Jump to content






Photo
- - - - -

Randian Axioms and Postulates in Metaphysics


  • Please log in to reply
11 replies to this topic

#1 Stephen Boydstun

Stephen Boydstun

    $$$$$$

  • Members
  • 1,767 posts
  • Gender:Male
  • Location:Virginia
  • Interests:Metaphysics; Theory of Concepts and Predication; Philosophy of Science and Mathematics; Philosophy of Mind; Foundations of Ethics; Physics; Mathematics; Biology; Cognitive Science

Posted 29 January 2010 - 10:20 AM

I take metaphysics to be the comprehension of widest existence, where the concrete existents encompass any and all concretes, physical or mental or whatever. Those of us who are physicalists take all concretes to be physical concretes.* Still, even for the physicalist, it remains that metaphysics is comprehension of widest existence, where the concrete existents encompass any and all: concretes of ordinary experience (including history and inventions) or concretes of this or that empirical science.

The topic of metaphysics, in my meaning, is concrete existence, whether those concretes are entities, attributes, actions, or relations, whether actual or potential. This conception of metaphysics is consistent with Rand’s views that all existents have particular and specific identities (AS 1016, 1035–37; ITOE 6, 240), that “everything that man perceives is particular, concrete” (ITOE 1, 199), and that “‘things as they are’ are things as perceived by your mind” (AS 1036).

To fully comprehend the concrete, we require the abstract. To fully comprehend the actual and the potential, we require the possible.

Rand conceived of logic as “the art of non-contradictory identification,” and she held that “logic rests on the axiom that existence exists” (AS 1016). Recall that Rand’s first axiom for metaphysics is the affirmation that existence exists. ($) Two further axioms are manifest to one in the act of grasping the statement “existence exists.” These are that something exists which one perceives and that one exists and possesses consciousness of existing things (AS 1015). We have the following:

(E) Existence exists.
(E1) Something exists which one perceives.
(E2) One exists and possesses consciousness of existing things.

I concur with the preceding, and to Rand’s E-axioms I would add a related supposition, which I call my epsilon-premise:

(ε) There is nothing in existence whose existence cannot be asserted.

If something exists, its existence can be asserted. Then because (E1) and (E2) are implicit in the act of grasping the assertion that any particular thing exists, there is nothing in existence that is not potentially the subject of acts of consciousness.**

Rand also takes as axiomatic that “to exist is to be something, as distinguished from the nothing of nonexistence, it is to be an entity of a specific nature made of specific attributes” (AS 1016). A thing is itself, it is what it is. An object is of one sort or another, it possesses certain attributes and not others, and its actions are certain ones and not others. In three words,

(I) Existence is identity.

Every existent is with identity. If something exists, then it is not without identity.

The identities of existents exist with them. Adding (ε), we may conclude (ι): all identities of existents can be asserted. Any identity of an existent that is so can be asserted to be so. Implicit in the act of grasping the statement that a certain identity is so, of a certain existent, is the fact that some identity holds which one perceives and that one exists as a consciousness of identities.

Concerning consciousness, taken as the act of perceiving that which exists, Rand poses the further axiom:

(C ) Consciousness is identification.

The art of non-contradictory identification is logic, in Rand’s conception of it, and “logic rests on the axiom that existence exists” (AS 1016). Then it is not a logical possibility that nothing exists (further).

Rand presents (E) expressly as an axiom. She does not present (I) and (C ) expressly as axioms, though she strongly suggests that they are axioms. They are introduced as completing the traditional law of identity, the principle that a thing is itself, or A is A (AS 1016; further – A, B). Rand takes (I) and (C ) to state primary facts and to be immediate and most important elucidations of the three concepts she takes to be axiomatic expressly: existence, identity, and consciousness (ITOE 55–56).

Rand takes the proposition “Existence is identity” to express a primary fact. This assertion is a fundamental proposition composed with the concepts existence and identity (further), which concepts, along with the concept consciousness, Rand takes as axiomatic. Other propositions expressing primary facts are “Existence exists” and “Consciousness is of existence not only itself.” Rand constructs arguments to show that these two propositions are indeed axiomatic (AS 1015–16, 1039–40).

I have come round to taking the proposition “Existence is identity” as an epigram encapsulating identity postulates on existence. A few years ago, led by Rand’s text, I began drawing forth specific identity postulates on existence contracted into “Existence is identity.” I have constructed arguments showing that some of these postulates are indeed axiomatic. I will state these propositions and my proofs of their axiomatic status in subsequent posts to this thread.

A philosophic axiom “is a statement that identifies the base of knowledge and of any further statement pertaining to that knowledge, a statement necessarily contained in all others, whether any particular speaker chooses to identify it or not” (AS 1040). Rand’s axioms, then, cannot be proven without circularity. Her axioms affirm such things as the fact of existence, the fact of one’s consciousness, and the fact of one’s existence. These cannot be proven without recurrence to themselves if Rand has found axioms that truly meet her requirement of absolute fundamentality in reality and in any knowledge of reality.

The E-axioms are implied in any conscious action, including in any claims denying those axioms (AS 1016). More generally, for any of her axioms, Rand maintains that one has to “accept it and use it in the process of any attempt to deny it” (AS 1040). One cannot deny the truth of these axioms without some form of self-contradiction. Pose any such axiom as false, there will be contradictions with what one retains as real or there will be contradictions with presuppositions of the activity of rational deduction.

(See further.)

