I'll start this by quoting my own comments in the youtube video embedded above,
- Well, this show was interesting on a number of fronts. On the one hand, they're trying to push James Burke's connections app(which I hope to get someday), and argue for a connectionists viewpoint . . . and, on the other hand, the interviewer mentions how "what we perceive about the world, is not necessarily what it is."
Jacob Bronowski, for one, notes(really quotes), in his "Science and Human Values" how the idea that science discovery comes from going around and jotting everything down is wrong. Science is not a collections of facts. If that's the way science works, we'd never get past noting that the Earth appears flat, that things don't evolve, and so on, and so forth. What's more, if the brain was just taking in all the sensory data like a camera and recording everything it sees, the head would explode. Or, it would be saturated . . . on sight(not to mention the hearing, the sense of touch, and taste and so on). The memory storage would be completely filled up as of when the eyes open. So, the brain can't work connectionist.
- I think it's the second or third post of this blog, where I posted one of my first efforts to show connections between some connections James Burke finds in his "Connections" video series/book, and Jacob Bronowski's ideas - from his "Science and Human Values" to his "Origins of Knowledge and Imagination."
I've shown some new findings on those connections in various posts(I even explain it in my latest Gospel of Truth redo - in the introduction. But, I actually introduce the new findings, for the first time, in my finite and infinit blog post). I point out that while their seems to be a similarity between new ideas generated in a technological setting(by means of damning up nature in some way. For instance, in agriculture, one clears the field, and at that point, one needs to introduce the ideas of irrigation/pest control, and I'm forgetting the third one. In mathematics, for instance, after numbers advance enough to construct sums like 1+2=3, one might ask 1+a=3 instead of the usual 1+2=a. Here the concept of a variable gets naturally introduced), and new ideas in mathematics. In mathematics, idealization corresponds to clearing of the fields. But what's the connection between the abstract world of mathematics and the concrete technological world?
The idea I point out comes from Susanne K. Langer's "Introduction to Symbolic Logic", which has the best introduction to how abstraction works I've ever seen in any logic, or mathematics book. What she points out is there's constituent relations and logical relations. The constituent relations are the relations between things on a concrete level. And the logical relations exist on the mathematical level.
I think that's all the new ideas since writing my original account of my findings between some of James Burke's connections and Jacob Bronowski's ideas. I actually don't explain the whole constituent/logical relations thing. I'd have to re-write the whole explanation of abstractions, which can be found elsewhere in the blog or in Susanne's book. I want to get on to some new ideas here.
- Maybe a little storytelling is o.k. here . . . I had hit on abstraction as the essence of mathematical discovery from probably junior high school times through high school. Well, I found Susanne's book in my first semester of college before heading off to the military for four years. I had already signed on for military. But, I had time to do a semester. So, I did. I thought I had found everything I needed to become a mathematician in Susanne K. Langer's Introduction to Symbolic Logic book. I thought abstraction was all you needed to understand to know how mathematics works. Of course, just because I find new stuff like Jacob Bronowski's "Origins of Knowledge and Imagination" doesn't mean her book doesn't retain usefulness.
I actually found Jacob Bronowski in an accidental way. I had heard of Jacob Bronowski before. Before there was Barnes and Nobles, with large science sections, there was smaller bookstores. I mean, in a Barnes Noble, you could find stacks after stack of science books section. Before them, bookstores only had one row of a single stack dedicated to science books. And yet, in almost every such bookstore was Jacob's "Ascent of Man." Just picking that book up would suggest an amateur science historian or limited value. It wasn't till my out of the blue introduction to Jacob's "Origins of Knowledge and Imagination" that I found the true value of Jacob Bronowski. It was my sister, working in a Los Angeles used book store, who gave me a copy for free. I don't remember if it was a birthday or christmas present. Or, she just flatly thought it looked interesting. But, just the title suggested to me, that everything I knew about Jacob Bronowski went out the door!
Jacob Bronowski's little book unified and explained the revolutionary view of knowledge posed from non-Euclidean geometric revolution of the early 1800s to Einstein's relativity theory and Quantum mechancs and Godel's theorem. It was published posthumously, and so it's not very well known. And, I've been trying to make sense of it ever since! I can't remember the date of when she introduced the book to me. I'd have to know when she worked at that used book store I suppose. She seems to have lost her head; and, we don't talk anymore actually! She doesn't even explain to me what her problem is! Anyways!
