1 : The problem of infinitesimal to define a limit

What is infiniteness? It has been an important philosophical problem since Zeno of Elea pointed out its paradoxical nature, which claimed attention again, when Isaac Newton and Gottfried Wilhelm von Leibniz established calculus in the 17th century. They, however, treated infinitesimal and infinity in a naïve way, when they defined the limit of a function.

Let us recognize what is the matter with the naïve way at that time by a simple example of defining the derivative of the squaring function, f(x)=x². The derivative is the value of the following difference quotient and was defined as follows.

We can now factor out the h to obtain

Now we can let h go to zero to obtain the derivative, 2x. So,

Although this sort of definition is still adopted for beginners, it is inconsistent because it first supposes h not to be zero to factor out and then supposes h to be zero to define the limit. Infinity causes the same problem, since 1/h approaches zero as |h| approaches infinity. In this article I use infiniteness or the infinite in a wider sense that includes both infinity and infinitesimal.

Newton published Philosophiae Naturalis Principia Mathematica in 1687 and in 1734 George Berkeley criticized the contradiction of supposing infinitesimal to be zero and non-zero^[1]. To avoid this criticism the epsilon-delta definition of limit was devised.

2 : The epsilon-delta definition of limit

The epsilon-delta (ε-δ) definition of limit was developed by Augustin-Louis Cauchy^[2], Bernard Bolzano^[3] and accomplished by Karl Weierstraß^[4] in 1861. This method is usually thought to define the limit of a function without infinitesimal.

Let us prove the previous example in terms of the epsilon-delta definition.　

For each real ε there exists a real δ such that

For all h we have

That is to say

So we can say by means of the epsilon-delta definition

This method does not use such phrases as “approach zero infinitely” or “come as close as desired to zero” and seems to succeed in eliminating infinitesimal. It is true that ε itself is not infinitesimal, but it is universally quantified with the sign of ∀. Therefore the epsilon-delta definition does not eliminate infinitesimal but just transforms the infiniteness of ε approaching zero into the infiniteness of substituting ever smaller value for ε.

The significance of the epsilon-delta definition consists in distinguishing the finite and the infinite of limit and transferring the latter from the attribute of the recognized object to that of the recognizing subject, although developers of the epsilon-delta definition themselves were not aware of it. This transference reminds us of the Copernican Revolution that Immanuel Kant brought about in The Critique of Pure Reason. The Copernican Revolution is achieved originally by looking for the observed movements not in the heavenly bodies, but in the spectator^[5] and in metaphysics by looking for the infinite not in the recognized object, but in the recognizing subject.

Now, if it appears that when, on the one hand, we assume that our cognition conforms to its objects as things in themselves, the unconditioned cannot be thought without contradiction, and that when, on the other hand, we assume that our representation of things as they are given to us, does not conform to these things as they are in themselves, but that these objects, as phenomena, conform to our mode of representation, the contradiction disappears: we shall then be convinced of the truth of that which we began by assuming for the sake of experiment; we may look upon it as established that the unconditioned does not lie in things as we know them, or as they are given to us, but in things as they are in themselves, beyond the range of our cognition.^[6]

As is indicated at the latter half of this quotation, Kant supposed things in themselves to be infinite (unconditioned). Now that we know nothing about things in themselves, as Kant claimed, we should not suppose them to be infinite or unconditioned. What I would like to support is the former half of the quotation, that is to say, we should assume that our cognition of the infinite does not conform to our objects as things in themselves, but conform to our mode of representation.

The question whether everything in the world consists of simple elements or not is the second antinomy of cosmological ideas in The Critique of Pure Reason. According to Kant, the principle of infinite division of space is not that of constituting the recognized object but that of regulating the recognizing subject. So, Kant regarded the antinomy of infinite division as the illusion produced by mistaking the constitutive principle for the regulative^[7].

Bolzano, one of developers of the epsilon-delta definition, would have opposed to this Kantian interpretation, because he was an objectivist whose theory of science (Wissenschaftslehre) believed in proposition in itself (Satz an sich). He wrote Paradoxes of the Infinite and put questions whether a half line plus a segment line is longer than a half line or whether a line is longer than a half line by infinite length, although they are all infinite^[8]. It was because he did not conduct or notice the Copernican Revolution of the infinite that he felt these paradoxical.

Infinity is a number that is bigger than any number. The Copernican Revolution turns this objective definition to a subjective one: infinity is a number that is too big for a subject to measure. The subject definition makes Bolzano’s paradox less paradoxical: a half line, a half line plus a segment line and a line are all too long for a subject to measure and therefore we cannot say which is longer than which.

The developers and introducers of the epsilon-delta definition believed that they could eliminate infinity or infinitesimal from limits of functions just because they presupposed that the truth can be attributed to the object independent of the subject. You can make infinity or infinitesimal invisible by transferring them from the object to the subject but they are not eliminated. The epsilon-delta definition was right in respect of separating the infinite from the finite but now that infinity and infinitesimal did not disappear we should make them visible again. This is why non-standard analysis came to the fore in 1960s.

3 : Non-standard analysis of limit

Non-standard analysis has received much attention as a new strict approach to infinity or infinitesimal since Abraham Robinsonin published his idea in 1966^[9]. You can also confer with Howard Jerome Keisler’s easy-to-understand textbook based on non-standard analysis, Elementary Calculus.

Non-standard analysis extends real number to hyperreal number to include infinity and infinitesimal. Positive and negative infinities are infinite hyperreal numbers, while the other hyperreal numbers that are between real numbers are finite hyperreal numbers. Infinitestimals are finite hyperreal numbers but not real numbers except zero.

The relation between a real number and a hyperreal number is similar to that between a real number and a complex number.
Just as a complex number consists of the real part and the imaginary part, a hyperreal number consists of the standard part and the nonstandard part, namely the real part and the non-real part. Infinity has no standard part. The standard part of a real number plus infinitesimal is equal to the real number. A limit is defined as the standard part of a difference quotient.

Non-standard analysis defines the derivative of f(x)=x² as follows.

Let h be infinitely close to but not equal to 0,

then we have

Taking standard parts, the derivative of the squaring function is

Non-standard analysis seems more similar to the classical method of Newton and Leibniz than the epsilon-delta definition, but it is not identical to the former in that it distinguishes real numbers from non-real hyperreal numbers. The relation between them is, however, not clear. There is no space between zero and the least real number where the infinitesimal number is plotted and there is no fixed position next to the largest real number where the positive infinity is assigned. The continuity of real numbers admits no space in a real number line for non-real hyperreal numbers. The non-real hyperreal numbers are not the objective numbers located anywhere on a real number line but subjective numbers above them. The “hyper” of the hyperreal number can mean “above object” namely “subjective”.

4 : The paradox of Achilles and the tortoise

The paradoxes of infinity were known before infinitesimal calculus was invented and the most famous is the paradox of Achilles and the tortoise devised by Parmenides or his disciple, Zeno of Elea. Let’s see how the Copernican Revolution of infinitesimal can resolve its aporia.

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Suppose Achilles allows the tortoise a head start of d metres. Suppose further the velocity of Achilles is v_a m/s and that of tortoise is v_t m/s, which is lower than v_a m/s. It takes d/v_a s for Achilles to get to the tortoise’s starting point, while the tortoise has run v_t* d/v_a meter distance. It will then take some further time for Achilles to run that distance, by which time the tortoise will have advanced farther and so on. Zeno claimed that because this catch up process should be repeated infinitely, Achilles could never overtake the tortoise.

Achilles actually has only to run a finite distance D to overtake the tortoise.

Because we suppose 0 < v_t< v_t,

Therefore this infinite geometric series converges.

Now that the distance is finite, the time to overtake the tortoise is finte.

So where did Zeno make a mistake? Aristotle said as follows.

Hence Zeno’s argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time. For there are two senses in which length and time and generally anything continuous are called ‘infinite’: they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite.^[10]

Would it have persuaded Zeno that he was wrong? He might have retorted that Achilles could never overtake the tortoise if time itself could be divided infinitely. To avoid such a misunderstanding we must distinguish the divided time and the time for division. If each division takes a certain length of time, infinite division will require infinite time. But it is the time for dividing subject and the divided time and the divided distance corresponding to it has a finite length. Thus the Copernican Revolution of attributing the objective infiniteness to the infinite activity of the subject can refute the quibble of Zeno.

5 : Is there an object corresponding to the infinite?

For the ancient Greek philosophers what is infinity or infinitesimal was the matter of physics. The physical questions whether we can divide matter into atoms, hadrons, elementary particles and so on infinitely or not, whether the universe has infinite size or not should be answered on empirical ground. Mathematical questions are different in that they can be answered theoretically.

The ideal object of mathematics is different from the real object of positive science. For example, a mathematically complete circle is not found in the real world. Infinity and infinitesimal are not the attributes of real objects or even ideal objects but the attributes of subjective activity on ideal objects. This is the conclusion of this article.

As our cognition is finite, we don’t know whether what we call infinity or infinitesimal has its counterpart in ideal or real objects. It is, however, because we don’t know it that we can have such concepts as infinity and infinitesimal. That is to say, we can recognize the infinite because our cognition is finite. In other words, God, who we imagine has infinite power, could not recognize the infinite.

6 : References

1. ↑ The Analyst: a Discourse addressed to an Infidel Mathematician (section) 15 (author) George Berkeley
2. ↑ Résumé des leçons données à l’École royale polytechnique sur le calcul infinitésimal, Oeuvres complètes d’Augustin Cauchy, Série 2, tome 4 (page) 47 (author) Augustin-Louis Cauchy
3. ↑ Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwei Werten, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (author) Bernard Bolzano
4. ↑ Differentialrechnung – Ausarbeitung der Vorlesung an dem Königlichen Gewerbeinstitut zu Berlin im Sommersemester 1861 von H. A. Schwarz (author) Karl Weierstraß
5. ↑ Kritik der reinen Vernunft (page) 14-15 (author) Immanuel Kant
6. ↑ Kritik der reinen Vernunft (page) 14 (author) Immanuel Kant
7. ↑ Kritik der reinen Vernunft (page) 460 (author) Immanuel Kant
8. ↑ Paradoxien Des Unendlichen (section) 19 (author) Bernard Bolzano
9. ↑ Non-standard Analysis (author) Abraham Robinson
10. ↑ Physics, Book 6 (author) Aristotle

1 : The performance of the new server

I have had a shared server of DreamHost host my websites but found it inappropriate to have a shared server host database-driven websites and am moving them to a new hosting server, Sakura Internet’s VPS with 4GB RAM, 120GB storage, 2.40GHz dedicated Intel(R) Core(TM)2 Duo CPU and 100Mbps bandwidth. Hosted by this new server, systemics websites are expected to get lighter and faster than now.