Notes
*That does not mean the sciences traditionally not counted among the physical sciences should be thought of as subdivisions of the physical sciences.
**There are things no longer in existence which were not subjects of consciousness during their existence and which are now no longer potential subjects of consciousness.


~~~~~~~~~~~~~~~~

Beyond the Axiomatic – Identity with Measurement

I have held with Rand:

(Im) All concretes have measurable relations to other concretes.

Furthermore,

(Cm) Cognitive systems are measurement systems.

The thesis (Cm) is, in my view, a claim to be set within the popular approach of analyzing cognitive systems as computational systems.

Rand’s measurement-omission analysis of concepts implies a distinctive magnitude structure for metaphysics. The argument supporting that claim was given in the sixth paragraph of my essay Universals and Measurement (The Journal of Ayn Rand Studies 5(2): 271–305). What is that argument? It is simple: All concretes can be placed under concepts. All concretes can be placed within some concept-class or another. For all concretes, some of those concept-classes will be of the Randian measurement-omission form. Then all concretes must stand in some magnitude relations such that the Randian form of conceptual rendition is applicable to them.

The preceding argument does not rely on a supposition that all one’s concepts are formed by a process of measurement omission. The argument says only that any concept you give me can be reformed into a measurement-omission form, or if not, at least this much is true: all the concretes falling under your concept fall under some concept(s) or other for which we can discover its measurement-omission form. That last, modest premise is all I need to conclude that Rand’s measurement-omission form of concepts implies that all concretes must stand in certain minimal magnitude relations.
Rand’s concept theory—not her formation theory, but her analysis theory—implies a specific, meager (but nontrivial) magnitude structure for all concretes, which is to say, a distinctive magnitude structure for metaphysics. In my “Universals and Measurement,” I uncovered that minimal structure and characterized it in three ways: by its automorphisms (#13), by its mathematical category, and by the types of measurement it affords.

What is meant by a magnitude structure? That means an ordered relational structure. These structures are not only abstractions. They can be concretely realized relations. The most accessible example is geometry. Not analytic geometry (with coordinate systems, calculus, and all that), but the plain old synthetic geometry such as one learns to weave in a high school geometry course. The various geometries are all ordered relational structures, but some have all the structure of others plus more. Here are some geometries in their cumulative hierarchy of structure:
{[(Ordered) Affine] Euclidean}
{[(Ordered) Absolute] Euclidean}
{[(Ordered) Absolute] Hyperbolic}
{[(Projective) Affine] Euclidean}
[(Projective) Elliptic]
These layers of ordered relational structure are an objective matter. They have been discovered, not arbitrarily constructed (a, b).

I should mention for a general educated audience that there are no mathematical formulas that cannot be translated into a natural language such as English. That is something that needs frequent mention. Similarly, the epigrammatic strings I improvised above concerning geometry are nothing but abbreviations of English statements. One true rendering of the string {[(Ordered)Affine]Euclidean} would be “Axioms that imply ordered geometry can be joined with certain further axioms to imply affine geometry, and those combined axioms can be joined with certain further axioms to imply Euclidean geometry.” Another true rendering would be “A Euclidean plane (or space) is composed of affine structure and specific additional structure, and the affine structure is composed of order structure and specific additional structure.”


The minimal magnitude structure implied (by measurement-omission concept analysis) for all concretes is metaphysical structure. It is structure beyond logical structure; it is constraint on possibility beyond logical constraint. Yet it is structure ranging as widely as logical structure through all the sciences and common experience.

So when I said above “to fully comprehend the actual and the potential, we require the possible,” the structure of possibility is not only logical, but mathematical. The contrast of concrete existence to abstractions over it is to be cashed, in terms of mathematical structures, as the contrast of synthetic geometry to analytic geometry.

In “Universals and Measurement,” I sought and found the specific minimal magnitude structure the world must have such that Randian conceptual rendition of the world is possible. To ask for such structure conditions for the possibility of conceptual rendition sounds suspiciously similar to Kant’s quest for the conditions of possible experience or his quest for the conditions of possible cognition (KrV B138). There is a great difference between what I was seeking in “Universals and Measurement” and what Kant was seeking in his famous questions. The magnitude structure I captured is not something that our cognitive system (specifically, our conceptual faculty) prescribes for the knowable world. Rather, however pervasively that structure is in the world, it is there independently of our cognitions.

Ours is not a Kantian program. We do not say that because our conceptual faculty works necessarily in such-and-such way we must find the world everywhere conforming to that way. We do not say that our concepts must all necessarily be susceptible to being cast in measurement-omission form, and that therefore we must find the world affording that form. Rather, we leave open to trial whether we shall find the world everywhere congenial to the measurement-omission form of concepts.

There are indeed some indispensable concepts we should not expect to be susceptible to being cast under a measurement-omission form of concepts. Among these would be the logical constants such as negation, conjunction, or disjunction. The different occasions of these concepts are substitution units under them, but the occasions under these concepts are not with any measure values along dimensions, not with any measure values on any measure scale having the structure of ordinal scale or above. Similarly, it would seem that logical concepts on which the fundamental concepts of set theory and mathematical category theory rely have substitution units, but not measure-value units at ordinal or above. The membership concept, back of substitution units and sets, hence back of concepts, is also a concept whose units are only substitution units. Indeed, all of the logical concepts required as presupposition of arithmetic and measurement have only substitution units. Still, to claim that all concretes can be subsumed under some concept(s) other than those, said concept(s) having not only substitution units, but measure values at ordinal or above, is a very substantial claim about all concrete particulars.

