The Lumber Room

"Consign them to dust and damp by way of preserving them"

On “trivial” in mathematics

One aspect of mathematicians’ vocabulary that non-mathematicians often find non-intuitive is the word “trivial”. Mathematicians seem to call a great many things “trivial”, most of which are anything but. Here’s a joke, pointed out to me by Mohan:

Two mathematicians are discussing a theorem. The first mathematician says that the theorem is “trivial”. In response to the other’s request for an explanation, he then proceeds with two hours of exposition. At the end of the explanation, the second mathematician agrees that the theorem is trivial.

Like many jokes, this is not far from the truth. This tendency has led others to say, for example, that

In mathematics, there are only two kinds of proofs: Trivial ones, and undiscovered ones.

Or as Feynman liked to say, “mathematicians can prove only trivial theorems, because every theorem that’s proved is trivial”. (The mathematicians to whom Feynman said this do not seem to have liked the statement.)

A little examination, however, shows that all this is not far from reality, and indeed not far from the ideal of mathematics. Consider these excerpts from Gian-Carlo Rota‘s wonderful Indiscrete Thoughts:

“eventually every mathematical problem is proved trivial. The quest for ultimate triviality is characteristic of the mathematical enterprise.” (p.93)

“Every mathematical theorem is eventually proved trivial. The mathematician’s ideal of truth is triviality, and the community of mathematicians will not cease its beaver-like work on a newly discovered result until it has shown to everyone’s satisfaction that all difficulties in the early proofs were spurious, and only an analytic triviality is to be found at the end of the road.” (p. 118, in The Phenomenology of Mathematical Truth)

Are there definitive proofs?
It is an article of faith among mathematicians that after a new theorem is discovered, other simpler proofs of it will be given until a definitive one is found. A cursory inspection of the history of mathematics seems to confirm the mathematician’s faith. The first proof of a great many theorems is needlessly complicated. “Nobody blames a mathematician if the first proof of a new theorem is clumsy”, said Paul Erdős. It takes a long time, from a few decades to centuries, before the facts that are hidden in the first proof are understood, as mathematicians informally say. This gradual bringing out of the significance of a new discovery takes the appearance of a succession of proofs, each one simpler than the preceding. New and simpler versions of a theorem will stop appearing when the facts are finally understood. (p.146, in The Phenomenology of Mathematical Proof, here/here).

For more context, the section titled “Truth and Triviality” is especially worth reading, where he gives the example of proofs of the prime number theorem, starting with Hadamard and de la Vallée Poussin, through Wiener, Erdős and Selberg, and Levinson. Also p. 119, where he feels this is not unique to mathematics:

Any law of physics, when finally ensconced in a proper mathematical setting, turns into a mathematical triviality. The search for a universal law of matter, a search in which physics has been engaged throughout this century, is actually the search for a trivializing principle, for a universal “‘nothing but.” […] The ideal of all science, not only of mathematics, is to do away with any kind of synthetic a posteriori statement and to leave only analytic trivialities in its wake. Science may be defined as the transformation of synthetic facts of nature into analytic statements of reason.

So to correct the earlier quote, there are three kinds of proofs from the mathematician’s perspective: those that are trivial, those that have not been discovered, and those niggling ones that are not yet trivial.

Written by S

Sat, 2010-04-03 at 14:40:30 +05:30

Posted in mathematics

Tagged with ,

8 Responses

1. “analytic triviality”

Is that an unintended oxymoron?

Anirbit

Mon, 2010-04-05 at 11:10:37 +05:30

2. Totally.

But I’d argue (at the risk of making a trivial point) that it’s a case of structure and organisation as much as of triviality – after about twenty definitions and lemmas the original theorem is just one more step – or am I missing the joke? Or are they?

beli

Tue, 2010-04-06 at 06:28:38 +05:30

• Yeah, you’re right. The point is that the definitions and lemmas are thought of as part of the (ill-defined) “understanding”, and once the theorem is “understood”, it is trivial.

