Q. I would like to create e-courses in mathematics. A professor of mathematics has asked me to help him out with this. What kind of formatting do I use to present the material which consists of lectures and questions and answers? Is Powerpoint a good solution? I would be very grateful for suggestions and pointers, and also addresses of web sites that could be of help. A big thanks to all those who are willing to advise me.
A. Check out MIT open courseware:
http://ocw.mit.edu/index.html
and this list of online math sites:
http://tutorial.math.lamar.edu/links.aspx
http://ocw.mit.edu/index.html
and this list of online math sites:
http://tutorial.math.lamar.edu/links.aspx
What is the difference between a sufficient condition and a necessary condition?
Q. In mathematics, what is the difference between a sufficient condition and a necessary condition? What is the definition of these two terms? Please give one or two easy to understand examples of how they are used.
Thanks in advance!
Thanks in advance!
A. Technically:
1. Philosophy and Conditions
An ambition of twentieth-century philosophy was to analyse and refine the definitions of significant terms � and the concepts expressed by them � in the hope of casting light on the tricky problems of, for example, truth, morality knowledge and existence that lay beyond the reach of scientific resolution. Central to this goal was specifying at least in part the conditions to be met for correct application of terms, or under which certain phenomena could truly be said to be present. Even now, philosophy's unique contribution to interdisciplinary studies of consciousness, the evolution of intelligence, the meaning of altruism, the nature of moral obligation, the scope of justice, the concept of pain, the theory of perception and so on still relies on its capacity to bring high degrees of conceptual exactness and rigour to arguments in these areas.
If memory is a capacity for tracking our own past experiences and witnessings then a necessary condition for Penelope remembering giving a lecture is that it occurred in the past. Contrariwise, that Penelope now remembers the lecture is sufficient for inferring that it was given in the past. What, then, is a necessary (or a sufficient) condition? This article shows that complete precision in answering this question is itself elusive. Although we can use the notion of necessary condition in defining what it is for something to be a sufficient condition (and vice versa), there is no straightforward way to give a precise and comprehensive account of the meaning of the term "necessary (or sufficient) condition" itself. Wittgenstein's warnings against premature theorising and overgeneralising, and his insight that many everyday terms pick out families, should mandate caution over expecting a complete and unambiguous specification of what constitutes a necessary, or a sufficient, condition.
Specifically:
1.) If I opened the door, I used the key.
2.) If you touch me, I'll scream
While in the case of the door, using the key was necessary for opening it, no parallel claim seems to work for (2): in the natural reading of this statement, my screaming is not necessary for your touching me. McCawley claims that the "if"-clause in a standard English statement gives the condition � whether epistemic, temporal or causal � for the truth of the "then"-clause. The natural interpretation of (2) is that my screaming depends on your touching me. To take my screaming as a necessary condition for your touching me seems to get the dependencies back to front.
Confused? Good. My work here is done....
I linked the complete article from Stanford on the subject below....
1. Philosophy and Conditions
An ambition of twentieth-century philosophy was to analyse and refine the definitions of significant terms � and the concepts expressed by them � in the hope of casting light on the tricky problems of, for example, truth, morality knowledge and existence that lay beyond the reach of scientific resolution. Central to this goal was specifying at least in part the conditions to be met for correct application of terms, or under which certain phenomena could truly be said to be present. Even now, philosophy's unique contribution to interdisciplinary studies of consciousness, the evolution of intelligence, the meaning of altruism, the nature of moral obligation, the scope of justice, the concept of pain, the theory of perception and so on still relies on its capacity to bring high degrees of conceptual exactness and rigour to arguments in these areas.
If memory is a capacity for tracking our own past experiences and witnessings then a necessary condition for Penelope remembering giving a lecture is that it occurred in the past. Contrariwise, that Penelope now remembers the lecture is sufficient for inferring that it was given in the past. What, then, is a necessary (or a sufficient) condition? This article shows that complete precision in answering this question is itself elusive. Although we can use the notion of necessary condition in defining what it is for something to be a sufficient condition (and vice versa), there is no straightforward way to give a precise and comprehensive account of the meaning of the term "necessary (or sufficient) condition" itself. Wittgenstein's warnings against premature theorising and overgeneralising, and his insight that many everyday terms pick out families, should mandate caution over expecting a complete and unambiguous specification of what constitutes a necessary, or a sufficient, condition.
