39]

Plutarch proceeds then to the example of Democritus’ paradox of the cone’s composition. If the circles making up the cone were equal, the sides would be even and smooth, but the shape would be a cylinder and not a cone. If instead the cone’s circles were unequal, the figure would come to a point, but its sides would be uneven, making its surface rough and giving it the appearance of a staircase. [See my  animate diagram below:]

democritus cone animation.25.complete

[Moving diagram by Corry Shores, made with Open Office Draw and Unfreez]

Plutarch seems to indicate that Chrysippus’ solution is to say that the cone is unequal, but the circles composing it are neither equal nor unequal. Plutarch notes that this means we have an unequal body that is made of parts which are not unequal, and this seems absurd. [Plutarch following reasoning is not clear to me, but I provide it below. He does at least seem to be saying that] if a body is unequal, that could only mean that its parts are unequal.

Here, that he may convince Democritus of ignorance, he says, that the superficies are neither equal or unequal, but that the bodies are unequal, because the superficies are neither equal nor unequal. [p. 415] Indeed to assert this for a law, that bodies are unequal while the superficies are not unequal, is the part of a man who takes to himself a wonderful liberty of writing whatever comes into his head. For reason and manifest evidence, on the contrary, give us to understand, that the superficies of unequal bodies are unequal, and that the bigger the body is, the greater also is the superficies, unless the excess, by which it is the greater, is void of a superficies. For if the superficies of the greater bodies do not exceed those of the less, but sooner fail, a part of that body which has an end will be without an end and infinite. For if he says that he is compelled to this, . . . For those rabbeted incisions, which he suspects in a cone, are made by the inequality of the body, and not of the superficies. It is ridiculous therefore to take the superficies out of the account, and after all to leave the inequality in the bodies themselves. But to persist still in this matter, what is more repugnant to sense than the imagining of such things? For if we admit that one superficies is neither equal nor unequal to another, we may say also of magnitude and of number, that one is neither equal nor unequal to another; and this, not having any thing that we can call or think to be a neuter or medium between equal and unequal.
[Plutarch Section 39]

Plutarch continues to say if this reasoning applies in this case, then why not for other types of figures?

Besides, if there are superficies neither equal nor unequal, what hinders but there may be also circles neither equal nor unequal? For indeed these superficies of conic sections are circles. And if circles, why may not also their diameters be neither equal nor unequal? And if so, why not also angles, triangles, parallelograms, parallelepipeds, and bodies? For if the longitudes are neither equal nor unequal to one another, so will the weight, percussion, and bodies be neither equal nor unequal. How then dare these men inveigh against those who introduce vacuities, and suppose that there are some indivisible atoms, and who say that motion and rest [p. 416] are not inconsistent with each other, when themselves affirm such axioms as these to be false: If any things are not equal to one another, they are unequal to one another; and the same things are not equal and unequal to one another?  [Plutarch Section 39]

Let’s look at how Plutarch formulates the issue in the following sentence, because it will help us integrate this puzzle with the history of calculus. He says:

But when he says that there is something greater and yet not exceeding, it were worth the while to ask, whether these things quadrate with one another. For if they quadrate, how is either the greater? And if they do not quadrate, how can it be but the one must exceed and the other fall short? For if neither of these be, the other both will and will not quadrate with the greater.  [Plutarch Section 39]

Here the wording is, one figure is greater than another yet does not exceed it (in magnitude). If we think of each partition of the cone as narrowing to the infinitely small, then one such circular partition is larger than one of its immediate neighbors, however, it does not exceed or go beyond the smaller one by any finite amount. In that sense one partition can both be larger than its neighbor without extending beyond it. [See further discussion by Katz and Sherry.] Also notice Plutarch refers to quadrating the figures. Perhaps this is like finding the quadrature of a figure. If so, that would be interesting again with respect to the history of calculus. Boyer describes Archimedes’ method of find the area of a parabola segment using a method of exhaustion, in Archimedes’ text “Quadrature of the Parabola.” Let me quote from an old source that explains quadrature and its place in the history of calculus:

