Wisdom Of The Week

Here are a few of Abbott's reasons for why mathematics is reasonably ineffective, which are largely based on the non-Platonist viewpoint that math is a human invention:

• Mathematics appears to be successful because we cherry-pick the problems for which we have found a way to apply mathematics. There have likely been millions of failed mathematical models, but nobody pays attention to them. ("A genius," Abbott writes, "is merely one who has a great idea, but has the common sense to keep quiet about his other thousand insane thoughts.")
• Our application of mathematics changes at different scales. For example, in the 1970s when transistor lengths were on the order of micrometers, engineers could describe transistor behavior using elegant equations. Today's submicrometer transistors involve complicated effects that the earlier models neglected, so engineers have turned to computer simulation software to model smaller transistors. A more effective formula would describe transistors at all scales, but such a compact formula does not exist.
• Although our models appear to apply to all timescales, we perhaps create descriptions biased by the length of our human lifespans. For example, we see the Sun as an energy source for our planet, but if the human lifespan were as long as the universe, perhaps the Sun would appear to be a short-lived fluctuation that rapidly brings our planet into thermal equilibrium with itself as it "blasts" into a red giant. From this perspective, the Earth is not extracting useful net energy from the Sun.
• Even counting has its limits. When counting bananas, for example, at some point the number of bananas will be so large that the gravitational pull of all the bananas draws them into a black hole. At some point, we can no longer rely on numbers to count.
• And what about the concept of integers in the first place? That is, where does one banana end and the next begin? While we think we know visually, we do not have a formal mathematical definition. To take this to its logical extreme, if humans were not solid but gaseous and lived in the clouds, counting discrete objects would not be so obvious. Thus axioms based on the notion of simple counting are not innate to our universe, but are a human construct. There is then no guarantee that the mathematical descriptions we create will be universally applicable.

For Abbott, these points and many others that he makes in his paper show that mathematics is not a miraculous discovery that fits reality with incomprehensible regularity. In the end, mathematics is a human invention that is useful, limited, and works about as well as expected.

For those who seek something more practical out of such a discussion, Abbott explains that this understanding can allow for greater freedom of thought. One example is an improvement of vector operations. The current method involves dot and cross products, "a rather clunky" tool that does not generalize to higher dimensions. Lately there has been a renewed interest in an alternative approach called geometric algebra, which overcomes many of the limitations of dot and cross products and can be extended to higher dimensions.

- Is mathematics an effective way to describe the world?