Rand took the thesis (Im) to be axiomatic in that she took it to be entailed by her axiom (I). A thing not measurable in any way “would bear no relationship of any kind to the rest of the universe, it would not affect nor be affected by anything else in any manner whatever, . . . in short, it would not exist” (ITOE 39). Rand is supposing that anything bearing some relationship to the rest of the universe bears some measurable relationship to the rest of the universe. I think that this supposition, which is tantamount to (Im) (all concretes have measurable relations to other concretes), is a postulate additional to the axiomatic postulate (I) (existence is identity). I do not regard the postulate (Im) to be axiomatic; unlike the axiom (I) (i.e., specific postulates that fall under the epigram and are axiomatic), the postulate (Im) can be denied without self-contradiction and is therefore open to possible restriction by counterexamples. Like Rand, however, I take (Im) to be an unrestrictedly true postulate.

Return now to the proposition (I) itself: Existence is identity. Rand demonstrated that some other propositions in which axiomatic concepts appear are axiomatic, but she did not demonstrate the axiomatic status of this proposition. She challenged anyone to make any claim of knowledge that does not presuppose (I). One cannot claim to know how to lift water to the roof of a skyscraper without knowing the identities, the natures, of gravity, water, and pipes (AS 1036). Let one “who does not choose to accept the axiom of identity, try to present his theory without using the concept of identity or any concept derived from it” (AS 1040).

If “Existence is Identity” (I) were axiomatic, then any knowledge presupposes it and no counterexample to it should be forthcoming. But inability to come up with a counterexample to a proposition is not enough for demonstrating the proposition to be an axiomatic one; absence of counterexample obtains for any postulate applicable to and true of all concretes, not only for the axiomatic postulates. To show the postulate to be axiomatic, we must show its denial to be contradictory to itself or its presuppositions.

Continued below
—Exclusions of Non-Contradiction: Entities
—Exclusions of Non-Contradiction: Actions
—Exclusions of Non-Contradiction: Attributes

Edited by Stephen Boydstun, 04 February 2010 - 09:02 AM.


#2 Stephen Boydstun

Stephen Boydstun

    $$$$$$

  • Members
  • 1,767 posts
  • Gender:Male
  • Location:Virginia
  • Interests:Metaphysics; Theory of Concepts and Predication; Philosophy of Science and Mathematics; Philosophy of Mind; Foundations of Ethics; Physics; Mathematics; Biology; Cognitive Science

Posted 29 January 2010 - 10:35 AM

—Exclusions of Non-Contradiction: Entities

Part 1 – Identity as to Kind

Rand states her finer structure for the law of identity as follows. “Whatever you choose to consider, be it an object, an attribute, or an action, the law of identity remains the same. A leaf cannot be a stone at the same time, it cannot be all red and all green at the same time, it cannot freeze and burn at the same time. A is A. . . . A contradiction cannot exist. An atom is itself, and so is the universe; neither can contradict its own identity; nor can a part contradict the whole” (AS 1016).

Rand’s law of identity entails that objects come in some exclusive kinds. Leaf and stone are kinds that are exclusive with respect to each other. Any object is also of kinds that are not exclusive of each other: a leaf is a kind of plant part, it is a kind of light catcher, and it is a kind of drain clogger. To say that an object is a leaf and a stone violates identity in Rand’s sense; it is a contradiction. But to say that an object is a leaf and a drain clogger is no contradiction. Objects come in some exclusive kinds, and it is sensitivity to these sets of kinds that is written into Rand’s conception of non-contradiction concerning the kind-identity of an object. (Cf. Plato’s Sophist 252e–54b.)

Rand clearly intends that what is here proposed for objects is to be generalized to entities. Every entity is of some kinds that are exclusive relative to other kinds of entity. Rand uses the term entity in the paragraph preceding the object examples of leaf and stone. That is, she uses entity in the initial statement of her law of identity: “To exist is to be something, . . . it is to be an entity of a specific nature made of specific attributes” (AS 1016). On that page, it is clear that she takes for entities not only what are ordinarily called objects such as leaf, stone, or table, but micro-objects such as living cells and atoms, super-objects such as solar system and universe, and substances such as wood.

Now we have a modest problem. If we say “to exist is to be an entity of a specific nature made of specific attributes,” we seem to say that attributes are either entities or are not existents. Consider for attributes “the shape of a pebble or the structure of the solar system” (AS 1016). To avoid the patent falsehood that the shape of a pebble does not exist, shall we say that not only the pebble is an entity, but its shape is an entity? Rand reaches a resolution by a refinement in her metaphysics nine years after her first presentation. In 1966 she writes “Entities are the only primary existents. (Attributes cannot exist by themselves, they are merely the characteristics of entities; motions are motions of entities; relationships are relationships among entities)” (ITOE 15). Let us say then that to exist is either (i) to be an entity of a specific nature made of specific attributes or (ii) to be some specific character in the nature of entities.

In ITOE Rand also makes the refinement of taking materials, physical substances, to be not fully specific entities. “Materials exist only in the form of specific entities, such as a nugget of gold, a plank of wood, a drop or an ocean of water” (ITOE 16). Materials, for Rand, would seem to fall under both (i) and (ii), and I do not see any defect in that.

Let us now expose self-contradictions that obtain in denial of the ramification of "existence is identity" that entities are always of some exclusive kinds. Suppose an entity exists and is not of any kind that excludes it being any other kinds. If the supposed entity is nothing but existence itself, then there is no contradiction; one is simply talking about existence as a whole. So suppose an entity exists and is not of any kind that excludes it being other kinds and is not existence as a whole.

Then the supposed entity could be one with any other entities that are of exclusive kinds (just as a leaf that is a drain clogger could be one with a leaf that is dead, maple, and wet). For it is not an entity of any kind excluding it being other kinds. But to say that an entity is not of any exclusive kind and that it is one and the same with another entity that is of some exclusive kind(s) is a contradiction. (Non-A is A.) Indeed if some entity were not of any exclusive kind, then it could be one with the person who supposes such an entity. Then to suppose an entity that is not of any exclusive kind is to suppose that one’s person could be an entity not of some exclusive kinds. But that supposition contradicts the presupposition that one is of the exclusive kind person, a person who makes the (errant) supposition. (Cf. Aristotle’s Metaphysics 1007b19–1008a28.)

So I have argued the axiomatic standing of “existence is identity,” where the existents are entities and the identity is kind-identity. All entities are of some exclusive kinds—a leaf cannot be a stone at the same time—and this postulate must be accepted on pain of self-contradiction.

