thankyou Michael... brings me back to one of my fave Heideggerian texts,
his deconstruction of Kantian metaphysics in 'What is a Thing'; OK, so the
mathemata are those characteristics of things whereof we know in advance;
mathematical thinking is that thinking that brings with it what is always
already known and pro-ceeds from there onwards (one way might be more and
more about less and less -- specialisation and its fragmentation of Being
into this and that, into regions of particular whatnesses, but never into
whatness it self [this being mirrored in the always hoped-for never
obtainable dream of getting back to whatness through something like a
'Unified Field Theory'; but the whatness that can be humpty-dumptied is not
the whatness that came before the division into parts {of whatness} -- the
advent of something like science and its mathematical thinking .... sorry,
gotten lost in my parentheses...
Breaking a-part what holds together in order to bring the parts back to the
Whole that has parts (but is no longer the Being that was not part-icular)
is the busy-ness of mathematical thinking; take the aristotelian line that
is not merely a collection of happenstance points but a co-l-lect-ing of
contiguities, a holding together, a logos... if we look at the line as does
bitmap graphics applications (in their logos) then we see exactly the
accidental happenstance of dots that cuddle up to one another but only in
separations that border one another... if we look at the line as does
vector graphics applications (in their logos) then we see lines-as-lines as
not de-composable into co-cuddling points but only de-finable in terms of
the two end-points that de-scribe the limits of the line (from here to
there), but, these end-points do not collect the line, they merely mark its
extremities and make its description (read -- printing, to screen or paper)
easier and more economical of code (and more gratifying in an aesthetic
sort of way)
How does this relate to the rationals and the reals? The rational numbers
are apparently derivable from the integers, themselves an extension of the
natural counting numbers (so say the mathematiicians). The counting numbers
(1,2,3,4...but not including zero) account for themselves in relations of
succession and precession, nextness and previousness, stepping forward and
back... arithmatical operators like addition and subtraction can only be
thought strictly in terms of succession and precession, I mean: if we try
to perform 1 + 2 with the counting numbers, we have to be asking what is
the next to the next of one... we are certainly in the sphere the orbit of
the already knowable (mathemata) and its extensions (we know what next
means and thus what next to the next means too... we can count on these
numbers, we can ac-count for them, we can live with them and they can
in-habit our lives)... but even here with the oh so familiar
counting-beings we can go further with this simple familiarity and ask the
uncanny question concerning some 'limit' some 'measure' of these
counting-beings, their 'number' so to speak: that is, we come up against
the uncannyness of some kind of 'infinity'... from within the known-already
comes along its path and within its orbit a knowable unknown, the
accountable uncountable, the ?. It stems from being able to 'apply' the
next-one relation (add, move-on, one step) 'indefinitely' over and
over...again-ness. This ? is not properly one of the counting-beings but is
a limit (an other being the 0, the zero) of them seen as a whole with parts
(each number-thing is a part of the infinite-whole of numberdom); it is not
one of them because one can not per-form a 'next' on it, it remains the
same when succession is attempted, it stops, nextness ends, terminates; but
without the ? the finite, the not-in-definite counting-beings would lose
their integrity in the indefinitely extendable relation of next-one-ness or
succession; the infinite defines the finite as that finite that needs its
infinite. If the mathemata is that realm in which the already-knowable
moves in its own orbit, then we must already know that infinity that
derives from counting and nextness...necessary for a 'continuity', that the
continuing can be continued ad... [like these ellipses, the etcetera
clause, we know what we mean]...
