What’s common between Roman Numerals, Chess, and Chemistry?
TLDR: It's my wishful thinking that somewhere out there exists a better notation for chess and chemical equations which would rescue them from the current lackluster notations much in the same way as Arabic\Indian numerals rescued Math from Roman numerals.
What is common between Roman Numerals, Chess, and Chemical equations? All of them suffer from a bad notation that stymied an otherwise potentially explosive growth seen in other fields which got the benefit of great notations. Curious? read on…
Those who are familiar with Roman Numerals, would agree how tedious it is to do even simple arithmetic with them, let alone representing and performing larger calculations like factorials, nth roots of numbers, etc.
For instance, MDCCLXXVI represents the number 1776 in this system. Imagine representing and calculating something as simple as the below sum in Roman numerals!!
One prime candidate for the tediousness of roman numerals is the lack of economy in representing even the small numbers.
In contrast, the same calculations can be performed more easily using the Arabic\Indian number system. The entry of Algebra and ideas like representing unknown quantities using alphabets x,y,z, etc. leads to the notion of polynomials which in turn has led to an explosive development in mathematics. The algebraic notation (and most of the notation from modern mathematics we use today) seems so natural in aiding our minds to represent complex mathematical ideas and even discover new ones, that it becomes easy to overlook the role of notation as a tool of thought. This quote from A.N. Whitehead aptly summarizes the value of good notation in this context:
By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race. — A. N. Whitehead
In his book, A History of mathematical notations, Florian Cajori, does a fantastic job of providing comprehensive details about the historical development of notations and I strongly recommend anyone interested in furthering their knowledge in this field.
Instead of simply repeating the details already covered by this book or re-listing the attributes of a good notation (which is covered in many other scholarly papers found on the internet), I refer you to the links mentioned in the reference section of this blog.
I would rather use this blog to highlight a few of the glorious examples of how a good notation helped drive further developments in the fields of math and science. I would further use it to drive the attention towards some other places which suffer weak notations. The notation for Chess for instance Following is one of the popular notations used to represent the state of chess pieces on the 8 x 8 chessboard: https://en.wikipedia.org/wiki/Algebraic_notation_(chess). While simple enough and gets the task done of effectively allowing human players and even computers to communicate the state of the game of chess, it adds very little value in terms of any more development in the game of chess.
This chess notation is effectively the Roman Numeral equivalent for the game of chess !!! it does not allow much further insight nor allow for comprehensive analysis / development for the the game of chess other than the current brute force methods of tree navigation used by the computer programs like stockfish and even the human players.
Similar trouble is applicable to the notation used for chemical equations. Anyone who has studied balancing the chemical equation might agree that the current notation with all its anomalies and exceptions, lacks the rigor offered by something like the differential equations or the cartesian coordinate system.
Sample Equation : 2 H2 + O2 => 2 H2O
The notation in chemical equations lacks the rigor offered by Differential equations or Cartesian Co-ordinate systems. This is the Roman Numeral equivalent for Chemistry!!
Following are some of the good examples where the notation helped further the development of math. If this helps drive enough motivation for the experts\hobbyist\enthusiasts to develop a better notation in other fields which currently suffer the weaknesses of bad notation (the notation for chess/chemical equations for example), I would think the purpose behind this blog will be achieved for me.
Example 1: Cartesian Co-ordinates
The cartesian coordinate system easily tops the list when it comes to the benefits of a great notation. The simple notion of assigning pair of numbers (x, y) to the points in a Euclidean plane bridged the gap between Geometry and Algebra simplifying the solution of many problems and proofs which otherwise would be tedious using the concept of Euclid Geometry alone. Consider the following example:
Given the pentagon as follows and the segment FG , find if the segment when extended intersects the polygon.
Using the methods of Euclidean geometry alone, this would be tedious proof and also may need some additional data about the nature of the polygon and the segment.
Enter the cartesian system!! It does exactly that, it provides additional data about the nature of these two shapes in a very simple \ generic way and simplifies the proof.
The Cartesian coordinate system does much more to the development of geometry which would be tedious/ or probably not possible by the method of Euclid geometry alone. By simply adopting the notation of depicting a point by a pair of numbers (x, y) it does the following:
- Offers a simple and generic method to provide information about points that can be actually used to solve geometrical problems which are not possible by naming points simply as P, Q, etc.
- It opens up scope for further development of geometry by paving way for the notion of equations of a line. Not just line but the equations of circle, ellipse, parabola, and possibly the equations of an infinite set of shapes in terms of polynomials and differential equations which further helps geometry benefit from the developments in algebra and helps build an intuitive idea in the development of Algebra from Geometry!!
- It opens up scope for thinking and solving interesting problems beyond the 2-dimensional geometry into 3d and higher dimensional (non-Euclidean geometries) which would have been too much of a cognitive load to think about, to begin with using only Euclidean geometry.
I could go on and on about the merits of the cartesian system, but you may already have got the gist.
Example 2: Feynman Diagrams
I won't pretend to be an expert in atomic physics (or any kind of physics to be honest 😁) and have mostly read about the merits of the Feynman diagrams in simplifying the understanding of the subatomic formulas. Following excerpt from the above wiki:
The interaction of subatomic particles can be complex and difficult to understand; Feynman diagrams give a simple visualization of what would otherwise be an arcane and abstract formula.
This certainly is a form of notation that helps reduce the cognitive burden of an otherwise tedious expression.
Example N: …
It is possible to come up with a much larger list of examples drawn from varied fields like computer programming, game theory, engineering, and even road signs where the choice of a good notation helped in a) simplify and b) lead to explosive progress in the related fields. I won't waste your time by trying to accumulate those in this blog
It is my wishful thinking that somewhere out there exists a better notation for chess and chemical equations which would rescue them from the current lackluster notations much in the same way as Arabic\Indian numerals rescued Math from Roman numerals thus making programs like stockfish much simpler.
On an ending note: I had initially decided to name this blog: “Notation as a tool of thought”. However, that title is already taken by this wonderful paper by Kenneth Iverson the winner of the ACM Turing Award in 1979. I highly recommend you to read it for some very insightful thoughts by Kenneth Iverson (who also happens to be the creator of the APL programming language). In fact, it is this work on notations and APL, that led him to be honored with the ACM Turing award.
References:
- Analysis of Mathematical Notations: https://digital.wpi.edu/downloads/2z10wq46j
- Notation as a tool of thought: https://www.eecg.utoronto.ca/~jzhu/csc326/readings/iverson.pdf
- A history of mathematical notations: https://www.amazon.com/History-Mathematical-Notations-Dover-Mathematics-ebook/dp/B00EZCA540