The following are the result of benchmarks tests performed on the new server. The new hosting service seem better than the other in the same price range (3980JPY/month).

• System Benchmarks Index Score: 2047.7
• Requests per second: 78.17
• Timing cached reads: 8634.30 MB/sec
• Timing buffered disk reads: 133.54 MB/sec
• Throughput: 136.811 MB/sec

2 : The result of UnixBench

Benchmark Run: Mon Dec 26 2011 02:38:39 - 03:06:34
4 CPUs in system; running 4 parallel copies of tests

Dhrystone 2 using register variables       46508632.5 lps   (10.0 s, 7 samples)
Double-Precision Whetstone                     9328.0 MWIPS (9.6 s, 7 samples)
Execl Throughput                              10171.1 lps   (30.0 s, 2 samples)
File Copy 1024 bufsize 2000 maxblocks        250721.1 KBps  (30.0 s, 2 samples)
File Copy 256 bufsize 500 maxblocks           75421.9 KBps  (30.0 s, 2 samples)
File Copy 4096 bufsize 8000 maxblocks        548661.2 KBps  (30.0 s, 2 samples)
Pipe Throughput                             6070815.5 lps   (10.0 s, 7 samples)
Pipe-based Context Switching                1190019.9 lps   (10.0 s, 7 samples)
Process Creation                              26912.5 lps   (30.0 s, 2 samples)
Shell Scripts (1 concurrent)                  13293.1 lpm   (60.0 s, 2 samples)
Shell Scripts (8 concurrent)                   2179.7 lpm   (60.0 s, 2 samples)
System Call Overhead                        5290285.9 lps   (10.0 s, 7 samples)

System Benchmarks Index Values               BASELINE       RESULT    INDEX
Dhrystone 2 using register variables         116700.0   46508632.5   3985.3
Double-Precision Whetstone                       55.0       9328.0   1696.0
Execl Throughput                                 43.0      10171.1   2365.4
File Copy 1024 bufsize 2000 maxblocks          3960.0     250721.1    633.1
File Copy 256 bufsize 500 maxblocks            1655.0      75421.9    455.7
File Copy 4096 bufsize 8000 maxblocks          5800.0     548661.2    946.0
Pipe Throughput                               12440.0    6070815.5   4880.1
Pipe-based Context Switching                   4000.0    1190019.9   2975.0
Process Creation                                126.0      26912.5   2135.9
Shell Scripts (1 concurrent)                     42.4      13293.1   3135.2
Shell Scripts (8 concurrent)                      6.0       2179.7   3632.9
System Call Overhead                          15000.0    5290285.9   3526.9
                                                                   ========
System Benchmarks Index Score                                        2047.7

3 : The result of ApacheBench

Server Software:        cloudflare-nginx
Server Hostname:        www.systemicswiki.com
Server Port:            80

Document Path:          /index.html
Document Length:        2769 bytes

Concurrency Level:      5
Time taken for tests:   1.279284 seconds
Complete requests:      100
Failed requests:        0
Write errors:           0
Total transferred:      328689 bytes
HTML transferred:       276900 bytes
Requests per second:    78.17 [#/sec] (mean)
Time per request:       63.964 [ms] (mean)
Time per request:       12.793 [ms] (mean, across all concurrent requests)
Transfer rate:          250.14 [Kbytes/sec] received

Connection Times (ms)
              min  mean[+/-sd] median   max
Connect:        8   10   2.0     10      20
Processing:    22   27   5.8     25      54
Waiting:       21   26   5.7     25      53
Total:         31   37   6.6     35      65

Percentage of the requests served within a certain time (ms)
  50%     35
  66%     36
  75%     39
  80%     40
  90%     47
  95%     57
  98%     60
  99%     65
 100%     65 (longest request)

4 : The result of hdparm

/dev/hda:
 Timing cached reads:   17256 MB in  2.00 seconds = 8634.30 MB/sec
 Timing buffered disk reads:  402 MB in  3.01 seconds = 133.54 MB/sec

5 : The result of dbench

Operation      Count    AvgLat    MaxLat
 ----------------------------------------
 NTCreateX    2616715     0.019    48.463
 Close        1922078     0.002     9.930
 Rename        110809     0.049    31.016
 Unlink        528470     0.073   362.068
 Deltree           62     5.670    27.625
 Mkdir             31     0.005     0.032
 Qpathinfo    2371866     0.011    42.817
 Qfileinfo     415512     0.002     2.664
 Qfsinfo       434905     0.106    10.264
 Sfileinfo     213161     0.037    47.427
 Find          916987     0.037    36.356
 WriteX       1303855     0.056    48.902
 ReadX        4102068     0.008    47.178
 LockX           8520     0.006     1.693
 UnlockX         8520     0.005     1.995
 Flush         183388    14.297  1123.166

Throughput 136.811 MB/sec  5 clients  5 procs  max_latency=1123.184 ms

6 : Systemics websites in 2012

The main theme in 2012 is the same as that in 2011 – philosophical study in history of science. I will especially bring the concept of entropy into focus, which is so important to systemics. I know natural science is not a popular field, but, since it is the fundamental field of systemics, I must take up this subject first.

1 : Three aspects of pre-established harmony

Although the philosophy of Gottfried Leibniz is often referred to as monadology, he had never used this term. Instead, he identified himself as “the author of the system of pre-established harmony”. For him arguments about monads are just preliminary arrangements for his system of pre-established harmony and not the ultimate goal of his metaphysics. This article, using “the system of pre-established harmony” as the proper name of his metaphysics, elucidates its three aspects, namely pantheism based on the ontological argument, monadology based on the law of continuity and optimism based on the principle of sufficient reason so as to clarify why Leibniz advocated the system, and points out its problem.

2 : Pantheism based on the ontological argument

Leibniz defines God as follows to apply the ontological argument.

God is absolutely perfect and perfection is nothing but the magnitude of positive reality, taken in the precise sense of leaving aside the limits or bounds in things that have them^[1]

According to Leibniz God is complete in possessing positive reality and must contain as much reality as is possible^[2]. The “reality (réalités)” stems from “thing (res)” in Latin, by which Leibniz meant not only existing things but also possible things. So, Leibniz’s God is not identical to the existing universe. He wrote, there is an infinite number of possible universes in the Ideas of God^[3], which indicates his God not only includes existing universe but also all possible universes. At this point his pantheism is different from usual pantheism that identifies God with the existing universe.

Leibniz was originally a mathematician and, as mathematics deals with possible worlds, he probably thought God would be incomplete if He did not include possible universes. In his article in 1696, Surprising expression of all numbers by means of 1 and o, representing the origin of things by God and nothingness or the mystery of the Creation^[4], Leibniz connected the binary number system with the genesis by God: since every number can be expressed by means of 1 and o, everything consists of not God and materials but God and nothingness. If you interpret 1 (unity) as “positive reality” and 0 (zero) as nothingness, you will make out the binary implication of his assertion that God has the unlimited magnitude of positive reality and there is nothing outside God.

In Monadology, Leibniz’s late representative work of metaphysics in 1714, he wrote, God alone is the primary unity^[5], identifying God with the primary source of 1. Here he called what he had called “metaphysical point” and so on before Surprising expression of all numbers by means of 1 and o in 1696 “monad”, which stems from a Greek word “μονάς” meaning 1 or unity. Thus he regarded all the creatures as unities derived from the primary unity of God.

While Leibniz defined God as omnipresence in terms of the ontological argument, he also respected the other traditional Christian definitions of God as omniscience, omnipotence and omnibenevolence. The property of omniscience presupposes that God has the cognitive power. How can omnipresent God have cognizance of objects? We usually think of cognition as acquiring information on the objects outside our consciousness through a window. If God were to know objects in this way, it should be against the ontological argument, because there would be objects to be known outside God and God would not be omnipresence. As God must “contain as much reality as is possible”, God should not have any window for cognition and His cognition must always be self-reflection.

3 : Monadology based on the law of continuity

Another aspect to understand the system of pre-established harmony is monadology based on the law of continuity. Leibniz is along with Isaac Newton the inventor of infinitesimal calculus and, since a function must be continuous to be differentiable, he postulated the following law of continuity as an axiom of his metaphysics.

When the cases (or what is given) approach continuously and at last overlap each other, the consequences or events (or what is asked) must do it as well. ^[6]

This formula is neither clear nor strict, but the example using an ellipse and a parabola that Leibniz gave to explain the law of continuity is easy to understand. So, let me introduce it. The graph below presents an ellipse in blue whose foci are F₁ (0,1) and F₂ (0,f) = (0,3).

The law of continuity applied to an ellipse, a parabola and a circle

As the value of “f” approaches infinity, the blue ellipse gets infinitely close to the red parabola. Now that the ellipse can approach the parabola continuously, we can apply the law of continuity to it: what is true of the ellipse is also true of the parabola. For example, any beam emitted from F₂ reflects the ellipse mirror and converges on F₁. Provided that F₂ lies at infinity, the beam from F₂ can be considered to be parallel to Y-axis and therefore converges on F₁.

Although this is not what Leibniz wrote, we can also say that the blue ellipse infinitely approaches the green circle, as the value of “f” gets infinitely close to 1. Now that the ellipse can approach the circle continuously, we can apply the law of continuity to it: what is true of the ellipse is also true of the circle. As for the previous example, two foci merge together into a center of the circle and any beam emitted from the center reflects the circle mirror and converges back on it.

Parabolas, ellipses and circles are usually considered to be different in kind but the law of continuity tells us that this is not the case. Here assign these three figuratively to God, humans and non-human creatures respectively. The limit of an ellipse is a parabola as the distance between foci approaches infinity and the limit of an ellipse is a circle as the distance between foci approaches zero. Likewise the limit of humans is God as the completeness of human cognition approaches infinity and their limit is lifeless creatures as the completeness of human cognition approaches zero. Still these three substances of cognition are equal in that they are unities or what Leibniz called monads.

A monad is a simple substance, which enters into compounds, and ‘simple’ means ‘without parts’ ^[7], so that there can be neither extension nor figure nor divisibility^[8]. This explanation might lead you to take it for a physical atom, but as the current science shows, an atom has extension, figure and divisibility. Leibniz distinguished monads as metaphysical points or those of substance from physical points and mathematical points.

Thus physical points are only indivisible in appearance; mathematical points are so in reality but they are merely modalities ; only metaphysical points or those of substance (constituted by forms or souls) are exact and real^[9]

As this quotation indicates, forms or souls constitute metaphysical points or those of substance. According to Leibniz, spirits as rational souls are more complete and therefore more near to God than other souls and forms in general.

