I missed the whole month of Might, for several reasons. But, Knowledge (the month of the Bab) is right around the corner, so we'll resume. After that, the Motivational comes to fore.
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When looking at the popular posts, we have the following order:
- Technical spirit (Sept 2, 2009) -- Motivated by articles in the ACM Communications. Turns out that this post has a similar sense (more below).
- We are not idempotent? (May 9, 2012) -- The concept can be applied to many situational themes, especially those related to the strangling mesh of a mathematical/computational origin that is being put over human kind. By whom? Those who have the wherewithal to do so. Now, the concern is not the reality of this type of thing with its being, naturally (and supernaturally), part of human nature. The problem is that Independent Investigation of the Truth applies here, with more connotative emphasis than some might have thought (way beyond the religious-based usages). Too, those doing IIofT must attune their heads and hearts, as needed. Part of this would be to properly filter processing of "residues" (see below).
- Christopher Hitchens (Jan 21, 2012) -- He was of comprehension, more than most. The lesson is that we don't comprehend (as said von Neumann) many things; yet, those who build constructively (even though they still allow the use of infinity) do not grasp the underlying issue. How can they when the universe seems to be reflecting right back to them what they are looking for? (see below)
- Audacious comment (Aug 3, 2013) -- Along the line of the prior bullet, on the BF&S page on Wikipedia, there is a listing of notions that seems to have appeared firstly in the Writings. The list ought to be (and will be, at some point) much longer. In fact, creative urges, unfounded otherwise, will result from the Words. However, one discussion, which is an example of how things can go awry, considered whether the Writings could be used logically (as in, from one view, embedded within the Writings, some string being able to support a reasoning chain). Well, given that humans are, in part, rationally based, the answer is yes. However, the modern trend is to discount that which cannot be Turing-based (not as hopeless as it might seem - see below).
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Now, the recent Communications (see first bullet) had an article that applies to the themes behind the blog. The article (by Mark Braverman, of Princeton University, "Computing with Real Numbers, from Archimedes to Turing and Beyond") is considered premium and is not available on-line (so, run down to the nearest library).
Mark uses the progression from the natural numbers upward to illustrate his points. We all have an intuitive sense for these numbers. The rationals may be a little more problematic. Some feel that they ought to be introduced earlier in the educational process. Going further, one gets into the reals and the more involved number systems, such as complex numbers.
As an aside, one thing that we see with the earlier types is a close correspondence between the thing and how we name it. For instance, for any lower-level number, we can easily say it and even demonstrate it with objects. Even with the more complicated numbers, there are times when we see "nice" ways to handle things. Mark uses an example of a simple harmonic oscillator. In fact, one finds closed-form opportunities in many places. Prior to computation's emergence last century, humans did quite well this way.
The point, so far? Names and Attributes are one thing that we'll continue to bring up. Too, computation brings in complexity; approximations allow us some leeway; yet, at the core, there are the lessons from the Valleys (underdetermination, of sorts).
Mark gets into more complex notions that result from trying to model natural processes (and God's creation). Too, in order to get beyond undecidability and its influence, we have seen many creative schemes applied. For instance, with the spread of computers and systems, all sorts of algorithms exist that allow people to compute, and display, Julia sets which are great for illustrating issues related to convergence and its opposite.
A key notion deals with equivalence which is a very hard problem that has been, in part, conquered. One might say that dropping the whole notion has been the operational mode (it was phrased this way, as, other things have undergone equivalent (yes) parsing - all the way up to ...).
After Mark discusses how bringing in parameters allows further control including the continued existence of problematic issues, he suggests that allowing for noise can help reduce things to a tractable state. In a sense, one might see this as a parallel seen in the success of fuzzy methods.
But, then we get back to probabilistic issues which come from another use of numbers. So, computation is hard; too, though, it ought to be instructive to mathematics in a quasi-empirical sense. Perhaps, that hardness can be, in the long term, indicative of the limits that we need to understand. That is, being cannot be encompassed by this means (or, by any other of our way); but, given its growing usefulness, modelling can be considered essential to our future.
So, we're at a type of cusp. Describing this seems to be in order; perhaps, we'll get there at some point.
Remarks:
09/30/2013 --
Modified: 09/30/2013
