Melegoturny Marygoraumd

Metaphysics - Book Nu - Sec. 6
Even if the composition of a thing is easily related to a specific number, it is not clear what good comes from this number. For instance, a mixture of honey and water is less wholesome when mixed in a ratio of 3:3 than mixtures of indefinite, but more dilute, proportion.

A problem arises from identifying different things with numbers sharing a factor: these thing should share the characteristics corresponding to that factor. Thus fire and water must be relatively prime. Furthermore, several unrelated things in nature share the same number. The number seven applies to the number of vowels, notes in the scale, stars in the Pleiades, and the age at which some animals lose their teeth. There is no difficulty in finding such analogies in both eternal and perishable things. The "good" characteristics of numbers that are associated with good existing things are just a coincidence: in every category of being an analogous item can be found, e.g. straight in length corresponds to plane in surface corresponds to white in color, etc.

The Ideal numbers are not the causes of musical phenomena, thus the mathematical nature of music does not require the existence of Ideal numbers. The fact that so much trouble arises when considering the generation of Ideal numbers indicates that numbers are simply not separable from sensible things.

---

The very last section of the Metaphysics attacks the notion that good is attached to specific numbers as well as attacking numerology as the result of people being able to find correspondences if they are looking for them.

This is a rather odd, anticlimactic note to end on. Apparently numerology was just as easy to ridicule in Aristotle's time as it is now. While the argument that fire and water can't have numbers sharing a factor is pretty amusing, the argument of this section isn't especially profound.

---

Well, that does it. Going in, I'd read in a few places that this work is a hodge-podge of related topics lacking any solid unifying principle. This was borne out quite clearly in my reading. This isn't to say it isn't a valuable read, even apart from its historical interest as one of the more influential works in philosophy. From my own scientist's perspective, it puts into clear relief how much we still rely on essences in our descriptions of nature, at the same time showing what sort of conclusions we are driven to by a strong adherence to this way of formulating things. At the same time, the examination of actuality and potentiality shows quite clearly how remarkable it is to have a mechanistic theory predicting seemingly goal-seeking processes.

I have to say I was surprised at the very limited amount of theological argument there was, and what there was largely repeated conclusions arrived at in the Physics. Maybe that's why it was so easy for Aquinas to reconcile this with Christian theology.

Current Mood: happy
Current Music: Radiohead - The Bends

Metaphysics - Book Nu - Sec. 5
It is not correct to infer that the principles of the universe are similar to those of animals and plants to say that the more complete always comes from the indefinite and incomplete as a plant grows from a seed. In fact, this is not even true of animals and plants, as seeds do not exist without complete plants.

Generating place along with the mathematical solids, and saying they exist somewhere that is not specified, is not satisfactory as a theory of generation. Nor is the idea that all things comes from elements and numbers are the first things generated from them satisfactory because it does not specify the method that numbers are generated. They cannot be generated by intermixture because the one would not be distinct from the other numbers. They cannot be generated by juxtaposition since that assumes that numbers, which are nowhere, have a place.

To say something comes from something else means in one sense that some of the elements remain in the products. If number comes from contraries, such as the one and plurality, one of these must persist. All things made of contraries eventually are destroyed because contraries destroy the compound; numbers, however, are not destroyed.

The type of cause numbers are of substances or being has not been identified. Some consider them formal causes, identifying numbers with shapes, or things with ratios in the same way a harmony is a ratio, but these theories do not explain how attributes are numbers. Number is more like matter, but substances are more like ratios of ingredients.

Number is thus neither a cause as agent, matter, or form. Nor can it be an end.

---

This section attacks the ideas that numbers can be the efficient, formal, or even material causes of things. That they can't be efficient causes follows from the fact that they cannot be generated themselves. Making them formal causes does not account for any attribute, except maybe the shape.

Overall, Aristotle's objection here seems to be that numbers are just not enough like anything they are supposed to be causes of to actually be causes of these things. This seems to be more the suggestion of a tactic for destroying theories of this sort: it seems like it would only be a clear argument on a case-by-case basis.

Current Mood: determined
Current Music: My Morning Jacket - Evil Urges (cont'd)

Metaphysics - Book Nu - Sec. 4
The thinkers who opine about the generation of numbers are not merely giving a theoretical account. (Since some say the odd numbers are not generated, and this implies that the even numbers are generated). That is more than theoretical is evident from the fact that they say the even numbers are generated by unequals when they are equalized, something that could not occur if they always had been equalized.

A primary difficulty with these theories is how the elements and principles are related to the good itself. In the first place, there is disagreement whether the good even exists from the beginning, or whether it requires progress in nature. For instance, the poets make Night or Chaos the first gods while they make the later born Zeus the ruler. On the other hand, the Magi, as well as Empedocles and Anaxagoras, make the good, in various guises, a first principle. While goodness cannot be denied of a first principle that is self-sufficient, it is impossible that the one or an element is good. If they were, all units would be species of good. Additionally, if all Ideas were only of goods, nothing would exist; if they were also Ideas of substance, all substantial things (including animals and plants) would be goods. Furthermore, plurality, being the contrary of one, would be bad, so all numbers but one would contain badness. The bad would be the space in which the good is realized, equating the bad with what is potentially good.

These objections follow from a combination of flaws in the positions of these thinkers: every principle is made an element; contraries are made principles; the one is made a principle; numbers are treated variously as first substances, separable, and Forms.

---

Aristotle further attacks theories that make (Ideal or mathematical) numbers the ultimate constituents of reality by arguing that these theories are incompatible with the existence of good, through creating too much good by attributing good to the nature of the elements at the same time as making the bad an attribute of far too many things. He ends the section by isolating the several flaws in these theories.

The good plays an enormous role in Aristotle's ontology as essentially the active principle of final causes. If it is assumed the good has no objective existence, a position taken by any mechanistic scientific theory, the central argument of this section goes out the window. Even in that case, it does contain the idea that there's nothing moral about math.

Current Mood: determined
Current Music: My Morning Jacket - Evil Urges

Metaphysics - Book Nu - Sec. 3

Those who believe that numbers are Ideas (and whose arguments do not conclusively prove the existence of numbers) treat each one separately and try to explain the existence of number. The Pythagoreans, observing properties of numbers in real things such as musical scales and celestial motion, took real things to be made of numbers. Those who say mathematical number exists alone oppose this, and have gone so far as to say that numbers cannot be objects of sciences.

