Impossibility & Humans,
Lead: Amy Klar & Paul Eichorn
Summary by Mark Knights:
The discussion was led by Amy Klar and Paul Eichorn and focused on the different types of impossibilities that John Barrow discusses in his book. After reviewing these impossibilities, we discussed different schools of thought regarding mathematics and the question of whether math was invented or discovered by humans.
The first type of impossibility that Amy and Paul discussed was mathematical impossibility. The basic idea of mathematical impossibility is summed up in Goedel's theorem, which is based on logical paradoxes such as "This statement is a lie." Essentially, what this says is that any mathematical theory is almost closed, but will never be closed completely. One must always start with some axioms that cannot be proven, but must be accepted as true (unconditionally). This applies to all of our logic as well as our mathematics. An example of this is Euclidean geometry, where, for instance, the idea that 2 parallel lines will never meet is an unconditionally true axiom. Mathematical impossibility has important implications for science as well, because science is based on math.
The second kind of impossibility discussed was Practical Impossibility. These are things that are possible, in principle, but not in any reasonable amount of time. For instance, there are equations that could potentially be solved but which would take longer to solve than the universe will exist. We then discussed Philosophical Impossibility, which is the idea that you cannot definitely know what the experience of another person is like.
Schools of Mathematical Thought
Amy and Paul then described what the two differing schools of thought on mathematics are. On one hand, there are the Formalists, who believe that we constantly experience mathematics, that math permeates our world, and if we can figure the math out, we can predict the outcome of events. In essence, they believe that math is natural and is built into our world.
On the other hand are the Platonists, who believe that mathematics is separate from what we experience, and lies in some sort of ideal or pure realm. They say that nature isn't connected to mathematics because of things like chaos and impossibilities.
This sparked a discussion by the group about the true nature of mathematics: is it inherent in nature or has it been invented by humans? We discussed the fact that a number of different civilizations developed pyramids at around the same time, independently of one another. The question was posed as to whether this type of complicated structure was invented by humans (coincidentally, in many areas of the globe at the same time) or if people had somehow derived it from nature.
Some suggestions, about what the explanations could be, were that the shape of mountains had inspired the structures, or that our brains are structured such that ideas of math (such as the pyramid shape) are built into them. Perhaps an advanced alien civilization showed humans how to build pyramids! We also wondered if perhaps alien beings' minds are structured so differently from ours that they would have different ideas of mathematics.
April 24, 2000