Monday, August 5, 2013

Mathematics I

Context: See Tru'eng anewfocus going forwardmathematics.



Our collective friend, Hitchens, said many things that we ought to think about now and then. And, based upon the websites that are quoting Hitchens, there are many who are doing just that. One of his quotes actually made it to the TED (deals with current, and future, technological issues) site.

In short (paraphrased), assertions without evidence can be dismissed without evidence. When I first saw that quote being reported in some web page from some source (actually, Google brings up several sites - but, my first awareness was several years ago), I felt that I would have responded to him, something like this (had I the quick wit under the spotlight and were in his presence): like axioms? You see, Hitchens was many times pushed beyond the point where he said that he didn't know. Yes, he said "didn't know" many times; but, either given his inherent talent at showmanship or mere frustration at putting idiots in a bottle or whatever other motive, Hitchens would go off on his erudition-showing response (sometimes, these were rants - however, well considered and stated).


Aside: Some attribute the quote to Richard Dawkins who made a similar statement. Wikipedia points to what is known as Hitchens' Razor with references. And, it's really a re-phrasing of an age-old slogan.


A constructive look:

Hitchens did not deal with mathematics, but he did respect results that were founded upon such. You see, that would be most of modern scientific models ("most" is arguable; however, we're talking 2000 plus and the fact that about every discipline has mathematized its knowledge base - many times to a large degree -- I ask, what discipline with any modicum of a quantitative does not use advanced statistical methods?). So, who of the modern domains has not used mathematics?

It turns out that there may be many who are not so learned (not meant pejoratively). As, this revolves around a central issue related to numeracy, or lack thereof, that has insidious implications for the future. We'll get back to that. Expect though, for me to show that numeracy limits us, many ways (not arguing against the great ideas founded upon numbers and the computational -- rather, consider, can our artifacts subsume Being?).

The basics:

We'll first have to deal with what exactly is this thing called "mathematics?" We can start with a few little pointers here and branch out. Please note, this approach is pseudo-constructive (albeit, not to the level of Bourbaki - yet, effective in its own way, as we will show).
  1. An encyclopedia is a good place to start: Wolfram's MathWorld comes to mind. I have watched this grow (and, even, to a small extent, contributed). As well, the Wikipedia view is phenomenal (actually, what I love about Wiki is that I can look at something (that is well-sourced, usually) without running down myriads papers and books). By studying this type of material, one could get a fairly good idea of what mathematics is, though several issues, such as motivation, would not be as apparent as could be. As Plato thought, arithmetic can be one of the starting points for mathematical education. Geometry, and its use in our world, is the next step. 
  2. We can also take a behavioral view and say this: mathematics is what mathematicians do (this little thing out of St. Andrews is wonderful - please look at the Mathematician of the day page). If we cared to study all that has been done, then we might have some idea of what this field encompasses. But, much of mathematics is mental (and, we don't normally expect people to read minds). One may see symbols and numbers, yet, their use is not always intuitively obvious. On the other hand, we can see what has been labelled under the concept, even if future perturbations, and extensions, would not be seen using this descriptive approach. However, there is one little thing that lurks: is mathematics more than what people do (next bullet)?  
  3. Ah, metaphysics coming in? Let's keep it simple, as we could discuss, for a long time, what might be called the differences between big "M" (Mathematics) and little "m" (mathematics). How can we circumvent that? You see, if you look at MathWorld (or what Mathematica does), you'll see all sorts of operative marvels (basis of STEM) with results that astound (our technological age) when looked at closely. Yet, just ask yourself if the reality behind whatever is being expressed (or computed or manipulated) by our artifact (assuming small m, okay?) is equivalent to what we understand via our use of the artifact? Another way to think of it is to consider the map-territory problem. First, extend "map" to encompass the sum total of our prowess with mathematics. The territory? Yes, we would really like to know (but it's not the former). In short, we tend to get these two merged (map-territory) as there seems to be no real way to do the differentiation (ability to note differences) many times (as in, that particular action is underdetermined - note, please, the poster is very much aware of proof-theoretic powers - yet, the poster sees the importance of being aware of quasi-empiricism and its issues).   
  4. Is mathematics that which can be built from axioms and proofs? We all know that mathematics involves theorems. Some of these are applicable to our lives; others are beyond the comprehension of most (if you listen to von Neumann, he would say beyond the comprehension of any - see first quote). Now, there are many ways to argue proof methods, but logic is involved. Then, that brings in knowledge and how we know. Also, we need to consider Russell's and Whitehead's efforts to put a firm basis under logic. It took hundreds of pages to get to proving (by terse equations) simple addition which we can teach early on to the human mind. Too, Russell, and later work, brought in computablilty issues which may or may not be problematic to the future of automation (ah, letting the cat out of the bag - the web is laden, heavily with pseudo-mathematical pursuits what need our scrutiny).           
Now, that listing is not trivial. But, the list does include the important things and has a progression that we'll be looking at in depth (basis for truth engineering's necessity and operational look).

