Seek clarity of thought? Here’s how

Are you struggling with a problem, but are unable to obtain clarity on a solution? Here’s an approach to get clarity that has worked for me: read a book with great clarity; that is, a book that expresses fundamental ideas in a simple, clear way.

I am on my fifth read of An Introduction to Mathematics by Alfred Whitehead. Every time I feel my thoughts getting muddied I pull out this book and read it. This book takes the complex subject of mathematics and describes its fundamental ideas with brilliant clarity. Reading this book brings clarity to my mind, in all of my thoughts.

Do you have a book that you enjoy reading because of its clarity of thought?

The importance of examples

Examples are essential for the stimulation of our thoughts.

— An Introduction to Mathematics by Alfred North Whitehead

Misty Profundity

We’ve all had this experience: you are listening to someone talk and think, “Wow, this is really profound, albeit vague, stuff”, and then later, while reflecting on it, you realize “That was a bunch of crap.” Alfred North Whitehead states this situation nicely:

“It is a safe rule to apply that, when an author writes with a misty profundity, he is talking nonsense.”

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Here’s a related blog: A Test of “Profound Statements”

The last thing to be discovered in any science is what the science is really about

Mathematician studied for centuries the abstract problem of drawing tangents to curves before Newton finally discovered what the subject really is about — rate of change.

It is a well-founded historical generalization that the last thing to be discovered in any science is what the science is really about.

Men go on groping for centuries, guided merely by a dim instinct and a puzzled curiosity, till at last some great truth is loosened.

— An Introduction to Mathematics by Alfred North Whitehead

Joint Discoveries Are Common

Joint discoveries are quite common in science.

Discoveries are not in general made before they have been led up to by the previous trend of thought, and by that time many minds are in hot pursuit of the important idea.

Examples of joint discoveries:

• Law of natural selection: Darwin and Wallace
• Discovery of Neptune: Adams and Leverrier
• Creation of differential calculus: Newton and Leibniz

— An Introduction to Mathematics by Alfred North Whitehead

Progress of the History of Thought

The progress of science is divided between

• periods characterized by a slow accumulation of ideas, and
• periods when, owing to the material of thought that has been patiently collected, some genius by the invention of a new method or point-of-view suddenly transforms the whole subject on to a higher level.

The genius who has the good fortune to produce the final idea which transforms a whole region of thought does not necessarily excel all his predecessors who have worked at the preliminary formation of ideas. In considering the history of science it is both silly and ungrateful to confine our admiration with a gaping wonder to those men who have made the final advances toward a new epoch.

— An Introduction to Mathematics by Alfred North Whitehead

See what is general in what is particular and what is permanent in what is transitory

Consider how all events are interconnected.  When we see the lightening, we listen for the thunder; when we hear the wind, we look for the waves on the sea; in the chill autumn, the leaves fall.  Everywhere order reigns, so that when some circumstances have been noted we can foresee that others will also be present.

The progress of science consists in observing these interconnections and in showing with a patient ingenuity that the events of this ever-shifting world are but examples of a few general connections or relations called laws.  To see what is general in what is particular and what is permanent in what is transitory is the aim of scientific thought.

— An Introduction to Mathematics by Alfred North Whitehead

What symbolism does ET use?

The simplest and universally known symbols are the integral numbers, represented in the Arabic notation by 0, 1, 2, …, 9, 10, 11, …, 98, 99, 100, 101, …

We are so used to these symbols that we think that they always existed.  However, it wasn’t until the 11th century when an Indian mathematician, Bhaskara, created them.

We see these symbols as obvious and natural.  And now we can’t imagine any other way of expressing numbers.

It’s interesting to ponder how a civilization on another planet symbolizes integral numbers.  It’s highly unlikely that they use the same symbols.  Perhaps they’ve created a symbology which makes things easier.  Perhaps it has enabled them to progress their mathematics and technology at a faster rate.  I wonder if their symbology uses base 10?  I wonder if we humans will discover an even better symbology than the notation we currently use?

“The symbols for integers illustrate the enormous importance of a good notation.  By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race.” [Whitehead]

“Before the Arabic notation was introduced multiplication was difficult, and the division even of integers called into play the highest mathematical facilities.” [Whitehead]

I wonder if we are creating today any symbolism that will “increase the mental power of the race”, and will endure for centuries or millenium?

Great Book = Interesting Information + Arguments

What is the longest argument you’ve ever made? What is the longest argument you’ve ever read?

By an “argument” I mean: collect together some information nuggets, show how they are related, and then draw a logical conclusion from them.

Most of the (engineering) books I read are oriented toward providing information and techniques, not toward forming arguments.

Recently, however, I have been reading two outstanding books:

— Wealth of Nations by Adam Smith

— Introduction to Mathematics by Alfred Whitehead

And through careful reading I have become aware of the arguments being made in these books.  I say “careful reading” because their arguments aren’t immediately obvious, at least not to me.

After reading a page I pause and reflect on the ideas presented.  Slowly I am seeing how the arguments are being constructed.

In Smith’s book the arguments are well contained; at the end of each chapter he ties together the various parts of the argument.  Whitehead’s arguments are more complex and subtle; they can span multiple chapters.

Whitehead’s book is on mathematics.  It occurred to me, “Why are there arguments in a book on mathematics?  Shouldn’t it just contain information and techniques, like my engineering books?”  I’ve been puzzling over why I like Whitehead’s book so very much, particularly since I am not especially interested in mathematics. Now I think I know why: because it contains both information and arguments.

The realization that I have come to is that I like books which contain both interesting information as well as arguments.

Whitehead was both a mathematician and a philosopher.  Smith was both an economist and a philosopher.

A philosopher is a master of arguments.

I think great books are those that contain interesting information and are also philosophical (i.e. contain arguments).

In our sound bite society we don’t see many long, elaborate, elegant arguments.  That’s a shame.

Measuring Time

Time is defined by reference to astronomical phenomena. Astronomical recurrences mark out equal intervals of time:

– a year is defined as one trip of the earth around the sun

– a day is defined as one rotation of the earth

Relegation of the determination of the measure of time to the astronomer arises from the consistency[1] of the recurrences with which they deal. If such consistency had been noted among the recurrences characteristic of the human body, we would have looked to the doctors of medicine to determine the measure of time[2].

[1] Example of a “consistent” recurrence: the number of days it takes for the earth to orbit the sun is 365.25 days, year after year … the recurrences are consistent. For all ordinary purposes of life on earth, the various astronomical recurrences may be looked on as absolutely consistent.

[2] The heart beat is periodic, but not consistent; it beats quickly when we are active, slower when we are resting; such inconsistent recurrences would not be useful for measuring time.

— Alfred North Whitehead, An Introduction to Mathematics

Question: how is “hour” measured? Is there an astronomical recurrence that indicates an hour? An hour is one twenty-fourth of a day, of course, but how did the ancients realize that one twenty-fourth of a day has elapsed?