This is why God governs spirits as a prince governs his subjects, and even as a father cares for his children; while he disposes of the other substances as an engineer manipulates his machines. ^[10]

Though there is a hierarchy of substances in order of completeness that consists of God, human spirits, souls of animals and forms of things in general, they as the same monads and the same substances are continuous just like parabolas, ellipses and circles. Following the law of continuity, we must think that what is true of God is also true of the other substances. Since God has nothing outside Him, He has no windows to interact with the outside. So are the other substances.

The Monads have no windows, through which anything could come in or go out.^[11]

The cognition of God is not the copy of the outside but the expression of Himself. Nor is our cognition the copy of the outside but the expression of God Himself. Leibniz wrote, no external cause acts upon us excepting God alone^[12] and there is absolutely no other external object which comes into contact with our souls and directly excites perceptions in us^[13]

The expression of the universe on monads becomes incomplete, that is to say, perspectively distorted in the order corresponding to human spirits, souls of animals and forms of things in general. They still express the same entire universe and therefore calleda perpetual living mirror of the universe^[14]

A monad is a unit in that it expresses the coherent and therefore unified university, while the expressed universe is diverse.

The passing condition, which involves and represents a multiplicity in the unit or in the simple substance, is nothing but what is called Perception, which is to be distinguished from Apperception^[15]

Leibniz’s metaphysics is different from that of Descartes in acknowledging animal souls or forms in general having perception, although he considered it quite different from apperception of human spirits. The Cartesian dualism of thought and extension resulted in the so-called mind-body problem. Descartes regarded pineal gland as the window through which the soul and the body interacted^[16], while Leibniz denied the window and tried to solve the mind-body problem through the approach of mentalism.

If a particular substance never acts upon another particular substance^[17], how can human spirits can control their bodies, as they will?

The soul follows its own laws, and the body likewise follows its own laws; and they agree with each other in virtue of the pre-established harmony between all substances, since they are all representations of one and the same universe. ^[18]

This is what Leibniz called pre-established harmony.This harmony is analogous to that of activity existing between two independent clocks whose ticking is perfectly synchronized without causal interaction and interference^[19]. If it were not for the pre-established harmony between monads, there would be no harmony in God that includes all monads, but it is contradictory to the definition of God as the infinitely complete being. Thus we can see that Leibniz’s system of pre-established harmony is founded on the omnipresence of God defined by the ontological argument.

4 : Optimism based on the principle of sufficient reason

Leibniz called the principle of contradiction and the principle of sufficient reason the two great principles. The principle of sufficient reason (principe de la raison suffisante) is such thatthere can be no fact real or existing, no statement true, unless there be a sufficient reason, why it should be so and not otherwise^[20].

Corresponding to the two principles, there are also two kinds of truths: those of reasoning and those of fact. Truths of reasoning are necessary and their opposite is impossible, while truths of fact are contingent and their opposite is possible. The former are justified by the principle of contradiction and the latter the principle of sufficient reason. To stop the search for reasons of the latter from plunging into regressus ad infinitum, the final reason must be in a necessary substance whose reason you cannot find any longer. Leibniz thought it must be God.

This is a cosmological argument for the existence of God. Since Leibniz used the ontological argument as well, God cannot be confined to the mere first cause that does not include the subsequent events. His ontological argument defines God as the substance that includes not only the entire existing universe but also all possible universes. He must have the sufficient reason He selected this universe rather than other possible ones.

Leibniz believed that God had selected the best of all possible universes and realized it, because He has omniscience, omnibenevolence and omnipotence. So, the actual existence of the best that wisdom makes known to God is due to this, that His goodness makes Him choose it, and His power makes Him produce it. ^[21]. This doctrine is called optimism (optimisme).

According to Leibniz, optimism is a kind of theory of pre-established harmony.

As we have shown above that there is a perfect harmony between the two realms in nature, one of efficient, and the other of final causes, we should here notice also another harmony between the physical realm of nature and the moral realm of grace, that is to say, between God, considered as Architect of the mechanism of the universe and God considered as Monarch of the divine City of spirits. ^[22]

His optimism incurred disrepute in those days: Why can we say that this world is the best of all possible worlds, although this world obviously has so much evil? He retorted that the world with a little evil is better than that without it, because it helps us appreciate good.

A little acid, sharpness or bitterness is often more pleasing than sugar; shadows enhance colours; and even a dissonance in the right place gives relief to harmony. We wish to be terrified by ropedancers on the point of falling and we wish that tragedies shall well-nigh cause us to weep. Do men relish health enough, or thank God enough for it, without having ever been sick? And is it not most often necessary that a little evil render the good more discernible, that is to say, greater? ^[23]

It was natural that Leibniz should not exclude evil from God. If he had attributed good to God and evil to non-God, his theory would have been morally rational and easily accepted, but then he would have been obliged to admit that God had what is not God outside Himself, which contradicts the omnipresence of God postulated by the ontological argument.

5 : Problems of the theory of pre-established harmony

The preceding analysis has clarified that Leibniz’s theory of pre-established harmony presupposes pantheism resulting from the ontological argument. Generally speaking, the being whose existence the ontological argument demonstrates has omnipresence but it is different from the Christian god that has omniscience, omnibenevolence and omnipotence. Leibniz thought God is intelligent enough to know the best of all possible universes, benevolent enough to select it and powerful enough to realize it as the existing one, but this being is more than the mere amount of universes whose existence alone the ontological argument can prove.

If omniscience just meant that the universe expresses itself as it is and if omnipotence just meant that the exiting universe is different from possible universes, then we could safely say that the being whose existence the ontological argument demonstrates had omniscience and omnipotence. Such a way of cognition and fulfillment, however, is different from that of us humans.

Leibniz supposes the monads as human spirits mirror the entire universe as God does and their intentions are fulfilled owing to the pre-established harmony of God. The fact is that what we regard absolutely true often turns out to be false and what we intend to fulfill absolutely ends in a failure. The theory of preformation that Leibniz believed in is today considered to be false and his attempt at Catholic-Protestant Reconciliation failed. If the harmony is not always pre-established in this way, doesn’t it follow that God contains contradiction in Himself?

Maybe Leibniz might defend his theory of pre-established harmony in the way to defend his optimism: if what we perceive is always true, we cannot appreciate the truth, so God often makes us make an error to maximize the value of the truth or if what we intend is always fulfilled, we cannot appreciate the fulfillment, so God often makes us experience failure to maximize the value of success. Such an explanation is arbitrary and not persuasive, because it just confirms the present situation.

In fact, we can draw the conclusion contrary to optimism by the same logic: it is not God but the Devil that governs this universe, that is to say, He chooses the worst of all possible universes and mingles evil with a little good in the chosen universe so as to awaken futile expectations in people and maximize their suffering from disappointment.

Leibniz founded a system of pre-established harmony on omnipresence of God resulting from the ontological argument. Descartes also adopted the ontological argument but unlike Leibniz he did not seriously examine how God knows or how He fulfills his intention. Leibniz is worth praising in trying to construct a coherent system of metaphysics based on the ontological argument, but as the presupposition is wrong, the more consistent with the presupposition he tried to make his theory, the more grotesque his metaphysics becomes. In such a case, radical doubt on the presupposition instead of making makeshift modifications to it is necessary.

6 : References

1. ↑ Monadologie (section) 41 (author) Gottfried Leibniz
2. ↑ Monadologie (section) 40 (author) Gottfried Leibniz
3. ↑ Monadologie (section) 53 (author) Gottfried Leibniz
4. ↑ Mira numerorum omnium expressio per 1 et 0, repraesentans rerum originem ex Deo et Nihilo, seu Mysterium creationis (media) Die philosophischen Schriften (band) 6 (editor) Karl Immanuel Gerhardt (author) Gottfried Leibniz
5. ↑ Monadologie (section) 47 (author) Gottfried Leibniz
6. ↑ Lettre de M. L. sur un principe general utile à l’explication des loix de la nature par la consideration de la sagesse divine, pour servir de replique à la reponse du R. P. Malebranche (page) 1 (author) Gottfried Leibniz
7. ↑ Monadologie (section) 1 (author) Gottfried Leibniz
8. ↑ Monadologie (section) 3 (author) Gottfried Leibniz
9. ↑ Système nouveau de la nature (section) 11 (author) Gottfried Leibniz
10. ↑ Système nouveau de la nature (section) 5 (author) Gottfried Leibniz
11. ↑ Monadologie (section) 7 (author) Gottfried Leibniz
12. ↑ Discours de métaphysique (section) 28 (author) Gottfried Leibniz
13. ↑ Discours de métaphysique (section) 28 (author) Gottfried Leibniz
14. ↑ Monadologie (section) 56 (author) Gottfried Leibniz
15. ↑ Monadologie (section) 14 (author) Gottfried Leibniz
16. ↑ L’homme et un traitté de la formation du foetus (page) 73 (author) René Descartes
17. ↑ Discours de métaphysique (section) 14 (author) Gottfried Leibniz
18. ↑ Monadologie (section) 78 (author) Gottfried Leibniz
19. ↑ Considerations sur les Principes de Vie, et sur les Natures Plastiques, par l’Auteur du Systeme de l’Harmonie préétablie (media) Die philosophischen Schriften (band) 6 (editor) Karl Immanuel Gerhardt (author) Gottfried Leibniz
20. ↑ Monadologie (section) 32 (author) Gottfried Leibniz
21. ↑ Monadologie (section) 55 (author) Gottfried Leibniz
22. ↑ Monadologie (section) 87 (author) Gottfried Leibniz
23. ↑ Essais de Théodicée (section) Première partie, 12 (author) Gottfried Leibniz

1 : Critique of Anselm’s argument

It was Anselm of Canterbury (Born: 1033; Died: 1109) who first proposed the ontological argument for the existence of God. According to him even atheists who say God does not exist but understand what God is acknowledge that God exists in their understanding.

Hence, even the fool is convinced that something than which nothing greater can be conceived exists at least in the understanding. For, when he hears of this, he understands it. And whatever is understood exists in the understanding. And assuredly that, a being than which nothing greater can be conceived, cannot exist in the understanding alone. For, suppose it exists in the understanding alone: then another being which is greater than it can be conceived to exist also in reality. Therefore, if that, than which nothing greater can be conceived, exists in the understanding alone, the very being, than which nothing greater can be conceived, is one, than which a greater can be conceived. But obviously this is impossible. Hence, there is no doubt that there exists a being, than which nothing greater can be conceived, and it exists both in the understanding and in reality. ^[1]

The Latin word maius that Anselm uses here is the comparative of a neutral adjective magnum. To use Hare’s terms^[2], it has not only the descriptive meaning “large” or “big” but also the evaluative meaning “noble” or “important”, just as the English adjective “great”. Suppose maius has only the descriptive meaning “larger” and the god (let me use lowercase) is larger than anything in this meaning. The god is then the whole world including reality and understanding. Were it not for the god thus defined, there could be nothing that exists. As this is against the empirical fact, the god as the largest must exist.

