Arnauld's Paradox, featuring gloriae, Skyman & kdb - now with maherarar too!

SirOracle is Zapata.
The previous discussion was about numbering systems and fractions - this wasn't raised "out of the blue"
zapata` Can you have a fraction where the top part is less than the bottom part, and have it equal to one where the top part is more than the bottom part?
Skyman not in the same notation, zapata`
zapata` Yes, you can, is the answer, Skyman
* Skyman waits for that fraction.
Skyman or those fractions.
zapata` What do you think about the fraction problem, Skyman? It's not that hard.
Skyman I gave up, zapata`
zapata` Skyman, check this out -5/6 = 5/-6
Skyman bah, zap, I'm unimpresed
zapata` Really, Skyman?
Skyman very.
zapata` What fails to impress you?
afekz more precisely, what should be impressive?
Skyman your "answer."
Skyman something really entertaining, that's what.
zapata` Well, it is a solution to the problem, isn't it?
gloriae I'm not impressed either, because you thought it was a legitimate problem as opposed to a tweaked riddle by fiddling with +/-
Skyman a trivial one zap.
zapata` It is a legitimate problem. The answer is a legitimate answer. This is mathematics.
zapata` Mathematics is mathematics. It works whether it impresses you or not.
Skyman Come up with something really clever, zapata`
gloriae we're never impressed by people who want to impress us
kdb well said gloriae
kdb dunno what the context is
gloriae zapata`s efforts
zapata` Sure, the truth often fails to impress.
* You were kicked from #mensa by gloriae (my turn to toss you, I guess)
Session Close: Sun Jun 22 00:18:26 2003

FootNote:
What I would have asked next, if they hadn't kicked me, is what "less than" and "more than" mean. If you ask them, I'm sure they can tell you.

This is called Arnauld's Paradox. Arnauld was a friend of Pascal's. Arnauld makes the point that the terms "greater than" and "less then" are meaningless when applied to negative numbers, and that proves that negative numbers can't exist. Arnauld is right.

Although non-existent, negative numbers can be useful in intermediate calculations as a means to an end.


SirOracle Are you a mathematician?

maherarar Yes.

SirOracle I wonder if you think this is funny (link to the above)

maherarar "Existence," sans qualifiers, is a ridiculous notion to apply to mathematics.

SirOracle Of course. You can have a positive number of apples

maherarar Can you have half a piece of chalk?

SirOracle yes, if you start with a definition of what a whole piece is. Like you can have a quarter of a cake

maherarar Ah, so you can define things so as to make what seems to be a nonexistent number exist. Now the whole gang rushes in the door. negative numbers, the square root of 2, transcendental numbers, the square root of negative 1, and last but not least all the quaternions and Z/pZ

SirOracle Only things which are tangible exist - let's say. It's obvious that zero things don't exist and a negative number of things can't exist. Pascal would have said that zero minus 5 is zero.

maherarar There are fewer than 10^80 particles in the universe. Does 10^81 exist?
Even if Pascal were that stupid, and I doubt he was, he would have been wrong.

SirOracle When we're talking about existence, we have to say what things we're talking about. 5 apples exist. 5 doesn't

maherarar Math got the retardation out of its system over a hundred years ago. So you're taking the position that numbers don't exist. That's at least consistent.

SirOracle No, Pascal wasn't stupid he was provocative, just as you and I are provocative.

maherarar By the way, do not attribute the thoughts of a friend of Pascal's to Pascal.

SirOracle I didn't.

maherarar Anyone who goes around declaring some mathematical constructs existent or nonexistent immediately reveals that he's ignorant of mathematics and is immediately ignored by all mathematicians.

SirOracle Do you have any problem with the argument as it was presented?

maherarar Which argument presented where?

maherarar also, you spelled "than" "then" on your crappy blog

SirOracle I often do misspell words. Why is it crappy?

maherarar Because it features a bogus and vague argument from a long-dead dude who lost said argument as well as a misspelling.

SirOracle Meaning, which statements are mathematically incorrect?

maherarar "Arnauld makes the point that the terms "greater than" and "less then" are meaningless when applied to negative numbers, and that proves that negative numbers can't exist."

SirOracle I asked you if you thought it was funny. My guess is that you don't. Or you might, but for different reasons that you think I do.

maherarar "Arnauld is right."

maherarar I don't. It's old and dull and wrong. It's like if I showed you a map of north america with a northwest passage. Would that really elicit a chuckle? Whatever.
People made mistakes in the past. They're making mistakes now.