It is clear that numbers do not exist apart; otherwise their attributes would not be observed in natural things as observed correctly by the Pythagoreans. An objection to their view is that they make, for instance, things with weight out of numbers which are weightless. Those who make mathematical objects separate believe that mathematical theorems do not apply to sensible objects.

Some thinkers have taken limits, such as points (which limit a line), lines (which limit a plane), and the like, to be real things. Extremes, however, are not substances, or else anything with a limit (e.g. motion) would have substance. Plus, extremes are always extremes of sensible objects, and therefore do not exist apart anyway.

Since the soul and sensible bodies would exist without magnitudes, and magnitudes would exist without number, the objects of mathematics contribute nothing to the existence of real things. Similarly, Ideal numbers contribute nothing to existence, and even worse, no mathematical theorem is true of them (unless a new, and arbitrary, sort of mathematics were invented for them.)

Those who posit both Ideal and mathematical numbers do not account for mathematical number, which they place between Ideal and sensible numbers. Their attempts to account for them either make them identical with Ideal numbers or require multiplying the number of elements. For instance, if the Ideal one and mathematical one are principles of Ideal and mathematical numbers respectively, they will share the property of unity. But at the same time, there is a plurality of them, and plurality, according to this view, requires the dyad to be generated.

The Pythagoreans also believe eternal things to have been generated. We should shift our attention from them, though, because they are more interested in formulating a theory of nature, which changes, while we are more interested in unchangeable things.

---

This section touches on what is actually solved by the various schools of thought regarding the existence of number at the same time it brings objections to each. The major points are that numbers do not have the properties of real things, and thus cannot be real constituents of them, and that numbers do not contribute anything to the existence of real things, because the real things would exist anyway without them.

This is one of those sections that initially seems to be one of Aristotle's stronger because of the clearness of its arguments. It doesn't take long to see most of them as specious at best. The argument about numbers contributing nothing to existence is possibly the most circular argument in the work. Perhaps speciousness, much like triviality, goes hand-in-hand with immediate clarity.

Additionally, the arguments of this section rehash arguments from Book Mu Sec. 8 in much less detail.

Current Mood: pensive
Current Music: Rautavaara - Cantus Arcticus, String Quartet no. 4

Metaphysics - Book Nu - Sec. 2
Eternal substances cannot consist of elements. Things that consist of elements have matter, which is potential. Something that is potential must be capable of not existing. Since the eternal is not capable of not existing, it cannot be composed of elements.

Some thinkers, seeing objections to making the relative an element, make the dyad an element along with the one instead of the unequal. These thinkers are faced with other objections. Such explanations arose from many sources. An important one was the thought that all things that exist are one (Being itself), so that the things that are many must be composed of that which is not, and thus it must be proved that that which is not is.

There are many objections to this. First, there are many senses of 'being,' namely, the categories, and it is impossible that a single nature could account for the various modes of being (substance, quality, quantity) of a single thing. Second, 'non-being' has as many senses as there are senses of 'being': for instance, not being straight is the non-being of a certain quality, and so forth. The false and the potential are types of non-being, and generation proceeds from the latter.

The things that are thus generated are number, lines, bodies, etc, and not substances. The dyad, for instance, cannot be the cause of there being a multiple number of shades of white, or else these shades would be numbers themselves. The same applies to substances. Nor does this line of thought explain the multiplicity of relatives (e.g. different ways of being unequal).

For each thing, something that is potentially it must be presupposed: the thinkers that have held the previous have supposed the relative to be a potential substance. It seems problematic that there should be multiplicities in categories other than substance. Quantities and qualities are many only because the substrate becomes many. There should be a matter for each, but it would be nonseparable from substances. Again, the question arises how there could be many substances and not one. If the quantitative and essences are not the same, explaining multiplicity in the former does not explain it in the latter.

Another question is what justifies the belief in the existence of number. Those who believe in Ideas think numbers are Ideas and are the causes of real things. But some of those who don't believe this assert the existence of mathematical number, which is not maintained to be the cause of anything; indeed, theorems of arithmetic apply to sensible things, anyway.

---

The bulk of this section is devoted to arguing that attempts to generate multiplicity out of unity and some other principle are inadequate. Specific objections based on the existence of the different categories are brought against this other principle being the dyad or being non-being.

I'm not really sure what the very first paragraph has to do with the current discussion of number. Maybe the argument is that number is eternal (as was suggested in Book Lambda) and therefore cannot be composed of elements. Book Mu, however, denied that number is substance, so this does not seem to be the intent.

The arguments in this and the previous section strike me as being more developed along the lines in previous books of the Metaphysics. There hasn't been any direct repetition of arguments from Book Mu, though the basic conclusion is essentially the same: number cannot be used as the basis of all things.

Current Mood: mellow
Current Music: Szymanowski - String Quartets; Webern - Langsamer Satz

Metaphysics - Book Nu - Sec. 1
While philosophers have traditionally made contraries the first principles, contraries are always predicated of an underlying subject. Substance has not contrary, so this is more of a first principle than contraries. The same thinkers always considered one of the contraries to be matter. For instance, the unequal is matter for the equal, or plurality is matter for the one. Others take the pair of contraries, e.g. the great and small, to be matter along with the one as form. Some take the more general route of naming that which exceeds and that which is exceeded as principles. All these views result in number in general being prior to specific numbers, including two. A problem arises for these thinkers when considering what is contrary to the one. Either it has none [and therefore does not enter into things], or the other principle, such as plurality, is contrary to the one, which would make the one a few, since a few is contrary to plurality.

The one denotes a measure, the unit of a type; it is not itself a substance, being predicable of all examples of the type. At the same time, number is a measured plurality.

The notion that the unequal (or the dyad of great and small) is one thing is impossible. Things like the many and the few are modifications of numbers, not substrata. Furthermore, each item in these pairs of contraries is relative and posterior to quantity and quality. The relative is the furthest thing from being a substance, as can be seen by noting the relative has no proper generation, destruction, or movement: without changing itself, a thing can change from greater to lesser by the change of what it is relative to. The matter of a thing must have the potentiality of that thing, and the relative is neither potential or actual. Finally, for number to consist of the few and many, either both of these or neither must be predicated of it, which clearly is not true.

