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[Xmca-l] Re: Objectivity of mathematics



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.