What is Math?

I was sitting in the way back. I stared at the empty pizza boxes and wished I had come earlier to the session “Math and Philosophy.” Q&A time came along and being the question-asking kind of guy that I am I raised my hand among the veteran thinkers that occupied the room. After a moment’s recognition I said, “So… I’ve been doing this thing for the last 13 years and I know this is a difficult question to answer but could you give me your best opinion on what exactly is math?”… He referred me to some recommended books…

I know its a tough question, but I wish he took a stab at answering it anyway.

So here is my shot on explaining what Math really is:

Math is a fundamental and powerful isomorphism of reality, based on the linguistics of quantity

Math is patterns orchestrated into theories.  And groups of theories can be formed into more grand overarching theories that guide us in our searches elsewhere within the field of mathematics and the field of reality. Theories are like watch towers built to survey the surrounding regions (or the zoom out button when you look at the planet of math)

Math is about relationships and it tells stories.

Imagine a story about Mary and Jerry. They were a happy young two who recently got back from their honey moon in Italy. They were a wonderful couple who particularly enjoyed dancing and playing video games. Jerry works as a chef, and Mary was a flight attendant and part time singer. One day Jerry was trying to get pasta stains out of his favorite Pokémon shirt when Mary waltzed in to the room with an exuberant smile on her face. Jerry wondered what would could be the cause of such joy while the unfortunatehood of his shirt glared back at him. Mary stopped her waltz suddenly, threw her hands in the air and exclaimed, “I’m pregnant with triplets!” Jerry fainted. Eight months later the three; Frank, Wanda, and Juan joined and created the happy family.

(note: an isomorphism is basically saying there are two different things with the same form – a lot like an analogy in that it describes a separate scenario with a similar form/theme)

We can make many isomorphisms off of this story. One isomorphism is to change the story into Chinese, another is to draw a picture of the story, or act out the story, or perhaps somehow moral meaning can be drawn from this story. Furthermore the story might look different if its retold by a videogamer as opposed to a dancer. Certain isomorphisms will focus on some elements of the story more than others. A drawing for example may never show the family with triplets, it might just show Mary excited and waltzing with Jerry shocked while holding his shirt. We could also write the story can be mapped and isomorphed onto logic such as “If Jerry is married to Mary then they will have 3 kids”

We can even map and isomorph the story onto mathematics. And depending on what the subject of interest is we could describe the story as

2 + 3 = 5

Huh?? This hardly tells a story, this gives no information about the emotions and color and social implications. It strips away nearly everything from the story but this also allows us to pursue one goal very well. If we needed to answer the question how many people are in the family it would be substantially easier to answer the question with the story isomorphed into relevant math information, then to have to read the whole story and then answer the question. Admittedly both ways of answering are fairly easy but as my brother once said, “we can do this the easy way, or the really easy way.” (doing things the hard way is not an option… well unless you translate it into Chinese and are not familiar with Chinese)

However what if the story is about a man selling 13 cars for 1,740$ each. We could talk a lot about the story and I could tell you about the man’s name, occupation, social status, favorite ice cream, first dog, and we could talk about the cars or how they were sold but all of this is rather irrelevant to the question of how much money did the man make. So we take this realistic situation and we isomorph or translate it into mathematics. Suppose the man was quite uneducated in mathematics. He may feel that is impossible for him to know how much money he has until he counts it up. Of course we know that this wouldn’t be too difficult but that is because we are so familiar with translating reality into math language. In fact so much of reality hovers around mathematics that we can start to analyze math by itself.

For example once we know that 13 X 1740 = 22620 we can answer both the question of how much the man makes, but we can also find the surface area in centimeters of a long flat line that happens to be 13 centimeters across and 1740 centimeters long. We can even answer hypothetical questions such as if we have 1740 workers being paid 13 dollars per hour; how much money would a company spend on its employees per hour. There are an infinite amount questions we can imagine that can be answered by simply knowing 13 X 1740 = 22620.

Heres where things get interesting. Math has been grown and nurtured by reality for so long we can cut math’s umbilical cord from reality we receive the birth of the mathematics were familiar with. That is the world of pure mathematics. Now we can stop thinking of reality as helping us understand math (It would be rather difficult to come up with the idea of multiplication abstractly without reality providing a host of examples that lead into the idea of multiplication, such as areas of land), Now we can start thinking of math as helping us understand reality. For example knowing how to multiply double digits together could help us find area’s of land much faster. Basically what I’m saying is that it’s almost as if humanity has created a machine (mathematics) that seems to be so intelligent that it has now taken over humanity.

