non-standard models of botany

Would it really kill you to spell that with a z?
On Jun 3, 3:27 pm, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx> wrote
As much as it would kill you to refrain from apparently uncontrollable
capitalised insulting.

It's reactive, dumbass, wherefore it does not even APPEAR
to be uncontrollable. All you have to do to avoid getting
capitalized insults is STOP INSULTING *ME* gratuitously.
YOU WERE PRESENT DURING THE PRIOR YEARS of
my argument with Torkel about whether random idiotic
interpretations of things were vs. weren't relevant&analogous
as refutations of the relevance of non-standard models.
THE POINT
is that the non-standard models ARE MODELS of the axioms
and CONFIRM ALL the THEOREMS of the theory!
The kinds of idiotic interpretations that you and TF were offering
as dismissable ARE NOT MODELS (of the axioms; though, of
course, EVERY structure is a model of SOME sentence).


You were talking, specifically, about "Apples grow on trees."
This is problematic because it seems to employ some natural-
language things that I don't know standard terms for, like non-
absolute generalizations; in natural language, "Apples grow on
trees" does NOT MEAN "Absolutely all apples grow absolutely
only on trees". But we are talking about first-order theories here
and the usual first-order translation would wind up meaning
basically that. But just forgetting about natlang details, let's
just go ahead and translate it is Af[ap(f)->gt(f)], where
f is a fruit, ap(.) means it's an apple, and gt(.) means it
grows on trees.

The dictionary and the botany textbook jointly either imply
this truth or they don't; it's either a theorem of this theory
or it isn't. Whether the natural language is Finnish or something
else is not even relevant: the dictionary IS relevant and its
constraints and stipulations ARE incorporated into the axioms.

In order for this to be a theorem, it would have to be true in all
models. It would have to be basically inconceivable that something
could be an apple and not grow on a tree.

For a long time, this was in fact the case.

Indeed, right about the time FOL got discovered in the 1880s,
botany began to get well-developed enough as a science for
this theorem-hood question to be meaningful: was it even POSSIBLE
for an apple to come into existence other than by growing on a tree?

I don't know whether people felt they had proved this or not.
Prior to the discovery of genes it would be hard to know what
they even thought of as the DEFINITION of an apple.
Then, as PRIOR TO THE DISCOVERY OF NON-EUCLIDEAN
GEOMETRY, the standard model was THE ONLY ONE THEY
WERE AWARE of. But that does NOT imply that anyone had
to bring in "heavy machinery" or import irrelevant non-standard
models in order to cloud questions of provability:
NON-STANDARD MODELS ALWAYS PRIORLY EXISTED,
EVEN if they hadn't been discovered yet. THEY ALWAYS PRIORLY
MET all the relevant criteria for satisfying the axioms. The axioms
ALWAYS, FROM INCEPTION, had the property that they could
be modeled by these wilder more confusing structures.
So you DON'T get to BLAME those who insist on the relevance
of these structures for bringing in irrelevant heavy machinery:
IT WAS ALWAYS possible for these structures to satisfy these
axioms.

Similarly it may have become possible (I'm not a botanist)
to grow apples in some sort of culture or petri dish with a
root system not including a whole tree, but that's non-standard,
but it STILL MATTERS.

.

Relevant Pages

  • Re: non-standard models of botany
    ... as refutations of the relevance of non-standard models. ... is that the non-standard models ARE MODELS of the axioms ... only on trees". ... could be an apple and not grow on a tree. ...
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    (uk.rec.gardening)
  • Re: Apple tree recommendations?
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    (uk.rec.gardening)