from the glacier

First, a preface to a preface. Do you ever have to take a step back from writing to get those "creative juices" flowing again? I do, and for those who don't know, on the glacier, that means cracking another beer. That having been accomplished, leads me to a preface (by way of a quote):

Maybe I should have title this "against Kodiak" for, although he has my respect, I think we have a disagreement of some proportion. Another preface would be to read his post, or more fully, the several comments to the post. So basically I am stepping out on a limb, speaking of neither math nor science (except maybe incidentally), but rather starting with some modern local history. Those of you who have become accustomed to my rants will have to swallow your surprise at this, and everyone else will have to take a big leap of faith that I may know something more than just equations. Again, the preface is way too long... maybe I should just leave off with a preface. . .

The year was 1997. The location was Southern California. The underground was the NMS People's Revolutionary Council. It rained incessantly. A great man (with me and a fifth of Vodka) created a plan. A nine month plan. Well, for those of you who remember, it met with mixed results, and ultimately, the plan disbanded the NMS.

That short trip down memory lane is meant to illustrate, comically, that the endeavors of humanity can frequently be short-sighted. Kodiak himself remarked that: "the nine-month plan turned into a five-year plan." We were poking fun at the communist five year plan, that never seemed to get accomplished (shades of Orwell). But I think that if we look back at this it will teach us, as a microcosm, a valuable truth in the macrocosm. (footnote - the problem with Catholics and politics - a later rant)

What have I learned? Don't drink? Hardly. The kernel of truth I found was that if you attack an effect of a greater cause in order to achieve an end, but the cause you attack is not the ultimate cause of the deficiency you are trying to correct, you will fail. I admit it, the NMS was a diversion that I invented based off of an old "Melvins" poster in my sister's house. I was trying to correct the feeling of ennui and being trapped in a "fishbowl", I was fighting the weather. But a "secret" society based loosely on parodying communists did not go to the heart of that ennui. It failed. Ennui came back. Look at the modern American regime. I hate the phrase "culture of death" so I will invent a new one: "unculture of bad math" or, to shorten it, "2=3". Since we live in a 2=3 world, many people - well meaning, pious, generally good people - try to fight it. They say "abortion is the only issue I vote on". Or, when confronted by a politician "is he pro-life?" Or they talk about the constitution, or the founding fathers, or natural law, or the rule of law. Yes, the falling away from all of these things has created a 2=3 regime. But I deny, and will affirm my denial, that these things are the ultimate cause. [time to get those creative juices flowing] Physicists and Mathematicians (see, I couldn't stay away) run the culture. Once science (or at least the habits of thought and common terminology of science) filters its way into society, society changes. So where does 2=3 come from? Not divorce, not abortion, not the "living constitution", not rock music, not gay rights, animal rights, right-wingers, left-wingers, not positive law, not legislating judges, not any other political ill you can name. What do you have historically in math and science at the beginning of the twentieth century? BAD MATH. Set theory. Quantum mechanics and relativity understood without philosophical grounding. Sentential calculus (symbolic logic devoid of meaning in the symbols). Truth statements. Geometry as analysis. Consistency becomes the Holy Grail of mathematicians - not truth. Lucidity fails. Quote Poincare - even subject to his own criticism (written in 1908):

All the efforts that have been made to upset this order, and to reduce mathematical induction to the rules of logic, have ended in failure, but poorly disguised by the use of a language inaccessible to the uninitiated.
Roughly seventy years later we have the 2=3 world coming to full bloom (with Roe v. Wade if you want to make that your starting point). So, what are we to do? Go to the ballot box and vote in only "good" people? Lobby the government to have the constitution place on an altar of unassailable footing, so that no "interpretations" counter to the founding fathers can be made? Do we form little towns hidden from the rest of the world (see Pennsylvania) so we can live isolated and not have to deal with the outside world? ("who is my neighbor?") Shoot your T.V.? No.

Absolutely not.

Fuck the nine month plan. Fuck the five year plan. Fuck voting in or out. But, most of all, fuck the mathematicians. Over, that is. We need to inundate ourselves in their inaccessible language. Learn it, breathe it, speak it - destroy it. Shift the footing of physics back to philosophy - by understanding it. Use their arguments against themselves. It can be done. Make Heisenberg kneel to Thomas. Einstein curtsy to Aristotle. And this, once completed, spreads: in our children, our schools and colleges, our media, our newspapers, our everyday speech. Instead of "survival of the fittest" in media you hear "for the natural end" or "for the common good".

Not a four year plan of an administration, but a SEVENTY year plan of global domination.

While reading Science and Method the other night, Poincare puts the versus in the field of mathematical speculative theory as a contradiction between Kant and Leibniz. Or, Bertrand Russell relying on logic alone versus an idea of induction and basis in common thought processes as the background for mathematical reasoning. Basically, the logicians have said that there can be no synthetic judgments a priori in mathematical reasoning. It has to be logic and the rules of inference (not including induction) simply. But, as Poincare notes, one logician say that a conjunction combines TWO things. This is all before he comes to a definition of ONE.

But I guess my question is, stated by example, does the statement " all circles are similar" [in Euclidean Geometry, because we all know they aren't in other Geometries] follow from the definition of circle, or from the definition of similarity? We could say that because circularity is defined as equidistant from a point, and also show that any other similar figures have a defined ratio to a point of all sides that circles are defined as similar figures. But so also with squares, but this doesn't seem to be as great of a revelation. Maybe because we can line up squares meeting at one vertex and the two sides will correspond (at least to the extension of one side) whereas when when have two circles line up at one point, there is not the same "overlapping" . . . Perhaps it is evident from induction that all rectilinear similarly equiangular and equilateral figures are similar, but it is not so evident with curves. Curves are "trickier".

