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[Xmca-l] Re: Objectivity of mathematics
The focus of this thread on "objectivity of mathematics" seems to be making a shift from what do we mean by "mathematics" to what do we mean by "objective."
But despite the prior focus, I'm not recalling anyone ever explicating what specific construction of mathematics is being discussed.
At the risk of stating what already is obvious to the discussants, here is a brief historical sketch of what "mathematics" has meant (according to my meager knowledge of it).
A) Prior to the Greeks, mathematics was an empirical domain; the circumference of a circle was about three times the diameter, because that's what measurement showed.
B) In the time of Thales, Pythagoras, and Euclid, the methodology changed. Starting from self-evident truths about the world (axioms and postulates), logic replaced empirical verification as the adjudicator of new mathematical knowledge. In this respect, mathematics is unlike natural sciences in which the empirical world adjudicates new knowledge through experimentation.
C) The methodology started to change again around 1830 in connection with Lobachevsky and Bolyai revisiting of Euclid's parallel postulate. This began the decoupling of axioms and postulates from the world, their primary virtue being self-consistency; all this leading to the formalist and logicist constructions of mathematics as determined solely by the internal relations of aspects of its discourse.
Of course, history does not erase the past. When a young child discovers empirically that starting with one quantity and adding another reliably produces the same result as starting with the second quantity and adding the first, she or he is still doing and learning mathematics. So, I wonder if the dispute in this thread simply reflects different participants focusing in on different aspects of this historically still evolving human practice of mathematics.
From: firstname.lastname@example.org [mailto:email@example.com] On Behalf Of anna sfard
Sent: Sunday, November 09, 2014 7:50 AM
To: firstname.lastname@example.org; 'eXtended Mind, Culture, Activity'
Subject: [Xmca-l] Re: Objectivity of mathematics
I'm happy to agree with you, Andy. As I do with Martin. I think the two of you are on the same page, more or less, but get distracted by subtleties of word use. For instance, none of you said explicitly what is the category of things to which the adjective "objective" can be applied. Words or worlds?
Being explicit on this point may help (as it helped me to understand that I have no use for this word, and this does not mean that I deny the existence of the/a "real world") anna
[mailto:email@example.com] On Behalf Of Andy Blunden
Sent: Sunday, November 09, 2014 3:27 PM
To: 'eXtended Mind, Culture, Activity'
Subject: [Xmca-l] Re: Objectivity of mathematics
A wonderful metaphor, Anna. But it isn't magic like it is in the fairy tale, is it? When the text gets out of control and unfolds according to "laws of its own," so to speak, that is *Nature*.
That is all I am saying when I say mathematics is not *just* a social convention. And the same applies to other natural sciences.
anna sfard wrote:
> There are two types of "validity" I think we can talk about here,
> Andy, the external - one that manifests itself in the fact that
> mathematics works for us in other things we do; and internal - the one
> that stems from strict adherence to the rules of the game (discourse).
> Re the latter, mathematical discourses are like the sorcerer's
> apprentice's broom: once put in motion, they get life of their own and
nothing can stop them.
> Can somebody stop ME please? :-)
> -----Original Message-----
> From: Andy Blunden [mailto:firstname.lastname@example.org]
> Sent: Sunday, November 09, 2014 3:02 PM
> To: anna sfard
> Cc: 'eXtended Mind, Culture, Activity'
> Subject: Re: [Xmca-l] Re: Objectivity of mathematics
> I think your quote expresses a truth, and an important truth.
> It is more precise, but it is what I meant when I said earlier that
> the unit of analysis "rotates.", with the mediator becoming the object.
> The statement is still kind of agnostic on the question, isn't it
> though, Anna? Mathematical relations are often only an approximation
> to things happening in the material world, and the validity of the
> mathematics is not thereby any the less for that.
> *Andy Blunden*
> anna sfard wrote:
>> No, Andy, I don’t think this was, or should be, said. I apologize in
>> advance for quoting myself, but it would be too much to try to say
>> things anew in the middle of work on an all different text:
>> "mathematical communication apparently reverses the developmental
>> order known from colloquial discourses: whereas these latter
>> discourses are created for the sake of communication about physical
>> reality, in mathematical discourse objects are created for the sake
>> True, also mathematical communication is supposed, eventually, to
>> mediate practical activities, and thus to pertain, in one way or
>> another to the world of primary objects that predate the discourse.
>> However, this fact may easily escape one’s attention. The
>> realization trees of mathematical signifiers [for the sake of the
>> present conversation, you may replace the "realization trees" with
>> "chains of signification"], although likely to have primary objects
>> or processes on such objects at their basis, may be too rich and
>> complex to be embraced at a glance. Leaving the concrete foundations
>> of such trees temporarily out of sight may thus be the condition for
>> the proficiency of
> mathematical communication."
>> Xmca-ing is addictive!
>> -----Original Message-----
>> From: email@example.com
>> [mailto:firstname.lastname@example.org] On Behalf Of Andy Blunden
>> Sent: Sunday, November 09, 2014 2:41 PM
>> To: eXtended Mind, Culture, Activity
>> Subject: [Xmca-l] Re: Objectivity of mathematics
>> "Nothing outside the text" is a way of saying that "the text alone
>> forms the object."
>> Would you agree, in the context of mathematics, that the text alone
>> forms the object?
>> *Andy Blunden*
>> Martin John Packer wrote:
>>> Who has said that there is nothing outside the text, Andy? Not
>> not Anna, not Huw, not me, not Ed, and not Luis so far as I can see.
>> If this is the question that is at issue for you here, I think you're
>> the only person for whom it is an issue.
>>> On Nov 9, 2014, at 3:52 AM, Andy Blunden <email@example.com> wrote:
>>>> Is there really *nothing* outside the text?