To say a proposition is false is to say its negation is true, so that another way to state this dilemma is that both sides believe they are arguing for the sake of truth.

One of two things can happen. The first, one side capitulates and comes to believe what he thought was true is indeed false, a happy ending. The second, and far more common situation, is that an impasse is reached. This is the truest test of personality.

I once had an argument with a full-grown, well educated, degree-holding woman about the numerical result of dividing any real number by one. I stated that the result is always the original number. She agreed, except in the case of 1 divided by 1, the result of which, she vehemently insisted, was 0. Why? Well, “You divide one into one and *you have nothing left*.” And nothing is zero.

No mathematical trick, example, demonstration, or appeal could shake this woman’s conviction. She patiently listened, at first anyway, to whatever I had to say, but always returned to the certain sure “fact” that if you divide one into one *you have nothing left*. She eventually gave up on me, dismissing my bizarre opinion as the result of an eccentric mind addled by overexposure to arcane books.

We parted on friendly terms. We had contact a few times afterwards where it became clear she was not going to write any op-eds about my recalcitrance, nor was she going to organize any protest, nor indeed was she going to plead for the government to restrain my speech so that I might not spread my error to the young. She decided to let me be, doubtless reasoning to herself that not everybody can know everything, that some inaccuracy is inevitable.

Some disagreements must necessarily lead to a parting of the ways. Murderer Kermit Gosnell and his counsel disagreed with the State of Pennsylvania over several propositions, and still disagree. Now the State will use force, not to impel Gosnell to believe what he does not, but to restrain and punish him for his actions. The distinction is important. Gosnell is not facing grief because of his belief, but his behavior.

Convincing adults that what they believe is false is always hard labor and often impossible. Many people, particularly those in positions of power, cannot abide dissension, so they use their power to squelch opposition. They create speech codes, restrict the press, implement “fairness” doctrines, or mute opponents physically. This happens on a smaller scale in homes or offices run by bullies.

It’s not the action of others which grates, but it’s that they won’t see reason. *Why can’t he just understand!* *What is wrong with him!* Some people are tenacious and will not let a point drop until his opponent lies about agreeing, or his opponent runs away to avoid harassment. Others are so passionate that they will not countenance the company of those with fail to march in step with them. Friendships are ruined over political disagreements neither party has much chance of influencing.

These days it seems the civilized standby “Let’s agree to disagree” is used less frequently, replaced by estrangements. The absence of the polite “out” is a predictor of tumultuous times. Camps are being drawn, sides taken. These things happen.

And now it strikes me that there is one more possibility than the two sides ultimately agreeing or disagreeing. Many years ago, I had to break up a bedtime fight between my two sons because one claimed, “Davy Crockett is too King of the wild frontier!” while the second took the opposite position. A détente was reached when I convinced them that if they didn’t shut up and go to sleep it wouldn’t matter what Davy Crockett was. Thus a proposition can be seen as uninteresting, too.

We are waiting for the other shoe to drop. Not in the picture but what was the degree major? A better argument for hours of fun is the answer to the square of i, which is the square root of minus one. Of course there is also that old stand by: what is zero divided by zero? Then there is Monty Hall and the goats. I see no end to possible disputes.

I’m not going to explain but your full-grown, well educated, degree-holding woman was right in a way. You were talking past each other. The key is in the different prepositions being used.

Couldn’t be truer! For instance, whenever I try to tell you that God does not exist and so on, even after saying so many reasonable things, you

insiston denying the obvious truth!Joking aside, I see your point all too well. It is very difficult to continue any discussion when you realise that other people will simply not “let it go”. In the internet it is even worse, with people trying to just shut down any dissenting words they see, shunning people who are saying distressing things to their tiny worldviews.

DAV,

Say, you’re not disagreeing with me, are you?

You might be right about lady, but don’t forget she agreed that 3/1 = 3, 2/1 = 2, etc.

Briggs,

Ok. Despite what I said (and I know you’re not going to believe me) I’ll try. What she said was outright hilarious. I almost spit out my coffee. I’ve done a lot of traveling and have come across people who talk and think like that — some even where I’m from. Since she’s educated, she was probably pulling our leg and left out the “Uh Yup!” just to watch you dance. Kinda like jumping into a hen house and yelling BOO! to watch the chickens run.