The god whose existence the ontological argument proves is just the largest in the descriptive meaning and not the greatest in the evaluative meaning. If the god were the greatest in the evaluative meaning, it could be the omniscient and omnipotent supreme being. Generally speaking, however, it is not necessarily true that the larger the better. The ontological argument confuses the descriptive meaning with the evaluative and demonstrates the existence of the largest being that is different from God.

2 : Critique of Descartes’ argument

René Descartes is the most famous of the modern successors of Anselm’s argument. He is different from Anselm in starting from the thinking ego that he thought of as the most certain being. According to Descartes God whose idea I have does not remain within me.

For though the idea of substance be in my mind owing to this, that I myself am a substance, it should not, however, be the idea of an infinite substance, seeing I am a finite being, unless it were given me by some substance in reality infinite. ^[3]

He not only says that God is the substance that transcends ego but also that the unlimited substance is the cause of the limited and therefore has more reality.

Now, it is manifest by the natural light that there must at least be as much reality in the efficient and total cause as in its effect; for whence can the effect draw its reality if not from its cause ? And how could the cause communicate to it this reality unless it possessed it in itself? And hence it follows, not only that what is cannot be produced by what is not, but likewise that the more perfect, in other words, that which contains in itself more reality, cannot be the effect of the less perfect; and this is not only evidently true of those effects, whose reality is actual or formal, but likewise of ideas, whose reality is only considered as objective. ^[4]

If it were true, the unlimited substance (God) that produced ego would have even more reality than the limited substance (ego) whose reality is beyond doubt. Is this argument really valid?

Causes precede their effects in terms of time. The further you trace back to the past, the more uncertainties you face. In this sense causes have less reality than their effects. When it comes to the relation between ground and conclusion instead of that between cause and effect, you can say the former is more certain than the latter. Even if you adopt this interpretation, you cannot say God has more reality than ego, because the order of Descartes’ Meditations on First Philosophy suggests the existence of ego is the ground and that of God is the conclusion drawn from it. If the existence of God were the ground and that of ego were the conclusion, why had not Descartes first demonstrated the former before demonstrating the latter?

Defining a god as “first cause” or “prime mover” and demonstrating his existence in terms of the causal relationship is called the cosmological argument. Descartes tried to combine the ontological argument with the cosmological, but since Descartes thought the existence of God could be demonstrated like mathematical truth, he should not have resorted to the causal relation.

The cosmological argument is a topic of another page. Let’s focus on Descartes’ ontological argument. According to Descartes, God is so complete that its essence includes its existence.

Indeed such a doctrine may at first sight appear to contain more sophistry than truth. For, as I have been accustomed in every other matter to distinguish between existence and essence, I easily believe that the existence can be separated from the essence of God, and that thus God may be conceived as not actually existing. But, nevertheless, when I think of it more attentively, it appears that the existence can no more be separated from the essence of God, than the equality of its three angles to two right angles from the essence of a rectilinear triangle, or the idea of a mountain from that of a valley; so that it is not less impossible to conceive a God, that is, a supremely perfect being, to whom existence is awanting, or who is devoid of a certain perfection, than to conceive a mountain without a valley. ^[5]

Granted that the essence of something includes its existence, does it follow that the object of the essence really exists? The concept “existence” means existence and we can safely say that the essence of this concept includes existence, but as “existence” is an ideal concept, the ideal object of the concept itself cannot exist. Existing things can exist, but whether what I think exists really exists or not cannot be decided just by examining the concept of the subject. So, the concept “an existing thing” just refers to an existing thing as a possible object and does not prove that it really exists. Similarly the concept “the existing god” just refers to an existing god as a possible object and does not prove that it really exists.

Of course, Descartes defined God not as an existing thing but as “a supremely perfect being (ens summe perfectum).” Although the descriptive meaning of “perfect” without evaluative meaning is different from that without it, Descartes confused these two meanings in the name of the ambivalent word “perfect”. As for the former there could be two meanings and therefore we can make the following two interpretations of why the supremely perfect being must exist.

1. Perfect means boundless, that is to say, unlimitedly large. In this case, Descartes’ ontological argument is equal to that of Anselm.
2. Perfect means complete with all sorts of predicates. In this case, the ontological argument is as follows: God has all predicates because perfect. Therefore He has the predicate “exist”. So, God exists.

The second interpretation is no less problematic as the first one. If God had all predicates, He should also have such predicates as “imaginary” or “incomplete”. Since most of predicates have their antonyms, God would be a being that includes all contradictions. If God tried to avoid contradiction, He would not be a supremely perfect being.

The first interpretation also incurs a similar problem. If God were so large as to include not only the entire real world but also the ideal world, contradicting thoughts in the ideal world would be united into one being, God. According to Hugh Everett’s many-worlds interpretation, the universe is a quantum superposition of infinitely many worlds. If we adopt this interpretation, contradicting quantum worlds would be united into one being, God. Thus God would be self-contradicting in both real and ideal worlds.

To sum up, which interpretation you may adopt, the supremely perfect being without the evaluative meaning has nothing to do with that with the evaluative meaning, namely the omniscient and omnipotent supreme being. Since intelligence and volition is based on selection of one among many possibilities, a whole being that embraces all possibilities and all predicates has the largest information entropy and cannot make any judgments. So, the supremely perfect being without the evaluative meaning is disorder that can neither know nor do anything – a being quite opposite to the omniscient and omnipotent supreme being.

3 : Critique of Kant’s critique

Kant criticized the ontological argument in Critique of Pure Reason. According to him, the judgment, “God exists”, is either analytic or synthetic, as every judgment is. In the former case the being of God remains within thought. In the latter case the truth of this judgment cannot depend solely on the law of contradiction and therefore the ontological argument that the judgment “God does not exist” is self-contradictory is invalid^[6].

The conclusion of the ontological argument, at least that of Descartes, is not analytic. First of all the judgment cogito ergo sum, which Descartes regarded as the most certain, is not analytic. The world without ego is theoretically possible, but now that the existence of thinking ego is a fact, it is absolutely impossible to think that there is no thinking ego. So long as the ontological argument is based on the experience of Descartes, his judgment should be considered to be synthetic.

Descartes believed that the object of thought existed as the real world outside the thought, but even if this supposition were false, the real world would necessarily exist. Thought can be unreal so long as the real world exists outside the thought. If there were no real world outside thought, the world within the thought would be no longer a fiction. In short if all were dreams, the dreams would not be unreal dreams but reality. Since the whole world including ego has at least partially reality, the god defined as the whole world proves to exist.

Kant took money as an example to emphasis the absurdity of the ontological argument and distinguish real objects from possible concepts. According to Kant “be (sein)” is not a real predicate and adding it to the subject does not change its concept.

In my financial position no doubt there exists more by one hundred real dollars, than by their concept only (that is their possibility), because in reality the object is not only contained analytically in my concept, but is added to my concept (which is a determination of my state), synthetically; but the conceived hundred dollars are not in the least increased through the existence which is outside my concept. ^[7]

Kant recapitulated his critique of the ontological argument as follows.

Time and labour therefore are lost on the famous ontological (Cartesian) proof of the existence of a Supreme Being from mere concepts; and a man might as well imagine that he could become richer in knowledge by mere ideas, as a merchant in capital, if, in order to improve his position, he were to add a few noughts to his cash account. ^[8]

Kant’s argument of 100 dollars is famous. It is, however, not appropriate to refer to money in order to distinguish real objects from possible concepts. Kant probably thought of a silver coin of Thaler as the real object of money, but the essence of dollar consists not in the metal mass but the promise to exchange anything worth the assigned value with it in the future. If a sincere man promises you to give you a present worth 100 dollars, your financial position amounts to possessing 100 dollars although you do not possess any real objects. On the other hand even if you possess 100-dollar coins as real objects, they will become mere metal mass when the government that issues them collapses. So, money is originally a possible concept rather than a real object.

Of course, I cannot make the possible concept of 100 dollars my property by just wishing I had 100 dollars. Still whether I possess 100 dollars or not depends not on whether they are real objects or possible concepts but on whether the possession is intersubjectively approved or not. Likewise whether a god exists or not depends not on whether it is a real object or a possible concept but on whether the god is intersubjectively approved or not.

Etymologically “exist” means “stand outside”, ex-sistere in Latin. Anselm, Descartes and Kant assumed the existence of the god to be standing outside consciousness, but what is important for the god to function as God and necessary to transcend the mere possible concept is to stand outside the consciousness of an individual person, that is to say, stand inside the social consciousness. The god does not need to stand outside the social consciousness.

It does not follow that religion must confine itself inside consciousness. Believers of a god often create its idol and worship it. The idol, however, just symbolizes the god and it is not the god itself. Similarly money is represented by a piece of metal or paper. Such a token just symbolizes money and it is not money itself. The essence of a god or a currency does not lie in any real object. When everyone stops believing in the god or the currency, it disappears and the real object that symbolized it becomes just a thing.

Both god and money have function of redressing the imbalance in exchange. So long as money is trusted, money can redress the imbalance of production and consumption in terms of assets or debts. So long as the god is trusted, he can redress the imbalance of good and bad conducts in terms of grace or punishment.

Grace or punishment should originally be carried out by secular powers. They are, however, neither omniscient nor omnipotent and as a result a good person often becomes unhappy while a bad person can become happy. To maintain moral order, a supreme being is required that is omniscient enough to observe every conducts and omnipotent enough to fulfill complete grace and punishment. This is why the god, the omniscient and omnipotent supreme being, is socially necessary.

The unlimitedness of god no more requires that he should be the whole world than the universal exchangeability of money requires that money should be identical with the entire commodities. Although money was one of commodities that had value for a specific usage at the primitive stage of economy, it has evolved into a universal medium without any intrinsic value since the birth of fiat money. Although a god was one of real objects at the primitive stage of religion, he has evolved into a universal medium without any visible reality since the birth of Judaism.

Kant gave up theoretically demonstrating the existence of God and morally postulated the existence of God so that those who abide by moral laws could be happy^[9]. Now that Kant regarded God as such, he should have ascribed the error of the ontological argument not to the confusion of an analytic judgment with a synthetic one but to identifying what is not God with God.

