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#76 Re: This is Cool » Zero !!! » 2008-10-29 17:02:05
It is not a mathematical failure that occurs, but a physical one.
It's a mathematical failure, not a physical one. Your assertion requires us to believe that because we don't currently understand how to do something in mathematics, that it cannot possibly be done.
Please, tell us, from the perspective of the photon, what is the distance between The sun and the Earth? From the perspective of the photon, how long does it take to travel between the sun and the Earth?
#77 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-29 16:45:11
Your view on theoretical physics is a bit odd. Or perhaps maybe it is mine which is odd, but let me explain. If we were to find that a hypothesis (again, let me state for clarity that a hypothesis is what is assume to be true for a conclusion to hold, this is the mathematical usage of the word) to Einstein's Theory of Relativity (ToR) is false, then eventually the ToR would be forgotten, it would be of no use.
Which is exactly why we no longer use Newton's laws of motion. Oh, wait, we still use them more often than relativity theory. Just as Euclidean geometry is "wrong" in the context of the real world, so are Newtonian mechanics. Yet, both are still extremely useful.
The razor in physics that is used is "does it work?". Work of course meaning does it allow us to make accurate predictions.
Whether or not it works has nothing to do with being right or wrong.
In mathematics, there is no such razor. We don't ask in mathematics whether or not it holds in the real world because the question alone does not make sense.
Nor does that question alone make any sense in any of the other sciences. The beginning assumptions can be incorrect, yet the theory can still be very useful.
The work above is mathematics in theoretical physics, but it is not theoretical physics. Theoretical physics is the work done before and after the equation is solved. If you can hand the work to a mathematician who knows absolutely nothing of physical interpretation of symbols, and he can solve the problem, then I would conclude that his work is mathematical in nature and not theoretical physics.
Theoretical physics on the other hand is discovering things like the Schrodinger equation, or coming up with experiments, or interpreting the mathematical work to it's physical meaning. We can thus separate the mathematics from the theoretical physics. What do you think of this?
You make it sound as though the difference between a mathematician and a scientist is like the difference between someone who is an expert at writing and someone who is not only capable of writing, but also knows something worth writing about.
Personally, I think mathematicians deserve more respect than that, but to each their own.
#78 Re: Puzzles and Games » How much would you need to invest? » 2008-10-29 16:19:17
What is the balance at the end of the nth year?
The balance at the end of the nth year is
#79 Re: Puzzles and Games » How much would you need to invest? » 2008-10-21 02:01:49
What is the balance at the end of the nth year?
If no one else solves this within a week or so, I will.
Perhaps you could crosspost (with mods' permission, of course) your formulas from this thread in the Formulas section, since you took the time to put them in LaTeX and explain them. 
#80 Re: Help Me ! » Got Another Tough One! » 2008-10-20 06:53:19
It appears that you're having a little bit of trouble with percentages.
Let's briefly examine the word percent.
Per indicates division.
Cent is from the Latin word centum, meaning hundred.
Thus, percent literally means "divide by 100."
For example, 50% = 50÷100 = 50/100 = 0.5.
When 50% is expressed as 0.5, we typically refer to it as a proportion. This applies to any percentage, not just 50% (i.e. 13% = 0.13).
To find the proportion of your exam that consists of questions from topic 3, we take the number of questions in topic 3, 27, and divide by the total number of questions on the exam, 150.
27 ÷ 150 or 27/150 <-- they mean the same thing
To convert the proportion to a percentage, we simply multiply the proportion by 100. This is most easily done by moving the decimal point two places to the right.
#81 Re: Help Me ! » Probability and Statistics » 2008-10-20 06:28:54
A. What is the average deaths per day?
B. What is the probability that there will be no deaths on a given day?
C. What is the probability that there will be 1 death on a given day?
D. What is the probability that there will be 2 death on a given day?
Edit to add: I just noticed that the town population is given. This makes it a binomial distribution instead of a Poisson distribution:
We'll let Y be the same random variable as above.
