Mathematics and Imagination Lessons

I had the opportunity this morning to review some of the notes I made about Immanuel Kant’s Critique of Pure Reason.  I realize on second reading of my highlights in the Kindle book, that I failed to understand what Kant was trying to say, my bad.  Of particular interest to me today was in the preface where Kant says

"A philosophical system cannot come forward armed at all points like a mathematical treatise, and hence it may be quite possible to take objection to particular passages, …”  He goes on to say that he is 64 years old and does not have the time to make his writing free of all mistakes and inconsistencies, he depends upon future generations to smooth the “rough edges” of his exposition.

              Kant, Immanuel. Complete Woks of Immanuel Kant (p. 30). Minerva Classics. Kindle Edition. 

I have been collecting evidence for imagination from other writers, to learn if what they have to say is the same as my own conclusions.  This has led me down some interesting side paths.  Most notably the lecture by Richard Feynman "The Relation of Mathematics to Physics" and a small book “Godel’s Proof” by Ernst Nagel and James R. Newman.  As a former math major this book really expanded my horizons and showed me connections that I had never contemplated.  

I haven’t found anyone that contradicts what I am thinking.  What surprises me is that imagination is such a common theme among mathematicians, scientists and engineers. I see now that all my reading of the past was always in search of an answer to a question, and if what I read didn’t seem to apply to my question, I would ignore it, or at least give it no attention.

I suspect that if I attempt to give you quick answers, I may find myself contradicting what I have to say at later date.

First, I define imagination as mental images, models of what is not before my senses.  Imagination includes memory, planning, what I have in mind to create (write, draw, say).  Intuition is a form of imagination.  I see a car, and my imagination using memories and logic tells me it is a 3 dimensional object with an internal combustion engine and wheels. My imagination may be wrong, it may be running on a battery, it may be clever picture of a car painted on the wall of a barn.  Similarly, we don’t “know” the person we are talking to, we only know our imagination of them.  

In Math, the number two is an abstraction from pairs of items that exist either in the world of in our imagination. The number two is a concept, perhaps it represents a process of pairing.  But you will never find the number 2 in the objective world, certainly not the number “-2”, nor the square root of "-2".

The awakening moment for me was realizing that the number line is pure imagination!  This insight came when I was working on a series of proofs about ordered fields.  Since I am interested in learning the difference between imagination and conceptual thinking I would write each proof imaginatively, drawing diagrams as needed to prove the point.  Then I would work to write a formal proof based on prior axioms, definitions, and previously proved theorems.

Example: If “a” is a member of an order field, numbers are an ordered field, and “a” > 0 then prove 1/a >0.

Imaginatively this is drop dead obvious, 

I imagine a number line: 

<——————————|————|—————————> positive direction

negative                        0             a

I imagine “a” some number >0 on the right hand side of the 0 in the drawing above.

I imagine 1/a as a point on the line that clearly is >0.  Taking the reciprocal of a number cannot be negative.

Notice that for a complete imaginative proof, I would need two diagrams, one for a < 1, another for a >= 1.

Formally, i.e. logically, it takes more thinking.  (The fringe benefit of formal methods is that they come closer to what may be programmed into a computer.)

1. Suppose that 1/a < 0 and “a” > 0, contrary to initial assumptions.  // I am thinking of proof by contradiction
2. Then a * 1/a = 1 > 0 by axiom of multiplicative inverse.
3. But a positive number times a negative number is < 0 (i.e. negative), by previously proved rule that -a=a(-1). // I did this in an earlier exercise, so I can build on the result.
4. Hence supposing that 1/a < 0 leads to a contradiction, that a negative times a positive is > 0, 
5. Therefore 1/a > 0.   QED

Comments, the power of abstraction (conceptualization) is that “a” stands for any number.  The number line is pure imagination, formally this number line represents the mathematical definition of an ordered field.  The image gives our minds something to “hang onto” in order to use the concept of “ordered field.”

Notice that I am using my imagination to manipulate abstract symbols, number line, zero, positive and negatived.  This is a step away from using my imagination to manipulate a pencil and paper to draw a picture.

With practice I have learned that the imaginative proof is frequently in incomplete, as seen above, and sometimes it is just plain wrong.  What seems clear and intuitive is not always correct.  A good example is the “Russell Paradox” of set theory.  

The incentive for creating things, (diagrams, paintings, essays, science, machines, tools, music, etc.), is that it brings my imaginations to life and embodiment.  At the same time creations make my imagination more detailed, complete, and accurate to the domain that I am imagining in.