
#37
Jun2408, 07:45 PM

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Infinity is not a number. Yet another thing every mathematician worth his or her salt knows.




#38
Jun2508, 01:16 PM

P: 121

I am suprised at you. Whether or not infinity is a number is a matter of definition, from the viewpoint of the field of mathematics. There are different definitions. It is a matter of preference, or perhaps of convenience.  When my oldest daughter Charlotte was six, she was scheduled to take an entrance exam to get into second grade in a private school. I had stong suspicions that she would be asked what was the largest number she knew. With mischief in my heart, I taught her to answer "infinity." The time for the test came. The tester reported to us that when she asked Charlotte what was the biggest number she knew, Charlotte smiled a big confident smile and answered, "Infinity! And, infinity is equal to ten, because ten is the biggest number I know!" :)  But, in all seriouness, how about this for a "universal" definition of a number  "A number is an answer to the question 'how many elements are in that set?' " DJ 



#40
Jun2508, 01:46 PM

P: 121

Yeah, Zeno's paradox is difficult for modern mathematicians to understand. I will try to explain the paradox and try to explain while it is difficult for us moderns to understand at the same time. I will do this by giving a modern formulation of the paradox that shows where Zeno thought differently from the way we do. The paradox is this. Suppose a hare is chasing a rabbit and the hare goes twice as fast as the rabit. The hare starts out 120 miles behind the rabbit and the hare runs at 60 miles an hour. The rabbit runs at 30 miles an hour. The race is track is four miles long. After one hour, the hare is one half mile behind the rabbit. After another half hour the hare is 1/4 mile behind the rabbit. The nth time period has length one hour / 2^n. As each time period passes, the hare halves his distance to the rabbit. To actually catch the rabbit, the hare would have to run for as long as it takes to transverse an infinite number of these intervals. But, in the real world, infinity does not exist. Hence the hare never catches the rabbit. But we all know that the hare passes the rabbit when they have both run for one hour, and after two hours, the hare crosses the finish line. This material is quasioriginal with me, so, it may well stand improvement. Please let me know if it does. There are too many dirrerent ways to explain this in modern terms to make a start on it. But, I think what I have written catches the essential idea and illustrates the different viewpoints of the relationship between mathematics and the real world. Actually, the viewpoint of mathematicians even two centuries ago was closer to the Greeks. DJ "By quasioriginal," I mean, "I claim no credit for being the original proporter of these ideas in case they should happen to be right, but, I accept full responsibility for them if they are wrong." 



#41
Jun2508, 01:57 PM

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#42
Jun2508, 02:27 PM

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#43
Jun2508, 07:16 PM

P: 121

Please accept my apology, DH. English is not a "contextfree" language, and I misinterperted the the context in which you were making your statment. In other words, it was "my bad," not "your lack," that resulted in my suprise. My last paragraph should have read: "But, in all seriouness, how about this for a "universal" definition of a (cardinal) number  "A cardinal number is an answer to the question, 'How many elements are in a set?' " I really would like to know your opinion on this proposal. For example, this proposal would exclude the cardnalities of classes from the collection of cardinal numbers. [A Note for Beginners: If I remember correctly, Bertrand Russel introduced "classes" into ZermeloFrankel (ZF) set theory for the purpose of getting his paradox (Russell's paradox) out of set theory. If I understand correctly, one can consider the "class of all sets" in Russel's extension of ZF. In other words, a "class" is like a "really big set." Nobody cared  (correction  no mathematicians cared)  except those working in mathematical logic  until recent years when category theory began to take over.] [Another Note for Beginners: The ubiquity of category theory in modern mathematics is a movement kicked off by Alexander Grothendieck's application of category theory to derive his generalization of the HirzebuchRiemannRoch theorem in the 1950's. Category theory was developed (mostly by Grothendieck) in the thousands of pages of tomes called EGA and SGA that stand for the French equivalents of "Exposition of"  and "Seminar in"  "Algebraic Geometry." There were other giants  such as Grothendieck's advisor Jean Dieudonne, the mysterious Jean Paul Serre, and the incredible Pierre Deligne  whose mathematical work played major roles in bringing category theory to the attention of the mathematical community.] Another example of the kind of thing I am asking about is whether or not the infinities in "nonstandard analysis" match up with the infinite cardinal numbers in the usual set theory that mathematicians use. In other words, did Robinson include  or try to include  the cardinalities of classes in his "nonstandard analysis." (I have the impression that there are logical problems that are best avoided by excluding the sizes of classes from the collection of cardinal numbers. I don't remember what they are, except, i have the impression that they are related to Russel's paradox.) (I'm guessing, here, that the collection of cardinal numbers form a class and not a set. I would like to know if that is correct.) 



