Political note

I generally avoid political commentary when I'm sober, but I found the quote below (written by a former managing director from UBS turned mathematician and lecturer) one of the most highly illuminating descriptions of the mechanics of political psychology that I've read. In particular, it brings into view the self-contradictory nature that generally pervades states that grant people the so-called right of "self-rule."

" If you want to see what I mean by the arbitrariness of categories, check the situation of polarized politics. The next time a Martian visits earth, try to explain to him why those who favor allowing the elimination of a fetus in the mother's womb also oppose capital punishment. Or try to explain to him why those who accept abortion are supposed to be favorable to high taxation but against a strong military. Why do those who prefer sexual freedom need to be against individual economic liberty?

I noticed the absurdity of clustering when I was quite young. By some farcical turn of events, in that civil war of Lebanon, Christians became pro-free market and the capitalistic system—i.e., what a journalist would call "the Right"—and the Islamists became socialists, getting support
from Communist regimes (Pravda, the organ of the Communist regime, called them "oppression fighters," though subsequently when the Russians invaded Afghanistan, it was the Americans who sought association with bin Laden and his Moslem peers).

The best way to prove the arbitrary character of these categories, and the contagion effect they produce, is to remember how frequently these clusters reverse in history. Today's alliance between Christian fundamentalists and the Israeli lobby would certainly seem puzzling to a nineteenth century intellectual—Christians used to be anti-Semites and Moslems were the protectors of the Jews, whom they preferred to Christians. Libertarians used to be left-wing. What is interesting to me as a probabilist is that some random event makes one group that initially supports an issue thus causing the ally itself with another group that supports another issue, thus causing the two items to fuse and unify . . . until the surprise of the separation. "

The Black Swan by Nassim Nicholas Taleb, page 16

I would recommend anyone who takes an active interest in politics to contemplate this passage and see to what extent their own political passions are controlled by the powerful force being described by Mr. Taleb. It may help one realize more precisely the extent to which politics is radically unreal, a realization that has a tremendously freeing effect on the soul.

Postscript to the preceding discussion

To those interested in the financial markets we may point out a partially related meditation resulting from our preceding discussion. Many (not all) financial models, having been developed by mathematically inclined persons used to thinking of continuous quantities, model asset prices as function taking place in continuous time. Hopefully we are all beginning to question now, given that we have shown that space is discrete, whether time is discrete as well (or whether these mathematical categories are even meaningful in relation to the essence of time and space). However, ignoring that point for the moment, we must point out as fact that time in the financial markets is not continuous, since the markets open and close at definite points, placing discrete cuts in the apparently continuous whole of market activity. This would be irrelevant if asset prices could not change after market close. However, we can observe, albeit rarely, enormous changes in value that occur (practically speaking) instantaneously at the market’s opening that result from accumulated orders placed overnight, usually as a result of news that came after the previous day’s market close.

This is especially important if we consider portfolios that attempt to remain neutral to market fluctuations through the use of derivatives. If asset prices are modeled using Brownian motion (the usual method), which was originally designed to model the behavior of particles of pollen suspended in water, then the model will not be able to appropriately price the risk of a discontinuous overnight movement, since the objects Brownian motion models cannot move discontinuously. This poses no problem to the original intended uses of Brownian motion, but it does pose a significant problem to attempts to hedge portfolio value, since it will result in the failure of such methods at precisely the point when they are most necessary. The financial engineers who designed such portfolios, instead of questioning the validity of their underlying theoretical assumptions, will all-too-often simply accept that the infinitely small probability of hedging failure predicted by their financial models had in fact transpired, effectively blaming fate instead of their own mathematically brilliant models, models that have the one drawback of not entirely corresponding to that which they seek to describe.

There are other issues with the usual formulas used to construct hedged positions, but these are beyond the scope of our present inquiry into the essence of space.

That space is composed of discrete units, and that continuous quantities cannot accurately describe space in its actuality

The previous argument we entered into, wherein we proved the existence of a finite and indivisible lower bound to space, had the helpful side effect of proving that space is composed of discrete units and is therefore not a continuous whole. However, this argument was not accepted by all, particularly those with some knowledge of calculus. Indeed, very often in discussions on this issue one encounters resistance precisely along these lines. Since the mathematical conception of space underlies the notion of space that many of us have come to accept, we naturally resist the notion that space does not completely correspond to its mathematical map, which has no finite lower bound. However, actual space cannot actually correspond to the mathematical representation in light of which we define the concept of space. Let us quickly revisit our initial argument to show why.