For Part B:
For Part C:
For Part D:
Notice how small the difference is for the results using Poisson versus binomial distribution. The Poisson distribution can be used to approximate the binomial distribution when n is large, p is small, and λ < ~7.
#82 Re: Help Me ! » Related rates problem. Please help! » 2008-10-19 16:34:54
Evaluating the expression at t=12 without taking the derivative first will only give you the height at which the ladder rests against the wall after 12 seconds, not the rate of change for that height.
#83 Re: Help Me ! » Related rates problem. Please help! » 2008-10-19 08:45:40
Use the Pythagorean theorem. C is constant (15 feet, the length of the ladder).
Let A be the distance between the wall and the ladders base.
Let B be the height at which the ladder rests against the wall.
B(t) = √(15^2-(10-t/4)^2) = √(225-(10-t/4)^2)
taking the derivative of B(t) with respect to t, then evaluating at t=12, will give you your answer.
#84 Re: Puzzles and Games » How much would you need to invest? » 2008-10-19 06:12:55
Suppose you have the opportunity to make a one time investment in an annuity that earns 10% fixed interest per year, compounded annually, from which you will receive annual payments beginning one year after the investment is made. The first year's payment is $50,000. Each year thereafter, the payment increases by 4% over the previous year's payment. The payments continue being paid annually until the balance is withdrawn. How much money would have to be invested initially?
This is clearly a growing (geometric) perpetuity.
And since this is the case, you will notice that the blackened portion of your problem statement is defective in the sense that the balance or fund will never be fully withdrawn or totally depleted.
I don't have time right now but my initial calculation shows that your initial investment should be
$833,333.333...I'll check back in later for a full explanation.
The balance will never be depleted from the annual payments, but will eventually be withdrawn by the owner, as a transaction separate from the annual payments. Typically, perpetuities don't really provide payments forever, they provide payments until the balance is withdrawn (or the financial institution collapses, etc.).
Your answer is correct. We could use the formula for the present value of a perpetuity with non-level payments, but a much easier way to obtain the answer is by recognizing that the principle of the perpetuity must have a net growth of 4% per year. Since the perpetuity is earning 10%, this implies that 6% interest (i.e. 60% of the total interest earned each year) is used to pay the annual payment. Recognizing that interest compounded annually is the same as simple interest at t = 1 year, we get the equation $50,000=.06x. Solving for x, we find the perpetuity's value one year before a $50,000 payment is $833,333.333 .
(In hindsight, I probably should have made the annual payments start at $150,000, in order to get a nice round number for the initial investment.)
#85 Re: Puzzles and Games » How much would you need to invest? » 2008-10-18 08:48:35
Nobody has solved this yet? Is a hint in order?
#86 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-17 09:17:42
You cannot preform science by pretending it exists in a mathematical vacuum.
I never said or implied that you can.
The equations in physics all have meanings, and this meaning is separate from their mathematical meaning.
That's ridiculous. The equations mean the same in science as in math. If they didn't, the equations would be useless, and mathematical proofs meaningless. It is precisely because the equations have consistent meaning that makes them useful.
We say a theory in physics is correct if it allows us to make accurate predictions about the physical universe. Do you disagree?
That statement is oversimplified to the extent that I cannot agree or disagree.
#87 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-17 04:37:48
Given the postulates Einstein started with, Special Relativity theory is absolutely correct.
I would like to argue this point in depth because it seems like this is the major divide. Given the postulates that Einstein started with, his work was arguably mathematically correct.
Given his postulates, his work is correct, not "arguably correct."
However this does not mean that it must carry over to the physical world.
No, it doesn't. But we can't hold his work to a higher standard than mathematicians' if we're going to compare the validity of science versus the validity of mathematics. If we can legitimately say that Einstein was wrong, then we can just as legitimately say that Euclid was wrong or that the Pythagorean theorem does not hold on a large scale in the natural world, and is therefore wrong.