{ this must needs be continued (sic) but later since I've got to attend
more quotidian matters }
Step Two next and shortly...
bye, thinkers of the mathemata
michael P
>Cologne, 14-Apr-2000
>
>michael of Kent schrieb:
>
>> Michael, { not drunk this time }
>>
>> I wonder if it might throw some light on this business of the
>> reals&rationals if we consider the proposition that the reals and the
>> rationals are 'simply' not at all the same kinds of beings; I mean that
>> from the 'perspective' of the real 'continuum', the rationals have these
>> "cracks", interstices, filled by infinite reals, but, could this be an
>> example of the 'wrong-thinking' exemplified precisely by Heidegger's notion
>> of mathematical/technological thinking: the reals have built an empire and
>> the rationals have been made over to be-come decent citizens of the new
>> real state.... as I said earlier, I think the invention of the reals
>> signals an incredible feat of modernity as it re-invents a notion of such
>> crowded continuity as to have the re-accomodation of the rational orbit as
>> having discontinuous gaps or cracks greedily filled by the monstrously
>> dense (beyond density itself) reals... interestingly, in all 'real'
>> measurement and certainly in the hollowed sphere of the digital, we only
>> have the real approximations (as if they were somewhat truncated reals....)
>> of rational numbers to play with... I mean, the counting number -- two [
>> the one after 'One'] -- is not the rational -- 2/1 or 4/2 or 2.000000 etc
>> -- and even less the real number --two -- [the limit of an infinite series,
>> it being not a memberof such a series!!!] ... these three (and we've left
>> out the integers too!!!) modes of numberdom are *radically* different
>> beings but the reals have 'conquered' the others as technological progress,
>> as updates of software, in the manner of the Gestell... whatcha think oh
>> Heideggerians
>>
>> think on
>>
>> michael
>
>Michael,
>
>In the first place I think mathematics has to be understood in the sense of a
>fundamental learning, _mathaesis_. And Heidegger is very _instructive_ here
>(Umlauts will probably be lost in the transmission):
>
>
>I translate the last part of this passage:
>
>"The mathaemata, these are things insofar as we take knowledge of them,
>take them
>into our knowledge as that which we, properly speaking, already know in
>advance:
>the body in its bodiliness, the vegetable in its vegetableness, the animal
>in its
>animality, the thing in its thingliness, etc. Proper learning is thus a highly
>remarkable kind of taking, a taking in which the taker only takes what he
>basically
>already has. To _this_ learning corresponds also teaching. Teaching is a
>giving,
>proffering; but in teaching it is not what is learnable that is proffered, but
>rather, what is given to the pupil is only the instruction to take for
>himself what
>he already has. If the pupil only takes over and adopts what is proffered,
>he will
>not learn. He will only come to learning if he experiences what he takes as
>something which he, properly speaking, already has. Only then does genuine
>learning
>take place, when the taking of what one already has is a
>_giving-to-oneself_ and is
>experienced as such. Teaching thus means nothing other than letting the others
>learn, i.e. bringing each other mutually to learning. Learning is more
>difficult
>than teaching because only someone who can genuinely learn and only as
>long as he
>can can genuinely teach. The genuine teacher is distinguished from the
>pupil only
>by the circumstance that he can learn better and wants to learn in the proper
>sense. In all teaching, the teacher learns most.
>
>This learning is the most difficult: to really and fundamentally take that
>into our
>knowledge which we always already know. Such learning which we are
>concerned with
>here demands that we stick continually to what is apparently most obvious,
>e.g. to
>the question, what is a thing. We ask persistently only _the same_, with
>respect to
>use, obvious _uselessness_: what is a thing, what is a tool, what is a
>human, what
>is an art work, what is a state, what is the world."
>
>Number is then mathematical not because it has to do with numbers, but
>because it
>is part of what we always already know fundamentally about beings, that
>fundamental
>knowledge without which we could not encounter any being as a being. There
>is no
>pro-gress, no step forward in this thinking. And H.'s thinking is even
>re-gressive,
>it ultimately takes the step back.
>
>Michael
>_-_-_-_-_-_-_-_-_-_- artefact text and translation _-_-_-_-_-_-_-_-_-_
>_-_-_-_-_-_-_-_-_-_-_-_-_-_-_- made by art _-_-_-_-_-_-_-_-_-_-_-_-_-_
>
http://www.webcom.com/artefact/ _-_-_-_-_-_- artefact@xxxxxxxxxx
>_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_ Dr Michael Eldred -_-_-
>_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_
>
>
>
>
>
>
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