How is math different from science? Why the distinction matters for scientific progress

Math and science are strongly related disciplines, but they differ in inherently different ways.

Math and science have plenty in common, such as the pursuit of objective truth, the use of logic, the forbidding of contradiction, and posing questions followed by the search for supporting evidence. 

And yet math and science are inherently different in very key ways. 

In principle, all of mathematics can be developed by a human brain and sufficient time to think. The superstructure of mathematics is derived via logical deduction from a set of appropriately selected axioms. A mathematical theorem can be proven by the human mind alone—although a chalkboard does come in handy.

Science differs from math in that science is inherently empirical. Science is driven by the scientific method: generate a falsifiable hypothesis, test predictions from the hypothesis with appropriate experiments, observe the results and draw conclusions whether the hypothesis should be accepted, modified, or discarded. Thus, the scientist needs far more than a brain and thinking time. The scientist needs experimental equipment, measurement devices, computers, and observed data to ply his trade. He probes the universe with scientific tools to answer his questions, unlike the mathematician who can probe mathematical truth just by thinking about it. 

Therefore the approaches that mathematicians and scientists use are intrinsically different from each other. Using only math to prove a scientific theory is not going to get you anywhere, and using only empirical scientific tools to prove a mathematical theorem also just isn’t going to work.

You cannot use scientific methods to pose a mathematical axiom, or to prove a mathematical theorem. Take the most basic geometric axiom: the shortest distance between two points is a straight line. If you wanted to prove this scientifically, you would scribble two points on a flat surface, then draw a line between them. Except in the physical world, each of the two points has a finite diameter, and the line has a finite thickness. So you can start and end your line in a nearly infinite number of spots within each point. Also, the line has a finite thickness, and as long as the thickness grazes each point, then the line is a connection between the points. Therefore, a scientific experiment would demonstrate there are infinite lines that connect two points! In the abstract mathematical world, it is understood that the points are zero-dimensional, and the line thickness is zero, thus the axiom is demonstrated to be true by logic alone. 

Conversely, you cannot use mathematical approaches alone to get to a scientific conclusion. Deduction and logic can be used exclusively to construct mathematics. But it cannot be used exclusively to construct scientific theories. This is because logic and the mind alone is inadequate to describe our complex world. We need to look outside of ourselves to see how the universe ticks. Consider our ability to predict planetary orbits, or our understanding of how to construct a transistor, or our expertise at developing medical therapeutics. There is no purely mathematical approach that can get us these real-world results. Of course, scientists use math, and use it a lot; however, empirical evidence is the basis for making a scientific claim. What I am getting at is that mathematics alone is not enough for scientific proof. René Descartes is renowned as a mathematician having invented the Cartesian coordinate system and laid the foundations of calculus; however, his delving into metaphysical physics led to some wild tangents because many of his contentions were not properly backed up by empirical evidence. The long-term reason for Albert Einstein to not deserve the Physics Nobel Prize for his work on relativity is because at the time it was a purely theoretical work and not backed up by experimentation until 1919 when gravitational lensing of the sun was observed. Instead, he won the Prize for his work describing the photoelectric effect (though he really should have gotten a second Nobel once relativity was solidly verified by evidence). If a scientific claim is extrapolated from current knowledge via mathematics, it has to eventually be backed up by empirical evidence.

Why this matters

In my work as a physicist I cannot make any conclusions without empirical evidence. In order to do science, I need to follow the scientific method. This means getting outside of my limited brain and measuring stuff in the outside world to try and validate hypotheses. Empirical evidence is king.

We have to be careful that we do not substitute empirical evidence with mathematical evidence. An example of this is the math performed by computers as governed by a computer model, with the output claimed to be scientific evidence. Another example is using mathematical formulas to theoretically extrapolate from known physical observations to some far out space that is unprovable by empirical evidence.

It is very tempting to say, “theoretically things should be this way, based on my logical and mathematical extrapolations from what we know about the world”, modeling that in a computer or an analytical equation and then calling it a day. We humans want to be able to know the world, which is one of our greatest strengths; however, sometimes there is overreach when we try to use our limited brainpower to extrapolate from known physical conditions with math alone. 

One important distinction is where a theoretical scientist develops a hypothesis with mathematics, and then later (even much later) it is proven via experiment. This was the case when Peter Higgs and others theorized about the existence of the Higgs Boson in the 1960’s. The particle was finally detected by CERN decades later in 2012 with the Large Hadron Collider, earning Dr. Higgs along with François Englert the 2013 Nobel Prize for Physics. Here, mathematics and the known physics in the 1960’s was used to generate the hypothesis, with physical experimentation providing the scientific evidence later on. The mathematical extrapolation from real world observations was testable and was eventually tested.

What we need to watch out for is the overreaching extrapolation from known physical conditions using mathematics, computer models, and blackboard equations and claiming that this is scientific evidence in and of itself. Empirical evidence is the standard by which scientific claims are judged.