Sunday, January 13, 2019

Everybody Loves Raymond...

I regularly re-run my favorite old Raymond Smullyan puzzle (that actually goes back to "Annals of the New York Academy of Sciences," 1979, Vol. 321, although my version is an adaptation from Martin Gardner's presentation in his Colossal Book of Mathematics), and recently realized I failed to do so in 2018, so may as well remedy that now. Skip over if you've seen it before, but down below I have newly-tacked on a video tribute to Raymond from a prior 'Gathering For Gardner' Celebration:

Imagine you have access to an infinite supply of ping pong balls, each of which bears a positive integer label on it, which is its 'rank.' And for EVERY integer there are an INFINITE number of such balls available; i.e. an infinite no. of "#1" balls, an infinite no. of "#523" balls, an infinite no. of "#1,356,729" balls, etc. etc. etc. You also have a box that contains some FINITE number of these very same-type balls. You have as a goal to empty out that box, given the following procedure:

You get to remove one ball at a time from the finite box, but once you remove it, you must replace it with any finite no. of your choice of balls of 'lesser' rank (from the infinite supply box). Thus you can take out a ball labelled (or ranked) #768, and you could replace it with 27 million balls labelled, say #563 or #767 or #5 if you so desired, just as a few examples. The sole exceptions are the #1 balls, because obviously there are no 'ranks' below one, so there are NO replacements for a #1 ball.

Is it possible to empty out the box in a finite no. of steps??? OR, posing the question in reverse, as Martin Gardner does: "Can you not prolong the emptying of the box forever?" And then his answer: "Incredible as it seems at first, there is NO WAY to avoid completing the task" (emptying out the box). [bold added]
Although completion of the task is "unbounded" (there is no way to predict the number of steps needed to complete it, and indeed it could be a VERRRRY large number), the box MUST empty out within a finite number of steps!
This amazing result only requires logical induction to see the general reasoning involved:

Once there are only #1 balls left in the box you simply discard them one by one (no replacement allowed) until the box is empty -- that's a given. In the simplest case we can start with only #2 and #1 balls in the box. Every time you remove a #2 ball, you can ONLY replace it with a #1s, thus at some point (it could take a long time, but it must come) ONLY #1 balls will remain, and then essentially the task is over.
S'pose we start with just #1, #2, and #3 balls in the box... Every time a #3 ball is tossed, it can only be replaced with  #1 or #2 balls. Eventually, inevitably, we will be back to the #1 and #2 only scenario (all #3 balls having been removed), and we already know that situation must then terminate.
The same logic applies no matter how high up you go (you will always at some point run out of the very 'highest-ranked' balls and then be working on the next rank until they run out, and then the next, and then the next...); eventually you will of necessity work your way back to the state of just #1 and #2 balls, which then convert to just #1 balls and game over (even if you remove ALL the #1 and #2 balls first, you will eventually work back and be using them as replacements).

Of course no human being could live long enough to actually carry out such a procedure, but the process must nonetheless, amazingly, conclude after some mathematically finite no. of steps. Incredible! (a pity Cantor isn't around to appreciate this intuition-defying, ‘see-it-but-can-hardly-believe-it’ puzzle).

Friday, January 11, 2019

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

(Sunday, my favorite Raymond Smullyan puzzle post goes up here again as a re-run, so one last call for any fuzzy logic info per request in prior post.)

Saturday, January 5, 2019


Request For Fuzzy Information….

via HERE
"...the pervasiveness of fuzziness in human thought processes suggests that much of the logic behind human reasoning is not the traditional 2-valued or even multivalued logic, but a logic with fuzzy truths, fuzzy connectives, and fuzzy rules of inference. — Lofti Zadeh
What makes society turn is science, and the language of science is math, and the structure of math is logic, and the bedrock of logic is Aristotle, and that’s what goes out with fuzzy [logic]
— Bart Kosko
I’m essentially a Non-Aristotelian, which is to say I don’t put much faith in the Law of Non-contradiction, Law of the Excluded Middle, nor even the Law of Identity — nice, rough approximations useful in the macro day-to-day world, but probably not accurate representations of reality (especially where human language is involved), and moreover giving rise to many of society’s problems! Much like Newtonian mechanics functioning as a useful tool for the macro world we experience, but not as a precise description of physical/quantum reality.