QUADRATURE, in geometry, denotes the reducing a figure to a figure. Thus, the finding of a square which shall contain just as much surface or area as a circle, an ellipsis, a triangle, &c. The quadrature, especially among the ancient mathematicians, was a great postulatum. The quadrature of rectilinear figures is easily found, for it is merely finding their areas or surfaces, i.e., their squares; for the squares of equal areas are easily found by only extracting the roots of the areas thus found. The quadrature of the curvilinear spaces is of more difficult investigation; and in this respect extremely little was done by the ancients, except the finding the quadrature of the parabola by Archimedes. In 1957, Sir Paul Neil, Lord Brouncker, and Sir Christopher Wren, geometrically demonstrated the equality of some curvilinear spaces to rectilinear spaces; and soon after the like was proved, both at home and abroad of other curves, and it was afterwards brought under an analytical calculus; the first specimen of which was given to the public in 1688 by Mercator, in a demonstration of Lord Brounker’s quadrature of the hyperbola, Dr Wallis’s reduction of a fraction into an infinite series of division. Sir Isaac Newton, however, had before discovered a method of attaining the quantity of all quadruple curves analytically by his fluxions before 1668. It is disputed whether Sir Christopher Wren or Mr Huygens first discovered the quadrature of any determinate cycloidal space. Mr Leibniz afterwards found that of another space; and in 1669 Bernoulii discovered the quadrature of an infinity of cycloidal spaces both segments and sectors &c. [Note, previous word ‘infinity’ illegible in my source, and might be incorrect.] [Encyclopædia Perthensis page 526]

In the Boyer text, he describes how for the parabolic segment, Archimedes inscribes a large triangle, then other within the remaining space, and others still.

image

(From Boyer, p.52)

By summing the areas of the series of triangles, we can find the total area.

image


[So perhaps if we were to quadrate one of the circles of the cone, we would obtain a polygon with infinitely many sides and whose area differs from the circle’s only by an infinitesimal amount. In that sense, it would be less than the circle but no part of the circle would exceed the polygon’s boundaries by any finite amount. It does not seem that Plutarch is comparing the size of the circle with its quadrature, but rather the quatradatures of two neighboring circles. However, it is interesting that he evokes the method of exhaustion. Again, see
Katz and Sherry for an excellent treatment of this cone example and Archimedes’ method of exhaustion.

Plutarch. De communibus notitiis adversus Stoicos. From The Perseus Digital Library

http://www.perseus.tufts.edu

which cites the material thusly:

Plutarch. Plutarch's Morals. Translated from the Greek by several hands. Corrected and revised by. William W. Goodwin, PH. D. Boston. Little, Brown, and Company. Cambridge. Press Of John Wilson and son. 1874. 4.

Particularly the pages:

Section 37:

http://data.perseus.org/citations/urn:cts:greekLit:tlg0007.tlg138.perseus-eng1:37

Section 38:

http://data.perseus.org/citations/urn:cts:greekLit:tlg0007.tlg138.perseus-eng1:38

Section 39:
http://data.perseus.org/citations/urn:cts:greekLit:tlg0007.tlg138.perseus-eng1:39

Boyer, Carl. The History of the Calculus and Its Conceptual Development (The Concept of the Calculus). New York: Dover, 1949.

Encyclopædia Perthensis Edinburgh: John Brown, Ancor Close, 1816

Viewed on Google Books:

http://books.google.com.tr/books?id=SGvfGB5X_ncC&printsec=frontcover#v=onepage&q&f=false






13 Apr 2014

Stoic Theory of Mixtures in Sellars’ Stoicism

by Corry Shores
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Stoic Theory of Mixtures


in


John Sellars


Stoicism


Ch.4 Stoic Physics


Stoics hold that two things can be mixed with one another in three different ways.