The amazing thing about mathematics is that it really seems like it has a mind of its own. Its hard to explain how this works, but it reminds me of ideas like artificial intelligence. When we do difficult math word problems we often leave the world of reality and instead translate the word problem into math concepts/formulas. Then we solve the problem in the math world and interpret our answer in the real world. For example, consider the following algebra problem

Hrothgar of the realm of Norwoodsian is selling apples and bananas. He sells Apples for $4/apple (these are some realll good apples) and he also sells Bananas for $6/banana (you ain’t never had such a delicious banana). He sold 40 items but forgot how many where bananas and how many were apples. He also made a total of $190. Can you help the dwarf figure out how many bananas and apples he sold? (No… why would I help you Hrothgar… this is your own problem buddy)

Go ahead and think about it for a moment

more moments

many moments of passing

Time doth continue in the way that is does (are we going backward or forward in time? How can you tell? Throwing a ball in the air looks the same as when the ball falls back down just run in reverse)

Time is up (What is even time?)

So you have been thinking about the solution to this problem. We could think about this problem in a logical manner or we could just be stupid and use math rules (yes, the point of following math rules is to be stupid… it allows you to not think as much. Computers are great at math, but they have no intelligence, they just blindly follow rules without questioning)

Lets do the logic way. Okie dokie. Hrothgar is in a pickle so lets help him out. We know he sold 40 items and that he made $190. We know that bananas are $6 and apples are $4. So how can we determine how many of those 40 items were bananas and how many were apples? (Critic: “seriously, i hate when math teachers respond to my questions with more questions… screw you Socrates.”)

Well we know there is a difference between apples and bananas. (much like the differences between apples and oranges) That is that banans are $2 more expensive then apples. Hmmmm, so then if we replace an apple with a banana then we gain $2. How much is 40 apples? (Critic: Stahhhhhhhppp)

40 apples is $160. If instead we had 39 apples and one banana we would have $162. Or 38 apples and 2 bananas would be $164. Every time we replace an apple with a banana we gain $2. So how many $2 bills must we gain to go from $160 (price of 40 apples) to $190 (amount Hrothgar made).

Well the difference between $190 and $160 is $30. How many $2 bills make up $30 (in other words whats 30 divided by 2).

The answer is 15.

15 means what? (critic: do you really want me to say bananas… okay fine, I’ll say the word)

15 is the amount of bananas! (Critic: yeahhhhh, lets all go crazy over a math answer… NO)

So then the amount of apples is 25.

Alright… were done with logically thinking about it. But we can also climb the mountain using a different path (we find this strange door at the base of the mountain that leads to a long hallway. At the end is an elevator that takes you all the way to the top of the mountain)

There are two parts to this journey. First we must translate the problem. Lets call the amount of apples X and the amount of bananas Y.

The total amount of money made from apples is price times quantity so 4X. The total amount of money made from bananas is 6Y. What can we do now? What information do we have to help us?

The total amount of money made is $190, the total money from apples is 4X, the total money from bananas is 6Y. Thus it seems fair to say

4X + 6Y = 190

Well thats cool but how do we solve for X and Y. Well if you know your Linear/Matrix Algebra you know you need at least two equations to solve two independent vairables. So we need another equation.

The total amount of items is 40. Can you use this information to make another equation

(so many dots………………………..)

All the dots…… I just love dots… dots are the silence after a phrase. It is difficult to listen and contemplate at the same time. So after a phrase, silence allows us to dwell on it……….

and were back. 40 items, we said quantity of apples is X, and quantity of bananas is Y.

Thus

X + Y = 40

and, noted previously

4X + 6Y = 190

There are a few ways to solve it from here but lets use one of my favorite mathematical properties… subsitution (I substituted Mr. Witherspoons wallet with mine… heh)

so X + Y = 40, but if this is true then Y must equal the difference between 40 and X. So then Y = 40 – X

Warning: what you are about to witness is very devious. We will now swap the Y in the equation of “4X + 6Y = 190” with the fact that Y = 40 – X

So 4 times the quantity X plus 6 times the quantity Y altogether equals 190. But Y also equals 40 – X. So with a slight of hand we have

4X + 6(40 – X) = 190

woah…

one equation… one variable…

Solvable.

