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Old 2005-09-11, 13:30   #1
mfgoode
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Lightbulb Algebraical Fallacies 1 and 2.


IN these days when cranks are appearing in this forum e.g. sghodeif it will be worth revising subtle axioms which, many may have forgotten, that were taught sometime in their formal education.
As a sequel to ‘Peano Postulates’ I present here two such algebraical fallacies.

Paradox 1: To prove that any number is greater than itself.
Assume a and b are positive such that a>b
Multiplying both sides by b we get a*b >b*b
Subtracting a^2 from both sides and factoring we get a (b-a)>(b+a) (b-a)
Now dividing both sides by (b-a)
We get a > (b + a).
Then since b is positive not only is 'a' greater than itself, but greater than any number greater than itself!!

Paradox 2: To prove that 1/8 > ¼
We make use of a logarithmic property.
3 > 2
Multiply both sides by log(1/2) Then
3*log (1/2) > 2 *log(1/2)
Or log (1/2)^3 > log (1/2) ^2
Whence 1/8 > ¼
Ref : ‘Riddles in mathematics’ by Eugene P. Northrop
Mally
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Old 2005-09-11, 14:16   #2
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One procedure when providing a proof goes something like this:

1) Make an assumption
2) Carry out operations leading to a contradiction
3) Realise that the contradiction proves the assumption is not true.

What you (or Mr. Northrop) have done is to use the contradiction to “prove” that your assumption is true. Therefore there is no paradox because any mathematician would use the contradiction to realise that the assumption is not true and therefore a is not > b.
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Old 2005-09-11, 15:21   #3
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Quote:
Originally Posted by mfgoode
:
Now dividing both sides by (b-a)
Dividing and taking square roots should always ring an alarm. I'd say just about all so-called paradoxes rely on division by zero or assuming that the square root of a number is a unique value.

In your case, dividing by (b-a) requires that a≠b, but this is not the problem here (for a change). If a>b, b-a is negative and we need to flip the inequality when we divide by it because the signs on both sides change. Since we assumed that a>b, the last line must read a < (b + a), which is just fine.

Edit: same thing in the other puzzle: log(1/2) is negative, flip the inequality when multiplying by it and all is well.

Alex

Last fiddled with by akruppa on 2005-09-11 at 15:24
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Old 2005-09-11, 17:23   #4
mfgoode
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Cool Algebraical Fallacies 1 and 2

Quote:
Originally Posted by Numbers
One procedure when providing a proof goes something like this:

1) Make an assumption
2) Carry out operations leading to a contradiction
3) Realise that the contradiction proves the assumption is not true.

What you (or Mr. Northrop) have done is to use the contradiction to “prove” that your assumption is true. Therefore there is no paradox because any mathematician would use the contradiction to realise that the assumption is not true and therefore a is not > b.
I am afraid Numbers that you have put the cart before the horse!
1) is NOT an assumption but a FACT
2) the operation seems valid but violates an axiom.
3) The result contradicts the fact (!) and is wrong. Therefor it is a paradox.
Mally
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Old 2005-09-11, 17:28   #5
mfgoode
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Thumbs up Algebraical Fallacies 1 and 2

Quote:
Originally Posted by akruppa
Edit: same thing in the other puzzle: log(1/2) is negative, flip the inequality when multiplying by it and all is well.
Alex
Astute observations indeed Alex. You are spot on!
I will take you on, on more difficult paradoxes if you dont mind.
Mally
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Old 2005-09-11, 19:00   #6
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Quote:
Originally Posted by mfgoode
Assume a and b are positive such that a>b
Pardon me for assuming that when you said assume you meant assume.
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Old 2005-09-12, 01:59   #7
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Quote:
Originally Posted by Numbers
Pardon me for assuming that when you said assume you meant assume.
I detect latent hostility. If I'm wrong I apologize, but if I am right, your attitude is unwarranted.

the first paradox isn't following a proof by contradiction outline, "a > b + a" is not presented as a contradiction to "a > b" (it can be continued to one, but it is not presented as such). The proof outline is a straightforward implication proof

P => Q

PF.
Assume P
.
.
.
Therefore Q
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Old 2005-09-12, 04:31   #8
mfgoode
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Default Algebraical Fallacies 1 and 2

