At 22:00 +0000 17/07/2002, billdukie2004 wrote:
> I have to admit that I'm not sure if I should be posting this
>message just yet (since I am currently reading FOR for the first time
>and still have some seventy pages to read), but I've been mulling
>over the concept of supposed mathematical certainty, and I find the
>issue somewhat troubling.
> My main point of contention involves the fact that mathematical
>abstractions are man-made constructs--i.e., "...a modern physicist
>would regard a mathematical circle as a bad approximation to the real
>shapes of planetary orbits, atoms and other physical things." Even
>though one could only gain pretty poor knowledge from the study of
>planetary orbits approximated as circular, the fact that orbits are
>not circular does not in any way invalidate any knowledge of ideal
>circles that one might have discovered in their errant pursuits of
>circular orbits. The nature of a simple circle is defined by man:
>two-dimensional, constant radius from a defined center, etc., so how
>could any newly obtained physical knowledge detract from what we know
>about the circle as currently defined?
> In FOR, DD discusses the instance of curved triangles having
>angles that sum to values other than 180 degrees, but this does not
>affect the fact that Euclidean triangles have angles that do sum to
>180 degrees: of this I am certain (based on the definition of a
>Euclidean triangle, a definition specified by man). In this case,
>newly obtained physical knowledge (i.e., curved spaces) forced our
>mathematics to broaden its scope (i.e., curved, non-Euclidean
>triangles), but our previous mathematical knowledge (i.e., related to
>Euclidean triangles) remained intact and correct.
> Obviously physical knowledge can greatly impact how we apply
>mathematics to physics, but how can mathematical certainty be an
>impossibility simply because we cannot prove physical theory? Based
>on the conventional definitions of addition and natural numbers, how
>can it not be certain that 1+1=2? I have defined what '1', '+', '=',
>and '2' mean, so '1+1=2' is most certainly true; how could it be
>otherwise?
I agree with you. Courageously (DD knows he will inflame at least the
mathematicians) Deutsch posit quasi-explicitly that physical reality
is *fundamental*.
But I have less doubt about "1+1=2" than about any physical laws.
Except perhaps SWE-like equations, just because, indeed, I am open
to the idea that those SWE are inside-views (by self-anticipating
number-theoretical machine) consequences of "1+1=2" and some more
complex arithmetical propositions.
This could answer partially to iambiguously.domain.name.hidden (alongside
with Charles' comment) when iambiguously writes:
> Here is the difficulty I have regarding whatever the
> laws of physics may or may not be: the Primordial Why?
> In other words, why does anything exist at all? Why
> does it exist as it does and not some other way? Does
> it really matter what science can tell us about "how"
> physical reality works if it has not a clue as to
> "why" it works as it does? How exactly do how and why
> entwine? I would imagine that, 100,000 years from now,
> folks will read the Fabric Of Reality and...and what?
The physical laws could be arithmetical necessity, or
mathematical necessity.
Of course someone can ask "why mathematical truth or why
arithmetical truth?". That's the unsolvable mystery. But we
cannot get the natural numbers without positing them (to
summarize the godelian failure of logicism).
Actually natural numbers can be used to justify why, without
them, you could not even ascribe any meaning to the word "why".
(Just try to define the notion of natural numbers without
using them implicitly, you will intuit the impossibility).
Bruno
--
http://iridia.ulb.ac.be/~marchal/
Received on Fri Jul 19 2002 - 08:49:59 PDT