One of the things I found attractive about “General Semantics” 5 decades ago was the anti-Aristotelianism of its founder Alfred Korzybski. I’ll skip over that history though because Korzybski can be a bit of a turgid, impenetrable read. But here are some easily-accessible reads from a popularizer that introduce G.S.:

I raise the subject at all because I recently read the 25-year-old volume “Fuzzy Logic” by Daniel McNeill and Paul Freiberger, having previously read Bart Kosko’s “Fuzzy Thinking” — these are two of the vintage (1990s) popular books on “fuzzy logic,” a multi-valued logic that moves away from binary Aristotelianism or classical black-and-white logic. In some sense fuzzy logic puts general semantics onto a firmer footing, and is surely a better representation of how the world works than binary logic.
If you’re completely unfamiliar with fuzzy logic here are a couple of handy Web links (the McNeill/Freiberger book is very good also):

The founder of fuzzy logic was an award-winning researcher Lotfi Zadeh who actually delivered the 1994 Alfred Korzybski Memorial Lecture:
(interestingly, many of F.L.'s early proponents were foreign-born/non-American, and F.L. was developed more in other nations before it was in the U.S.)

A related concept that may also be worth exploring is Eleanor Rosch’s “prototype theory” for how we cognitively categorize things (i.e. there are a lot of gradations and overlaps, not simple clean categories). In fact, fuzzy logic also relates back to Sorites paradoxes that I've written about previously here. In short, the world is full of gradations and vagueness, yet as a practical matter we treat it as much more discrete.

One thing the McNeill book makes clear is that fuzzy logic caught on much faster (being built into appliances, manufacturing, control systems, etc.) in Japan than in America where it faced a lot of opposition. Some find the very term ‘fuzzy logic’ to be uncomfortable if not oxymoronic; personally, I like it because it immediately gets at what is wrong with classical logic, namely that even symbolic logic derives from words and words ARE inherently ambiguous or fuzzy; we need a multi-valued logic that deals with that, not that largely evades it, pretending, for example that all statements are true or false (“true” and “false” are themselves ambiguous terms). There are very deep unacknowledged problems in traditional syllogisms like the following (even if you convert it to pure symbols):
All men are mortal
Socrates is a man
Therefore Socrates is mortal

(All these terms, “all,” “men,” “mortal,” “Socrates” need be precisely defined, but cannot be — we are simply so accustomed to employing and interpreting language that we bypass the given vagueness and imprecision). We are like a fish in water that is oblivious to the wetness.
By the way, as an interesting side-bar, here are a couple of syllogisms the McNeill book used to make a point:

All oak trees have acorns.
This tree has acorns.
This tree is an oak tree.


All pro basketball players are very tall.
Bob is very tall.
Bob is a pro basketball player.

Of course the readers of this blog are so brilliant they probably didn’t fall for it ;)  …or did you?

Both syllogisms are logically identical and FALSE, but generally speaking, more people will blunder with the first one and think it true because contextual knowledge/cues leads them astray.

If you’re not familiar with fuzzy logic, in simplest form it merely says that many statements don’t neatly fall into true or false categories. Is ‘John is tall’ a true statement? Depends of course how one defines “tall.” If we say it means 5’10” or above and John is 5’9.995” then do we round up or conclude that John is not tall? What if John is a woman; do we have a different definition of “tall” for a woman? Heck, what if John is a duck, suddenly “tall” needs an entirely different criteria. Or, if John is 5’11” and thus tall, at what moment did he “become” tall having previously been say 5’4” at some point in life? And what if someone is 6’10” — is he still tall, or is he now ‘very tall,’ or some other new category? It’s all complicated. Context counts. Yet most people hearing “John is tall” think they're hearing a simple declarative, meaningful sentence, despite all that fuzziness. In fuzzy logic, a statement is assigned a ‘value’ somewhere between 0 and 1, rather than the simple 0s and 1s of traditional logic (i.e., John might be viewed as 0.85 tall when all the contextual variables are factored in).

