Is Math a Discovery or an Invention?

Thomas Kando 

(Note: the title of this article is similar to an article by Mario Liovo from which I quote extensively, below) 

I have been thinking about the relationship between the universe and mathematics. Humans have measured and given numbers to various objects. The earth’s circumference is 40,000 kilometers. The speed of light is 300,000 kilometers per second. When I first learned about the earth’s size in elementary school geography class, I thought “wow! How neat. How come the earth is exactly 40,000 kilometers in size? What a coincidence. Such a simple and memorable whole number.” 

Of course, I was putting the cart before the horse. I did not understand that “kilometer” is not an a priori characteristic of nature. The earth is (approximately) 40,000 kilometers in circumference because humans decided to use as their basic unit of length one forty millionth of the earth’s circumference, however long that is. They called it the meter, of which one thousand added together make a kilometer. 

The metre was originally defined in 1791 by the French National Assembly as one ten-millionth of the distance from the equator to the North Pole along a great circle, so the Earth's polar circumference is approximately 40,000. (metre) 

But it was later determined that its length was short by about 0.2 millimetres because of miscalculation of the flattening of the Earth, making the prototype about 0.02% shorter than the original proposed definition of the metre. Regardless, this length became the French standard and was adopted by most of the rest of the world. So the polar circumference of the Earth is actually 40,008 kilometres, instead of 40,000. (Earth’s Circumference) 

Then there is the speed of light: Light travels at (nearly) 300,000 kilometers per second (in a vacuum). How convenient! I thought as a youngster. Like earth’s circumference, the speed of light is also a neat, simple and memorable quantity. 

Of course humans must deal with natural phenomena in order to survive, so we have developed measurement systems. These are arbitrary, but hopefully as practical and as scientifically usable as possible. We must measure everything - time, temperature, electricity, weight, distance, speed, you name it. A decimal system seems to be advantageous over alternative systems, as exemplified by temperature: water freezes at zero and boils at 100. Neat and easy. A liter of water weights a kilo, which is a thousand grams. Simple. 

But how about time? To measure this variable, the world uses the neat sexagesimal system of seconds, minutes and hours, inherited from the Babylonians and the Sumerians many thousands of years ago. This is a combination of decimal and heximal (based on six). 

IF the earth’s circumference were exactly 40,000 kilometers and the speed of light exactly 300,000 kilometers per second, we would have to believe one or more of the following things: 

1. Light and the earth were created by a rational force that recognizes the meter (and the entire decimal system) as primordial, as a priori, as existing in and of themselves before the universe was created. God? Or:

2. The speed of light and the size of earth are somehow, causally, related. 

And: 

3. The length of the second is also a natural a priori - not human invention. 

Of course, the first of these beliefs is medieval, the second one is far-fetched and undocumented, and the third one is unbelievable, and somewhat a restatement of the first one. 

In fact, the speed of light is not exactly the neat 300,000 kilometers per second. It is a little bit below 300,000. Thus none of the quantities and measurement units I have mentioned - the meter, the second, the temperature degree, the decimal system - are a priori parts of nature. They are all human inventions. The universe is not inherently decimal. 

Are there any numbers or systems of numbers that are embedded in nature, prior to human design? Well, there is Pi (π). This is the ratio of a circle’s diameter to its circumference, and when expressed in decimal language, its value is approximately 3.14. As such, it is called an irrational number, in that after its decimal point, the digits go on forever, in no particular order. 

* * * * * * * 

So the question is: Is math invention or discovery? Are some numbers embedded in nature/the universe a priori Is God a mathematician? 

I came across a brilliant article by Mario Livio: “Is math invented or discovered?” This leading astrophysicist’s answer to this question is: Both. 

Since I am way out of my depth here, let me quote Livio at some length: 

Mathematics is the language of science. Einstein pondered, "How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?" Math captures the natural world in an uncanny way. Is math an invented set of tools, as Einstein believed? Or does it actually exist in some abstract realm, with humans merely discovering its truths? 

Einstein's school of thought is called Formalism. The opposite view is Platonism: The view that “ideas” exist independently from the physical world. According to Formalism, mathematics is a human invention. According to Platonism, it is a discovery. 

We humans use numbers, and numbers are amazingly good at describing nature. How did this come about? 

(Our species) possesses an innate talent, called subitizing, for instantly recognizing quantity, which undoubtedly led to the concept of number. We are very good at perceiving the edges of individual objects and at distinguishing between different shapes - abilities that probably led to the development of arithmetic and geometry. 

Imagine if the intelligence in our world resided not with humankind but rather with a singular, isolated jellyfish, floating deep in the Pacific Ocean. Everything in its experience would be continuous, from the flow of the surrounding water to its fluctuating temperature and pressure. In such an environment, lacking individual objects or anything discrete, would the concept of number arise? If there were nothing to count, would numbers exist? 

Not only do scientists cherry-pick solutions, they also tend to select problems that are amenable to mathematical treatment. But this careful selection of problems and solutions only partially accounts for mathematics' success in describing the laws of nature. Such laws must exist in the first place! Luckily for mathematicians and physicists alike, universal laws appear to govern our cosmos: an atom 12 billion light-years away behaves just like an atom on Earth; light in the distant past and light today share the same traits.... Mathematicians and physicists have invented the concept of symmetry to describe this kind of immunity to change. 

The laws of physics seem to display symmetry with respect to space and time: The same laws explain our results, whether the experiments occur in China, Alabama or the Andromeda galaxy--and whether we conduct our experiment today or someone else does a billion years from now. If the universe did not possess these symmetries, any attempt to decipher nature's grand design--any mathematical model built on our observations--would be doomed because we would have to continuously repeat experiments at every point in space and time.

Mathematics is an intricate fusion of inventions and discoveries. Concepts are generally invented. However, mathematics would not work at all were there no universal features to be discovered. 

Why are there universal laws of nature at all? Or equivalently: Why is our universe governed by certain symmetries and by locality? I truly do not know the answers, except to note that perhaps in a universe without these properties, complexity and life would have never emerged, and we would not be here to ask the question. 

The title of Livio’s 2009 book is suggestive: Is God a Mathematician?.-

To conclude, here are some examples of mathematical discoveries: 

1. The aforementioned number for Pi ( π), i.e. the ratio of a circle’s diameter to its circumference. 

2. Pythagoras’ theorem: In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides. 

3. Archimedes’ formulas for the surface of a sphere: 4 π r square,   and for the volume of a sphere: (4/3) π r to the third power.

And a few examples of numerical inventions: 

1. The meter, and all other measurement systems, such as the gram and kilo for weight, the degree for temperature, the “big four” for electricity, our decimal system in general. 

2. Other more awkward measurement systems (used primarily in Anglo-Saxon countries) such as inches, feet, ounces, pints, quarts, gallons, etc.) 

3. Our sexagesimal temporal system, which means that it is based on six and ten - seconds, minutes, hours, days, weeks, months, seasons, years, centuries). 

Note that our measurement of time is a hybrid of invention and discovery when it comes to the duration of a year: We have no choice but to recognize that a year is approximately 365.2422 days long, the time it takes earth to evolve around the sun. 365 is a fact of nature, not a human convention. 

4. The mathematical symbol “zero:” The ancient Greeks and Romans did not recognize the number “zero.” Ancient cultures which did use it include India and the Mayans. To be sure, the Greeks and Romans did recognize nothingness, doing so verbally with words such as “nulla.” They were able to add, subtract, multiply and divide. leave comment here