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[Xmca-l] Re: Objectivity of mathematics



Yes, Ed, that is probably a better way to illustrate the objectivity of mathematics. We agree here,
Andy
------------------------------------------------------------------------
*Andy Blunden*
http://home.pacific.net.au/~andy/


Ed Wall wrote:
Mike

    Yes that is the meaning of invert v (or caret). Long ago, it also took me awhile to figure that out (smile).

You want to say, 'a necessity within the integers such that'? I mean it *must* be that 5 x -1 = -5 so as maintain consistency within the integers. The integers are what is termed an extension of the counting numbers (and zero). If 5 x -1 did not equal -5, there is a sense in which arithmetic would fail to operate as before with the counting numbers as now embedded in the integers. That is, the integers are an extension of the counting numbers that maintain for the counting numbers their usual properties. This forces certain mathematical behavior on the non-counting number integers. Foe instance, negative times positive must be negative and negative times negative must be positive. Although it is often the case that teachers teach these later operations as 'social conventions', they aren't, they are consequences of the mathematics.
Ed
On Nov 3, 2014, at  7:49 PM, mike cole wrote:

That is really a great addition to Andy's example, Ed. Being a total duffer
here i am assuming
that the invert v is a sign for "power of" ?

You, collectively, are making thinking about "simple" mathematical
questions unusually interesting.

The word problem problem is really interesting too.

mike

PS - I assume that when you type:  There is, one might say, a necessity
within the integers is that 5 x -1 = -5.   you mean a SUCH not is?
mike**2
:-)


On Mon, Nov 3, 2014 at 12:29 PM, Ed Wall <ewall@umich.edu> wrote:

Andy

      I have often used this for various reasons (smile).

       There are some problems with this example and the way, perhaps,
you are using it. The are certain conceptual tricks at play which cause, I
think, some of the problems. Let me illustrate.

1. Let a= b + k where k is zero.
2. a^2 = ab + ak
3. a^2 - b^2 = ab + ak - b^2
3. (a-b)(a+b) = (a-b)b + ak
4. (a+b) = b + ak/(a-b)
5. However, a-b = k, so a+b = b + a.

The is a rule in school, and it is not exactly a 'mathematical' rule, that
you can't divide by zero. However, is a convention, You can, indeed, divide
by zero, but you need to think about it a little. In algebra, one might
argue, the slope of a vertical line is roughly of this problematic form,
but that is a problem of representation. As plots 1/x near the origin, one
sees a discontinuity caused by this problematic, but that is expected. The
interesting cases occur, however, in calculus where one considers, in a
sense, 0/0. These can be somewhat undecidable without a little more
information (as case, perhaps, of we can't know). However, in a sense, x/x
at zero is just 1 and x^2/x at zero is just 0 (and, in fact, your example
is somewhat of this uncertain nature).
   So the rule isn't necessarily 'objectively' introduced if I understand
you correctly; nonetheless, one needs to be careful and preserve a sort of
thoughtful consistency in mathematics (by the way, a certain inconsistency
can often observed in the historical records in the development of
mathematical topics). Anyway, I think you may be saying is that
'objectivity' is something that needs to be introduced when consistency of
the discipline is threatened. Perhaps, in mathematics the difference is
that this is done internally rather than externally. If that is the case,
here may be an example:

5 x 5 = 25
5 x 4 = 20
5 x 3 = 15
5 x 2 = 10
5 x 1 = 5
5 x 0 = 0
5 x -1 = -5

There is, one might say, a necessity within the integers is that 5 x -1 =
-5.

Ed



On Nov 2, 2014, at  1:17 AM, Andy Blunden wrote:

Some people will be familiar with this:

1.  If  a = b                                    2. then a^2 = ab
3. then a^2 - b^2 = ab -b^2
4. then (a-b)(a+b)=(a-b)b
5. then a+b=b
6. then, because a=b, 2b=b
7. then 2=1

