On false assumptions and Popes

Mike posted a comment on a previous post, in which the topic of the irrationality of the statement “2 = 1” came up. (That is, the irrationality of 2 = 1 as a general principle, not as something that could sometimes be true with certain qualifiers, like 2 pints = 1 quart.)

This reminded me of one of my favorite math “proofs” demonstrating the crucial and fundamental importance of assumptions in understanding and comprehension (a theme of mine which regular readers of the blog may recognize). It is from a story familiar to many, though the “proof” has been attributed to various great mathematical thinkers, such as G. H. Hardy or Bertrand Russell — I’m not sure who really originated it, but my money’s on Hardy. I will abstractify (wow, that was a fun word to write!–much more fun than “generalize”…) the tale with generic names and simplify it a bit:

Mathematician: “If you begin with a false assumption, you can prove the ‘truth’ of just about anything you so desire.”

Challenger: “Oh, really? Well, assume that 2 = 1 and use it to prove that you’re the Pope.”

Mathematician: “Well, OK… I am one. The Pope is one. Therefore, the Pope and I are two. And, thus, the Pope and I are one.”


You really do have to watch out for those assumptions! Jeremiah 17:9 will always come back to bite you if you’re not careful. There are few things, if any, so false that any man would find it impossible to convince himself of their truth given the time and the motivation.

And, as prophecy indicates, there is a time coming when lies will be preferred over truth (2 Thess 2:8-12, cf. Rom. 1:24-25).

Frankly, I’d say that time has already gotten a foot in the door

11 thoughts on “On false assumptions and Popes

  1. The Sophists of ancient Greece loved this sort of argument. Consider this citation from EVERYDAY LIFE IN ANCIENT GREECE by Cyril Edward Robinson (reprinted by AMS Books, New York), p. 144:

    “Here is a sample of sophistic logic. ‘Your father is a dog,’ says one. ‘So is yours,’ says another. ‘Answer my questions and I’ll prove it,” says the first. ‘Come now, have you a dog?’ ‘Yes, and a sorry one too.’ ‘He has puppies, I take it?’ ‘He has.’ ‘The the dog is a father.’ ‘So it certainly seems.’ ‘Well, and he is yours, is he not?’ ‘Yes.’ So then he is a father and yours; so the dog is your father, and you are own brother to his puppies.'”

    Ah, the joys of equivocation…;)

    One of the pillars of evolutionary philosophy is meaningful equivocation of this sort. “Evolution” is attributed loosely to the coalescence of astral bodies and planets, their inevitable decay and change thereby, the abiotic formation of organic compounds, the (unproven and practically impossible) abiotic origin of life from said compounds, the adaptive radiation of life, and the (likewise unproven and practically impossible) naturalistic development of life into more and more complex forms. The equivocation lies in that none of these things are really the same as any of the other things, especially when one considers the last two (sometimes called – by the more honest – “microevolution” and “macroevolution”).

  2. Well, math and logic are two different genres; so I would be skeptical of a mathmetician using the language of logic.

    Nonetheless, I agree. People can “prove” anything they want to. In the ‘scientific method’ you’re suppose to develop a hypothesis. That often means picking the conclusion in advance. Then the researcher says, “Hey, I knew that I was right!” Big surprise.

    You’re quote of Jeremiah was a very good point. Very good. Yeah, no kidding.

    You reminded me of something Mr Armstrong once said. How mankind has the ability to acquire material knowledge, but how the Bible provides the spiritual knowledge otherwise unaccessible to mankind.

  3. Thomas

    As an extension the following statement is also true:
    If Bertrand Russell is the Pope then the Pope loves to go to the pub.

    This brings us to the following extremely important principle:
    Let there be a non-empty pub. Then there exists a person such that if that person is drinking then everyone is drinking.
    Case 1 If everyone is drinking then it is true that for each person if that person is drinking then everyone is drinking.
    Case 2 Let there be at least one person who is not drinking. For the sake of argument let that person be the Pope. Then it is also true that if the Pope is drinking then everyone is drinking.
    From this we can re-state the Drinking Principle as:
    If the Pope drinks then everyone drinks. Now, there’s power for you.

    Thanks to Raymond Smullyan for bringing the Drinking Principle to light in his classic exposition of logic “What is the Name of this Book?”.

  4. Steve: Thanks for your comment, but did you actually say, “I would be skeptical of a mathematician using the language of logic.” What?!? No, no, no, no… The mathematician is your friend! Frankly, I would hold quite the opposite view, and would be skeptical of anyone claiming to argue logically who could not argue mathematically — the two fields are so closely related as to be virtually two sides of the same coin! In fact, for my money the purest languages for logical discussion I have ever seen are always mathematical in flavor, if not in character. (And, if someone considers himself a good logician, I would argue that he probably would be good at most mathematics, too, if he were properly shown.)

    Check out “Mathematical Logic” on Wikipedia. Good stuff.

    Any further discussion would eventually lead us to Bertrand Russell, who was mentioned in the next comment by…

    Thomas: Thanks for the comment! In particular, I am pleased that you have given me a new sentence I plan on using as often as possible: “For the sake of argument, let that person be the Pope.” That’s hilarious!

    I can already see it now…

    Judge: Mr. Smith, now why exactly were you driving faster than the posted speed limit?

    Smith: Well, Your Honor, consider a case where a person is driving down the street and is too busy fiddling with his car radio. For the sake of argument, let that person be the Pope…

    Also, Thomas, I will have to look into that book. Believe it or not, I was just discussing with my children about a month ago the classic “Who shaves the barber?” and the “List of Lists That Do Not Include Themselves” paradoxes, and Amazon says that Smullyan was a leading expert on Gödel’s incompleteness theorem, which is one of my favorite topics.

    Thanks, again, both of you for your comments!

  5. Mr Smith! I failed to express myself as well as I should have. That’s my fault. At any rate, I wasn’t dismissing mathmatics, not in the least!

    I was taking college level math as a junior in highschool. Physics was the most fascinating class that I ever took. I have tremendous respect for math. In every way.

    Logic uses different script notations than math. And where they use the same script, the meaning of those symbols are usually different. That’s the thing I intended to express.

    They’re two sides of the same coin in much the same sense as chemistry and biology. Both heavily overlap. Yet the two are different. That’s all.

    I’ll check out the Wikipedia article you mentioned. I suspect that I’ll run into a Kantian approach. (He was a professor of mathmatics and logic at… Berilin University, if I remember right).

    Anyway, I can’t believe that I’m writing a comment this long. Feel free to delete it. The bottom line? I agree with everything you said.

  6. Howdy, again, Steve —

    I hope I haven’t stressed you out! I’ve considered the discussion a lot of fun, and I certainly haven’t meant to come across as combative — please forgive me if I’ve goofed.

    Thanks for your input!
    Wallace Smith

  7. Pingback: An actual request for hypocrisy? « Thoughts En Route

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