DEE WILSON CONSULTING
Book Review:
Mathematical truths, human or stand-alone
Pi in the Sky; Counting, Thinking and Being
John Barrow, 1992
I've been reading Pi in the Sky; Counting, Thinking and Being (1992) by John Barrow, which is an entertaining discussion of the history and philosophy of mathematics. At several points in the book, Barrow returns to the question of whether mathematical truths exist independent of human minds and are discovered, or are a set of logical rules invented by humans, which turn out to have surprising properties and consequences. Great mathematicians and logicians have held strongly felt opposing views regarding this issue. One of the strengths of Barrow's book is that he clearly sets forth the opposing views and arguments for both positions, before outlining the most interesting version of Platonism I've encountered. Before describing Barrow's conjectures, I should say that I do not believe that this version of Platonism is true, probably true or hopefully true, but it is a fascinating example of how a philosophical debate that a few people care about passionately, while most people (including mathematicians and physicists), could care less can be transformed by reversing perspective.
The following, titled "A Computer Ontological Argument," is contained in pages 280-284 in Pi in the Sky.
Barrow begins with a quote from William Mace: "Ask not what's inside your head but what your head's inside of."
Barrow imagines a computer simulation of galaxy formation. "We program the computer with the laws of Nature that govern the way gravity acts between particles of matter and how gas and dust respond to gravitational forces, how heating and cooling take place, and so forth. Then we follow the computations of the computer as it provides us with an unfolding history of events which would develop from a particular, perhaps random, starting distribution of particles." Eventually, assuming the computer program contains a complete set of Nature's laws (that have a mathematical structure), the world will come to contain planets, organic molecules, living organisms, conscious biological humans who create myth, theology, science, etc.
Barrow states: "Let us now examine this state of affairs a little more closely. The computer simulation is just a means of carrying out very long sequences of intricate mathematical deductions according to particular rules commencing from a given starting state." Though Barrow does not engage in this speculation, large numbers - or an infinite number- of computer simulations could be generated simultaneously, simulations which have slightly different starting conditions or different mathematical rules. Barrow continues: "if we remove the prop of the computer hardware .. the entire sequence of events that unfold in the simulation ( i.e., stars, planets, etc.) are all just mathematical states.... So if we think of the simulation as at web of mathematical deductions spanning out from the starting state ... we will eventually come across the mathematical structures that correspond to self conscious beings. ... This means that we can think of the mathematical formalism as containing self conscious states -- 'minds' -- within it. This speculative line of reasoning turns the Platonic position inside out. ... We exist in the Platonic realm itself. We are mathematical blueprints."
Barrow asserts: "This approach has all sorts of interesting ramifications. It means that anything that can happen - anything that is a possible consistent statement in the language of mathematics -- does happen in every possible sense of the word."
Barrow comments that the consequence of this world view is that understanding the physical world (or perhaps the world itself) may be a means of resolving seemingly impossible mathematical problems such as in set theory. " ... it means that the past emphasis on applying mathematics to sciences like physics is reversed. We should study the physical world in order to determine the nature of the most basic mathematical structures" because "these powerful abstract structures seem to be manifested in the unobservable part of physical theory, that is in the laws of Nature ..." Of even greater interest is the implication for cosmology, i.e., that the cosmos did not originate in the Big Bang, but in the mathematical structures found in the laws of Nature. How could such structures be created, and what would it mean to assert that they exist independent of a physical world?
-- Dee Wilson