The previous post, Halt! (or not), described the Turing Halting Problem, a fundamental limit on what computers can do, on what can be calculated by a program. Kurt Gödel showed that a very similar limit exists for any (sufficiently powerful) mathematical system.
This raises some obvious questions: What is calculation, exactly? What do we mean when we talk about a program or algorithm? (And how does all of this connect with the world of mathematics?)
Today we’re going to start exploring that.
But my brain is full!
You may have noticed that, in a number of recent posts, the topic has been math. The good-bad news is that there’s more to come (sorry, but I love this stuff). The good-good news is that I’m done with math foundations. For now.
To wrap up the discussion of math’s universality and inevitability — and also of its fascination and beauty — today I just have some YouTube videos you can watch this Sunday afternoon. (Assuming you’re a geek like me.)
So get a coffee and get comfortable!
Take a moment to gaze at Euler’s Identity:
It has been called “exquisite” and likened to a “Shakespearean sonnet.” It has earned the titles “the most famous” and “the most beautiful” formula in all of mathematics, and, in a mere seven symbols, symbolizes much of its foundation.
Today we’re going to graze on it!
In the recent post Inevitable Math I explored the idea that mathematics was both universal and inevitable. The argument is that the foundations of mathematics are so woven into the fabric of reality (if not actually being the fabric of reality) that any intelligence must discover them.
Which is not to say they would think about or express their mathematics in ways immediately recognizable to us. There could be fundamental differences, not just in their notation, but in how they conceive of numbers.
To explore that a little, here are a couple of twists on numbers:
Oh, no! Not math again!
Among those who try to imagine alien first contact, many believe that mathematics will be the basis of initial communication. This is based on the perceived universality and inevitability of mathematics. They see math as so fundamental any intelligence must not only discover it but must discover the same things we’ve discovered.
There is even a belief that math is more real than the physical universe, that it may be the actual basis of reality. The other end of that spectrum is a belief that mathematics is an invented game of symbol manipulation with no deep meaning.
So today: the idea that math is universal and inevitable.
We sometimes say that dogs are living in the now. Sometimes we say that of people who live in the moment and don’t think much about the future (or about the consequences). Whether we mean that as a compliment — as we generally do with dogs — or as an oblique implication of shallowness depends on the point we’re making.
There is the tale of the ant and grasshopper; it divides people into workers who plan for the future and players who live in the now. The former, of course, are the social role models the tale holds heroic. The grasshopper is a shifty lay-about, a ne’er do well, a loafer and a moocher, but that’s not the point.[1]
The point is our sense of «now» and of time.
In a discussion a while back I mentioned in passing that humans sense wetness and time. That was challenged on the basis that we don’t sense time at all and — when it comes to wetness — sense only pressure and temperature. There is some truth to that. We don’t have an actual time sensor, nor do we have specific “wetness” sensors.
I’ve been thinking about this ever since (not constantly; you know, on and off). A key question is whether wetness can be reduced to pressure and temperature and remain wetness. And time is a topic all on its own!
For the record: Here is my final answer…
I mentioned recently that I intended to write some “For The Record” (FTR) posts setting down — once and for all — my views on certain oft-debated topics. “Once and for all” is misleading, though. My opinions evolve over time, and no controversial topic is ever truly closed. “Here and for now” would be a better phrase.
This one will certainly draw a sand line where some will stand on my side and others — people I like and respect — will stand on the other. I’m not sure I believe there is a right answer here; it really depends on your worldview. If nothing else, this seeks to explain my rationale as well as my opinion.
So, for the record: here we go on guns!
It was a number of years ago that the book you see pictured here on the right caught my eye. I was wandering around a bookstore, as booklovers do, seeing what there was to see (and possibly buy). This may surprise you, but I’ve always enjoyed a good debate, so the book’s topic seemed attractive and a nice change of pace from baseball and science books or SF novels.
Plus: Aristotle, Lincoln and Homer Simpson! Who could resist that? A glance at a few of the pages showed an easy and breezy open writing style that went down nicely, and the bits I read were quite intriguing. I snagged it thinking it would be right up my alley, and that I’d enjoy it thoroughly.
I never got more than a third of the way through it!
This is a long-winded comment in reply to Mike Smith’s recent blog post (and comments therein), Steven Weinberg’s new book on the history of science. We got to discussing a personal letter written by Albert Einstein about a year before his death in 1955. The letter — which seems to present religion as “childish” — surfaced in the public eye when it was sold at an auction in 2008. Given Einstein’s generally expressed views about religion, the letter appears to undercut those views.
Or does it? Atheists and theists alike have tried to claim Einstein as their own, but his views are complex enough to resist a clear victory by either side. The letter seems a point in favor of atheism, but that may be an over-simplification.
In any event, my reply ran long (and was getting kind of off-topic), so I decided to use it as an excuse to try to get back into blogging again…
Posts pertaining to programming!
Logos con carne 