---

Section one of Book Nu begins again the attack on the generation of numbers by principles like the one and the dyad, this time arguing against all attempts to generate them out of contraries. In the first place, contraries presuppose substance. Secondly, it is not clear how a unit can fit into a pair of contraries while retaining its unity. Indeed, the unit is not substance but is a standard for measurement of a class of things, and number is the result of measurement. Thirdly, the contraries proposed as principles of number are nothing more than attributes of number, not to mention relative.

Book Nu seems, at its start, to be independent of Book Mu. To me it seems more in the spirit of the central Books, perhaps following the development of the nature of contraries in Book Iota. The arguments are slightly more general, and they attack the adequacy of the proposed first principles to describe the nature of all numbers rather than show that such principles lead to pathological (but at that, not seemingly impossible) conclusions.

Current Mood: exhausted
Current Music: Hindemith - various pieces

Metaphysics - Book Mu - Sec. 10
There is a problem that faces both those who believe in Ideas and those who do not. Namely, for anything to be capable of separate existence, substances must be separable, but if they are, what is the nature of their principles and elements? If the principles and elements are individual (as opposed to universal), there will only be as many substances as there are elements, and since only universals are knowable, the substances will not be knowable. On the other hand, universals are not substances, as demonstrated previously. Thus, if they are the principles of substance, nonsubstance would be prior to substance. These difficulties are not avoided by making Ideas out of elements and claiming these to be separate unities from the substances with the same form.

To say that knowledge is universal and therefore the principles of things are universal is in a sense true and in a sense not. The resolution lies in the fact that knowledge has two different meanings. On one hand, there is potential knowledge that deals with the universal and the indefinite; on the other, there is actual knowledge that deals with definite objects.

---

The last section of Book Mu is a sort of appendix, bringing up an issue related to universals that is independent of whether Ideas exist or not. Namely, how can knowledge, which is universal, be squared with the existence of actual things. Aristotle's answer is that only potential knowledge deals with universals, while actual knowledge deals with individual objects.

This section would seem to have fit a little better when actuality and potentiality were being treated, but it is not entirely out of place here.

The problem here is very close to Hume's and Kant's main problem. Whereas for the first two, the question is how certain knowledge of things can be possible without experience (Hume says it's not possible at all; Kant says it's possible when the conditions of knowing leave no other choice), Aristotle's is roughly how knowledge of individual things of experience can be possible when it is only of universals. Aristotle answers by classifying theoretical knowledge as potential. For instance, the knowledge that the sum of the angles in a triangle is always 180 degrees, being a universal statement, is potential, and only becomes actualized for an individual triangle when that individual triangle is encountered.

This answer seems to me to beg the question. It's fine to say that the objects of knowledge are only actual things, but how is the leap to universality ever made? Maybe the treating of abstracted attributes leads to universality- something like Husserl's ideation- but this treads very closely to giving the universals separate existence.

Well, that wraps up Book Mu, and from outlines I've seen, Book Nu contains more about mathematical objects. There are only six sections, so hopefully this commentary can be wrapped up before I head to Puerto Rico for my brother's wedding.

Current Mood: calm
Current Music: Ween - La Cucaracha

Metaphysics - Book Mu - Sec. 9
Since discrete numbers do not have contact with one another but are related to by succession, there are questions of whether the units in two or three succeed the Ideal one or not and whether the two or the units in the two are prior.

Similar problems arise about the reality of posterior mathematical objects such as lines, planes, or bodies. These are thought by some to be constructed of long and short, broad and narrow, deep and shallow, respectively, which are all forms of the great and small. Then, either the pairs of principles are isolated from one another, or one pair implies the other. [In the former case, planes would not contain lines, and bodies would not contain planes.] In the latter case, lines, planes, and bodies would all be the same. Furthermore, none of this explains angles or figures. In fact, these principles are really attributes of magnitude; it does not follow magnitudes are composed of them.

A difficulty that arises by making Ideal numbers separate arises more generally when universals are made separate. Namely, is the separate universal present in a real instance of the universal. For example, is the Ideal animal in a particular animal? Similarly, is the one in itself present in a two? None of these problems arise if the universals or numbers are not supposed to be separate.

Another approach to explaining magnitudes is to generate them from a point and some other matter, where the point is like, but not the same as, one, and the other matter is like, but not the same as plurality. This approach has the same difficulties as mentioned above: either lines, planes, and bodies are all the same, or the higher dimensional objects do not contain the smaller. Similarly, the same objections apply to generating number from one and plurality as those that apply to generating it from one and the indefinite dyad. Mainly, this scheme does not explain the origin of each unit. They must come from the one-in-itself and plurality, but making plurality part of a unit is a contradiction since a unit is indivisible. Indeed, plurality is just another number, and the question arises whether it is finite or infinite. The same questions can be asked about points used in generating magnitudes. They cannot be generated by the Ideal point and a distance because points are indivisible and distances are not.

From all of these objections follows the conclusion that numbers and magnitudes cannot exist separately from things. The wide disagreement of thinkers on the nature of numbers indicates that the alleged facts from which they start are themselves incorrect. Those that posit only mathematical number did so in response to the difficulties that arise in the theory of Forms. Those that identified Ideal and mathematical number did so to try to make Forms numbers but could not see how to make mathematical number exist apart from Ideal. Those that separated the two did not see these problems. Those who believe in Ideas followed the same sort of thinking, inspired by Socrates who formulated universal definitions but did not separate them from things. His successors, thinking it was necessary to have separate eternal, nonsensible substances, and not seeing anything besides the universals, made them separable.

The views of thinkers on the first principles of sensible substances were discussed in the Physics, and are outside the scope of the present work. The views of those that admit substances other than sensible must be discussed next.
---

The discussion on the problems with making Ideal numbers separable from things concludes in this section. The basic arguments involve the difficulties in explaining the great number of attributes of numbers (including especially units) from only two generating principles. An analogous difficulty in making universals separate is pointed out, and the rationales behind Aristotle's predecessors' theories are discussed.

