[EMAIL PROTECTED] wrote: > I did point out in my last post that there appears to be no simple way > to make such reductions (between math concepts and classes of specific > things). For instance no one has yet succeeded in showing how math > concepts such as infinite sets and transfinite sets (which are precise > math concepts) could be converted into physical notions. A also > pointed to David Deutsch's excellent 'Criteria For Reality':

I fail to see any 'knock-down' character in this argument. Peter says that mathematical concepts don't refer to anything 'external', and on one level I agree with him. But they are surely derived from the contingent characteristics of experience, and AFAICS experience in this context reduces to the contents of our brains. So 'infinite sets' is just a model (brain material at another level of description) which IMO counts as a 'physical notion' unless you start off as an idealist. Put simply, you can't think mathematical thoughts without using your brain to instantiate them - and you don't literally have to instantiate an 'infinite set' in the extended sense in order to manipulate a model with the formal characteristics you impute to this concept. In fact, the inability to convert infinite and transfinite sets into physical notions is excellent empirical evidence that they *don't* exist in any literal sense - they don't need to, as their usefulness is as limit cases within models, not as literal existents (nobody has ever literally deployed an infinite set). This is a thoroughgoing contingentist position, and I don't see that it can be refuted except by rejecting contingentism and starting from idealism. But then you've begged what you're trying to prove. > 'If according to the simplest explanation, an entity is complex and > autonomous, then that entity is real.' ('The Fabric Of Reality', Pg 91) Autonomous of what precisely? In what sense is a mathematical concept autonomous of your brain, or the collection of brains and other recording devices that instantiate it? Remember that we're talking about mathematical *concepts* - i.e. things we can grasp - it's merely a metaphor to claim that these models *refer* to autonomously existing platonic realities. Either a metaphor, or the presumption of such platonic reality, not its proof. > As Detusch points out, mathematical entities do appear to match the > criteria for reality: 'Abstract entities that are complex and > autonomous exist objectively and are part of the fabric of reality. > There exist logically necessary truths about these entities, and these > comprise the subject-matter of mathematics.' Truths are only equivalent to 'existents' for an idealist. Fair enough, but then this has to be accepted axiomatically, or not at all. I can't honestly see why this is so hard to grasp. David > >But why can't it be reduced to classes of specific physical things? How > >can you show that it is necessary for anything corresponding to this > >description to 'exist' apart from its instantiations as documented > >procedures and actual occurrences of their application? > >David > > I did point out in my last post that there appears to be no simple way > to make such reductions (between math concepts and classes of specific > things). For instance no one has yet succeeded in showing how math > concepts such as infinite sets and transfinite sets (which are precise > math concepts) could be converted into physical notions. A also > pointed to David Deutsch's excellent 'Criteria For Reality': > > 'If according to the simplest explanation, an entity is complex and > autonomous, then that entity is real.' ('The Fabric Of Reality', Pg 91) > > > As Detusch points out, mathematical entities do appear to match the > criteria for reality: 'Abstract entities that are complex and > autonomous exist objectively and are part of the fabric of reality. > There exist logically necessary truths about these entities, and these > comprise the subject-matter of mathematics.' > > > >Language, logic, and math are human inventions just as chair is, c.f. > >William S. > Cooper "The Evolution of Reason". > >That chair would continue to exist even if all > humans were wiped off the Earth - but the concept of 'chairs' wouldn't > and neither > would '2'. > >Ontology is invented too. > >Brent Meeker > > I distinguish between two kinds of abstract concepts - abstract > concepts of universal applicability, which I think are objectively real > and abstract concepts of limited applicability, which are clearly human > inventions. You don't accept the distinction. But I pointed out that > for abstract concepts of universal applicability, there appears to be > no difference between cognitive and ontological categories, where as > for abstract concepts of limited applicability, there clearly is a > difference between cognitive and ontologic categories. > > So I would tend to say that the concept of '2' is clearly 'out there', > where as the concept of 'chair' is 'in our heads' and quite possibly > even the concrete instances of a 'chair' is 'in our heads' as well! > After all, is it really the case that a chair is an object 'out there' > with definite objective physical dimensions like length? Isn't it > actually the case that all that's 'out there' is a 4-dimensional > 'chair' world-time? - which I point out to you as really a > *mathematical construct* ;) > > > > >Actually, it's an arguement against doing so. If mathematical > >terms referred to particular things, they would not be universally > >applicable. > >They are universally applicable because they don't refer to anything. > > >1Z > > Math concepts are super-classes or abstract classes being used to > classify *other* astract classes. I pointed out three different > ontological catgories: > > (1) Abstract entities of universal applicability (like math concepts) > (2) Abstract entities of limited applicability (human constructs like > alphabets or a chair concept) > (3) Concrete instances (like a particular example of a chair) > > I'd say you can make a good case that the entities in (1) are the only > real objective reality. It's (2) and (3) that are actually 'in our > heads'! --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. 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