Read Latex

Thursday, September 01, 2022

Dinner with Andrew: The Sequel

So, we're at the gym again. A couple of laps and I chat with Dr. Chiang a tenured Ph.D. Professor in the department of Computer Science.

Me: I've found a new 1500-page book on Probability and Machine Learning. Well, its actually two books. But it comes with code!

Dr. Chiang: I got paid today!

Me: I have to warm up around the track. I go upstairs to the 1/8-mile track. I'm thinking of Gödel again. He had read Russell and Whiteheads Principia Mathematica in its entirety and confronted Russell about everything in it being provable. His attention to detail was incredible and I would mention this to Andrew who appears later in the conversation. Gödel believed there was a way that the US could be transformed into a dictatorship. Einstein and Morgenstern had great difficulty restraining him from announcing this at his US citizenship hearing. (See page 12 here)

I returned downstairs to the quad machine and managed to get through three sets of eight, increasing the weight in twenty-pound increments as I warmed up. Then I made it to the sitting bench press which I have undertaken to perform at three angles with various seat heights and grip positions. After a set, Andrew appears, carrying a notebook which he is eager to show me.

I would be remiss to provide only my own recollection and I am fascinated by the Rashomon phenom of the same event witnessed by two people being perceived differently. I have asked Andrew to provide a quick sketch of his recollection, quick, because as a student he has bigger fish to fry than recounting a random gym conversation, but it is interesting and he has graciously provided a recap, which I have worked into this thread.

Me: That is excellent. Please show me. But first, you are taking four courses. Tell me how they are going.

Andrew: Electronics course. Ohm's Law. The Linearization of V = IR in I-V space to determine R as the slope of the curve.

Me: Aha! But if you are determining slope then you must make TWO measurements to obtain a result, say if you wanted to do an automated determination of resistance. That is so interesting that one measurement is not sufficient.

Andrew: (Presenting the sketch book. I have been doing some sketches to hone my ability to express myself [in diagrams that you might find in a book]. In order to practice my arm control for drawing, I drew various straight lines and connected them in arbitrary ways.

Me: They look like a Voronoi diagram. Constructions like these have unexpected utility I have toyed with:

Me: This is excellent. Nothing is as portable as pencil and paper, but you are going to the next level, to quote reddit. There are a set of very useful tools that will amplify your efforts as well. Like Maxima for symbolic mathematics, like Geometry Expressions™ for symbolic geometry. The Shovel vs. the Bulldozer.

Andrew: Behold - The introduction of Euclid's elements. The foundation of all classical Greek mathematics. The beginning of the exploration of the 5 postulates.:

Then like a magician producing a deck of cards Andrew pulls out a black notebook bound with black wire. The first few pages are various doodles and patterns which then turn to the money shot, a page showing Euclid's Five Postulates on Geometry.

Andrew: Some of these are tasks, some are proofs. The distinction is Q.E.F vs Q.E.D:

QEF: Quod Erat Faciendum (that which was to be done) vs.
QED: Quod Erat Demonstrandum (that which was to be proven)

Covering these basics would yield unexpected fruit.

Andrew: Consider Euclid Postulate 1. A straight line segment can be drawn joining any two points.

Andrew: Given a line segment, construct an equilateral triangle. This was done by drawing two intersecting circles with the given line as the radius. Thus, the equilateral triangle was constructed based upon the equality of the line lengths rather than the equality of the angles.

Me: The act of drawing the point in the first place induced the existence of a pair of coordinates to locate that point. The same for its partner. Drawing line created a relationship between the two points called length. Length is a disembodied quantity, an attribute, a scalar whose value depends on the coordinates. If I hand you the length, I have not handed you the points from which the length was determined. Length is a Doppelgänger.

Our goose bump moment was observing that the assumption that we can connect two points lies in the notion that they are a finite distance apart. We had previously discussed spherical distortions of the function 1/x that made it appear like a baseball moving points at infinity to the other side of a sphere.

This seemed like a lot of baggage for postulate one, so we moved on while I did another set of reps on the sitting bench press.

Euclid Postulate 2. Any straight line segment can be extended indefinitely in a straight line.

Andrew paraphrased this as, "A ray can be extended from a given point.", which I liked.

Andrew: Given a line, duplicate the line at another point. Draw more lines extending them as in postulates 1 and 2, and then directly called upon the result of proposition 1 to accomplish its task.