Intriguingly, some of the objections Aristotle comes up with for generating magnitudes with dimension from zero-dimensional points underlie the axiomatic formulation of point-set topology in terms of open sets. The relationship between the discrete and the continuous really is a sticky issue; Leibniz identified it as a "labyrinth." Aristotle's cautious intuitions about the continuum seem largely correct, as far as the standard topology on real numbers is concerned.

The other objections again deal with how generating real numbers from Ideal numbers lead to them having properties that don't seem compatible with their mathematical properties (ordering, for instance), as well as issues of distinguishing things that are mathematically indistinguishable (see Leibniz again).

The concluding discussion of the motivations of the previous thinkers in formulating these theories would have been quite valuable at the outset of the demolition of the theories, at least to make it clear what Aristotle was making objections to. Ultimately, as is made clear here, Aristotle's concern is to eliminate the duplication of aspects of reality, whether they be qualities duplicated by separate universals, or quantities duplicated by Ideal numbers. In each case, the message is that believing in all these things leads to more difficulties than they solve, if they indeed solve the difficulties they were meant to solve.

Current Mood: awake
Current Music: Royals at Yankees

Metaphysics - Book Mu - Sec. 8

What are the differentia of number? Numbers differ from each other, qua numbers, as quantities. If units differ, they must either as quality or quantity. If units differed in quantity, equal numbers would differ from each other in number of units. It does not seem they could differ in quality since no attribute could be attached to a unit. Quality in numbers would have to come from the one or the dyad, but the first has no quality, and the second only provides quantity (in doubling).

As has been shown, if Ideas are numbers, the units can be neither comparable nor noncomparable. However, is is also not correct to say that numbers are real things, and the ideal one is the starting point of them. The existence of an Ideal one should imply the existence of an Ideal two or three from which real twos or threes are generated. The most problematic view is the one that identifies ideal and mathematical numbers, suffering from both the fact that mathematical numbers cannot be Ideal, as well as the problem that Ideal numbers do not behave like mathematical numbers (as was shown in the previous section.) The Pythagorean view that only mathematical numbers exist, not separably, but as magnitudes, at least does not suffer the problems that occur by accepting the separate existence of numbers. It runs into trouble when considering indivisible numbers since no magnitude is indivisible. In summary, for number to exist self-subsistently, it must do so in one of these ways; since all these ways have been shown to give rise to impossibilities, number cannot exist self-subsistently.

Since units do not differ from each other, it cannot be said that some units come from the great while others come from the small. Nor can a unit be said to come from the great and small equalized. If it could, then a two could not be distinguished from a unit.

If a unit and a two were Ideal, since the unit is prior to the two, it would be an Idea of an Idea.

If it exists separately, number must be either infinite or finite. Clearly the former is not true, since number must be odd or even, and infinite number is neither. Also, as an Idea, infinite number must be the Idea of something, but this is not reasonable. At the same time, if number is finite, where does it stop? Some say it stops at ten and generate things such as the void or proportion within the ten (other things like movement are assigned to the principles). This is clearly not enough to function as Ideas for everything that exists: it would follow that some things come into existence without Forms, and thus Forms are not causes. Similarly, it is paradoxical to assume that neither eleven nor any of the successors of ten are Ideas, though ten is. Also, every instance of the same number must be a Form for the same kind of thing: if three is the Idea of a man, since there are an infinite number of threes, there will be an infinite number of man-in-himselfs.

Is one prior to two or three? As numbers greater than one are made up of ones, it would seem the one is prior. On the other hand, if the universal or form is prior, the numbers, as forms for units, would be prior. In this latter case, which applies to real things, the one cannot be seen as the starting point, unless by this the unity of the form is meant. Some try to make unity in both senses the starting point- units are the matter and the unified number the form- but this is impossible, giving incompatible characteristics to a single thing. Also, since a two is divisible, the unit must be more like the Ideal one and vice-versa, making both prior to the two; however, the people who hold this view generate the two first. Similarly, since both two and three are unities, the pair must again be a two, but from what is this two produced?

---

Aristotle continues picking apart the theories that numbers function as Ideas or even can exist separately by showing that each theory leads to absurdities or impossibilities. The basic strategy is to show that the nature of numbers known through mathematics is in some way incompatible with the nature of real things, which variously must have differentia, be finite or infinite, and have a distinct priority.

The lengthiness of my summaries of this and the previous section are a symptom of the level of detail of the arguments Aristotle makes. Because these sections are so devoted to minutiae of theories that today seem absurd, there's not much connection I can make to them. (The Pythagorean view does remind me of a guy who once told me that compact discs stored music in the microscopic numerals that were actually written on them.) I suppose the success of mathematics in describing the natural world in the modern sciences might tempt some to give an ontologically causal role to mathematical objects, but I haven't ever heard such an argument made (outside of New Age publications, of course).

Current Mood: tired
Current Music: Portishead - Third

Metaphysics - Book Mu - Sec. 7
Whether or not, or to what degree, mathematical units are comparable, undesirable consequences follow for those who would make them causes of things.

In the case the units are all comparable and indistinguishable, the only numbers are mathematical numbers. These cannot be Ideas, since no single instance of a particular number has any better claim to be an Idea than any other instance of that number. On the other hand, Ideas must be numbers if they exist, since the principles and elements- the one and the indefinite dyad- are principles of number, and Ideas are neither posterior nor prior to number.

In the case no unit is comparable with any other, these numbers are not mathematical, which require units to be comparable. Nor are they Ideal, since the sequence of Ideal numbers is generated by the one and the dyad, but a two with noncomparable units cannot be generated that way. Also, from an Ideal number, there is a sequence of the same numbers generated from the Ideal: for instance, from the ideal one comes a first one that follows it, then a second and third one, and so forth. These units must be prior to the numbers that name them. Furthermore, an ideal two or three could not exist if all the units are noncomparable, because numbers are counted by repeatedly adding one. This method of generation implies that numbers cannot be generated by the one and the dyad.

Finally, in the case where units are comparable within a number but not between numbers, problems arise when considering the parts of numbers greater than one. For instance, while ten is made of ten units, it is also made of two fives; the Ideal ten must be made of Ideal fives, in which case the units that compose it must differ (otherwise the fives would not differ). If they differ, there must be other fives, and what sort of ten do these make up given that there is only one ten in the Ideal ten.

