TDDPirate’s Ideas

Few months ago, I at last bought my copy of “A New Kind of Science” by Stephen Wolfram (ISBN 1-57955-008-8). I expect to finish reading the entire book few months from now, and then go on to reading other books.

The book fulfilled my expectations of being interesting and intellectually stimulating book.

The first observation, which I made from reading the book was that (more…)

There is a book called “On Intelligence” by Jeff Hawkins, and it is about yet another theory of the human brain’s operation. The author pointed out several gaps in current researches in neurology and AI.

However, the book dealt with static aspects of the brain’s structure. It omitted the dynamics. In particular, the following points were not covered:

1. Factors affecting formation, destruction and strength of synapses connecting two neurons to each other (see also: Hebbian learning).
2. Growth of axons and dendrites - where do they prefer to grow?
3. Initial wiring in a fetus’ brain.
4. Existence and effect of any chemical means for non-local communication between neurons.
5. How are new neurons created and how do they integrate into the existing network?

The only dynamic aspect of brain operation, which was dealt with by the book, was the firing pattern of neurons.

According to the article Mathematical proofs getting harder to verify, it is now very difficult and sometimes impossible to be certain about the correctness of mathematical proofs.

I can envision the rise of the special profession of mathematical patching. It would work as follows.

1. A fundamental theorem is proved, but its proof is difficult to verify.
2. A lot of mathematics is being based upon that theorem.
3. Another fundamental theorem is proven and receives similar honorable status in mathematics.
4. A contradiction is found, which means that both theorems cannot be both true.
5. In order to save the rest of mathematics, the theorems are patched.

Patching, in this context, means adding qualifications to the theorems, so that fully-qualified versions of the theorems do not contradict each other. The qualifications will be based upon the actual way the theorems are used in subsequent mathematical development, which is normally less than the full generality of the theorem.

Neuroscience Tutorial - an illustrated guide to the essential basics of clinical neuroscience.
Neurosciences on the Internet.
There are also other related Web sites.

http://www.aip.org/pnu/2005/split/747-2.html

Best Practices for Time Travelers

1st Joint Conference Between Earth Physicists and Epsilon Uridani Physicists

5. Discussion about Quantum Mechanics

E.U. present their model of the microscopic universe and their key thought experiments.

Earth present Quantum Mechanics.

E.U. physicists ridicule it - citing quickly all those paradoxes (such as EPR, quantization of gravitation, etc.). All paradoxes except for two are already familiar to Earth physicists.

E.U. criticize also the thought experiments, starting from Stern-Gerlach experiment. Their attack is on the fact that the abstractions have not been properly constructed. Some of the neglected details are, in fact, very important.

Then, a review of the histories of the ideas is made.

E.U.: WHAT?! Your physicists do not learn epistemology?!!!

Summary

The Earth physicists took advantage of playing with symbols without referents: their mathematics is very developed. E.U. can use several mathematical concepts developed on Earth for their physics research.

Yes, I have seen* it!

*given a suitable definition** of “seeing”

**It is not advisable to look directly at the Sun under any circumstances. Therefore the Venus transit can be safely viewed only via some instrument such as a telescope. I saw it via a more complex instrument, which consisted of a telescope, camera, Internet connection (http://www.astronomy.org.il/) and my PC. The fact that I did not see it at real time does not really matter.

There is an interesting discussion about large numbers in the Wikipedia and in Robert Munafo’s article.

Munafo describes several notations for concise expression of very large numbers. The basic idea is to define special operators (symbols) to allow one to express a very large number using a small number of characters (digits, operators/punctuation marks). Since a short string of characters, which represents a very large number, can be represented in a small number of bits (say, 16 times the string’s length in characters) - what we are really doing is an attempt to implement an “hash function” from large numbers into their representations, while doing this in inverse.

In other words, a mathematical notation for concisely expressing a very large number corresponds to a data compression algorithm, like the algorithms used in gzip or bzip2. I am not saying “equivalent” here, because I cannot demonstrate a way to derive a compression algorithm from a mathematical notation definition or vice versa. The data compression algorithm takes, as its input, a string, which represents the very large number in some straightforward form. The algorithm’s output is another string, which expresses the same number by means of a suitable mathematical notation.

It is well known that every compression algorithm compresses a subset of the strings presented to it, but expands another subset of the potential input strings. The point of using a compression algorithm is that the strings, which interest us, get compressed, while only “garbage” strings get expanded.

Therefore, for any mathematical notation for representing very large numbers, there is a subset of numbers, whose representation in that notation is shorter than a straightforward form; and a subset of numbers, whose representation is longer.

What if?

Let’s say that the straightforward form, which I mentioned above, is the base-2 representation of a number. The corresponding representation, using a mathematical notation, is a string of N characters. We can consider it to be a base-2 number which is 16N+1 bits long (the extra bit is needed to guarantee that the number starts with ‘1′).

Let’s try a new way to represent very large numbers. We take a number, say 42, and represent it as binary number. We interpret the binary number as a mathematical notation, and translate it into the represented number, typically a very large number. An equivalent way of saying this is that we uncompress the string, which is the binary representation of 42.

Then, we take the very large number and read it as a mathematical notation, which denotes another very large number. In other words, we uncompress again the binary string. We can iterate this process several times.

The following questions, for which I don’t have answers, follow from the above discussion:

1. Are there sequences of infinite length? Or do such iterations always repeat the same strings after a sufficiently long cycle?
2. What is the largest number, which we can reach from 42 using this approach? What mathematical notation (uncompression algorithm) would allow us to reach the largest number from 42 by means of a finite (but very large) number of iterations?
3. Can a similar methodology be applied also to transfinite numbers?

The data structures in Scheme (and for that matter, also in Lisp) can be used to model different realities. Functional programming corresponds to closed systems, which evolute in time without interaction with their surroundings beyond initial conditions and harvesting of computation results. Imperative programming corresponds to open systems, which interact with their surroundings and their state contains a record of such interactions which occurred in the past.

So I am wondering whether additional models of reality can be investigated by means of Scheme. Such as open systems, which hold memory of both past and future events.

In my Google search, I found only the following:
SCMUTILS Reference Manual, which is referred to by Christopher Browne’s Web Pages. However SCMUTILS is not what I am looking for.

Source of inspiration: chapter 3 of the SICP 2nd edition.