[seqfan] Re: Two "dumb" sequences and a question

Tom Duff eigenvectors at gmail.com
Sat Dec 3 17:03:56 CET 2022 

I believe the new edition is a Dover reprint of the original.

On Sat, Dec 3, 2022 at 7:57 AM jean-paul allouche <
jean-paul.allouche at imj-prg.fr> wrote:

> Does someone know whether the new editions of The World of Mathematics
> (2000/2003) got some update, or whether they are identical to the 1956 one?
>
> thank you
> jp
>
>
>
> Le 03/12/2022 à 02:10, Allan Wechsler a écrit :
> > Tom Duff mentions that *The World of Mathematics* is a bit dated.
> >
> > If you wanted to recommend a more modern book, that gives a good overview
> > of modern mathematics for laypeople, what would it be?
> >
> > If it doesn't exist, what should it look like? How should it be
> organized?
> > What topics should it cover? I have my own ideas but would like to hear
> > other opinions.
> >
> > On Fri, Dec 2, 2022 at 2:48 PM Tom Duff <eigenvectors at gmail.com> wrote:
> >
> >> It occurs to me (after spending a couple of hours on this long-winded
> >> message)
> >> that a favorite "non-mathematician" reference on a lot of mathematical
> >>   topics is
> >> "The World of Mathematics", edited by James R Newman. It's a four-volume
> >> set
> >> of essays about a broad variety of mathematical topics. My parents gave
> me
> >> a
> >> copy when I was in high school. I kept it by my bedside for years, and I
> >> still find it
> >> delightful and illuminating. Be aware that it's pretty dated. It was
> >> published in 1956
> >> and perforce misses a lot of important modern mathematics -- no
> coverage of
> >> the
> >> proofs of the 4-color theorem, Fermat's last theorem and the Poincare
> >> conjecture,
> >> or of computational complexity (e.g. the existence of NP-complete
> problems)
> >> and
> >> other computer science topics, etc.
> >>
> >> On Fri, Dec 2, 2022 at 11:33 AM Tom Duff <eigenvectors at gmail.com>
> wrote:
> >>
> >>> (Sorry if this is veering off-topic. I think Ali Sada's query contains
> an
> >>> important, implicit, question that is broadly relevant to the seqfan
> >> list,
> >>> which is home to a lot of people with mathematical interests who,
> >>> nevertheless, don't consider themselves mathematicians.)
> >>>
> >>> Ali Sada asked for "materials on set theory (for non-mathematicians)."
> >>> I think a pretty good place to start for the specific questions Ali is
> >>> asking is Wikipedia's Paradoxes of Set Theory page:
> >>> https://en.wikipedia.org/wiki/Paradoxes_of_set_theory
> >>> and looking at topics it refers to.
> >>>
> >>> As a rule of thumb, Wikipedia is mostly pretty good on mathematical
> >>> topics, but you have to be ready to skip a lot of rambling if you have
> >>> a specific need, and the level of sophistication required to understand
> >>> any topic varies greatly.
> >>>
> >>> While I was looking at that Wikipedia page, it got me wondering
> >>> what Ali means by "non-mathematician". Almost everyone has at least
> >>> a little mathematical knowledge. When I was six or seven years old,
> >>> there was a boy up the street who, when asked "What's 2+2?", would
> >>> cry and run away. (He was most often asked the question as a way
> >>> of taunting him for his general intellectual disability, as children
> >>> will, so he was most likely reacting to the insult rather than the
> >>> mathematical content of the question.) But other than that and similar
> >>> cases, most people have at least a little mathematical understanding.
> >>>
> >>> So the question is, what is the threshold? What bit of knowledge do
> >>> mathematicians cleave to that non-mathematicians don't? And I don't
> >>> think there is such a thing. Possible candidates include:
> >>>
> >>> long division
> >>> Euclidean geometry and the axiomatic method
> >>> mathematical induction and the axiom of infinity
> >>> the Chinese remainder theorem
> >>> Xeno's paradox and its resolution (limits, epsilon-delta arguments)
> >>> Russell's paradox and its resolution (axiomatic set theory)
> >>> Uncountability of real numbers and Cantor's theorem
> >>> Banach-Tarski paradox and the axiom of choice
> >>> Modularity theorem (Taniyama-Shimura conjecture)
> >>>
> >>> I think this list is roughly in order of mathematical sophistication.
> >>> You can make the list as long as you want, but I think it's foolish
> >>> to point at a spot in the extended list and say "you need to be this
> >>> sophisticated to be a mathematician". Indeed, there are great
> >>> mathematicians who don't believe (or profess not to believe) in
> >>> some of the topics on my little list.  Doron Zeilberger is an
> >> ultrafinitist
> >>> (https://en.wikipedia.org/wiki/Ultrafinitism) and (I think) denies
> >>> the validity of mathematical induction, though he certainly knows
> >>> how it works.  I have talked to professional mathematicians whose
> >>> position is that the Banach-Tarski paradox just shows that the axiom
> >>> of choice is bunk.
> >>>
> >>> I think, rather, that what makes you a mathematician is a willingness
> >>> to proceed into the mathematical unknown and find the beauty hiding
> >>> there. A lot of the world's greatest mathematicians have spent their
> >>> careers in latching on to a problem and seeing where it takes them.
> >>> Andrew Wiles has said that he was drawn to mathematics by Fermat's
> >>> Last Theorem -- and when he saw a glimmer of light (the
> Taniyama-Shimura
> >>> conjecture) he went after it, and after 7 or so years searching (plus a
> >>> long
> >>> career studying number theory and algebraic geometry in general) he
> >>> beat it! My dear friend Martin Davis latched on to a suggestion by
> >>> one of his professors (Emil Post) that Hilbert's tenth problem
> >>> begged for an unsolvability proof, and spent the next 25 or so years
> >>> working towards a solution -- and he and a few colleagues beat it.
> >>>
> >>> But being dragged around by a big problem is not the only way to
> >>> go. John Conway spent a lot of time on the characterization of
> >>> finite simple groups, but it wasn't the only focus of his career.
> >>> He was ready to look at just about any problem, large or small, and
> >>> push into the unknown in the direction it suggested. I like to think
> >>> that that's the path I've followed. Mostly I've tried to shine a
> >>> little light on different pretty objects that fell in my path.
> >>> Though, heaven knows, there are a few problems that I've spent years
> >>> on, but because I'm bull-headed and slow to understand, not because
> >>> they were big, important problems.
> >>>
> >>> Another thing to realize is that being "not good at math" is,
> >>> paradoxically, not a reason not to pursue math. Plenty of world-class
> >>> mathematicians will tell you that if their careers depended on being
> >>> able to get the right answer to simple arithmetic problems they'd
> >>> be in the streets, begging.  I like to joke that my life's goal is
> >>> to die having committed an even number of sign errors.  Faced with
> >>> not understanding something, a mathematician's attitude is not "I'm
> >>> not good at this", but rather "Can I figure this out?". Having
> >>> watched a good number of mathematicians at work, I can say that
> >>> many spend most of their time getting the wrong answers or heading
> >>> down dead ends. The thing that makes them mathematicians is not
> >>> being discouraged by that.
> >>>
> >>> Non-mathematicians think that math is too hard. Mathematicians think
> that
> >>> math is too hard, but they want to figure it out anyway. The challenge
> is
> >>> the attraction.
> >>>
> >>> On Fri, Dec 2, 2022 at 8:35 AM Ali Sada via SeqFan <
> >> seqfan at list.seqfan.eu>
> >>> wrote:
> >>>
> >>>>   Thank you all for the informative responses. And Tom is right. I
> didn't
> >>>> intend to submit these sequences. I am sorry for not making this
> clear.
> >>>> I just wanted to understand how logical structures are spontaneously
> >>>> generated from a few simple rules and how these structures normally
> >> prevent
> >>>> the creation of paradoxical sets. I would really appreciate it if you
> >> could
> >>>> share with me materials on set theory (for non-mathematicians).
> >>>>
> >>>> On another note, I respectfully disagree with Brendan. "Useless"
> >>>> sequences might be a burden on the OEIS editors, but I don't think
> they
> >>>> would harm the efficiency of the OEIS search function. How many