This proof appears to follow the "social conventions" that kids are
taught in algebra, but arrives at an absurdity.
The reason is that the deduction from 4 to 5 is an error. "Cancelling
out the (a-b)" doesn't work if (a-b)=0.
The limitation that the divisor be not zero, is obviously not there just
for a social convention. It *objectively,* has to be introduced because
otherwise, the common factor rule has the capacity to destroy the entire
system. The rules governing the above operations must conform to objective
constraints which belong entirely to the world of mathematics, and have
nothing to do with the world beyond the text. It doesn't matter if a and b
represent lengths or baseball scores.
Despite what the advocates of discourse theory believe mathematics is
subject to a whole range of constraints, and reveals a whole lot of
relationships and symmetries, which have nothing to do with the world
outside of mathematics, of "applications." Social conventions have to
adhere to these objective constraints to make good mathematics,
irrespective of whether they reflect material interactions.
Andy
(PS I don't know the above equations are going to look after going
through the mail server. Let's just hope for the best.)
------------------------------------------------------------------------
*Andy Blunden*
http://home.pacific.net.au/~andy/


Ed Wall wrote:
Andy

   Yes, I recall Piaget's claim (Piaget gathered a cadre of
mathematicians around him in his later years). Mathematicians tend to come
back and do 'foundations' after they have been playing around for a number
of years. Beginning at an end point (which was, in large measure, what the
curriculum of the 1980s did) may have been a large part of the problem.
     I've been thinking and realized that I may not be clear about
what you mean by 'objective relations.' What makes a relation objective
versus the opposite?
      The journey of discovery that mathematics offers is, I think,
reasonably exciting. However, so many children (and adults) find it dead
boring and intellectually repulsive.  Some of that may be social convention
(I remember a mother of one of my algebra students complaining because her
daughter was 'too' involved in doing mathematics; it wasn't 'girl like') ,
but some of it may be teaching/curriculum. There is an amusing article
"Lockhart 's Lament" which sort of touches on this (Devlin gives the link
in a brief intro):
    https://www.maa.org/external_archive/devlin/devlin_03_08.html

Ed

On Oct 30, 2014, at  6:28 PM, Andy Blunden wrote:


:) So many issues.

Ed, do you recall Piaget's claim of ontogeny repeating history in
mathematics. I read it in "Genetic Epistemology" but I am sure he would
have formulated the idea elsewhere. I found the formulation in that book
highly unconvincing at the time. Interesting in that respect is that the
logical sequence of relations within mathematics is opposite to the
historical sequence, and how (in my experience) the efforts in the 1970s to
make ontogeny follow logic proved so unsuccessful. But maybe this failure
was due to contingencies, I don't know.
And apart from Hilbert/Godel/Turing's demolition of Principia
Mathematics, Russell's demolition of Frege logicism was also very profound.
Both of course proved that mathematics is constrained by objective
relations and social conventions have to conform to that objectivity or
they fail.
The first thing, in my view, is to establish that, whatever this or
that group of people believe to be the case, mathematics is a science which
is constrained by objectively existing relations and reveals those
relations, which are nothing to do with the laws of physics, space-time,
and the infinite complexity of reality, etc. I think the journey of
discovery which is available to children has the potential to both teach
and endear mathematics for people. Learning social conventions is dead
boring and politically repulsive (if separated from natural necessity), to
my mind.
Andy

------------------------------------------------------------------------
*Andy Blunden*
http://home.pacific.net.au/~andy/


Ed Wall wrote:

Carol and Andy

      In the historical record, the first time I see the distinction
being clearly drawn between mathematics and the material world is with
Aristotle. However, he did not write in a vacuum and the paradoxes of Zeno
clearly indicated some problems (for instance, the story of Achilles and
the Hare). You can also see in the stories of Socrates that mathematics was
being done for the sake of the mathematics (for instance, the Meno). There
are some indications of the same fascination early on in India (the Chinese
tended to be fairly pragmatic).
     I was raised, in a sense, to observe the distinction Andy makes
below concerning 'objective.'. Godel muddied the water a bit with his first
and second incompleteness theorems, capsizing, so to speak, the monumental
work - Principia Mathematica - of Russell and Wittgenstein. Anyway, I
wonder (and I am searching for words here) if children 'naturally' realize
such objective validity, but are rather immersed in talk of such and come
out brain washed (a little of 'which comes first the chicken or the egg').
I find it hard, I guess, to draw a fine line between objective validity and
social convention (and, as to this latter, I have in mind the interactions
among 'mathematicians.') For example, when a mathematical paper is
referred, the logic of argument is not 'completely' analyzed by a reviewer
with standing in the mathematics community although it is presumed if
necessary it could be (I, unfortunately, know of cases where a paper has
been reviewed approvingly and then later found to be flawed and withdrawn).
Further,arguments of proof have themselves been debated as to their logical
standing over time (proof by contradiction is one such). Nonetheless,
mathematicians do see, as Andy noted, a world of mathematics with necessary
and sufficient relations. Children, in the K-12 mathematics curriculum
(immersed in text created by mathematicians and like-minded) may well be a
different matter.
     Carol, I am fine if you post this conversation. My historical
contributions were a little hurried and I glossed over exact dates and
probably too quickly interjected details. I am certain that those with an
eye for such things will notice mistakes and I apologize in advance. I have
also have noticed the K-8 curriculum seeming as if ontogeny is
recapitulating phylogeny. This raises large questions for me as it took
smart men and women a long time to make the leaps which are, in effect, now
tacitly presumed. One conclusion I have come to is that children are very,
very intelligent (smile). Another is that  the standard mathematics
curriculum may be a little outworn.
Ed