On the other hand, if we say that Euclid and other mathematicians were correct, given his postulates, then we must extend that same courtesy to scientists' work if we are to compare their relative accuracy.
Just because I can subtract 5 feet from 3 meters per second does not mean it has a physical meaning.
Sorry, you cannot subtract 5 feet from 3 m/s. You may as well try to subtract 5 kilometers from 7 liters.
I would like for you to illuminate exactly what you consider the postulates of his theory to be.
For special relativity, Einstein started with two postulates:
1. The laws of physics are the same in all inertial reference frames.
2. The speed of light in a vacuum is equal to the value of c, independent of the motion of the source.
Oh and by the way, from the United States National Academy of Sciences:
Some scientific explanations are so well established that no new evidence is likely to alter them. The explanation becomes a scientific theory.
A theory, at least with respect to physics, quantum mechanics, cosmology, etc. are the mathematical constructs used to make predictions. Explanations are the words used to describe those theories. If the theory is once removed from reality, then the explanation is twice removed. All that is required of an adequate explanation is consistency with the mathematical theory. A small modification to the theory, such as special relativity provided for Newton's laws of motion, can cause major problems with the explanation.
Scientists are concerned with theories, while laymen and young school children are typically perfectly satisfied with explanations.
#88 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-16 15:44:30
"Not yet falsified" is the highest level a scientific theory can get, and the same is not true in mathematics.
You still aren't getting it. Let's take Special relativity as an example. Scientists know it's not correct, and don't claim otherwise, despite your contrary and incorrect assertions.
Given the postulates Einstein started with, Special Relativity theory is absolutely correct. The uncertainty is in the postulates. That's no different from Euclid's work. It doesn't apply in the real world, but given his postulates, it holds absolutely true. The difference lies in mathematicians having the luxury of defining their working environment. Scientists don't have that luxury. They have to describe the world in the best terms they can. If those terms are wrong, it keeps their theories from being as accurate as they otherwise could be. However, given the environment they described and thought they were working in, their theories remain as valid as anything mathematicians come up with.
I didn't say that they do!
Good. Then you finally recognize that falsifiability is not the single defining characteristic of science. We're making progress.
You're trying to extrapolate on my words, and you're putting things in that aren't there. Of course for the scientific community to accept something it has to go under rigorous testing. But once this happens it still gets the label "not falsified". When the mathematical community accepts something, it gets the label "proved".
It is proved only for the given conditions. If those conditions change, the validity of the proof may or may not hold, just like any other science.
No, they aren't. When Einstein first came up with Relativity, he threw in some constants because otherwise his theory would imply that the universe was expanding. When Hubble showed that the universe was actually expanding, he took them out. Now we know the universe is not only expanding, but expanding at an accelerating rate. We must now put some constants back into Relativity equations so that they still hold. Each of these show an old idea being falsified and replaced with a new.
Each shows a theory being modified to better model reality as our understanding of reality becomes more precise. The conditions changed.
And a theory is an explanation.
Nothing could be farther from the truth. Well, okay, some things could, but you're completely wrong with that assertion.
The origin of the moon was actually shown to be from a meteor hitting earth, rather than a satellite being trapped in the Earth's orbit. This was an overturning of a previous accepted idea, a revolution.
You certainly have low standards for what defines revolution. 
New theorems have been made, but the old theorems (even if no longer used) are just as true.
Only some have been replaced with more accurate approximations. Just like any other science.
You think a mathematician can check against all the numbers that only numbers equivalent to 1 modulo 4 can be written as a sum of two squares? Absolutely not.
Are you really being this disingenuous with your argument?
In all cases outside of discrete mathematics, mathematicians can not check a conjecture against all known possibilities. Indeed, once they come up with a proof, they don't check against possibilities as they know it would be fruitless.
Hmmm If we assume you are correct here, then it would imply that an absolute proof would not be possible. Yet you've claimed that such proofs are what differentiate mathematics from the other sciences. I guess we can consider the discussion complete.
