On 31 Jul 2011, at 14:15, Craig Weinberg wrote:
Reblogging myself here, but curious to see what you think of the idea
that 1 cannot be proven greater than 0.
In which theory?
The notion of proof is theory and definition dependent. (contrary to
computability, which is absolute, by Church thesis).
If you agree to define x < y by Ez(z+x = y) "E" = "It exists". I
assume classical logic + the axioms:
x+0 = x
x+s(y) = s(x+y)
0 denotes the number zero, and s(x) denotes the successor of x, often
noted as x+1. Cf the whole theory I gave last week. I use only a
subset of that theory here.
So we have to prove that 0 < s(0). By the definition of "<" above, we
have to prove that Ez(z + 0 = s(0))
But s(0) + 0 = s(0) by the axiom x + 0 = x given above.
So Ez(0 + z = s(0)) is true, with z = s(0). (This is the usual use of
the existence rule of classical logic).
Of course we could have taken the theory with the unique axiom "1 is
greater than 0". For all proposition we can always find a theory which
proves it. The interesting thing consists in proving new fact in some
fixed theory, and change only a theory when it fails to prove a fact
for which we have compelling evidences.
Bruno
Someone’s comment on the previous chart mentioned the difficulty
(impossibility?) of proving that 1 > 0. It’s an interesting kernel
there, and it reminds me of the whole “time does not physically
exist”
realization. On one level, I can think of zero as having no different
relation to 1 than it has with any other number. Zero does the same
thing to any number as it does to one and should be thought of more
properly as the hub of the decimal spiral.
I’m no mathematician, but I suppose that 0 is also formally defined
as
an integer between 1 and -1 or something. Still it exposes the
question of whether the elemental underpinnings of our ability to
count is really anchored in anything at all other than our own
anthropological conventions of counting. Beyond numbers themselves, it
appears that the whole quantitative notion - of greater than or less
than, and of ‘equal’ is nothing but a figment of our feelings
about
order. There may not be any inherent moreness to something than the
absence of something. If it’s the same thing, it actually seems more
palatable to see the absence of something being a condition predicated
upon the things’ a priori presence, no?
Even if we want to get into quantum atopoietic craziness where things
come out of nothing, rendering such a possibility discretely seems to
threaten the whole notion of mathematical coherence. If any or all
quantities, variables, and formulas can be generated arbitrarily from
0, then 0 would seem to be the same thing as ∞, and greater than 1
or
any other arithmetic expression.
Anthrodeximal Numberline
Maybe it’s time to create a new numberline, without all of the
repetitive decimal numerals. Instead there could be a Wiki of new
quantitative symbols and names which anyone can add to and own as a
permanent vector in the schema. It would be easy to translate them to
and from Arabic numerals online and some interesting possibilities for
informal encryption and unanticipated mathematic-linguistic
synchronicity.
By removing the aspect of repetition, we would unmask the semantic
bias of the math logos and arrive at a pure generic linear calibration
defined only in it’s own idiosyncratic a-signifying terms. Sort of
like breaking the mantra of math, it’s trance-like rhythms that
disguise it’s human neurological origin from us. By adding more
unique
qualitative sense to the thing, the quality-flattening power drains
out and the system seems to disqualify itself.
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com
.
For more options, visit this group at http://groups.google.com/group/everything-list?hl=en
.
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to
everything-list+unsubscr...@googlegroups.com.
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