Is it possible that a number is different than the units that compose it, for instance, that two is different than the pair of units of which it is made? If so, this either occurs if the number shares in the set of units in the same way a white man shares in both white and humanity; or it occurs if one is a differentia of the other. The set of units that makes up a number, though, does not have a unity based on contact, intermixture, or position.

A consequence of holding that numbers are Ideal is that there are prior and posterior versions of each number. For instance, the twos that generate a four are prior to the twos that generate an 8. These all are Ideas, however.

In general, the attempts to differentiate units from one another are absurd. Number must only be equal or unequal, and equal numbers cannot be differentiated. If any two units added together make two, this will be true for a unit taken from a two added to a unit taken from a three. Is this two prior or posterior to the three? It should be prior, given that the unit from the two is prior, and the unit from the three is simultaneous with the three. Furthermore, either the Ideal three is not greater than the Ideal two, which would be weird, or it is greater, in which case it contains a number equal to the Ideal two, which thus must be the Ideal two, but this is not possible if there are a first and a second number.

Ideas cannot be numbers if the units are comparable, since Forms are unique but numbers are not. Also, if an Idea is a number, then one Idea will be in another, and all Forms will be parts of one Form.

---

In this section, Aristotle argues that every possible hypothesis regarding the comparability of units in Ideal numbers leads to impossibilities.

I'll admit I have a hard time seeing what is impossible about a lot of these conclusions. Some seem to be impossible merely by conflicting with the desire to use the one and the indefinite dyad as first principles. Others are objectionable, apparently, by giving unexpected priorities to the various numbers. The two clearest lines of thought for me in this section are the following. First is that the indistinguishability of mathematical numbers make them unsuitable as candidates for Ideals. Second is the argument against making units in one number noncomparable to those in another by noting that ten can be broken up into two fives.

So far, then, I am able to appreciate this section more as a demonstration that Ideal numbers, if they exist, don't quite behave the same way as mathematical numbers.

Current Mood: calm
Current Music: Fleet Foxes - Fleet Foxes

Metaphysics - Book Mu - Sec. 6

There are a number of views that can be held by those who believe numbers are separable substances and first causes.

For one thing, if numbers are substances, there are three possibilities regarding their comparability. First, each number in succession is different than each previous one: all units are non-comparable. Second, the units that make up each number are all equal to each other: this is characteristic of mathematical number. Third, the units within one number are comparable with each other but not with those that make up another number: thus, the two units that the number two comprises are equivalent, but they are not the same as the three that make up the number three. These last are Ideal numbers.

Another issue is whether numbers are separable from things, and again there are three different possibilities: no number is separable; all numbers are separable; some numbers are separable and others are not.

Most of the possible combinations of these possibilities have been held by previous thinkers. None who consider numbers to be only substance of all things have held that all numbers are noncomparable. Some have held that both mathematical and Ideal numbers exist, as separable substances. The Pythagoreans held that only nonseparable mathematical numbers exist (which furthermore were spatial magnitudes.) Another says only the numbers as Forms exist. Similar things hold for lines, planes and solids.

These are all the ways that numbers could exist as substances, and they are all impossible, though some are more impossible than others.

---

After attacking the theory that Forms are substances, Aristotle prepares to attack the theory that numbers are substances by comprehensively listing the (six) possible ways numbers could exist.

This section is almost completely tautological, the only substance (heh heh) being vague indications of what has historically been held. Nevertheless, it is often convenient to exhaustively list all possibilities for the sake of future reference.

Current Mood: contemplative
Current Music: Texas vs. Rice - NCAA Div I Baseball Regional

Metaphysics - Book Mu - Sec. 5

What do Forms contribute to sensible things, anyway? They don't cause change; they don't help as far as knowledge of them; they don't even contribute to their existence (if they are indeed separate from the individual things). Anaxagoras' view that they are causes, in the same way white is a cause for a white thing, is easily refuted. To say Forms are patterns for things requires some being to use them in the manufacture of the sensible things, or else this is mere metaphor. Even if this could be established, a different pattern would be needed for each universal attribute of a sensible thing. To say that Forms are substance, but that they exist apart, leads to the absurdity that the substance of a thing and the thing are separate. To say that Forms are causes of both being and becoming (as it is in the Phaedo) leads to questions of why some things with Forms do not come into being, while some things without Forms do come into being.

---

Aristotle continues with objections to the theory of Forms, mainly claiming that they are nothing better than extra ontological baggage.

There's really not much to comment on here. The argument that Forms do not contribute to exactly the areas they were meant to address (e.g. knowledge) is philosophically devastating, which is why a similar tactic is frequently employed in philosophical discussions.

Current Mood: calm
Current Music: Pink Floyd - two versions of ASOS

Metaphysics - Book Mu - Sec. 4
The Ideas must be examined in the same way mathematical objects were examined; that is, they must be examined in and of themselves. Ideas were first proposed to explain how knowledge could be possible in the Heraclitean doctrine of an ever-changing world. Socrates, in considering issues of character, came to Ideas in connection with the universal definitions or essences which he needed as the starting point of deductive argument. Unlike his followers, Socrates did not make the ideas exist apart.

(Socrates can be said to have introduced two things to philosophy- inductive arguments and universal definitions. Democritus, with his hot and cold, and the Pythagoreans with their numerical treatment of e.g. opportunity, justice, and marriage, though, considered universal definitions in a limited sense.)

The followers of Socrates, following his arguments, and driven to find the causes of sensible things, extended the domain of the Ideas to cover all things spoken of universally. In doing this, they greatly multiplied the number of Ideas needed. However, no proof of the existence of the Forms is convincing, either being non sequiturs or leading to the existence of Forms for things that are thought should have no Forms. For instance, some arguments lead to the various conclusions that negations, perishable things, or relations have Forms. Other arguments suffer from the problem of the third man. At the same time, these arguments also lead to undesirable (to their proponents) conclusions about priority of existence: for instance, the dyad is prior to the unit, and relations are prior to self-sufficient things.

There is a contradiction in how Forms apply to actual things. On one hand, there should be Forms of anything that is one in concept, including many nonsubstances. On the other hand, for Forms to have any bearing on everyday things, they must only apply to substances. Since those things share in Forms as in something not predicated of something else, the Forms must be substance. Thus, the things called by the same name as the Forms must also be substance, unless this is simply homonymy, in which case the Forms explain nothing about things.

