[Xmca-l] Re: Objectivity of mathematics
Ed Wall
ewall@umich.edu
Mon Nov 3 21:06:13 PST 2014
Mike
I will take a look, thanks..
Ed
On Nov 3, 2014, at 10:50 PM, mike cole wrote:
> We had a long discussion of the "minus times a plus"
> issue a few years ago, Ed. Should make amusing reading for you! :-)
> mike
>
>
> On Monday, November 3, 2014, Ed Wall <ewall@umich.edu> 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.
>>
>>
>>
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