SirOracle You still haven't said what's wrong with it.

maherarar R is an ordered ring in that it admits a relation <> 0.

SirOracle All numbers are abstract. No numbers exist. Only numbers of things can exist. Negative numbers of things cannot exist. A zero number of things doesn't exist either.

maherarar So it's a red herring to claim that negative numbers don't exist. Can half of a thing exist?

SirOracle Yes, if you have a prior definition of a whole thing. We covered that.

maherarar The problem is that we have prior definitions of a negative thing and the square root of a negative thing.

SirOracle Half a rock can't exist except in relation to a whole rock

maherarar You speak from ignorance and instead of attempting to rectify your ignorance, you attempt to justify the conclusions you arrived at via your uneducated though processes.

SirOracle Very good! What's your resolution of Arnauld's Paradox then?
Imagine you'd been in the chatroom. What would you have said?

maherarar Arnauld doesn't have a paradox. He doesn't have a clue. That's why he's been forgotten. I probably would have said "Morons." and chatbombed. It's not a discussion worth a serious response.

SirOracle I'll copy what I said into this window and you answer it.
Can you have a fraction where the top part is less than the bottom part, and have it equal to one where the top part is more than the bottom part?

maherarar Yes.

SirOracle What does "less than" and "greater than" mean?

maherarar less than means neither equal to nor greater than. a is greater than b if and only if there is a sum of squares that can be added to b to yield a. ooh. add to the last one that a cannot equal b.

SirOracle Then how can something that is lesser be equal to something that it greater?

maherarar It can't be.

SirOracle But you just said it was

maherarar Where?

SirOracle 8 statements back <maherarar> Yes.

maherarar That was a different question.

SirOracle Okay. you see no inconsistency in what you just said. We're not going to get any further with that then.

maherarar I suppose not, unless you want to actually learn math. I really see no need to click on a link of yours if you're unwilling to learn the mathematics behind arnauld's mistake.

SirOracle I'm quite willing to learn, I thought you weren't willing to explain

maherarar OK. a < 2 =" b.">

SirOracle Before you start, I'm going to tell you something you won't believe. I've discussed this problem with Bertrand Russell.

maherarar That's all you need. Fine, I don't believe you. If you can, using real numbers x_1 through x_n and your previous hypotheses, show that this definition is not antisymmetric, I'll give you ten billion dollars.

SirOracle How does it relate to the subject under discussion? I am willing to learn the mathematics behind Arnauld's mistake, which is what you offered to teach me.

maherarar It's a definition of "less than." If you can work out a contradiction, you win. If not, I win.

SirOracle No. Just explain Arnauld's mistake

maherarar There's no mathematics behind a mistake. He's just wrong. How can I justify wrongness?

SirOracle If you won't then I'll draw the conclusion that you can't

maherarar What could you possibly mean by that? It's like if I said this :
maherarar sfiweh
maherarar sdngfiowntoiwehnr
maherarar xighowehilwenroiw
maherarar s8y3948hy5983ih5
maherarar Therefore 1 = 2.
maherarar Prove me wrong.
maherarar His claim doesn't follow from his premises. He's wrong. Simple as that.

SirOracle I say that Arnauld was right, in that terms "less than" and "greater than" are meaningless when applied to negative numbers (of things if you like) and therefore
negative numbers of things don't exist

maherarar You would say that, SirOracle, because you are a stupid person unable to grasp the concept of mathematical proof. This is why I suggested earlier that you not attempt to ally yourself with me any further.

SirOracle Well, thank you for taking the trouble to try to enlighten me.

"Blaise Pascal was convinced that numbers "less than zero" couldn't exist. Gottfried Leibniz admitted that they could lead to some absurd conclusions, but defended them as useful aids in calculation."
The Encyclopedia of Astrobiology, Astronomy, and Spaceflight

Envoy Wed Oct 05 21:07:13 2005

maherarar I did not authorize publication of my remarks.

SirOracle Who said you did?

SirOracle if a is greater than b is a divided by b greater than unity?

maherarar Not necessarily.

SirOracle give me an instance when it is not

maherarar -2/-4 = 1/2 <>SirOracle So -2 is greater than -4?

maherarar Yes.

