I cringe when hearing "Math teaches you to think".

It's a well-meaning but ineffective appeal that only satisfies existing fans (see: "Reading takes you anywhere!"). What activity, from crossword puzzles to memorizing song lyrics, doesn't help you think?

Math seems different, and here's why: it's a specific, powerful vocabulary for ideas.

Imagine a cook who only knows the terms "yummy" and "yucky". He makes a bad meal. What's wrong? Hrm. There's no way to describe it! Too mild? Salty? Sweet? Sour? Cold? These specific critiques become hazy variations of the "yucky" bucket. He probably wouldn't think "Needs more umami".

Words are handholds that latch onto thoughts. You (yes, you!) think with extreme mathematical sophistication. Your common-sense understanding of quantity includes concepts refined over millenia (zero, decimals, negatives).

What we call "Math" are just the ideas we haven't yet internalized.

Let's explore our idea of quantity. It's a funny notion, and some languages only have words for one, two and many. They never thought to subdivide "many", and you never thought to refer to your East and West hands.

Here's how we've refined quantity over the years:

- We have "number words" for each type of quantity ("one, two, three... five hundred seventy nine")
- The "number words" can be written with symbols, not regular letters, like lines in the sand. The unary (tally) system has a line for each object.
- Shortcuts exist for large counts (Roman numerals: V = five, X = ten, C = hundred)
- We even have a shortcut to represent emptiness: 0
- The position of a symbol is a shortcut for
*other*numbers. 123 means 100 + 20 + 3. - Numbers can have incredibly small, fractional differences: 1.1, 1.01, 1.001...
- Numbers can be negative, less than nothing (Wha?). This represents "opposite" or "reverse", e.g., negative height is underground, negative savings is debt.
- Numbers can be 2-dimensional (or more). This isn't yet commonplace, so it's called "Math" (scary M).
- Numbers can be undetectably small, yet still not zero. This is also called "Math".

Our concept of numbers shapes our world. Why do ancient years go from BC to AD? We needed separate labels for "before" and "after", which weren't on a single scale.

Why did the stock market set prices in increments of 1/8 until 2000 AD? We were based on centuries-old systems. Ask a modern trader if they'd rather go back.

Why is the decimal system useful for categorization? You can always find room for a decimal between two other ones, and progressively classify an item (1, 1.3, 1.38, 1.386).

Why do we accept the idea of a vacuum, empty space? Because you understand the notion of zero. (Maybe true vacuums don't exist, but you get the theory.)

Why is anti-matter or anti-gravity palatable? Because you accept that positives could have negatives that act in opposite ways.

How could the universe come from nothing? Well, how can 0 be split into 1 and -1?

Our math vocabulary shapes what we're capable of thinking about. Multiplication and division, which eluded geniuses a few thousand years ago, are now homework for grade schoolers. All because we have better ways to think about numbers.

We have decent knowledge of **one noun**: quantity. Imagine improving our vocabulary for structure, shape, change, and chance. (Oh, I mean, the important-sounding Algebra, Geometry, Calculus and Statistics.)

Caveman Chef Og doesn't think he needs more than yummy/yucky. But you know it'd blow his mind, and his cooking, to understand sweet/sour/salty/spicy/tangy.

We're still cavemen when thinking about new ideas, and that's why we study math.

## Other Posts In This Series

- How to Develop a Mindset for Math
- Developing Your Intuition For Math
- Brevity Is Beautiful
- Learning To Learn: Embrace Analogies
- Learning To Learn: Pencil, Then Ink
- Intuition, Details and the Bow/Arrow Metaphor
- Finding Unity in the Math Wars
- Why Do We Learn Math?
- Math As Language: Understanding the Equals Sign

Nice one.

Maths is the most fundamental science in the actual meaning of the word ‘science’ – the knowledge. It is this fundamental nature of mathematics that makes us ‘math aware’ and you have described this in a nice way.

The energy-mass relation found by Einstein or the anti-particle discovery by Dirac’s solutions are examples of how maths is far ahead of our ideas and if we truly respect the fundamental nature of maths we can explore the vast information however non-intuitive it may seem to be.

Happy Math

Thanks Harish. Yep, math is pretty fundamental to the world. Is math an approximation of reality, or is reality an approximation of math?

How could the universe come from nothing? Well, how can 0 be split into 1 and -1?

As usual … so insightful. This one line will probably occupy my week.