Distributing the 6 to the 40 and the X we get

4X + 240 – 6X = 190

rearrange

4X – 6X + 240 = 190

combine like terms

-2X + 240 = 190

Subtract 240 from both sides of the equation (we can do anything we want to one side of an equation as long as we do it to another). The reason we do this is so that we can isolate and find X

-2X + 240 – 240 = 190 – 240

-2X = -50

divide both sides by -2

X = 25

X + Y = 40

so Y = 40 – X

and if X = 25 then Y must be 15

Hence X = 25, Y = 15

Now we must wake up from our math world and come back to reality. X is quantity of apples and Y is quantity of bananas. There you go Hrothgar

*Hrothgar hands you the axe of king Toramus as a token of his thankfulness… +8 attack damage gained*

Wow…

That was a lot to digest (much like the amount of mac&cheese I am about to digest).

If you followed the second method carefully then all of the steps we made hopefully make sense. But it still feels so strange that we never really intuitively thought about how to answer the question, but simply by using math properties we managed to solve it perfectly. We ignored reality all together once we established our equations.

How peculiar. I like this problem because its simple enough that you can reason through it logically without too much trouble, but the intuitive reasoning still remains hidden as we solve it mathematically. But higher level math problems can be incredibly difficult to solve intuitively.

For example consider the following word problem that uses differential equations:

 

“A 1000 gallon holding tank that catches runoff from some chemical process initially has 800 gallons of water with 2 ounces of pollution dissolved in it.  Polluted water flows into the tank at a rate of 3 gal/hr and contains 5 ounces/gal of pollution in it. A well mixed solution leaves the tank at 3 gal/hr as well.  When the amount of pollution in the holding tank reaches 500 ounces the inflow of polluted water is cut off and fresh water will enter the tank at a decreased rate of 2 gal/hr while the outflow is increased to 4 gal/hr.”

Determine the amount of pollution in the tank 200 hours after the intial conditions.

Go!

… (Reader: “Nope!!!”)

Yeah… i know.

all the information we need is here. Theoretically we could logically reason through everything and find the answer. But this is extremely daunting. So most people wouldn’t attempt this question unless they had a firm understanding of differential equations.

Its really fascinating how powerful math is. It still confuses me why we can take reality translate it into a mathematical language, follow a series of rules, and then come out with an answer to whatever real world question we had.

here is one last example perhaps closer to home. Assuming you remember algebra 2, one of the things you were repeatedly force fed was the quadratic formula (an odd looking ugly creature)

Though the formula has quite a lot of intuition underneath its very easy to miss it all. Here is a good example of a simple problem that would be rather hard to solve with knowing about quadratics. This is so strange since this is just a real world problem. but again we translate the problem into these weird symbols to follow weird rules with weird formulas and wind up with the precisely correct answer

So here is the problem:

“A 3 hour river cruise goes 15 km upstream and then back again. The river has a current downstream of 2 km an hour. What is the boat’s speed?”

It looks fairly innocent. Maybe its hard to solve, but the question itself is pretty straightforward. You could almost imagine someone reasonably asking a question like this in the real world. And yet if you try to just “think about it” you may find that its really difficult. Or instead you can just translate it, put it into a quadratic formula and solve away…

here is the solution for those interested (look near the bottom of the page):  solution

So you’ve managed to read through this entire article (trying to understand my terribly confusing writing) because some part of you is curious like me as to what math is. So let me give you my final thoughts

Essentially my conclusion is that math is a language born from the fundamental properties of the universe. So math is in a way human-made, but since it is so intertwined with the properties of the universe itself, it also has a life of its own.

Its as though we invented English, but then happened to find that all the stars in the sky form constellations and series of constellations that make perfectly coherent words and phrases.

Pragmatically speaking however, math is an isomorphism. It is an analogy of reality confined to subject of quantity.

Ultimately… math is weird

And math is connected to the universe

So the universe is weird

And you are a part of the universe

Which explains your life question of why you are so weird

And your life is a story

Which means the universe is a story

In fact, I do believe in a strange way math itself is actually a story…

Remember what I said about those dots…

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