Finally, if each element in the formula of a thing is an Idea, but what the thing is must be added on top of all this, isn't this latter a vacuous addition? For instance, if all the elements, like "plane figure" in the definition of a circle are Ideas, does their need furthermore to be an idea of a circle itself?

---

Aristotle moves on from his consideration of mathematical existence to a similar consideration of the Forms. While there is certainly some sort of existence belonging to mathematical objects, it is not so clear that the same can be said about Forms, which must be multiplied greatly in order to be capable of explaining the causes of sensible things, and even then fail to explain what they are supposed to.

Much of this section repeats attacks Aristotle has previously made on the Theory of Forms, but there are a few new arguments here. One shows that the theory of forms comes into conflict with his own theory of substance as what exists self-sufficiently. Another, of more interest, attacks the notion of Form as being vacuous.

I say this is of more interest because it gets at the heart of the conflct between holism and reductionism: if a complete definition is given in terms of simpler things, the idea of the more complicated thing is in some sense superfluous, perhaps only formulated for the convenience of communication. The trouble with Aristotle's argument here is that he just above expressed skepticism about the existence of relations, while the relations between the elements within a definition are a critical part of its completeness. For instance, a square can be defined as a closed plane figure of four equal-length line segments. The four equal-length line segments are elements in Aristotle's mind, but it's not clear whether the closedness of the figure is, even though it is exactly this closedness that distinguishes a square from a crooked path. Bertrand Russell insisted on the reality of relations for, I think, exactly this sort of reason.

Current Mood: awake
Current Music: Haydn - String Quartets 41, 37, 38

Metaphysics - Book Mu - Sec. 3

A science need not treat separately existing objects or even objects that exist only unto themselves. For instance, the science of motion does not have as its subject something that exists separate from sensible objects, nor does it treat some special object that exists within sensible objects; it only treats sensible objects qua moving. Similarly, the mathematical sciences may treat sensible objects but not qua sensible.

Mathematical attributes are attached to things in much the same way maleness or femaleness attaches to animals without there being a separately existing maleness or femaleness. These attributes are prior in formula and simpler than the objects they belong to, and as such, the sciences that treat them are simpler and more accurate. A science, like the universal part of mathematics, that abstracts qualities from specific magnitudes, is more precise than one that takes into account the specific magnitudes.

Supposing things to be separated from their attributes when they are considered in that capacity is not an error. Arithmeticians, for instance, only treat things qua indivisible. Therefore, they make statements about existing things, but the type of being they consider is fulfillment as opposed to matter.

Because of this concentration on abstraction, mathematics does treat both the beautiful and the good (which are different, the latter always involving conduct). The chief forms of beauty are order, symmetry, and definiteness, all three attributes considered at length by mathematics. Since these attributes are causes of a sort, mathematics must treat causes.

---

The objects of mathematics exist as inseparable attributes of sensible objects. Through abstraction, sciences validly consider attributes as separate things, but the statements made about such attributes apply to the things they were abstracted from.

Aristotle solves the problem of how mathematical objects exist by claiming they are attributes abstracted from actually existing objects- they are part of the fulfillment of those things. This, at the same time, shows how statements about mathematical things actually can apply to sensible things.

While this seems a tenable explanation for numbers and geometrical objects at first glance, I'm not sure how it addresses what I'd call the perfect circle problem, namely, that nothing in nature is a perfect circle, but that's just what geometry talks about. In fact, Plato's theory of Forms seems to have been invented simply to solve exactly this sort of problem. Another objection to Aristotle's conception is that it is difficult to see how it could apply to many of the objects of modern mathematics. The general objects - things like groups or manifolds- can probably be seen as abstractions of abstractions, but concrete versions of these that simply could not exist as an attribute of a sensible object (an 18-dimensional manifold, say) would seem to fall outside this scheme.

Current Mood: calm
Current Music: Terry Riley - A Rainbow in Curved Air;

Metaphysics - Book Mu - Sec. 2

It is impossible for mathematical objects to exist in sensible objects. This is not only because two solid objects cannot occupy the same space at the same time, but it is also because points are not divisible. To understand this latter statement, note that if mathematical objects are in bodies, a body must be divided at a plane; the plane must be divided at a line; the line must be divided at a point. Thus, if the point is not divisible, a body cannot be divisible, which it clearly is.

It is also impossible for mathematical objects to exist separate from sensible objects. This is because such a supposition gives rise to an absurd number of classes of things. For instance, if mathematical solids exist separately from sensible solids, so do mathematical planes, lines, and points. Since a mathematical solid is composite, its parts must exist prior to it, and thus planes, lines, and points that are different from the first class of mathematical planes, lines, and points. Similarly, the second class of planes gives rise to a third class of lines, and points, and so on. For similar reasons, there will be separate things that are treated mathematically, for instance the objects of astronomy, geometry, harmonics, or optics. Since the latter two treat objects of sense, there must be separate senses that sense these separate sounds or objects of sight. Also, there are universal mathematical theorems that extend beyond these substances, indicating a substance intermediate between Ideas and intermediates that aren't numbers, points, spatial magnitudes, nor times.

Another proof that mathematical objects do not exist as separate entities follows from the observation that they must then be prior to spatial magnitudes in substance. We have already seen that incomplete spatial magnitudes are posterior to complete spatial magnitudes. Since the order of substance is opposite the order of generation, and the line is generated before the plane (and so on), the body must be prior to the plane in the sense of substance. Indeed, bodies may be animated by souls, but who can imagine a plane being animate? Furthermore, mathematical objects are quantities and divisible; substances must hold together as one. A body is a sort of substance, being in a sense complete, but the same cannot be said about planes or lines. Things prior in formula (such as whiteness in a white man), need not be prior in substance, since it cannot exist separately.

In summary, mathematical objects can neither exist in nor separately from sensible substances. Neither are they substances in a higher sense than bodies are. Thus, mathematical objects either do not exist or exist in some special, qualified sense of existence.

