# This Is Mathematics. This Is NOT Mathematics???

Posted on: August 3, 2022

This is Science. This is Not science?

Tereza Tizkova

July 13, 2022

Mathematics allows us to quantitatively understand our Universe.

It is the building block for everything in our daily lives, including software, architecture, engineering, art, finance, weather predictions, cryptography…

But what if an alien asked you what is mathematics? How would you explain it?

For some reason, the questions I am getting about studying and doing mathematics are repetitive. My friends, family, colleagues, a random guy at a party, paramedic inside of an ambulance, they all seem to succumb to a few myths.

The standard college dictionary definition of mathematics is something like: “The study of quantity, form, arrangement, and magnitude; especially, the methods and processes for disclosing, by rigorous concepts and self-consistent symbols, the properties and relations of quantities and magnitudes, whether in the abstract, pure mathematics, or in their practical connections, applied mathematics”.

This lengthy description is the first red flag, pointing out the lack of a brief, yet cogent definition.

If there is not a consensus about what is mathematics even among scientists, no wonder that there exist

This is a selection of reactions I heard at least once.

Please take the above with a pinch of salt. I am not saying everyone should know or care about all other fields than their own.

I just think every field would get its own funny stereotypical reactions.

1. Mathematics is NOT just numbers and calculations.

Source: Reddit

This is common at job interviews. Hearing things like “I would expect that you calculate this faster, considering you are a math student.”

Seeing my professors algunas veces making the most trivial mistakes only confirms to me that being a human calculator and a math genius probably isn´t the same thing.

Well, but if mathematicians don´t do just some calculations, what do they actually study?

Two things typical for pure mathematics are studying mathematical structures and formulating and proving theorems.

Mathematical Structures

Mathematical structure is a set endowed with some additional features. Imagine for example a bunch of numbers that interact certain way (e.g. you can add them up, multiply them, say which of any two numbers is bigger etc.)

Or imagine a set of points where we can define how far the points are from each other. (This means that we can define a metric structure.)

Or it is known that elliptic curves have a so called “group structure” which is a special feature that points on the curve have that makes it such a cool choice for cryptography btw.

There are tons of other really interesting structures.

Theorems and Proofs — We start with axioms (rules that are assumed true without discussion) and from them we built some claims (theorems) that have to be proven rigorously.

Sometimes it takes a lot of work and a lot of known theorems to prove something new.

New theorems are built on what is already known, like a Lego blocks.

Two examples of how theorems may look like. Do you see any numbers?

Source: [1]

[2] Math is NOT just a language. Italian astronomer and physicist Galileo Galilei is attributed with the following quote. “Mathematics is the language in which God has written the universe.”

Why is mathematics so often called the language of science?

Well, it certainly does seem like a different language when mathematicians talk to each other and I have seen people claiming that mathematics is a language.

I personally disagree and have yet to meet a mathematician that agrees.

The main differences between math and language:

Math is restricted in scope to abstract concepts, their properties and relations between them.

Language is universal in scope. We are able to describe by language that the sky is blue and that we feel happy, sad or hungry.

While with math, we would need to first define “sky” or “blue” to make such statements.

Math is exact. Language allows us to express the same thing by different description or a metaphor.

Of course, there are different mathematical expressions describing the same statement. But even if you can say that “manifold is intuitively just a wiggly R^n”, writing this as a formal definition to your paper would not be met with understanding.

Math expresses relationships and those relationships must be proven to be at least self consistent with the rest of the body of mathematics.

On the other hand, languages are arbitrary in nature and not self consistent, there is not a “right” or “wrong” way to transmit knowledge, feelings or ideas, it either works or it doesn’t.

Mathematics is not a language, it just uses a language to claim and prove theorems?

What I find interesting is that all the “mathematical language” should even be omitted. Even the Pythagorean theorem can be written without a single word, with just expressions (this way of writing is rather painful).

The simple example is that we can write the claim “When two numbers are added, their order is not important” in mathematics as “a + b = b + a”.

Of course, mathematics has some points in common with languages though.

Math and Language are both a way we have to organize our abstract thinking.

Both help us to “name” the physical world and to communicate it

Math is NOT perfect or “always true”?

Mathematics intends to be rigorous, beautiful, truthful and precise, thus, in a sense, it aspires to a certain perfection.

I have mentioned the axioms — the mathematical assumptions that are absolute truths and the rest is built upon them. We create something called “a formal system” (or “an axiomatic system”) from the axioms, according to a given set of rules.

You may have heard of some mathematical paradoxes, such as the Barber paradox.

In 1931, Kurt Gödel discovered in 1931the famous incompleteness theorem.

In layman terms, the theorem says that if you have a consistent formal system (i.e., a set of axioms with no contradictions) in which you can do a certain amount of arithmetic, then there are statements in that system which are unprovable using just that system’s axioms.

This could be interpreted as an imperfection of mathematics.

Also, in some formal systems, there can exist statements that are both true and false. Seems not quite right, huh?

Mathematics is NOT a science?

This one is maybe a bit controversial and requires a deep dive into philosophy. I will refer to a beautiful article “Is Mathematics a Science?” if you want to read more about this.

The answer of course depends on what one means by “mathematics”, and what one means by “science”.

Science is a form of the theory of knowledge — epistemology. A proper science must have a means of validating its claims as well as a means of identifying and rejecting false claims.

One of the primary things that distinguishes science from other epistemologies is that it is

a) systematic and b) nondogmatic.

However, the axioms in mathematics are very close to dogma, and it seems not to care about truth or falseness, or even the outside world for that matter.

There are proofs, but there seems to be no experimentation .

So what is the definition?

I have not of course debunked all the misconceptions, e.g. my personally favourite — that girls lack “math cells”.

There are other ways to interpret math, for instance as logic, science of patterns or torture of students.

I don´t agree with either. Saying that math is logic only allows you to analyze a very small part of mathematics. The latter is attributed to Lynne Steen who referred to mathematics as the science of patterns.

Pattern is a repeated arrangement of numbers, shapes or other entities, so yes, patterns can suggest things to study mathematically, but in my opinion, it is better to say that mathematics is the study of structures hidden behind patterns.

In the end, my favourite definition is that Mathematics is a discipline of abstract entities and relationships between them.

However, if an alien asked me what is math, I probably wouldn´t say this definition. In fact, I would probably run.