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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 17:49:50 -0800*In-reply-to*: <8EB7C6A3-BA03-469A-A2A5-94274633BE10@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>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca-l@mailman.ucsd.edu>*Sender*: <xmca-l-bounces@mailman.ucsd.edu>

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>

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

**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>

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