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Re: Non-Euclidean Arithmetic
Posted:
Sep 12, 2012 3:02 PM
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On Wed, Sep 12, 2012 at 9:27 AM, Paul Tanner <upprho@gmail.com> wrote:
> On Wed, Sep 12, 2012 at 11:34 AM, kirby urner <kirby.urner@gmail.com> wrote:
>> On Tue, Sep 11, 2012 at 8:54 PM, Paul Tanner <upprho@gmail.com> wrote:
>>
>> << snip >>
>>
>>> I dealt with the act that you are the one who said that almost
>>> everyone never needs to deal with mathematics that deals with
>>> non-computable objects - when as we all know everyone does need to
>>> deal with all the reals in any and all mathematics based on the reals,
>>> which is essentially all the math everyone learns in high school and
>>> beyond.
>>>
>>
>> That shouldn't be the case though. Confining our numeracy and
>> alpha-numeracy training to the real numbers and its subsets is a huge
>> deficiency in today's mediocre curriculum,
>>
>
> I did not say confine things to real numbers.
>
Right, that would be daft. Nor should the multiplication operation be
confined to numbers in general.
E.g. from a hypothetical high school abstract algebra segment, digital
math track:
>>> from perms import anyperm, P
>>> permA = anyperm()
>>> permA
{' ': 'g', 'a': 'y', 'c': 's', 'b': 'p', 'e': 'u', 'd': 'd', 'g': 'q',
'f': 'h', 'i': 'n', 'h': 'z', 'k': 'm', 'j': 'x', 'm': 'f', 'l': 'o',
'o': 'j', 'n': 'e', 'q': 'c', 'p': 't', 's': 'i', 'r': ' ', 'u': 'k',
't': 'a', 'w': 'r', 'v': 'v', 'y': 'l', 'x': 'w', 'z': 'b'}
>>> permB = anyperm()
>>> pA = P(permA)
>>> pB = P(permB)
>>> pC = pA * pB
>>> pC
P({' ': 't', 'a': 'd', 'c': 'm', 'b': 'e', 'e': 'z', 'd': 'x', 'g':
'q', 'f': 'p', 'i': 'h', 'h': 'v', 'k': 'c', 'j': 'n', 'm': 'o', 'l':
'l', 'o': 'i', 'n': 'w', 'q': 'y', 'p': 'r', 's': 'k', 'r': 'u', 'u':
'g', 't': ' ', 'w': 'j', 'v': 'a', 'y': 's', 'x': 'f', 'z': 'b'})
>>> permB
{' ': 'u', 'a': ' ', 'c': 'y', 'b': 'b', 'e': 'w', 'd': 'x', 'g': 't',
'f': 'o', 'i': 'k', 'h': 'p', 'k': 'g', 'j': 'i', 'm': 'c', 'l': 's',
'o': 'l', 'n': 'h', 'q': 'q', 'p': 'e', 's': 'm', 'r': 'j', 'u': 'z',
't': 'r', 'w': 'f', 'v': 'a', 'y': 'd', 'x': 'n', 'z': 'v'}
>>>
You can see that ' ' (space) goes to 'g' in permA whereas 'g' goes to
't' in permB. In their product, ' ' goes directly to 't'.
The complete source code is less than 20 lines of code, with a
language phasing in no later than 8th grade:
from string import ascii_lowercase as letters
from random import shuffle
letters = letters + " " # add space
def anyperm():
perm = {}
target = list(letters)
shuffle(target)
return dict(zip(list(letters), target))
class P:
def __init__(self, mapping):
self.mapping = mapping
def __mul__(self, other):
newmap = {}
for c in self.mapping:
newmap[c] = other.mapping[self.mapping[c]]
return P(newmap)
def __repr__(self):
return 'P('+str(self.mapping)+")"
(using plaintext view to get the spacing right -- syntactically meaningful)
> We can avoid real numbers only by avoiding just about everything
> taught in Algebra I and beyond. Avoiding just about everything taught
> in Algebra I and beyond is just dumb.
Not trying to avoid numbers, just not wanting to use them exclusively.
The idea that mathematics is only about numbers is a stereotype that
leads to serious forms of bigotry, difficult to overcome in later
life.
Playing chess is a mathematical activity. The Time-Life series
entitled 'Mathematics' is what recruited me to the field, and it had
lots about computers.
I'm concerned about "bait and switch". They promise studying math
will pay off, but then they withhold all the good stuff, saying "oh
no, that's computer science".
Kirby
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