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

[Tom Duff]

The Princeton Companion to Mathematics (2009, ed. Tim Gowers) is a possible
candidate. Not as “general interest” as World of Mathematics, but a
beautiful book with broad coverage. Moderately pricy ($99?), but so is a
Dover reprint copy of World of Mathematics. I’ve had a copy for a few
years, but, no longer being possessed with the energy and focus of youth,
have not read it thoroughly yet.

On Sat, Dec 3, 2022 at 7:49 AM Allan Wechsler <acwacw at gmail.com> wrote:

> 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/
> > >> > >
> > >> > >
> > >> > > --
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> > >> > >
> > >> > > --
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> > >> > >
> > >> >
> > >> > --
> > >> > 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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