4 : References

1. ↑ Anselmus Cantuariensis Proslogion, 2 Quod vere sit Deus (author) Anselm of Canterbury
2. ↑ The Language of Morals (author) R. M. Hare (page) 111-126
3. ↑ Meditationes de prima philosophia, Meditatio III, 23 (author) René Descartes
4. ↑ Meditationes de prima philosophia, Meditatio III, 14 (author) René Descartes
5. ↑ Meditationes de prima philosophia, Meditatio V, 8 (author) René Descartes
6. ↑ Kritik der reinen Vernunft (author) Immanuel Kant (media) Akademieausgabe (volume) 3 (page) 400
7. ↑ Kritik der reinen Vernunft (author) Immanuel Kant (media) Akademieausgabe (volume) 3 (page) 401
8. ↑ Kritik der reinen Vernunft (author) Immanuel Kant (media) Akademieausgabe (volume) 3 (page) 403
9. ↑ Kritik der praktischen Vernunft (author) Immanuel Kant (media) Akademieausgabe (volume) 5 (page) 124

1 : The rise and fall of phlogiston theory

The phlogiston theory is an obsolete chemical hypothesis that tries to explain why combustion changes substances. According to this theory, all flammable materials contain an element called phlogiston and, when a substance is burned, its phlogiston is released and the remaining ash is held to be its true form.

The phlogiston theory was first advanced by Johann Joachim Becher, a German chemist and physician, in 1667^[1]. It was developed and popularized by Georg Ernst Stahl, a German chemist and physician, in 1697^[2]. Becher named the flammable element terra pinguis, fatty earth in Latin, and Stahl renamed it phlogiston, a combustible element in Greek. The phlogiston theory received strong and wide support among the Western scientists throughout the 18th century.

Johann Joachim Becher (left) and Georg Ernst Stahl (right), partially quoted. Source: The Engines of Our Ingenuity and Polarlys

According to this theory, substances lose weight when they are burned to ash because they have released phlogiston. Metals, however, turned out to gain weight when they were burned. Some phlogiston supporters insisted that phlogiston should have negative weight in this case, but it is too arbitrary to assume the weight of phlogiston positive or negative according as the sort of substances.

In 1783 Antoine Lavoisier, who proved the principle of conservation of mass, refuted the phlogiston theory^[3] and proposed the oxygen theory of burning. Still some scientists such as Joseph Priestley adhered to the phlogiston theory. Why was this theory so attractive to those in the 18th century?

2 : The relation to witch-hunts

Investigating the historical background of phlogiston theory, we will notice that the German witch-hunt reached its zenith when Becher advanced the phlogiston theory in 1667. The peak years of witch-hunts in southwest Germany, where he was born and lived, were from 1561 to 1670^[4]. The large-scale witch trials and executions waned thereafter and the death sentence for witchcraft ended in German in 1775. Both the phlogiston theory and witch-hunts in Europe were over at the end of the 18th century.

The alleged witches were burned at the stake in Germany and most of other areas. The picture below depicts execution of witches in Switzerland in the 16th century. Those in the 18th century would thought the smoke ascending from witches should include phlogiston.

Burning of three witches in Baden, Switzerland (1585) by Johann Jakob Wick. Source: Dietegen Guggenbühl: Hexen (medium) Sandoz-Bulletin 24 (date) 1971 (page) 38

Death by burning had been the official punishment for heresy since the Roman Catholic Synod of Verona legislated in 1184. It was also the traditional punishment for women found guilty of treason (murder of her husband) in the United Kingdom. The Malleus Maleficarum, the most famous treatise on witches, first published in Germany in 1487, says that witches are burned at the stake because the majority of them are women^[5].

Burning wounds was an Ancient medical treatment to disinfect them and stop bleeding. Since crime tears open a wound of a social system, it is symbolically significant for the power holder to burn the criminals at the stake so as to stop the aggravation of the wound and show off the recovery of the healthy social order. Fire has been the symbol of the male sex from ancient times and it is even more symbolically significant for the paternal religion such as Christianity to burn heretics at the stake who worship the maternal religion and women guilty of treason, especially female witches that meet these two conditions.

We can find the idea of purifying fire (purgatorius ignis) also in Catholic doctrine of purgatory, where souls could be purged from venial sins in life and achieve the holiness necessary to enter heaven.

An image of souls being purified by flames in purgatory (Die Stadtkirche in Bad Wimpfen) Source: Peter Schmelzle

The witches were thought to be polluted by Satanism and persecutors burned them at the stake in order to purge them from Satanism. So we can say it is a kind of exorcism. An exorcism is the act of driving out Satan, demons, evil spirits and the like from possessed persons.

Saint Francis Borgia performing an exorcism painting by Francisco Goya, partially quoted. Source: Gerald Farinas

If we interpret burning witches at the stake as an exorcism, we can find the origin of the phlogiston theory here. Burning witches at the stake expels Satanism from them, lightens their sins as well as their physical weight and reduces them to their true forms before they sold their souls to Satan. Generalization of this idea result in the phlogiston theory: burning materials expels phlogiston from them, lightens their weight and reduces them to their true forms before they are possessed by phlogiston.

3 : The relation to antiphlogistic bloodletting

Another notable background of the phlogiston theory is bloodletting that was in fashion in those days. Bloodletting or phlebotomy is the withdrawal of blood from a patient in order to cure or prevent disease.

A photo of bloodletting in 1860, partially quoted. Source: Rare and Unusual Photos and Images From the Burns Archive (publisher) Newsweek

Note that both of the proponents of phlogiston theories, Becher and Stahl, were physicians. Physicians in those days attributed various sorts of diseases to excessive blood and surgeons or barbers were apt to resort to bloodletting in order to heal them.

It was observed that local infections healed when they burst and bled, so lancing was used to reduce local infection, referred to as an excess or “plethora” in Hippocratic terms. Similarly, bloodletting used to reduce extreme fevers was called antiphlogistic bloodletting. Antiphlogistic bloodletting and therapeutic seasonal bloodletting were common to many early traditional systems. Bloodletting fell out of practice in the West in the early twentieth century as a consequence of overzealous practice: the sicker the patient, the more the letting.^[6]

Fever and inflammation are analogous to burning and phlogiston corresponds to blood. Becher and Stahl might think just as bloodletting lightened fever and inflammation and reduced patients to their true healthy forms, letting phlogiston out would lighten materials and reduce them to their true forms, nonflammable ash.

When a body is infected with pathogens such as bacteria and viruses, it causes inflammation, which is the response of the organism to remove the pathogens rather than a symptom caused by infection and results from the increased movement of plasma and leukocytes from the blood into the injured tissues. It is conceivable those who did not have such a medical knowledge attributed the red swell to an excess of blood and tried to allay inflammation by means of bloodletting. Discharging pathogens as well as blood might heal infections. It is natural that they applied bloodletting to lower fevers in general.

Of course it has no medical rationale that bloodletting is effective in the treatment of fevers in general. Bloodletting is used today in the treatment of only a few diseases such as hemochromatosis and polycythemia. Its practice was, however, performed by doctors from antiquity up to as late as the end of the 19th century because its catharsis effect in addition to a mere placebo effect had a positive mental influence on patients. A patient suffering from a high fever could be under the illusion that, seeing blood whose color is the same as that of fire spout from the wound, he or she was discharged from fever in combination with blood. The same logic was applied to other alleged causes of diseases as if bloodletting had been a panacea.

4 : The age of catharsis

Witch-hunts hysteria reached its zenith during the 17th century, the phlogiston theory during the 18th century and bloodletting during the beginning of the 19th century. The period between the 17th century and the beginning of the 19th century coincided with the Little Ice Age when the Western people suffered from cold weather. A crop failure because of low temperature fomented social unrest. People made scapegoats of witches, purged them from the community (or strictly speaking, Satanism from them) and tried to reduce social entropy by the catharsis effects. As witch-hunts were too unscientific, they faded out in the 18th century. Still the desire for catharsis did not fade out and pseudo-scientific bloodletting thrived. Those who felt uneasy about causes of diseases being invisible made a scapegoat of visible blood and purged it from their bodies. The background of the phlogiston theory was the desire for catharsis exemplified in witch-hunts and bloodletting in the early modern era.

5 : References

1. ↑ Physica subterranea, 6.5 De Decompositis terreis, siccis & liquidis, pinguibus & macris (author) Johann Joachim Becher
2. ↑ Zymotechnia fundamentalis sive fermentalionis theoria generalis (author) Georg Ernst Stahl
3. ↑ Réflexions sur le phlogistique pour servir de suite à la théorie de la combustion et de la calcination (author) Antoine-Laurent Lavoisier
4. ↑ Witch Hunting in Southwestern Germany, 1562-1684: The Social and Intellectual Foundations (author) H. Midelfort (page) 71
5. ↑ The Malleus Maleficarum, Part I, Question I (author) Heinrich Kramer, Jacob Sprenger
6. ↑ Integrative Medicine (author) Benjamin Kligler, Roberta Lee (page) 184

1 : Who first invented the Cartesian coordinate system?

A Cartesian coordinate system is a mathematical system that identifies the position of each point by its distance from a set of perpendicular lines that intersect at the origin of the system. The following figures show examples of Cartesian coordinate systems for 2D (left) and 3D (right)

Illustration of the Cartesian coordinate systems for 2D (left) and 3D (right). Source: K. Bolino and Jorge Stolfi

The adjective Cartesian is attributed to René Descartes, whose family name was Cartesius in Latin. Thanks to this name Descartes is usually thought to be the first inventor of the Cartesian coordinate system and even a plausible legend flourished that Descartes, lying on the bed, thought of the coordinate system so as to track the movements of a fly on the wall.

But Descartes did not use the term coordinate nor did he explain the idea equivalent to it explicitly in his writings. Of course it is true that together with Pierre de Fermat he is the founder of modern analytic geometry, which is often called Cartesian geometry. Although geometry developed highly in Ancient Greece, geometrical laws they discovered was scarcely analyzed algebraically before Descartes. Descartes used letters from the beginning of the alphabet, such as a, b, c…, for constants and those near the end of the alphabet, such as x, y, z…, for variables, and expressed a line as an equation composed of these alphabets. This practice continues today.

Let me introduce an example quoted from The geometry of Descartes in 1637.

The geometry of Descartes. Source: La Géométrie (author) René Descartes (page) 321

Suppose the length of AG, KL, NL, AB, BC is respectively a, b, c, x, y, of which a, b, c are constants and x, y are variables that can vary according to a rotary movement of the ruler GL around G. The following function of x and y indicates that the locus of the point C is a quadratic curve.

If you regard AK and AG as x-axis and y-axis that intersect at the origin A, this equation is the function of the locus of the point C identified by a Cartesian coordinate system. But Descartes himself regarded x and y as just a variable length and he did not locate the point by reference to any Cartesian coordinate system. It was Gottfried Wilhelm Leibniz’s article in 1692 that first treated a line as a function (functiones) specified with reference to coordinate (coordinata) with x-axis (absciss) and y-axis (ordinata) ^[1].

