X Marks the Spot

The Lost Inheritance of Mathematics

Richard Garfinkle, David Garfinkle

• Publisher: A K Peters/CRC Press
• Year: 2021
• ISBN:
  9780429588624
• Edition: 1
• Topic: Mathematics
• Subtopic: Games in Mathematics
• Index: Mathematics

X Marks the Spot is written from the point of view of the users of mathematics. Since the beginning, mathematical concepts and techniques (such as arithmetic and geometry) were created as tools with a particular purpose like counting sheep and measuring land areas.

Understanding those purposes leads to a greater understanding of why mathematics developed as it did. Later mathematical concepts came from a process of abstracting and generalizing earlier mathematics. This process of abstraction is very powerful, but often comes at the price of intuition and understanding. This book strives to give a guided tour of the development of various branches of mathematics (and what they're used for) that will give the reader this intuitive understanding.

Features

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• Treats mathematical techniques as tools, and areas of mathematics as the result of abstracting and generalizing earlier mathematical tools

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• Written in a relaxed conversational and occasionally humorous style making it easy to follow even when discussing esoterica.

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• Unravels how mathematicians think, demystifying math and connecting it to the ways non-mathematicians think and connecting math to people's lives

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• Discusses how math education can be improved in order to prevent future generations from being turned off by math.

1

Why This Book?

If we said that this was a math book, how many people would run screaming into the night rather than read it?

If we said that this was a book about how mathematicians think, how many would toss it aside because who cares how they think?

But if we said that this is a book about something mysterious, invisible, and barely understood that is used to control and manipulate your life, what would you do?

And if we said that this is a book about a rightful inheritance, a wealth of tools for remaking the world as you see fit, that for many of you was lost through no fault of your own, wouldn’t you want it back?

Even if that inheritance was stored in a box with a complicated lock and the name “Mathematics” on it?

Math is the common heritage of all people, a power over the world that early in life we have a sense of and a desire for.

Consider:

The first complaint a child makes is “I don’t want to!”

The second is “It’s not fair!”

The first comes from human nature, the second from mathematics. Without a sense of equality, how can we have the wounded fury that rises from seeing someone else have more than we do?

Once we get past this primal arithmetic judgment, many of us lose this basic grasp of mathematical principles. We are soon taught that math is a mysterious other way of thinking, that the people who practice it are strange, aliens dwelling among us. But in that second cry of a resentful child can be seen how deeply math is ingrained in our everyday thinking. The aliens are not just among us; they’re in our heads! (cue fleeing crowds).

But what is this lost inheritance, and why is it so tightly bound up with basic human thinking?

Consider the following process of thought that everyone follows:

Step 1: Observation. A person looks out at the world and sees something.

Step 2: Internalization. A thought of that thing is created in the person’s mind.

Step 3: Imagination. The person manipulates that thought, changing it or placing it into a story so that a different thought related to the initial thought is generated.

Step 4: Externalization. The person takes the latter thought and tries to treat the real-world object we started with as if it had to act according to the thought we imagined.

In most cases, this process does not work. We cannot, just because we imagine a witty dialogue with another person, make that person play the right part, saying their lines at our behest and accepting our conclusions when we have made them. We cannot, just because we envy them, fly like birds. We cannot make clouds edible just because they look like marshmallows. We cannot force the world to conform to our imaginations.

This process of thought, of attempted control of reality by act of conception, is sometimes called magical thinking and it does not, in general, do what we want.

But there is a region of thought and action where we can make this process work. It is a region that must be entered with care and navigated with precision. In this region, we:

1. 1. Observe with great care, measuring accurately and properly the characteristics of the real-world object.
2. 2. Internalize those characteristics into a mental object that works as a model of the physical object.
3. 3. Transform this internalized mental object using processes of thought which preserve the truth and accuracy of our modeling to produce other thoughts that we:
4. 4. Apply out in the real world using new knowledge of the object.

Done correctly, this more controlled and delicate use of magical thinking actually works. It allows for a cycle of interaction between world and mind in which mind takes from the world, alters what it takes, and then puts it back in order to change the world.

In this way, we can find the weights of buildings, the heights of mountains, the distances of stars, the odds of winning and losing games, and the reason written words appear on a computer screen as we tap keys.

This region, this strange region of thought where magic works, is mathematics.

It may seem that a region of magic where you have to work at it and take care isn’t magical enough. After all, what’s the point of wishing for something if it takes ages to prove whether or not you can have your wish? But it seems to us worth the effort. It’s mind-boggling that there can be such a correspondence between the real world and our minds, and that yes, one may have to work at it, but at least some of our wishes can be granted if we learn the proper ways and are willing to take care with how we formulate our wishes.

