“Mathematics, a veritable sorcerer in our computerized society, while assisting the trier of fact in the search for truth, must not cast a spell over him.”
– California Supreme Court, overturning a criminal conviction based entirely on fraudulent use of statistics.
Let’s start with an analogy.
Question 1. Can you safely drive a car without understanding any mathematics of vehicle dynamics ?
A: Of course. Every day Hundreds of millions of people drive cars in full control and safety — without understanding any math.
Obviously they learned vehicle dynamics physics using a “language” other than math.
Imagine if your government refused to let you drive — until you could pass tests about the specific mathematics of vehicle dynamics : mass, motion, moments of inertia, centrifugal and centripetal forces, Hooke’s law, geometry, friction, nonlinear differential equations, understeer and oversteer.
While a lot of mathematicians might be pleased . . . the general public would be outraged.
Question 2. Can you repeatably mix an alcoholic drink without understanding the mathematics of fluid flow, boundary layers, Navier–Stokes equations, turbulent flow and mixing ?
A: Of course. Millions of people easily mix drinks without understanding any math.
Question 3. Can you repeatably bake a loaf of bread without any understanding of the mathematics of thermodynamics, Joule–Thomson effect and gas expansion ?
A: Of course. You can mix and bake a loaf of bread without understanding any of the higher maths that could be used to describe some of the dynamics involved.
You get the idea. It’s simply silly to assert that anyone needs to understand any math to understand principles of motion, dynamics, fluid flow, thermodynamics or cause and effect.
The same holds true for Cosmology. As a Cosmology expert who’s written one of the the largest glossaries on the subject and taught the principles to many non-scientists, it is my opinion that most people can understand most, if not all, of Physical Cosmology’s principles of motion, dynamics, fluid flow, thermodynamics and so on — without understanding any math.
But too many math advocates in charge of setting University physics curriculum currently refuse to let you study physical cosmology principles unless you have an enormous higher math background !
As an example — please take a moment to examine this seemingly reasonable list of pre-requisites to start studying Physics (that’s before adding math fields specific to astrophysics and cosmology) prepared by Nobel Laureate Gerard ‘t Hooft.
His excellent starting point list includes nineteen major categories (19 categories) of subjects and dozens of subtopics. Every one of them is a fascinating facet in understanding our fabulous fields of physics.
(Though it laudably includes a course in the English language, alarmingly it lacks any hint of study of scientific method, logic and reasoning, or fallacies. That omission is sadly typical in physics degrees.)
Innumeracy — pages of equations overwhelm most students.
The problem is every one of Gerard ‘t Hooft’s subjects and links are drenched in mathematics with very few photographs, diagrams, or graphs or even just charts; just page after page after page of equations.
Some people thrive on equations and even learn fastest using them. However, most people do not.
This avalanche of math overwhelms, intimidates and discourages huge numbers of people who were otherwise fascinated by astrophysics or cosmology.
In the words of Harvard’s esteemed E.O Wilson —
“During my decades of teaching biology at Harvard, I watched sadly as bright undergraduates turned away from the possibility of a scientific career, fearing that, without strong math skills, they would fail. This mistaken assumption has deprived science of an immeasurable amount of sorely needed talent. It has created a hemorrhage of brain power we need to stanch.”
Thankfully, there are occasional minor counterexamples like this undergraduate Cosmology course at Northwestern in Illinois. But as mentioned, that is an undergraduate course that only skims the surface of cosmology.
Sadly, this gratuitous math barrier prevents millions of students from enjoying physics, astrophysics and cosmology. This makes the math requirement a de facto example of a Priesthood (even if unintentional or unconsciously).
When this is done deliberately it is called Obscurantism. That is where subject knowledge is a fiercely guarded secret revealed only to those who jump past barriers that are wholly unnecessary and often highly complex or expensive in time or personal sacrifice. This is not unlike the outrageously harmful and sometimes deadly college Hazing required to obtain fraternity admission; or how the legal industry uses Latin terms to obscure legal ideas from us ordinary mortals.
Oddly, its often the non-math fluent administrators who enforce such math barriers. The best Physics professors typically take great pains to increase the approachability of physics to maintain the student’s enthusiasm.
