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

Fri Dec 2 20:33:23 CET 2022

(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/
>
> --
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>
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