Thursday, November 16, 2006

Mathematical Thanks

Having used up nearly my whole day fighting a rather intractable problem for work, I now cannot find my notes on what I wished to write about. I can tell you that it was going to be something about Logic, and I hope to deal with it sometime in the future. Logic, after all, is required for philosophy, and also computing - and is as basic to both as addition - or reading.

But since there is just one more week to go before the day we Americans spend in Giving Thanks To God, it seems good (and logical!) to contemplate the mathematics of thanksgiving.

Let us, then, turn to our collection of GKC and read this short little technical prayer called:

I cannot count the pebbles in the brook.
Well hath He spoken: 'Swear not by thy head,
Thou knowest not the hairs,' though He, we read,
Writes that wild number in His own strange book.

I cannot count the sands or search the seas,
Death cometh, and I leave so much untrod.
Grant my immortal aureole, O my God,
And I will name the leaves upon the trees.

In heaven I shall stand on gold and glass,
Still brooding earth's arithmetic to spell;
Or see the fading of the fires of hell
Ere I have thanked my God for all the grass.
[GKC Collected Poems CW10:209]

Naming leaves on trees - wow, I almost forgot to call your attention to this line! We actually do this in computing, and things like the IP addresses of the INTERNET and even the telephone numbers use the technique - we always start with a "root" name, then at every branch, add on the name of the branch as a suffix to the name we've already gotten - sooner or later we get to the leaf , and we have its name!

Even stranger is the idea of counting, not stars or sand, but hairs of one's head: See Matthew 10:30 - "But the very hairs of your head are all numbered." If God tracks such trivial details as this, how much more, then, will God care for us in the things that matter - especially when we say, as the priest does in each Holy Mass: "Gratias agamus Domino Deo nostro" = "Let us give thanks to the Lord our God." Let us all respond "Dignum et justum est" = "It is right and just" - today and every day.


  1. Dr. Thursday,

    Interesting food for thought. Chesterton admits that there are limits to our human abilities:

    “I cannot count the pebbles in the brook.”

    Yet, God can, and apparently does, count everything in the universe, including the last grain of sand and, presumably, the tiniest sub-atomic particle (Or does He count each and every sub-atomic particle and quantum?):

    "But the very hairs of your head are all numbered."

    This leads to all sorts of interesting conclusions, such as

    1) Our universe is of discrete nature; yet

    2) As humans we cannot perform all these “calculations”, we are limited in how we have to think about things. When we count, we can count only so much, before we start loosing the meaning of the counted thing. For example, most people start loosinng meaning, or don’t understand the implications, when numbers get very large or very small.

    One could even formulate a law - The larger/smaller the counted number is, the less meaning it carries for us humans. Perhaps that is why we have 10 fingers to count with. And the corollary - The infinite is totally mysterious or incomprehensible to humans.

    3) Among other paradoxes of nature, we are caught in the discreteness vs. continuity paradox. This affects how we reason about things. We either think using discrete models or “analog” or “continuous” models.

    A good example, for those who are old enough to remember them - we can have either a digital or a “discretely counting” computers, but there used to be also “analog” computers. But the key question still remains - At the lowest “quantum” level, do even the analog computers work discretely? Or, is God or an angel able to perceive and “name” each and every quantum as the calculation on an analog computer is being performed?

    Wild Goose


    “There are 10 kinds of people, those who understand binary, and those who don’t.”

  2. "...GKC admits there are limits..." Admits? Rather, I would say that he brags that in this life we are limited - it's Thomistic of him as well as scientific!

    However, the whole point of the second verse of his poem is how we shall transcend that limit, and be able to count everything properly. (Ooh, I get thrills every time I read this.) It's such an elegant demonstration of... well, I'd better not get too technical here, but in an advanced programming class I had to do something like that naming of the leaves on an infinite tree... wow, what a thrill (or horror), and a foretaste of eternity.

    And the word eternity brings up the topic of the knowledge of God: not only does He know all conceivable physical values, but also all the integers, and all the reals, and all the curves in the plane, etc. since He knows everything. It's not possible for there to be something He does not know. His knowledge is direct, because He knows their being, and not through measurements. (Wow, a new term for Heaven: the place where ontology and epistemology meet.)