Specifically:
1.) If I opened the door, I used the key.
2.) If you touch me, I'll scream
While in the case of the door, using the key was necessary for opening it, no parallel claim seems to work for (2): in the natural reading of this statement, my screaming is not necessary for your touching me. McCawley claims that the "if"-clause in a standard English statement gives the condition � whether epistemic, temporal or causal � for the truth of the "then"-clause. The natural interpretation of (2) is that my screaming depends on your touching me. To take my screaming as a necessary condition for your touching me seems to get the dependencies back to front.
Confused? Good. My work here is done....
I linked the complete article from Stanford on the subject below....
Can we really say that anything can extend infinitely?
Q. In mathematics, we are taught that many functions, relations, and concepts can extend infinitely? But isn't the distance of such objects limited by the current size of the universe?
I'm going to ask this in both the Mathematics section and the Astronomy section to see if I get different answers.
I'm going to ask this in both the Mathematics section and the Astronomy section to see if I get different answers.
A. The distance of anything is limited by the size of the universe.
However, the size of the universe could be... infinite.
There is no evidence, yet, that imposes an upper limit to the size of the universe. At best, what we found so far is evidence for a lower bound (a minimum size) of at least three times the portion we can see.
The attempts by WMAP to find "curvature" to the universe ends up with showing a "flat" universe. By itself, this result does not prove that the universe is infinite (there exists finite but edgeless spaces that are geometrically flat), however it does leave the possibility open.
There is still a possibility that the universe is both flat and infinite; it could be flat and finite (but without an edge); it could be made of finite sections, each section connecting to the next one by a "fold" -- the soccer ball analogy of 2008; it could even be curved (and finite) -- but our measurements are simply not accurate enough to show the curvature.
Right now, you can use whichever one suits your idea, but you cannot prove the other ones are wrong. At least, no yet.
There are many scientists who are working on what could have existed before the Planck Time (the earliest moment we can explore with our present knowledge). For most of their ideas to make sense, the universe has to be finite in spatial extent (and it must be embedded in higher-dimension spaces).
But, so far, that is only wishful thinking. And that is not sufficient (yet) to discard the possibility that the universe could be infinite in spatial extent.
Of course, if the universe is infinite now, then it was already infinite at the Planck Time and, although not impossible, that does raise more questions.
To paraphrase a beer commercial (beer and astrophysics are compatible):
Stay curious my friend!
However, the size of the universe could be... infinite.
There is no evidence, yet, that imposes an upper limit to the size of the universe. At best, what we found so far is evidence for a lower bound (a minimum size) of at least three times the portion we can see.
The attempts by WMAP to find "curvature" to the universe ends up with showing a "flat" universe. By itself, this result does not prove that the universe is infinite (there exists finite but edgeless spaces that are geometrically flat), however it does leave the possibility open.
There is still a possibility that the universe is both flat and infinite; it could be flat and finite (but without an edge); it could be made of finite sections, each section connecting to the next one by a "fold" -- the soccer ball analogy of 2008; it could even be curved (and finite) -- but our measurements are simply not accurate enough to show the curvature.
Right now, you can use whichever one suits your idea, but you cannot prove the other ones are wrong. At least, no yet.
There are many scientists who are working on what could have existed before the Planck Time (the earliest moment we can explore with our present knowledge). For most of their ideas to make sense, the universe has to be finite in spatial extent (and it must be embedded in higher-dimension spaces).
But, so far, that is only wishful thinking. And that is not sufficient (yet) to discard the possibility that the universe could be infinite in spatial extent.
Of course, if the universe is infinite now, then it was already infinite at the Planck Time and, although not impossible, that does raise more questions.
To paraphrase a beer commercial (beer and astrophysics are compatible):
Stay curious my friend!
What are your thoughts concerning poltergeists and air, fire, earth, and water?
Q. Would it not seem that the so-called "poltergeist" paranormal phenomenon (with its movement of inanimate, sometimes extremely heavy, objects; opening and closing of doors and windows by no visible means; unexplained noises; destruction of garments and household objects; materialization, dematerialization, and teleportation of matter; sudden appearance of fires and pools and flows of water; malfunctioning of electrical equipment, and even physical assault) would be the best choice for scientific analysis given the extreme concentration and intensity of physical evidence?
