[Xmca-l] Re: Objectivity of mathematics
mike cole
mcole@ucsd.edu
Mon Nov 3 17:49:50 PST 2014
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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