When is a proof? (A**-Clown, Double Barf Bag ALert, CODE RED)

Mathematics Association of America ^ | 1 June 2003 | Keith Devlin

[.cnI redruM]

Eviscerate at your leisure. There is plenty here worthy of fulsome contempt.

[Numbers Guy]

I saw Derbyshire's link to this piece a few minutes ago. Since the author is so in love with fuzzy left-wing proofs, that means he must doubt his own argument.

[Dr. Frank fan]

Honestly my only reaction to this is eye-rolling. The more I read the more I roll my eyes, to the point where I doubt I'll be able to finish reading this.

For starters, I have no idea what the hell the things he is talking about have to do with "left-wing" and "right-wing". There is a real issue to discuss here but the author just seems obsessed with labelling things he doesn't like "right-wing".

[Luke Skyfreeper]

Frankly, I don't see it.

The author's point -- that even mathematical proof, traditionally regarded as quite objective, ultimately contains a subjective element -- is valid, and he does a good job of explaining it.

[BlueNgold]

Typical of the left ... using a circular argument to prove a circular argument is a circular argument...

[Luke Skyfreeper]

For starters, I have no idea what the hell the things he is talking about have to do with "left-wing" and "right-wing".

Those are simply terms he produced, rather out of thin air, to describe the concepts he was aiming at describing. True, you may not like his choice of words. Personally, I'm not hyper-sensitive to the point it bothers me.

There is a real issue to discuss here but the author just seems obsessed with labelling things he doesn't like "right-wing".

On the contrary: it's the things that he LIKES, and that all mathematicians like (certainty) that he labels "right-wing."

He's just using words he chose because he couldn't think of any more descriptive words. Sheesh, don't take it so personally. There's no political agenda here. It's mathematics he's talking about, for God's sake.

[Luke Skyfreeper]

And YES, I know, just because you're paranoid doesn't mean they aren't out to get you...

[.cnI redruM]

I don't make the 1 to 1 relation between leftism and valid subjective thought. Nor do I quite get the 1 to 1 relation between objective and right wing thought.

[hellinahandcart]

If I'd stuck with math back in high school, would this article make sense to me now?

[Luke Skyfreeper]

I don't make the 1 to 1 relation between leftism and valid subjective thought. Nor do I quite get the 1 to 1 relation between objective and right wing thought.

Well, from my perspective, conservatism generally has to do with both clearly-thought through principles and facts. Liberalism has to do with how it feeeeeeeeeeels. So I "get" the connection, agree with it, and find the association between hard, verifiable logic and conservatism to be a compliment.

[sjmiller]

As a professional mathematician who knows all of the mathematicians mentioned but two (the author and one other), I feel somewhat suited to comment.

I'll leave the statement on "like right-wing policies" for others. For me, I consider the following "a proof". One has a set of definitions, axioms, and assumptions, and the rules of logical implication. For example, if I know that P implies Q (P --> Q) and I know that P is true, the Q must also be true.

I consider something to be proved if by applying the rules of logic it can be deduced from my axioms and assumptions. This leads us to the concept of relative proof, namely, we can show the result follows from our initial assumptions, but we don't know if our initial assumptions are valid.

I'll give two standard examples. Let one of my axioms be that A and not A are both true (A an arbitrary statement). Then it turns out that, using the laws of logic, you can prove ANY statement B is true. Why, it turns out that A and not A both being true is quite powerful. Thus, we have an inconsistent axiom structure, and it isn't surprising that we can deduce anything.

Another example is Russel's Paradox. Mathematicians used to think that given any property P, one can form the set of all x having property P. Russel considered the following: Let R be the collection of all objects x such that x is NOT an element of x. If R is a set, then R is an object having some elements in it. We can ask: is R an element of R? From the definition of R, an object x is in R if and only if x is NOT an element of X. So, if R is not an element of R, then R is in R; if R is an element of R, then R is not in R! Contradiction! Such a set cannot exist, we cannot just collect any group of objects to be a set.

The point to get from this is that there can be lots of surprising subtleties. Things that "look" or "feel" convincing are often false. I would say two major ways for something to be false is that (1) it just doesn't follow from the axioms / assumptions (and you made an argumentative mistake); (2) the original axioms / assumptions are bad / inconsistent, and what you are studying now highlights that.

For the proofs of Fermat and small prime gaps, we weren't finding mistakes with our axioms. We were finding deductive mistakes in very long, very technical, complicated arguments (at least for Fermat, I'm not too familiar with the small gaps paper).

There is a lot more to the subject. It turns out there are many results which can be proven to be undecideable from a set of axioms / assumptions.

But I hope this gives the general idea. He is right that there are different levels of proofs. We have much more "rigorous" proofs than a few centuries ago. We have lots of cases with strong heuristics indicating what "should" be true. The problem is the field is getting very technical. I know -- I write papers in the field, and referee. I've found mistakes by papers from big names where all the arguments are fine, but the lower boundary term of an integral was dropped, and that made all the difference. However, I don't DOUBT that papers like this COULD be successfully refereed, if people took the time. Hales' proof of the Kepler Conjecture (on sphere packing) is a lot harder, as the computer checked numerous cases. So, while some things can be done by hand, the universe may collapse before we finish.

