[Xmca-l] Re: Objectivity of mathematics
mike cole
mcole@ucsd.edu
Mon Nov 3 20:50:07 PST 2014
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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