One of the mathematical writing strength that differs with other fields is its adherence to incredible brevity.

If one compares a Ph.D. thesis from someone in humanities, it would be unacceptable if its length were less than 300 pages, on the contrary, it is acceptable for a math doctoral student to submit a thesis with less than 100 pages.

The usual criterion for a doctoral thesis (or indeed from any scholarly work) is that it should be “new, true, and interesting”. If one can prove that in a single page – they’ll probably hand over the sheepskin.Lander and Parkin's paper about a conjecture by Euler is the best example of mathematical brevity. Short but profound.

Lander, L. J., & Parkin, T. R. (1966) Bulletin of the American Mathematical Society, 72(6), 1079.

How is this great brevity achieved?

By using symbols in place of a whole paragraph’s worth of words!

One of the symbols, in particular, that has immense power is the so-called relational symbols. If you place a relational symbol between two expressions, you created a sentence that holds “true”. In other words, when you write down a mathematical sentence involving a relation, you are asserting the relation is True (the capital T is intentional).

It’s not okay to write “3 < 2”, you say “2 < 3”. The symbol “>” is an example of a relational symbol. When we have variables in a relational expression, things become more complicated slightly.

If you’ve read my previous posts, these are the symbols of relation we’ve seen so far:

These symbols, when placed between numbers, produces a statement that results in a Boolean state, either true or false. For false results, we don’t usually write them down instead, we state the relations that doesn’t hold by negating the relation symbol. This is done by slashing through it, but some of the symbols above are negations of others.

The illustration is the best visualization of what relational symbols do:

relation symbols

These days, there is a slight tendency to define functions as being a special class of relations. This tendency is slightly silly but it's not wrong because functions are a special type of relation, but it is the least intuitive approach. There is, then, a need to define what a relation is,

a relation is any set of ordered pairs with a stated restriction on the ordered pairs (which may be in a relation if it is to be a function.)

References:

1. https://blog.paperpile.com/shortest-papers/
2. http://www.mathnstuff.com/math/spoken/here/1words/r/r25.htm
3. A Gentle Introduction to the Art of Mathematics by Joe Field

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