#89 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-16 14:21:07
Not in the least! Falsification is a method under which an idea goes strenuous test after test and we hold it true so long as it continues to pass tests. Mathematicians do not accept ideas which have only resisted tests, mathematicians only accept ideas which have been proven. This is why we have not concluded that the Riemann Hypothesis is true, no matter how many of the zero's we have verified fit.
Both discard ideas once there is evidence to suggest they are not true. Mathematicians have not abandoned the Riemann hypothesis, because there is no evidence to suggest it is not true. Scientists do not claim relativity to be correct. Scientists do not claim Newton's laws of motion to be correct. They recognize them for what they are, models that approximate reality closely under particular conditions.
You keep making assertions about scientists that aren't true.
You're either not understanding my point, or you're ignoring it. When a mathematician says something is true, he means for it to be proven so in the system he is studying. When a scientist says something is true, he means that it has not been falsified yet. These two ideas are entirely different.
I understand your claim. I just recognize it for what it is: incorrect. Again, you are making unsubstantiated claim about scientists in order to falsely differentiate between them and mathematicians, who are themselves scientists. Scientists don't accept just any idea that hasn't yet been disproved. That's simply not how science works.
Truths in biological and geological sciences are not far apart at all. They both rely on falsification, so it is perfectly fine to conflate the two.
As do mathematicians, so why not conflate scientists and mathematicians? There's far fewer differences between physicists, cosmologists and mathematicians than there are between biologists and physicists or biologists and cosmologists.
Someone hasn't be paying attention to string theory, the theory behind planetary evolution, expansion of the universe, the existence of dark matter, the existence of dark energy, the existence of dark flow. Relativity was certainly revolutionary, it showed for one thing that gravity was a force which altered spacial dimensions rather than a "pulling" force, as Newton thought of it. That turned physics up on it's head! Of course, find all the elementary particles was also revolutionary since before hand the atom was thought to be an indivisible piece of matter.
Those are all evolutionary advances in science. Are you familiar with the events and discoveries that led to them? Einstein, for example, just took Mach's work a small step further with relativity. With regard to gravity, you are confusing the theory with the explanation.
Of course, find all the elementary particles was also revolutionary since before hand the atom was thought to be an indivisible piece of matter.
You'll notice that I mentioned quantum physics as being revolutionary. Thanks for reiterating the point.
Your comparison is rather ridiculous. You seem to be attempting to conflate iteration and recursion with the iterative method of science. The theorems and algorithms in numerical analysis don't change, but the theories in the sciences do. Every day.
Are you saying that numerical analysis has not been refined over the course of its existence? Interesting. And wrong.
What you seem to be missing is that even though a mathematician can, in a relatively short amount of time, check a hypothesis against all known possibilities, while the scientist must conduct experiment after experiment, or make observation after observation, which may take years (or decades, or longer), they are doing the same thing. They are testing their hypotheses for flaws. It is the very act of logically testing hypotheses that makes something a science. It doesn't make any difference whether it takes five minutes, or two centuries. As long as mathematicians continue to rely on logic in proofs, mathematics will remain a science. It's really that simple.
#90 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-16 09:10:48
Take any subject which is agreed by virtually everyone to be a science, and I can guarantee you the fundamental principle at the heart of it is falsification.
This idea of falsification is not shared by mathematicians. Indeed, mathematicians prove statements don't hold by counterexamples, but this is very far removed from falsification.
It's not far removed from falsification, it is falsification.
When a mathematician says a mathematical statement is true, it does not simply mean "not falsified yet". A mathematical theorem being regarded as true is a much stronger statement then saying a theory in physics is true.
In both disciplines, conditions are set implicitly or explicitly that must hold in order for the statements to be true.
By saying mathematics is a science, you conflate the idea of mathematical truth with scientific truth, when in fact these two are quite far apart.
Not at all. That's like saying biological truths are conflated with geological truths because they are both sciences.