---

Aristotle begins the investigation into mathematical objects by showing that they neither exist in or separate from sensible substances. This means that they cannot be said to exist in any unqualified way, and in particular, that they are not substances of the sort that could be postulated to underlie other kinds of substance.

From its clearness in intent and conclusions, this has been one of the easier sections in the Metaphysics to follow. On the other hand, the arguments against the existence in or separate from sensible objects seem to involve certain leaps of logic, undercutting the impact of the section. In the first case, the assumption that dividing at a point implies the divisibility of the point is made. Modern mathematics allows the point to stay within one of the two resulting segments (treating the end of the other segment is consequently a little more subtle- for any point near the end, there is always another point even closer to the end.)

The first argument against the separate existence of mathematical objects is even more bizarre. It's not clear to me at all why the first separability implies the separabilities of all these higher order objects. Plus, even if you accept this line of argument, Aristotle rejects it simply because it is too baroque. He's on slightly better footing when objecting to the separate, nonsensible harmonies and objects of sight (how can you have a nonsensible object of sense?)

A very interesting result of this latter argument, though, is Aristotle's assertion that mathematical objects cannot be substances in any sense more than a body is a substance. Here he explicitly gives "completeness" as an explicit, though explicitly vague characteristic of substances. It seems to be very similar to the concept of self-sufficiency, but I'd always found that a pretty vague characteristic, too. This, on the other hand, has more intuitive appeal. We understand what it is for a body to be complete in a way that planes and lines somehow only exist within bodies. It's tempting to use this example as a basis for a description of substance by analogy.

Current Mood: calm
Current Music: Portishead - Third

Metaphysics - Book Mu - Sec. 1
Having dealt with the substance of sensible things in the Physics, and here with substance that has actual existence, the question is raised whether or not there exist any immovable and eternal substances, and if so, what have others said about them. There are two common opinions on the subject: some say that mathematical objects and Ideas are substances, either of the same or different natures, and some say that only mathematical objects are substances. Thus, the present inquiry should begin by considering whether or not mathematical objects exist, and if so in what way they exist, without presupposing they are Ideas, substances, or principles. Similarly, the question of whether the substances and principles of existing things are mathematical objects or Ideas must be considered.

To begin, do mathematical objects exist in sensible objects, or separate from them, or in some other way?

---

In the opening section of Book Mu, Aristotle identifies that the subject of discussion will be the nature of mathematical objects and opens the investigation with the question of how they exist.

It's pretty clear from the outset that Book Mu and Book Lambda do not belong in the same set of works: Lambda identifies the first mover(s) as eternal and immovable substances, and here it is an open question whether any such things exist. In fact, mathematical objects were given as an example of such at the beginning of that book. The introduction of Mu, on the other hand, seems to put it most naturally following Book Theta, which dealt with potentiality and actuality.

The nature of the existence of mathematical objects is still a lively topic in modern philosophy and one I'm especially interested in. Mathematics seems to give us a basis for knowledge completely separate from anything we have experienced (Kant pointed to mathematical statements as examples of synthetic a priori knowledge). At the same time some of the most arcane mathematics explains some very deep observations that have been made in nature. I'm not hoping that Aristotle will offer much insight into questions related to these points-- otherwise people would undoubtedly be quoting them endlessly-- but this discussion should shed some light on his own ideas about substance.

Current Mood: aggravated
Current Music: Sleepytime Gorilla Museum - In Glorious Times

Metaphysics - Book Lambda - Sec. 10
Does the universe contain the highest good as something separate or as the order of its parts? The answer is probably both are true, in the same way the good of an army exists both in its leader and in its order. All things in the world are connected, ordered together to one end.

Thinkers who held different views than our own face paradoxes and impossibilities. For instance, many thinkers have held that all things are made out of contraries, but they do not indicate how, which is troublesome because contraries are not affected by one another. While, our resolution of all this lies in the recognition of a third factor, other thinkers try to make one of the contraries matter, e.g. the many is matter for the one. However, "the matter which is one is contrary to nothing." Plus, all things (excepting the one) under this view have some component of evil. One of these schools of thinkers do not even recognize good and evil as principles. Another recognizes good but does not explain in what way. Furthermore, no other school explains why some things are perishable and others are imperishable. Some make things out of the non-existent; others make them all one.

Empedocles' view that the good is love is paradoxical, since love is both a principle of movement and matter. Also, his view that strife is imperishable is paradoxical since he considers it bad. Anaxagoras recognizes the good as a motive principle. It is paradoxical to not assume a contrary to the good exists.

No one explains the cause of becoming. Those who suppose two principles must suppose a third. All other thinkers must deduce that there is something contrary to Wisdom, but there is nothing contrary to what is primary (since all contraries contain matter.)

If nothing but sensible objects exist, there is no first principle, but each principle would have a principle before it, as the mythologists indicate.

How is extension created from unextended parts? Number, as mover or form cannot produce a continuum. Nor can there be a contrary as mover or form, since it would be possible for it not to be, and thus the world would not be eternal. Making number a first principle makes the universe depend on a series of principles, but "the rule of many is not good; let there be one ruler."

---

The final section of Book Lambda considers in what way the universe contains the good, and then criticizes previous philosophical positions that create everything in the universe from contraries as being paradoxical.

Aristotle ends this heavily theological book weakly with rushed attacks on previous philosophies. He doesn't even bother to specify who held many of these positions, a stark contrast with the earlier parts of the Metaphysics. Many of the paradoxes he mentions here aren't developed well enough to see, at least at first glance, what is so paradoxical about them. Perhaps a more careful reading of this last section would make this clearer to me, but it seems like such an afterthought at this point that it does not seem worth it at the moment.

Current Mood: tired

Metaphysics - Book Lambda - Sec. 9
When considering divine thought, problems are raised. First, what does it think? It can't be nothing, or else the divine thought would be no better than sleeping. If it thinks something other than itself, it cannot be what is best, because its value derives from its thinking. Furthermore, what it thinks must be what is most divine and precious and must not change: any change would be a change for the worse, not to mention a movement in what is supposed to be unmovable. Also, is the divine thought the capacity of thought or the act of thinking? The former implies that the act of thinking is wearisome to the divine thought, and also implies the existence of something more valuable- the object of thought. All this leads to the conclusion that the divine thought thinks itself, and its thinking is a thinking on thinking. One objection to this is that knowledge usually has something else as its object. However, knowledge can sometimes be its own object, as in the productive and theoretical sciences.