I for one, would like to lay down the following LAW: (Beitia's first law of mathematics)

1: Induction is necessary.

2: Mathematical objects exist prior to the science of mathematics.

3: Mathematics truly is (with the Philosopher) the study of magnitude qua magnitude.

3a: not a relationship of formal symbols (an arithmetic formal calculus)

4: Man does not make mathematics, he discovers it.

Thus, math is as much art as science, and a view toward the beautiful guides a mathematician as sure as a poet. Re-read Euclid II. 14. He proves the anunciation without even constructing that which is his goal. Beautiful. Genius. Examine II. 11 - the golden ratio. He constructs the golden ratio along the vertical side to cut the horizontal in such ratio - all without having even defined ratio yet. Beautiful. Genius. Examine Godel's inconsistency proof (for those with some training in mathematical logic). The way he turns Russell on his head and forces the sentencial calculus to talk about itself - Beautiful... Genius. . . [for those who cannot read mathematical logic, he constructs within system X a statement that translates "I cannot be proved in system X" - pure genius]

Beauty guides all science

I found this neat little essay on number theory at a Barnes and Noble in the used pile (why there was a used pile at a new bookstore is mysterious, perhaps it was fortune...) called The Continuum by Hermann Weyl, an early Twentieth Century mathematician [the book was published originally in 1918]. In addition to the standard nearly unintelligible garble, there were a few points worthy of notice. First, Weyl decides to ground number theory (including natural number, rational number and real numbers) solely in the ideas of a few logical principles of inference and two axioms. Instead of using the Euclidean definition of number as a multitude of units, Weyl follows the modern adaptation by using the idea of "immediate successor". Thusly, two axioms: every number has one unique immediate successor, and "one" is not the immediate successor of any number. From this and his rules of inference he proceeds to deduce the rule of number theory.

Unfortunately, number theory is vastly boring to those with no strong interest in it. So I can't go into that great of depth. It just seems important to note that there is a difference since the classical period of mathematics by changing multitude to immediate successor, and also by using rule of inference on strings of purely symbolic logic. In that sense the manipulations he does with these strings do not necessarily have to be about numbers. I digress. Additionally, he uses the same method for deriving other numbers (than counting ones) that I have toyed around with doing, even classifying them as "second degree" "third degree" etc. . . (for the rational numbers). But this is also an aside. Two other things are of note before I mention his antimony.

First, he thinks that a derivation of purely formal number theory is PRIOR to Geometry - which I have not been able to firmly refute thus far (but I just finished the book this morning). But, not to contradict himself, he says that the continuum formed by real numbers is a continuum of individuals different from and not applicable to "real" continua - time and space. Hmm. . . food for thought.

Secondly, and quote:

Finally, I'll finish with Weyl's antimony (Bertrand Russell has a similar one, etc. etc. in the realm of mathematics, they all boil down to Epimenides' self-referential statement "this sentence is false").

Some adjectives are what they describe ("brief" when said is brief) - these are autological.

Some adjectives are not what they describe ("long" is only four letters, and one syllable - and therefore not long) - these are heterological.

What about the adjective "heterological" which one is it???

That, as a preface, provides us with a good point for embarking on a little mental journey. As we all know, Heisenberg's Uncertainty Principle is the basis for quantum mechanics in the sense that it is the ground. It basically say that the position and speed of a particle are mutually unknowable. It also provides for the "virtual paths" I rambled about in the Feynman Diagram diatribe in an earlier post. Even in the transcribed lecture of Paul Davies that T.A.C. so diligently mailed to me (how they found my address I will still never know) had the author asserting that quantum uncertainty destroys the causal argument for the existence of God - since things come into and out of being spontaneously. I digress (again). I really do have a point. So, the principle of uncertainty has been extended to say that there is uncertainty in the things, but it began by saying that the uncertainty comes about because of a problem of observation. The simple looking at the sub-atomic thing changes the thing by looking. Davies, in God and the New Physics, goes so far as to say that if there were a cat in a box with a deadly chemical that will be released by a chemical reaction having fifty-fifty odds, that there is in fact a "virtual live cat" and a "virtual dead cat" in the box at the same time until an observer opens the box, and thus vanishes the virtual leaving only the real live or dead cat. He even has nifty hand drawings of dead/live cat to illustrate his point (I'm not kidding). Forgive the second digression as well. Glean one think from the rant - uncertainty is because of the observer.

Flash to Thomas, De Ente et Essentia, to be precise. In his observations on the being and essence of intellectual substances he shows that for God, to be and to be God are on and the same thing. Additionally, he shows that in angelic substances the essence is not the same as the act of existing, and so that even without matter, there still exists potentiality in angels, etc. Then he comes to lowly man. Again, for an intellectual substance there is some overlap between essence and act of existing, which is to say, in some limited sense, that to be a man and to be are somewhat the same. Without going into greater detail (for few people will bother to have read this far anyway, even if I assure the reader that I will eventually have a point), it is enough to say that if existence is somehow necessary to be a man, then there is a sense in which being and intellectual being affects the world around us in a way different that a simple unthinking causal reaction (like a falling rock). Permitting a course analogy, it is as if the world were a sheet of paper with iron filings on it. But every intellectual substance is like a magnet, of greater or lesser power depending on the kind of substance it is, and its mere presence affects the world around it. hmm.......

Could we then say that Heisenberg's principle has its causal root in Thomas' definition of man as intellectual substance? Is it possible that the basis of quantum physics is rooted in the "empty" and "unenlightened" dark age philosophy of a Schoolman? A rather appalling thought for a amateur mathematician and physicist (theoretical only) such as myself. I really don't know if I will be able to sleep tonight. But, to end with a quote from the same essay as the first quote: