Saturday, August 4, 2012

Weakness in modern pure mathematics

Several posts, of late, have talked about the need for a re-look at technical issues. This will continue in a, hopefully, coherent fashion with measurable effectiveness (being cheeky).
All along, there has been a suggestion here that things are awry because of foundation'al issues. And, this is because of more than the concerns about quasi-empiricism. These issues go way back.


There is a chain, folks, that can explain (partly) the source of the problem. Dawkins, in fact, has used the argument. In the sense of science, let's put it this way (paraphrase) -- the biologist talks to the physicist (and chemist), the physicist talks to the mathematician, and to whom talks the mathematician? God. The adage is age-old but worthy of attention.

Aside: What goes along with this is that only some of a cohort set are the ones worthy (of value) enough to push back the horizon. This was fine when things were 'pure' and simple-minded (yes, people, the classic joke of the unaware thinker -- do we really need that?). But, given those considered worthy (yet, they have feet of clay as do the rest) a computer and the web and watch out (yes, people, the crap that we see now is of this ilk -- how did reasoning, smart folks allow such a bad state to develop? - Harvard isn't off the hook - yet, as said before, the theory of multiple intelligence comes from that institution -- so, numeracy (as in, applications of mathematics -- all types) is not the epitome (STEM, a misdirection, of sorts) by any means -- well-rounded-ness? ever heard of that?).


Aside: In several posts, there may be use of undecidability and quasi-empirical in the context of computational issues of note. The discussion of this topic will continue under the context of Computability in all of the related blogs.


So, the quakings that we see (covered in postings) do deal with mathematics, with those who do it, and with its applications. We will continue to go on about that as the growing computational frameworks are troublesome in very many ways. As I've said, there is no timeline that I'm adhering to (as of yet). We'll follow things to wherever facts and features lead.

In that vein, I just ran across a lecture that is saying similar things, essentially. I'm going to be listening to this and related discussions. The lecture is titled Logical weakness in modern pure mathematics (see on youtube) and is given by Prof. Wildberger at UNSW (New South Wales). What caught my eye was that on the first few frames of Part II we find this list of problems:
  • Inconsistent rigour
  • Problematic definitions
  • Reliance on 'axioms'
  • Computationally weak
  • Impoverished examples
Then, the Prof shows a partitioning into the troublesome areas and those not so much. The list includes: calculus/analysis, set theory/logic, geometry/topology, probability/measure theory and parts of algebraic geometry.


Logical weakness in modern pure mathematics
Aside: The lecture topic has 'pure' in it which study does not generally include applications (that's the whole point, removal via abstraction from anything of consequence). But, the Prof mentions  early on that we need to have examples in order to provide a basis that is more solid (paraphrase - pun intended, to boot). As well, though, the argument here is that those who apply are resting themselves, many times, on what the theorists tell them. In a sense, we cannot have pure without its mappings to usage (I know, philosophically arguable) if we are to overcome the weaknesses alluded to by the Prof. Being cheeky, I feel that a peripatetic approach is required, albeit virtually founded. This type of thing is very much in the interest of the techies to pursue (rather than bigger pockets).


This post is a marker as I expect things to get interesting as I follow his arguments. It's real nice to run across this. Why? Many times the problems are as I've mentioned here. Teach a manager a little math and watch out. They'll run amok. Lots of what we see on the web (yes, Google, you, too, mathematicians as you claim to be) is this type of thing. I marvel at how far computational methods have been pushed despite the fact that they have the slimmest of supporting theory.

You question this? Ah, open up your code and let us see.

In another realm, we have program trading. This is the highest type of idiocy possible, even if those propagating the madness have their Doctorates (sheesh, you guys/gals, ever consider foundations?).


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.

08/08/02012 -- On effectiveness. It's there, and we take advantage of it (even if we have no clue as to its origin -- and let hubris reign more than humility). NASA is a very good example with Curiosity Rover. But, the recent Russian failure to launch a satellite says that things can go awry at any time despite good efforts. NASA has had its failures, too. The main issues are at the boundaries (to be discussed) and when extrapolations go way beyond what the basis will support (many examples, we'll get there). But, as the Prof showed with his rational model, we can extend, indefinitely, twixt two rationals. So, the boundary (reference) above has, at least, a dual meaning.

Aside: The Prof mentions several times his antipathy to thoughts about the infinite and how we may have conquered it (ala Cantor's work, et al). The Prof's approach can go toward the very large, and small, yet it would not be a convergence (another concept that he doesn't like). What would be a good term to use?

08/07/2012 -- MF77 deals with object oriented vs expression oriented. The former has some relations with category theory, etc. The latter one might think of as 'code' related. We can think of demonstrating something computationally rather than jawboning (it's good to hear that this is what mathematicians do). The Prof seems to be pushing computer orientation which raise other issues, but we'll keep listening. If he can keep computers from becoming an even bigger mystery, then all would be well. Otherwise, we're talking the development of a monolith, in a certain sense, that would become troublesome, indeed.

08/06/2012 -- We'll jump to the meat and go through the sequence on Rational Trigonometry. His book and papers are available at the UNSW site.

08/04/2012 -- MF3, at 7:38, shows why 'New Math' failed.

08/04/2012 -- Just went through MF1 and MF2. In essence, the Prof is establishing a foundations of mathematics that is amenable to anyone. The analog? Think about the intelligence tests that are supposedly without any cultural bias. The Prof's approach ought to end with a mathematics that will be accessible to any person who wants to make the effort to learn it, not just the elite class. The consequence: we may find discoveries far beyond our imagination since cultural biases are the most limiting especially for those who are of the advanced types whose talent causes them to be forced into (or reinforced toward) the highest (purported, remember?) cultural status.

08/04/2012 -- The first video of the series (count as of now, 101). As of today, the last video.

At 22:56 of MF87
08/04/2012 -- Will do some notes here as we go along. On finishing up MF87, it was nice to hear him say that the youngsters can smell when things are awry. They may not be able to put their finger on what is the problem, but that there is a problem is, somewhat, obvious to them. I'm not going to follow the lectures sequentially but will bounce around by interest and applicability to my work. Notes will be here; there will be occasional summaries posted with comments about how it all applies to the topic of this blog (and its associates).

Aside: von Neumann said that we can't understand mathematics but get used to it. So, that's by use, repetition, etc. Is the Prof arguing the Hilbert side?

Not being critical at this early stage, as I want to hear what he has to say in its entirety. See 22:56 (image), which is right on!

Modified: 06/25/2015

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