Our argument consisted (in brief) of pointing out that any real partition of space must contain some amount of space (this is clear from the definition). Since this is the case, there must necessarily be a smallest unit of space. We proved this by examining the contrary, namely, the belief that there is no smallest unit of space and that space is infinitely partitioned. We showed that, if this was so, there must be an infinite number of partitions in any given finite section of space, each containing some actual amount of space (else they would not be divisions of space at all), and therefore leading to the conclusion that there was an infinite amount of space in a finite amount of space. Other than being inherently contradictory, this argument would also make motion impossible, which is absurd, since things do move. Therefore, the contrary—that space is composed of finitely small units—must be true.

Distinguished interlocutors pointed out that any of these finite segments, even the smallest possible unit that was proved to exist, could be further divided into another segment, thereby proving that the smallest possible unit was not the smallest possible unit—showing an apparent contradiction in our position. The appearance of contradiction arises because we insist on our ability to further subdivide the smallest possible unit that the preceding argument firmly established. But by what right can we actually do this? It is certainly true that, given any real number, one can always find a smaller one through simple division. This also shows that the 3-dimensional real “linear” space that we intuitively think of (constructed by extending the real number line in all three dimensions) can always be divided further, since it is composed of real numbers. However, at this point the mathematical model breaks down in relation to actual space, since we have already demonstrated that there must be a smallest actually existing unit of space. If one takes this smallest unit and mathematically divides it into halves, one has not created a smaller unit of space, but has created a unit of a unit. This construct, while mathematically reasonable, and physically existent insofar as it is a division of a unit, is not in itself an actual unit of space.

To explain this (perhaps abstract) point by reference to a concrete example—suppose one takes a glass of water and sets to dividing it in halves. One will first have two half-glasses, then four quarter-glasses, and so on. Each new division we create will be a division of the water, since there is still water in each new division. However, when we finally have divided our glass of water into millions of separate glasses, each containing only one molecule each, what now can we do? Our mathematical method of procedure has let us divide the water by two, each time producing new divisions. Now, however, if we were to slice up the molecule of water even further, we would find that we no longer had any water at all, but simply a scattering of oxygen and hydrogen atoms. This is because water cannot be actually divided into any smaller piece than one water molecule. One could draw a water molecule on a piece of paper and proceed to slice it up in a million mathematically distinct segments—in fact, given infinite time and infinite paper, one could even infinitely divide it. But these divisions, while being no doubt impressive, would in no way exist.

The point of what we have gone through is to establish firmly that space is composed of a finitely-bounded quantity of discrete units. This conclusion comes analytically from our usual concept of space. Additionally, if space is itself composed of discrete units, then nothing in space can itself be continuous. Therefore, we must assert, despite their usefulness, that continuous quantities do not exist in any actual sense. That is, assuming that what we usually refer to as “space” is itself actual and not simply different than what we think it to be but essentially different than what we think it to be…

That space, in its actuality, must be finitely subdivided (with a nod to Zeno)

There either is or is not a smallest actual unit of space. If there is not a smallest actual unit of space, then space is infinitely subdivided. If there is a smallest actual unit of space, then space is not infinitely subdivided, nor is it infinitely subdivisible. This matter cannot be settled by a direct appeal to empirical evidence, since it has already been established that there is a lower bound to the observable universe. The existence of this bound does not, however, prevent the potential existence of smaller unobservable particles—perhaps an infinitely large number of smaller particles (implying smaller subdivisions).

If there is not a smallest actual unit of space, then it follows that there are an infinite amount of subdivisions of space actually existing in any given finite region of space. If this is so, then it also follows that there is an infinite amount of space in any finite region of space. Now, it would seem that it would take an infinite amount of time to move through an infinite amount of space, unless one has the capacity to move at an infinite velocity. Since human beings do not seem to possess the means to means to travel at an infinite velocity, it follows that no one would be able to cross even a finite distance, since to do so they would have to move through the infinite subdivisions of the finite amount of space. In fact, nothing unable to move at an infinite velocity would be able to move at all, since it would have to cross infinite space to reach a point even infinitesimally farther away from where it started.