Some might imagine that Descartes presupposed the coordinate system to set up the equation without explaining it explicitly in his writings. Even if it should be the case, it would be doubtful whether Descartes was the first inventor of the Cartesian coordinate system, because similar systems had been used before the age of Descartes.

For example, the system of latitude and longitude that Eratosthenes used in his world map created in the year -193 is an orthogonal coordinate system. As the Earth is a sphere, the identification system of latitude and longitude itself is a non-Cartesian orthogonal coordinate system. Eratosthenes, however, recognized that the Earth is round and he created a 2D map, projecting the sphere surface on the plane. The following is Eratosthenes’ map of the known world and the cylindrical projection made his map a Cartesian coordinate system with latitude and longitude.

The modern reconstruction of Eratosthenes’ map of the known world. Source: World map according to Eratosthenes

Descartes did not first invented the Cartesian coordinate system nor did he first establish analytic geometry on the ground of the Cartesian coordinate system. Should we stop calling the coordinate system Cartesian? No. It deserves its name, because we can consider the Cartesian coordinate system to be an application of Cartesian philosophy to mathematics. In order to recognize this, let’s look back upon the philosophical method of Descartes.

2 : How did Descartes establish the foundation for knowledge?

Descartes proposed four precepts to reach the truth in Discourse on the Method.

The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.

The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.

The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.

And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted. ^[2]

The order of the first and the second should be reverse. Our research starts from uncertainty. It is not until we encounter uncertainties that we divide them into elements in order to reduce uncertainty and spread certainty step by step to the whole. The process of methodological skepticism followed the second precept, the discovery of “I think, therefore I am” was the result of the first, the proof of the existence of God and the foundation of the scientific knowledge on it resulted from the third and its verification put the fourth into practice.

Let’s see this process in detail. What Descartes thought the most certain was the ego as res cogitans. The existence of the self-conscious ego is evident to itself in spite of and because of the radical skepticism and Descartes chosen it as the starting point of his philosophy and science. But, if what he found was that the existence of Descartes’ consciousness was evident to Descartes’ consciousness, it was too trivial to be called the truth. So, starting from the ego as res cogitans, he tried to prove the existence of God and thereby establish the foundation of scientific knowledge.

According to Descartes, if the existence of the finite ego is evident, the existence of the infinite, namely God, must be even more evident.

And I must not imagine that I do not apprehend the infinite by a true idea, but only by the negation of the finite, in the same way that I comprehend repose and darkness by the negation of motion and light: since, on the contrary, I clearly perceive that there is more reality in the infinite substance than in the finite, and therefore that in some way I possess the perception of the infinite before that of the finite, that is, the perception of God before that of myself.^[3]

This is an ontological argument for the existence of God and the proof is not correct, but let’s put it aside here. Descartes doubted even the mathematical truth, supposing a cunning demon that had conceived him. If, however, the existence of God is proved, we have no reason to suppose such a demon and can believe the sincerity of God.

For, in the first place, I discover that it is impossible for him ever to deceive me, for in all fraud and deceit there is a certain imperfection: and although it may seem that the ability to deceive is a mark of subtlety or power, yet the will testifies without doubt of malice and weakness; and such, accordingly, cannot be found in God.

In the next place, I am conscious that I possess a certain faculty of judging [or discerning truth from error], which I doubtless received from God, along with whatever else is mine; and since it is impossible that he should will to deceive me, it is likewise certain that he has not given me a faculty that will ever lead me into error, provided I use it aright. ^[4]

In this way, he started form the ego as a thinking thing and reached the truth of the world through the medium of the Almighty God.

3 : Why does the Cartesian coordinate system deserve its name?

We have seen how Descartes established the foundation for scientific knowledge in general. The Cartesian coordinate system applies a similar method to geometry, which explains why the Cartesian coordinate system is Cartesian.

The starting point of Cartesian philosophy, the ego, corresponds to the origin of the Cartesian coordinate system. This correspondence is not arbitrary, because the origin of the origin is the ego. That is to say the origin of the visual 3D space was originally located at the ego and, even if it is not physically at the position of the ego, one understands it, imagining transferring it from the position of one’s ego to the standpoint of alter ego.

Human beings have made a conceptual distinction between the top and the bottom, the front and the rear, the right and the left before Descartes or Eratosthenes. This indicates that we have located qualitatively, if not quantitatively, the position in 3D space around us in reference to a naïve Cartesian coordinate system where the ego forms the origin, the gravity direction the top/bottom axis, the direction of sight perpendicular to it the front/rear axis and the direction perpendicular to both the right/left axis.

As the visual image we have is based on a perspectively distorted projection of 3D space onto 2D retina, it does not correctly represent the object. Although the Almighty God could perceive the entire object immediately and correctly, it is impossible for human beings with limited recognition ability. All we can do is approach the objective truth, transferring the origin and thus changing the view.

I will explain it by means of a simple example. There are black and red coordinate axes in the graph below and the coordinates of the triangle ABC and its center of gravity (mass center) G differ in relation to the axes.

The choice of the axes

Suppose vectors of the points whose origin is the black O are a, b, c and g. Since O, A, B are on the same line, the equation below is true.

Since G is the center of gravity, the equation below is also true.

If vectors of the points whose origin is the red O are a, b, c and g, the former equation is false, but the latter is true. If your eyes are located at the black O, you might have an illusion that A and B are identical, which you will find false, when you see them from the viewpoint of the red O. Although the coordinates of A, B, C and G vary in relation to an origin O, the relation between the triangle and its mass center is independent of it. Still, as you must set the origin and axes anywhere to prove it, the origin and axes are methodologically necessary.

If the ego as the limited is the origin, God as the unlimited is the coordinate axes that have unlimited extension and the world that the ego can recognize by means of God is the space that the origin can specify by means of axes. The unlimitedness of axes enables us to link the limited origin to any point in the unlimited space. This unlimitedness, however, is not that of God. To approach the complete unlimitedness of God, the ego must transcend its singularity as the origin and reach the universality of the truth.

Descartes pursued the universality beyond the singularity of starting points not only in philosophy but also in geometry. The quadratic equation is based on the assumption that AK is the x-axis and AG is the y-axis in the figure in La Géométrie, but the locus of the point C is a quadratic function no matter where the origin and axes are set up. Descartes’ assertion that curves should not be classified according to tools for drawing them but according to their order is still valid today.

If Descartes had applied his method for founding his philosophy to geometry, he might have been the founder of the Cartesian coordinate system in reality as well as in name. In reality, however, he was not aware of the Cartesian coordinate system and it was Leibniz who succeeded to Cartesian geometry as well as Cartesian philosophy and founded Cartesian geometry on the Cartesian coordinate system.

4 : References

1. ↑ De linea ex lineis numero infinitis ordinatim ductis inter se concurrentibus formata, easque omnes tangente, ac de novo in ea re Analysis infinitorum usu (author) Gottfried Wilhelm Leibniz (journal) Acta Eruditorum, vol.11 (date) 1692 (page) 168-171 (medium) Mathematische Schriften, Bd. 5 (editor) Carl Immanuel Gerhardt (page) 266-269
2. ↑ Discours de la méthode, Deuxième partie (author) René Descartes (page) 141-142
3. ↑ Meditationes de prima philosophia, Meditatio III (author) René Descartes (page) 24
4. ↑ Meditationes de prima philosophia, Meditatio IV (author) René Descartes (page) 2-3

1 : Humean skepticism on objectivity of causality

What is essential for an object to be a cause of another? According to Hume, contiguity, succession and necessity are essential^[1]. Hume doubted the objectivity of causality. In fact we cannot perceive the tie by which causes and effects are united. We can infer, if not directly perceive, contiguity and succession from what we perceive more easily than necessity. From what can we then infer the necessity of causality? Hume thinks it is a frequent repetition of an object after another.

For after a frequent repetition, I find, that upon the appearance of one of the objects, the mind is determin’d by custom to consider its usual attendant, and to consider it in a stronger light upon account of its relation to the first object. ‘Tis this impression, then, or determination, which affords me the idea of necessity.^[2]

If the relation between causes and effects is a subjective connection, its necessity is not objective and a cause might produce an effect different from the usual one.

When I see, for instance, a billiard-ball moving in a straight line towards another; even suppose motion in the second ball should by accident be suggested to me, as the result of their contact or impulse; may I not conceive, that a hundred different events might as well follow from that cause? May not both these balls remain at absolute rest? May not the first ball return in a straight line, or leap off from the second in any line or direction? All these suppositions are consistent and conceivable. Why then should we give the preference to one, which is no more consistent or conceivable than the rest? All our reasonings a priori will never be able to show us any foundation for this preference.^[3]

Is this Humean skepticism right? I think the necessity of the connection is not a necessary condition to be the causal connection but a sufficient condition not to be the causal connection. It is not the case that the connection between causes and effects is uncertain because of its subjectivity but that the subject must divide an object into causes and effects because of its uncertainty. If a connection between a cause and an effect gets necessary, the cause and the effect will be regarded not as a cause and an effect but an object.

For example, although pushing a button of a remote controller causes TV broadcasting, we usually describe it just “turning on a TV”. When do we then divide this one object into two, pushing a button of a remote controller and the start of TV broadcasting? It is when we face uncertainties, that is to say, when we are not accustomed to a remote controller, when we have it repaired and so on.

Division into two is sometimes insufficient. According to Hume, causes must be contiguous to their effects. This means a chain of causes must link an effect to a cause, if the latter is remote from the former.

Tho’ distant objects may sometimes seem productive of each other, they are commonly found upon examination to be link’d by a chain of causes, which are contiguous among themselves, and to the distant objects; and when in any particular instance we cannot discover this connexion, we still presume it to exist. ^[4]

How far must we divide the chain and how contiguous must the minimum unit of a cause and an effect be, then? To a scale of an atom or quark? Instead of being lost in such a scholastic maze, we should reconsider the original motive of our dividing causes and effects. Now that the division is motivated by reduction in uncertainty, the chain should be divided until we can dispel it.

Let’s return to the example of the remote controller. When pushing a button of a remote controller stops turning on a TV and it is not what you expect, you experience an increase in uncertainty and you must divide the object into a chain of causes in order to find a true cause of malfunction. Thus you divide the simple chain,

into a more complex one,

and confine the true cause of uncertainty to a smaller segment of the uncertain whole, as if, when an infectious disease is prevalent, we try to find the infected and isolate them to prevent the disease from spreading.

Uncertainty is the state that can be otherwise than is expected. In order to reduce the uncertainty our information system called intellect divides the object that can be otherwise than is expected into a chain of causes, verifies that each causal connection cannot be otherwise than is expected and finds the part that is otherwise than is expected. If you notice the diode is broken, you can reduce uncertainty by replacing it with a new one. Once the situation cannot be otherwise than is expected, the chain of causes shrinks like an accordion and turns into a single object “turning on a TV”.

