On 27 May 2012, at 00:06, meekerdb wrote:

On 5/26/2012 9:35 AM, John Mikes wrote:Brent wrote:1. Presumably those true things would not be 'real'. Only provablethings would be true of reality.Just to be clear, I didn't write 1. above. But I did write 2. below.

Ah OK. Sorry. I have been wrong on that.

2. Does arithmetic have 'finite information content'? Is the axiomof succession just one or is it a schema of infinitely many axioms?Appreciable, even in layman's logic.In '#1' - I question "provable" since in my agnosticism an'evidence' is partial only, leaving open lots of (so far?) unknown/able aspects to be covered. In the infinity(?) of the "world" alsothe contrary of an evidence may be 'true'.As Bruno said, "Provable is always relative to some axioms and rulesof inference. It is quite independent of "true of reality". Whichis why I'm highly suspicious of ideas like deriving all of realityfrom arithmetic, which we know only from axioms and inferences.

`We don't give axioms and inference rule when teaching arithmetic in`

`high school. We start from simple examples, like fingers, days of the`

`week, candies in a bag, etc. Children understand "anniversary" before`

`"successor", and the finite/infinite distinction is as old as humanity.`

`In fact it can be shown that the intuition of numbers, addition and`

`multiplication included, is *needed* to even understand what axioms`

`and inference can be, making arithmetic necessarily known before any`

`formal machinery is posited.`

Bruno

#2 is a technically precise formulation of what I tried to expressin my post to Bruno.IFF!!! "anything" (i.e. everything) can be expressed by numerals,the information included into arithmetic IS infinite,I see no reason to suppose that. Everything ever expressed so farhas been done with a finite part of arithmetic. Assuming everyinteger has a successor is just a convenience for modeling things;you don't have to worry about running out of counters. There is abook "Ad Infinitum, The Ghost in Turing's Machine" by Rotman thatproposes what he calls "non-euclidean arithmetic" which does notassume the integers are infinite. I can't really recommend the bookbecause most of it is written in the style of Frenchdeconstructionist philosophy, but the Appendix has some interestingideas.however as it seems: in our (restricted) view of "theworld" (Nature?) there seem to be NO numbers to begin with.In our human 'translation' we see 1,2, or 145, or a million "OFSOMETHING" - no the (integer?) numerals.Axioms? in my vocabulary: imagined things, necessary for certaintheories we cannot substantiate otherwise.Axioms are just part of a logical, i.e. self-consistent, system.Mathematicians don't even care if they are "true of reality". Theymay or may not refer to imagined things; they are just assumed truefor some inferences. I could take "I am typing on a keyboard" as anaxiom, which I also happen to think is true, or I could take "I am aprojection in a Hilbert space" which might be true, but is much moredubious.In another logic than human, in another figment of a "physicalworld" different axioms would serve science.Logic is about the relations of propositions, statements inlanguage. Humans already have invented different logics.2+2=4? not necessarily in the (fictitious) "octimality" of the'[Zarathustran' aliens in the Cohen-Stewart books(still product of human minds).2+2=11 Brent"The world consists of 10 kinds of people. Those who think inbinary and those who don't.--You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-list@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com.For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.

http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.