The Mathematician's Mind

Jacques Hadamard was a well know French mathematician who undertook a monograph length exploration of "the psychology of invention in the mathematical field" (as the subtitle has it).  The essay is particularly interesting because it combines introspection into his own working process with reports from a variety of other mathematicians to concoct a basic theory of how it feels to discover a new mathematical idea.  The upshot is that most mathematician's seem to work with various visual, auditory or even motor (in the case of Einstein) images at a pre-verbal level.  Hadamard takes these images to be conscious representatives of unconscious processes that get linked together as so many stepping stones that later guide linguistic or conventionally symbolic arguments.  In short, there's nothing deductive about mathematical invention, which in Hadamard's view is not a substantially different process than what we might more readily associate with poetic invention.  The conscious mind prepares the field with a long immersion into the subject.  Then the mathematician sleeps on it, so to speak, and their unconscious tries out various combinations until a successful one bubbles to the surface in the form of concrete images.  Finally, these images must be consciously and painstakingly translated back into a form that will stand up to the scrutiny of logical communication.  I would reserve judgement on whether Hadamard's is an adequate description of all possible types of invention; surely the question has been studied in much greater depth in the 70 years since his publication.  But it is a coherent and interesting one that definitely fits with my own experience of writing.

Cutting Through Spiritual Materialism

I picked up Chögyam Trungpa's classic about three years ago and found I couldn't get into the first few chapters.  I'm not sure what I didn't like back then because this time I found the book really insightful and engaging.  These transcriptions of talks he gave to Western students back in the seventies fall into two sections.  In the first half, Trungpa explains what the spiritual path is not, and in the second he begins to progressively explain what it is.  

The title of the book stems from the message of the first part.  The spiritual path is not another attainment that we can "get".  It's not another skill or notch in our belt or collectible display item for the modern home.  The idea that the path leads to a goal is what stops us from walking down it.  Trungpa spends many chapters warning about the various illusions and self-deceptions we will accidentally create for ourselves as we try to use our concept-addicted ego to navigate.  It's only when we give up this ego's desire to possess wisdom, when we surrender and open up to our inability to grasp things that we begin get an idea of where we are going.  The overall message is very similar to The End of Your World -- let go of your expectations about how things should be.  Since giving up is a lot harder than it sounds, it's nice to hear many of these same lessons repeated.

The second half of the book contains Trungpa's own presentation of the path from the perspective of his Kagyu lineage.  Since this is a Vajrayana tradition, the presentation works its way up through the three vehicles.  His discussion of the Hinayana tradition begins with a particularly interesting chapter on dependent arising ("The Development of Ego"), passes through the four noble truths, and gives a great reading of our circulation through the six realms.  Then there are a couple of chapters devoted to the core ideas of the Mahayana tradition: the Bodhisattva and Emptiness.  Finally, his view of the path culminates in the Tantric practice of transmuting energy.  It's probably more of a commentary on my own practice than the book to observe that this chapter seemed somewhat vague and ripe for abuse.  In Tantra, everything is legal, so to speak.  When seen properly, anger, lust, etc ... are all just various flavors of awake energy that need not be avoided.  While this is a satisfyingly non-dual philosophical conclusion, it leads to some obvious practical difficulties.  Trungpa seems to want to instill in us a confidence that everything we do from true awareness is correct and appropriate.  While he himself acknowledges the danger this attitude represents, he points out that this teaching only occurs at the end of the path, when the student should know how to use it properly.  Given that anyone with an internet quickly discovers Trungpa himself arguably fell victim to some of these dangers, it seems relevant to ask if his "crazy wisdom" was something we should learn from rather than emulate.