Number Line in Synthetic differential geometry

  • #1


Hello! I just start looking at SDG and I'm already having difficulties with a few concepts as expressed by A Kock as:
"We denote the line, with its commutative ring structure (relative to some fixed choice of 0 and 1) by the letter R"
"The geometric line can, as soon as one chooses two distinct points on it, be made into a commutative ring, with the two points as respectively 0 and 1"

1 - What is this choice of the points 0,1 for? Sure they are the two unit elements of the ring R, but are they used to generated the other elements in R ? because from these 0,1 it seems he constructs other elements of R as: 1+1, 1+1+1...
2 - But if this is the case, are the integers the only elements of R ?
then he "promotes" the ring structure of R as an algebra over the rationals because it's required for the element 1+1, 1+1+1 to have an inverse.
3 - so.. is because of this Q, given all we have is 0,1, that it become possible to include the rationals in R?
4 - what about the irrationals then? Is \pi included in R? if so how to obtain \pi from 0,1 and Q?

My general understanding is that the real numbers are include in R, which moreover, in order to include the nonzero and nilpotent elements, has to loose its field structure in favor of a ring structure (with the subring of the rational). In others words my four questions above seem all "bad questions" but I really don't understand his comment about what the points 0,1 are for, how they give a ring structure to the geometric line or if/how they are used to generate others elements in R. thanks much ;)
  • #2
1.) The choice of 0 defines geometrically the origin of the line as a point space and of course defines the additive identity. They don't themselves generate the elements. Your elements are already existent in the given point set on the line. They establish the scaling and center for the definition of the ring operations of + and * on those already existent points.

This is what he meant by the second sentence in the first paragraph:
This is a decisive structure on it, already known and considered
by Euclid, who assumes that his reader is able to move line segments
around in the plane (which gives addition), and who teaches his reader
how he, with ruler and compass, can construct the fourth proportional
of three line segments; taking one of these to be [0,1], this defines the
product of the two others, and thus the multiplication on the line.

One is not constructing the points analytically, one is taking the geometric points, all of them, and giving them algebraic structure.

2.) As your presumption in 1. is not the case...

3.) See 2.

4.) See 3.

Geometrically you presumably have the equivalent of Dedekind cuts in the geometric axioms. Some form of "any division of the line into two connected sets must have the boundary point on exactly one of those sets" or the equivalent.
  • #3
Nice, thanks much: all the points in the geometric line are contained in R. And so what about his comment in the foot note:
1.) why does he consider the elements written as 1+1, 1+1+1 for?
2.) are these elements in 1.) the only multiplicative invertible in R? what about the irrationals? are they contained in R and if so are they multiplicative invertible?
I'm trying to understand which elements are or are not multiplicative invertible in R, after all that's why R is a ring and not a field. The the basic concept is the set $$D=[x:x^2=0]$$
3.) Are those ##x\in D## the only elements of R which do not have an inverse?
I really would like to ask ##10^6## more questions... but I appreciate any comments.
  • #4
Looking further at the text, I would suggest you run away very fast!

It looks like he's trying to define something along the line of surreal numbers, and non-standard analysis. That is to say he's trying to give an axiomatic construction to infinitesimals. It would take me no few hours of deep study to see if he's being coherent and consistent. Possibly he's got a brilliant new way of approaching the subject but it is, IMNSHO not the place to start learning about differential geometry. You should get a solid foundation in traditional analysis, topology, differential geometry, and category theory before trying to tackle this speculative bit of mathematical work. It is definitively "off the mainstream".

I would certainly say that his "synthetic" construction is non-intuitive to say the least (note that intuitionistic and intuitive are not synonyms) and should be approached with utmost caution. I however am not at the forefront of mathematics research so you may also want to seek a second opinion from a proper mathematician.

That having been said, with regard to your questions I believe my first answer was inaccurate because he is deviating from the classic definitions. I think he is working in a category where the completeness axiom is neither specified nor negated but replaced with his additional axiom.

Note that if R' is a subring of R then R can be viewed as an algebra over the subring R'. His observation about the ring R being an algebra over the rationals is merely a statement that the rationals form a subring. I don't see how the statement is explanatory. He is, verifying with the 1, 1+1, ... observation and the construction of reciprocals in the ring that the rationals do indeed form a subring. But he is denying that the ring is a field by replacing the completeness axiom with his "local differential structure axiom" Axiom 1. This set D is his construction of a tangent space at 0, or space of differentials=infinitesimals. This will then be extended, I presume, to higher dimensional manifolds.

Mind you, I have some problems myself with the way we usually define the tangent structure to differential manifolds and there are some very sloppy definitions where difference quotients are glibly tossed around when the function arguments are geometric points on a manifold for which no addition has been defined and thus f(x+dx) is nonsense. But this work is addressing it in a way I do not find clarifying or helpful. That is my opinion and again I do suggest you seek other opinions as well.
  • Like
Likes jim mcnamara
  • #5
Oh yes, SDG is definitely not maintream but it does posses some very interesting features: intuitionistic logic, constructive proofs and of course those ##x\in D: x^2=0## which are: nihlipotent, non invertible and different from zero (if I got it this right). That's pretty much all there is in SDG, but then they (Kock, Lawvere, ...) are able to construct an entire new geometrical setting to do differential geometry. As you said, most of their research is done using category theory (= good luck to me) but category theory is not necessary to express the results of SDG. The book "Basic concepts of synthetic differential geometry" by R. Lavendhomme is the simplest introduction to SDG I found, where the tools of category theory are only considered in the last chapter.

I'm ok with the statement (R is an algebra over the rational Q) is the same as (R is a ring with Q as a subring), but still what Kock is referring to when using those 1,1+1,1+1+1 is very much unclear: if it's a way to verify the subring structure for the rationals in R, then what about the irrational numbers in R, don't they have an inverse in R? weird..
  • #6
I don't see the virtue in it. The D subset of nilpotents is just a Grassmann algebra axiomatically tacked on rather then the usual construction to constitute the tangent space. The point is to "synthetically" construct the tangent bundle but there's not really any new mathematics that I can see. One can be intuitionistic by recognizing the classic constructions as conceptual constructions and not "objective realities". I don't (as yet) see how this construction adds anything to the mathematics or to its application.

I believe that there is a more straightforward way to treat differentials than treating them as "infinitesimal" quantities. The modern interpretation as (finite) auxiliary linear variables (and for manifolds tangent space coordinates) works find for me and keeps things finite thus well defined.

All Kock et all are doing that I can see is taking the tangent vectors and axiomatically embedding them within the manifold itself. There's still the 1 to 1 mapping between their construction and the traditional one and one still has to do the same calculations to solve e.g. Einstein's equations.
  • #7
hilbert created a technique called segment arithmetic, for going from a geometric line to an algebraic field whose elements correspond to points of that line. It is detailed in the book of hartshorne, geometry: euclid and beyond. essentially a pair of segments represents a (positive) number. hence choosing a single segment, and fixing it, means then every segment corresponds to a number. moreover choosing the segment with an ordering on its endpoints allows one to distinguish positive from negative numbers. hence all one has to do to start the process is choose a zero and a unit point.
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