Faraday established the basis for the concept of the electromagnetic field in physics, inspired Maxwell’s equations and essentially invented electric motors. He also discovered the principles of electromagnetic induction, diamagnetism (the levitating frog trick), and the laws of electrolysis.
A quick web search amazingly turns up a bunch of logical fallacies arguing that you are forbidden from discussing or understanding cosmology until you have expertise in half a dozen higher maths. But in perusing the arguments, I have yet to find even one valid (fallacy-free) argument supporting that view. Here’s a concise analysis of a few of them.
Myth 1: “Physics, by definition, is the subset of Mathematics which pertains to our universe.”
Truth: Physics is not a subset of mathematics, nor vice versa.
Physics is about understanding physical reality using experiments and observations.
Math is merely a language, one of many languages to help understand natural phenomena. Math helps some people understand the physical laws of nature. However, in no sense is math an experimental science. Many would argue that Math is not a science at all.
Myth 2: “Mathematics is the language of physics and you can’t learn a subject without learning its langauage (sic).”
Even Nobel Prizewinner Richard Feynman subscribed to this myth by writing :
“To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in“.
Truth: Sorry Richard, I must most respectfully disagree. Mathematics is only one language of physics; one of several ways to understand physics principles. Just like one can learn to drive a car without any vehicle dynamics math, or follow a recipe there are other ways to understand physics and cosmology; other “languages.”
Myth 3: “As long as you don’t have math (and hence can’t make quantitative descriptions and predictions), you are not doing physics – you are merely making up stories.”
Truth: This is a two part strongly misleading claim. There is a gigantic gulf between using advanced math (e.g. Calculus or Algebra) and using quantitative descriptions.They are two separate things with nearly zero overlap.
Of course, using quantitative descriptions is highly helpful in assisting understanding physics and cosmology. But measuring things does not inherently require using math beyond arithmetic. Just like driving a car — one should understand that (not how) vehicle speed is quantified by speedometer and understand how to compare that measurement to speed limit signs.
You could get me to agree that this example uses Arithmetic, but comparing one quantity to another quantity is not really using math. In any case using quantitative descriptions is certainly not higher math.
Myth 4: “If you don’t have a mathematical description of something, you don’t really understand it.”
Truth: One can understand physics phenomena, even extremely complex multi-variate, non-linear phenomena at a world class level without using any mathematical description.
Here’s an example: Returning to the car driving and vehicle dynamics analogy — at most, only a handful of world class racing car drivers understand any math of vehicle dynamics that their lives and careers depend upon.Yet they are paid tens of millions of dollars due to their exquisite and precise command of vehicle dynamics that most
people mathematicians can only dream about and admire. (aka the Fifth stage of competence.)
While traveling at 140 to 180 mph (225-290 kph) and more, while turning so sharply their body is subjected to 2 to 3 sideways Gs (2 to 3 times their normal weight pressed against their “seat’s” side) they can place a car to within an inch or so of the same spot on a sharply curved road lap after lap after lap.
These drivers unarguably “really understand” vehicle dynamics – while few have any more than the faintest whiff of mathematical understanding of it. They absolutely fulfill Prof Feynman’s poetic appeal they they have “a real feeling as to the beauty, the deepest beauty, of nature” for automobile dynamics.
If you push me, I could easily show persuasively these drivers understand the physics far better than mathematicians. This is in no small part because math comes to a grinding halt with non-linear six degrees of freedom dynamics – which you and I employ to drive a car every day.
And if you persist I’ll insist we discuss Gödel’s incompleteness theorems – which prove that for all but the most trivial circumstances maths cannot be relied upon to be self-consistent. That would clearly include the General Relativity Field equations which Big Bang is based upon . . .
To resume, my experience has shown me that for Cosmology enthusiasts it is wholly unnecessary to have any advanced math expertise to understand Cosmology principles.
I’ll close with a thought from Professor Einstein —
“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.“
What do you think ?
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1. Here’s an illuminating Youtube video with Physics Noble Laureate Richard Feynman explaining the difference between math and physics.
2. To some mathematicians merely asking this article’s title question is blasphemy (Religious irreverence). The mere allusion that science could have blasphemy makes me queasy. Heresy is OK (differing opinions), but blasphemy – Youch! Carl Sagan provides a useful guide for this —
“There are no forbidden questions in science, no matters too sensitive or delicate to be probed, no sacred truths.”