[lit-ideas] Re: Beg to differ, say, about fractals


--- On Tue, 14/12/10, Eric Yost <mr.eric.yost@xxxxxxxxx> wrote:

> From: Eric Yost <mr.eric.yost@xxxxxxxxx>
> Subject: [lit-ideas] Re: Beg to differ, say, about fractals
> To: lit-ideas@xxxxxxxxxxxxx
> Date: Tuesday, 14 December, 2010, 7:16
> 
> 
> On 12/13/2010 7:53 PM, Donal McEvoy wrote:
> > This still leaves open, it seems to me, whether the
> right-angled triangle somehow existed from the beginning and
> whether all we did was discover this entity and its
> attendant structural properties - all of which were there
> from the beginning. Or did we invent the right-angled
> triangle, which not only was never perfectly physically
> embodied but never existed prior to our development of this
> mathematical 'object' or 'concept'?>
> 
> The right triangle, from before any beginning, was a set of
> relations such that the truth of
> a-squared+b-squared=c-squared, in reference to right
> triangles, was always the case.

This would seem to posit the right triangle as a mathematical object 
constituted by "a set of relations", the truth about which "was always the 
case".

If so, does that not entail that the "set of relations" "was always the case"? 
And, if so, does that not entail that the right triangle, constituted by that 
set of relations, "was always the case" - or was somehow there from the 
beginning?

Let us say it is true that "e = mc2". This "e = mc2" describes a structural 
property of the universe. This 'structural property' we might say is a physical 
or natural law. Such a law of physics is posited as holding anywhere in the 
physical universe, from the beginning until the end - it describes an 
invariant. We can conceive logically possible universes where this invariant 
does not obtain, so this invariant is not a law of logic that must hold in all 
possible worlds. In the case of "e = mc2" we do not think that the invariant it 
describes is our invention, rather it is our discovery. The proposition "e = 
mc2" is our invention but the invariant it describes is not. Nor is the truth 
of the proposition our mere invention: as the truth of the proposition depends 
on its correspondence with an invariant that we do not invent. We invent or 
develop a means to describe something invariant that pre-exists, and exists 
independently of, any such description of
 it.

Let us say it is true that "2 + 2 = 4". This "2 + 2 = 4" describes a structural 
property - an invariant. So, if true, does "a-squared+b-squared=c-squared in 
reference to right triangles". We might say these invariants are mathematical 
laws rather than physical laws. It seems we might nevertheless say that we do 
not invent these 'laws' but discover them, just as we do not invent but 
discover the physical invariants or laws of the universe (our invention being 
limited to the means of describing them, though we might say even this is more 
of a 'discovery' than invention).

We are soon back with our problem (or a variant of it). This problem disappears 
if we view it as entirely unproblematic that mathematical invariants were there 
from, or pre-existed, the beginning of the physical world: for any physically 
possible world must also be a mathematically possible world and both must also 
be a logically possible world. When we talk of the law-like structures of the 
universe, we are talking about structures within structures - with the most 
general being the law-like structures of logic that must hold in any possible 
world. But surely, even if it is correct, it is not entirely unproblematic that 
mathematical invariants were there from, or pre-existed, the beginning of the 
physical world? Surely there are meaningful questions we might pose here and 
even attempt to answer?

> However Donal also certainly knows that the word "exists"
> is not a predicate, nor are its verb-form declensions
> meaningful verbs. (Ask Robert Paul, which I have already
> done by asking you to do so.) "Contrite unicorns exist."

Donal is open-minded as to whether "exists" can (or cannot) be deployed as a 
"predicate" (fearing the answer may turn into a rather empty one hingeing on 
the definition of "predicate"), but does not see how this much answers or 
tackles the problems here.

> The questions seems to be, "Where does the truth of
> a-squared+b-squared=c-squared, in reference to right
> triangles, exist?"
> 
> Am I asking the right question?

I am not sure. A simple answer to this question, following Tarski, might be 
that the truth of "a-squared+b-squared=c-squared" exists in the correspondence 
between that proposition and the existence of an invariant that it describes. 
Whatever the problems with this, another more fundamental problem is to address 
the character of "the existence of an invariant" of this mathematical sort - an 
existence that we understand can never be (perfectly) physically embodied.     

Donal
London             



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