Mathematics is viewed by many to be an objective science. There are independent co-discoveries in this highly abstract field. People discover the same mathematical things and relations despite having not met. This gives credit to the position that math is indeed objective. In this article, we are going to the mathematical concept we normally call the Pythagorean theorem; a concept that is not purely Pythagorean at all. It was discovered more than 1000 years before the philosopher Pythagoras was even born by the Babylonians. It too was independently discovered by the Chinese. The field of mathematics with all its sub-fields and specialties is widely seen to be independent of the other sciences and, more so, the daily life. The mathematician’s pursuit of “mathematical truth” is seen by many to have an intrinsic value without any practical use. What’s startling though is that when it is applied to other fields of studies, mathematics lends them great power. Something that biologists, physicists, or economists can’t say back to the mathematicians.
Game theory was used to make us better understand evolution. It’s the same with the use of statistics by Ronald Fisher for biology. But there can never be a time that some mathematical conjecture or theorem can be solved by some experiments or some theoretical work in biology.
James Robert Brown, in his book Philosophy of Mathematics, spells this out:
“A revolution in physics can cause a subsequent revolution in chemistry or biology. Yet never in the entire history of mathematics has a result elsewhere (in the non-mathematical sciences) had any impact on our evaluation of mathematics. That is, no mathematical belief was ever overthrown by a discovery made in the natural sciences.” – Brown, Philosophy of Mathematics, p. 29
The Objectivity of the Mathematical Image
Also in the book, Brown outlined ten elements of the mathematical image. These are claims about mathematics that are commonly accepted–the standard conception. They are, according to Brown, what most working mathematicians, most educated people, and what most philosophers of mathematics likely believe to be true.
He himself doesn’t endorse them all. But he happily does one: that math is objective.
Unlike discoveries in the natural sciences (let’s say of theories that are susceptible for getting rejected, changed, or abandoned), the discoveries in mathematics (like theorems) seem to be objectively independent of space, time, and culture.
If such is the case, it’s not hard to imagine how different people who lived in different ages who spoke different languages and used different symbols in doing mathematics would come up with the same discovery. This was the case with the Pythagorean theorem. Let’s take a look at two cultures and their relationship with the theorem.
The Babylonians, the Chinese, and the “Pythagorean” Theorem
The Babylonians, as it turned out, had their own system of mathematics. It’s quite different from ours but they influenced us in some way. One astonishing facet of their numerical system is that it was sexegesimal–based on 60. We usually count with our fingers. They counted with their twelve knuckles on one hand and five fingers on the other.
What’s also impressive is that they were fascinated with the square root and were well-versed with quadratic equations. You can read more about it here.
We know a lot about their mathematics thanks to the archaeologists and mathematicians who reconstructed them. There’s this famous find called the Plimpton 322, that is a clay tablet which is one of many where they recorded their financial transactions. Aspiring scribes also used them for practice in their schools.
The mathematician and historian of science Otto Neugebauer and Assyriologist Abraham Sachs described the Plimpton 322 to be “the oldest preserved document in ancient number theory”.
This artifact suggests that as early as 1900-1600 B.C.E., more than a thousand years before Pythagoras was born, they already knew of the Pythagorean Theorem: they knew that the square of the hypotenuse (side opposite of the right angle) is equal to sum of the the square of the other two sides. It is expressed by this famous equation:
Joran Frinberg, the Swedish math historian, stated that it’s not only Plimpton 322 that testifies that the Babylonian mathematicians knew of the Pythagorean Theorem. In the 1980s, Christopher Walker discovered a fragment of a tablet named BM 96957 that is a direct join to VAT 6598. This was passed on to Eleanor Robson, a Professor of Ancient Near Eastern History in University College London. After some work there, she published a paper about the Pythagorean Theorem in the Journal of Cuneiform Studies called Three Old Babylonian Methods for Dealing with “Pythagorean” Triangles.
Here’s a translation in the paper about a Babylonian method for dealing with the “Pythagorean” triangles:
“[The breadth is 2 cubits, the height 0; 40 (cubits). What is its diagonal? You: square 0;10, the width. You will see 0;01 40. Square 0;40, the length. You will see 0; 26 40. Add it to 0;01 40. You will see 0; 28 20. What is the square root? the square root is 0;41 … … .The diagonal]”
This excerpt of Robson’s translations show that they know that the length of diagonal (the hypothenuse) can be inferred from adding squares of the width and the height respectively! The writer in this tablet seems to be a teacher instructing students how to find the diagonal.
