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Prometheus


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Joined: Sun Aug 07, 2011 8:58 am Posts: 309

I've got to write a short essay on 'any' subject in mathematics. I was thinking of writing about one of the Greek mathematical paradoxes, maybe Zeno's 'Achilles and the tortoise' (which i understand the solution to be a proper understanding of series).
I'd thought i'd just throw it out here though, see if any one knows of any other good Greek paradoxes worth writing about...





iNow


Original Member
Joined: Thu Aug 04, 2011 11:40 pm Posts: 5672 Location: Iowa

Maybe the current sovereign debt crisis they are facing...
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"[Time] is one of those concepts that is profoundly resistant to a simple definition." ~C. Sagan





DrRocket


Original Member
Joined: Fri Aug 05, 2011 2:22 am Posts: 477

Quote: I've got to write a short essay on 'any' subject in mathematics. I was thinking of writing about one of the Greek mathematical paradoxes, maybe Zeno's 'Achilles and the tortoise' (which i understand the solution to be a proper understanding of series).
I'd thought i'd just throw it out here though, see if any one knows of any other good Greek paradoxes worth writing about... There are several "Zeno paradoxes". They tend to be rather silly, as they are a combination of flawed logic and lack of appreciation for simple scientific observation. The Achilles and tortoise "paradox" for instance clearly fails to recognize the rather simple fact that Achilles (or surrogate) regularly overtakes and passed tortoises. That ought to be have been a sufficient clue for the poser of the paradox to find the flaw in the mathematical logic.The flaw of course is that the "intervals" considered are not of uniform length and in fact decrease rapidly and approach zero quickly. There is an even sillier Zeno paradox  the arrow paradox http://en.wikipedia.org/wiki/Zeno%27s_paradoxes The real puzzle is that anyone ever considered these lapses of logic as paradoxes. You might consider writing an essay on the fact that virtually all of mathematics arises from a very small set of axioms. In rigorous speak the basis is the ZermeloFraenkel axioms plus the axiom of choice. In practice this means only that one assumes the ordinary arithmetic of the natural numbers (the Peano postulates in may text books) plus the ability to select one element from each member of a family of nonemply sets. Most people are willing to accept these tenets, but from them arise essentially the entire edifice of modern mathematics  one actually constructs the real and complex numbers and from that point on it is just a matter of definitions and logic.
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Last edited by DrRocket on Fri Jul 06, 2012 1:17 am, edited 1 time in total.





Prometheus


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Joined: Sun Aug 07, 2011 8:58 am Posts: 309

DrRocket wrote: The Achilles and tortoise "paradox" for instance clearly fails to recognize the rather simple fact that Achilles (or surrogate) regularly overtakes and passed tortoises. That ought to be have been a sufficient clue for the poser of the paradox to find the flaw in the mathematical logic. Diogenes the Cynic is said to have answered the 'paradox' by silently walking across the room. I guess they were paradoxes at the time because there wasn't a mathematical argument to refute them, but since there is now, it should be considered a puzzle. I've actually come across a few sources which claim the paradox hasn't been resolved, but i don't even understand their reasoning let alone answer them. http://paulconnor.org/2009/03/06/beyondinfinityrussellandzeno/The lamp paradox has got me though  in an infinitesimal sequence of turning on and off a lamp will the lamp be on or off after a minute. DrRocket wrote: You might consider writing an essay on the fact that virtually all of mathematics arises from a very small set of axioms. In rigorous speak the basis is the ZermeloFraenkel axioms plus the axiom of choice. In practice this means only that one assumes the ordinary arithmetic of the natural numbers (the Peano postulates in may text books) plus the ability to select one element from each member of a family of nonemply sets. Most people are willing to accept these tenets, but from them arise essentially the entire edifice of modern mathematics  one actually constructs the real and complex numbers and from that point on it is just a matter of definitions and logic. Looking up Zeno's praradox i began reading about Bertrand Russel's ideas on it  which led me to set theory and then to the ZermeloFraenkel axioms. I need to study set theory much more before i write an essay on it though.