    Speaking of meetings - you bring up the "paradox" between discrete and continuous: Physicists like to talk about a ramp beside a staircase - but the first step of the staircase might be the square root of two, an irrational number, though easy enough to construct. The fact that energy is packaged in quanta does not constitute a "discrete" (integral) measure - for all we know, every quantum is a trancendental - which somehow would make a real nice joke on God's part. Hee hee.

    Also, about analog computers working discretely at the quantum level - that does not thereby convert them to digital - it just makes them intractable. For one thing, typical analog computers do not "count", though it is easy enough to make such things, and they do work perfectly according to their design. (I'll post instructions for one on my own blogg one of these days.) But the electronic forms of analog computers perform work in higher mathematics, like integration or solution of differential equations. Their computations are accurate to within the precision at which they are built; the interesting issue of course is in our ability to read what result they have found! (They don't come with printers, you see, nor CRTs, at least not character-generators as these are.)

    Finally, the "infinite" is not totally mysterious or incomprehensible - as an idea. Golly. Mathematicians deal with it in quite a number of different forms, and in fact one needs to be able to handle it well if one wants to approach certain problems... for example, what IS the tangent of pi over two? Or, somewhat simpler, what is one over zero? There's a quick "dirty" answer, and a longer precise one - and it is not incomprehensible. Just draw a straight line that goes up-and-down - and you have it! Strange to say, even some very intelligent people have gone quite wrong on this, claiming that infinity cannot be represented on a computer. That's just missing the point, as you can see from the previous sentence. Then there's the very interesting idea called "closure of the free monoid over a fixed alphabet" - another form of infinity, but now I AM getting technical, but I do hope to write about that too, and soon, because it's important.

    So - some very interesting questions you have raised, especially about analog machines. I have an interesting book on - er - well, I'll say "programming" them, but that is quite misleading. In any case, I hope this helps our readers a little.

    By the way - do you think I've used enough puns in this comment? Well, bit by bit, I work at it, whether by analogy, or discretely. Hee hee.

  3. Dr. Thursday,

    I don’t mean to be too argumentative, because most what we are discussing is a grey area, or rather the hairs we are trying to split are too fine. So, knowing my limits, (here is my pun :-), and admitting that there is a lot of speculation involved, I have no real problem with anything you are saying here.

    This whole idea with quanta and discrete occurred to me some time ago when I was thinking about how God or angels could interfere with natural processes and change the course of nature if they wished to do it in such a way that it would be undetectable by us humans. I think this is how the universe was designed on purpose, and how it could work.

    I am aware of GKC’s fondness of limits, but I am not sure if the word “bragging” is appropriate. The frame isn’t more important than the picture, although it is an important, or even essential, part of any picture. In practical terms, we need the frame, or picture limits, before we even start painting.

    We don’t admire the frame, but Mona Lisa. We only, perhaps “cheerfully” accept the necessity of limits, but only if we realize why they exist. Besides, limits are not as obvious in other art forms, be it music, sculptures, architecture, or mathematics.

    About God - I am familiar with St. Thomas’ Summa, especially with the beginning where he talks about God. Still, if “it is not possible for there to be something He does not know,” such a statement is arrived at only through our limited knowledge of Him, so it is like circular reasoning of a limited human mind about something unlimited or unbounded. (Which was the original Greek definition of infinity/eternity, or apeiron.) We can reason about it, but do we really know? Never mind the physicist, (who shouldn’t worry about infinity too much), but the big question I challenge you with, should you accept the challenge - How much does the mathematician really know about infinity? OK, how do you represent infinity on a computer? (Just like randomness? Which isn’t really true randomness, but pseudo-randomness.)

    About measurement, quantum and trascendental physics - it is a Divine joke, just like all modern science is. I am not sure what Chesterton would have said about modern quantum physics, perhaps he would consider it too esoteric, just like he thought about Einstein’s mathematics and physics. Really, do we want our science and mathematics to be “transcendental”? Isn’t it causing problems? If quanta exist only as probabilities until an observation or measurement is made, (which affects them), what are they other than the construct of our imagination or fancy? I agree that there is definitely a paradox at the root of this problem as well, but I think that every measurement is of discrete nature, perhaps in more sense than one.

    Naturally analog computers don’t work in the same way digital computers do, i.e. via “counting” with discrete amounts of electricity using binary math. But they do count in the quantum way - a capacitor charges or discharge via energy levels, and is therefore subject to quantum energy levels, even though our benchtop instruments may not be accurate enough to count each and every quantum that is being added or removed from the charge.

    Wild Goose


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