(1) Juxtaposition: mixed parts remain distinct and independent despite their proximity: “grains of the two entities are mixed together but remain distinct from one another, as in the case of salt and sugar mixed together in a bowl.” (Sellars 88d)

(2) Fusion: parts are mixed, lose their independence, but not their identity, and form a composite: “a new entity is created out of the two enti- | ties, which cease to exist independently, as when using a number of ingredients when cooking.” (Sellars 88-89)

(3) Total blending: parts mix together homogenously while maintaining individual properties, as one can still be extracted from the mixture: “the two entities are mixed together to the point that every part of the mixture contains both of the original entities, yet each of the original entities retains its own distinctive properties and can in theory be extracted from the mixture. For instance, it is reported that if one mixes wine and water in a glass it is possible to extract the wine out of the mixture by using a sponge soaked in oil – and this has been supported by experimentation (Stobaeus 1,155,8–11 with Sorabji 2004: 298–9). Although the wine and water are completely mixed, in a way that the grains of salt and sugar are not, it is still possible to separate the two liquids.” (Sellars 89)

Total blending also implies that were one to put a drop of wine into the sea, it will distribute itself into every part of the sea. (Sellars 89)

In totally blended mixtures, (a) the two original entities are destroyed in created the third new entity, the mixture, and yet (b) “this new entity contains within it the qualities of the two original entities, and so it is possible to extract the original entities from the mixture (in a way that is not possible in the case of fusion, the second kind of mixture).” (Sellars 89)



From the original text [the following is quotation]:

The Stoics’ own theory of mixture is relevant here. They suggest that two material entities might be mixed together in three different ways (see Alexander, Mixt. 216,14–217,2). The first of these is “juxtaposition”, in which grains of the two entities are mixed together but remain distinct from one another, as in the case of salt and sugar mixed together in a bowl. The second is “fusion”, in which a new entity is created out of the two enti- | -ties, which cease to exist independently, as when using a number of ingredients when cooking. The third the Stoics call “total blending”, in which the two entities are mixed together to the point that every part of the mixture contains both of the original entities, yet each of the original entities retains its own distinctive properties and can in theory be extracted from the mixture. For instance, it is reported that if one mixes wine and water in a glass it is possible to extract the wine out of the mixture by using a sponge soaked in oil – and this has been supported by experimentation (Stobaeus 1,155,8–11 with Sorabji 2004: 298–9). Although the wine and water are completely mixed, in a way that the grains of salt and sugar are not, it is still possible to separate the two liquids. One slightly paradoxical consequence of this theory of total blending that the Stoics appear to have accepted was the thought that if one added a single drop of wine to the sea then that single drop of wine would have to mix with every part of the sea, in effect stretching itself out over a vast area (see DL 7.151). The Stoics described this third kind of mixture as a process in which the two original entities are destroyed and a new third entity, the mixture, is created. However, this new entity contains within it the qualities of the two original entities, and so it is possible to extract the original entities from the mixture (in a way that is not possible in the case of fusion, the second kind of mixture). [Sellars 88-89]


From:

Sellars, John. Stoicism. Durham: Acumen, 2006.


Katz and Sherry’s [Pt.4.1] “Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.1 ‘Critique of Nieuwentijt’, summary


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

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Mikhail G. Katz  and David Sherry


“Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”


4. Cum Prodiisset


4.1 Critique of Nieuwentijt



Brief Summary:

Leibniz begins Cum Prodiisset by criticizing Niewentijt’s position that the product of two infinitesimals is zero.


Summary

Niewentijt “defended a conception of infinitesimal according to which the product of two infinitesimals is always zero.” [KS 577] In his Cum Prodiisset, Leibniz begins by criticizing this argument. Regarding Niewentijt’s positions, “Leibniz rejects nilsquare and nilcube infinitesimals, which are altogether incompatible with his approach to differential calculus, as we will see in Sect. 4.6.” [577]



Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174


The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

 

Katz and Sherry’s [Pt.4] “Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4 ‘Cum Prodiisset’, summary


summary by Corry Shores
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Mikhail G. Katz  and David Sherry


“Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”


4. Cum Prodiisset


Brief Summary:

In the following subsections, the authors will focus on Leibniz’ Cum Prodiisset, because it is of crucial importance for understanding Leibniz’ fundamental stance.