Of course, over the decades or so, I've made connections between James Burke Connections and Jacob Bronowski's ideas, as I posted in the first three posts of this blog. I think I may have found some new ideas.
- Maybe what I've been bothered about is the nature of symbols, and that mathematics is symbolic. Susanne K Langer, of course does perhaps the best job of explaining the nature of symbols and their usage in her "Intro to Symbolic Logic" book. She notes the difference between natural language and logical symbolism. In natural language, a single word can sometimes be a noun, and other times be a verb. Wife in one sentence is a noun - His wife cooks dinner. In another sentence, wife can be a verb - Katherine is the wife of "fill in the blank". And, so, in natural language, words are very surface level - almost like the Earth is flat initial perceptions. Logical symbolism is underlying structural relations, that linearize, or put things in their logical order. But, they are also underlying structural relations. These are structural relations that comes from looking behind ones initial sense perceptions. The classic example is the discovery of the solar system. Noticing the retrograde motion of the planets, and how Venus and Mercury follow the sun at sunrise/sunset a lot more than the outer planets of Mars/Jupiter/Saturn. One can pattern your ideas/concepts based on vague language; or, one can pattern them logically. Below, I find some more about patterning with logical ideas than vague ideas.
Various phenomenon of mathematics seemed to be something more than what I've explained(and Jacob Brnowski first in his Origins book). It seemed that picturing new things in terms of old transferred properties that allowed the old concepts to do thing to new concepts, and allowed one to do those things in another, new more general level. For instance, in classical algebra, one performs all the basic arithmetic operations on letters. A+A= 2A. A-A=0 actually!(one of the hardest initial conceptual steps from arithmetic to algebra), and A/A=1. 1 and 0 are what's called identity elements in even higher abstract algebra. These are things that seem easy today; but, when not pointed out, are as hard as discovering the sun centered solar system. But, also different numbers and how they work gets transferred to another more general mathematical setting. For instance, fractions, irrational numbers(kind of hidden under radical signs and gets handled with laws of exponents), imaginary numbers.
There's also geometric to algebra, and algebra to geometric transfers of properties, and what's one can do is then made possible in the other domain. Trigonometry and vectors are geometrically inspired algebra. Trigonometry is the geometry of triangles transferred to an algebraic domain. While vectors are a polygonal geometry transferred to an algebraic domain. There's also the example of the geometric algebra in Euclid's Elements(which really goes back to Babylonian days). The a^2+2ab+b^2 is geometrically shown in Euclid's Elements books 2, prop 4. - and likewise for solving quadratic equations. But, one can go the other direction as well. One could argue the Babylonian discovery of the Pythagorean theorem was by means of playing with the number theory of square - geometrically.
What all this and more is pointing to is that certain mathematical domains act as diagrams for new, sometimes different(such as from algebra to geometry), or more general domains(such as from arithmetic to algebra), or logical pictures. Because each is constructive, or generates concepts from concepts, and they are underlying structural relations . . . this is what gives mathematics its power of learning nature.
I picture the Minoan ruins below because the Minoan temples were mazes - or a labyrinth.
generates. The famous Minoan mythology is that of getting lost in the Minoan labyrinth, and using a clue, or a ball of string to be able to find ones way out. Otherwise, you get eaten by the Minotaur, a half man half bull beast.
Well, this is the throne room, which is the center piece of the Knossos city/temple labyrinth.
There's also many Minoan coins with labyrinths on them; so, you know they thought these were interesting,
I point out the Minoan fascination with labyrinths because, when one is studying the underlying structural relations, one needs a clue of how to view that reality that is not what your first sensory perception tells you. In mathematics, clearly, one transfer the abilities already established from one domain to another - giving new possibilities, and putting the old possibilities in their place.
- So, how does all this relate to my connections between James Burke Connections and Jacob Bronowski's "Origins of Knowledge and Imagination"? Mathematical concepts are diagrams of how to solve a problem. But, they are symbolic diagrams. They idealize first. To get at nature at all, one has to idealize. Idealization is a first step. When one then transfers that ability to a new domain, one certainly gains initial new abilities. But, also, one gets new problems. For instance, when numbers are transferred to a geometric setting, one gets holes between the discrete number reality and the continuous geometric reality. One has to fill in those holes, first by rational numbers, then by real numbers.