On 10/6/2014 9:11 PM, 'Chris de Morsella' via Everything List wrote:
*From:*[email protected] [mailto:[email protected]] *On
Behalf Of *meekerdb
*Sent:* Sunday, October 05, 2014 5:57 PM
*To:* [email protected]
*Subject:* Re: Is mathematics human thinking? WAS [generalizations_of_islam -
God Matter]
On 10/5/2014 4:34 PM, 'Chris de Morsella' via Everything List wrote:
Mathematics is human thinking, we are smart to have mastered SOME of it
(not all, as
the progression of math shows).
John M
John one question that comes to mind then is: if math is the cultural
accumulated
product of human thought over the arc of the history of recorded culture,
then what
about all the mathematical and geometric patterns that appear and reappear
in nature
quite apart from any human cultural input. For example how ratios such as
the golden
ratio (e.g. 1·618034 approximately), or the Fibonacci series manifest in
things as
diverse as conch shells to the spiral arms of [spiral] galaxies.
>>But notice that they appear approximately and finitely - quite different than the
mathematical abstraction a idealization.
Isn’t the approximation though itself, an artifact of the impossibility of expressing
certain ratios in our number system. Pi for example is NOT 3.14159, but that is an
accurate approximation of it to some pretty high degree.
Or are you instead stating that natural examples of such patterns and ratios -- as the
Fibonacci Series and the Golden Ratio -- are themselves approximate (of course I agree
with that)… nor would I expect anything other than that for emergent natural systems,
such as a spiral galaxy, or living organisms for example.
And in geometry the ratio of a radius to a circumference has been very closely
approximated by human cultural achievement, but this ratio certainly is not a human
cultural invention… is it?
Again, "approximately" and by our best current theories space is not Euclidean
Sure, agreed. [not making the case for the eternal supremacy of Euclidean spacetime
<grin>… though it remains a useful simplification of the four dimensional manifold of
spacetime down into just three idealized dimensions of space], but then other geometries
of spacetime… like Minkowski (or the exotic tightly curled dimensions of String Theory)
don’t they also have their own maths as well?
>>and maybe not even a continuum.
If spacetime is pixelated those recent experimental results from ESA seem to rule out
any graininess in spacetime down to a scale trillions of times smaller than the Planck
scale.
There exists a large number of such ratios in geometry, math and in nature itself.
Certainly these precisely defined relationships existed before there were hominids on
this planet…
>>What exists is theory dependent.
Not sure I understand how the circumference of a circle being somewhat more than six
times the radius is theory dependent? Wouldn’t it not remain unchanged in the universe
we inhabit absent any theory at all?
Within Euclidean geometry there is a line passing through any two points - and always
has been. What the mean about "nature itself" is a different question; one that depends
on operational definitions that interpret the relationships. What is a "point"? a "line"?
Each of those questions really opens things up <grin> Euclidean geometry is an idealized
mathematical construct for space. I would agree that a lot of mathematical systems
develop impressive internally consistency, for this they are useful tools… and so even
if they are approximate simplified representations – ex. : Euclidean space is NOT the
spacetime we very much seem to inhabit.
>>How does one determine whether a "line passes through a point"?
I agree (we can’t, precisely)… without being able to define what the meaning of a point
is – and just saying it is some infinitesimal place in space… IMO, rather misses the
point of “the point”!
I think you are making the error of taking our theories to be facts and then expressing
amazement at how good our theories are at describing the facts.
I think our theories are our cultural products, they are the accumulated result of the
evolution of human thinking brought up to the present day. On one hand this is so, and I
agree with you. Mathematical, geometrical systems, logic, and automata and the
theoretical underpinnings around them are synthetic systems we have evolved – over
history. But to a degree these systems of thought, these theories, provide us with what
(to us at least <grin>) seems like a pretty good model of our universe.
That we have come upon a lot of our theories and math through cultural means cannot not
however be used to therefore assert that the relationships, ratios, behaviors etc.
expressed by those theories only exist within them and do not have any existence outside
of them.
Sure, we got to them via cultural mechanisms, and the model **is not** the thing it
models, but can’t one argue that this is a process really in which – via cultural
Darwinian means -- we are discovering a kind of meta country… that our theories, models
etc. are more maps we have made of a real “country”.
in fact can you even conceive of a time or universe where these basic mathematical
ratios do not hold true? Perhaps you can, but it would be a bizarre universe utterly
unlike the one in which we inhabit.
In the sense you mean such a universe cannot exist. In the sense I mean, it doesn't
exist. The "basic mathematical ratios" are true the way "red is a color" is true. Their
"truth" is a tautology derived from axioms and rules of inference we've adopted to keep
our language from confusing us. They are facts that "exist" in our universe, they are
true relations among concepts we've invented as part of our theories.
Brent
Even the most basic stuff… say the concept of the set. Is this just a human cultural
invention? Certainly on one level it is, we have developed a theory of sets and
incorporate and manipulate sets at so many levels of human activity, but does this fact
of our cultural discovery of set theory and wide employ of the techniques and structures
it provides us with translate into the much more fundamental claim that set theory
itself only exists in so far as humans have invented it. Would not some alien culture
(biological or with some artificial substrate) come discover the same set theory as we
have? If not… then why? I am arguing that there is something fundamental about an
abstract something such as a set… even an empty set. The kinds of operations the manner
in which it selectively includes “likes” while excluding “unlikes”.
What about fractals? Purely a human artifact? Then explain how fractals show up all over
nature from ferns to snowflakes?
And… the infinite set of countable natural numbers [e.g. 1, 2, 3… N]? Is this purely a
human cultural invention with no independent existence outside of human culture?
>>Would it make any difference if there were only 10^10^10 particles in the universe?
Wouldn't it be an inconvenience only in some mathematical proofs?
Brent
If I could answer that with absolute certainty I wouldn’t be here <grin>… to the best of
my limited knowledge… I would hazard a no it would not make a difference in reality even
if it demolished some mathematical crystal palace of abstract proof. And I agree with
your point – if I understood what you were saying that is -- that abstract systems may
demand – for reasons of their own internal consistency – conditions to exist that may
not reflect actual reality.
-Chris
As you can see from my questions… If that is I understood your position of course
<grin>… I think that there is strong evidence for many kinds of mathematically precise
relationships in nature, that many quite clear patterns exist and repeat across many
scales and domains in the natural universe (outside of human culture).
It seems to me that math is better defined as our accumulated human cultural achievement
in understanding basic fundamental laws and patterns of the universe we inhabit. It is
our human cultural discovery of something a lot deeper and vaster than what can possibly
be contained in the meager store of our species accumulated musings over the last
handful of millennia.
Cheers,
Chris
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
