Archive for April, 2008

Writing Big Numbers, Practically

I was reading a book yesterday called The Emergence of Number by J. N. Crossley, and it got me wondering: Is there a way to write any natural number, no matter how large, inside of a fixed finite writing space (say a 1 by 1 cm square) using only a single notational convention?

For example, you could write the number $100$ as $10 \times 10$, and that would be one convention. Or you could write it as $10^2$, and that would be another. Clearly the first wouldn’t work. It would eventually lead me to a number that runs over the side of my 1 x 1 box. For example, I might get ${10}\times{10}\times{10}\times{10}\times{10}\times{10}\times{10}\times{10}\times{10}$. What about the second? For example, I might get something like $10^{10^{10^{10^{10^{10^{10^{10}}}}}}}$. Depending on what the ratio of size of the exponent to the size of the base numeral is, it may be possible to answer yes to my question. There is a practical problem, though. Once you get smaller than the atomic scale, how would you represent the succeeding exponents? Even assuming that you could reduce the size indefinitely, limiting toward zero, this answer seems unsatisfactory. It’s like proposing to make an automobile more aerodynamic by simply shrinking it to the size of a marble. Well, it’s not really useful for driving anymore, so what’s the point?

I wonder, is there a convention which automatically rescales naturally as the value of the number in question increases? Besides the obvious, of course, which is to simply invent a new convention when the number hits the side of the box. My gut feeling is no, such a convention couldn’t exist. Maybe an evolutionary algorithm of some kind would work. But that seems to me to be stretching the rules of the problem–it’s not really deterministic in the everyday sense. I couldn’t rederive such a convention, for instance.

What if we alter the question so that an arbitrary, finite number of conventions can be used? Is it possible then? I don’t think so. We’d have to cycle through the conventions eventually. Even if we take the set of conventions and derive new conventions combinatorially (i.e. cycle once through the $n$ conventions one by one, and when you reach the one you started with, combine it some way with a second convention to form a convention $n+1$.), we’d eventually run out of combinations.

This is giving me a headache, honestly. :p

Pirates, Kitties, and People Talking to CCDs

Honestly, who watches TV anymore when you can watch YouTube instead? Aside from Battlestar Galactica last Friday, I haven’t turned a TV on once in the past three weeks. And here’s another reason to perpetuate that habit.

There’s a bloke on YouTube named Paul Harrison. He’s a former Christian (as of six years or so ago, I believe), and he runs a channel where he discusses his former Christianity, atheism, and intelligent design, and reviews Christian apology books. Take a look see.

Wow, They Can Disagree

First I learn from Tyler DiPietro that DaveScot (also lovingly referred to as “DaveTard”) managed to utter something reasonable for once in his life.

Then I go searching on Uncommon Descent and find DaveScot actually arguing (however placidly) with the Uncommonly Dense over the connection (or more accurately, lack thereof) between natural selection and Nazism.

More on Scientific Theories

In the prelude to his quantum mechanics textbook Principles of Quantum Mechanics, R. Shankar describes several features of the progression from older scientific theories to newer ones, and how the older ones are related to the newer ones:

• There is a domain $D_n$ of phenomena described by the new theory and a sub-domain $D_0$ wherein the old theory is reliable (to a given accuracy).
• Within the sub-domain $D_0$ either theory may be used to make quantitative predictions. It might often be more expedient to employ the old theory.
• In additional to numerical accuracy, the new theory often brings about radical conceptual changes. Being of a qualitative nature, these will have a bearing on all of $D_n$.

This is a nice way of thinking about it, I think.

Logic: You’re Doing It Wrong

Mark Chu-Carroll, over at one of my favorite blogs, Good Math, Bad Math, writes about the discussion over Expelled::

…what strikes me is that we haven’t paid enough attention to something even more important than whether or not there’s a link between Darwin’s theory of evolution and the nazis.

Suppose that it was true that Darwin’s writings about evolution were the primary thing that motivated the Nazi’s genocide against the Jews, the Romany, and all the other “undesirables” that they killed. Forget, for a moment, that the linkage is a crock. Pretend that it’s the truth.

What difference does it make?

Does the truth become less true because some idiot used it to justify something awful?

Scientifically Proven

In what may be the stupidest attempt at research using the Internet in history, I decided earlier today to google around and try and find the origins of the phrase “scientifically proven”. Yes, I hear you guffawing.

What I found instead were scientifically proven aromatherapy remedies, hangover remedies, fitness remedies, alternative “medicines”, anti-vaccination testimonies, baldness remedies, cosmetics, “junk” (i.e. boob, vag, and schlong) enhancers, and a BBC story on the hue of Jesus’ skin. I can has masochism?