On Mon, Dec 17, 2018 at 11:31 PM Bruce Kellett <[email protected]> wrote:
> On Tue, Dec 18, 2018 at 4:12 PM Jason Resch <[email protected]> wrote: > >> 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? >> > > 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. > > > > >> 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? > > I mean that neither Peano nor Robinson brought integers into existence, not that integers are defined ostensibly. But regarding ostensibility of mathematical objects, I would say our knowledge of them comes from simulation (interacting and playing with objects of the mathematical world in our heads). Brains of mathematicians, via experience (empiricism) and simulation via their own brains gain a sense and intuition for these properties. It can be a sense not unlike sight or hearing. Which is why one mathematician can point out mathematical facts to another and others can see and appreciate that fact. If I tell you 28 is a perfect number <https://en.wikipedia.org/wiki/Perfect_number>, and you accept that fact and agree, how is it you come to agree? Is it through some human "math sense"? 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.