To take an example, the binomial theorem may look intimidating to a school student, but it is arguably trivial once you understand it. To take a better example (not my own), Spivak says in Calculus of Manifolds that [Full disclosure: example pointed out by another user at the reddit thread where I posted a comment that later became this post. :P]:

Stokes’ theorem shares three important attributes with many fully evolved major theorems:
1. It is trivial.
2. It is trivial because the terms appearing it have been properly defined.
3. It has significant consequences.

I think “fully evolved” is a good phrase here: when theorems have theories built about around them, they become very well understood, and trivial in hindsight. [Or at least, mathematicians would like all theorems to be trivial, but in reality, many aren’t.]

At least that’s the mathematician’s usage; physicists may still call the binomial theorem or Stokes’s theorem not trivial. :-)

S

Wed, 2010-04-07 at 02:01:19 +05:30

• The Binomial theorem is a trivial statement that people messed around with to prove another trivial statement, that heat flows from hot stuff into cold stuff, using a vague but trivial (atomistic) assumption along the way, thereby ending all debate of said assumption ;-) (warning – not entirely accurate)

It’s not totally contextual but on a side note etc i gotta mention the axiom of choice along with a certain theorem by Banach and some other guy

beli

Thu, 2010-04-08 at 04:17:43 +05:30

• @Beli: if you are referring to the Banach Tarski theorem as trivial, i would disagree… it took me ages to even *get* it :)

srivatsan

Fri, 2010-04-09 at 02:52:53 +05:30

• no no just an example of a totally intuitive statement leading to a totally non- intuitive result. i dunno the math so it just looks messed up to me :)
just a tongue-in-cheek comment to a tongue-in-cheek article. if i have to spell it out :)

beli

Sun, 2010-04-11 at 01:53:35 +05:30

3. Prof. Devlin,

I truly enjoyed your post with Michael Jordan. I totally failed to enjoy your post on trivialization. Smart people who are well-balanced suffer from a character flaw. I have studied two subjects in particular, law and most areas of history, that I find building on these areas very simplistic. Since I lack and egotistic personality, I find myself asking your trivialization question. If I can learn it, why cannot you. Various answers present itself depending on the situations. The most obvious being my 25 year old nephew nephew, born with down syndrome, and has a 50 word vocabulary. The more difficult cases deal with professionals who lack the years of study in my areas of strength.

I usually get a giggle out of them when I tell them that as a matter of education and not intelligence that allows me to view the law differently than they do. Most intelligent people understand the difference education makes. It moves you away from trivializing their education and background , and rightly places the emphasis on my education and training .

I would recommend substituting trivialization with the following concepts: learned intuition. At age 35. I played my first soccer game. And all the sports I played, football, baseball, and lacrosse. I always played a defensive position. No one wanted to play soccer goalie , so I volunteered. Between my football and lacrosse skills , I intuitively understood playing goalie. I found it easier to kick a soccer ball, then a football were catching and throwing a lacrosse ball.

Summer football practice had multiple purposes, one of which was to intelligently teach football reactions so that they became intuitive. After breaking the initial code , I understood that all became intuitive. Perhaps because I started from a different base point than most students. I look back and recognize my accomplishments. I had reading disability in early grade school. I had speech disability and early grade school. I overcame both , but I recognize the learning process involved to make the skills intuitive.

At age 65. I discover life is a learning experience. To steal imagery from Andrew Greeley, PhD, University of Chicago, sociology. The baby must die to become a toddler, the toddler must die to become a child, etc. I like the imagery and contextually Greeley used it to symbolize the dichotomy of the various ages of maturation. Again, people with normal self-confidence generally fail to recognize the significance of their achievement and trivialize them.
My terms:
learned skills.
Taught learned intuition-tlii
learned intuition

Thank you for your consideration and reading my thoughts. I accurately set forth my background on the future learning site.
Greg Keer

Greg keer

Tue, 2014-11-11 at 21:17:19 +05:30

• Just a note: I am not Prof. Devlin. You seem to have left the comment at the wrong webpage.
Regards,
Shreevatsa.

S

Tue, 2014-11-11 at 21:41:33 +05:30