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.

 

What Does Literature Mean to Me?

Jorge Luis Borges: Conversations by Richard Burgin
My rating: 5 of 5 stars

I have just started reading this collection of interviews. I am struggling to come to terms with literature. I hope this book is useful for my quest. I read the introduction, first and last interviews and skipped around a bit with the help of the index. I don't think I every finish this book. I did go read a few of his stories. They were good, but I believe Ray Bradbury, or Roger Zelazny are better.

Page 7 "Writing's the kind of activity between thinking and dreaming. You have a dream at the outset and then somehow you have to pin it down." This is interesting to me because it is the model for mathematical work, we image a theorem that needs to be invented, then we pin it down by checking it for contradictions against previously invented axioms and theorems. Both writing and and mathematics are an effort to explore the gap between what we know and what may be.

Page 14 he says "... I think of reading a book as no less an experience that traveling or falling in love. I think that reading Berkely or Shaw or Emerson, those are quite as real as experiences to me as seeing London, for example."

I discovered Borges from a scene in the Netflix movie "Squid Game." He was the author of a book on the desk of a missing person.

In one of his interviews he says the purpose of literature is pleasure. Ok, my 7th grade English teacher, Mrs. Ryan, told me the same thing. I keep suspecting that there is something deeper because of works such as "The Brothers Karamazov," not a pleasant story. So my quest to understand "why literature" is still continuing. Azar Nafisi in her book "Reading Lolita in Tehran" said that the purpose of literature to to explore new ways of being human. I get this. In one of his books Richard Dawkins said that instead of reading great literature, we now read popular psychology books to learn about the human life. Interesting idea and consistent with my own reading until I came to the conclusion that humans are immensely malleable and the varieties of people are even greater that the varieties of songs to be sung using the finite scale of notes given us.

Borge likes American movies, especially westerns because they are modern epics, and he like detective stories because of their solid plots. Ok, so my tastes in reading and viewing are redeemed.



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The Book of Why: The New Science of Cause and Effect by Judea Pearl
My rating: 3 of 5 stars

My notes are for the sake of future research that I may do. Statistics is field of endeavor I have avoiding out of my preference for axioms and theorems of pure mathematics. I gained most of my statistical experience from college physics experiments, the classic "least squares fit" that economists call "regression analysis."

This book is fascinatng. When I first read about Bayesian statistics about 1981, it was part of our fire control software, I judged it to be nonsense. Now I understand better. Judea Pearl does not mention Kalman filters, but they were critical to our success in the Navy.

His annotated bibliography is excellent. It is organized by corresponding chapters. I thought this adds to the difficulty of looking up specific items, but it is still manageable. An appendix of terms, acronyms and definitions would have been useful. The index is OK to assist with these kinds of questions.

His case studies included examples from genetics, cholera, smoking, college admissions, nurture vs nature, cause of scurvy, "algebra for all" education policy, use of tourniquet on the battle field, and others were all fascinating. Some of them were scary because they showed how good people could get things terribly wrong.

I didn't take the time and extra research to learn his methods but I found a lot of good information in his book:

Ladder of causation page 28.
Level one is observations e.g. data collection, this is classics statistics. What are the variables, how are they related?
Level two is intervention, what can we do to alter the observations? e.g. Would aspirin help my headache?
Level three is counter-factual, what if different choices or circumstances had existed before we made our observations? e.g. did X really cause Y? what if X had not been present?

Casual diagrams blew my mind! The power of such a simple tool to organize ideas, discover statistical bias, and understand their possible relationships. He clarifies from the findings of the courts the difference between discrimination and bias. Discrimination is an act of will, bias is something that happens and needs to be discovered.

Different variables in a casual diagram:
Confounding, deconfounding,

chain junction, fork junction, collider junction

The puzzle of Simpson's paradox where aggregated results contradict results from partitioned subsets. The lesson is that aggregating or partitioning data needs to be done with an awareness of the process that generated the data. {Personally I have seen too many software engineers and data people dive right in with their aggregating and partitioning tools without questioning the suitability of their tool. A good example is HADOOP and MapReduce.}

"Back-door adjustments" are not easy for me to describe, but vital to deciding interventions. Apparently the back-door adjustments is a means of eliminating confounding variables, i.e. "all else being held constant ...." (another phrase economists seem to like to use).

Moderating factors, interactions between possible causes before the effect is observed.



