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#21 Flappie

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Posted 20 June 2005 - 10:59 AM

I often wonder how the Romans managed to multiply, but that's a bit offtopic.

Your mind could do with filling a bit more Flappie.

For some reason people here like to stick Roman numerals on their houses, it keeps reminding me. :shrug:
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#22 Adanac

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Posted 20 June 2005 - 11:06 AM

Does anybody know how the Russians multiply two numbers together? Put the numbers in two columns. With the first number half it, rounding down if necessary and write the number below it. Continue this until you get down to 1. With the second number double it and put the result below and continue this until you have the same number of rows. Then cross out any number in the second column for which there is an even number in the first column. E.g. – how do you multiply 69 by 92?

69 92
34 184
17 368
8 736
4 1472
2 2944
1 5888

Having crossed out 184, 736, 1472 and 2944 because they are opposite even values we add up the right-hand column to come to the answer:

92 + 368 + 5888 = 6348 which is 69 x 92.

Edited by Adanac, 20 June 2005 - 11:07 AM.


#23 mordecai_*

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Posted 20 June 2005 - 11:09 AM

I think skeptic is missing the point, it's how mathematics is used to describe relationships, and in what sense mathematics is "absolutely true". In other words when we use it to describe empirical relationships which can be tested, in that sense, that relationship is absolutely true in for the given context given our ability to measure things. Since measuring completely accurately you would end up with unwieldly infinite values to describe quantities or relationships that would take many human lifetimes to communicate, so humans deal in generalities in every day conversation.

Skeptic could argue that language is not absolutely true, that any word that has ever been defined is not objectively true simply on the basis of the fact that words and their definitions are made up and defined by people, they are themselves axiomatic i.e. the word "apple" when referring to an apple doesn't need any more explanation, when I say "the apple fell to the ground", no one doubts my definition of the word apple, they take it that it is absolutely true by definition.

The whole of science would go belly up if not for mathematics, this computer I am writing to you would have been impossible to design without mathematics, so saying something is not "objectively true" does not mean anything when it has empirical applications. The fact that math works is self-evident truth, we accept a degree of error in our meaurements because in principle 'perfect' measurements are either impossible or there is a point beyond which it becomes useless for us to talk about it in it's application to our lives.

For instance we don't talk about objects in absolute technical terms, i.e. there is an apple, made of substance x, which is made of atoms, which are in turn made of protons, neutrons and electrons, etc, etc. We speak in terms of generalities, contexts and relationships.

Edited by mordecai, 20 June 2005 - 11:21 AM.


#24 mordecai_*

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Posted 20 June 2005 - 11:24 AM

I have one for skeptic:

Are measurements objectively true? And if not, when I buy a bag of chips and it says 255grams and I weigh it and it turns out to be 255g. Then the measured amount of chips in the bag is not objectively true?

Edited by mordecai, 20 June 2005 - 11:25 AM.


#25 medazelim_*

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Posted 20 June 2005 - 05:29 PM

I dont know much about philosophy in fact I dont even think I know what the word means exactly. Having difficulty at first trying to figure out what on earth this thread is about it became much clearer when skeptic wrote this:

So, we have established that it is valid to say 1+1=2, right.

Now, what if I were to ask you to say whether this is valid: 1+1=10....?

Of course not! Not if I am using the decimal* system anyway!

But if I was using the binary system it would be completely valid to say 1+1=10...!

Enough said?


But two in binary is denoted by "10" and it means double of one unit not ten times. In binary instead of counting 9 digits and then starting another column one place to the left(as we do in the decimal system) we start a new column after we've counted two digits(or something like that).Its not a case of 1+1 =2 is not absolute truth it still remains true in binary the only difference is the language used. In the same way un +un = deux is the same as one + one = two.

Does anybody agree with that or is philosophy just way above my understanding?

Edited by medazelim, 20 June 2005 - 05:31 PM.


