Re: Current Issues in the Philosophy of Practice of Mathematics & Informatics

[Patrik Eklund <peklund@cs.umu.se>]

On 2015-08-09 05:10, Fred E.J. Linton wrote:
> Not wishing to broadcast my illiteracy in the matter ...
> So I ask you now, in public, where my shame can be greatest: what do
> you mean by "lativity"?

Thank you, Fred, for your questions. We were actually nervously waiting
for somebody to ask that question, so we will remember you always for
having done that.

In logic we typically have signatures, terms, sentences, structured sets
of sentences, entailment, models, satisfactions, axioms, theories and
proof calculi. We cannot e.g. define entailment before we have a notion
of sentences, and we should not define sentences before we have a notion
of terms. The latter is a bit more controversial. In first-order logic I
would see P(x), where P is a "predicate symbol", as a term, and not as a
sentence, whereas putting a quantifier in front of it, Ex.P(x), makes it
no longer a term. This is why I have difficulties e.g. to accept that
the two 'not's in expressions like "not Ex.P(x)" and "Ex.not P(x)" would
be the same. I am starting to think they are only informal as symbols, a
bit similar as Church said lambda is and informal symbol, so actually
not part of the formal syntax. Am I wrong or am I wrong?

In logic we indeed need a signature (sorts and operators) in order to
construct the categorical object of terms. Construction is important. We
need terms to categorically construct sentences, which appear because of
a sentence functor not being extendable to a monad. Otherwise sentences
are terms, aren't they, because then we could substitute sentences
within sentences, and that does not comply with our traditional view of
sentences.

Traditional first-order pretty much doesn't care, and neither did
Aristotle about these things. Aristotle's and first-order logic are
therefore very "illative" and also very unsorted, I would like to add.

You are not broadcasting illiteracy at all, and your shame couldn't even
be small because no shame whatsoever is justified to exist, at least not
on your side. There may, however, be some of it now or eventually on my
side, but let us see what happens if/when/how this dialogue develops. It
may indeed turn out that at least some members of this catlist will see
me just as a devoted soldier "seeking the bubble reputation, even in the
canon's mouth".

When we were searching for a name describing what we try to explain, we
wanted to be careful not to use a "reserved word" that is more easily
misunderstood than not well understood. In the latter case, we thought
we can always try to explain, as I am about to do now. In the former
case it would be a differentiation, which is more tricky.

So here goes.

'Lative' is related to motion, and more specifically, motion 'to' and
'from', so when terms appear in sentences, terms 'move into' sentence,
and 'appear within' sentences. At the same time, sentences 'move away
from' terms, and separates terms from sentences. In comparison,
'ablative' is motion 'away', and nominative is static. The lative
locative case (casus) indeed represents "motion", whereas e.g. a
vocative case is identification with address.

We want to underline the need not to have "mixed bags", so that we can
ensure that a term does not appear in the bag of sentences, or a
structured set of sentences would appear in the bad of entailment. I
shouldn't compare with waste sorting, because then somebody might say
"Patrik Eklund said Kurt G??del didn't care about waste sorting".
Obviously, I do respect the work of G??del, even if at the same time I do
find his approach "illative". From categorical point of view, G??del also
complies only with the underlying category of sets, but as we have shown
(e.g. in
http://www.sciencedirect.com/science/article/pii/S0165011413000997), we
may have other underlying categories for the the term monad. The term
constructor and the use of category theory as the metalanguage for logic
is here important. Logic developed "hand-in-hand" with set theory and as
being a metalanguage for category theory should then not be confused
with the lativity of logic we explain using categorical notions.

Let me also again underline that nomenclatures and classifications in
health care is one of our motivation areas of examples and applications.
Nomenclatures exist also elsewhere, but he ones appearing in health we
find very motivating.

At this point of our "research program" we believe we have a fair
understanding of the lativity as related to signatures, terms and
sentences, and we hope we have a fair intuition about how we now proceed
to investigate the lativity of that with respect e.g. to to entailment
and models.

Thanks again, and indeed, possible shame in whatever form or magnitude
is all mine.

Best,

Patrik



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