Perhaps an even more clear criteria is the way in which mathematical knowledge and scientific knowledge grows. The only way a revolution in mathematics can occur is if someone makes a statement which is not true. In order to do this, a mistake must be made, and it must have gotten passed all the people in the mathematical community that said "yea, that's right". That is to say, revolutions (overturning past ideas) are not expected in mathematics.
Of course not, because everyone in the mathematical community has said "yea, that's right." Unexpected does not imply unlikely.
On the other hand, revolutions are not only expected in the sciences, but they are demanded.
Demanded by whom? The last revolution of science (restricting my statements to Physics, Cosmology, and Quantum Theory) was the development of quantum theory. Before that, Newton's laws of motion and calculus. Everything else, including relativity, has been more evolutionary than revolutionary.
We fully expect that theories in physics will eventually be overturned, and even more so in biology or archaeology, because that's how science works. It's an approximation to the truth that only gets better and better.
Sounds a lot like numerical methods. Oh, wait
#91 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-16 07:18:16
All_Is_Number wrote:Mathematics is no more different from Science than cosmologists are different from biologists. They all have fundamental differences, but those differences are outnumbered by their fundamental similarities.
I'd have thought that it doesn't matter how many similarities there are, if there is a difference then they are different.
Ducks have a beak like a goose, have feathers like a goose and swim like a goose.
But ducks quack and geese honk, so ducks aren't geese.
By that logic, physicists and cosmologists can't both be scientists, because they are different.
Your analogy would be more accurate, if we equated scientists to birds, mathematicians to ducks, physicists to geese, cosmologists to quail, etc. Different species of birds are different, yet they are all species of birds.
#92 Re: Jai Ganesh's Puzzles » General Quiz » 2008-10-16 04:48:38
#851. After Greenland, which is the biggest island in the world (in area)?
Trick question?
#93 Re: Puzzles and Games » Math Questions (Easy?) » 2008-10-16 01:51:56
I see Skitt's law is alive and well. Apparently JaneFairfax does not understand recursion, either.
The only example of recursion so far in this thread has been gmsc's link. Placing the word that answers the question within the question doesn't make that question recursive. It might make it clever, perhaps, but it does not make it recursive.
Examples of recursiveness:
The first two examples are finitely recursive, while the third example is infinitely recursive.
Do you see the pattern yet, Ms. Fairfax?
#94 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-16 00:36:25
I see a distinction between logical and physical.
Interesting. Could you please elaborate?
#95 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-15 13:57:08
Mathematics transcends what I normally think of as Science, as it investigates things far beyond the physical universe ... in fact it is seldom concerned with the physical universe.
There is no information within the universe that is not a part of the universe, so I'm not sure how mathematics can investigate things beyond the universe.
From Wikipedia's Mathematics entry:
"The mathematician Benjamin Peirce called it 'the science that draws necessary conclusions'. Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere."
#96 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-15 12:40:18
This is a great argument for why mathematics is not science.
Not at all. It just illuminates that mathematician define their working environment to investigate the properties of their tools, while scientists use the known properties of the tools to investigate the characteristics of their working environments.
That's just a PC version of "it's true".
No, it's not the same thing. Scientists are generally very careful to distinguish between the two things.
I could spend an hour with each post rewording all my statements to be technically correct, but I'm not going to do that. Obviously truth in my previous quote was referring to scientific truth.
It wasn't how you wrote it, it was what you wrote.
Are you saying that you believe rigorous peer review can solve the problem in falsification as illustrated by Duhem?
No, I'm saying that "I don't disagree with Duhem's argument as you've expressed it" (as of your previous post). I've not researched exactly what Duhem's assertions were, so beyond your short description, I haven't enough knowledge to form an opinion about them. Having said that, rigorous peer review is an important component of scientific (including mathematical) research.
Comparing mathematics and science is like comparing a toolmaker with a mechanic or craftsman. Without each other, they would both need to find alternate work. Their relationship is of a symbiotic nature.