#26 DJP

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Posted 20 June 2005 - 05:59 PM

Skeptic's example is clearly an example of ambiguous notation. Mine comments about multiplication are possibly examples of ambiguous notation, I was sort of arguing in that direction.

I don't know to what extent notation should be considered mathematics.

DJP

#27 Fortigurn

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Posted 20 June 2005 - 07:10 PM

So, we have established that it is valid to say 1+1=2, right.

Now, what if I were to ask you to say whether this is valid: 1+1=10....?

Of course not! Not if I am using the decimal* system anyway!

But if I was using the binary system it would be completely valid to say 1+1=10...!

Enough said?

What you have done here is committed the fallacy of equivocation. All we have here is an argument from ambiguity, not anything actually factual.

This is actually semantics, not mathematics.

#28 DJP

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Posted 20 June 2005 - 07:29 PM

It was a good attempt at an understandable example. The problem is that to make it "understandable" he had to pick something that is hardly an axiom.

If you want real examples, I consider this easy to get into:
http://en.wikipedia....allel_postulate
(but to get the point of the discussion you have to get to the end and think about non Euclidean geometries which is pretty hard)

This is also not too hard:
http://en.wikipedia....d_(mathematics)
(it's got a steeper learning curve but is actually much easier than talking a whole lot about curved spaces)

Anybody who's done first year maths should be able to get into it.

DJP

#29 Fortigurn

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Posted 20 June 2005 - 07:35 PM

It still doesn't change the fact that the original argument was predicated on a logical fallacy.

#30 DJP

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Posted 20 June 2005 - 09:41 PM

It still doesn't change the fact that the original argument was predicated on a logical fallacy.

What are you talking about?

Skeptic's given an analogy by way of example, and being an analogy, it wasn't perfect. I imagine Skeptic would agree. You've objected to his analogy, on the same grounds that I did. It doesn't prove that the "original argument was predicated on a logical fallacy." I don't even think that Skeptic has really presented an argument, except perhaps that he doesn't like mathematical realism.

And anyway, who says that semantics isn't mathematics?

DJP

#31 Fortigurn

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Posted 20 June 2005 - 09:43 PM

It still doesn't change the fact that the original argument was predicated on a logical fallacy.

What are you talking about?

I'm talking about the original argument.

And anyway, who says that semantics isn't mathematics?


Oh of course, how could I have missed it?

#32 DJP

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Posted 21 June 2005 - 01:04 AM

I still don't know what the original argument is. I'm interested to find out.

DJP

#33 Skeptic

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Posted 21 June 2005 - 01:31 AM

Fortigurn

So, we have established that it is valid to say 1+1=2, right.

Now, what if I were to ask you to say whether this is valid: 1+1=10....?

Of course not! Not if I am using the decimal* system anyway!

But if I was using the binary system it would be completely valid to say 1+1=10...!

Enough said?

What you have done here is committed the fallacy of equivocation. All we have here is an argument from ambiguity, not anything actually factual.

This is actually semantics, not mathematics.


With hindsight, yes: in trying to provide an example that people, who had difficulty with the original argument, could follow, I fell into the trap of using a completely irrelevant example.

I retract that example and thank you, medazalim, for pointing out the problem with it. DJP, thanks for providing links to better examples.

However, I stand by my original argument, which is that at the level of axioms, mathematics cannot be said to be "true".

Fortigurn, do you also disagree with my original argument? It is clear that my original argument isn't based on equivocation or any other fallacy which I can identify, so if you disagree with that, you have to point out why. Remember, I said that at the level of axioms, mathematics cannot be said to be true. Dealing with mathematics at the level of axioms takes us to the Philosophy of Mathematics, which is a meta-perspective. Remember, we know that non-logical axioms refer to a priori propositions that are only valid for a given mathematical theory. They are theory specific. They do not "work" in other, equally valid theories - theories, which, taken togetherm kae up the body of knowledge of mathematics. Non-logical axioms do not satisfy all conditions in all possible universes. Non-logical axioms outnumber logical ones in modern mathematics.

