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

Fri Dec 2 20:47:53 CET 2022

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/
>>
>>
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>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>