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

• By what factor do the following change if lengths are increased by a factor of n?
• The length of a line?
• The area of a 2D surface (e.g., a square or the surface of a sphere)?
• The volume of a solid (e.g., a cube or sphere)?
• The 4D 'hypervolume' of a 'hypersolid' (e.g., a hypercube or hypersphere): see http://www.math.union.edu/~dpvc/math/4D/welcome.html

As the (2D) surface of a (3D) cube can be unfolded (on the left) so the (3D) 'surface' of a (4D) hypercube can be unfolded (on the right). To form the surface the faces must be joined to enclose a region of 4D space!

• Imagine a fixed quantity of stuff – 1Kg of gold, say – spread evenly through each of the following. By what factor does its density change if lengths are increased by a factor of n if it is spread through ...
• ... a line?
• ... a plane figure (e.g., a square or surface of a sphere)?
• ... a solid (e.g., a cube or sphere)?
• The 4D 'hypervolume' of a 'hypersolid' (e.g., a hypercube or hypersphere)?

• A hypersphere in N-dimensional space is the collection of all points a fixed distance from the center of the hypersphere – it is (N-1)-dimensional.
• In 1D space two (0D) points
  
• In 2D space a 1D circle
• In 3D the 2D surface of a sphere
• In 4D the 3D 'surface' of a hypersphere
• Etc

• Imagine a fixed quantity of stuff in N dimensions expanding evenly from a point, forming a series of concentric hyperspheres:
• the hyperspheres have N-1 dimensions
  
• so, if the distance from the center increases by n times 
• then the hypersphere gets n^N-1 time bigger
• and the density gets 1/n^N-1 times smaller – density is proportional 1/(distance)^N-1

• So if gravity is like a quantity of stuff – 'attractive power' – being emitted from a body at a constant rate, then we should expect the force of gravity to be proportional to 1/(distance)^N-1, where N is the number of dimensions of space. 
• That the force of gravity is proportional to 1/r² is evidence that our space is 3-dimensional.

2. Why 3 Dimensions?

• Fact: planetary orbits are unstable if gravity is proportional to 1/r^M if M>2.
• What's the greatest number of dimensions in a universe with stable planetary systems?
• What's the greatest number of dimensions in a universe in which intelligent life evolves?

• Why might intelligent life be impossible in 2D?

• Anthropic explanation of the 3-dimensionality of space: only 3D space can support intelligent life, space is 3D because intelligent life exists, because there are creatures who can ask 'why 3Ds?'
• But what question does this actually answer?