Let’s go through it really quick but let’s circumvent the “weird” numerical system saying that the square of 0; 40 is 0; 26 40. Let’s just make it good old 1,600.
We square 10, the width. That gives us 100. Then the tablet writer instructed to square 40, the length. We get 1,600. We add it to the square of the width (0;01 40 in the translation). The result is 1,700. Now, the writer asked: “What is the square root?” Good thing we have our handy calculators and perform the operation there, we get 41.23105625617661. What did our Babylonian math teacher arrived at? 0; 41. That’s 41 without the decimals!
The teacher is not Pythagoras at any stretch; it’s a different person that the theorem was named after!
Robson noted that with the values given in the text, the error for their methods is just 2.15% depending on the length of the hypotenuse and the proportion of the length to the hypotenuse. The text also has different methods for solving for the hypotenuse.
Same mathematical theorem, same technique, different person. Sounds far fetched? Sounds like a co-discovery of an objective thing. Now, on with the Ancient Chinese’relationship with the same theorem.
The Chinese, of course, have a different name for the Pythagorean Theorem. It’s called the Gougu Theorem. The ancient Chinese used some abstract manipulation of shapes (mainly the 3:4:5 Pythagorean triangle) to show why the square of the hypotenuse is equal to the sum of the square of the other remaining sides.
This theorem appeared in the Zhoubi Suanjing or The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven. This is the earliest surviving Chinese writing on mathematics.
Li Yan and Du Shiran in their history of mathematics book Chinese Mathematics pointed out that in the first part of the Zhoubi Suanjing, a particular case of the theorem is mentioned. Gou, the width, is said to be three. The height, Gu, is four while the Xian or hypotenuse length is five. This is the 3:4:5 “Pythagorean” triangle.
But, this is not so “Pythagorean” anymore, right?
In the latter part of the text, a general application of the theory can be found according to the authors. Translated, it reads:
“Gou, Gu each multiplied by itself and added and then taking the square root, we get the hypotenuse (Xian)” – Chinese Mathematics, p. 30
This can be expressed as this:
It seems like the Chinese independently discovered the same mathematical fact! They, like the Babylonians, put it to great use, such as for surveying. It is even said that ‘Emperor Yu’ could rule the country because of the Gougu Theorem. With regard to that, the authors of Chinese Mathematics quoted ancient Chinese mathematician Zhao Shuang in his commentary about the Zhoubi Suanjing:
“Emperor Yu quells floods, he deepens rivers and streams, observes the shape of mountains and valleys, surveys the high and low places, relieves the greatest calamities and saves the people from danger. He leads the floods east into the sea and ensures no flooding or drowning. This is made possible because of the Gougu theorem…”
Chinese Mathematics, p. 30
Is Math Objective?
The Babylonians had this theorem. The Chinese did too. Pythagoras (or his followers) also came up for the theorem and created their own proof for the theorem as well. Euclid. Even President James A. Garfield discovered a proof.
Just in this website Cut The Know alone, there is a collection of 121 proofs. That’s 121 ways of proving something to be true.
So, what does this show?
This shows that mathematics can be pretty objective.
For one, the “Pythagorean” theorem seem to exist beyond time, space and culture, that it is something that is discoverable rather than just a convention. The Babylonians discovered it independently from the Chinese; Pythagoras from them and Bhaskara, an Indian mathematician and astronomer. They may have done it differently but they all found the same “mathematical truth”.
From this, Brown and others quip that if the there were Martians and they do math, their math will be similar. They’d be gazing upon the same “truths” or, at least, the same mathematical image. If there were intelligent life out there beyond our solar system and they do mathematics, it shouldn’t surprise us if they have their own “Pythagorean” theorem.
So, is math objective? There’s still a lot of discussion to be had among mathematicians and philosophers of mathematics. For now, this serves as an evidence that math could really be objective; it’s something that “exists” beyond the confinements of space, time, and culture–things that it could help describe, represent, model, and even simulate.
Bogomolny, A. (n.d.). Pythagorean Theorem and its many proofs in Interactive Mathematics Miscellany and Puzzles. Retrieved April 26, 2018, from https://www.cut-the-knot.org/pythagoras/
Li, Y., & Du, S. (1987). Chinese mathematics: A concise history (pp. 25-28). Oxford: Clarendon press.
Robson, E. (1997). Three Old Babylonian methods for dealing with” Pythagorean” triangles. Journal of Cuneiform Studies, 49, 51.