What happens when you take the only thing left and divide it into something. Why, pieces of course! and you no longer have that only thing left — you are left with nothing.

Uh, yup!

Bet she had a twinkle in her eye that was lost on you.

DAV,

It’s always possible that she was having me on, of course, but I wasn’t the only one in the audience and it seems that this lady really does believe what she professes.

The mathematically adept often cannot understand how people fail to see what is so clear to them.

Briggs,

Could be. We were pounded silly in school for saying “divide into” instead of “divide by”. Now I think I know why.

Wonder how you ended up in the argument in the first place. I’ve had many conversations in my lifetime but don’t recall ever having a discussion of division by one.

I have got to try this divide-by-one thing on my three year-old grand daughter. I use cookies when playing counting games with her, and it will be interesting to see what she says when we start to divide one cookie without breaking it, and she doesn’t get the cookie.

You cannot win an argument with this little woman.

The lady seems to have morphed her understanding of division into what actually is subtraction for the special case of 1/1. Quite understandable for people who are not math-intuitive since division is just repeated subtraction. What she forgot is that nothing disappears after the operation. Cutting an object in two leaves two halves still on the table; cutting it in “one” still leaves one lying there, although for her it is gone. Can you appreciate her anxiety trying to resolve this Schrodinger anomaly?

The absolute best along these lines is the heavy boots story.

http://www.phys.ufl.edu/~det/phy2060/heavyboots.html

I doubt if the woman in Briggs story was joking but DAV may be on to something as to the source of the confusion. Words just do not mean the same thing to everyone, as seen in my link. There is also the story, which I cannot find a link to, about members in the audience of a scientific talk for the general public hearing the term “vacuum of space” but thinking household appliance.

Briggs, surely the best way of explaining this is to ask “do you realise that 4/1 means how many times does 1fit into 4, answer, 4……….ergo, 1/1 means how many times does 1 fit into 1, answer,1?

I feel for you Dr. Briggs. Computers do arithmetic with an adder (not the snake). They add by adding, subtract by adding, multiply by adding and divide by adding. Try explaining computer arithmetic to some one who insists you can’t subtract by adding.

Since division is just repeated subtraction it is perfectly obvious that if you divide 1 by 1, then you subtract the 1 from 1 and you are left with 0. That ought to be obvious to the meanest intelligence. However, she forgot the next step. How many times did she subtract the 1 from 1 before reaching 0? Exactly 1 time and that is the answer.

All of these explanations for 1/1=1 are ignoring one simple, inescapable fact: if you divide one into one YOU HAVE NOTHING LEFT.

Cheers. 🙂

If it is true that when you divide one into one, you have nothing left, what do you get when you subtract one from one?

“Why canâ€™t he just understand! What is wrong with him!”

I ask myself that question about many people here and vice versa.

Simple. Divide 3 into 1 (or 1’s) you have 3 1’s, divide 2 into 1’s you have 2 1’s, but divide 1 into 1’s, well you can’t split 1 and still have a 1, so since we are dealing with integers, you have to round down to zero. So 1 / 1 = 0. Just sayin’.

No need to argue, just give her an abacus.

As the consequences from their irrationality are negative, we should strive to make our fellow citizens more rational. An approach that should be tried, in my opinion, is to teach logic to people during their years as students. By about the 7th grade, I suspect, students can be grounded in the classical logic. In subsequent years, they can be taught the more mathematically challenging inductive logic.

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There’s always the money argument. If her salary, which is one salary, is divided only by herself, then she should receive zero salary.

A wise person once told me that people are more inclined to accept something if they think they thunk it up themselves. As soon as you detect intransigence of thought, try switching from advocacy to seed-planting. On occasion, it’s amazing what can sprout up from even the most seemingly infertile mental soil, given time and proper fertilizing.

Conrad,

I have an abacus and it uses biquinary arithmetic. Try explaining that to the lady.

Computers also used different kinds of arithmetic. The CDC machines used binary, the Burrows machines used octal and IBM used hexadecimal. I used to have a calculator that could convert numbers from radix to another.

“Gosnell and his council”

Surely not.

” You are right ” is music to my ears!

^_^