Re: Is mathematics human thinking? WAS [generalizations_of_islam - God Matter]

Tue, 07 Oct 2014 16:24:04 -0700 



________________________________
 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]] 
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

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. 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
>

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