#44
Jun2508, 10:20 PM

P: 121

However, I think that the viewpoint that I propose has a different spin than the viewpoints expressed in your previous posts. I am proposing that the crux of Zeno's reasoning (expressed in modern language) is that he didn't believe that an infinite quantities of things really existed in the real world. Assuming that is what he was thinking, he might have been right of course, in which case one explanation of his paradox is that you can't keep dividing time in half. When you get down to a certain granularity, you take an instantaneous jump accross the atomic length of time. If I remember correctly, the Greeks did believe that matter was not infinitely divisible, that's what they (e.g., Lucretius?) meant by by "atom," the smallest, indivisible, unit of matter. Granted (a correction to my previous post) they did think that time had no beginning, so, my statement that they did not believe that infinity existed in the real world was a little hasty. But, it does not seem unreasonable to me that they thought that an infinity of intervals of time did not exist. This seems more reasonable to me than the supposition that they thought that the sum of the geometric series diverged. More accurately, that they thought that it was impossible for an infinite number of terms to have a finite sum. Even though that is "literaly" what Zeno said, I suspect he took his "paradox" as a demonstration that infinite quantities do not exist i nature. What do you all think? I know that at least some of yon haave thought about it. Here is a pointer to one of your posts that suggests a different intrepertation than the one that I am purporting. http://www.physicsforums.com/showthr...no%27s+paradox Yours, DJ P.S. It's important to consider what the ancients thought. They were excellent mathematicians (Archimedes is credited with discovering the essence of the calculus) and they had a different "worldview" than we do. So, from an objective viewpoint (like for example from the viewpoint of Bayesian probability theory), there is a nonzero probabiity that they wer more right than we are in their conception of realiity. 



#45
Jun2508, 10:42 PM

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My second thought is that Zeno simply thought any series with an infinite number of terms had to diverge, and yet it obviously doesn't. Paradox. Not surprising since he predates the concept of a convergent infinite series by a mere 2100 years or so. 



#46
Jun2508, 11:52 PM

P: 121

I guess you're right about getting a little "off topic." Time to move on! DJ 



#47
Feb909, 10:57 PM

P: 328

The next person to say "1 / infinity = 0" gets a complimentary punch in the face.
You might as well say "1 / applesauce = 0" Because applesauce is just as much a number as infinity. 



#48
Feb1009, 04:18 PM

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#49
Feb1009, 04:37 PM

P: 328

So here is a similar problem, given an continuous uniform distribution over the interval [0,1], what is the probability that any number picked at random is rational?




#50
Feb1009, 05:14 PM

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0, naturally.




#51
Feb1009, 06:03 PM

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The point being that 0 probability on a continuous probability distribution does NOT mean "impossible" nor does probability 1 mean "certain to happen".




#52
Feb1009, 09:30 PM

P: 328

I mean applesauce!!! 



#53
Feb1009, 11:31 PM

P: 328

If 1/infinity = 0 then 1 = 0 * infinity. If 1/infinity = 0 then 2*1/infinity = 2*0, also known as 2/infinity = 0 If 2/infinity = 0 then 2 = 0 * infinity If 1 = 0*infinity AND 2 = 0*infinity Then 1=2 Reductio ad absurdum. 



#54
Feb1109, 07:32 PM

P: 270

You might also have heard of Transfinite numbers. 


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