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

Tom Duff eigenvectors at gmail.com
Sat Dec 3 17:11:23 CET 2022 

I should point out that WoM is pretty good on the first 10,000 years of
Mathematics, and only misses the last 60 or so. Though it’s biographical
sections lean heavily on Eric Temple Bell’s Men of Mathematics which is now
widely regarded as anecdotal and unreliable (on Galois and Cantor
particularly.)

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