Me: Euclid was the earliest effort at the task of creating a unified system of geometry that Russell and Whitehead attempted symbolically with Principia Mathematica some two plus millennia later. 

We agreed that the level of dedication and attention to detail was worthy of admiration, and that it was great sadness that Gödel starved himself to death. We later agreed that this presented the notion that there is no preferred basis in Euclidian space,

Me: This makes the assumption that straight lines remain straight, that space if flat, except in regions of high gravity and the cosmological constant has two values depending on whether it is measured via cosmic background radiation or Hubble's standard candles.

I adjusted the seat and did another set of sitting bench presses. Am I conflating abstract space with real space? We review dimensionality and how the amount of wiggle room grows as we venture from the number line, to the real plane, to three dimensional space, finally adding time.

Andrew: Consider Euclid Postulate 3. Given any straight lines segment, a circle can be drawn having the segment as radius and one endpoint as center.

We talked about how this is just one way to specify a circle. One can also specify a circle by three points that live on its edge. This brought up the idea of constraints as a way to present geometry and the multiple ways of saying the same thing shape-wise. We talked about Phil Todd, the Portland genius who created GeometryExpressions™, used to create some of these figures. We talked about implicit equations of a circle which require a search and explicit forms that will represent the top or bottom half and how reexamining assumptions like this can lead to new work. This led to the code for the infinity sign below.

Andrew: Given a line segment and a short line segment, cut off a part of the longer line equal to the line segment. Euclid cheated a bit because he drew the short line segment in connection with the longer segment, rather than sticking with a compass and straightedge construction. 

Me: I am displeased with the discrepant notation of using a single character for the short line segment's length while notating the end points of the other line segments  as full (x, y) citizens in the sketch.

Andrew (post facto): For all I know, this could be an issue of translation, as the copy of the elements that I have is based upon Heath's translation. You stated that referring to the short line segment as a magnitude of length only, left it without any documentation of its location. Thus, a scalar would be suspended in space.

Me: It was at this point that I used the term Doppelgänger or disembodied scalar to refer to a length floating in space, as compared to an anchored point drawn with specific (x, y) intention.

Euclid Postulate 4. All right angles are congruent (the same).

Me: This one sounds like a tautology, as in, it is what it is, though granted, 'is-ness' is a lot larger space that the set of 90° angles.

Me: What about the relationship of Cartesian grids to hexagons?

Andrew: There is proof of the impossibility of trisecting an angle with a compass and straight-edge. The proof is here and uses a cubic polynomial.

Me: This is example of symbolic representations providing insight into visual representations of geometric relationships.

Andrew: I am interested in the fact that Euclid dared to purport that all right angles were equal before our modern notion of the dot product allowed us to extend the concept of orthogonality into higher dimensions.

Me: Doug Gilmore at the University of Illinois once used the construct wherein two linguistic statements were orthogonal to each other once.

Andrew: We agree that this is a fascinating notion to introduce orthogonal statements that neither support nor oppose each other but rotate the discourse by 90° into a higher dimension of understanding. These orthogonal statements could form the "basis" or "basis functions" for a view of the universe.

Me: [the term 'basis functions' is a trigger for me] Very interesting.

We talked about circles as basis functions for a drawing:

  • One circle of fixed radius that couldn't move.
  • Many circles of fixed radius that could move.
  • Many circles of varying radius that could move.

About this time, a young man was violently upset was being escorted from the gym after an altercation on the basketball court. It was hard to remain focused on our conversation, but we did our best, our limbic systems nonetheless activated for a fight or flight situation occurring before us. We remained on task, hoping that he would be back in better spirits.

It was time to do another set of bench presses of the upright and orthogonal variety.

We resumed talking about how ideas, facts and statements can be orthogonal to each other. Sometimes redundancy of expression cements an idea. For terseness orthogonal statements cover the most territory in the least words.

Me: There is also a symbolic definition of dot product being the definition of orthogonality, specifically the formula:

The question arose as to whether giving someone the symbolic representation of something is the same as giving them the geometric representation. It isn't. Both are needed. Two faces of the same coin.

We talked about the appearance of angle as being another disembodied scalar like length, those comprising the two fundamental aspects of dimension. We wondered if the 12 dimensions of string theory, including the tiny curled up dimensions aren't conflating rotation with length. The mean streets all sit orthogonal to each other.