Summary

 

Around 1701, Leibniz published, Cum Prodiisset, which is “of crucial importance in understanding Leibniz’s foundational stance”. [KS 577] The authors will focus on it in the following subsections.



Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174


The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Katz and Sherry’s [Pt.3] “Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 3 ‘A Pair of Leibnizian Methodologies, summary


summary by Corry Shores
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Mikhail G. Katz  and David Sherry


“Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”


3. A Pair of Leibnizian Methodologies


Brief Summary:

Leibniz had two methodologies, which the authors call the A-methodology (using Archimedes’ exhaustion) and the B-methodology (using infinitesimals). Recent Leibniz scholars either acknowledge both of Leibniz’ methodologies or just the first type. The authors believe those in the second camp are misreading Leibniz’ notion of the infinitesimal’s fictionality.


Summary


Leibniz had two infinitesimal calculus methodologies: one by exhaustion and one using the law of continuity. (KS 575) The first relies on Archimedes’ exhaustion method, and the authors call it the ‘A-methodology’. The second uses infinitesimals and is called the ‘B-methodology’.


Leibniz considered infinitesimals as fictions. In his time, this was a controversial position, especially for some of his disciples, like Bernoulli, l’Hôpital, and Varignon. Accordiing to Ferraro, “Leibniz’s infinitesimals enjoy an ideal ontological status similar to that of the complex numbers, surd (irrational) exponents, and other ideal quantities.” (576)


The authors will now examine how commentators attribute either both A and B methodologies or just the A-methodology. They first quote from Leibniz’ 1702 letter to Varignon.

Here Leibniz outlines a geometrical argument involving quantities c and e described as ‘‘not absolutely nothing’’, and goes on to comment that c and e [KS quoting Leibniz:]

are treated as infinitesimals, exactly as are the elements which our differential calculus recognizes in the ordinates of curves for momentary increments and decrements (Leibniz et al. 1702, pp. 104–105). [KS 576]

Jesseph argues that Leibniz proposes both A and B methodologies. Like Bos, Jesseph emphasizes Leibniz’ law of continuity and regards it not as a mathematical principle but rather as a “a general methodological rule with applications in mathematics, physics, metaphysics, and other sciences’’ (KS 576 quoting Jesseph ibid p.21).


Recent work on Leibniz’ calculus is divided into two camps: 1) Those who recognize both methodologies (Bos, Ferraro, Horváth, Jesseph, and Laugwitz), and 2) those who have a syncategorematic interpretation that only recognizes the A-methodology. The authors believe that the second reading “is due to an incorrect analysis of Leibniz’s fictionalism.” (KS 577)



Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174


The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Katz and Sherry’s [Pt.2] “Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 2 ‘Preliminary Developments’, summary


summary by Corry Shores
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Mikhail G. Katz  and David Sherry


“Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”


2. Preliminary Developments


Brief Summary:

Indivisibles are not infinitesimals, although they are often confused. Indivisibles have one dimension less than what they divide. Infinitesimals have the same dimensions as what they are a part of.


Summary

 

We should first distinguish indivisibles from infinitesimals.


Leibniz first uses the term ‘infinitesimal’ in 1673, but he credits the coinage to Mercator.


There is more than one conception of the infinitely small. Many commentators do not differentiate them. For example, Boyer seems to imply that Archimedes infinitesimal and kinematic methods provided the basis for Leibniz’ differential calculus. (Boyer, The concepts of the calculus, p.59.) (KS 573d).  However, Archimedes’ infinitesimal method uses indivisibles and not infinitesimals. His indivisibles are the limits of division, and thus they have one dimension less than the areas they are dividing (they are one dimensional while what they divide is two dimensional).

