[hist-analytic] Materialism and mass as a unit of measurement
Baynesr at comcast.net
Baynesr at comcast.net
Tue Feb 1 12:12:42 EST 2011
Scott,
Brief reply. I wanna move on to some philosophy. Your points on inequalities are well taken. However, I took this into account. I've raised this elsewhere and the response is generally the same: Weirstrass inequalities in the definition of a limit. I am familiar with this, although I appreciate your taking the trouble. I do not believe that this approach gives a suitable definition of 'as close as you please'; it gives the right answer once you stipulate how close you want to get for any value of 'x' in 'f(x)' but this is different. My inquiry is into the meaning of '<' which I don't believe has ever received much clarification owing to its seeming simplicity. In some ways it is; but in other ways it is not. If you look, carefully, at how the df. of a post Weierstrassian goes, you will see that it isn't quite as simple, although now, ostensibly, there is a reasonable algebraic characterization. For more details see my remarks on the nature of comparatives and inequalities on the hist-analytic web site under the search term 'comparatives'.
Also, on the partitioning of intervals in a Riemann sum, etc. Yes! The partitioning case is, if I am right, exactly the same sort of thing as the limit case. I stick to Riemann sums because I want to give an example not involving limits, per se, which come in once you introduce the integral. But, again, the point is: How DO we define 'getting as close as you want'. I can get a number as big as I want by way of the successor function, but the successor function will not afford an acceptable definition in my opinion of 'as large as you want'. Underlying much of this is that calculus is a creature of physics, involving rates of change.
Rather thank trivially disagree or look for disagreement, let's stick to where I think you are exactly right: the problem is in the nature of inequalities and eventually quantities. Here is where the philosophy resides. Russell is good on this in Principles of Mathematics. On hyper-reals, let's defer this just a bit; I will bring this up later in the context, probably, of Zeno etc. For now I want to do some phil. of mind.
By the way, the business of indeterminacy and uncertainty will enter. I tipped my hand a bit early. Give me some time to catch up.
Regards
Steve
----- Original Message -----
From: "Scott Holbrook" <scott.holbrook at gmail.com>
To: Baynesr at comcast.net
Cc: "hist-analytic" <hist-analytic at simplelists.co.uk>
Sent: Sunday, January 30, 2011 7:52:57 PM
Subject: Re: Materialism and mass as a unit of measurement
I know you wanted to defer comments, but here are some things you may want to consider when getting this clearer.
I think the your using the epsilon-delta definition. It does have inequalities and I'm not aware of any other defitnions using inequalities. But, it should be noted that epsilon-delta defitnion IS the defintion that says we can get as close as we like to some number (viz. the limit). (For any epsilon there exists a delta such that, for all x, if 0 < |x - c| < delta, then |f(x) - L| < epsilon). So, that's why it seems that the element of choice is also in the "inequality defintion"...because the two are the same.
But, what I want to point out is that in the now accepted non-standard treatment doesn't use any inequalities which, I think , would eliminate the choice that Steve is talking about. The reson that "Steve's choice" is there has, I think, more to do with inequalities than some intrinsic feature of Calculus. Inequality solution sets have, most often, more than one element. So, in effect, we could choose any element in the solution set and get a right answer. I think the Reimann sums are more of the same. The choosing of the rectangle width is more less "geting as close as you want."
(the inequality business has actually been the standard for about 200 years or so, with the work of Bolzano and Cauchy being "arithmetized (i.e., formalized) by Weistress)
However, non-standard analysis defines a system of hyper-reals and then sets the limit EQUAL to the function evaluated with a "non-standard" part. So, if it's plausible that the "choice" in Steve's post is an artifact of the inequalities (as opposed to intrinsic to Calculus), then I'm not sure how this "choice" is different from the conventionalist's "choice of system." But, if it is something differnt, something connected with that particulr way of defining limits, then something needs to be said for the apparent lack of a similar choice in the non-Standard analysis.
I think there are actually 2 levels of choice here. One would be the choice of a system. I can choose either "epsilon-delta" or "non-standard" approaches to resolving my Calculus homework. But this doens't seem to have much to do with the math itself. Then there would be choices within a system which would be constrained by the math itself. I think Steve's choice might fall here. Once I decide to use epsilon-delta calculus, then the inequalities give rise to this "choice." But If I choose to use non-standard analysis, then it's gone (and I don't, at present, see where it may be hidden).
By the way, the business of indeterminacy and uncertainty will enter. I tipped my hand a bit early. Give me some time to catch up.
Scott
P.S. Incidentally, I think there may be some relation here and to his earlier posts about indeterminacy (the stuff about arrows and not being able to know where you started). But I'll need to re-read those posts to make the connection more lucid to myself.
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