SirOracle There is an article on the web that you may find helpful. http://nti.educa.rcanaria.es/penelope/uk_confboye.htm

SirOracle It starts with this sentence. When you are a beginner in teaching mathematics, you will perhaps not classify the concept of negative numbers as one of the most difficult for your pupils to acquire.

SirOracle and ends with this one This history shows easily that it is possible to acquire a certain facility, even an operational virtuosity, formally, without having comprehension of what one handles; when the interrogations appear, the obstacle is then created; let us retain the reflections of Carnot, who posed fundamental problems: it is not possible that -1/1 = 1/-1 except if one gives up some laid down rules. The "negatives" are not "numbers" like the positives.It is perhaps necessary to get convinced that mathematics are useful to solve theoretical or abstract problems, and not concrete problems. The difficulty lies in the relations between physical reality and its mathematical modeling.

maherarar Do you think you've made some sort of point here?

SirOracle Is it possible that you can't see the point?

Not only is he unable to form a satisfactory mental model of the situation, he is unable to appreciate that there is a problem, even when it's spelled out to him.

He thinks that what chumps like Arnauld, Pascal, Leibnitz, Carnot and all mathematicians since have found to be a real and difficult issue is a fiction, the truth of which only a mathematician of his calibre can grasp.

Oh, well. ...

Session Close: Wed Oct 05 21:07:13 2005

A further and fuller explanation
Before mathematics became an Artform, mathematicians were primarily concerned with how well their models matched the real world. Maths is useless unless it is modelling reality. In pure mathematics you can postulate anything, no matter how preposterous: in applied mathematics you can't.

In the real world numbers mean quantities of real things. Richard Feynman was a great scientist fundamentally because his first concern was the real world. He was horrified by students who had been taught maths and physics but were unable to apply the disciplines to explain real phenomena. Albert Einstein was a poor mathematican, he said so himself and employed a mathematician to help him, but I have no doubt that he did the tensor calculus that resulted in the marvellously succinct e=mc^2. His concern was also to explain real-world phenomena, and his great triumph was predicting that, in the solar eclipse, stars close to the sun would apparently move because the sun's gravity would bend the light coming from them. They moved as he predicted. After that the sceptics sat up and took notice.

In the real world, negative numbers can't exist and zero doesn't exist. If your Mom bakes you cake and you eat half of it, what you have left is half a cake. If you then eat the other half then you have a zero quantity of cake. Zero doesn't exist: you've got no cake. You can't have your cake and eat it.

In the real world, existence matters a great deal. Negative numbers represent something notional, a debt, for instance. If you borrow $50 it's because you want $50 that really exists. In exchange you might be willing to undertake to repay that amount plus interest, say 10%, making $55. You will at some time need real money that exists to discharge the notional debt. Practical men are willing to accept that sometimes $50-$55 = $0, simply because $50 is real and present and -$55 is
neither real nor present.

Lending money at interest mattered so much in the real world that Christianity forbade it, calling it usury. That's why money-lenders were Jewish. This meaning of usury proved impractical so it was tacitly modified to mean "excessive interest". This was just one side-effect of the existence of positive numbers and the non-existence of negative numbers.

Shylock, from the Merchant of Venice, was a Jewish money lender. My nick, SirOracle, comes from the Merchant of Venice. Such links and motivation for conduct are a long way removed from mathematical modelling.

The Last Word.
I offered maherarar the last word on the matter, He says that he answered my last question with "no". He stands by what he said already. He says that I deleted the link to the website that expressed his views. Here it is:

maherarar Here's the mathematical "reality" you request :
http://planetmath.org/encyclopedia/OrderedRing.html

Nobody seems to understand the problem

Are there absurdities?

lets try again. What I believe is what I philosophers and mathematicians believe: negative numbers are not real, they are merely the combination of the negation sign with a number, because it's convenient in calculations. If mathematical equations throw up imaginary answers, then they are discarded.

In looking for another way to demonstrate that negative numbers creates absurdities, perhaps you might like to consider this one.

Take any positive real number a. divide it by successively smaller numbers and the result tends to infinity. It's easy to see that a/0 = positive infinity.

Now divide a by successively smaller negative numbers. The result tends to negative infinity. Or, to put it another way, a/-0 = negative infinity.

Are you really comfortable with that? Are you happy that -0 is different from +0? If not, what's wrong with the argument?