Another question that is raised is whether the object of the divine thought is composite. It cannot be, otherwise the thought would change as it passed from part to part. Nothing that has no matter is divisible. As human thoughts of composite are in a certain period of time (the time in which it takes to attain the whole), the divine thought is throughout eternity.

---

The divine thought, being the best of all beings, has itself for an object which is furthermore a simple object.

This section expands on a consequence of the nature of the divine being mentioned briefly in Sec. 7, namely that God is a self-thinking thought. Any other object of thought for God would obviously be beneath him. The simplicity of His nature is deduced from his immateriality.

While it's pretty clear how much influence these past few sections had on medieval Christian theology, these past two sections seem to point more toward a Deistic theology. In the previous section, you have a god whose influence is essentially transmitted through celestial spheres and which stands off as something to be attained. In this section, you have a God whose perfection is such that He can't be bothered with anything but Himself.

Current Mood: calm
Current Music: Pink Floyd - Atom Heart Mother (4 June 1971)

Metaphysics - Book Lambda - Sec. 8

The existence of an eternal, unmoved mover has been demonstrated, and its nature as substance has been discussed, but the question of how many such movers exist has not been addressed. The theory of Ideas identifies such substances with numbers but sometimes claims numbers are unlimited and sometimes claims there are only 10 substantial numbers (with no explanation why just that many.) Given unmovable first movers are substances, the fact that eternal movement requires an eternal mover, and that each different movement requires a different mover, we see that each different movement of the stars corresponds to a different substance.

Thus, how many unmovable substances exist is a question for astronomy, the branch of mathematics closest to philosophy. We can compute this number ourselves to give us a concrete number to consider, but a more exact computation by astronomers would be preferred. To begin, Eudoxus supposed that the motions of the sun and moon each required three spheres: the sphere of fixed stars, one moving along with the center of the zodiac, and another moving across the zodiac, inclined at different angles for the moon and sun. Each planet requires four spheres, two the same as the first two for the sun and the moon, and two more. Callippus enumerated all the same that Eudoxus did, but added two each to the sun and moon, and one to all the other planets but Jupiter and Saturn. We note that to explain all motions, there need to be spheres that counteract the motions of each of the more outer spheres, bringing us to a total of fifty-five (or forty-nine if the additional movements of sun and moon are not added.) That this number is necessary is left to more powerful thinkers.

If no spatial movement is not involved in the moving of a star and if every substance having attained the best state is an end, there can be no other unmovable substance than those listed above. Since a movement cannot be for the sake of another movement without introducing an infinite regress, the end of every movement is a divine body moving through heaven.

There is only one heaven, both in form and being. If there were a different heaven, say, for each man, the prime mover of each heaven would share a form with the prime movers of other heavens, and thus would require matter to distinguish it from the others.

We note that the tradition handed down by our forefathers that the bodies in the heavens are gods, stripped of the mythological embellishments used to persuade the multitude, was an inspired utterance.

---

The number of prime movers that exist is the same as the number of motions required for the bodies in the heaven, of which there is only one. It is up to astronomers to compute this exactly, but it looks like there must be fifty-five of them. All motion has as its end one of the divine celestial bodies.

Aristotle follows up his most profound theological speculations with a quirky section that claims there are exactly fifty-five (or maybe forty-nine) gods, at least if he's counting correctly, in addition to providing a couple of corollaries to the previous section. The first corollary, that the end of every motion is a divine celestial being, seems to be a justification for astrology. The second, that heaven is unique, seems to be aimed at Protagoras' "man is the measure of all things."

I guess it's the same as with many intellectual endeavors in history. The formulation of a great founding principle is quite often followed up with a filling-in of details that can be quite bizarre. Specifically, the exact number of first movers, whether 55 or 49, seems so extraneous to anything Aristotle has talked about so far.

Current Mood: calm

Metaphysics - Book Lambda - Sec. 7
It is plain in fact that the first heavens move with an unceasing circular motion. Thus, these heavens must be eternal. Also, since that which is moved must be moved by something, there must be another, eternal mover that is, by virtue of being eternal, substance, and actuality, moved without being moved. Objects of desire and objects of thought are unmoved movers. The primary object of thought and desire is the apparent good. Thus, that for the sake of which is unmovable, and it causes things to move toward it by being loved.

Things that move are capable of being other than they are; this even applies to the heavens at least insofar as they are moved. The first mover produces this motion. It is necessary in the sense of being contrary to impulse, in the sense that all motion without a good is impossible, and in the sense that it cannot be otherwise.

This principle underlies the heavens and the world of nature. Its life is the best we can enjoy and only for a short time. In contrast, since it is actuality, it is always in this best state. Thought, in itself, deals with that which is best in itself. Thought thinks itself since thought shares the nature of objects of thought. That which is capable of receiving thought is substance; that which possesses the object of thought is active. The act of contemplation is what is best. Thus, God is always in an act of contemplation. Since the actuality of thought is life, God is a living being, eternal and most good.

Pythagoras and Speusippus, who say that supreme beauty and goodness were not present at the beginning of the universe, since they are effects instead of causes, are wrong. As the seed comes from a complete being, the first thing must be complete.

It has been shown that there is an eternal, unmovable substance separate from sensible things. This substance, being simple, cannot have any magnitude. Since it has infinite power, causing movement through time, it cannot be finite. But there are no infinite magnitudes.

---

There is an unmoved mover. It moves in the same way an object of thought moves thought, and this is by being loved. This unmoved mover is God. Since God is constantly in the act of thought, he is a living being.

This has to be one of the most remarkable sections in the whole Metaphysics. The existence of an unmoved mover was the subject of the last couple books of the Physics, but this section makes a great theological leap and identifies this with God. Furthermore, the nature of God is described as a self-thinking thought, simply since the act of contemplation is the most pleasant thing that humans can engage in (as far as Aristotle can imagine), and that thoughts move without themselves being moved.

No matter how elegant this argument is, and it is elegant, it's still difficult to not laugh at the idea that the heavens move in a circle eternally because the heavens love God.

Current Mood: annoyed
Current Music: Yankees at Rays

Advertisement