~~~~~~~~~~~~~~~~

Beyond the Axiomatic

Philosophers often use the term entity to mean any item whatever. That is one customary usage and perfectly all right. Rand wanted to take entity into her technical vocabulary as something more narrow. In her sense, entity is much like Aristotle’s substance, but wearing identity on its sleeve.

In her 1966-67 treatise ITOE, she writes that "entities are the only primary existents" (15). She goes on to name some things that cannot exist without connection to entities: attributes, motions, and relations. She takes all of these as genuine existents. Into the pool with attribute, motion, and relation, Rand also places event, locomotion, action, and activity of consciousness (7-8, 29-33, 39).

All of those characters are, for Rand, concrete existents; but they are not entities in the way she intends her concept entity. To qualify as an entity, an entity has to do more than be able to stand as the subject of predication (or as the argument of a propositional function). Running can be the subject of predicates, but it is an action, not an entity.

I conjecture that concrete existents are best conceived as entities only if standing in some measurement relations above ordinal (viz., interval and ratio and multidimensional spaces of these).

Edited by Stephen Boydstun, 18 August 2011 - 01:26 PM.


#3 BaalChatzaf

BaalChatzaf

    $$$$$$

  • Members
  • 11,360 posts
  • Gender:Male
  • Location:Currently residing in New Jersey, the Bad-a-Bing State.
  • Interests:mathematics, physics, alternative energy sources.

    I am also involved in preparing recorded books for blind and dyslexic folks.

Posted 29 January 2010 - 01:11 PM

I should mention for a general educated audience that there are no mathematical formulas that cannot be translated into a natural language such as English. That is something that needs frequent mention.



I also mention to the same general audience that some mathematical statements translated into "ordinary" or street language are so syntactically convoluted as to be incomprehensible. To check this out, I recommend to the read Euclid's Elements Book V The Heath English translation from the Greek. See how well you do with natural or ordinary language as compared to an algebraic expression of the same propositions. Or read Newton's -Principia Mathematica-. Some of the non-mathematical sentences (even in English) are rather convoluted.

If that does not convince you try translating a proof that Tychinoff's theorem is equivalent to the Axiom of Choice into street language.

Stephen, are you up to the challenge?

Ba'al Chatzaf
אויב מיין באָבע האט בייצים זי וואָלט זיין מיין זיידע

#4 Stephen Boydstun

Stephen Boydstun

    $$$$$$

  • Members
  • 1,767 posts
  • Gender:Male
  • Location:Virginia
  • Interests:Metaphysics; Theory of Concepts and Predication; Philosophy of Science and Mathematics; Philosophy of Mind; Foundations of Ethics; Physics; Mathematics; Biology; Cognitive Science

Posted 31 January 2010 - 12:20 PM

—Exclusions of Non-Contradiction: Entities

Part 2 – Identity as to Particularity

I have argued the axiomatic standing of “existence is identity,” where the existents are entities and the identity is kind-identity. All entities are of some exclusive kinds—a leaf cannot be a stone at the same time—and this postulate must be accepted on pain of self-contradiction.

My argument began by accepting that some entities are of some exclusive kinds, and it concluded that all entities are of some exclusive kinds. The conclusion is reached by avoidance of self-contradiction. One might deny the premise that there are some entities that are of some exclusive kinds. That would be a contradiction with the presupposition that the point of one’s theoretical inquiry is truth, “the recognition of reality” (AS 1017). It would be the contradiction of a consciousness with its objects. The rule against self-contradiction in one’s affirmations rests on the rule against contradiction of a consciousness with its objects. Non-contradictory identification is at root fidelity to existence as it exists.

That some entities exist, that some entities are of exclusive kinds, and that some entities are particulars excluding other particulars, are facts given in perceptual experience. So it goes too for the facts that some entities have actions, have attributes, and stand in relationships to other entities or to parts of their own whole. However one transitions to all from the some in the two preceding statements, the all-statements will inherit the ultimate basis of perceptual experience.

Now let us leave aside the kind-identity of an entity for a while and consider only the particularity of the entity. Surely it is some particular entity and not some other particular entity. Surely all entities are exclusive particulars. If some were not, then these could be one with other particulars that are exclusive particulars. To say that an entity is not an exclusive particular and that it is one and the same with another particular that is an exclusive particular is a contradiction. Indeed if some entity were not a particular excluding it being any other particular, then it could be one with the particular person who supposes such a non-excluding particular. Then to suppose a particular entity that is not also not any other particular entity is to suppose that one’s person could be not a particular entity excluding it simultaneously being some other particulars. But that contradicts the presupposition that one is a particular making a particular subject supposition.

So I have argued also the axiomatic standing of “existence is identity,” where the existents are entities and the identity concerns particularity. To say “I hold a leaf in my left hand and a leaf in my right hand, and they are the same particular leaf” is a contradiction.

We use this sort of principle in mathematics as well as in everyday action. In an indirect proof in mathematics, to show that from a thesis it follows that two is one (2=1) is to show the thesis is in contradiction of the mathematical postulates we are taking for true. And to show that from a thesis it follows that one is one (1=1) is to show consistency of the thesis with our postulates.