My mind flashes back to a conversation where a 6th grade teacher was bemoaning her students apparent inability to get the concept of “number” place. And yet the “gamers” in class I presume had already moved on to interpreting the various effects of changing entries within a matrix the started to delve into the Xbox infused space time continuum (infinite lives).

@mark: Heh, awesome :). Yeah, I was really surprised by that “something from nothing” relation [0 = 1 + (-1)]. I think there may be a difference between true “nothingness” and “no net difference” (i.e., two forces which are there, yet cancel).

Good one Kalid. I would go for the latter one – reality might be an approximation of math!

As for Mark’s comment on nothing, let us think about the Big Bang. The theory goes that there was nothing – no space, no matter and no energy before the event. If it was ‘something’ then it has to be ‘nothing’! And everything that is today is out of that ‘nothing’. Even the word ‘before’ can’t be defined in the context, its like zero time (as is zero Kelvin). So, the universe came out of ‘nothing’ as 1 and -1 come out of zero! Wow! satisfying example of ‘reality being an approximation of math’

Thanks Harish. Yeah, I love thinking about these types of things — and math gives us the tools to do it. Would anyone ever consider that something could come from nothing without the 0 = 1 + (-1) example as a starting point?

I’m trying to understand part 3.7 – 3.8 on measuring distance in Maths Better Explained, the one about Rambo, Bambi and Seinfeld. I just can’t figure how you arrived at 6.7 and 13.34 respectively out of those other figures you gave. Could you please help me out

Hi Kevin! For that example, we want the 3d-distance, which is sqrt(x^2 + y^2 + z^2). I put the numbers into this calculator:

http://instacalc.com/5199

Hope this helps!

I like this article’s presentation about the function of mathematics. I think it’s an interesting way of using an analogy of how we thinks with words and more words. But I also think that mathematical ideas were shape by our intuitive sense of direction and space as well. Zero is a math idea; but “emptiness” probably had existed before zero were discovered.

Words have layers of meaning. Some meanings are lay terms but the same word may also be assigned to specialized meanings as well – this means that some words or meanings are more on the side of abstraction. And this is probably also true for math idea. Lawyers are the worst students of mathematics (I’m a lawyer), but even they use fancy terms like “denominator” when they want to borrow the idea of division (just want to sound smarter). But most mathematical terms goes to the side of more abstraction – at least I thinks mathematicians want to keep them that way. In the end, very little math will become intuitive enough to be internalized. What I think is the problem with math is that math is always taught in school – not as a concept of thinking – but as a problem solving science/art. Most people are not problem solving geniuses so they fall apart and hate math. These same people don’t actually hate abstract thinking; they can be very intelligent individuals. But for them, learn to think in abstract with words is easier and more accessible. Whenever one turns to maths, one faces problem-solving challenges.

I think this is what turn people away from math – and lose its benefit as a tool for abstract thought.

Your insights and intuitions that underscore mathematics are a joy to read!

However, I can’t say I concur with the analogy that the universe coming into being from nothing might be likened to splitting 1 and -1. This may work for abstract objects but we should be rightfully suspicious when transposing such a notion across to physical reality. True nothingness has no properties, not even abstract ones, and as such cannot undergird a formula let alone anything tangible. It would seem then that the Lucretian principle of “ex nihilo nihil fit” still holds true… Out of nothing, nothing comes.

Whilst I’d never say “maths teaches you to think”, I think it is entirely true (and apposite) to say “maths teaches you to *reason*”: after your first undergraduate analysis course you never again make unwarranted generalisations or assume properties that can’t be proved; you learn to make clear, structured arguments that are logically sound; you learn how to proceed where intuition fails.

In fact, I suspect that once the “think” to “reason” refinement is made, the rest of your article becomes redundant!

where are the references of the text

The references are you understanding that kalid is a bright individual. If you want “worse explained” then go to http://www.wikipedia.org.

your explanation is still complecated and hard to understand. Can you give a simple concept of matematics that could inspired and give motivation for childern and adults. Why people should learn matematics, what is the direct benefit for living, for small business and for house wife to save the money. Make it simple and factual.

@gunsa

Mathematics is first and life is next. It rules not only our lives but even that of animals. Its right there before we opened our eyes, not in the analytical sense but in the basic sense. A child calculates how much to cry for food, and then how much to cry when in pain. Even an illiterate person has to know price of goods if he wants to survive (even if he barters). Now think about animals. Even they have to know how to count! A bird has to figure out how many days will it take to build its nest, if it doesn’t then it will not survive. A predator must have a sense of how many steps it should be behind its prey and how much force it needs to apply on the ground so that it can catch its prey.