Thank you Travis.
It was an odd remark I agree.
I am here for the pursuit of truth which can never be stamped out!
Its a case of 'take it or leave it'
Mally
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Old 2005-09-12, 07:46   #9
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Quote:
Originally Posted by TravisT
I detect latent hostility. If I'm wrong I apologize, but if I am right, your attitude is unwarranted.
Any one who has read more than two posts in this forum will know that I am not and do not claim to be a mathematician. Most of the mathematicians in this forum have forgotten more maths since breakfast time this morning than I will ever know, so I will not try to cross swords with anyone on the maths. I come here to learn maths. But what I do know something about is the English language.
Quote:
Originally Posted by mfgoode
Assume a and b are positive such that a>b
Has he or has he not asked the reader to make an assumption that a > b?
He has specifically not said a is bigger than b. He said, let’s start by assuming it is.

Then I made my post in which I pointed out that his argument had not followed the normal course. In his reply to this he could have said what you did, that this is not a proof by contradiction it is a proof by implication (something I had not previously heard of). Instead of doing that he shouted (capital letters are normally inferred as shouting) that he absolutely had not asked me to make an assumption, he had stated a fact.
Quote:
Originally Posted by mfgoode
1) is NOT an assumption but a FACT
Well I’m afraid that this is so not true. He clearly and unambiguously asked the reader to make the assumption that a is bigger than b. How can he possibly then say that he did not do this, he stated a fact. Under the circumstances, I think that what I said was quite restrained and I am sorry you think my attitude is unwarranted but I do not at this time feel that I have anything to apologise for.
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Old 2005-09-12, 23:06   #10
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Quote:
Originally Posted by Numbers
Under the circumstances, I think that what I said was quite restrained and I am sorry you think my attitude is unwarranted but I do not at this time feel that I have anything to apologise for.
What is there to restrain? Is this really *that* important?

To be clear, I'm not asking you to apologize, I don't think anyone's feelings have been hurt, but I'm simply at a loss to see what's so important about the fact that he said "assume" instead of "choose". Pick your battles, man.

BTW, I think one of the most interesting facets about the English language is that roughly 93% of the population did NOT learn it as a first language. It's a good thing to keep in mind.
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Old 2005-09-13, 08:44   #11
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Quote:
Originally Posted by TravisT
Is this really *that* important?
Travist, You say that you are at a loss to see what is so important about choice of words. So let us put the whole thing into some kind of context.
Math (as I’m sure I do not need to remind you) is a science that depends upon logic for its foundations, for its axioms and for the conclusions we can draw and for the correct solution to its problems; both invented problems like Mally’s and real world problems.
Mally presented his logical paradox as a counter to the, to use his phrase, “cranks who have been appearing in this forum”. The “logic stream” (for want of a better expression) of his paradox is then found to be at fault because the very first word of his paradox is found to be incorrect.
I have in fact just finished an essay for my course in which I discuss this very issue, and present a number of examples where, for the answer to a question to have any meaning the question has to be framed so that it is two orders of magnitude more accurate than the required answer. The conclusion I draw at the end of my essay is that we as mathematicians have a duty to ensure that the questions we attempt to answer are framed in such a way that the answer has some meaning. If we fail to do this then we are giving others the opportunity to trivialise and demean our work. So, yes, I do happen to believe that in the context in which it was presented it is important that the question is framed correctly.
To those who would counter this point of view by saying that it was only a trivial quiz in a math forum I would say that if we allow ourselves to adopt sloppy logic, sloppy thinking and to skip gracefully over the odd wrong word or two in such “trivial” situations on a regular basis then we lose the habit – or maybe never acquire it in the first place – of thinking in the rigorously logical way required by the science we purport to practice.
That does not of course mean that we should pounce like avenging angels on every single typo, or wrong word slipping in from someone for whom English is not their first or even their second language. But I don’t think that I did “pounce like an avenging angel”. I merely made an innocuous, one-line comment, that would have drawn the attention of anyone who was sufficiently interested to think about it, to the fact that the “logic stream” of Mally’s paradox was at fault. I am sure that there are other points of view, other ways of looking at this, but since you asked this is, for the moment, mine.
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