We have that all entities are of exclusive particularity and are of some exclusive kind(s). These postulates must be accepted on pain of self-contradiction.


~~~~~~~~~~~~~~~~

Beyond the Axiomatic

When I shift between the word entity and existent in the following, it is never accidental. A shift to existent means that the statement applies not only to entities, but to actions, attributes, and relations.

I distinguish between the particular and specific identities of an existent. Particular identity answers to that and which, to where and when, and to how much. Specific identity answers to what, to character such as kind, form, capability, or susceptibility. Every existent consists of both a particular and a specific identity. Existence is particular identity together with specific identity. Consciousness is identification particular and specific (Boydstun 1991, 43–46).

A distinction between particular and specific identity may be implicit in Rand’s writings addressing identity, as when she writes of the child’s identity stage of “awareness of specific, particular things” (ITOE 6).* I find the distinction convenient and natural for precise thinking about identity and contradiction of identity (1995, 105, 110). I shall use it in my development of Rand’s metaphysics.

It is convenient to divide the how much particular identity into two sorts. There is on the one hand what I shall call the item-measure particularity of the existent. That would be its possibilities for being placed in sequences, for being counted, and for being placed in frequency distributions, discrete or continuous. On the other hand, there is what I shall call the trait-measure particularity of the existent. I am using trait as a short cover for attributes, actions, and dimensioned relations (including classical and quantum states). The relations implicated in item-measure particularity are not relations along dimensions; they are purely numerical relations.

Item-measure coincides with the type of measurement now called absolute in the measurement literature. Trait-measure includes the types the experts call ordinal, hyper-ordinal, interval, and ratio. Trait-measure includes also all the multidimensional forms of measurement, such as the use of topological vector spaces in physics or the use of trigonometry in making drapery. Measurement restricted to the sense of trait-measures is what Rand presumes when she writes that “entities (and their actions) are measured by their attributes (length, weight, velocity, etc.)” (ITOE 7). I should note that where and when are trait-measures, though I spotlight them apart from all other trait-measures.

Sameness “applies in the most strict sense to what is numerically one” (Topics 151b28–29). A belly is one and the same as a tummy. A two-footed terrestrial mammal is one and the same as a man. That man assisting in the childbirth is one and the same as the philosopher Socrates.

Leibniz would add that a triangle is one and the same as a trilateral (NEU 363), and Frege shows us a less obvious geometric example of particular identity (1879, 21). All of these identity statements may be represented by the schema X=Y, where the relational sign (=) is the logical identity sign, at present restricted to the that and which and item-measure portions of particular identity.

We learn that the morning star is one and the same celestial object as the evening star, namely the planet Venus. The fact that Venus is Venus (A=A) is the bottom explanation for the fact that the morning star is the evening star (X=Y). But for the particular identity of Venus to support the truth that the morning star is one and the same celestial object as the evening star, we are presuming not only that Venus is numerically one with itself. We are further presuming, with reason, that Venus is in a determinate sequence of locations across particular times and not in any other sequence. Venus has a determinate world line (or avenue) in spacetime and not any other world line.

It is only a fuller particular identity of Venus, answering not only to that and which and item-measure, but to where and when, that allows the particular identity of Venus to adequately support the truth that the morning star is one and the same celestial object as the evening star. The morning star, evening star, and Venus are all in spacetime in (the same) excluding ways. Venus rising from one’s eastern horizon excludes Venus simultaneously rising from one’s horizon due north, and so forth. Similarly, the spacetime ways of leaves exclude the one in my right hand being the one simultaneously in my left hand.

In the first part of this post, I moved from the axiom that all entities are exclusive particulars to saying, as an example of that axiom, that it would be a contradiction to say “I hold a leaf in my left hand and a leaf in my right hand, and they are the same particular leaf.” We see now that that is not an example purely of the axiom and nothing more. It is an example of the axiom, but with further features of the particular identities of leaves. The example sports more of the full suite composing any concrete particular identity than only that and which and item-measure; it sports where and when.

I take it as a postulate that all concrete particulars (short of the universe itself), whether physical or mental, have delimited loci in spacetime. This is how I shall divide concrete particulars such as stones, throws, chills, clouds, rainbows, and thoughts from, on the other side of the division, abstract particulars such as sets, mathematical groups, and operator actions on Hilbert spaces. Concrete particulars have particular histories in the space and time had by the physical universe. In thinking about abstract particulars as such, we suspend consideration of any potential exemplification they may have in actual space and time.

This postulate, that all concrete particulars have delimited loci in actual spacetime, accords fully with ordinary experience and modern science. But I do not have a proof showing that if one denies that all concrete particulars are somewhere and somewhen, then one ends in contradiction. So I take the thesis that the particular identity of any concrete existent includes an answer to where and when as a well-founded postulate and definition, not as an axiom.

My proof that every entity is a particular excluding it being other particulars relied only on the mere that and which and item-measure portion of particular identity. I shall now argue that there is no entity whose particular identity consists only of those portions; particularity is always accompanied by delimitation in space and time or by some other trait-measures.

Suspend for now the question of whether the following entity has a delimited locus in spacetime. In other words, leave aside whether this entity is concrete or abstract. Suppose an entity of exclusive that, which, and item-measure particular identity and suppose that one of the item-measures this entity affords is countability in elementary counts using the natural numbers 1, 2, 3, . . . . The entity can be item 5, for example, in a countable arbitrary collection of 17 items. But then this entity also affords countability in collections of its own kind (which are non-arbitrary collections). Indeed it affords countability in a subcollection of its own kind in which members have all the same traits as one another. How can the individuals in this last sort of collection be countable, leaving aside any potential for keeping the individuals distinct by exclusive locations in space (spacelike hypersurfaces of spacetime)?

I see only one way. If some of the traits common to members of this subcollection of the kind are measurable and if these traits have different sets of particular measure values, then these individuals can have distinction in a count fixed by distinction in those sets of trait-measure values.

The preceding bit of reasoning will go through for all forms of item-measures in the exclusive numerical identity of an entity. I used the case in which the entity’s appropriate item-measures included counts using the natural numbers. The same conclusion will obtain for cases in which the appropriate item-measures include one-one and onto mappings to the rational, real, or complex numbers.

The general situation is that the exclusive particular identity of an entity does not obtain without the entity having uniquely delimited loci in space and time or a unique set of trait-measure values for some non-spatiotemporal traits. This squares with Rand’s thinking of the measurement of entities (and their actions), for purposes of her measurement theory of concepts, as being “measured by their attributes, . . . and the standard of measurement [being] a concretely specified unit representing the appropriate attribute.” (ITOE 7). The measurement in mind here is measurement above absolute scale, so measurement at ordinal scale and above. That general situation squares also with Rand’s thought than attributes are separable from entities only in thought, although I would caution against saying “an entity is its attributes in every way,” for an entity has the capability of entering item-measures independently of its attributes and their trait-measures. (See ITOE 276, 278–79, and Sciabarra 1995, 146–47). (One does not get credit for knowing the principles of counting until one understands that items of any mixture of kinds can be counted #40.)

I arrive, then, at a version of the Identity of Indiscernables, where the able is an affordance of the entity, a character of the entity. Here is the version of the principle of the Identity of Indiscernables now at hand: Multiple entities alike in all their traits and alike in all the particular measure values of their traits, including all their loci in space and time, are not multiple entities. Rather, they are one self-same entity. (Further)

Let me collect the results from my magnification of Rand’s axiom (I) “existence is identity” as it comprehends “a leaf cannot be a stone at the same time.” Of the primary form of existence, entities, we have the following thus far:

Axiom (Iek): Every entity is of a kind excluding it being some other kind of entity.

Axiom (Iep): Every entity is a particular excluding it being other particulars.

Corollary of (Iep): Every entity has (i) uniquely delimited loci in space and time or (ii) a unique set of trait-measure values for some non-spatiotemporal traits or both (i) and (ii).

Postulate now taken for Definition: A concrete particular (whether entity, action, attribute, or relation) is a particular having some delimited locus in the spacetime of the actual world.

Continued below—
—Exclusions of Non-Contradiction: Actions
—Exclusions of Non-Contradiction: Attributes

Edited by Stephen Boydstun, 18 August 2011 - 01:29 PM.


#5 Stephen Boydstun

Stephen Boydstun

    $$$$$$

  • Members
  • 1,767 posts
  • Gender:Male
  • Location:Virginia
  • Interests:Metaphysics; Theory of Concepts and Predication; Philosophy of Science and Mathematics; Philosophy of Mind; Foundations of Ethics; Physics; Mathematics; Biology; Cognitive Science

Posted 05 February 2010 - 09:33 AM

—Exclusions of Non-Contradiction: Actions

Rand’s law of identity entails that actions come in some exclusive kinds in the following sense. Burning of a leaf and freezing of a leaf are kinds of actions that are exclusive with respect to each other. However, to say that a leaf is burning and floating is no violation of identity, no contradiction. Some actions of objects—burning and freezing in the case of leaves—are exclusive with respect to each other. Rand’s conception of non-contradiction concerning actions pertains to these. (Cf. Republic 436b, 436e; Metaphysics 1061b35–62a1.)

Rand’s law of identity also entails that every entity that has actions has certain actions and not others. A green leaf manufactures chlorophyll; a stone does not. Rand’s conception of non-contradiction concerning actions pertains to these exclusions as well. “The nature of an action is caused and determined by the nature of the entities that act; a thing cannot act in contradiction to its nature” (AS 1037). (Cf. On Generation and Corruption 338b15)

I want to prove axiomatic the truth that every action-bearing entity bears certain kinds of action and not others. Suppose an entity could bear any kind of action without restriction of the kind of action. Then it could bear all the acts of a leaf and a stone. Indeed, it could bear all the acts of all the kinds of entity there are. Such an entity would be the conjunction of all the kinds of entity there are with respect to their possible actions.

It would be more. Not only could this super-acting entity bear all the actions of, say, a leaf. It could also burn and freeze at the same time. Yet, having all the possible actions of a leaf, its burning excludes its freezing at the same time. Our super-acting entity is capable of burning and freezing at the same time, and it is incapable of burning and freezing at the same time. Our super-acting entity can float on water, like a leaf, and yet, like a stone, it cannot float on water. These are contradictions. No such entity can exist. There is no entity that can bear any kind of action without restriction of the kind of action.

Moreover, let a person suppose there could be an entity that could bear any kind of action without restriction of the kind of action. Such an entity could bear the act of supposing its existence, just as a person might do. But unlike a person, the super-acting entity could suppose at the same time that such an entity is impossible. But this contradicts the presupposition of a person that contradictories are false.

So I have argued the axiomatic standing of “existence is identity” where the existents are action-bearing entities and the identity is restriction of the kinds of actions of those entities. Rand’s thesis that any entity that exists has a specific nature [“to exist . . . is to be an entity of a specific nature . . .” (AS 1016)] has now been proven to be axiomatic insofar as the action nature of entities is concerned. The postulate that every action-bearing entity bears certain kinds of action and not others must be accepted on pain of self-contradiction.

It should be noticed that I have not proven that, for every action-bearing entity, some of the kinds of action it bears are exclusive with respect to each other. I leave open the possibility that some kinds of entities can bear all the kinds of actions in their repertoire simultaneously. Certainly a leaf is not such an entity.