I'm rambling, so I'll stop here. Basically, I look at a "proof" as something that has been shown (step by step by step) to follow from a given set of axioms and assumptions and the rules of logical inference. Just because someone checks every step does NOT mean they checked correctly. Many times we have results that we "accept" as true, without having really done all the details (doing enough so we see the general picture, it looks similar to things we've seen before, so ....). As time is finite, often one reads papers and says "that statement seems plausible", and one keeps reading. If anyone wants to chat more, email sjmiller@math.ohio-state.edu.

[dead]

I understand his left wing approach to the essentially right wing problem of subjective vs. objective truth, but Euclid is only tangential to this theorem and have no idea why he would even be dragged, kicking and screaming, into this discussion.

On a more didactic level, the conundrum can be better expressed as a fluid rather than a concrete problem, for all that’s worth. Of course, one can easily envision a world where all liberal outlooks are expressed in liquid form and all conservative outlooks are expressed as solids, but of course that would depend on the average temperature of such a world, obviously.

I think the Hungarian physicist, Gotig Boldevitch, expressed it best, when he said, in his 1837 Theorem on Negative Intuity, “Sometime in the future, somebody named dead is going to invent me and thrust my imaginary self into some rambling but important sounding incoherence he posts on the internet, whatever that turns out to be.”

Despite his relatively fictitious nature, Boldevitch was very insightful and would only laugh at the evanescent state of truth in a world where political machinations constantly shift the meanings of words rendering “proof” profoundly ethereal, as it were.

[Luke Skyfreeper]

It would be interesting for someone to do a survey of the political views of:

a) mathematicians, and

b) art professors.

Wonder what we would find?

[Cogadh na Sith]

Left wing proof:

Hypothesis "Y" is true because it makes me feel good to believe it is true.

Q.E.D

[Ernest_at_the_Beach]

Keywords seem about right!

[Luke Skyfreeper]

I understand his left wing approach to the essentially right wing problem of subjective vs. objective truth, but Euclid is only tangential to this theorem and have no idea why he would even be dragged, kicking and screaming, into this discussion.

On a more didactic level, the conundrum can be better expressed as a fluid rather than a concrete problem, for all that’s worth. Of course, one can easily envision a world where all liberal outlooks are expressed in liquid form and all conservative outlooks are expressed as solids, but of course that would depend on the average temperature of such a world, obviously.

I think the Hungarian physicist, Gotig Boldevitch, expressed it best, when he said, in his 1837 Theorem on Negative Intuity, “Sometime in the future, somebody named dead is going to invent me and thrust my imaginary self into some rambling but important sounding incoherence he posts on the internet, whatever that turns out to be.”

Despite his relatively fictitious nature, Boldevitch was very insightful and would only laugh at the evanescent state of truth in a world where political machinations constantly shift the meanings of words rendering “proof” profoundly ethereal, as it were.

Green, man, very green.

[Luke Skyfreeper]

Keywords seem about right!

On the contrary: mathematically speaking, he has something (very) meaningful to say; and he says it.

[.cnI redruM]

Not imho. That has something to do with my characterization of its author as an A**-Clown.

[.cnI redruM]

>>>>>>I'll leave the statement on "like right-wing policies" for others.


Wouldn't it be sweet if Dr. Devlin could show the same level of professional courtesy to those with whom he disagrees.

[Dr. Frank fan]

Those are simply terms he produced, rather out of thin air, to describe the concepts he was aiming at describing. True, you may not like his choice of words. Personally, I'm not hyper-sensitive to the point it bothers me.

Actually, he didn't "produce" those terms (let alone "describe" anything with them - at best, he used them as inept labels, not descriptions). They already exist and are in common usage in the area of politics. I'm surprised you didn't know that ;-)

Seriously, the point is that to borrow the terms "right-wing" and "left-wing" for the notions he is describing is misleading unless the analogy is clear (which it is not). Misleading readers is the opposite of good explanation.

On the contrary: it's the things that he LIKES, and that all mathematicians like (certainty) that he labels "right-wing."

Would that include the part where he says, "Like right wing policies, for all that it appeals to individuals who crave certitude in life, the right wing definition of mathematical proof is an unrealistic ideal that does not survive the first contact with the real world."?

I see your point about how (what he calls) "right-wing" is meant to stand for, The traditional way mathematics is done. However, I don't think it's a correct interpretation of this article to say that it's "right-wing" mathematics which he "likes". It seems like what he's really saying is, Things are changing (becoming "left-wing") in mathematics and you ought to get used to it, because I have.

He's just using words he chose because he couldn't think of any more descriptive words.

I'll have to take your word for it that you know what was in the author's mind when he wrote this, I guess. As far as the supposed inability to think of more descriptive words, he was doing fine with the "right-or-wrong" vs. "fuzzy" dichotomy. What exactly did the terms "right-wing"/"left-wing" add? To fail to understand that these terms are loaded is a failure of the author.

Sheesh, don't take it so personally.

Who's taking it "personally"? What I said was that it was an eye-roll-inducing and misleading use of terminology that distracted me from the (valid) point of the article and (almost) prevented me from reading it.

No skin off my nose. *His* loss, not mine, if his inept and gratuitous terminology reduces his readership.

There's no political agenda here.

I am not as convinced as you are of that.

It's mathematics he's talking about, for God's sake.

Well, except for the gratuitous and unsupported crack about "right wing policies", of course.

I'd have liked this essay much more if it had actually focused on the mathematics as much as you seem to think it did.