Perhaps the most prevalent area of inexactitude in science is in its use of mathematical modeling.

Mathematics is not a science per se. It is a very useful related discipline that provides scientists and engineers with extremely useful tools—statistics being just one of them. Nevertheless, the point needs to be understood at the outset that mathematics does not and cannot prove anything in relation to reality.

Albert Einstein said, famously; 

“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” 

He got it exactly right. There is only one area where mathematics aligns almost completely with reality. That is in the act of counting. You might argue philosophically that, say, “three apples” in the real world only embody the concept of “threeness,” but it is splitting hairs to distinguish that from their embodying the concept of “appleness.” When there are three apples, there are three apples. The counting numbers—the natural numbers, as mathematicians call them, can reasonably be taken to denote a reality of the real world.

Beyond that, when we talk in terms of negative numbers, real numbers, irrational numbers, or complex numbers, we are manipulating abstract concepts that cannot be demonstrated to exist in reality. They are inventions of the mind of man that can, nevertheless, be put to good use to model reality. And that’s just for starters. We can introduce algebra, geometry, calculus and all the various fields of mathematics, most of which can also be put to excellent use in modeling reality.

Contemporary science and engineering employ mathematics to model reality and, time and again, the models turn out to be so close to reality that it predicts the real world accurately. In some instances, the very laws that science proposes can be expressed mathematically—as for example with Newton’s Laws of Motion.

Indeed, Newton serves as an excellent example of the productive use of mathematics, since his gravitational theory and its associated equations are pretty much all you need to spray space shots around our solar system. He formulated it more than 200 years before the first space shot.

And none of that proves Newton’s gravitational theory correct. In fact, nowadays his gravitational theory is regarded as incorrect and has been superseded by Einsteinian gravitation. The mathematics did not and never could prove the theory correct, but it created a very close-to-accurate model of reality.

The Map is not the Territory

There are several things to be concerned about with mathematics. The first is to note that in terms of the models it can create, mathematics can be divided roughly into two parts: discrete mathematics and continuous mathematics. 

Discrete mathematics is the study of mathematical structures made up of separate components, individual items, like quanta. So the objects studied in discrete mathematics, such as integers and statements in logic, have distinct, separated values. 

In contrast, continuous mathematics deals with objects that vary smoothly, without gaps. An example of this continuity is the simple equation y = 2x. The two linked variables, y and x, in the equation are continuous.

We can ask the question: Is reality continuous or discrete? 

The evidence from quantum mechanics is that reality is discrete. So, for example, a spectrometer viewing the light emissions form a particular substance shows us spectral lines rather than a continuity of wavelengths. This, incidentally, accords with the objective science view of reality. However the way our minds model the world is necessarily that way. We see, for example, an iron bar. It is a thing and hence discrete. However we see it and conceive of it as continuous from end to end. We can imagine that it is composed of atoms—discrete things—but we do not know for sure. Even if we possessed a microscope that was powerful enough, and could clearly see the atoms, we could not know for sure whether the space between them was really empty rather than containing some kind of material or energy.

While it is the case that discrete and continuous mathematics can be used together in some contexts, a mathematical model of a real situation always depicts the world as one or the other, discrete or continuous. If the models work well, it will be valued. For example, the mathematical models that were used to calculate artillery tables were valued by the military because there were very accurate within practical parameters, but they were not perfect. 

We adopt the same attitude to the problem of infinity—a concept that emerges in both continuous and discrete mathematics. We require the concept, for example, to establish calculus. We cannot demonstrate infinity in the real world, we can only presume it. Nevertheless, if we’re careful in using it, we can employ it productively in mathematics, and employ the mathematical models we create productively. Mathematics can be right within its own context of modeling and mapping. However the map is not the territory. 


Mathematically, extrapolation is where you extend a method (say a formula or a technique) outside the range of proven real world applicability, and logically deduce that it applies to all areas outside the range. Mathematics even has a specific kind of proof (the inductive proof) that works by extrapolation. 

This is fine in the domain of mathematics, as it does not need or even care for real world confirmation. It is correct axiomatically and thus an inductive proof applies all the way to infinity. All mathematical extrapolation is valid for the mathematical map, but the map is not the territory.

