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Date: Sat, 28 Jan 2023 07:43:56 +0200
From: Patrik Eklund
To: Pedro Resende
Cc: Steven Vickers , categories list
Subject: categories: Re: Terminology for point-free topology?
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Quantales are interesting examples also from the viewpoint how they may
be used for many-valued truth in logic. There are certain mechanisms
where "points can be recovered", but in my view this is not a sufficient
justification for working only algebrally in logic, or algebraically in
topology for that matter.
Propositional two-valued logic is a bit similar. The boolean values are
not pointfree but "term-free", if the analogy is allowed. We can do
something with propositional logic, but representation of "condition"
or "state" requires "insideness, in the sense of inside points". Points
as such representing states to me makes no sense in practice. We can
surely create fancy examples, but we cannot define things like "asthma"
or logically differentiate between Alzheimer's and vascular dementia
using pointfree topology.
Pointfree or not, term-free or not, I think it is important to justify
freeness whenever the calculation machinery allows it, but at the same
time refrain from being overenthusiastic about pointfreeness in the
sense of "I can work totally without points". Such things I would call
not just pointfree but indeed pointless, in particular as such
pointlessness kind of intentionally shuts out any possibility for
real-world application.
Some parts of theoretical mathematics is about seductive tricks, and
some mathematicians fall for it. Potential practicality of even "deepest
theoretical theory" keeps feet on the ground, even if practicality is
not realizable or desirable. But my view is that we must keep
"real-world applicability" at least as a "general burden" in the sense
that all science must useful, in one way or another. Science should
never be just "aus liebe zur Kunst".
Patrik
On 2023-01-27 19:55, Pedro Resende wrote:
> Hi Steve,
>
> Sorry for the radio silence, it???s been a hectic week.
>
> Concerning your question about a less derogatory expression??? I think I
> like `algebraic reasoning??? versus `point-based reasoning??? (which to me
> sounds better than `pointwise', I don???t know why).
>
> This is analogous to commutative algebra versus algebraic geometry.
>
> In any case, am I right that it seems to be somewhat consensual (in
> this thread) that `pointfree topology??? is the appropriate terminology
> for the kind of topology that *can* (but not necessarily has to) be
> studied without reasoning in terms of points?
>
> Incidentally, in my mind the `pointfree' terminology should also apply
> to more general notions, such as quantales, or at least some classes
> of them. For instance, inverse quantal frames are `the same' as
> localic etale groupoids, and they have associated etendues.
>
> Best wishes,
>
> Pedro
>
>> On Jan 23, 2023, at 1:47 PM, Steven Vickers
>> wrote:
>>
>> Dear Pedro,
>>
>> Of course, that's the very reason why I wanted to transfer it to the
>> style of working without points.
>>
>> That's slightly unfair, in that in many cases of reasoning
>> algebraically, without points, it's not at all clear how to do it
>> pointwise.
>>
>> You and I have certainly experienced that in our work on quantales,
>> which are much more purely algebraic gadgets. Our approach via localic
>> suplattices (algebras for the lower hyperspace monad) gives a more
>> point-free approach to the subject, but it takes effort - I think
>> you'll agree - to work with the hyperspaces in a pointwise manner.
>>
>> Do you think there's a less derogatory term for the style of reasoning
>> without points?
>>
>> All the best,
>>
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