However, since clearly things do move, it follows that space cannot be actually infinitely subdivided. If space is not actually infinitely subdivided, it must be finitely subdivided; therefore, it is necessary that there be a smallest unit of space. If there is a smallest unit of space, then it follows that there must also be a smallest possible unit of matter or energy, though it is entirely possible that this unit may be too small to be detected.

very particular

What makes a particular a particular?

Our average experience of the world does not consist in the appreciation of particularity, but of universality. This is where the empiricist philosophers went wrong, thinking that the immediacy of our experience is the experience of particulars, when in fact the primal element of our experience is the experience of universals.

However much this tempts me to suggest that particulars are not particularly anything, I know that this isn't quite true. I'm not sure how true this always is of individual objects, but there certainly are some objects which are in fact different than other objects of their class. I may rejoin that these objects that are different than other similar objects are simply closer to their universal than others, but I wonder whether this is enough of an explanation.

At any rate, it certainly seems that particular people can have some meaning in their particularity, but the manner in which this particularity exists remains mysterious.

More numbers

(1, 2, 3) ---> (3 + 2) - 1 = 4

I haven't figured out how to move from 4 to 5 yet, so if you have any suggestions please let me know.

With a nod to the set theorists

All numbers arise from 0, from which spills forth equally number and numeration.

From 0 comes 1, unity, which comes from the placement of 0 within a context.

1 = 1 = 2, or, to put it better, {0, {0}} = 2. Though since 1 x 2 = 2, it may seem that 2 is higher than 1, the existence of 2 depends entirely upon 1 and without 1 it would be nothing at all.

2 comes into being immediately upon the definition of 1. This same movement makes possible the emergence of 3. The motion towards 1 (present in all numbers) in 2 impels the separation of 2 into both itself and that of itself that turns back towards one. Thus,

2 = (1 = 1)
2 = {0, {0}}

However, these two formulations are not strictly the same, for the first stresses the unity towards 1 in 2 while the second stresses the manner in which 2 is spun forth from 1. The dissimilarity-within-itself of 2 gives rise to three, since

2 = (1 = 1) = 1
1 + {0, {0}} = 1 + 2 = 3

Dimension

A dimension is a manner in which that which always stands beyond expression comes to expression.

People usually speak of three dimensions (height, width, and depth) and may additionally append a fourth (time).

In reality, the three spatial dimensions are not dimensions at all. Extensionality itself can be spoke of as a dimension in certain situations, but in the average experience of most people, the so-called domain known as "space" is a manner in which other dimensions (generally time) are experienced.

From "East Coker," by T.S. Eliot

IV.

The wounded surgeon plies the steel
That questions the distempered part;
Beneath the bleeding hands we feel
The sharp compassion of the healer's art
Resolving the enigma of the fever chart.

Our only health is the disease
If we obey the dying nurse
Whose constant care is not to please
But to remind of our, and Adam's curse,
And that, to be restored, our sickness must grow worse.

The whole earth is our hospital
Endowed by the ruined millionaire,
Wherein, if we do well, we shall
Die of the absolute paternal care
That will not leave us, but prevents us everywhere.

The chill ascends from feet to knees,
The fever sings in mental wires.
If to be warmed, then I must freeze
And quake in frigid purgatorial fires
Of which the flame is roses, and the smoke is briars.

The dripping blood our only drink,
The bloody flesh our only food:
In spite of which we like to think
That we are sound, substantial flesh and blood—
Again, in spite of that, we call this Friday good.

from Faraway (so close!)

Emit: "Long ago there must have been a golden age of harmony between heaven and earth. High was high, low was low, inside was in and outside was out.
But now we have money. Now everything's out of balance. They say time is money...but they got it all wrong! Time is the absence of money. Wouldn't you agree, Karl?"

Cassiel: "What can I add to that? Time is running away from me, Mr --"

Why Life is prior to space

We may temporarily permit ourselves a foray into a somewhat unclarified to point out now that the priority of time over space necessitates us to consider life as prior to space. I will endeavor to point out why in the following sections:

We have stated previously that time is the medium in which and through which change was experienced, and that, furthermore, it is the core condition for the possibility of any experience at all. Change, however, ought not be defined as a mechanical but as a formal process. We can show this is correct via the following reasoning:

Nothing that changes remains exactly what it was
A thing that has not altered in any way that is potentially observable (whether immediately or via posterior consequences) has not changed.
No change is potentially observable if there are no observers.
Conversely, if there are no observers there is no change.
Therefore, change only exists if potentially observable (whether immediately or via posterior consequences)

Now, observers only ever view formal characteristics. The concept of non-formal observable objects is absurd, since such objects would be formalized by experience and are therefore self-contradictory. Since observers can only view formal characteristics, we must state that change, which only exists in relation to a (potential) observation, must always be formal.