> >>>> milliseconds would a thousand of these sequences add to the search
> time?
> >>>> And I am a Hardy's fan. I don't think "useless" is a bad word at all
> >> when
> >>>> it comes to mathematics!
> >>>>
> >>>>
> >>>>
> >>>> Best,
> >>>> Ali
> >>>>
> >>>>
> >>>>      On Thursday, December 1, 2022 at 03:26:29 AM GMT+1, Joseph Myers
> <
> >>>> jsm at polyomino.org.uk> wrote:
> >>>>
> >>>>   And we do in fact already have A053873 and A053169.
> >>>>
> >>>> --
> >>>> Joseph S. Myers
> >>>> jsm at polyomino.org.uk
> >>>>
> >>>> On Wed, 30 Nov 2022, Tom Duff wrote:
> >>>>
> >>>>> I don't think Ali Sada seriously wants to add these sequences. He's
> >>>> trying
> >>>>> to understand an OEIS-driven version of Russell's paradox. The
> >>>> resolution
> >>>>> of the paradox is that not everything you claim is a sequence really
> >> is
> >>>> a
> >>>>> valid sequence as far as the OEIS is concerned, just as in ZF, the
> >>>> rules of
> >>>>> set construction preclude the Russell's paradox "set" from being
> >>>>> constructed. OEIS's rules aren't as rigorous as ZF's, because our
> idea
> >>>> of
> >>>>> what's a submittable sequence is an evolving thing.
> >>>>>
> >>>>> The point of Russell's paradox is that a wild-west attitude to set
> >>>> theory
> >>>>> (i.e. that the objects satisfying any predicate at all define a set)
> >> is
> >>>>> just asking for trouble.
> >>>>>
> >>>>> On Wed, Nov 30, 2022 at 5:55 PM Frank Adams-watters via SeqFan <
> >>>>> seqfan at list.seqfan.eu> wrote:
> >>>>>
> >>>>>> Another problem is that the content depends on the current state of
> >>>> our
> >>>>>> knowledge. This is unacceptable.
> >>>>>>
> >>>>>> Franklin T. Adams-Watters
> >>>>>>
> >>>>>>
> >>>>>> -----Original Message-----
> >>>>>> From: Brendan McKay via SeqFan <seqfan at list.seqfan.eu>
> >>>>>> To: seqfan at list.seqfan.eu
> >>>>>> Cc: Brendan McKay <Brendan.McKay at anu.edu.au>
> >>>>>> Sent: Wed, Nov 30, 2022 7:07 pm
> >>>>>> Subject: [seqfan] Re: Two "dumb" sequences and a question
> >>>>>>
> >>>>>> This is like the "all numbers are interesting" proof: If some
> >> numbers
> >>>>>> are not
> >>>>>> interesting, then there is a smallest non-interesting number, which
> >> is
> >>>>>> clearly
> >>>>>> an interesting property.
> >>>>>>
> >>>>>> Regardless, I hope that neither sequence is added to OEIS. The value
> >>>> of
> >>>>>> OEIS
> >>>>>> as a research tool is diluted every time useless made-up sequences
> >> are
> >>>>>> added.
> >>>>>>
> >>>>>> Brendan.
> >>>>>>
> >>>>>> On 30/11/2022 10:58 pm, Ali Sada via SeqFan wrote:
> >>>>>>> Hi everyone,
> >>>>>>>
> >>>>>>> Please consider the two sequences below:
> >>>>>>>
> >>>>>>> 1) Sequence AX contains all OEIS sequences where the A number is a
> >>>> term
> >>>>>> in the sequence itself. For example, A000027 since 27 is a positive
> >>>> integer.
> >>>>>>> 2) Sequence AY contains all OEIS sequences where either:
> >>>>>>> a) the A number is not a term in the sequence (e.g., A000040,
> >> since
> >>>> 40
> >>>>>> is not a prime number),
> >>>>>>> or
> >>>>>>> b) we don’t know if the A number is a term in the sequence or not
> >>>> (e.g.,
> >>>>>> A329697).
> >>>>>>> The question here is: Where should the number Y go? If we put it
> >> in
> >>>>>> sequence AY, then we know where it belongs and that contradicts the
> >>>>>> definition of AY.
> >>>>>>> Also, it couldn’t be part of AX because Y is not a term of AY.
> >>>>>>>
> >>>>>>> I’m trying to have some basic understanding of set theory and I
> >>>> would
> >>>>>> really appreciate your feedback.
> >>>>>>> Best,
> >>>>>>>
> >>>>>>> Ali
> >>>>>>>
> >>>>>>>
> >>>>>>> --
> >>>>>>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>>>>
> >>>>>> --
> >>>>>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>>>>
> >>>>>> --
> >>>>>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>>>>
> >>>>> --
> >>>>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>> --
> >>>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>>
> >>>>
> >>>> --
> >>>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>>
> >> --
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

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