Andrew: Consider if in two triangles, two sides and the angle subtended by them are equal, then the triangles are equal. The proof is a joining of the triangles along a common side and folding them upon each other. The proof presented by Euclid used a translational superposition of the triangles.

Me: The process of proof could be encoded within a trio of affine transformation matrices. In this way, truth could be containerized and exploited later. It is not until the truth is instantiated, at the point of the application of knowledge, that value is obtained. No symbolic expression achieves its full value until it is populated by actual numbers that express actual truths.

Andrew: So the truth would not be observed until the transformations were applied to a specific situation, although the true nature of the transformation would still be securely encoded.

Me: It is interesting that we have geometric expressions of truth and symbolic expressions of the same truth.

Andrew Postulate 5: Parallel lines never meet. Biased lines will eventually intersect. The devil of Euclid, that hyperbolic geometers love to ride on about.

Euclid Postulate 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Andrew then went on to illustrate the theorem of the isosceles triangle having extended legs, which produced a 12-angle complexity pop. I mentioned that if one went through all of Euclid's theorems (of which there appear to be 468) looking for complexity 'pops' (sudden increases) that one might be able to find a path of most escalating complexity and possibly the next big thing.

Andrew: If an isosceles triangle has its congruent sides extended, the angles subtended by the base and the congruent sides are equal, as are the angles between the base and the extended lines.

He recounted later that this proof obviously exploded in complexity or sophistication due to its length. It is observably longer and more detailed that the four propositions that precede it, managing to occupy an entire page in the book. You mentioned that this being the first burst of complexity, it would be prudent to watch out for and trace the pathway of similar bursts of complexity, and that following such a path would lead to the next node in the path of discovery.

Andrew then showed me a sketch where he had mapped every point in a grid from the origin to the grid and made a clever deduction about the null space of linear algebra.

Andrew: The singularity. To further practice drawing lines, I attempted to map an imaginary grid from the center of the page. A straight line does not have any intrinsic directional characteristics, so that my intention to map the center point to the grid points could be equally described as a mapping of the grid points to the center point.

Me: You have expressed the relationship between the Cartesian and polar coordinate systems, and how there existed finite angles with rational tangents that I could represent and a large number of continuous angles I could not represent.

Andrew: Another curious outcome of my drawing exercise, seemingly more artistic than mathematical, in which I was reminded of the Null space in Linear Algebra, where the entirety of a space is mapped into the zero vector, thereby vanished into nothingness. By coincidence, since I started drawing lines at the center of the page, the center point of the page had the highest accumulation of graphite and was therefore the darkest point on the page
as if it were a dark hole pulling in all of space.

Me: The origin is a singularity and if lines of zero width, that there would be no darkening, but that there would still be a singularity as a result of the density of lines at the center.

We talked about the density of the lines in such a graph and how as one moves closer to the origin the density increases until at the origin you have a black singularity. I initially said that the density was an artifact of the finite width of drawn lines but recanted that in favor of a definition of a circular region of fixed radius that we could count the number of line crossings in to obtain a line density metric. His sketch showed the intrinsic difference between a Cartesian and radial coordinate system, with the existence proof, obtainable by inspection that a finite grid with lines draw from each grid point to the origin can only represent a subset of the possible angles in a continuous space. Further the number of angles in the discrete set are always smaller than the number of angles that cannot be represented, implying that in the limit that the number of grid angles on an infinite grid is smaller than the number of continuous angles, which informally makes the case for different sizes of infinity.

Then I mentioned Euler's formula for points (vertices), edges and faces.

Andrew: 'haven't yet gone that far into graph theory...

Me: I think this precedes graph theory, but there is a connection. When I think of graph theory, I think of the Koenigsberg Bridge Problem:

Euler sure got around.

Collaborations that are highly disciplined can be rewarding. Michael Lewin writes about a collaboration between Amos Tversky and Daniel Kahneman. Ruthlessly revisiting items considered basic can yield new truth and bring old truth to life via exercise. Like a kata, a well-rehearsed prearranged form that the practitioner already knows, its repetition builds mastery.

There was a war on. This did not stop Freeman Dyson from applying himself to the problems of the day, to eventually winning a Nobel Prize. He had talent, but he also got back to basics and mastered them. Read this book when you have time. It is a set of letters through his life that provide a story, the hooks upon which to hang the practices of a physicist/mathematician.

Me: Watch "When the Curtain Falls", by Greta Van Fleet.