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

[jean-paul allouche]

Does someone know whether the new editions of The World of Mathematics
(2000/2003) got some update, or whether they are identical to the 1956 one?

thank you
jp



Le 03/12/2022 à 02:10, Allan Wechsler a écrit :
> 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/
>>>>
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>> Seqfan Mailing list - http://list.seqfan.eu/
>>
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