On Oct 30, 2014, at  4:34 AM, Andy Blunden wrote:


It's up to you, Ed.
I was fascinated with your maths history that Carol shared with me.
I learnt about the history of mathematics at some point in my undergraduate
life, but it is all a long time ago now, and I too found your observations
quite engaging.
I don't know at what historical point, mathematicians began to get
interested in mathematical relations without regard to any conclusions
being drawn from them about the material world itself. But I would be
interested to know. And I presume you, Ed, understood this distinction at
the time our xcma discussion broke off.
Andy

------------------------------------------------------------------------
*Andy Blunden*
http://home.pacific.net.au/~andy/


Carol Macdonald wrote:

Hi Andy  and Ed again

I really think this is a very important distinction.  Between you
and Ed, guys I urge you to copy these messages to XMCA.  They seem to be
significant contributions - principles - that we take us a good step
further in our discussion on the listserve.
If you want me to do this for you, please let me know.

Best
Carol

On 30 October 2014 08:32, Andy Blunden <ablunden@mira.net <mailto:
ablunden@mira.net>> wrote:
Just thinking ...
There is an important distinction between the objective validity
of any application of mathematics, and the objective validity of
mathematical processes themselves. Propositions about the real
world established by mathematics are always, at best, relative
truths. But the world of mathematics is no social convention in
itself, but necessary relations.

Andy

------------------------------------------------------------------------
*Andy Blunden*
http://home.pacific.net.au/~andy/
<http://home.pacific.net.au/%7Eandy/>


Carol Macdonald wrote:

    Read this Andy - it's totally intriguing.  I asked Ed to post
    it on the listserv. It seems it it ontogeny reciplating
phylogeny.
    Carol ---------- Forwarded message ----------
    From: *Ed Wall* <ewall@umich.edu <mailto:ewall@umich.edu>
    <mailto:ewall@umich.edu <mailto:ewall@umich.edu>>>
    Date: 29 October 2014 19:53
    Subject: Re: Apologies
    To: Carol Macdonald <carolmacdon@gmail.com
    <mailto:carolmacdon@gmail.com> <mailto:carolmacdon@gmail.com
    <mailto:carolmacdon@gmail.com>>>