Not a bad comparison, but certainly not complete. Mathematics doesn't just make tools for scientists to use, we do much more than that. But from this, I would conclude you are of the opinion that science and mathematics are fundamentally different. Is this correct?
Mathematics is no more different from Science than cosmologists are different from biologists. They all have fundamental differences, but those differences are outnumbered by their fundamental similarities.
#97 Re: Dark Discussions at Cafe Infinity » Religion with science » 2008-10-13 18:25:46
Hmm, that was a bad example. Even though no one knows what the destiny is, it's still already written down.
OK, so now instead of a book, you have some beetles and recording equipment.
The beetles run around crawling in holes and doing beetley things, and you record them doing it.
Nobody knew what they would do the first time, but on watching the video back you would naturally know every action that took place.Free will stealing?
Wouldn't the video images be representative of the beetles' past? I don't think we can draw any conclusions about the existence of freewill based on the ability to know the past.
#98 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-13 18:10:56
Relativity theory was established based on the observation that the speed of light has a constant value no matter how fast the object emitting that light is traveling. The entire mathematical framework rested upon a thought experiment, but that thought experiment could only be played out because of observations.
More precisely, one of Einstein's two postulates from which relativity theory was derived was consistent with (but not based upon) the lack of observation of the speed of light varying with respect to the observer's frame of reference relative to the ether. Observation of such a difference had been expected, based on the maths. Let's not forget that the Michelson-Morley experiment was a failure from the perspective of the experimenters.
Of course mathematical theorems fail when you remove the hypotheses, there is nothing special in this.
I'm not sure what you are referring to with hypothesis. A mathematical theorem (as well as a scientific theory) is typically well beyond the hypothesis stage.
However, it does not bridge the gap between giving evidence for something in science and proving something in mathematics.
That gap will always exist, for mathematics exist conceptually, while science is the application of those conceptual tools in the real world. Conceptually, all possibilities can be tested. That's not usually possible in the real world. When working in the real world, it is not often possible to invent a space in which certain problems do not exist. In maths, we can model a particle with a velocity of zero. In reality, a particle in such a state could never be observed. Scientists do not have the luxury of being able to define the environment in which they work. Scientists are continually discovering properties of the environment in which they work.
The fundamental different is still there: science says "this happens every time we've seen it, so it's always true" where as mathematics says "this is always true".
It would be more accurate to say that science claims their models closely resemble reality, based on predictions repeatedly matching subsequent observations. Most scientists are under no illusion of knowing the Truth. Even Einstein recognized that relativity is not absolutely accurate. However, most every scientist recognizes that relativity theory models reality more accurately than Newton's Laws of Motion.
Duhem argued that at the edge of science, observations almost always rest up underlying theories. The result is that if you "prove" something is not true that requires an underlying theory, then your "proof that X is not true" is only as sturdy as the theories it relies on. Therefore, it is not possible do disprove a theory. Indeed, it may be your disproof that is incorrect.
I don't disagree with Duhem's argument as you've expressed it. (It was actually the soundness of the design of the experiment that resulted in the failed Michelson-Morley experiment becoming one of the most famous physics experiments ever.) This is one of the reasons that rigorous peer review is so important in science. By the same token, there are many, many invalid mathematical proofs available, such as those that "prove" 1=0. Of course, the errors may not be so obvious when not made intentionally. This is one of the reasons that rigorous peer review is so important in mathematics.
Comparing mathematics and science is like comparing a toolmaker with a mechanic or craftsman. Without each other, they would both need to find alternate work. Their relationship is of a symbiotic nature.
#99 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-13 16:48:48
But have you heard of the problem of the 'Mutilated Chess Board'?
Yes, although I must admit that I don't immediately recognize the relevance.
#100 Re: Help Me ! » differentiation » 2008-10-12 17:44:13
Can someone please help me to get the 1st and second derivative of the following:
Hint:
Use the chain rule, the division rule, and the property that
.