  1. What? We know that we have mathematics and its use; such is abundantly evident. Much work in mathematics has been done over the years. The web has allowed an acceleration of means to get access to information about mathematics and provides a common platform for discussion and use. Marvelous techniques exist. The trend seems to be toward automation of some of these. The truth, folks, is that higher-order computation (as in advanced mathematics) requires, generally, human involvement in two, and perhaps more, areas. One is input (see the qualification problem as an example -- numeric processes have oodles of decision states prior to execution - in other words, setting up a solution attempt is a creative task). Then, we have output issues dealing with more than interpretation. For instance, ramification is important. 
  2. Who? We see people doing mathematics. Only some of those are involved with computation. But, certain classes of problems are computationally oriented and will continue to be so as technological progress improves computational artifacts and methods (of a very wide variety). That there is motivation to learn and do mathematics seems obvious, too. We have not seen a decline in the interest. More later. 
  3. Where? As we use mathematics, the larger picture does not go away. But, we'll defer all of that til later. 
  4. How? As, there is controversy about hubris and its appearance. It's almost impossible to know this up-front. In retrospect, we see it, or think that we do, many times. Unfortunately, right now, we have the finance community running off after multitudes of seemingly smart approaches; we need to push some type of sand box (or, at least, now, some notion thereof) into the awareness of these, folks; this we would like to demonstrate.

As stated at the start, there was some nod to Hitchens in this post albeit we are dealing with mathematics. The content here takes a broad view and tries to identify the major pieces. Getting into all of the nuances will be on-going, albeit the looks will be more broad than deep (for awhile, at least - at some point, we'll have to take on a few of the more important points in a technical manner).  


06/25/2015 -- ACM Communications had an article (Created Computed Universe) that suggest that our computional prowess ought to lead to agnosticism rather than to anything else. Of course, my initial remark: So many modern minds conjure and contort in order to introduce what is not much different than what some knew many millennia ago in the desert.

01/23/2015 -- Software? Well, we are talking more than apps (latest craze). We are dealing with fundamental questions which, then, gives rise to normative issues in mathematics (and, by extension, to the computational).

01/05/2015 -- Added context line at top. We're at a renew point.

03/03/2014 -- Acknowledgements, including math pedigree, will be expanded.

08/06/2013 -- Investigative journalism likes its Five Ws. That is fine for their topical views. To get technical, we have to add in, at least, an H (How?). Too, we need to change the order, according to various factors: domain, focal area, and more.

So, the order of the above list is "What?, Who?, Where?, How?."  There are other orders. For instance, "How?" have could come before "Where?" in the list. That, though, would have changed the emphasis (recall, truth engineering is the overarching theme).

We're ignoring Why? (why not?) and When? (ah, history of mathematics -- the future, too?).

08/05/2013 -- In this interview, Hitchens uses "horribly un-reflective" (around 12:00) in response to his Iraq stance's outcome (turmoil amongst those who wondered where he was coming from?) and whether he had, or not, tempered the position that he taken at the time. So he admitted to being reflective, which we would all have known anyway. As said above, lots of his positions were forced by reacting to idiots and their attempts to master him.

Modified: 06/25/2015

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