On Tuesday, December 18, 2018 at 5:31:06 AM UTC, Bruce wrote: > > On Tue, Dec 18, 2018 at 4:12 PM Jason Resch <[email protected] > <javascript:>> wrote: > >> On Mon, Dec 17, 2018 at 10:54 PM Bruce Kellett <[email protected] >> <javascript:>> wrote: >> >>> On Tue, Dec 18, 2018 at 3:38 PM Jason Resch <[email protected] >>> <javascript:>> wrote: >>> >>>> On Mon, Dec 17, 2018 at 12:26 PM Brent Meeker <[email protected] >>>> <javascript:>> wrote: >>>> >>>>> >>>>> But mathematical objects are completely defined by their axioms. >>>>> >>>> >>>> Are they? >>>> >>>> Two is a mathematical object. >>>> One of the properties of two is the number of primes it separates. For >>>> example "3 and 5", "5 and 7", etc. >>>> >>> >>> Definitions do not necessarily specify all the relationships into which >>> things can enter -- if that was necessary for a definition, no definition >>> would be possible. Clearly, common ostensive definitions do not have to >>> specify all the properties of an object, or even what it is made of: "That >>> is a rock" is a useful ostensive definition, but it does not commit one to >>> a full geological understanding of rocks, their formation and properties. >>> >> >> But shouldn't the system that "completely defines" some mathematical >> object, allow one to learn and discover those properties? Would you be >> satisfied with a physical theory that billed itself as completely defining >> all any physical phenomena, but couldn't tell us what the mass of the >> electron was? >> > > Definitions may be the starting point for a theory, but they are not the > complete theory. > > > If mathematical objects are completely defined by their axioms, then >>>> shouldn't this property be defined and known for two? Yet we don't even >>>> know the answer to this question, we don't know if it is infinite or >>>> finite. It might even be that no proof exists under the axioms we >>>> currently use. >>>> >>> >>> Mathematical objects may be completely defined by their definitions, in >>> that the definition corresponds to that unique object. But that does not >>> commit one to knowledge of all the relationships that might be true about >>> that object. You are requiring too much from a definition. >>> >> >> Let me update my example: >> Instead of considering "2", consider "Object T" which is "The number of >> primes separated by 2". >> Wouldn't this be a mathematical object that might be undefined by the >> axioms? >> > > The concept of twin primes has a simple definition. And the definition > uniquely specifies the object -- one would know one whenever one met one, > and that without ambiguity or error. But that does not mean that the > definition, of itself, should tell you whether the number of primes that > satisfy this definition of twin primes is finite or infinite, or even if > there are any such pairs of primes. > > But we are talking about definitions of objects, not axioms of a theory. > We know that any axiomatic theory will necessarily be incomplete -- there > will be formulae in the theory that are neither theorems nor the negation > of theorems. >
*Based on the examples I previously offered, that QM and SR are axiomatic theories, can we conclude they're incomplete? AG* > > > >> There is no possibility of ostensive or empirical definition. That's the >>>>> strength of mathematics; it's "truths" are independent of reality, they >>>>> are >>>>> part of language. >>>>> >>>>> But in any case, the axioms don't define arithmetical truth, which is >>>>> my only point. >>>>> >>>>> >>>>> No, but they define arithmetic, without which "arithmetical truth" >>>>> would be meaningless. >>>>> >>>> >>>> Was the physical universe meaningless before Newton? >>>> >>> >>> The physical universe is defined ostensively -- neither Newton not >>> Einstein brought it into existence. >>> >> >> I think that of the Integers. I'm open to any arguments you have that >> could change my mind. >> > > What is an ostensive definition of an integer? > > Bruce > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