2 : Humean skepticism on validity of moral judgment

Hume is also famous for skepticism on deriving “ought” from “is”. The assertion quoted below is said to take the lead of the 20th century meta-ethics arguments that makes a sharp distinction between fact and value.

In every system of morality, which I have hitherto met with, I have always remark’d, that the author proceeds for some time in the ordinary ways of reasoning, and establishes the being of a God, or makes observations concerning human affairs; when all of a sudden I am surpriz’d to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not. This change is imperceptible; but is however, of the last consequence. For as this ought, or ought not, expresses some new relation or affirmation, ’tis necessary that it shou’d be observ’d and explain’d; and at the same time that a reason should be given; for what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it. But as authors do not commonly use this precaution, I shall presume to recommend it to the readers; and am persuaded, that this small attention wou’d subvert all the vulgar systems of morality, and let us see, that the distinction of vice and virtue is not founded merely on the relations of objects, nor is perceiv’d by reason.^[5]

Humean skepticism on the conventional moral philosophy that have tried to derive “ought” from “is” of necessity is similar to that on the conventional natural philosophy that have tried to derive effects from causes of necessity and we can find a similar reversion here. That is to say, Hume thinks that, since “is” and “ought” are quite different and the connection between them is subjective, it has no objective necessity. In fact, it is the other way around: since its necessity is called into question, the subject must divide it into “is” and “ought” and recombine them to reduce its uncertainty. If you feel the connection between “is” and “ought” necessary, they must keep intuitive unity.

To take an illustration, murder is a bad thing beyond question for ordinary people, who cannot kill others without trembling and agony. At this intuitive level, the fact “It is murder” is inseparable from the duty “I ought not to do it.” In a special situation, however, say, on the battlefield where your hesitation in killing the enemy is likely to result in death of your side, the intuitive unity of the fact and the duty collapses and you must separate the fact from the duty.

But this separation increases uncertainty, because it follows that murder is sometimes wrong and sometimes not. To reduce this uncertainty, you must divide the fact that can be otherwise than is expected into segments, verifies that each connection with the duty cannot be otherwise than is expected and finds the part that is otherwise than is expected. If you conclude that we must not kill others unless the killing can prevent killing us, you can reduce the uncertainty.

3 : Uncertainty precedes the division into elements

It is not certainty but uncertainty that makes us conscious of the connection between causes and effects and the division into them itself is motivated by the reduction in uncertainty. It is not certainty but uncertainty that makes us conscious of the connection between “is” and “ought” and the division into them itself is motivated by the reduction in uncertainty. Generally speaking, an increase in uncertainty motivates a conscious system to divide a single whole object into elements to reduce the uncertainty and, after the division is complete, the system tends to think that first there are elements and then complexity (uncertainty) arises from the combination between them.

I know that the ontological relation between cause and effect corresponds to the deontological relation between means and purpose and the axiological relation between fact and value is similar to the epistemological relation between fact and truth. In this sense my comparison between Humean skepticism on objectivity of causality and his skepticism on validity of moral judgments may not be strict. But generalization of Humean reversion makes this difference unimportant.

We can find Humean reversion in the conventional explanation by systemics researchers. They have explained uncertainty in terms of complexity, the number of combination of elements. This explanation forgets the original arche of philosophy. It is not the case that an increase in elements increases the complexity and makes systems to reduce the complexity but that an increase in uncertainties makes systems to divide uncertainties into elements in order to reduce them.

In the Middle Ages alchemy was in fashion. They tried to transform copper and other base metals into gold, only to fail. The uncertainty of the world for those who believed in magic and alchemy was high. Modern chemistry reduced this uncertainty by dividing chemical compounds into chemical elements and showed that alchemists can only change compounds and cannot make new elements like gold from other elements. First there were not elements but uncertainties and the division into elements reduced the uncertainty of predicting chemical reactions.

It is true that the number and sorts of elements increases complexity but the complexity has only the reduced uncertainty, which is much lower than that before the division into elements. When systemics researchers talk of complexity, its uncertainty has been already reduced to a considerable degree. We should not forget that the original arche for systemics is not elements but uncertainty.

4 : References

1. ↑ A Treatise of Human Nature, Book1. Part3. Section2. (author) David Hume
2. ↑ A Treatise of Human Nature, Book1. Part3. Section14. (author) David Hume
3. ↑ An Enquiry Concerning Human Understanding, Section4. Part1. (author) David Hume
4. ↑ A Treatise of Human Nature, Book1. Part3. Section2. (author) David Hume
5. ↑ A Treatise of Human Nature, Book3. Part1. Section1. (author) David Hume

1 : The philosophy of origin and the origin of philosophy

The term “arche (ἀρχή)” is an ancient Greek word meaning a beginning, origin, first cause, the first place or power, sovereignty, dominion, command and so on^[1]. It was an important theme for the ancient Greek philosophers what the arche was.

The first philosopher in the world is usually thought to be Thales of Miletus. As he left no writings, we cannot know the details of his theory, but according to the reports of his contemporaries he was surely the philosopher that tried to explain rationally that water is arche of the cosmos. Thus philosophy began as the philosophy of the beginning.

Thereafter Anaximenes regarded arche as air, Xenophanes earth, Heraclitus fire, while Empedocles adopted these four elements (water, air, earth and fire) as arche. Aristotle succeeded to the Empedocles’ theory and thanks to Aristotle’s authority the theory of the four elements had dominated the European intellectuals for a long time until it was abandoned in the early modern times.

It was, however, more abstract theory, atomism, proposed by Leucippus, the successor of Empedocles, and Democritus, Leucippus’ student, that modern scientists succeeded to. Today we no longer consider atoms to be indivisible as the Greek word “ἄτομος” originally means, but divisible into elementary particles. Are the elementary particles such as the quarks, leptons, and gauge bosons the ultimate arche, then? No one except naïve materialists would approve of it, because we do not think of this world as a mere gathering of materials.

Of the four classical elements fire alone is non-material. Werner Heisenberg interpreted Heraclitus’ arche, fire, as energy and sought some common ground between Heraclitus’ philosophy and modern physics.

Since mass and energy are, according to the theory of relativity, essentially the same concepts, we may say that all elementary particles consist of energy. This could be interpreted as defining energy as the primary substance of the world. It has indeed the essential property belonging to the term ‘substance’, that it is conserved. Therefore, it has been mentioned before that the views of modern physics are in this respect very close to those of Heraclitus if one interprets his element fire as meaning energy. Energy is in fact that which moves; it may be called the primary cause of all change, and energy can be transformed into matter or heat or light. The strife between opposites in the philosophy of Heraclitus can be found in the strife between two different forms of energy.^[2]

It is true that energy conserves the quantitative identity as substance because of the first law of thermodynamics, but energy is still not all of the world. The world includes information besides matter and energy. Information is the lowness of entropy and entropy, unlike matter, cannot be reduced to energy. So, we cannot say energy is arche.

Since physics is an empirical science and its theory changes with the times, the theory of arche based on physics also can change with the times. Can’t we think of the philosophical arche that never changes with the times? Thus thinking and searching for the ultimate arche, a philosopher made a beginning for Modern Philosophy – namely René Descartes.

2 : Philosophy begins with wonder and skepticism

Descartes, who found the traditional science based on uncertain principles, sought for “the first philosophy (prima philospohia)” or “the philosophical principle (principia philosophiae)” that was absolutely certain and could be the foundation of all the knowledge. The word “principle” is the combination of “first (prim-)” and “take (capere)”. So, both prima philospohia and principia philosophiae meant philosophical arche to Descartes. Not only Ancient Philosophy but also Modern Philosophy began with the investigation into arche.

Descartes adopted methodological skepticism to find the philosophical arche. The first philosopher that adopted this method was not Descartes but Parmenides. Like an ancient Descartes, he asked himself what, if anything, it was impossible not to believe; and to him the answer was est: something exists^[3]. Descartes was different from Parmenides in attributing arche not to the recognized object but the recognizing subject. He doubted everything, not only the incredible knowledge based on sense but also mathematical truth, which he had believed to be the most certain. The radical skepticism led him to the conclusion that the existence of the doubting ego became the more certain, the more he doubted. So, he made ego his philosophical arche.

You might be tempted to say that arche for Modern Philosophy from Descartes via Hegel to Husserl was the thinking subject, because they all considered the thinking subject to be the substance that made all the knowledge true, but let’s look back on the process of Descartes’ contemplation. He began his contemplation with skepticism. If he had asserted his conclusion from the beginning without the process of doubt, he would not have made himself understood.

Skepticism is trial to believe the opposite of what you have believed. It is consciousness of uncertainty. Can’t we say then that arche for Descartes’ philosophy is uncertainty? If the world behaved as ego expected, the ego is inseparable from the world. Actually, the world often turns out to be different from what ego expected, which makes the ego conscious of the difference between the world in consciousness and the actual world, that is to say, the recognizing subject distinguished from the recognized object. It is not so much Descartes could not doubt the existence of ego in spite of the radical skepticism as he could be clearly conscious of the existence of ego beyond doubt thanks to the radical skepticism. In this way, Modern Philosophy is founded on the consciousness of uncertainty.

You might think that, as arche originally meant material substance in ancient Greece, ego or uncertainty cannot be arche, but it is not true even of Presocratic philosophers, because the Pythagoreans regarded arche as number. Aristotle wrote concerning the Pythagoreans as follows.

Well, it is obvious that these thinkers too consider number to be arche, both as the material of things and as constituting their properties and states. ^[4]

Since they did not even think of the Aristotelian distinction between matter and form itself in Pythagoras’ days, it is not right to confine the extension of arche to material substance.

Our interpretation of arche as uncertainty applies to ancient Greek Philosophy. While Aristotle succeeded to the Empedocles’ theory of the four elements as we noted, he also said that philosophy began with wonder.

It is through wonder that men now begin and originally began to philosophize; wondering in the first place at obvious perplexities, and then by gradual progression raising questions about the greater matters too, e.g. about the changes of the moon and of the sun, about the stars and about the origin of the universe. ^[5]

As is evident from this quotation, Aristotle thought the philosophical discussion as to “the origin of the universe”, whether arche was water, air, earth or fire, began with wonder. Although Aristotle identified arche with an element (στοιχεῖον), he should distinguish the former from the latter, because the former precedes the latter.

Plato, the master of Aristotle, also said a similar thing in the name of Socrates and interpreted the name of Thaumas, the father of Iris, the messenger of heaven, as “wonder (θαῦμα)”.