The McNeill volume has an interesting chapter comparing/contrasting Bayesian probability (which has gained even further favor since the volume was written) with fuzzy logic, and another interesting chapter on polymath Bart Kosko one of the major proponents of fuzzy logic. By the way, here are a couple of my favorite Kosko pieces from the Edge where he doesn’t specifically mention fuzzy logic but does criticize other statistical approaches:
My limited reading indicates ongoing controversy or conflict between Bayesian thinking (which of course is booming these days) and fuzzy logic that hasn’t taken off as much (there’s also fuzzy set theory, fuzzy systems, fuzzy probability, and other subtleties). Here are some old posts (with plenty of comments) by Mark Chu-Carroll trying to sort it out a bit.

Further, the McNeill book discusses hostility between AI theorists and fuzzy logic proponents, and again I'm not certain how much, if at all, that has changed in recent years.  Same for friction between neural network supporters and fuzzy theorists. Some of the discussion reminds me of the early dominance of behavioral or Skinnerian psychology in learning theory, which worked fairly well for pigeons, but ran into major difficulties explaining or duplicating human behavior. A century from now will a lot of AI theory/techniques seem as primitive as Skinnerian pigeon conditioning does?
Without ever bringing up fuzzy logic, a recent George Dyson piece ("Childhood's End") touches on the inadequacy of binary, digital approaches/computing, and predicts the ascent of analog computing:

What I don’t quite get from all this is a good feel for where fuzzy logic stands today in both the academic and applied world.
Sooooo, I’d be interested to hear from anyone more directly involved in it who can comment on the place of fuzzy logic these days. I could interview someone on the subject (sending out a set of questions), or someone can say whatever they wish (succinctly) in the comments, or even send me a longer piece as a guest post. IF there are any takers, here are some of the things I’m wondering:

1)  How much is fuzzy logic being applied in manufacturing, engineering, computer science/programming, and the like these days? I would think that fuzzy logic would be a necessity for the complex control of something like driverless cars -- is that a fair guess? What about speech-recognition programs or AlphaZero or medical diagnosis programs; how much fuzzy logic there?
2)  How widely available are classroom courses in fuzzy logic now?
3)  What are some good introductory books on the subject you would recommend to laypersons? And how about textbooks for the more academically-inclined?
4)  What can you say comparatively about the use of fuzzy logic in other countries (China, Japan, Russia, Europe…) versus the U.S.?
5)  What can be said about the current popularity (and any similarities) of Bayesian techniques and probability versus fuzzy logic? (I’ve seen differing accounts, that they compete or overlap, or even that fuzzy logic subsumes Bayesianism???)
6) What can one say about differences between fuzzy logic and other forms of multi-valued logic?


I'll end quoting these words from the end of the (1993) McNeill book:
"...Western civilization has overcome biases inherited from Aristotle in the past, and without the economic goad. And fuzzy logic is practical in the highest sense: direct, inexpensive, bountiful.  It forsakes not precision, but pointless precision. It abandons an either/or hairline that never existed and brightens technology at the cost of a tiny blur. It is neither a dream like AI nor a dead end, a little trick for washers and cameras. It is here today, and no matter what the brand name on the label, it will probably be here tomorrow."

Friday, January 4, 2019

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

(Sunday, will have a longish post up here concerning "fuzzy logic")
[aaaack!, by mistake this got posted earlier on Saturday :( ]

Tuesday, January 1, 2019

I Hereby Resolve…..

                 Happ Ne Year!  2019

Well, it’s January, and it’s a tradition, sooooo,
  I resolve to:

1)  start acting my age (chronological, not mental).

2)  sleep more, exercise more, and eat more salad, but not all at the same time.

3)  drink NO more coffee after 3pm. in the afternoon… unless perchance I catch its seductive aroma wafting through the air.

4)  NOT do in real life what I incessantly daydream of doing to Donald Trump.

5)  consume fewer carbs (…er, uhhh, wait, is chocolate a carb???).

6)  dance like no one is watching (…or, perhaps in my case, watching and puking).

7)  work more on my abs and less on the Riemann Hypothesis.

8)  floss more (….hahahaaaa, that one’s just a joke, you know, for all the dental students who are devotees of this blog).

9)  remain awed and amazed by prime numbers, paradoxes, recursion, and the minds of mathematicians unraveling it all!

....and lastly,