As soon as you apply extrapolation in science you are on shaky ground. Consider, for example, an activity as fundamental to geology as estimating the age of rocks. Such ages are calculated on the basis of radioactive decay. For example, the element Uranium 238 decays to become Thorium, which in turn decays until it becomes Lead. There are many steps to this process. The geological dating of a specimen can be achieved by estimating the original content of Uranium 238 and all the elements and associated isotopes in the rock sample when the rock was formed. The rock’s age is deduced according to the quantities of those elements and isotopes. The known proven-in-the-laboratory pattern of decay of Uranium 238 is applied. This is an extrapolation.

The accuracy of the calculation suffers from three problems: 

1. The estimate of the original content of U238 and the elements and radioisotopes it decays into could be incorrect.

2. The rock could have been contaminated during its long life in a way that altered the ratios.

3. The normal (predictable) process of radioactive decay could have been accelerated or decelerated by unusual conditions some time during the lifetime of the rock. 

One example of an anomaly is sufficient to demonstrate this problem. Radioactive dating on recent (roughly 50-year-old) lava flows at Mt. Ngauruhoe, New Zealand, have yielded a rubidium-strontium “age” of 133 million years, a samarium-neodymium “age” of 197 million years, and a uranium-lead “age” of 3.9 billion years. In each case, the dating method gives a wildly incorrect result and, as you can see, they are not even close to agreeing among themselves.

But what is the geologist to do? There are no better methods for estimating the age of rocks. It may even be that some of these estimates are correct in some cases. However, there is good reason to doubt.

Cognitive Bias and Mathematical Manipulation

In formulating hypotheses and proposing scientific models of real-world events, scientists almost always encounter the fact that their carefully designed experiments do not produce the hoped-for result, but may provide something that is close to the hoped-for result. In this area we encounter the problem of “cognitive bias.”

The term “cognitive bias” describes errors in thinking processes caused by holding on to individual preferences in the face of contrary evidence. This could be described as “unintentional dishonesty,” in that the individual affected by it is completely unaware of their bias. Where it crops up in scientific experimentation (outside of psychology, where it is an area of study), it is described as “confirmation bias.” It is the tendency to interpret experimental results in a way that confirms one’s cherished hypotheses or even pre-existing beliefs. In science pre-existing beliefs are often just fashionable theories.

The scientific method is supposed to eliminate such bias by the process of peer review. Other experts in the field review the published results produced by a specific scientist or scientific team and offer critiques. However, peer review is only effective if the reviewers are not also suffering from the same confirmation bias.

As we review some of the theories of modern physics in the coming pages, we will encounter the existence of “adjustable parameters.” We can explain by example:

Consider the trajectory of a ball thrown at an upward angle through the air. It will follow a parabolic curve almost exactly, rising in the air at first and then falling. It’s position in the air at any point will depend on the initial upward angle of its trajectory and the time elapsed since it was thrown. If there were no other forces affecting the ball it would move in a perfect parabola. However, the resistance of the air to the movement of the ball inevitably distorts the parabola. 

If we adjust the mathematical equation by adding an “adjustable parameter,” we can compensate for the air resistance. Adding a fixed parameter might do the trick, but air resistance can vary. It will be different at sea level than on a high mountain, and hence the parameter will need to be adjusted, for context.

This does not mean that the theory of the parabolic movement is incorrect, just that we need to adjust the model. The scientific problem is not that adjustable parameters are necessarily wrong—they may not be. But if you cannot assign a cause to the adjustable parameters in a model, then the model is clearly suspect. You can usually make inconvenient results look respectable by resorting to adjustable parameters.

Scarcity of data

Finally, we need to highlight the problem of data scarcity. In some areas of scientific study, there is insufficient data to offer strong support to any theory. If we take cosmology as an example, the field of study is handicapped because we can only make observations of the universe from the Earth or from satellites within the solar system. The accurate data that has been gathered is confined to a relatively short period of time—a few hundred years at best—less than the blink of an eye in the life of the universe.

Similarly, paleontology, the study of ancient life, is restricted to what can be discovered via the fossil record. Data is confined to the specific times when fossils were created. And the fossil record from any given era is only a minuscule snapshot of that time. This leaves huge gaps in the evidence for any theory in this field. 

Other areas of science are not so constricted. For example, with modern instruments, zoologists can examine both living and recently dead specimens of a species in fine detail and gather very large amounts of data to support or oppose any specific theory.