tag:blogger.com,1999:blog-3188764845452746737.post8040444715294858427..comments2021-02-03T06:04:48.471-08:00Comments on deretour: What are fractions?Hans Georg Lundahlhttp://www.blogger.com/profile/01055583255516264955noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-3188764845452746737.post-50481474130999493432009-12-04T09:25:30.893-08:002009-12-04T09:25:30.893-08:00Maths in English:
1 A quote from Aristotle 2 Two ...Maths in English:<br /><br />1 <a href="http://o-x.fr/tpqc" rel="nofollow">A quote from Aristotle</a> 2 <a href="http://o-x.fr/2lzf" rel="nofollow">Two more than two make four, but why?</a> 3 <a href="http://o-x.fr/gal" rel="nofollow">What are fractions?</a>Hans Georg Lundahlhttps://www.blogger.com/profile/01055583255516264955noreply@blogger.comtag:blogger.com,1999:blog-3188764845452746737.post-77627569700403348052009-11-01T12:13:47.864-08:002009-11-01T12:13:47.864-08:00Which, ALSO, illustrates once again: number, size ...Which, ALSO, illustrates once again: number, size and prooportion are three different things. Not one and the same.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3188764845452746737.post-77503821901486935592009-11-01T12:12:27.220-08:002009-11-01T12:12:27.220-08:00One thing was implied but not stated; as numbers a...One thing was implied but not stated; <b>as numbers and sizes and shapes and things are absolute, proportions are relative, competitive:</b><br /><br />three (3) is three whether you compare it to nothing or to something, whether you compare it to one (1) or to three million (3.000.000); <b>but</b> if you compare it to one, it is the proportion thrice (3/1), if you compare it to three million it is the proportion one millionth (1/1.000.000).<br /><br />A pizza slice remains same shape, same size and same proportion to its origin, the as yet uncut pizza, BUT when two slices are added, it diminishes its part of total amount of pizza still available from 1/8 to only 1/10, if you then eat six slices, each remaining slice increases from 1/10 to 1/4.<br /><br />How is that an increase? Well, remember that a slice 1/4 of a pizza is bigger than the next cut, 1/8 slice.<br /><br />The one thing is: 1/4 of a whole pizza is twice 1/8 of a whole pizza.<br /><br />The other thing is: 1/4 or for that matter 1/8 remains same proportion whatever the sizes or numbers involved (remember the 80 slices from 10 pizzas).<br /><br />So: 1/4 of half a pizza (=1/8 of a whole) is same proportion, though only half the size of 1/4 of a whole pizza.<br /><br />A proportion may thus increase or decrease at one end for two reasons: either because that end itself increases or decreases, OR because the other end does the reverse.<br /><br /><b>Proportions are competitive.</b>Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3188764845452746737.post-87864210308941078332009-10-31T08:17:28.380-07:002009-10-31T08:17:28.380-07:00But of course, there are people who say:
a) all n...But of course, there are people who say:<br /><br />a) all numbers are written with numerals<br /><br />b) all fractions are written with numerals<br /><br />c) therefore all fractions are numbers<br /><br />which is as stupid as:<br /><br />a) all cats are mammals<br />b) all dogs are mammals<br />c) therefore all dogs are cats<br /><br />"written with numerals" or "mammals" are what is called the middle term - the one you find in both premisses and which ties the other terms together or separates them before the conclusion sums up their relation<br /><br />Now, for a middle term to do its work, it must be univocal, at least to the extent that something can be clearly identified as being the middle term in one and same sense, one and same instance in both premisses. Like if one premiss says "all" or "none" about middle term, that includes all instances of the middle term in the other premiss too.<br /><br />But a predicate is not said "all" of, that is reserved for subject. And a predicate of a negative sentence is of course one you say "none" of. But neither premiss is negative.<br /><br />Therefore the middle term "mammals" and "written with numerals" may be applicable in quite different parts of theirs to the subject "cat" or "number" and the subject "dog" or "fraction". And the conclusion does not follow.<br /><br />Why don't they teach logic in these schools?Hans Georg Lundahlhttps://www.blogger.com/profile/01055583255516264955noreply@blogger.comtag:blogger.com,1999:blog-3188764845452746737.post-20688696092597086392009-10-31T07:41:42.225-07:002009-10-31T07:41:42.225-07:00Yes, there is a hinch: what if the added two slice...Yes, there is a hinch: what if the added two slices are from a bigger pizza?<br /><br />Well then, in that case, each of the 10 slices is still 1/8 of the original pizza it was mde from, AND still 1/10 of the number of slices, BUT the ones from the smaller pizza are less than 1/10 of the amount of pizza and the ones from the bigger pizza are more than 1/10 of the amount of pizza.<br /><br />Which illustrates once again, number (as number of slices) and quantity as for stuffing a stomach or for weighing on a balance is not the same thing.Hans Georg Lundahlhttps://www.blogger.com/profile/01055583255516264955noreply@blogger.comtag:blogger.com,1999:blog-3188764845452746737.post-15371895181770488322009-10-31T06:55:43.739-07:002009-10-31T06:55:43.739-07:00And (no, I won't log in right now) the great d...And (no, I won't log in right now) the great difficulty in math teaching nowadays comes from making operations (like -2=take away two) or qualities (like -2=two of a negative quality) or symbols which in themselves mean nothing (like 0) or things other than uncompared wholes, and therefore proportionals (like 1/8 or pi) the same as numbers.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3188764845452746737.post-43679251665650600032009-10-31T06:36:18.198-07:002009-10-31T06:36:18.198-07:00And a whole pizza is as much pizza as eight slices...And a whole pizza is as much pizza as eight slices, but not as many slices as them, since the whole is not sliced. And the slices of a pizza are as much as the whole pizza was, but not as many pizzas, since each is not a round pizza, but a slice.Hans Georg Lundahlhttps://www.blogger.com/profile/01055583255516264955noreply@blogger.comtag:blogger.com,1999:blog-3188764845452746737.post-7999605256034592352009-10-31T06:33:08.392-07:002009-10-31T06:33:08.392-07:00This means of course that a proportion, as a fract...This means of course that a proportion, as a fraction, is not a number. Numbers answer the question "how many" not only involve it as in "how many in proportion to how many others" but directly answer it.<br /><br />Two (2) is a number, whereas twice (2/1) is not a number but a proportion. However, since the number two is twice the one, one says that 2 is equivalent to 2/1.Hans Georg Lundahlhttps://www.blogger.com/profile/01055583255516264955noreply@blogger.com