Re: Is a function a relation?

"David BL" <davidbl_at_iinet.net.au> wrote in message news:e67c12ff-a6e9-4fc9-a8bd-fea0e16a5b73_at_f38g2000pra.googlegroups.com...
> On Jun 23, 4:34 pm, Cimode <cim..._at_hotmail.com> wrote:
>> On 23 juin, 08:14, David BL <davi..._at_iinet.net.au> wrote:> On Jun 23,
>> 1:35 pm, David BL <davi..._at_iinet.net.au> wrote:
>>
>> > > Yes that's one way of looking at it.
>>
>> > I'll expand on what I mean by that. It seems to me that one could use
>> > special conventions to "show" that just about any type can be regarded
>> > as a specialisation of a relation. E.g. one could say that a whole
>> > number in [0,255] is a relation by introducing symbols to represent
>> > 1,2,4,8,...,128 and the relation records a set of symbols that are
>> > then interpreted in the manner of an 8 bit unsigned representation.
>>
>> Relations is a possible construct that can represent *any* type if we
>> are to consider that a type is a set of values. Nevertherless, a
>> logical computing model (to define among other things the physical
>> reprentation of domain values) must be defined first (that is what I
>> spent the last 10 years working onto)...Hope this helps...
>
> It could be thought that the logical can only exist as an abstraction
> over the physical. However I don't believe that's a useful way to
> think. In fact I suggest it misses the idea behind physical
> independence. What I mean is that the logical doesn't need to be
> "realised" or "reified" by the physical at all!
>
> This could be seen as just a metaphysical comment (more specifically
> in favour of mathematical realism), but what I really mean is that
> pure mathematical systems can for example define things like the
> integers in a way that's unique up to isomorphism through the
> axiomatic approach, and that perspective is all one needs at the
> logical level. I don't see how the physical comes into it at all.
> Putting it another way (using the language of a mathematical realist
> in denial), database values don't exist in time and space!

While I would agree that the physical representation of the symbols and combinations of symbols that compose formal language terms is irrelevant, if what is in the Universe of Discourse can exist in time and space, then database values can exist in time and space. A value is the result of applying for a given term the valuation function which maps terms expressed in a formal language to things in the Universe of Discourse under an interpretation. So from one point of view, a value /is/ the thing in the Universe referred to by a given term, but just at the instant of interpretation. More importantly, if the micro-world that the database is supposed to model includes things than can occupy a locus in time and/or space, then the language, the logic and the intended interpretation must reflect that; however, correct me if I'm wrong, but isn't a pure mathematical system composed entirely of abstract objects, and aren't abstract objects by definition not located in time or space and thus not subject to change? How, then, can a pure mathematical system be a sufficient logical model for things that can change?

> It's not clear that type systems particularly help in this purist
> mathematical endeavour. I note that the usual axioms of set theory
> completely ignore any concept of type. I'd be interested to know
> whether modern mathematicians that have researched type theory believe
> it's important to mathematical foundations. My understanding is that
> Russell only investigated type theory with the aim to avoid paradoxes
> by preventing loops, but his work was made redundant by axiomatic
> systems like ZFC which is believed to be free of paradoxes.
>
Received on Wed Jun 24 2009 - 06:44:07 CEST