The amazing applicability of mathematics to the real world has caused many mathematicians, philosophers, and physicists to pause throughout history. How can something as abstract and ideal as mathematics apply to the real world?

In 1960, Nobel physicist Eugene Wigner (1902-1995) (right) wrote an essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” It meanders widely through a multitude of observations. But he touches on several essential, surprising observations about the relationship between physics and mathematics:

1. That humans can engage in the process of mathematics and develop extensive mathematical ideas which are free from contradiction

2. That the same mathematics applies to physical theories

3. That the human mind can extract mathematical relationships from such a small sampling of data which correctly model the data

4. That when the same mathematics tells us something we did not expect about physics, we can often run an experiment and find that the mathematics was correct in its description of the physics

Observations 1 & 3 are actually more about the surprising abilities of humans than about mathematics. In fact, 1 & 3 are major reasons why most contributors to Mind Matters News do not consider humans to be mere machines. It’s not that humans are quantitatively better than machines (they often outperform us on specific tasks), but rather that we are qualitatively different from machines (there is an entire set of tasks that humans can do that machines are incapable of performing).

Observations 2 & 4 are about the relationship between mathematics and reality. It is best expressed by the conclusion of Wigner’s essay, which states, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”

In the latest issue of Communications of the Blyth Institute, Gordon Mullings presents his account of why that connection exists:

Abstract: The Nobel Prize winning Physicist, Eugene Wigner, famously posed a powerful challenge (1960) by asking why is mathematics so effective, especially in the physical sciences. It is possible that the reason for the effectiveness of mathematics is not because mathematics is in any way causative, but instead because mathematics studies the structure of logical possibility and constraint. When plugged into a possible world, mathematics gives us the tools to analyze the logically possible outcomes. Therefore, when a possible world that is expressed mathematically sufficiently aligns with reality, mathematics becomes effective at expressing relationships and outcomes.

According to Mullings, the reason for the association between mathematics and physics is not that the mathematics is causative but rather that mathematics studies the logical structure of possibility and constraint. As a result, if the mathematics successfully captures the possibilities and constraints in the real world, it will provide a tool for further analysis of potential possibilities.

Additionally, Mullings shows how the principles of mathematics spring automatically if the philosophy of being is used. He demonstrates that the very fact of a distinction between two possible worlds opens up the necessary toolkit for all of mathematics. Because of this, the deep connection between mathematics and physical reality should not be surprising, though it is indeed a gift for which we should be thankful.

Even more impressive is the ability of the human mind to understand these connections. For any given set of data, there is an infinite number of possible explanations. The fact that humans can reliably choose explanations whose descriptive power extends indefinitely through more and more observations is truly amazing.

Also, introducing another article from the current issue of Communications: Are divergent series really an “invention of the devil”? The real villain in the piece is horrendously non-specific concepts of infinity. But that can be fixed. It turns out that hyperreal numbers (i.e., infinities that obey algebraic rules) resolve many of the paradoxes that previously plagued conceptions of divergent series. It is now possible to assign specific values to them. (Jonathan Bartlett)