# Are Leibnizian infinitesimals thought to be logical fictions by Leibniz scholars?

Japanese scholar Hide Ishiguro published a book in 1990 entitled "Leibniz's philosophy of logic and language" (second edition). Of particular interest, as far as the history of mathematics is concerned, is her Chapter 5.

Here she presents an interpretation of Leibniz's infinitesimals in the spirit of a conceptual framework developed by Russell, involving the so-called "logical fictions". This involves intepreting infinitesimals as non-designating terms, which correspond to propositions quantified over ordinary Archimedean magnitudes. Ishiguro's interpretation usually goes under the name "syncategorematic".

I was wondering about the current status of Ishiguro's interpretation of Leibnizian infinitesimals, among Leibniz scholars. Would it be accurate to affirm that this is the dominant interpretation as far as Leibnizian infinitesimals are concerned?

Note 1. HOPOS (Journal of the International Society for the History of Philosophy of Science) just published our rebuttal of syncategorematist theories that seek to sweep Leibnizian infinitesimals under a Weierstrassian rug.

• Hi katz, nice to see you around here. I've always read your posts on the mathematics SE's with great interest!
– Danu
Dec 28, 2014 at 14:36
• Who would you call "Leibniz scholars"? Dec 28, 2014 at 21:34
• @HDE, Leibniz scholars are people who go read Leibniz and try to understand what he said. These are more often historians and philosophers than mathematicians, but some mathematicians have been interested in what Leibniz said as well. I am not sure I answered your question so feel free to elaborate. Dec 29, 2014 at 9:03
• Doesn't your own paper address this? arxiv.org/abs/1205.0174
– user466
Jan 9, 2015 at 1:20

## 3 Answers

My impression is that Leibnitz usually offers several approaches that are alternatives, and are not consistent. (Not surprising, since most of the papers are his notes, written "for the desk", not intended by him for publication.)

Wallis in his book on integration, which is prior both Newton and Leibnitz, uses the concept of a right hand limit of a quantity that goes to zero. So it's arbitrarily small but never null.

Leibnitz in a published short paper, on what infinitesimals are, constructs them in a similar (but not always quite the same) way. Perhaps this may be taken as his "official" opinion, seeing that he published it. (But he treats them inconsistently across his entire body of work.)

Basically, he constructs a right triangle in cartesian coordinates, and intersects it with a geodesic in two places. He moves the geodesic continuously toward a vertex of the triangle, so creating another triangle whose sides get shorter and converge to zero (if the geodesic touches the vertex). But he defines a fraction where the numbers are lengths of the sides of the triangle so constructed.

In this case: the denominator cannot be zero, if the system is to be consistent, so the motion of the geodesic is restricted in that it cannot touch the vertex. Whatever can be constructed by such a motion is an infinitesimal, he says.

He defines each infinitesimal as a fraction constructed by such motion. Both denominator and numerator become smaller as the geodesic approaches, but never become null. And because the angle of the geodesic where it intersects each side of the triangle is NOT in general the same, the sides of the triangle constructed are NOT, except in a special case of the construction, equal.

Obviously, not all infinitesimals are identical. Infinitesimal to him, therefore, as far as his published work is concerned, refers to any quantity always decreasing, in the same way as Hinchin in his 1950's calculus textbook presented the matter. To be an infinitesimal, as far as Leibnitz is concerned, is to be constructed in a certain way by a certain series of changes in another function, but it says nothing of the quantity itself. (Hinchin nods in approval.)

However, in his discussion of modelling quality by quantity, via continuity, infinitesimals are arbitrarily small quantities, eventually becoming zero, at which point, one quality transforms into another.

I suggest treating his opinion of the ontology of infinitesimals as not substantially different from that of Wallis (limit of a variable quantity, going to zero, from the right hand side) which presumably inspired it, and is based on right hand limits of functions that decrease as their input increases. Merely he is not consistent across all his papers.

EDIT:

I am looking through one of my archive trying to find my translation of the important passages of Wallis's Arithmetica Infinitorum. From memory, Wallis wrote in one place the equivalent of

$\underset{x\rightarrow 0+}{\lim}x$

as what he meant by infinitesimal, in modern notation. This is the thickness of lines in his diagrams. Summing an infinity of such lines required to fill the figures gave the areas of the figures. But, according to Beeley, in his later communications to Leibniz, he writes they -- the lines -- have literally null thickness, he has decided. Leibniz disagrees. Leibniz's own idea in modern notation is closer to Wallis's first statement, and more precisely the above conception. Where the $x$ is input to a function (the infinitesimal), such that $dx = \frac{f(x)}{g(x)}\rightarrow 0+$ there where $x\rightarrow 0+$.

• Very interesting. Which paper are you referring to exactly? This construction sounds a bit like the "horn angle" but I would like to see the paper. As far as Wallis is concerned, it is interesting that Beeley has a paper (2008) pointing out differences between the conceptions of infinitesimal in Leibniz and Wallis. Dec 30, 2014 at 5:46
• Leibniz wrote: "Arithmetica infinitorum mea est pura, Wallisii figurata." This is cited in Beeley. The text is freely available in the Akademia edition, VII 3 A, pp. 61-110. Dec 30, 2014 at 6:02
• Just looked through P. Beeley's chapter, thanks for the recommendation: I didn't know that Leibniz corresponded with Wallis. Apparently Wallis changed his opinion on infinitesimals in the 1690's and argued with Leibnitz. I guess he didn't stick with his opinion in his first book, according to the letters cited by Beeley. Dec 30, 2014 at 6:10
• Good, maybe you can explain to me what they were arguing about, and which opinion Wallis changed from and which to. I couldn't really follow the argument in Beeley's presentation. Dec 30, 2014 at 6:12
• updated my answer. I need to look up my Leibniz references file to give proper citation to Leibniz's paper. Dec 30, 2014 at 7:25

The mainstream interpretation of infinitesimals is in the frame of "non-standard analysis". It gives a completely rigorous justification.

• Sasha, thanks for your comment. But how does this answer the question? Dec 29, 2014 at 9:00
• The issue is not so much rigor (see my other comments) as historical interpretation. Jan 13, 2016 at 9:21

I haven't read Ishiguro, so I'll just give my impressions of the history from a couple of other secondary sources:

Boyer, The History of the Calculus and its Conceptual Development, https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment

Blaszczyk, Katz, and Sherry, Ten Misconceptions from the History of Analysis and Their Debunking, http://arxiv.org/abs/1202.4153

The relevant part of Boyer is pp. 210-212. In Boyer's account, Leibniz is a ditherer who isn't quite sure how to interpret his own dx's. Sometimes he interprets them in a way that makes them sound like modern differential forms, but various difficulties always lead him back to talking about them as infinitesimals. I'm not sure how much I trust Boyer's interpretation, since he was writing before NSA, and his depiction of the history of calculus is a story in which the bad, evil infinitesimals are driven out by the good, honest limits.

Blaszczyk's misconception (or question, actually) #10 is "Is there continuity between Leibniz and Robinson?" This to me seems to swing a little too far in the opposite direction and makes out Leibniz to be overly prescient. However, they do make some very interesting points about close analogies between Leibniz and NSA.

My take on the whole thing is that Leibniz's attitude toward and understanding of his infinitesimals was probably analogous to Euclid's relationship with the parallel postulate. It was probably an uneasy relationship, and the definitive clarification wasn't to come until centuries later.

• BTW, katz is the author of the second source you cited, and is asking the question ;) Dec 29, 2014 at 21:00