~~~~~~~~~~~~~~~~

Beyond the Axiomatic

In Rand’s scope of the concept action for metaphysics, include as actions the phenomena dealt with in dynamics. Include also the phenomena of statics and strength of materials. Include the formation of stars, planets, oceans, and organisms. Include chemical reactions and phase changes. Include as actions, too, organic growth, locomotion, and the activities that are consciousness.

Rand writes “Existence exists—and the act of grasping that statement implies two corollary axioms: that something exists and that one exists possessing consciousness, consciousness being the faculty of perceiving that which exists” (AS 1015). Grasps in acts of consciousness are actions. Making statements are actions. The fact of consciousness is implicitly affirmed in making and grasping statements about existence. The fact of consciousness is the fact of an activity, a fact of living activity. One does not wait on education in biology to know one is alive, to know the self of self-awareness is a living being and that awareness of existence is a living action. Implied in the act of grasping any statement about existence is the fact of living action. Further, when Rand writes “I am, therefore I’ll think,” the existence of the thinking self is a living existence (AS 1058).

If existence and consciousness are axiomatic concepts, why are not living existence and living action also axiomatic concepts? (Cf.) Well, in part, they are. Consciousness is living action from the inside, from the side of the intender. Self-consciousness is living existence from the inside, from the side of the intender.

What about living action and living existence from the side of the intended, living action and existence when these are not acts of consciousness, but contents of consciousness? Are these concepts implied in the act of grasping the statement “existence exists”? No. In the immediate corollary (E2) the existing conscious one is known axiomatically to be living from its living acts of consciousness. The “I am” of “I am, therefore I’ll think” contains an identification of the particular live thinking self with something known additionally (not from acts on content, but) in informational content: one having life as an object in the world (and requiring thought for protection and sustenance).

In her 1957 exposition of her philosophy, Rand maintains that in motion there is always a thing that moves. This thing that moves Rand calls an entity, and she says of it that "without the concept of entity, there can be no such concept as 'motion'." More generally, "change presupposes the concepts of what changes, from what to what, . . ." (AS 1039).

Change presupposes entity, but does entity also presuppose change? If in affirming existence exists, one is ascribing an act of existing, then one is ascribing some sort of change to existing per se. There is no passage of time without change. There is no existing without passage of time. Even something existing an instant without duration exists within duration before and after its existence. Every existent that is not the totality of all that exists is in existence while something else exists. In that way, every existent is in time. Approaching time in this way, the usual idea that without change there is no time can be retained, but with the weak sense that an existent is at least in the company of changing things if not itself changing.

There is no need to ascribe act or change to existing itself. Spinoza thought of existence as an act of power. Joining that premise to the premise that whatever exists exhibits the nature of existence in a certain determinate way, he concluded: “Nothing exists from whose nature some effect does not follow” (E IP36).

Rand once remarked informally: “Everything existing is capable of some form of action” (ITOE 270). This is hand-and-glove with Spinoza’s IP36. Rand and I would reject Spinoza’s argument because we would reject the premise that existing is itself an act. To affirm “existence exists” sets a first and final home for the living acts of consciousness, but it does not imply, on pain of contradiction, that every existent is capable of some form of action.

