From what feels like out of nowhere, I’ve developed a hobby of mathematically expressing unique experiences I’ve had and other things I’ve learned about in my life. Since coming from an artistic background, it’s been pretty hard. There is still so much I need to learn. Though I have to say, I am enjoying the ride.
However, one thing in particular that I’ve run into in my journey through the field of mathematics has genuinely concerned me. The problem I’m referring to has to do with how mathematical expression work. The more I’ve learned, the more I’ve considered it a somewhat substantial roadblock. Yet, I don’t know for sure if I’m correct in my perspective. Due to this uncertainty, I am writing this article to present a solution I think I’ve found. I am also trying to get advice and critique from those interested and know more about math. I think I may be simply ignorant of what I’m going to present in this article. But without further ado, let’s jump into it.
The problem with continuity and bound
When trying to represent something mathematically, one can create an expression that mathematicians define as unbounded and continuous. Continuous as defined in calculus, having no discontinuities over particular intervals. Also, continuous in the sense that they extend indefinitely even while being (un)bound. Put another way, meaning that no matter what value for x you put in — assuming you have defined the other variables in your equation — you will get a solution for “y” or that function that does not represent a jump or undefined point.
Equations being continuous or unbounded in one or another direction works pretty well for a lot of things, but what I’ve come to realize is that, in life, there is plenty of binary variables that are at play. — variables that function like on and off switches. — For instance, you may ask, how fast can you run through a 500ft hallway per min? You can calculate your rate per min in both a linear or non-linear way (accounting for acceleration). You can then take this rate and do a further calculation to see how long it would take you to clear a 1,000ft or 3,450ft long hallway. What is not so easily calculable are the effects certain things, such as the lights being off or a security guard being present who prevent you from running down the hallway, have on your speed. They don’t have a rate at which they affect your speed per se. They are neither constants nor varying. In some cases, they are either on/off, present, or not.
To deal with these kinds of binary variables, mathematicians invented statistics and probability. Alongside statistics, there are piecewise functions, sigmoidal functions, and other tools useful when dealing with such variables. For the most part, these four mathematical tools work pretty well. However, they do have their problems.
Statistics and probability, at least at an elementary level, do not deal with absolutes but, more or less, averages. This perspective is true besides concepts like dummy variables.
When it comes to piecewise functions, in my opinion, they use a cheap workaround by explicitly stating when a function is expressed differently according to what interval of that function we compute. In some cases, this workaround is the equivalent of hedging an argument with caveats and disclaimers at the outset. In my opinion, it lacks a rigorous mathematical basis.
Finally, sigmoidal functions are fine but are at times too powerful when added to a particular expression. In some cases, they indiscriminately turn up or down too many variables in an expression when you want to affect just one or so. It’s like hitting the switch on your home’s breaker box when you want to turn off the lights in the kitchen.
As an aside, the basis for the computer science field was partially due to this failure to carry out such calculations mathematically. This is where algorithms find their value.
In any case, here, to me, is where subscripts come into play.
A new kind of variable
Beyond just being a label for terms within a particular domain/set, In my opinion, subscripts may solve the problem of dealing with more binary variables and functions, bounding and discontinuing them when appropriate.
To see how subscripts can help tame certain expression and functions, First, as always, axioms have to be laid.
The first axiom that I think is important goes as follows. No two variables with different subscripts can have any mathematical operation performed on them. This axiom means that regular variables with varying subscripts, such as A1, A2, Dn, and Kap, can’t be added, subtracted, multiply, etc., with each other until the subscripts are matching. Perhaps there needs to be an additional notation for this axiom. For now, we will use this notation to keep things simple.
The basis for this first axiom is to prevent subscript calculations from interacting with an entire expression unintentionally. It is also based on my own perspective that a+b=d ≠ a+b = c, where c=d, but c=[a,b]. Considering a set of the object and applying the tools of category theory and Stirling numbers of the second kind to them can further illustrate this point.
What the first axiom calls for is a procedure to match different variable’s subscripts with each other. This problem is solved by considering subscripts, no longer as mere identifiers of members of a set or function, but variables in themselves. So, for instance, take the variable N_r, where N = x+1, r = x². Any variable with a subscript that equals the value of r can be added, subtracted, etc., with N_r. Otherwise, they cannot.
For overall equations, they now resemble complex expression— expressions that include real and imaginary numbers. But now, instead of just real numbers, it will incorporate, for lack of a better phrase abstract/subscript numbers.
To see what all this looks like, we can begin with the examples. Take the functions and their corresponding graphs below:
Here, the function
is the abstract form, where
makes up the real number portion with a subscript constant 5, and
make up the abstract number portion with their respective subscript variable k.
In the first graph below, the variables A5 and k are denoted by the colors red and blue. The green line constitutes A5 subscript constant, 5. As you can see, where A5 subscript constant intersects with k are the points of discontinuity. The second graph depicts the abstract function G(x) and its respective discontinuities. From here, further analysis can be done. For now, this is sufficient enough to move on to the second example.
Here in example two I want to focus on sustained discontinuities and short-term bounding of functions over a particular interval.
The abstract function is,
where K is now defined as
and is both a real and subscript variable.
So here in the graph below the jump discontinuity occurs at x = 10 and 20. Between these two points, H(x) is bound both above and below. Overall the equation is unbounded above. The purple and red line represents k and H(x).
Below is H(x) written as a piecewise function in Desmos. As you can see, we were able to achieve the same results implicitly, instead of explicitly defining the constraints.
Summing it all up
Hopefully, these two examples showed the advantages of using subscript variables over piecewise functions and other mathematical tools. Perhaps not in all cases, but where the pros outweigh the cons, this may be another useful tool to keep in our mathematical tool-belt.
All in all, I think there is some potential here. However, I don’t know how far this potential extends.
A few interesting application of subscript variables is to perhaps use them to:
1. Cycle through variables of a set using trigonometric functions like the one I used in the example and other tools.
2. See if it’s possible to apply them in a more algorithmic way.
3. See how recursive functions manage when subscript variables are in use.
Beyond these three use cases, I can see their application in the social sciences, the field of probability and statistics, and perhaps game theory. Yet, this is just my first impression of the Idea of subscript variables.
In all honesty, I don’t know what can be done with subscript variables, if anything at all, but I do have high hopes for them. For those that have read this far, I hope this piece was worthwhile for you. Thanks so much.