On 5/30/2012 4:45 AM, Bruno Marchal wrote:

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

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 <mailto:aaloks...@gmail.com>> wrote:

    It is impossible to consider common properties of elements of
    an infinite set since, as is known from psycology, a man can
    consider no more than 7 objects simultaneously.

That's just about the number of distinct "chunks" of information you can hold in working memory, so that you can name the distinctive features of each one after they are removed from your sense experience (see http://www.intropsych.com/ch06_memory/magical_number_seven.html ). But I'm not talking about actually visualizing each and every member of an infinite set, such that I am aware of the distinctive features of each one which differentiate them from the others. I'm talking about a more abstract understanding that a certain property applies to every member, perhaps simply by definition (for example, triangles are defined to be three-sided, so three-sidedness is obviously one of the common properties of the set of all triangles). Do you think it's impossible to have an abstract understanding that a large (perhaps infinite) set of objects all share a particular property?

A single finite and faithful (to within the finite margin of error) representation of "triangle" works given that definition. This is there nominalism and universalism come to blows....

    Your remarkable objection that "*if two mathematicians
    consider two different arbitrary objects they will obtain
    different results"* demonstrates that you are not a mathematician.

Huh? I didn't write the phrase you put in quotes, nor imply that this was how *I* thought mathematicians actually operated--I was just saying that *you* seemed to be suggesting that mathematicians could only prove things by making specific choices of examples to consider, using their free will. If that's not what you were suggesting, please clarify (and note that I did ask if this is what you meant in my previous post, rather than just assuming it...I then went on to make the conditional statement that IF that was indeed what you meant, THEN you should find it impossible to explain how mathematicians could be confident that a theorem could not be falsified by a new choice of example. But of course I might be misunderstanding your argument, that's why I asked 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
    impossible thing : to observe an infinite set of objects
    simultaneously* considering then all their common properties
    at a single really existing object. Therefore two different
    mathematicians will necessarily obtain the same result.

So you agree mathematicians don't have to make an actual choice of a specific element to consider? Then how is free will supposed to 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" itself that is prohibited by the definition.

He didn't refer to a specific property but to a specific choice of element, which is what Loskin says entails the magic ability to select one among an infinite number. He apparently thinks of it like the complement of the axiom of choice: to pick an element you need to say,"Not this one. Not this one. Not..." an infinite number of times.

Hi Brent,

Yes, that is a very good point! The axiom of choice is a suspect here. Banach and Tarsky proved a paradox of the axiom of choice, it is the "scalar field" of mathematics, IMHO; you can get from it anything 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.


The axiom of choice allows for violation of conservation laws, if it where to be a physical law.



"Nature, to be commanded, must be obeyed."
~ Francis Bacon

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