On 10/7/2014 4:23 PM, 'Chris de Morsella' via Everything List wrote:
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*From:* meekerdb <[email protected]>
*To:* [email protected]
*Sent:* Monday, October 6, 2014 9:26 PM
*Subject:* Re: Is mathematics human thinking? WAS [generalizations_of_islam -
God Matter]
On 10/6/2014 9:11 PM, 'Chris de Morsella' via Everything List wrote:
*From:*[email protected] <mailto:[email protected]>
[mailto:[email protected]] *On Behalf Of *meekerdb
*Sent:* Sunday, October 05, 2014 5:57 PM
*To:* [email protected] <mailto:[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”!
But we easily define it within a model, e.g. in Euclidean 3-space it's a triple of
numbers. To say what it means for nature requires some operational procedure for
determining the three numbers, and when you try to make that precise you discover how much
idealization and abstraction it takes.
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.
And even of biological evolution before humans. It's well known that crows can
count to six.
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's my point. They are models. We define them to reflect what we suppose to facets of
nature. But they are still our inventions. That is clear from the way models have been
overturned and displaced. Our picture of the world since the development of quantum
mechanics is drastically different than the picture in 1890. So was the universe ever
really a clockwork?
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
I see your point, though I am not sure I agree with it. And again I go back to the
examples of mathematical relationships appearing in nature. The specific spiral shapes
that are quite common in nature and that closely approximate idealized mathematical
relationships such as the golden ratio indicate to me that nature itself uses math. And
why wouldn't it. The fractal patterns evident in say a fern are so economical and can
emerge from elegantly simple mathematical formulas. Ferns seem to use math... so do
flocks of birds. Algorithms based on the sequential application of various sometimes
surprisingly simple formulas pop up over and over in nature.
I think the evidence for this natural existence of mathematical ratios implies that math
(at least some math) is more than just a tautology that is consistent only because it
was defined in such a manner as to be internally consistent.
Chris
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.
But why not recognize that they are fundamental to our thinking and our communication with
one another, i.e. language.
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?
But they only approximate the ideal fractals of mathematics.
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?
I'd say it's an inevitable invention. Given that we see discrete countable objects and we
have limited discernment so that we easily abstract away individuality and put things into
classes it is inevitable that we would have the idea of "how many there are". But
"infinite"? I think not. Remember George Gamow's example of the Amazonian tribe that
counted "One Two Three Infinity".
>>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).
But what exactly does it mean for a pattern to exist in nature? Doesn't it just mean that,
to some approximation, it matches a theoretical model we have. The pattern is literally
something in our model. It is approximately and figuratively in some instance in nature.
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
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