This is the lost legacy of mathematics.

To be fair, there are two perspectives on this process of removing from the world, working in mind, and putting back: the mathematical, which largely focuses on the work of imagination, and the scientific, which is equally concerned with the back-and-forth between the worlds.

Thus, understanding math as an inheritance to be recovered requires not just looking at mathematics, but at science as well. We wrote a book, Three Steps to the Universe (University of Chicago Press, 2008), in which we tried to explicate the way scientists think in the course of discussing some of the stranger aspects of modern astrophysics.

Unlike that book, the book in your hands does not have the flare and flash of the sun, the destructive images of black holes, nor the hidden picturelessness of dark matter and dark energy. This book is almost completely concerned with things that go on in the human mind. But those things are closer to hand than the material objects we discussed in our first book.

Look below and count the dots:

• . . . . . . . . . .
• 1 2 3 4 5 6 7 8 9 10

The dots are spots of ink on a page, or spots of light on a screen. The numbers printed below the dots are also patterns of ink or light. But the counting, that’s a thing in your head, a purely theoretical thing.

If you read Three Steps to the Universe (and, of course, we selfishly think you should), the word “theoretical” will set off bells in your minds. We talked about scientists seeing the world as made of three universes:

The perceived universe, containing what is actually observed.

The detected universe, containing what can be discerned by experimentation.

The theoretical universe, containing the explanations for what is perceived and detected as well as the means by which predictions of later perceptions and detections are crafted.

Mathematics is one of the vital elements that make up the theoretical universe of the sciences, particularly in physics. Math makes that universe rigorous. This sounds kind of dull, but it is the rigorousness, the exactitude, and the care with which mathematical ideas are transformed from one to another that preserves the truth that enters the theoretical universe from the detected, so that what emerges at the end of the process is as true as what came in.

That’s actually pretty amazing. A path of thinking that preserves truth is unusual, to say the least. Truth is easy to lose in the forest of associations, emotions, memories, desires, fears, images, and catchphrases that fill up people’s minds.

How can this preservation of truth within the bewildering forests of mind work?

We’ll be looking into that a lot in this book, but let’s start by looking at the theoretical universe.

This is the universe where our understanding of things lies. In its shallowest regions, there are names of things. We look at something and think “chair.” That’s a theory of what the object is. Looking closer, we might find out that it’s a papier-mâché model of a chair or a picture of a chair or that it actually is a chair, but it’s too small for us to sit in.

We examine and we test and we refine our theories based on reality. But we can also use our theories to discern what the nature of the objects that are real are. It’s a chair; I can sit in it. You sit down and something snaps; low comedy ensues. We look more closely and see that the wood of one of the legs is rotten. We refine our theory further. I might be able to sit in the chair, depending on how sturdy it is.

And now we hit science and math. “How sturdy is that?” is a scientific question with a mathematical answer. In order to answer it, we need to define sturdiness, measure whatever contributes to sturdiness, and calculate an answer to the question. That answer has to be a helpful one.

We need a measure of sturdiness that will allow us to know what we can safely pile on the chair (starting with us). The uses to which we will put the answer help us form the question properly and guide us through making the theoretical and detecting processes we can use to find that answer.

This backward creation from needed answer to means of calculation and needed experiments is one of the ways that math and science are created and one of the reasons they actually fit the world of mind and the world of fact together.

The point is that the deeper we want to look into the real universe, the deeper we need to look into the theoretical universe. Inherited estates in the latter give us estates in the former. Gain ground in theory, gain it in reality.

Mathematicians reading the above may object to the subordination of math to science that is implied. That’s because mathematicians and physicists are like siblings sharing the same work. They know a lot about each other, and while they work together well, they are also strongly aware of how different they are each from the other, and they are very insistent on those differences, sometimes to the point of loud and occasionally silly arguments (much to the amusement of other family members).

Seriously, there is a fundamental difference between scientific and mathematical thinking, and while it will show up throughout this book, we won’t get to it until quite late (Chapter 12: The Smith and the Knight). Before we can explore those differences, we need to survey the world of mathematics in its assumptions and applications.

Why read this book? We’ve admitted that the flare and flash of new discovery isn’t here. The cool stuff from science books isn’t going to jump off the pages. So what about the subject of mathematics itself?

It is the exploration of a world that is simultaneously the shadow of our world and that casts our world as a shadow. It is a reflection that reveals the thing reflected. It is intimately bound up with reality in our minds, and to a great extent we cannot ever leave it. We can learn to know this vast estate or hide in one small room wondering what the noise is around us.

Unfortunately, to most people, mathematics is something that was forced upon them in school without any of this connection to reality. It was an arbitrary se...