How do you formulate a vague notion into a mathematical expression?

Question

I am a software engineer wanting to learn math. I also do a little bit of drawing.

What I am wondering is, how do you formulate a vague notion of something you're trying to model into a mathematical expression? Here is some background to give you more context of what exactly I mean...

When I write code, I have a vague notion of something I want to build. Say I want to build an ORM (a database basically), which is something I am currently doing as an open source project (it is building on ideas already discovered about database modeling, but missing from all the current solutions and the next evolution of a database). I often then look through some open source code for reference, to see how other's have done similar things. Then I sketch out the code a few different ways in a text file, and see if I can write pseudocode that will be simple and accomplish the objective. Then once I like the general code architecture, I write the real code.

When I draw, I have a vague notion of something I want to draw. Then I find some reference work perhaps, or look at something. Then I sketch out different ways of drawing that thing. Then once I like a sketch, I draw the real picture carefully based on that sketch.

So, it seems there should be a similar process for math, but I have been unsuccessful so far in figuring that out. I tried asking this and that question as a specific problem to solve, but the questions don't seem to resonate with the way mathematicians approach problems, so this is me asking a different way :)

When I first was learning to code, I had an idea in mind of what I wanted to build, and so stumbled through tutorials and such that showed me how to model things. After a few months, I could easily take an abstract concept and formulate it into code (functions, classes, variables, etc.).

When I was first learning to draw (still kinda in that phase), everyone says "just draw or just practice". But that doesn't work for me, I have to have a sensible method behind it, and motivation. Motivation comes from having a specific example of something to draw. For me that was nature drawing, I wanted to draw a leaf. And it took a while to find someone (a nature illustrator) who could show me how they think about taking an actual leaf from nature, and abstracting it into a drawing.

Now I'm wondering the same thing for math. How do you take an abstract, vague notion of something, and convert it into a mathematical expression? The two questions I linked to were specific examples I wanted to use to try and understand, but maybe they aren't good examples. I would try asking another specific question but I am a little disheartened.

I have read most of How to Prove It, which has been very helpful. But, it doesn't provide much guidance on the process of turning vague notions (like the ones linked) into mathematical expressions.

The main thing is, I am not learning anything by doing textbook practice problems that say "Given you have a set A = {1, 2, 3} and B = {4, 5, 6}, then the union A∪B = {1, 2, 3, 4, 5, 6}" type thing. They are very demotivating. I need a real-life example to learn from, and those two questions were a problem I thought could be formulated into a mathematical expression. Since those were bad questions, the general question is, how do you formulate some vague notion into a mathematical expression?

I am mainly looking for a specific example of how you take some problem/pattern/model, and start sketching out variables and mathematical expressions (using the symbols like $\sum$ and such) to try to describe the pattern/model. It doesn't have to be something you completely solve (or that is even solvable), just the process of how you model things out using mathematical symbols.

Answer 1

I'm far from the most experienced person here, but i'd like to share my take.

The first thing i would be asking is whether said vague notion is question about some sort of pattern.

For example, you asked a question regarding how to model a tree. Whilst there are many ways to mathematically describe a tree, by itself the tree is not a pattern.

However, if you were asking 'why do the branches of a tree grow in a particular fashion?', then i would say that is a pattern which you could describe mathematically. Similarly, if you were asking how fast would it take a tree to spread it's seed and grow in to a forest, that would also be a pattern which you could describe mathematically.

I would argue that we call a description of such a pattern a model.

Furthermore, i would say that the different questions lead to different techniques in modelling. In the first example, one might use something like an L-System to construct the model, where as in the second, one might use a Markov chain or a PDE model.

These models can then attempt to answer particular questions about the patterns.

Now, most textbooks have these abstract problems as exercises and examples because they teach the techniques used to work with those particular models. Think of it like learning a particular programming language. Just as a learning Java gives you the tools to write a Java app, you still need an idea of what you want your app to do. Learning set theory gives you the tools to answer set theoretic problems, but you still need to turn you vague notion into a question about a pattern which set theory can work with.

As far as your text books are concerned, i would ask around and try and find a textbook that works for you. No textbook works for everyone, and there is surely some around that speak to you. Maybe even favor fields that are inherently applied, like differential equations or probability theory.

Hope that helps.

Answer 2

I will try to give an example. Imagine you want to formally define the notion of comparaison between objects. Indeed, in daily life, we most of the time gives values to things and comparing these values could be useful. We untuitively know how to compare real numbers. What would it mean to compare sets, vectors, matrices,functions,... ?

A good start is to list the property such a comparison would have. You have to keep in mind that a model is an approximation of the reality, so you need to distinguish which property are essential, which one are only here to simplify the problem, which are not essential but are not here for simplification. Here is one tentative (there are many possibilities):

1. We can compare only similar objects, hence objects living in the same set $X$.
2. We can only compare two objects together (surely a simplification property) and decide which one is the best, so a comparison could be a function $f$ on $X\times X$ with value in $\{0,1\}$, $f(x,y)=1$ meaning that $y$ has a better or the same value than $x$; and $f(x,y)=0$ otherwise.
3. $x$ has always the same value than $x$, so we should have $f(x,x)=1$ for any $x$ in $X$.
4. If $x$ has less value than $y$ and $y$ has less than $z$, then $x$ has less value than $z$. It means: $$(f(x,y)=1 \wedge f(y,z)=1) \Rightarrow f(x,y)=1$$
5. Non essential property (it is debatable): if $x$ has less value than $y$ and $y$ has less value than $x$, then $x=y$. It means $$(f(x,y)=1 \wedge f(y,x)=1) \Rightarrow x=y$$

If properties from $1$ to $5$ are satisfied, it leads to the well-known notion of total order on a set. Total means that in defining $f$ on all of $X\times X$, we implicitely assumed that two elements of $X$ can always be compared. If the domain of $f$ is relaxed to a subset of $X\times X$, it is called partial order.

---

An alternative solution: property $2$ above may appear to be artificial and we would like to compare more elements of $X$ at the same time. A possibility:

1. We assume that all elements we want to compare live in the same set $X$.
2. Simplification property: given any non empty subset of $X$, we can decide which one is the best. This means that the best element always exists and is unique. This can be modelized by a function $B$ from $P(X)^*$ into $X$ satisfying $B(A)$ is an element of $A$. Here, $P(X)^*$ denote the set of all non empty subsets of $X$.
3. With this definition, we have $B(\{x\})=x$, so the property $3$ of the previous list is automatically satisfied.
4. Property $4$ above translates to $$(B(\{x,y\})=y \wedge B(\{y,z\})=z)\Rightarrow B(\{x,z\})=z$$
5. To find the best element of the union of two sets, it is enough to compare their two respective best elements, so $$B(A_1\cup A_2)=B(\{B(A_1),B(A_2)\})$$
6. If a set is contained in a larger set, then the best of the larger set is better than the best of the smaller set (a world champion should be better than your typical regional champion). Then $$A_1\subset A_2 \Rightarrow B(\{B(A_1),B(A_2)\})=B(A_2)$$ This property $6$ is actually a consequence of property $5$.

---

I hope this helps. It was fun to type anyway.