Rand writes that if something were to "bear no relationship of any kind to the rest of the universe, it would not affect nor be affected by anything else in any manner whatever, it would enact no causes and bear no consequences—in short, it would not exist" (ITOE 39). A thing that bore no consequences could not exist. If it was not subject to change, it could not exist, according to Rand’s thought here.

From the context of the preceding quotation, it might seem that Rand could count a concrete existent simply having its attributes or relationships measured as occasions of an entity bearing consequences. However, we should be wary here of my little word simply. Did Rand think that in order for something to be measureable, it had to be changeable? If not (i.e., if we may slip in that little simply), then Rand’s thought here does allow that the concept entity does not include that every entity is subject to change.

Would the proposition that every entity is capable of change follow once we add the postulate (Im), the postulate that every concrete existent stands in measureable relations? No. We should have to add the further postulate that every entity is capable of being a detector in some possible measurement operation. Detectors are necessarily changeable.

As I stated in the initial post of this thread, Rand took the thesis (Im) to be axiomatic in that she took it to be entailed by her axiom (I). A thing not measurable in any way “would bear no relationship of any kind to the rest of the universe, it would not affect nor be affected by anything else in any manner whatever, . . . in short, it would not exist” (ITOE 39). Rand is supposing that anything bearing some relationship to the rest of the universe bears some measurable relationship to the rest of the universe. I think that this supposition, which is tantamount to (Im) (all concretes have measurable relations to other concretes), is a postulate additional to the axiomatic postulate (I) (existence is identity). I do not regard the postulate (Im) to be axiomatic; unlike the axiom (I) (i.e., specific postulates that fall under the epigram and are axiomatic), the postulate (Im) can be denied without self-contradiction and is therefore open to possible restriction by counterexamples. Like Rand, however, I take (Im) to be an unrestrictedly true postulate. In my view, then, the thesis that all concrete entities bear at least the consequence of being susceptible to measurement is to be taken as a postulate that is not axiomatic.

It is plain, however, that Rand took all concrete entities to be susceptible to bearing consequences in a stronger sense than that. Her thought is that to be measurable, a thing must be changeable. The obvious class of counterexamples would be cases in which an invariant quantity of some action or attribute or relationship is measured. Entities coming quickly to mind having invariant attributes, or standing in invariant relationships, also have variable attributes and stand in variable relationships. An electron, for example, is invariant in its charge, spin, and rest mass. But its speed and direction of motion, its dynamic mass, and the amount of electromagnetic radiation it is generating are variable. The thesis that every entity is subject to change is very widely true.

Suppose we count as an entity anything that bears attributes and stands in relationships. Empty space is such an existent. Today, from quantum field theory, we understand the emptiest space to possess nonzero minimum energy and negative pressure. Without those dynamic aspects, it seems not right to think of physical space as a concrete entity. It is a concrete relational structure, but not a concrete entity. Rand addressed a family of entities we could call not fully specific entities, or better, not fully particular entities. The members of this family are physical substances such as gold, wood, or water (ITOE 16). Following Rand’s example of occasioning substances as fully particular entities, I say a particular volume of space (considered apart from its vacuum quantum dynamics) in one’s taxicab is a concrete existent fully particular, but not a concrete entity fully particular.

It seems thus far that being an entity requires being a dynamic participant in interactions. That is, to be a concrete entity, an existent must bear attributes and stand in relationships, it must bear changes in these, and it must be capable of acting and being acted upon.

Emptiest spacetime is such a thing. It is an entity, a concrete entity. It is the minimal entity in the family of not fully particular entities. I say a particular volume of space in one’s taxicab through a particular duration on one’s wristwatch is a fully particular entity. Material substance, including both matter and fields, is the general contrast, within the family, to the not-fully-particular entity that is spacetime.

Spacetime consists of a manifold (a, b) bearing attributes such as metrics, curvatures, and linear-connection structure. It bears changes in those, which, by Einstein’s field equations, are coordinate with changes in the distribution of matter and fields. Not only are these changes of curvature characteristics coordinate with changes in matter and (non-gravitational) field distributions, they are causal back and forth: source matter and fields structure spacetime in certain ways, and those spacetime structures constrain how other matter and fields can move or propagate.*

Let stand the postulate that every concrete entity bears consequences and enacts causes. I let it stand, though not as a postulate whose denial brings self-contradictions.

Though actions include more than actions that are changes (actions includes statics; think of a loaded column), not all changes are actions, in my view. I want to give some examples of changes that are not actions: the purely kinematical aspects of a dynamical situation; inertial motion; undriven uniform rotation; increase of entropy; and passage of time. Here are some changes that are actions: step or speech or perception of a man, gust of wind, tensing of a rope, and ticking of a clock. The changes I am taking as actions require causes. Changes that are not actions require no cause.

Between a desire and its fulfillment, between a perception and an emotion, between elevations of rising water (AS 1021, 1036), there are definite specific links of nature, connections Rand calls causal connections (1037).*(Cf.) There are no loopholes to the law of causality, no miracles; and there are no causal loops, no eating cake before having it (1037). Those are exclusions by the law of identity, for “the law of causality is the law of identity applied to action” (1037; cf. Meyerson 1908).

That last quoted statement situated Rand’s concept of the law of causality and its exclusions within her concept of the law of identity and its exclusions. Rand could speak of a law of causality, relying on readers to have some general idea of what that common phrase means and giving some indication of her meaning by her examples. It was not until years later that Rand gave her own definition of the law of causality:

All the countless forms, motions, combinations and dissolutions of elements in the universe—from a floating speck of dust to the formation of a galaxy to the emergence of life—are caused and determined by the identities of the elements involved. (MvMM 25)

That statement in 1973 resonates with these statements in 1957:

An atom is itself, and so is the universe; neither can contradict its own identity; nor can a part contradict the whole. (AS 1016)

The nature of an action is caused and determined by the nature of the entities that act; a thing cannot act in contradiction to its nature. (AS 1037)

Rand’s statement of the law of causality means that her concept of cause was broader than mine, if she was taking all motions to have efficient causes, not only material causes (a, b). I do not take all objectively given temporal connections to be linked by efficient causality. Consider a satellite orbiting the earth freely, without propulsion. For two adjoining time intervals that are short enough, the satellite is arbitrarily close to inertial motion; to make it closer, reduce the durations of those time intervals further. Rand would have it that two such successive intervals of the moving satellite are connected by being motions of a single object that endures through those times. That connection does indeed hold, but it is a connection of only material causality.

