Cont: Proof of Immortality, V for Very long discussion

Status
Not open for further replies.
We've already had a multipage side-argument, that got rather heated if I recall, about whether infinity is a proper number or simply a mathematical concept.

Gentleman we cannot let ourselves be distracted from our mission of being completely ignored. If we let our guard down for a second Jabba might listen to us.
 
JoeBentley said:
We've already had a multipage side-argument, that got rather heated if I recall, about whether infinity is a proper number or simply a mathematical concept.

Gentleman we cannot let ourselves be distracted from our mission of being completely ignored. If we let our guard down for a second Jabba might listen to us.
Click to expand...

Infinity is a concept. It is not a specific number; there are different sets of infinity.

Example: The set of all prime numbers, which is part of and smaller than the set of all integers.

Hans
 
Argumemnon said:
Still looks like chinese to me, but with parentheses.
Click to expand...

Don't stumble over the notation. Divide 1 by a number bigger than 1 and you get some number smaller than 1. Divide 1 by a still bigger number and you get a number that's smaller than the previous results -- a number closer to zero. Keep doing this repeatedly with ever-larger numbers. Your results become ever-smaller numbers, approaching zero. The bigger the divisor, the closer the result gets to zero, in a clearly visible way. Now if there were some ultimately large number and you divided 1 by it, the trend suggests that the result would be zero. This is most visible if you graph it.

That's the beginning of calculus. Calculus says, "We know you can't slice things up into infinitely small slices. But is there a way to tell what would happen if you could? And is this actually useful for anything?"

The classic example is your car speedometer. You measure speed (a rate) as distance traveled in a specific time. If it takes you 2 hours to drive 100 miles, you can normalize that by reckoning it as 50 miles per hour -- on average. That's a grossly determined average rate. In practice you sped up and slowed down. You can even reckon it as just under one mile in a minute, if minutes are your preferred time unit. But that still fails to capture slowdowns and speedups that occurred over less time than a minute. Your car speedometer (in a modern car, anyway) measures the distance traveled (by counting how many times your tire turned and guessing its radius) in a tiny fraction of a second and constantly positions the needle to show you a rough approximation of your speed when you consult the gauge. It's rough because you're still averaging over tiny time increments, and your tire rotation rate may vary slightly up and down within that increment and not be detected. But it's accurate enough to keep you from getting a ticket.

Calculus waves the wand and asks, "What if we could determine how far your tire turned in a slice of time that was so small it effectively is zero?" And in practical terms you'd say that's impossible since, if time doesn't actually pass, the wheel doesn't actually turn. Your rotations per unit time are unknown since "per unit time" in this case is zero. But calculus lets you look at the trend as you slice your time into finer and finer increments and say, for the smallest conceivable increment, that the normalized rate actually approaches a useful number.

Then calculus shows that, if you embrace the behavior of things predictably trending even over very, very small divisions like this, you can adopt certain definitions that now let you understand this "instantaneous" rate for phenomena you can model abstractly as equations. Instead of actually counting tire rotations, you can use the new algebra with its special-case definitions to manipulate symbolically the relationships among quantities and reason about their instantaneous rates of change, or -- in the reverse -- about their accumulated quantities, in a way that actually corresponds helpfully to reality.

That's the real-world check on what admittedly looks like a lot of Chinese. This stuff actually works. It's what lets us compute, for example, the exact center of gravity and moment of inertia for arbitrarily-shaped objects. So it's not just the domain of pimply-faced nerds writing incomprehensible gibberish.
 
JoeBentley said:
Gentleman we cannot let ourselves be distracted from our mission of being completely ignored. If we let our guard down for a second Jabba might listen to us.
Click to expand...

I'm going to post my secret award-winning chili recipe to see whether Jabba is actually reading and intentionally ignoring what I say.
 
JayUtah said:
Don't stumble over the notation. Divide 1 by a number bigger than 1 and you get some number smaller than 1. Divide 1 by a still bigger number and you get a number that's smaller than the previous results -- a number closer to zero. Keep doing this repeatedly with ever-larger numbers. Your results become ever-smaller numbers, approaching zero. The bigger the divisor, the closer the result gets to zero, in a clearly visible way. Now if there were some ultimately large number and you divided 1 by it, the trend suggests that the result would be zero. This is most visible if you graph it.