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The World Beyond Your Head: On Becoming an Individual in an Age of Distraction by Matthew B. Crawford
My rating: 1 of 5 stars

Reads like an undergraduate research paper under the review of a post-modernist philosopher.
The book has one wonderful chapter on the business and craft of repairing, building pipe organs.
The the remainder of the book is treating a narrow slice of academic life, lectures, cafeteria, gymnasium, and social media as if they are the essence of modern culture.

An important distinction that he overlooks is the difference between culture and nature. He treats to idea of resentful surrendering to the person of a ruler, authority, the same as surrendering to a fact of nature. He claims that "people" (I wonder whom he is referring to?) resent surrendering to the facts of nature as an extension of the ideas of learning to think for themselves. He considers independent thinking to be a dismissal of thinkers of the past, but independent thinking is coming to understand what other thinkers say, not ignorance of them, (except maybe August Comte who reportedly thought he would not bother to study Aristotle and invent all of philosophy on his own).

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Quote from W.E.B. Du Bois, "Black Reconstruction," 1935

Quote from W.E.B. Du Bois, "Black Reconstruction," 1935

Page 702.

The effect of caste on moral integrity of the Negro race in America has thus been widely disastrous; servility and fawning, gross flattery of white folk and lying to appease and cajole them; failure to achieve dignity and self-respect and moral self-assertion, personal cowardliness and submission to insult and aggression; exaggerated and  despicable humility; lack of faith of Negroes in themselves and in other Negroes and in all colored folk; inordinate admiration for the stigmata of success among whiter folk; wealth and arrogance, cunning dishonesty and assumptions of superiority; the exaltation of laziness and indifference as just as successful as the industry and striving which invites taxation and oppression; dull apathy and cynicism; faith in no future and the habit of moving and wandering in search of justice; a religion of prayer and submission to replace determination and effort.

Review of Sam Harris's "Waking Up: A Guide to Spirituality Without Religion"

The value of this book for me is that reading his words leading to "no self" suddenly clicked for me in terms of my own deeper sense of what this is about and being able to live in "no self" at any moment I want. Would I have gotten it without having years of daily meditation, self-remembering exercises, and periodic silent retreats sponsored by various spiritual organizations? I don't know. Perhaps it simply comes with maturity and time? I believe that some disciplines call Harris's "no self" the true Self with capital "S" or the "impersonal Self" shared by all beings. 

This book is a quick and easy read. Some of his high level remarks seem skewed, but he explains and elaborates later in the book, e.g. logical conflicts between religions. I believe that he occasionally confused poetic metaphor with assertion of facts, e.g. his critique of Gurdjieff's cosmology, the moon eats us, the old and new testaments of the Bible. He seems to be blind to the distinction between sermons for the students and sermons for the "coarse" folks (the "coarse" adjective from the Confessions of Saint Augustine). This same distinction has been made by commentators about the teaching of the Buddha, some sermons for the common folks, different sermons for the students.

Harris's discussions about neurology are fascinating. He stays in the essence of the problem. Many researchers, such as Daniel Dennett, Roger Penrose, get detoured into non-essential, but easier to access phenomena. On the other hand the only book I read by Dennett was "Consciousness Explained." Dennett suggested that if readers wanted to get to the really essential questions of consciousness, they should read his "The Intentional Stance." I haven't read this yet.

George Orwell's Essay "Looking Back on the Spanish War"

Timely quotes from England 78 years ago!

"As far as the mass of the people go, the extraordinary swings of opinion which occur nowadays, the emotions which can be turned off and on like a tap, are the result of newspaper and radio hypnosis.  In the intelligentsia I should say they result rather from money and mere physical safety."  (Page 145.)/(last paragraph of section 1)

"Recently I drew up a table of atrocities during the period between 1918 and the present [1942], there was never a year when atrocities were not occurring somewhere or other, and there was hardly a single case when the Left and the Right believed in the same stories simultaneously.  And stranger yet, at any moment the situation can suddenly reverse itself and yesterday's proved-to-the-hilt atrocity story can become a ridiculous lie, merely because the political landscape has changed." (Page 146.)/(second paragraph of section 2.

References

George Orwell, Facing Unpleasant Facts, Narrative Essays, Compiled by George Packer, Harcourt Inc. 2008.

https://orwell.ru/library/essays/Spanish_War/english/esw_1