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*To*: "eXtended Mind, Culture, Activity" <xmca-l@mailman.ucsd.edu>*Subject*: [Xmca-l] Re: Objectivity of mathematics*From*: mike cole <mcole@ucsd.edu>*Date*: Mon, 3 Nov 2014 20:50:07 -0800*In-reply-to*: <89D3E7A7-8790-4A13-BA94-7EFDC45B3225@umich.edu>*List-archive*: <https://mailman.ucsd.edu/mailman/private/xmca-l>*List-help*: <mailto:xmca-l-request@mailman.ucsd.edu?subject=help>*List-id*: "eXtended Mind, Culture, Activity" <xmca-l.mailman.ucsd.edu>*List-post*: <mailto:xmca-l@mailman.ucsd.edu>*List-subscribe*: <https://mailman.ucsd.edu/mailman/listinfo/xmca-l>, <mailto:xmca-l-request@mailman.ucsd.edu?subject=subscribe>*List-unsubscribe*: <https://mailman.ucsd.edu/mailman/listinfo/xmca-l>, <mailto:xmca-l-request@mailman.ucsd.edu?subject=unsubscribe>*References*: <CAGVMwbUV9Jdf2XpAshWE0dCNcT1kHRTrpxGy=Ci2M3xR9N3rhQ@mail.gmail.com> <9A0CD8DC-7A92-4561-A550-EF5900B9B1CE@umich.edu> <CAGVMwbUcnDxvMbpfha4QVHKtU6mi+igRvxZhzFFj8hyynkDiwg@mail.gmail.com> <5883D78B-B4BC-4EB7-987E-06C3C2D22EB1@umich.edu> <CAGVMwbVugt4Mqh_jLosUdHNyRmg19dEaULa7weLj8y7jHSXn6g@mail.gmail.com> <5451DB76.7010007@mira.net> <CAGVMwbWSmg+0X=VvUQMojAtX911Hg6jmfYqkx3a2EoTjCq2OxA@mail.gmail.com> <54520634.9080302@mira.net> <CD06F23C-D0F7-4169-AB99-E025679FDA4A@umich.edu> <5452C9B0.70505@mira.net> <32EF9A39-CDDA-4258-BBDE-7D7658C85A03@umich.edu> <5455DA75.9040906@mira.net> <8EB7C6A3-BA03-469A-A2A5-94274633BE10@umich.edu> <CAHCnM0CGp_qJU5mVEt0ieiK8r-A7qvg0N=5-Fd5eqjOkaTqWqA@mail.gmail.com> <89D3E7A7-8790-4A13-BA94-7EFDC45B3225@umich.edu>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca-l@mailman.ucsd.edu>*Sender*: <xmca-l-bounces@mailman.ucsd.edu>

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. > > >

**Follow-Ups**:**[Xmca-l] Re: Objectivity of mathematics***From:*Ed Wall <ewall@umich.edu>

**References**:**[Xmca-l] Re: Apologies***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Objectivity of mathematics***From:*Andy Blunden <ablunden@mira.net>

**[Xmca-l] Re: Objectivity of mathematics***From:*Ed Wall <ewall@umich.edu>

**[Xmca-l] Re: Objectivity of mathematics***From:*mike cole <mcole@ucsd.edu>

**[Xmca-l] Re: Objectivity of mathematics***From:*Ed Wall <ewall@umich.edu>

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