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I am trying to understand the following introductory passage in an early lecture by the philosopher/mathematician Gottlob Frege because I am interested in how Frege conceived of the role of geometric intuition in mathematical reasoning.

One of the most far-reaching advances made by analytic geometry in more recent times is that it regards not only points but also other forms (e.g., straight lines, planes, spheres) as elements of space and determines them b means of coordinates. In this way we arrive at geometries of more than three dimensions without leaving the firm ground of intuition. The geometry of straight lines for example is a four dimensional one, and so is the geometry of spheres. But there is a difference between the two, in that a sphere can always be determined unequivocally by four numbers, whereas it would seem that this is not possible in the case of straight lines. we determine a straight line by an equation between six quantities with a quadratic equation holding between them. We express this peculiarity of the geometry of straight lines by calling it a second order one, whereas the geometry of spheres is of the first order.

The geometry of pairs of points in the plane, with which we will here be concerned, is four-dimensional and of the third order.

How are we to understand "dimension" and "order" in this passage? I guessed that the dimension is the number of coordinates required (six for lines) minus the order of the equation required to hold between the coordinates (two for lines). To get a better grip on the basic idea I worked the algebra on the equations for lines in three dimensional space.

$x = x_{0} + at$

$y = y_{0} + bt$

$z = z_{0} + ct$

So that:

$t = \frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z - z_{0}}{c}$

And:

$abct^{3} = xyz - xyz_{0} - xy_{0}z + xy_{0}z_{0} - x_{0}yz + x_{0}yz_{0} + x_{0}y_{0}z - x_{0}y_{0}z_{0}$

Substituting the original equations into that, and doing a bunch of algebra, gives a quadratic in t with coefficients expressed in $x_{0}, y_{0}, z_{0}, a, b, c$. So I'm trying to make sense of Frege's description of the geometry of lines is dimension four and order two by thinking of the dimension as the number of coordinates minus the order of the equation required for those coordinates to characterize a line. But, I can't quite make that fit with what he says about the geometry of spheres.

Frege's early work in geometry is influenced by the analytic projective geometry in the style of Plucker, using homogeneous coordinates. I've consulted some internet sources and used the undergraduate textbook "Modern Geometries" to better understand homogeneous coordinates, but I have been unsuccessful finding definitions of dimension and order for this context. What does Frege mean by dimension and order in the above quotation?

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I guess that the modern transalation of the example of lines is the following: the Grassmannian $\mathbb{G}(1,3)$ of lines in $\mathbb{P}^3$ can be described by a single equation of degree $2$ in $6$ homogeneous parameters. In other words, it is a quadric hypersurface in $\mathbb{P}^5$; hence Frege is simply talking about Plucker embedding. The second example must be something similar related to $\textrm{Sym}^2({\mathbb{P}^2})$, but I do not see it now. –  Francesco Polizzi Nov 22 '11 at 21:59
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May I ask which words Frege originally used for "dimension" and "order" (in German)? That can sometimes clarify the meaning. Also, is there a link or reference where I can read the whole passage (in German)? –  Giuseppe Nov 22 '11 at 22:24
    
So I don't quite have the right idea on the first example. That helps though. Thanks. –  Jeremy Shipley Nov 22 '11 at 22:34
    
Giuseppe - That is a good question. I just have an English translation ready to hand. It might be clearer in the original German. The translation "we determine a straight line by an equation between six quantities with a quadratic equation holding between them" is likely a bit muddled. I think that Francesco's suggestion will help me to reconstruct what he has in mind for the first example though. –  Jeremy Shipley Nov 22 '11 at 22:39
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It's the lecture ""Geometrie der Punktpaare in der Ebene", and can be found on Google Books at books.google.com/books?id=nbcUAAAAYAAJ, page 98. The words for "dimension" and "order" are Dimension and Ordnung. The statement about the quadratic relation is "wir bestimmen den Strahl durch eine Gleichung zwischen 6 Grössen zwischen denen eine quadratische Gleichung besteht," for which Jeremy's translation seems fine. –  macbeth Nov 22 '11 at 23:07

1 Answer 1

up vote 5 down vote accepted

As Francesco Polizzi remarks, the idea is to identify $\text{Sym}^2(\mathbb{P}^2)$ with a cubic hypersurface in $\mathbb{P}^5$. Frege's lecture (see link in comments) actually goes on to explain how this is done. The cubic hypersurface is (in the six variables $s_1,s_2,s_3,t_1,t_2,t_3$),

$\text{det} \begin{pmatrix} s_1 & t_3 & t_2 \cr t_3 & s_2 & t_1 \cr t_2 & t_1 & s_3 \end{pmatrix}=0$

and the identification of $\text{Sym}^2(\mathbb{P}^2)$ with this hypersurface is

$\{[x_1:x_2:x_3],[y_1:y_2:y_3]\}\mapsto [x_1y_1:x_2y_2:x_3y_3:\frac{1}{2}(x_2y_3+x_3y_2):\frac{1}{2}(x_3y_1+x_1y_3):\frac{1}{2}(x_1y_2+x_2y_1)]$.

Quite possibly, someone more knowledgeable than me can motivate this!

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I guess this must be related to the variety $\textrm{Sec}(V_{2,2})$ of secant lines ("chordal variety") to the Veronese surface $V_{2,2}$ in $\mathbb{P}^5$. Indeed, the Veronese surface is $1$-defective and its secant variety is a cubic hypersurface in $\mathbb{P}^5$. Note that any secant line to the Veronese surface intersects it in two points (possibly coincident) so $\textrm{Sec}(V_{2,2})$ must have something to do with $\textrm{Sym}^2(\mathbb{P}^2)$. –  Francesco Polizzi Nov 22 '11 at 23:59
    
Indeed, the determinantal equation you wrote is exactly the equation of $\textrm{Sec}(V_{2,2})$, see [Harris, Algebraic Geometry, p. 145]. –  Francesco Polizzi Nov 23 '11 at 0:03
    
Thanks, Francesco, just what I wanted! –  macbeth Nov 23 '11 at 1:08
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Sorry, I should have been more explicit. This answer is addressing why the geometry of pairs of points in the plane is "third-order" and "four-dimensional": it's because the set of pairs of points in the plane (i.e., $\text{Sym}^2(\mathbb{P}^2)$) can be described as a degree-3 hypersurface in projective 5(=4+1)-space. Similarly, Francesco Polizzi's comment describes the set of lines in 3-space (i.e., $\mathbb{G}(1,3)$) as a degree-2 hypersurface in projective 5-space -- hence "second-order" and "four-dimensional". [To be continued.] –  macbeth Nov 24 '11 at 3:29
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As for the "geometry of circles", a circle in 3-space is determined by four quantities without any relation between them (its radius and the three co-ordinates of its centre.) But four quantities without any relation between them is "the same as" five quantities with a linear relation between them, i.e. a degree-1 hypersurface in projective 5-space. Hence we have a consistent interpretation of Frege's notion of a "geometry's" "order". –  macbeth Nov 24 '11 at 3:33

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