Archimedes’ infinitesimal method employs indivisibles. For example, in his heuristic proof that the area of a parabolic segment is 4/3 the area of the inscribed triangle with the same base and vertex, he imagines both figures to consist of perpendiculars of various heights erected on the base (ibid., 49–50). The perpendiculars are indivisibles in the sense that they are limits of division and so one dimension less than the area. Qua areas, they are not divisible, even if, qua lines they are divisible. In the same sense, the indivisibles of which a line consists are points, and the indivisibles of which a solid consists are planes. We will discuss the term ‘‘consist of’’ shortly.
(KS 574a, boldface and underlining mine)


[See Boyer’s treatment of Archimedes here, and see the original Archimedes’ text] However, Leibniz’ infinitesimals have the same dimension as the figures they make-up, and thus they are not like Archimedes’ indivisibles.

Leibniz’s infinitesimals are not indivisibles, for they have the same dimension as the figures that consist of them. Thus, he treats curves as composed of infinitesimal lines rather than indivisible points. Likewise, the infinitesimal parts of a plane figure are parallelograms. The strategy of treating infinitesimals as dimensionally homogeneous with the objects they compose seems to have originated with Roberval or Torricelli, Cavalieri’s student, and to have been explicitly arithmetized by Wallis (Beeley 2008, [Infinity, Infinitesimals, and the Reform of Cavalieri: John Wallis and his Critics. In Goldenbaum and Jesseph Infinitesimal Differences: Controversies between Leibniz and his Contemporaries] p. 36ff).
(KS 574)


Democritus used both indivisibles and infinitesimals. (KS 574)


Plutarch notes a puzzle of Democritus’ that would arise for indivisibles but not for infinitesimals. The puzzle is this. A cone can be thought as being made of many surfaces parallel to the base. If they are all equally, it would be a cylinder. But if they were all different, it would look like a staircase. However, we do not encounter this problem when we use the concept of the infinitesimal.  “This puzzle need not arise for infinitesimals of the same dimension, with an infinitesimal viewed as a frustum of a cone rather than a plane section.” (KS 574) So we are to think of each slice as being infinitesimally thin, but the outer part is at an angle. Thus one slice picks up where the prior leaves off, and we have an infinity of slices all making a smoothly tapering cone. (see diagrams of frustums below)

File:Frustum of a cone.jpg

(thanks wikimedia commons)

frustrum.wiki.Frustum_of_a_Decagonal_Pyramid.svg
(thanks wikimedia commons)

[We give a more detailed treatment of the Plutarch text here. From that entry, here is a moving diagram for the puzzle, including the distinction between the indivisibles and infinitesimals:]

democritus cone animation.25.complete

[Moving diagram by Corry Shores, made with Open Office Draw and Unfreez]


Zeno’s “metrical paradox proposes a dilemma: If the indivisibles have no magnitude, then a figure which consists of them has no magnitude; but if the indivisibles have some (finite) magnitude, then a figure which consists of them will be infinite.” (574) There is a further problem for indivisibles. They are boundaries of what they limit. But this means they are not immediately up against one another. So we seem unable to concatenate them in order to increase a magnitude.

If a magnitude consists of indivisibles, then we ought to be able to add or concatenate | them in order to produce or increase a magnitude. But indivisibles are not next to one another; as limits or boundaries, any pair of indivisibles is separated by what they limit. Thus, the concepts of addition or concatenation seem not to apply to indivisibles.
(574-575)


These problems do not apply to Leibniz’ infinitesimals. They do not have a zero magnitude, so they do not have the problem of being unable to add up to a larger magnitude. However, their magnitude is not finite, so an infinity of them is not infinitely large. Infinitely many infinitely small magnitudes make up finite magnitudes. This also allows us to perform arithmetic operations on them, which distinguishes them from Archimedes’ methods.