That's the beginning of calculus. Calculus says, "We know you can't slice things up into infinitely small slices. But is there a way to tell what would happen if you could? And is this actually useful for anything?"

The classic example is your car speedometer. You measure speed (a rate) as distance traveled in a specific time. If it takes you 2 hours to drive 100 miles, you can normalize that by reckoning it as 50 miles per hour -- on average. That's a grossly determined average rate. In practice you sped up and slowed down. You can even reckon it as just under one mile in a minute, if minutes are your preferred time unit. But that still fails to capture slowdowns and speedups that occurred over less time than a minute. Your car speedometer (in a modern car, anyway) measures the distance traveled (by counting how many times your tire turned and guessing its radius) in a tiny fraction of a second and constantly positions the needle to show you a rough approximation of your speed when you consult the gauge. It's rough because you're still averaging over tiny time increments, and your tire rotation rate may vary slightly up and down within that increment and not be detected. But it's accurate enough to keep you from getting a ticket.

Calculus waves the wand and asks, "What if we could determine how far your tire turned in a slice of time that was so small it effectively is zero?" And in practical terms you'd say that's impossible since, if time doesn't actually pass, the wheel doesn't actually turn. Your rotations per unit time are unknown since "per unit time" in this case is zero. But calculus lets you look at the trend as you slice your time into finer and finer increments and say, for the smallest conceivable increment, that the normalized rate actually approaches a useful number.

Then calculus shows that, if you embrace the behavior of things predictably trending even over very, very small divisions like this, you can adopt certain definitions that now let you understand this "instantaneous" rate for phenomena you can model abstractly as equations. Instead of actually counting tire rotations, you can use the new algebra with its special-case definitions to manipulate symbolically the relationships among quantities and reason about their instantaneous rates of change, or -- in the reverse -- about their accumulated quantities, in a way that actually corresponds helpfully to reality.

That's the real-world check on what admittedly looks like a lot of Chinese. This stuff actually works. It's what lets us compute, for example, the exact center of gravity and moment of inertia for arbitrarily-shaped objects. So it's not just the domain of pimply-faced nerds writing incomprehensible gibberish.
Click to expand...

Jay, you're working way too hard on my education for the little I'm paying you!
 
Argumemnon said:
Jay, you're working way too hard on my education for the little I'm paying you!
Click to expand...


Probably the best layman's introduction to calculus, infinity, and zero I have ever read is Charles Seife's Zero: The Biography of a Dangerous Idea. There are very few equations, it was written for normal people, not mathematicians ;), and goes into the history and why these things were needed, not just the math. It is extremely well written, highly entertaining, and will help you understand in plain English how it all works.
 
MRC_Hans said:
Infinity is a concept. It is not a specific number; there are different sets of infinity.

Example: The set of all prime numbers, which is part of and smaller than the set of all integers.

Hans
Click to expand...
Back in my number theory days (many decades ago), I got my hand slapped for making a similar analogy and using the word "smaller". It was "imprecise". You can map the primes onto the integers 1:1. Yes, there are integers that are not primes, but IIRC they are considered the same "class" of infinity because of the mapping. For the record, I still think of it the same way as you describe.

None of the above should be construed as exempting Jabba from his need to read the thread, pay attention to his critics, and stop manufacturing "agreement" out of whole cloth.

CT
 
You can elect me to whatever status you want. Doesn't mean Jabba will pay attention to it. Last I recall, my posts were too complicated for him to read. So he doesn't read them.
 
JayUtah said:
You can elect me to whatever status you want. Doesn't mean Jabba will pay attention to it. Last I recall, my posts were too complicated for him to read. So he doesn't read them.
Click to expand...

No, no, but for the rest of us.

Ok, then. In the name of God, Saint-Michael and Saint-George, I give you the right to bear knowledge and the power to mete out education. Rise, Sir Jay!
 
JayUtah said:
I'm sure I am, but if you check your orders of chivalry you'll find that foreigners don't receive the Accolade. So I'll probably have to settle for the beer.
Click to expand...

I've created a whole new order of chivalry, the Order of the Knowledgeable Fellow-Skeptics of Hitchens and of the Temple of Argumemnon, and you're now our first knight, called a knight-skeptic.

You get your beer once you come by the home office to receive your membership card.
 
Status
Not open for further replies.

Back
Top Bottom