For this feeling of wonder shows that you are a philosopher, since wonder is the only beginning [arche] of philosophy, and he who said that Iris was the child of Thaumas made a good genealogy. ^[6]

Why then does philosophy begin with wonder? Wonder is a feeling arising when you find the world more than you expected. So, Plato’s and Aristotle’s recognition is equivalent to our proposition that philosophy begins with the consciousness of uncertainty.

According to Aristotle, however, philosophy is a kind of recreation or pastime and philosophers can pursue science for the sake of knowledge, and not for any practical utility, because practically all the necessities of life are supplied to them^[7]. To be sure, Miletus was rich, when it produced the earliest philosophers such as Thales, Anaximander and Anaximenes. Croesus, king of Lydia, which ruled Miletus at that time, was so rich that his name became synonymous with the wealth, as is expressed in the phrase “as rich as Croesus”. If however the wealth is the only condition for philosophy, why didn’t the rich before Thales in the world began philosophy? We must search for conditions for philosophy from the viewpoint of not only ability but also necessity.

Miletus was a large maritime empire possessing many colonies and traded with foreign countries ranging from Egypt to those around the Black Sea. Trading with various nations in various places, people in Miletus must have felt strongly that what they took for granted in Miletus is not necessarily a matter of course among other nations in other places. Thales became the first philosopher, not only because he was of aristocratic origin and all the necessities of life were already supplied, but also because he went to Egypt to have instruction from priests and experienced different cultures. The stronger one feels the uncertainty of culture, the stronger motivation he or she will have for seeking for certainty that transcends the cultural difference, that is to say, for philosophy.

Descartes mentioned his motive for doubting everything and inquiring into philosophical principle in Discourse on the Method.

So that the greatest advantage I derived from the study consisted in this, that, observing many things which, however extravagant and ridiculous to our apprehension, are yet by common consent received and approved by other great nations, I learned to entertain too decided a belief in regard to nothing of the truth of which I had been persuaded merely by example and custom; and thus I gradually extricated myself from many errors powerful enough to darken our natural intelligence, and incapacitate us in great measure from listening to reason. ^[8]

If Descartes had led a scholastic life detached from the world, he would not have tried such a radical skepticism. He traveled widely throughout Europe to read the great book of the world (le grand livre du monde)^[9] instead of paper books in his study and experienced cultural relativity. Exposure to uncertainty drove him to the investigation into arche.

3 : Arche as apeiron

The first philosopher who used “arche” as a philosophical term was not Thales but his pupil, Anaximander. He claimed that arche is not empirical objects such as water, air, earth and fire, but apeiron (ἄπειρον), that is to say, an indefinite or boundless substance. Anaximander’s work is mostly lost and we can know his theory from a few existing fragments, of which the following is the most famous.

Anaximander, son of Praxiades, a Milesian, the successor and pupil of Thales said that arche, the element of existing things, was apeiron, being the first to introduce this name of arche. He says that it is neither water nor any other of the so-called elements, but some other apeiron nature, from which come into being all the heavens and the worlds in them. And the source of coming-to-be for existing things is that into which destruction, too, happens “according to necessity; for they pay penalty and retribution to each other for their injustice according to the arrangement of time,” as he describes it in these rather poetical terms. It is clear that he, seeing the changing of the four elements into each other, thought it right to make none of these the substratum, but something else besides these; and he produces coming-to-be not through the alteration of the element, but by the separation off of the opposites through the eternal motion. ^[10]

Why is apeiron arche, then? Aristotle demonstrated it as follows.

Everything is either arche or derived from arche. But there cannot be arche of the infinite (apeiron), for that would be a limit of it.^[11]

Does the infinite cease to be infinite if it has its limit? There are, for example, an infinite number of real numbers between 0 and 1. It indicates that the infinite can have its limit. Aristotle’s demonstration holds better, if arche is the whole instead of the infinite. In that case his demonstration would be as follows.

This proposition is just a tautology, “The whole is the whole.” It remains empty unless the content of the whole is identified. Moreover, his interpretation of apeiron as the infinite whole is historically doubtful.

No thought is irrelevant to the traditional thought. Thales, who regarded water as arche, and Anaximander, who regarded apeiron as arche, reflected the influence of the mythical cosmogony of Hesiod, who regarded chaos as arche, and further retrospectively the Babylonian cosmology, “Enuma Elish (Enûma Eliš)”, which described the earliest stage of the universe as watery chaos represented by two gods, Apsû and Tiamat. Taking this tradition into consideration, Anaximander’s concept “apeiron” is not the unlimited whole but chaos without boundary. So, our interpretation of arche as uncertainty has historical legitimacy.

You might still be reluctant to accept this interpretation, for the previous philosophers such as Descartes attributed arche to the principle that negates uncertainty, but the arche of the various concepts of arche is uncertainty and taking what denies uncertainty for arche has come from forgetting the original arche, uncertainty.

It applies not only philosophy but also religion. “Enuma Elish (Enûma Eliš)” described the earliest stage of the universe as follows.

When the sky above was not named,
And the earth beneath did not yet bear a name,
And the primeval Apsû, who begat them,
And chaos, Tiamat, the mother of them both,
Their waters were mingled together,
And no field was formed, no marsh was to be seen;
When of the gods none had been called into being. ^[12]

This myth was formed in Mesopotamia, when matriarchal religion was dominant. The matriarchal religion usually models its cosmogony after a fetus growing in amniotic fluid. The Babylonian cosmogony, where Tiamat is a primordial goddess of the ocean, is also matriarchal. Although Judaism is a patriarchal religion, “Genesis” leaves some traces of matriarchal cosmogony.

1:1 In the beginning God created the heaven and the earth.
1:2 And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.
1:3 And God said, Let there be light: and there was light.^[13]

At the beginning of “Gospel of John” the matriarchal vestiges are wiped out from its cosmogony and the Word (Λόγος) as God is declared to be arche.

1:1 In the beginning was the Word, and the Word was with God, and the Word was God.
1:2 The same was in the beginning with God.
1:3 All things were made by him; and without him was not any thing made that was made. ^[14]

As is evident from these, the precedence of darkness (chaos) over light (order) has been reversed with the times. Thales and Anaximander attributed arche to maternal chaos, while Heraclitus to paternal fire as logos. The relation was completely reversed in Platonic idealism. The true arche of Descartes’ philosophy, uncertainty, was forgotten, when he discovered ego as the philosophical principle, and the uncertainty was denied by the logos of the thinking subject. In spite of this reversal and because of this reversal, arche is uncertainty.

If we are allowed to interpret apeiron as uncertainty, we can find a point of agreement between Anaximander’s philosophy and our systemics. Uncertainty is the unselected mixture of the opposites such as hot versus cold, dry versus wet. Selecting one and excluding the other option result in limiting the unlimited uncertainty and forming the structure of the system. The basic idea of our systemics is that a system is the subject of selection and through the selection a system differentiates its structure from its environment.

Anaximander “produces coming-to-be not through the alteration of the element, but by the separation off of the opposites through the eternal motion.” Simplicius noted, the opposites mean hot and cold, dry and wet etc. ^[15] The combination of the two opposites result in hot + dry, hot + wet, cold + dry, and cold + wet. These four combinations correspond to the four elements, fire, air, earth and water. When something new comes into being, it does not result from altering one element into another, but from choosing one and discarding the other option.

According to Anaximander existing things that came into being from apeiron are destructed and return to apeiron. Most of scientist cannot accept his personified description, “they pay penalty and retribution to each other for their injustice according to the arrangement of time”, but there still remains the room for rational interpretation. Systems sustain themselves through selection, but when they select the wrong option, they fail to sustain themselves, that is to say, they cannot differentiate themselves from their environment and return to uncertainty as arche. Anaximander’s proposition reads as the personification of this.

4 : Arche as uncertainty

Plato and Aristotle were right in recognizing that philosophy begins with wonder. Uncertainty as arche is not the intra-horizontal uncertainty that is conceived of as a probability and does not surprise the expectant. Once conceived in the horizon of consciousness, uncertainty is greatly diminished. Uncertainty as arche is the trans-horizontal uncertainty that surprises the expectant. Because the function of consciousness systems is negation of uncertainty, the world can enter consciousness only via the negation of uncertainty. In this sense uncertainty, arche of the world, must be the trans-horizontal uncertainty that is still not internalized in the horizon of consciousness.

“Arche” has been confused with “element (στοιχεῖον)” or “substratum (ὑποκείμενον)” since the age of ancient Greece. But elements are derivatives made by reducing uncertainty and substratum is the subject that makes its structure of elements to reduce uncertainty. Both elements and substratum come out of the vast ocean of uncertainty, float on it and then, failing to maintain themselves, sink into it. History of systems is the repetition of this.

You might think that, if uncertainty brings about wonder, certainty without wonder should precede the uncertainty and therefore the certainty should be arche. But, because the certainty preceding the original uncertainty does not enter consciousness, it is as good as nonexistent. Certainty is only conceivable in the relation to uncertainty and the alternative of certainty or uncertainty produces uncertainty at the meta-level. The concept of certainty that precedes consciousness does not precede consciousness. It is a derivative concept derived from the original uncertainty.

The previous systems theories have treated elements as arche, combined elements into complexity and explained systems in terms of complexity. As a result the researchers have been bogged down over the problem such as what the ultimate element of every system is or what the element proper to this or that system is. Such a debate puts the cart before the horse. As elements are just means to reduce uncertainty, which element to select depends on which uncertainty to reduce. Our systemics is different from the previous systems theories from the start.

5 : References

1. ↑ An Intermediate Greek-English Lexicon, ἀρχή (author) Henry George Liddell, Robert Scott
2. ↑ Physics and Philosophy: The Revolution in Modern Science (author) Werner Heisenberg (page) 71
3. ↑ A History of Greek Philosophy: Volume 2, The Presocratic Tradition from Parmenides to Democritus (author) W. K. C. Guthrie (page) 20
4. ↑ Metaphysics, Book 1, section 986a (author) Aristotle
5. ↑ Metaphysics, Book 1, section 982b (author) Aristotle
6. ↑ Theaetetus, section 155d (author) Plato
7. ↑ Metaphysics, Book 1, section 982b (author) Aristotle
8. ↑ Discours de la méthode (section) Première partie (author) René Descartes
9. ↑ Discours de la méthode (section) Première partie (author) René Descartes
10. ↑ Commentary on Aristotle’s Physics, p.24, 13-25 sq (author) Simplicius
11. ↑ Φυσικά (author) Αριστοτέλης
12. ↑ Enûma Eliš, Tablet 1
13. ↑ בראשית א כתיב
14. ↑ Κατά Ιωάννην
15. ↑ Commentary on Aristotle’s Physics, p. 150, 24 sq (author) Simplicius

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