# The Problem with Math, and a Book Review, from Bruno Ombreux

May 7, 2007 |

There are two issues. One is description of reality. The other is math.

I have just finished the book "Understanding Scientific Reasoning" by Giere, Sickle and Mauldin. It is shedding some light on the first issue. This book has two main parts:

- Theoretical Hypotheses
- Statistical and Causal Hypotheses

Throughout the book, it is made clear that science doesn't pretend to describe any form of absolute reality, out of Plato's cave, but rather to produce approximate models of world phenomena. It is also made obvious that science is not something static, but rather an always-evolving process.

Reading the first part of the book is enough. The second part about statistics is less interesting. The authors somewhat fail to make it clear that statistical hypotheses are a subset of theoretical hypotheses. Perhaps the reason why the authors are devoting so much of their book to trite statistics is that this subject is difficult for their students. Anyway, the first part is great and providing the following summary for any scientific process:

fit/no fit
REAL WORLD      —————————–

—-    MODEL

|                                                                |

|                                                                |
|                                                                |
observation
reasoning/calculation
experimentation                        with experimental setup
|                                                                |
|                                                                |
|                                                                |
V
V
DATA           <———————————–>   PREDICTION
agree/disagree

There is a "real world" from which data is observed. There is a model describing part of the real world. From the model, a prediction about real world data is derived. The prediction must agree with the data in order for the model to be deemed fitting. A statistical model is nothing but a model expressed in statistical terms, e.g., "real world population is normal". Its predictions are what samples should look like if said model is true. The authors are providing a city map analogy. A map is a model of a city, not the city itself. It makes predictions: "Street A is intersecting Street B". These predictions can be checked by walking in the city; that is collecting real world data.

The terms "models" and "hypotheses" are used interchangeably, because they are one and the same thing. A model can be one hypothesis or a collection of hypotheses. "Hypothesis" makes it even clearer that science is not pretending to touch the essence of whatever the "real world" is.

The book is not dealing with the second issue, mathematics. We can ask ourselves where does math fit in?

So far, the description of scientific processes has not involved math. It doesn't need to. Science doesn't need math. Science has but one constraint: to be grounded in logic. Logic can be deployed in math, but also in language, that is rhetoric, or in graphical illustrations, drawn following strict rules.

Compare these 3 versions of Newton's Law of Gravitation:

1. Plain English

"Any two objects exert a gravitational force of attraction on each other. The direction of the force is along the line joining the objects. The magnitude of the force is proportional to the product of the gravitational masses of the objects, and inversely proportional to the square of the distance between them."

2. Graphical (assuming proportions are respected in the drawing)
______________ r _________________

F -F O—————————-o M m

3. Mathematical

F = g * M * m / r^2

One can see immediately the huge advantage of mathematics: Concision. Economy of space. Economy of ink. It is entirely possible to deal with science in plain English without a single formula, but books would be at least 20 times weightier.

Mathematics are a concise lingua franca. But everything written in the language of math can be rewritten in any human language or in the form of graphical descriptions / geometrical figures; think Chinese ideograms or Egyptian hieroglyphs.

The reason why mathematical descriptions do not correspond to the real world, is that they are used in the process of scientific discovery, which itself never applies to the real world, only to the formulation of hypotheses about this world.