Phil 105: Science and Philosophy - Fall 2004 Lectures

!You will be sorely disappointed if you think that these are a substitute for attending class!

Topic Two. The Shape of Space

1. Experiments:

• Suppose that we find many parallels, and 'long' diagonals – what should we conclude about the geometry of space?

• Is there another explanation?
  

2. Poincaré's Space

• Suppose space is Euclidean, with a finite radius – like Aristotle's space – but that everything shrinks by the same factor as it approaches the edge: as before, something 0 ≤ 1/n < 1 of the way from the centre to the edge, is 1- 1/n² of its length at the centre.

1/n	1-1/n2	My height	My stride
0	1	1.8m	1m
1/10	99/100	1.78m	99cm
5/10	3/4	1.35m	75cm
9/10	19/100	34cm	19cm
99/100	199/10,000	3.6cm	1.99mm

• There's a very nice applet, NonEuclid, which allows you to play around with such a (2D) space. (Note that NonEuclid is restricted by the resolution of the computer screen: you can't get closer to the edge than a single pixel, so we can't consider rulers that are as close to the edge as we like.You'll have to imagine things shrinking smaller than a pixel.)
  

• Suppose we use objects, say .2m rulers, in the disk to measure distances:
• How big is the disk measured to be? i.e., how many rulers can be lined up end-to-end from centre in a straight line?
  
• What will the lines measured to be shortest – i.e., measured to be straight – look like?
• How many measured straight lines will there be through a point parallel to a given line?
• What will the length of the diagonal of a triangle (with measured straight sides) be?
• What will the geometry of the space be measured to be?

• So if experiment tells us that there are many parallels, or that diagonals are 'long', what possible conclusions about the geometry of space can we draw?