DrRocket


Original Member
Joined: Fri Aug 05, 2011 2:22 am Posts: 477

Quote: Diogenes the Cynic is said to have answered the 'paradox' by silently walking across the room. I guess they were paradoxes at the time because there wasn't a mathematical argument to refute them, but since there is now, it should be considered a puzzle. I've actually come across a few sources which claim the paradox hasn't been resolved, but i don't even understand their reasoning let alone answer them. http://paulconnor.org/2009/03/06/beyond ... andzeno/The lamp paradox has got me though  in an infinitesimal sequence of turning on and off a lamp will the lamp be on or off after a minute. There is no paradox in any of the Zeno paradoxes, or in the piece by Paul Connor at your link, just philosophical nonsense. Fuzzy thinking that its worst. There is no puzzle either, since there is no question posed. All that is given is a premise, some mumbled nonsense, and a false conclusion. Diogenes response is the only bit of rational thought in evidence. Neither is there any paradox in Thompson's "lamp paradox". Again just misunderstanding and misuse of the concept of the infinite, a concept handled properly by mathematicians several time a day. The "lamp paraadox" deals with a function of time (on/off) that is discontinuous and has no limit at one minute, so there is no answer to the question, nor should there be any answer. There is also the issue of an attempt to pose a physical problem in terms that clearly cannot be achieved in the physical world (you cannot flip a real switch arbitrarily quickly or reverse that action in an arbitrarily short period of time)  and when you pose the question in terms of the limit of the associated on/off function the resolution is obvious. In short, when properly posed in terms of mathematics or physics there are no paradoxes in evidence. The "paradoxes" arise only in terms of philosophical nonsense.
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Last edited by DrRocket on Fri Jul 06, 2012 1:17 am, edited 1 time in total.





Prometheus


Original Member
Joined: Sun Aug 07, 2011 8:58 am Posts: 309

Could the puzzle  whatever  also be explained in terms of set theory? Still very new to this, so this might be jumbled, plus i don't know the correct notation...
Could we say that the infinities contained within the distance between the runners is a set which has infinite members (set A)? Could we also say that set A itself is entirely contained within another set (set B) which would make it A a subset of B. Now this is the bit i'm less sure of  would it then be fair to say that because set A is contained in set B, set A is then countable (you have enough members in B to count A plus more).
Don't know if that makes sense, I'm just trying to get my head around set theory. I can see why the Greeks had difficulties though  its a slippery concept.





DrRocket


Original Member
Joined: Fri Aug 05, 2011 2:22 am Posts: 477

.
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Last edited by DrRocket on Fri Jul 06, 2012 1:18 am, edited 1 time in total.





Prometheus


Original Member
Joined: Sun Aug 07, 2011 8:58 am Posts: 309

I'm learning maths to help with my science; i'm not sure how practical set theory would be. But for some reason it does appeal to me.





DrRocket


Original Member
Joined: Fri Aug 05, 2011 2:22 am Posts: 477

Prometheus wrote: I'm learning maths to help with my science; i'm not sure how practical set theory would be. But for some reason it does appeal to me. Basic set theory, as in the book by Halmos, is fundamental to learning other mathematics. It is rather simple stuff and not something that requires a formal class, and in fact it would be rather difficult to find enough material for a formal class. Get the book and read it. It is quite short and quite clear. You can read it over a weekend.
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GiantEvil


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Joined: Sat Aug 06, 2011 10:19 am Posts: 786

The Halmos book is good, I have a 61' edition. The paperback is the best deal. It isn't a large book and could be read in a weekend, but I've been intentionally digesting it slowly in small bits. For maximum comprehension. I also recommend this book on set theory.
_________________ It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence. W. K. Clifford





Prometheus


Original Member
Joined: Sun Aug 07, 2011 8:58 am Posts: 309

I'll check them out  see if the library has got copies first.





iNow


Original Member
Joined: Thu Aug 04, 2011 11:40 pm Posts: 5672 Location: Iowa

_________________ iNow
"[Time] is one of those concepts that is profoundly resistant to a simple definition." ~C. Sagan





