On 30 May 2012, at 08:12, Stephen P. King wrote:

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On 5/30/2012 12:06 AM, meekerdb wrote:On 5/29/2012 8:47 PM, Stephen P. King wrote:On 5/29/2012 5:18 PM, Jesse Mazer wrote:On Tue, May 29, 2012 at 4:38 PM, Aleksandr Lokshin <aaloks...@gmail.com> wrote:It is impossible to consider common properties of elements of aninfinite set since, as is known from psycology, a man canconsider no more than 7 objects simultaneously.That's just about the number of distinct "chunks" of informationyou can hold in working memory, so that you can name thedistinctive features of each one after they are removed from yoursense experience (see http://www.intropsych.com/ch06_memory/magical_number_seven.html). But I'm not talking about actually visualizing each and everymember of an infinite set, such that I am aware of thedistinctive features of each one which differentiate them fromthe others. I'm talking about a more abstract understanding thata certain property applies to every member, perhaps simply bydefinition (for example, triangles are defined to be three-sided,so three-sidedness is obviously one of the common properties ofthe set of all triangles). Do you think it's impossible to havean abstract understanding that a large (perhaps infinite) set ofobjects all share a particular property?A single finite and faithful (to within the finite margin oferror) representation of "triangle" works given that definition.This is there nominalism and universalism come to blows....Your remarkable objection that "if two mathematicians considertwo different arbitrary objects they will obtain differentresults" demonstrates that you are not a mathematician.Huh? I didn't write the phrase you put in quotes, nor imply thatthis was how *I* thought mathematicians actually operated--I wasjust saying that *you* seemed to be suggesting thatmathematicians could only prove things by making specific choicesof examples to consider, using their free will. If that's notwhat you were suggesting, please clarify (and note that I did askif this is what you meant in my previous post, rather than justassuming it...I then went on to make the conditional statementthat IF that was indeed what you meant, THEN you should find itimpossible to explain how mathematicians could be confident thata theorem could not be falsified by a new choice of example. Butof course I might be misunderstanding your argument, that's why Iasked if my reading was correct.)Arbitrary element is not an object, it is a mental but non-physical process which enables one to do a physically impossiblething : to observe an infinite set of objects simultaneouslyconsidering then all their common properties at a single reallyexisting object. Therefore two different mathematicians willnecessarily obtain the same result.So you agree mathematicians don't have to make an actual choiceof a specific element to consider? Then how is free will supposedto be relevant if there is no actual choice whatsoever being made?--Why do you keep insisting on a "specific" property to the"choice" while being shown that the a priori "specificity" itselfthat is prohibited by the definition.He didn't refer to a specific property but to a specific choice ofelement, which is what Loskin says entails the magic ability toselect one among an infinite number. He apparently thinks of itlike the complement of the axiom of choice: to pick an element youneed to say,"Not this one. Not this one. Not..." an infinite numberof times.Hi Brent,Yes, that is a very good point! The axiom of choice is a suspecthere. Banach and Tarsky proved a paradox of the axiom of choice, itis the "scalar field" of mathematics, IMHO; you can get from itanything you want.

`Banach and Tarski proved an amazing theorem with the axiom of choice,`

`but it is not a paradox, in the sense that it contradicts nothing, and`

`you can't get anything from it.`

Bruno

The point is is that what ever the choice is, there are ab initioalternatives that are not exactly known to be optimal solutions tosome criterion and some not-specified-in-advance function that"picks" one.??? The function is specified in advance, e.g. "triangles" is afunction that picks out things with three sides meeting pairwise asthree vertices. But I have no idea what you mean by "optimality".What does that word mean? Try this from http://encyclopedia2.thefreedictionary.com/Optimality1. (mathematics) optimal - Describes a solution to a problem whichminimises some cost function. Linear programming is one techniqueused to discover the optimal solution to certain problems.2. (programming) optimal - Of code: best or most efficient in time,space or code size.Is that helpful? -- Onward! Stephen "Nature, to be commanded, must be obeyed." ~ Francis Bacon --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.

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