    Carol

            As I started to answer, I realized that my reply would
    need to be a little more complicated than I had realized (so
    as I am writing I am thinking it through). Part of the problem
    is that I have been talking with Andy about converting
    'real-life' into something symbolic on, in a sense, an
    'advanced' level (for instance, physics or engineering).
    However, the very use of number is a converting of 'real-life'
    into symbols and that happened very early in a pictographic
    sense. This means, in a certain sense, the mathematics that
    happens now early on in schools and on the playground mirrors
    in an operational sense (as versus a social sense) what
    happened very early in history.
           Something happened around 400 - 300 BC (there are
    indications the it had been percolating for awhile) in how
    mathematics was viewed. Up until l that time mathematics -
    which was most usually  for commerce and calendars - was sort
    of done by recipes. People noted that if you did this and that
    you would get accepted answers, but arguments about 'validity'
    usually were of the form, "If you get some other answer, show
    me and I might believe you." Answers hinged on, on might say,
    the personal authority of the 'scribe' or 'teacher' (again
    reminiscent  of what goes on in schools today). Anyway, about
    300 BC Euclid published his Elements. This was, one might say,
    a geometric algebra, but more importantly, arguments within
    this work had a certain absolute nature; that is, if you do it
    this way, it is right no matter what anyone says otherwise
    (i.e. the best way I can say it is that 'within' mathematics
    the social convention became that social conventions had no
    force as regards the arguments). This was very, very different
    than what had gone before and Aristotle was moved to say that
    essentially there was no connection between mathematics and
    'real-life' and physicists who tried to make some connections
    were just wrong (there were also a series of paradoxes put
    forth by Zeno around 400 BC that indicated there were problems
    with making direct connections - they still have really never
    been resolved).
          Well, Greek thinking of mathematics (some of which has
    been called, n part, a rhetorical algebra) slowly faded from
    the scene, and people largely went back to 'experimental'
    arguments as regards things mathematics. However, in Arabia
    some of that thinking was preserved and the was a sort of
    rebirth. According to the historical records a group of people
    engaged in what was termed al' gebar became active. These were
    people who basically were generating and recording
    mathematical 'recipes' and who had developed ways of moving
    back and forth between recipes doing what is now called a
    syncopated algebra). In about 780 AD one such person wrote a
    book termed roughly Completion and Balancing. It is unclear,
    again what happened, and whatever it was it was different than
    the Greek geometric algebra, but again accepted arguments were
    socially assumed to not be vested in personal authority. Also
    there wasn't really a symbolic notation, but abbreviations
    were used.
           This seems never to have really caught on. In Europe,
    until around the 16th century mathematics had roughly the
    status of authoritative recipes although syncopated algebra
    was beginning to catch on (the Arabic influence). Again
    something happened and certain people began to symbolize
    'real-life' somewhat as it is done in modern times. One of the
    principal thinkers was a Vičte. Drawing on the Greek Pappus
    (290 AD), he distinguished three stages (1) Find a equation
    between the magnitude sought and those given; (2) Investigate
    as to whether the equation is plausible; and (3) produce the
    magnitude. This might look for a word problem as follows: (1')
    hypothesize a series of operations to generate the answer;
    (2') check to see if this is reasonable (students don't always
    do this, but physicists, engineers, and mathematics tend to do
    this); (3') calculate the answer. Vičte also created a
    symbolic notation.

         It is argued by some that there are some problems in all
    this. That is, it may be the case that the modern mathematics
    that underlies the sciences limits, in a sense, access to
    'real-life. I have been wondering - since the
    social/historical leaps taken although not obvious in the
    curriculum,are tacitly assumed in texts and by teachers -
    whether some of this (i.e. the leaps) may be limiting the
    access  to mathematics instruction. It is as if we are
    exposing children to a mathematics which operationally
    resembles that practiced long ago, but expecting them to
    'leap' to a view of mathematics that tacitly underlies the
    mathematics of today. I apologize for not being clearer, but I
    am yet working these ideas through.

    Ed

    On Oct 28, 2014, at  3:16 AM, Carol Macdonald wrote:

Ah Ed

Now you do need to explain that to me - that's no doubt the
    heart of the matter.
Carol

On 27 October 2014 20:26, Ed Wall <ewall@umich.edu
    <mailto:ewall@umich.edu> <mailto:ewall@umich.edu
    <mailto:ewall@umich.edu>>> wrote:
Carol

     If I understand what you mean by the conversion, it is
    quite interesting. There is a historical sense in which the
    possibility in the pre-modern times seems to have been
    realized twice. Now, it is, perhaps unfortunately, taken for
    granted.
Ed

On Oct 27, 2014, at  2:43 AM, Carol Macdonald wrote:

Ed, I see I repeated what you said - it was in the other
    conversation. The conversion is the heart of the matter.
Best
Carol

--
Carol A  Macdonald Ph D (Edin)
Developmental psycholinguist
Academic, Researcher,  and Editor
Honorary Research Fellow: Department of Linguistics, Unisa






--
Carol A  Macdonald Ph D (Edin)
Developmental psycholinguist
Academic, Researcher,  and Editor
Honorary Research Fellow: Department of Linguistics, Unisa






    --         Carol A  Macdonald Ph D (Edin)
    Developmental psycholinguist
    Academic, Researcher,  and Editor Honorary Research Fellow:
    Department of Linguistics, Unisa




--
Carol A  Macdonald Ph D (Edin)
Developmental psycholinguist
Academic, Researcher,  and Editor Honorary Research Fellow:
Department of Linguistics, Unisa





--
It is the dilemma of psychology to deal with a natural science with an
object that creates history. Ernst Boesch.