To summarise: mathematics rely on logical and non-logical axioms. Non-logical axioms aren't universally true; they are only "true" within a given theory. Non-logical axioms constitute the majority of axioms in modern mathematical theory. What does that tell you?

Mordecai and others, who keep pointing out that mathematics work, are all correct. But the issue of whether mathematics works or not, was never in dispute. At the level of pragmatics it is abundantly clear that mathematical models that satisfy the requirements of mathematical validity (even if they contradict other equally valid mathematical models) are all "true" (although I prefer the word "valid").

At the level of axioms, mathematics cannot be said to be "true".

Edited by Skeptic, 21 June 2005 - 01:42 AM.


#34 Fortigurn

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Posted 21 June 2005 - 01:44 AM

However, I stand by my original argument, which is that at the level of axioms, mathematics cannot be said to be "true".

Fortigurn, do you also disagree with my original argument?

With that statement? It may be true, it may not be true. It may be true and not true simultaneously. I have no idea.

It is clear that my original argument isn't based on equivocation or any other fallacy which I can identify, so if you disagree with that, you have to point out why.


I was referring to the original hilarious (and incredibly relevant), argument that 1+1 /= 2.

#35 Fortigurn

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Posted 21 June 2005 - 01:46 AM

At the level of pragmatics it is abundantly clear that mathematical models that satisfy the requirements of mathematical validity (even if they contradict other equally valid mathematical models) are all "true" (although I prefer the word "valid").

At the level of axioms, mathematics cannot be said to be "true".

It was the deliberate confusion of these two which constituted the original fallacy of equivocation.

#36 Skeptic

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Posted 21 June 2005 - 02:00 AM

Fortigurn

However, I stand by my original argument, which is that at the level of axioms, mathematics cannot be said to be "true".

Fortigurn, do you also disagree with my original argument?

With that statement? It may be true, it may not be true. It may be true and not true simultaneously. I have no idea.


OK, that's an honest answer and completely acceptable. :clap2:

#37 Skeptic

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Posted 21 June 2005 - 02:12 AM

DJP

I still don't know what the original argument is.  I'm interested to find out.


Here it is:

Dealing with mathematics at the level of axioms takes us to the Philosophy of Mathematics, which is a meta-perspective.

First set of statements:
1) Modern mathematics consist of a collection of theories which together constitute the body of knowledge of mathematics.
2) These mathematical theories can be valid individually, even whilst contradicting other, equally valid theories in mathematics

Second set of statements:
3) Non-logical axioms refer to a priori propositions that are only valid for a given mathematical theory; they are theory specific; they do not "work" in other, equally valid theories
4) Non-logical axioms do not satisfy all conditions in all possible universes.
5) Non-logical axioms outnumber logical ones in modern mathematics.

Based on the the above, the conclusion is: at the level of axioms, mathematics cannot be said to be true.

Anyone who can show any of the statements (propositions) above to be false, or the conclusion to be false, can falsify the argument.

#38 Fortigurn

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Posted 21 June 2005 - 02:24 AM

Fortigurn

However, I stand by my original argument, which is that at the level of axioms, mathematics cannot be said to be "true".

Fortigurn, do you also disagree with my original argument?

With that statement? It may be true, it may not be true. It may be true and not true simultaneously. I have no idea.


OK, that's an honest answer and completely acceptable. :clap2:

:thank:

#39 DJP

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Posted 21 June 2005 - 02:52 AM

I was referring to the original hilarious (and incredibly relevant), argument that 1+1 /= 2.

Would you consider 2 = 0 to be a hilarious idea?

DJP

#40 DJP

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Posted 21 June 2005 - 03:34 AM

2) These mathematical theories can be valid individually, even whilst contradicting other, equally valid theories in mathematics

Disagree. The same could be said of physics, but what a physicist would [try to] do is construct a bigger theory that contains both. Why not regard maths the same way?

DJP




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