On Mon, Dec 17, 2018 at 10:54 PM Bruce Kellett <[email protected]> wrote:
> On Tue, Dec 18, 2018 at 3:38 PM Jason Resch <[email protected]> wrote: > >> On Mon, Dec 17, 2018 at 12:26 PM Brent Meeker <[email protected]> >> 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? > > > 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? > > > >> 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. Jason -- 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.
