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

Андрей Заболотский zabolotis at mail.ru
Sat Dec 3 16:56:21 CET 2022 

Ian Stewart’s «The Great Mathematical Problems» is good. Although it’s mostly text.
Etienne Ghys’s «A Singular Mathematical Promenade», available for free here  https://arxiv.org/abs/1612.06373  , is a more difficult reading but has great illustrations. It has a main topic, which is relatively narrow, but provides many excursions into various fields.
  
>Суббота, 3 декабря 2022, 15:50 UTC от Allan Wechsler <acwacw at gmail.com>:
> 
>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/ 
 
 
 
 

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