In inertial motion, a body is at rest or in motion in a straight line at constant speed. This motion requires no efficient casual explanation. Forces (nonzero net forces) are the efficient causes of deviations from free, inertial motion. That is elementary physics since the light of Newton.

I restrict motion-applications of the general Randian concept of action to motions not inertial. It is these that require efficient causal explanations. This restriction allows us to rightly say that all actions require causes, which is implied by Rand’s thesis that “all actions are caused by entities” (AS 1037).

All actions are borne by entities, and all actions of entities are caused by entities. (See further a, b.) That all actions are borne by entities would be very widely accepted as plain. “Without the concept of entity, there can be no such concept as 'motion'" (AS 1957). “Motion presupposes entities that move” (Branden 1962). There is, however, something more restrictive being asserted by Objectivists making such statements than might be realized, because they are using entity with Rand’s restricted sense indicated in the initial post of this thread. Rightly understanding what is meant by action and entity, the two claims in the first sentence of this paragraph are postulates going beyond what must be accepted under pain of self-contradiction by their denial.

Let me collect the results from my magnification of (i) Rand’s axiom “existence is identity” as it comprehends “a leaf cannot freeze and burn at the same time” and (ii) Rand’s further applications of identity to action:

Axiom (Iak): Every action-bearing entity bears certain kinds of action and not others.

Postulate (Ia): Every concrete entity bears attributes and stand in relationships; bears changes in those, including changes that are actions; and is capable of acting and being acted upon. (Therefore, every concrete entity bears certain kinds of action and not others.)

Postulate now taken for Definition: A concrete entity is a particular existent having some delimited locus in the spacetime of the actual world and satisfying the conditions in (Ia) and in the Corollary of (Iep). (It follows that physical spacetime as identified in our physics is a concrete entity.)

Postulate (Ac): All actions are actions of entities and require efficient causes by entities.

Continued below—
—Exclusions of Non-Contradiction: Attributes

Edited by Stephen Boydstun, 05 February 2010 - 09:18 PM.


#6 Stephen Boydstun

Stephen Boydstun

    $$$$$$

  • Members
  • 1,767 posts
  • Gender:Male
  • Location:Virginia
  • Interests:Metaphysics; Theory of Concepts and Predication; Philosophy of Science and Mathematics; Philosophy of Mind; Foundations of Ethics; Physics; Mathematics; Biology; Cognitive Science

Posted 28 March 2012 - 11:10 AM

—Exclusions of Non-Contradiction: Attributes

Part 1 – Locke

We take up now Ayn Rand’s law of identity in application to attributes, such as color. “Existence is Identity, Consciousness is Identification. / Whatever you choose to consider, be it an object, an attribute or an action, the law of identity remains the same. A leaf cannot be . . . all red and all green at the same time . . .”(1957, 1016).

The logical character of the proposition that a particular surface cannot be all red and all green at the same time has been controversial. In Book 1 of his Essay concerning Human Understanding (EU), John Locke held “principles of demonstration ‘Whatever is, is’ and ‘It is impossible for the same thing to be and not be’” (EU 1.1.4) as unnecessary for the acquisition of all knowledge. “These maxims are not in the mind so early as the use of reason . . . . How many instances of the use of reason may we observe in children, a long time before they have any knowledge of this maxim, ‘That it is impossible for the same thing to be and not be’?” (EU 1.1.12).

In Locke’s view of development, one first gets ideas of particular things through sense impressions, then general ideas by abstraction from the particular ideas. Before one has learned that it is impossible for the same thing to be and not be, indeed, before one has learned to speak, one has learned that bitter is not sweet, white is not black, and red is not blue. Upon that and upon coming to speech, one learns that wormwood and sugarplums are not the same thing, that a rod and a cherry are not the same thing, that a square is not a circle, and so forth. Upon the same grounds that one came to know those differences, one later comes to know that it is impossible for the same thing to be and not be (EU 1.1.15-20; 1.3.3–4; 2.1.6, 22–26; 4.7.9–10).

Turning from development of knowledge to its structure, Locke goes on to say in Book 4 that in apprehending that white is not black or that a circle is not a triangle we do so directly. Knowledge that our ideas of these things are mutually exclusive is not known by demonstration, hence they require no principles of demonstration such as the principle of noncontradiction. We grasp the exclusivity of black and white in a self-evident perception, which Locke calls intuition. Like Plotinus, Anselm and many others before him, Locke thinks of all other knowledge as dependent on intuitive knowledge. The latter is the ultimate source of all certainty in knowledge. “A man cannot conceive himself capable of a greater certainty than to know that any idea in his mind is such as he perceives it to be; and that two ideas, wherein he perceives a difference, are different and not precisely the same” (EU 4.2.1; see also 3.8.1; 4.7.4, 19).

Is the sum of the angles of a triangle the same or variable from one triangle to another? Seeing the sameness of that sum and the sameness of that sum to the angle of a half-circle for all triangles in the Euclidean plane is not self-evident, but requires demonstration (EU 4.2.2). Each step of a demonstration, in Locke’s view, requires intuitive knowledge (EU 4.2.6). The mind can perceive immediately the agreement or disagreement of each step in the demonstration just as the eye can immediately perceive that black and white are distinct and whether a white paper is entirely so or contains some black marks (EU 4.2.5).

It is intuitive knowledge alone, in Locke’s sense of the concept, that is at the base of human knowledge, certain or probable (EU 4.2.8; 4.2.19). It enables the certain demonstrations in geometry, demonstrations with the ideas of “extension, figure, number, and their modes” (EU 4.2.9; a triangle is a relatively simple mode: 2.12.4; 2.31.3; 3.3.18; 3.9.19; 4.4.6; 4.7.9).

We enjoy accuracy in making and discerning differences in our simple ideas of extension, figure, and number because they are quantitative. (On simple ideas, see EU 2.2.1; 2.3.1; 2.5; 2.7.7-9; 2.30.2; 2.31.2, 12; 2.32.9, 14–16; 3.4.11, 14; 3.8.2; 4.1–2; 4.3.1–21; 4.4.4.) