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very cool and concise proof. but minor soapbox rant: practically every other proof I've read while casually studying number theory and group theory (I didn't ma

very cool and concise proof. but minor soapbox rant: practically every other proof I've read while casually studying number theory and group theory (I didn't make it very far, especially for GT) seems to use the word "clear". While the proof is supposed to be the nuts and bolts holding together this important, indisputable logic, they gloss over how "clearly" some supporting element is (without proving it). It was irritating to me because it defies the spirit of proving something. Even in this proof, the author says "clearly" the right hand side is between 0-1. That's not automatically clear to me. It seems to be true, but at a glance, I can't automatically be sure that the statement is always true for all values. It got really bad in the group theory book, because the "clear" sub-proofs were very primitive concepts, important things (sorry I can't recall an example) but that we normally take for granted. So when I saw them gloss over it with "clearly" I always felt suspect, like they were hand-waving.

andars 10y ago
This proof's use of "clear" is sorta reasonable, since they are probably assuming the reader is familiar with alternating series with decreasing terms. I would agree that it is sometimes fairly annoying. "Proof by intimidation" A: "it is clear that X" B: "Sorry, why is it clear?" A: "No, it is CLEAR that X!"
zodiac 10y ago
It's a general fact that any series satisfying the list of conditions in http://calculus.subwiki.org/wiki/Alternating_series_theorem http://calculus.subwiki.org/wiki/Alternating_series_theorem is bounded by any two consecutive partial sums; the author probably expects everyone to have seen the proof of this before
sweetjesus 10y ago
when I read it it wasn't clear to me either, so I looked at what was written and saw that the fractions on the right side were clearly getting smaller and smaller fast because each successive denominator consisted of the multiplication of a clearly increasing number of clearly increasingly large integers so within a matter of seconds it was clear to me that within that small number of terms that each additional term would have a vanishingly small effect on the (alternating sign) summation so what they said was had to be true. So, I do sympathize with your overall point, but this isn't really a good example of it. It's less clear to me what they said about the left side of the (in)equality because that expression to me is much more eye-glazing...but it doesn't take long to convince yourself of that one too. I think it's fair for them to say it's "clear", but you have to not let yourself be put off by the large number of algebraic symbols strung together.
- deleted 10y ago
  [deleted]
sp332 10y ago
"According to the Nobel Prize-winning physicist Richard Feynman (Feynman 1997), mathematicians designate any theorem as "trivial" once a proof has been obtained--no matter how difficult the theorem was to prove in the first place. There are therefore exactly two types of true mathematical propositions: trivial ones, and those which have not yet been proven." http://mathworld.wolfram.com/Trivial.html http://mathworld.wolfram.com/Trivial.html
Certhas 10y ago
Anecdote has it that a professor at my alma mater was presenting a proof to students, stating that X was obvious, before halting and wondering aloud: "Is this really obvious?" He examined the blackboard and started pacing for a good 15 minutes mumbling and thinking, before exclaiming with a triumphant expression on his face: "Indeed it IS obvious!" That said, obviously no proof can function without assuming some mathematical background knowledge. This is a paper from a research journal, not a textbook and the readership of the journal where it was published would presumably find this clear.
- stablemap 10y ago
  The site doesn't seem to mention it but the article is from the Monthly [1], which has a focus on exposition. Still, assuming some analysis seems fair. [1] http://www.jstor.org/stable/2308411 http://www.jstor.org/stable/2308411
- chx 10y ago
  This professor is a widespread urban legend and there's no source to prove it actually ever happened.
  - wfo 10y ago
    It (or something like it) has happened to most working mathematicians.
ska 10y ago
The problem is, if you actually do ever single 'atomic' step even very simple proofs become huge (see formal methods). To avoid this, we use "clearly" to mean something like "obvious t anyone with sufficient training, ie the target audience. There are two problems with this - if you are trying to obtain that training it may no yet be obvious. Worse, sometimes subtle problems slip through while you are nodding your head!
taeric 10y ago
I take "clearly" to be a flag to come back to that point later. Especially if it is not clear to me. It is often a flag for points that the author feels contributes to the intuition of the proof. Which is not that it is easy. Just that it is an educational part of the process.
chx 10y ago
This is why the Fermat's Library website is awesome: the notes on the margin explain this to you.
- digler999 10y ago
  totally missed that, whoops. I'll review it. thanks
ecesena 10y ago
To be fair, the author says that the value is between 0 and 1 since _the alternating series clearly converges_ to a value between the 1st and 1st+2nd. So in reality it explains pretty well why it's between 0 and 1, and leaves the "clearly" to something which in his opinion is common knowledge: https://en.wikipedia.org/wiki/Alternating_series_test https://en.wikipedia.org/wiki/Alternating_series_test
pvg 10y ago
The proof is quite terse but did you find the 'clearly' bit harder to digest and verify than the series expansion itself? You know a is an integer >= 1 and every time you subtract a term bigger than the one your subsequently add so there's no way to get past the sum of the first two terms. The bit of algebra and the series expansion is shown in the margin note and not touched upon at all in the text - it seems not nearly as simple to check by simply staring at the thing.
GFK_of_xmaspast 10y ago
> the author says "clearly" the right hand side is between 0-1 No, the author says 'the alternating series clearly converges to a value between its first term and the sum of its first two terms', which is a late-first-year-calculus sort of thing.
blahblah3 10y ago
Really nice proof! The "clearly" thing isn't too hard to see even if you aren't aware of the "alternating series theorem": Consider the sequence a1, a2, a3... with the sum being a1 - a2 + a3..., note we can write the sum as: a1 + (-a2 + a3) + (-a4 + a5) hence the sum is less than a1. We can also write it as: (a1-a2) + (a3-a4) + (a5-a6) .... hence the sum is greater than (a1-a2).

andars 10y ago

This proof's use of "clear" is sorta reasonable, since they are probably assuming the reader is familiar with alternating series with decreasing terms. I would agree that it is sometimes fairly annoying. "Proof by intimidation" A: "it is clear that X" B: "Sorry, why is it clear?" A: "No, it is CLEAR that X!"

zodiac 10y ago

It's a general fact that any series satisfying the list of conditions in http://calculus.subwiki.org/wiki/Alternating_series_theorem http://calculus.subwiki.org/wiki/Alternating_series_theorem is bounded by any two consecutive partial sums; the author probably expects everyone to have seen the proof of this before

sweetjesus 10y ago

when I read it it wasn't clear to me either, so I looked at what was written and saw that the fractions on the right side were clearly getting smaller and smaller fast because each successive denominator consisted of the multiplication of a clearly increasing number of clearly increasingly large integers so within a matter of seconds it was clear to me that within that small number of terms that each additional term would have a vanishingly small effect on the (alternating sign) summation so what they said was had to be true. So, I do sympathize with your overall point, but this isn't really a good example of it. It's less clear to me what they said about the left side of the (in)equality because that expression to me is much more eye-glazing...but it doesn't take long to convince yourself of that one too. I think it's fair for them to say it's "clear", but you have to not let yourself be put off by the large number of algebraic symbols strung together.

deleted 10y ago
[deleted]

sp332 10y ago

"According to the Nobel Prize-winning physicist Richard Feynman (Feynman 1997), mathematicians designate any theorem as "trivial" once a proof has been obtained--no matter how difficult the theorem was to prove in the first place. There are therefore exactly two types of true mathematical propositions: trivial ones, and those which have not yet been proven." http://mathworld.wolfram.com/Trivial.html http://mathworld.wolfram.com/Trivial.html

Certhas 10y ago

Anecdote has it that a professor at my alma mater was presenting a proof to students, stating that X was obvious, before halting and wondering aloud: "Is this really obvious?" He examined the blackboard and started pacing for a good 15 minutes mumbling and thinking, before exclaiming with a triumphant expression on his face: "Indeed it IS obvious!" That said, obviously no proof can function without assuming some mathematical background knowledge. This is a paper from a research journal, not a textbook and the readership of the journal where it was published would presumably find this clear.

stablemap 10y ago
The site doesn't seem to mention it but the article is from the Monthly [1], which has a focus on exposition. Still, assuming some analysis seems fair. [1] http://www.jstor.org/stable/2308411 http://www.jstor.org/stable/2308411
chx 10y ago
This professor is a widespread urban legend and there's no source to prove it actually ever happened.
- wfo 10y ago
  It (or something like it) has happened to most working mathematicians.

ska 10y ago

The problem is, if you actually do ever single 'atomic' step even very simple proofs become huge (see formal methods). To avoid this, we use "clearly" to mean something like "obvious t anyone with sufficient training, ie the target audience. There are two problems with this - if you are trying to obtain that training it may no yet be obvious. Worse, sometimes subtle problems slip through while you are nodding your head!

taeric 10y ago

I take "clearly" to be a flag to come back to that point later. Especially if it is not clear to me. It is often a flag for points that the author feels contributes to the intuition of the proof. Which is not that it is easy. Just that it is an educational part of the process.

chx 10y ago

This is why the Fermat's Library website is awesome: the notes on the margin explain this to you.

digler999 10y ago
totally missed that, whoops. I'll review it. thanks

ecesena 10y ago

To be fair, the author says that the value is between 0 and 1 since _the alternating series clearly converges_ to a value between the 1st and 1st+2nd. So in reality it explains pretty well why it's between 0 and 1, and leaves the "clearly" to something which in his opinion is common knowledge: https://en.wikipedia.org/wiki/Alternating_series_test https://en.wikipedia.org/wiki/Alternating_series_test

pvg 10y ago

The proof is quite terse but did you find the 'clearly' bit harder to digest and verify than the series expansion itself? You know a is an integer >= 1 and every time you subtract a term bigger than the one your subsequently add so there's no way to get past the sum of the first two terms. The bit of algebra and the series expansion is shown in the margin note and not touched upon at all in the text - it seems not nearly as simple to check by simply staring at the thing.

GFK_of_xmaspast 10y ago

> the author says "clearly" the right hand side is between 0-1 No, the author says 'the alternating series clearly converges to a value between its first term and the sum of its first two terms', which is a late-first-year-calculus sort of thing.

blahblah3 10y ago

Really nice proof! The "clearly" thing isn't too hard to see even if you aren't aware of the "alternating series theorem": Consider the sequence a1, a2, a3... with the sum being a1 - a2 + a3..., note we can write the sum as: a1 + (-a2 + a3) + (-a4 + a5) hence the sum is less than a1. We can also write it as: (a1-a2) + (a3-a4) + (a5-a6) .... hence the sum is greater than (a1-a2).