The paradox may not apply to infinitesimals in Leibniz’s sense, however. For, having neither zero nor finite magnitude, infinitely many of them may be just what is needed to produce a finite magnitude. And in any case, the addition or concatenation of infinitesimals (of the same dimension) is no more difficult to conceive of than adding or concatenating finite magnitudes. This is especially important, because it allows one to represent infinitesimals by means of numbers and so apply arithmetic operations to them. This is the fundamental difference between the infinitary methods of Archimedes (and later Cavalieri) and the infinitary methods of Leibniz and his followers.
(575)


Not rigorously making this distinction has led to misleading claims being made about 17th century calculus. In the following, the authors will “say that a magnitude consists of infinitesimals just in case the infinitesimals and the original magnitude have the same dimension. Otherwise, we shall use the term indivisible.” (575)

 



Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174


The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

 

 

Image credits:

Frustrum:

http://commons.wikimedia.org/wiki/File:Frustum_of_a_cone.jpg

http://commons.wikimedia.org/wiki/File:Frustum_of_a_Decagonal_Pyramid.svg

12 Apr 2014

Euclid, Elements [B1, D3] Book 1, Definition 3


by Corry Shores
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[The following is quotation, unless given in brackets]



Euclid


The Elements
Στοιχεῖα
 


Book 1
Βιβλίον I


Definitions
Ὅροι


Definition 3



Definition 3:
D3. The extremities of a line are points.

(Byrne, p.xviii)


γʹ. Γραμμῆς δὲ πέρατα σημεῖα.

(Heiberg)



[From Mendell’s commentary:]

Is this a separate definition of point (as in Aristotle) or a property of lines?

(Mendell)



 
 

Sources:

Euclid; Oliver Byrne. The First Six Books of The Elements of Euclid, in which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners. London: William Pikering, 1847.

Available at:

https://archive.org/details/firstsixbooksofe00byrn

https://archive.org/details/firstsixbooksofe00eucl



 

Euclid; J.L. Heiberg; Richard Fitzpatrick. Euclid’s Elements of Geometry. Ed. Richard Fitzpatrick. Online edition. 2007. Available at:

http://farside.ph.utexas.edu/euclid/elements.pdf



 

Euclid; Heath.  The Thirteen Books of the Elements. Vol. 1. Books I-II. 2nd Edition. Ed. and Trans. Thomas L. Heath. Cambridge: Cambridge UP, 1968.

Euclid; Heath.  The Thirteen Books of the Elements. Vol. 1. Books I-II. 2nd Edition. Ed. and Trans. Thomas L. Heath. New York: Dover, 1956.

Available at:

https://archive.org/details/euclid_heath_2nd_ed

Euclid; Heath.  The Thirteen Books of the Elements. Vol. 1. Books I-II. 1st Edition. Ed. and Trans. Thomas L. Heath. Cambridge: Cambridge UP, 1908.

Available at:

https://archive.org/details/thirteenbookseu02heibgoog


Euclid; Heath.  The Thirteen Books of the Elements. Vol. 2. Books III-IX. 2nd Edition. Ed. and Trans. Thomas L. Heath. New York: Dover, 1956.

Available at:

https://archive.org/details/euclid_heath_2nd_ed

Euclid; Heath.  The Thirteen Books of the Elements. Vol. 2. Books III-IX. 1st Edition. Ed. and Trans. Thomas L. Heath. Cambridge: Cambridge UP, 1908.

Available at:

https://archive.org/details/thirteenbookseu00heibgoog


Euclid; Heath.  The Thirteen Books of the Elements. Vol. 3. Books X-XIII. 2nd Edition. Ed. and Trans. Thomas L. Heath. New York: Dover, 1956.

Available at:

https://archive.org/details/euclid_heath_2nd_ed

Euclid; Heath.  The Thirteen Books of the Elements. Vol. 3. Books X-XIII. 1st Edition. Ed. and Trans. Thomas L. Heath. Cambridge: Cambridge UP, 1908.

Available at:

https://archive.org/details/thirteenbookseu03heibgoog


 

Mendell, Henry. Euclid. In Vignettes of Ancient Mathematics:

http://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/VignettesAncientMath.html



Joyce, David E. Euclid Elements. With diagrams, java applets.
http://aleph0.clarku.edu/~djoyce/java/elements/toc.html


Online Greek. Dimitrios E. Mourmouras:

http://www.physics.ntua.gr/~mourmouras/euclid/index.html