# [FOM] Leibnizian principles

Robert Rynasiewicz ryno at lorentz.phl.jhu.edu
Fri Nov 14 21:05:00 EST 2014

```According to Leibniz, if we assume that space is something in itself, besides the mere order of bodies among themselves, then:

“L’Espace est quelque chose d’uniforme absolument; & sans les choses y placées, un point de l’Espace ne differe absolument en rien d’un autre point d’Espace.  [Space is Something absolutely Uniform; and with the Things placed in it, One Point of Space does not absolutely differ in any respect whatsoever from Another Point of Space.]”

(The quote is from Leibniz’s third paper in the Leibniz-Clarke correspondence, as originally published by Clarke in 1717, p. 58, available on Google Books.  The English is Clarke’s translation.)

Thus, according to Leibniz, if there really were such things as points and lines, the points on a line would count as indiscernibles.  This would lead to a violation of the principle of sufficient reason, since there would be no sufficient reason to place something at one point rather than another.  For Leibniz, there are no geometric objects, only geometric relations obtaining between material bodies.

Although Leibniz does not explicitly formulate it this way, the test for indiscernibles is the presence of symmetries: a and b are indiscernible iff there exists a non-trivial automorphism f s.t. f(a)=b.

Robert Rynasiewicz

> On Fri 14.11.14, at 3:16 PM, Charlie <silver_1 at mindspring.com> wrote:
>
>
> 	   It’s always seemed to me that Leibniz had to have packed relations into his properties.   How else would one object having only properties, be able to reflect all properties about all other objects in the same world.   For L., everything is relational.  So, more or less as Blum says, I think it’s right that if point x is in position A, and there’s a point y on the same line to the left of x, they are different points, for I think Leibniz would consider it a “property” that
> “Point y is to the left of point x”.  I should add the qualification that I’m no Leibnizian scholar and that I’m sure I’ve oversimplified (or misinterpreted) much.
>
> On Nov 13, 2014, at 1:32 AM, Alex Blum <Alex.Blum at biu.ac.il> wrote:
>
>> A points position on a line is an essential property of a point despite it being relational, hence it must be counted a property.
>> -----Original Message-----
>> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of F.A. Muller
>> Sent: Wednesday, November 12, 2014 9:18 AM
>> To: <fom at cs.nyu.edu>
>> Subject: [FOM] Leibnizian principles
>>
>>> Message: 1
>>> Date: Tue, 11 Nov 2014 18:39:33 +1300
>>> From: W.Taylor at math.canterbury.ac.nz
>>> To: W.Taylor at math.canterbury.ac.nz
>>>
>>> If one doesn't wish to go to a hierarchy of predicate variables, and
>>> is prepared (largely ignoring 1st/2nd-order distinctions) to treat
>>> objects and properties on the same footing, then Leibnitz' law has an
>>> interesting "dual" form, which might be useful.
>>>
>>> IDENTITY OF INDISCERNABLES:    (Leibnitz)
>>>
>>> [all x,y]  x = y  <=>  [all P] Px <=> Py
>>
>> If true, a line has only one point, because all points share their properties.
>>
>>>
>>> IDENTITY OF INDISCRIMINABLES:      (dual)
>>>
>>> [all P,Q]  P = Q  <=>  [all x] Px <=> Qx
>>
>> If true, "having a unique successor" is identical to "having a unique prime decomposition" (when x is a natural number variable).
>>
>> --> F.A. Muller
>>
>> Sent from my iPhone
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