Contemplation/Signs of the Apocalypse: What if Our Understanding Math is Broken?

“The most severe implications are philosophical. The result means that the rules we use to manipulate numbers cannot be assumed to represent the pure and perfect truth. Rather, they are something more akin to a scientific theory such as the “standard model” that particle physicists use to predict the workings of particles and forces: our best approximation to reality, well supported by experimental data, but at the same time manifestly incomplete and subject to continuous and possibly radical reappraisal as fresh information comes in.” ~Richard Elwes

That is the general question in the New Scientist article “To infinity and beyond: The struggle to save arithmetic,” by  Richard Elwes.  The basic contemplation in the article is that while in general we can safely assume that 1+1=2, and always will, there may be underlying factors in mathematics that cannot be so definitively stated, which, when taken in consideration of the big picture of math, creates a profound degree of uncertainty.

I am not all that great with math, I never have been.  There are times that I greatly regret this, especially considering my great interest in the sciences, which so often rely on at least a moderate degree of mathematical prowess.  As such the implications of a potentially broken arithmetic is somewhat lost on me in the sense that it may carry more meaning to a mathematician.  But from a purely philosophical standpoint there is certainly a value in the consideration of such issues in math.

In a sense the working of mathematics has long been considered a steady and solid stone on which all further scientific and logical approaches to understand the universe can rest.  The fact that 1+1=2 will always be true, as a number of mathematical equations can also be said to be true, creates a basis for all subsequent logical axioms.  Even as we extend outward, beyond purely testable sciences and directly mathematical logical statements, we can come back to the truth inherent in numbers and their interactions.  But if the matters in Elwes’ article are to be taken seriously then it would suggest that the steadiness and solidity of that stone called math may in fact be uncertain and illusory at best.

These are the tiny apocalypses that keep mathematicians, physicists, and philosophers awake at night.  Perhaps we exist in a reality where the very concept of certainty is little more than abstract wishful thinking.  There is a startling parallel in the idea of a broken arithmetic to much of what is inconclusive in the study of physics.  There is the way that Einstein’s lovely theory of relativity does not quite synch up with quantum mechanics.  There is the unknowable nature of physical states as presented in Heisenberg’s Uncertainty Principle.  There are all those questions about what exists in the empty spaces between an atomic nucleus and the random fluctuating electron cloud that surrounds it.  And even considering that famed physicist Freeman Dyson says “But mathematics and physics are both open systems with many uncertainties, and I do not see the uncertainties as being the same for both,” there is certainly a sense of parallels.

So what then are we to do?  On one hand we can keep striving in the face of the infinite, keep wondering and asking ourselves, where does it all fit together?  Or we can reject it, we can take the stance of Doron Zeilberger and state “Infinite mathematics is meaningless because it is abstract nonsense.”  Or maybe we strive for a comfortable accomidationalist middle ground, simply affirming that we don’t know that much, things might be uncertain, but they still seem to work out brilliantly.  As Mr. Elwes put it “The clocks won’t stop or apples cease to fall just because there are questions we cannot answer about numbers.”

I tend toward the latter of the three in a philosophical sense.  I am reserved to the fact that there will always be things that I do not know or understand but that all and all the world seems to work consistently and to a fairly apparent (if not perhaps illusory) sense of certainty.  Perhaps this is my absurdist mindset at its very core.  A realization that discerning the truth of all things is not a possibility, but persevering nonetheless.

Human kind has an amazing capacity for asking questions and nitpicking away at them for a long time (just consider how long people have been thinking about what the real nature of things are).   I don’t believe there is a human alive (now, in the past, or yet to come) who does not experience a few of their own tiny apocalypses, where the question is framed and yet there are no definitive answers to be found.  Perhaps this is the cursed reality we are bound too, regardless how ceaseless our curiosity may be, there will always be that next implication, that +1 to every assumed final number.  This is the infinite.  We are finite beings.  We are born, we live for a while, and then we die.  And regardless of how fantastic the hardware of our brains may be, there may be realities that they are not equipped to understand.

This might come across sounding a bit fatalist but honestly I like to think of it on the flip side.  Instead of our insatiable curiosity being a curse we  should take it as a blessing.  Our capacity to keep asking questions and our daring to imagine further and further answers provides our lives with a sense of excitement and just maybe creates what one might call purpose and meaning to brief existences.  I might never fully grok the inner workings of atomic particles or how large cardinals fuck with arithmetics, but the opportunity to think about them seems vastly superior to the possible alternative; not to think at all.

~ by Nathaniel on August 19, 2010.