Formality, however, can exist independently of space. Indeed, formality by definition must be prior to space simply conceived, since space is itself a formalization. Time, however, as we pointed out before, is not necessarily a formalization, and only seems susceptible (at least as far as I am presently able to conceive of it) to a limited and partially extrinsic formalization.

Since we have already brought up (somewhat needlessly) the point that only observers formalize, and that space is a formalization, it is clear that there were observers prior to the formation of space. Observers by definition must be alive (though clearly not necessarily in any traditionally conceived biological sense, if our argument is correct). If there are living things then there is life.

THEREFORE, Life is prior to space.

Further proof of this may be had if one considers the discussion of change above. Since we have stated that time effectively changes into space, it is clear that there must be observers prior to the existence of space.

The reader endowed with forethought will already have realized that we are hurling ourselves, with breakneck speed, against Kant's antimony of the eternity and non-eternity of the world. Perhaps we will soon have gathered enough force behind us to shatter this cave wall and break through to the light of day beyond.

Continued

We must immediately clarify the manner in which we are speaking of some forms as prior and others as derivative. We stated previously, for instance, that time is prior to space, and that furthermore space owes its existence to time (at least in a qualified manner). However, we did not specifically analyze the manner in which this derivation takes place, and what it entails. We will now try to look at this briefly.

Derivation is participation. Plato makes this clear throughout his writings. However, the manner in which this participation takes place is what is typically not thought through with any manner of care. I believe that most, when they attempt to think through participation, have an overly mathematical construct in front of them. Participation is not the subset relationship; that is to say, the whole essence of a form that is participating in a derivative manner in another form is not necessarily contained in this form from which it is being derived.

To apply this to our previous example -- even though space relies wholly on time for all its original essential characteristics, this is not to say that the true essence of space is contained in time. To understand how this operates, we must come to a proper understanding of privation. Privation has traditionally been viewed as something strictly negative. Our immediate physical experience often confirms this. However, we must endeavour to understand the manner in which a thing may be reduced and in this reduction expanded.

Space has its origin in time. Since space is not time, space must either have properties that time does not have, or lack properties which time does have, or both. Time is the medium in which change is experienced. It is not, however, simply a condition of experience, but is in its core the condition for the possibility of any experience at all. Because of this, it is impossible to seize on time as something which we can control or even fix at a given point. Space is the crystallization of time. As such, it is inherently a privation when viewed from the standpoint of time, since it lacks the capacity for flow which is the defining characteristic of time. Thus from an originary standpoint space is a privation of time. Conversely, however, from the standpoint of space it reveals itself to be the truth of time, in that it is that towards which time always tended from the beginning. Furthermore, precisely because it is less than time from the standpoint of time, space has certain essential capacities that time lacks. The very statis that defines space allows us to hold onto it in a manner that is positive. Thus its very privation leads to wholly new possibilites that were not contained in its origin.

More to follow.

New wine into new wineskins

It has long been thought by those who know that every form is subservient to a higher form, and so on up to the very Form of forms; but it is equally true that each form must contain within itself its superior form, and even the Form of forms itself. For if a form did not contain the Form of forms, in virtue of what are we calling it a form? The matter of this containment needs to be clarified, but it is clear at the outset that all negative opposition of image and idea is immediately superfluous.

Because of this time can both be a condition of all existence as well as a property of any given object and therefore contingent upon that which it conditions. However, this contingency does not imply that there is anything necessarily spatial about temporality in and of itself, but merely demonstrates that the essence (and not simply existence) of anything that may be said to have an essence relies upon its further development into that which it conditions. This is where its contingency lies.

Since this is the case it is furthermore true that time can have a definite beginning, since it is from an absolute view prior to the existence of any object. Paradoxes involving the succession of events outside of time that may cause time can be solved by pointing out that the cause of time must be some variety of existence that, while certainly not temporal in and of itself, contains in and of itself that from which time springs.

Therefore, it is clear that what is to come has come before. But this may not yet be the end of the story. More to follow.