Instinct? yes, but its just another name of built in analytical engine inside living things.

Even non living things are ruled by mathematics. If you look around and see nature itself shows how important mathematical equations are. We throw a stone and its governed by the equation of parabola! And who does not know the amazing occurrence of Fibonacci series in nature!

Kalid…I have been teaching Developmental Math at the community college level for 13 years. My approach has always been to teach from a viewpoint that math is a language. ALL of my presentations springboard from the vocabulary used in that topic. What a delight to read this article written by YOU, Kalid! Thank you!

Hi Erin, you’re more than welcome, that’s awesome to hear! I really think the “math is language” metaphor works — we need to focus on the message being communicated, not the grammar rules! Of course grammar is necessary — but it’s not the focus, what’s the big idea?

Math is such a lonely subject. How do you deal with that feeling?

Sometimes, Math makes you feel small, either that, or mathematical knowledge feels so insignifigant or unimportant that you’d wonder, “have I been wasting my time doing it”

That is probably why I think it’s a lonely subject.

Hi Brian, I think it depends on your perspective. I actually like feeling small and insignificant compared to the ideas, the same feeling you might get when looking at the stars or a sunset. It’s something cool to be experienced! (And therefore rarely a waste of time :)).

Thanks, and I do like the sunset metaphor. (I consider seeing the sunset to be eternal gratification just because it’s so relaxing and calming and you can look at almost like forever if that’s even possible)

PS: Even number theory, E = mc^2 and fermat’s last theorem have hidden signifigance?

If you tell me that

profit = revenue – expense

and

profitability = profit / expense

and stuff like that, I can see how useful they are but for something as abstract as H’s uncertainty principle, I don’t see it’s use.

Mathematics is very important subject in our life.Math makes you feel small, either that, or mathematical knowledge feels so insignifigant or unimportant that you’d wonder, “have I been wasting my time doing it”.Good one Kalid. I would go for the latter one – reality might be an approximation of math!

As for Mark’s comment on nothing, let us think about the Big Bang. The theory goes that there was nothing – no space, no matter and no energy before the event. If it was ‘something’ then it has to be ‘nothing’! And everything that is today is out of that ‘nothing’. Even the word ‘before’ can’t be defined in the context, its like zero time (as is zero Kelvin). So, the universe came out of ‘nothing’ as 1 and -1 come out of zero! Wow! satisfying example of ‘reality being an approximation of math’ :)Even non living things are ruled by mathematics. If you look around and see nature itself shows how important mathematical equations are. We throw a stone and its governed by the equation of parabola! And who does not know the amazing occurrence of Fibonacci series in nature!

I understand that there are things which we don’t understand yet, we call them noise, because we cannot process them. It is such a wonderful thought that we’ve not yet known and bred in our bone what the concept is, but I wish we could be reminded of it every time we balk on finding a contradiction instead of vehemently denying it.

For example, when you said that ‘How could the universe come from nothing? Well, how can 0 be split into 1 and -1?’, I was like, ‘aha! There’s the catch, WE made up zero to represent nothingness and -1 and 1 as quantities for something, maybe matter and anti-matter. It is a mathematical manipulation, a convenient one albeit’.

It doesn’t strike me in the first instance when my instincts are reacting that maybe somewhere someone just just the same thing with universe, which we don’t understand. Until we find out, I guess we’ll never know. But yes, maths is not to be feared, for it is a tool to express things that we observe.

Cheers,

Blasphemous Aesthete

What would have happened if we had not studied mathematics?

As a thinker and a math teacher, so glad I ran across this article and blog.

Very nice articles!

Thanks so much for sharing this article, and this website. Your idea of sharing mathematical ideas through assessing their basic relation to human intuition is the style of learning that I’ve been looking for all my life. I’ve never really been satisfied with building upon principles I don’t understand: finally a website that makes intuitive sense! Thanks Kaled!

Thanks W E-G, glad the approach resonates!

If only you were my math teacher all those years back in school!

Very neat post!

The something from nothing comment made me wonder. If that’s the case then an infinite number of worlds exist because ie -2 and 2 or -0.7 and 0.7 etc come from nothing. However it doesn’t seem to make sense since we cannot have a true infinite number of things in the material sense, otherwise it would be countable?

Secondly I’d love to hear your thoughts on: Is Math a Feature of the Universe or a Feature of…: http://youtu.be/TbNymweHW4E

Excellent explanation

Now i started to love math