Modern Math Is Full Of Symbols. Is This Really Necessary?

It is no secret that modern math is full of symbols. To the uninitiated, this presents a tall barrier to entry. Often, the plethora of confusing symbols is one of the main reasons many young students drop mathematics in favour of some of the more (subjectively) attractive options.

Apart from that, even at practitioner’s and expert’s end, the symbol-count starts presenting issues. Experts from one field might need to unlearn/lose symbol-maps from their source field and learn/pick-up new symbol-maps for the new field that they wish to enter.

So, one really does have to ask THE question:

“How much is too much?”

In this essay, I start by covering the origins of some of the most basic mathematical symbols. Then, I present some examples of symbols used in modern mathematics.

From thereon, I look at some of the problems they present and explore potential solutions and consequences. Let us begin.

This essay is supported by Generatebg

The Origins of the Basic Mathematical Symbols

The ‘+’ symbol and the ‘−’ symbol share their origin in the Latin language. Before they came into existence, the symbol ‘p̄’ (più) was used for ‘plus’ and ‘m̄’ (meno) was used for minus.

It was Johannes Widmann who first introduced the ‘+’ and ‘−‘ symbols in his 1489 treatise “handy and pretty arithmetic for all merchants” (a rough translation from old German). But even then, the ‘+’ symbol meant ‘surplus’ and the ‘−’ symbol meant ‘deficit’.

Robert Recorde was the first person on record (pun totally intended) to communicate the ‘+’ and ‘−’ symbols in their modern forms.

Recorde also happened to be tired of writing “is equal to” repeatedly. So, he ended up inventing the ‘=’ symbol to replace the three words. He just happened to choose two horizontally parallel and equal line segments.

The division symbol ‘÷’ is known as the obelus. Johann Rahn used the obelus first as a symbol for division in his algebra book “Teutsche Algebra”. Before that, some mathematicians used the obelus for subtraction.

One of the first uses of the ‘×’ symbol for multiplication was in the appendix of John Napier’s book “Mirifici Logarithmorum Canonis Descriptio”. This appendix, in turn, is attributed to William Oughtred.

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Modern Math is Full of Symbols. But Why?

I just covered the origins of some of the most fundamental symbols in mathematics. Were you able to make out why mathematicians invented such symbols? You see, mathematicians are a lazy bunch.

Let’s be honest, most humans are. If you had to write “is equal to” a hundred times in an hour, would you put up with it? The answer to that question lies at the core of mathematical symbols.

A mathematical symbol conveys compressed information. In this sense, mathematics can be seen as a language that compresses information. Why is this? It is because the mathematicians are a lazy bunch who would like to save space and increase efficiency.

If mathematicians used conventional words instead of symbols, what we currently write in one page would expand to several pages. Needless to say, the cumbersome sentences might even lead to confusion.

Having said this, this symbolic-compression of mathematics leads to some inherent issues. Before we cover them, let us take a look at some examples of modern mathematical symbols.

Examples of Symbols Used in Modern Math

Check out the following mathematical expression:

What you see here is a tensor product of two vectors expressed using the decomposition of their bases. You need not understand this expression. I merely presented it here as an example of symbols we use in modern mathematics.

Take a look at the symbol between ‘x’ and ‘y’. The circle with a cross inside it denotes the tensor product.

Since we ran out of conventional symbols, we had to invent something like this to denote the tensor product, which is a relatively newer mathematical abstraction.

I just happened to present the tensor product as an example because I have worked with tensor calculus in the past. However, you can be certain that all cutting-edge newer sub-fields in mathematics sport such ‘fancy’ new symbols.

It could be the case that the same symbol denotes something else in some other mathematical sub-field. I just do not know. With that thought, we arrive at the core issue I would like to discuss in this essay.

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Modern Math is Full of Symbols. Is this Really Necessary?

Earlier in the essay, I mentioned that mathematics can be seen as a language that compresses information. Having said this, mathematics also differs from conventional languages ever so subtly. But the consequences are not so subtle.

Imagine that someone added new alphabets to the English language at the end of each year. You could never claim that you have mastered the language.

If you are away from the language for a few years, you might not even recognise it any more. Such is the case with mathematics.

Most outsiders have the false assumption that mathematics is a complete venture; that it is clearly established and set in stone. This is anything but true!

The Problem — Mathematics is a Growing Language

Mathematics is a growing discipline with researchers and mathematicians pushing the boundaries every single day. In this sense, mathematics is an inherently growing language. But the rate of growth is so vast that new symbols are needed all the time.

This, in turn, makes the complete grasp of mathematical symbols impossible for any single mathematician or human being (I’m discounting, of course, the ranked geniuses as exceptions).

With time, as more and more information accumulates under the term “mathematics”, more and more symbols are unavoidable. The upper bound is infinity.

Do we have a real problem here or am I being needlessly pessimistic? Also, where do we go from here with mathematics?

A Potential Solution

The issue here I feel is that students are taught “mathematics” as a subject in school. This gives the false impression that it is a subject that can be mastered by any single human being given enough time.

The ‘state-of-the-art’ in any given sub-field might be within the reach of any single human being. But as time flows, the ‘state-of-the-art’ in mathematics as a subject is likely to be out of reach for individuals.

It makes sense to me to clearly define the symbols and their mappings for any given sub-field. Then, the said symbols may be reused in other sub-fields (with their own clearly defined maps).

So, the meaning of each symbol becomes dependent on the context of the sub-field used. With such an approach, we would run into the issue of symbol conflicts in cross-field interactions.

Final Thoughts

I admit that I do not have all the answers to this problem in this essay. I am merely pondering the possibilities here. A lot of work will be needed.

I also admit that mathematicians/researchers out there are already partially using/implementing what I propose here. I am just voicing a unified structure to the process.

But on the other side, if we keep the unstructured growth of symbols unchecked, we run the risk of even experts becoming recursively beginners within the same field and dropping the ball.

This is perhaps unavoidable. Who knows? But as the field of mathematics grows, it becomes less and less approachable for individual students of the “subject”.

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Further reading that might interest you:

• Bourbaki – The Story Of The Rockstar Mathematician Who Never Lived
• How To Really Make Sense Of Hotelling’s Law?
• How To Mentally Square Any Number Ending In 5?

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