I have also posted this question in the Science & Mathematics > Physics section.
AIR (unexplained sounds or "movements" of air)
http://www.disclose.tv/forum/scientific-evidence-of-poltergeist-knocking-t25776.html
http://www.ghosttheory.com/2010/07/01/scientific-evidence-of-poltergeist-knockings-found
FIRE (spontaneous outbreaks of fires)
http://www.prairieghosts.com/al_fire.html
http://www.cryptomundo.com/cryptozoo-news/poltergeist-girls
EARTH (materializations of stones, mud, et cetera)
http://www.prairieghosts.com/millville.html
WATER (materialization of water or water gushing out of nowhere)
http://paranormal.about.com/od/earthmysteries/a/aa070306.htm
http://www.anomalist.com/reports/water.html
http://www.prsne.com/water_poltergeist.htm
The "air," "fire," "earth," and "water" references cleverly illustrate (with links offered as examples) four consistent physical manifestations displayed by our so-called "poltergeists." That these four variations coincides with the four "elements" is a curiosity that I find amusing. The intrinsic concentration and intensity of physical evidence might give "Science" a bit more to analyze if they took more time and effort to look and less time to ridicule and �explain away� without proper investigation.
I have also posted this question in the Science & Mathematics > Physics section.
AIR (unexplained sounds or "movements" of air)
http://www.disclose.tv/forum/scientific-evidence-of-poltergeist-knocking-t25776.html
http://www.ghosttheory.com/2010/07/01/scientific-evidence-of-poltergeist-knockings-found
FIRE (spontaneous outbreaks of fires)
http://www.prairieghosts.com/al_fire.html
http://www.cryptomundo.com/cryptozoo-news/poltergeist-girls
EARTH (materializations of stones, mud, et cetera)
http://www.prairieghosts.com/millville.html
WATER (materialization of water or water gushing out of nowhere)
http://paranormal.about.com/od/earthmysteries/a/aa070306.htm
http://www.anomalist.com/reports/water.html
http://www.prsne.com/water_poltergeist.htm
The "air," "fire," "earth," and "water" references cleverly illustrate (with links offered as examples) four consistent physical manifestations displayed by our so-called "poltergeists." That these four variations coincides with the four "elements" is a curiosity that I find amusing. The intrinsic concentration and intensity of physical evidence might give "Science" a bit more to analyze if they took more time and effort to look and less time to ridicule and �explain away� without proper investigation.
A. If you can find one that is consistent, then yes it would be ideal to study. Unfortunately most of them are not, so full time recording is difficult.
I only know of one that is fairly consistent, and that is McKenzies poltergeist in the Black Mausaleum in Greyfriars Cemetary in Edinburgh UK.
Mercat tours operate a guided walk of the cemetary - http://www.mercattours.com/paranormal.asp
It is interesting to note that the place was owned and managed by the town council, but felt the necessity to close the place due to the number of injuries sustained by visitors. Mercat tours turned this to their advantage by issuing a disclaimer, and taking photographic evidence of those injured.
These photo's now adorn the gatehouse of the cemetary for you to see. Most injuries are scratches and cuts, but there have been cases of the gate closing unexpectedly, and some more serious injuries.
Regarding the elements and Elementals, it's possible for them to cause problems, but also easy to dismiss it as natural events - unless you are expecting something.
I only know of one that is fairly consistent, and that is McKenzies poltergeist in the Black Mausaleum in Greyfriars Cemetary in Edinburgh UK.
Mercat tours operate a guided walk of the cemetary - http://www.mercattours.com/paranormal.asp
It is interesting to note that the place was owned and managed by the town council, but felt the necessity to close the place due to the number of injuries sustained by visitors. Mercat tours turned this to their advantage by issuing a disclaimer, and taking photographic evidence of those injured.
These photo's now adorn the gatehouse of the cemetary for you to see. Most injuries are scratches and cuts, but there have been cases of the gate closing unexpectedly, and some more serious injuries.
Regarding the elements and Elementals, it's possible for them to cause problems, but also easy to dismiss it as natural events - unless you are expecting something.
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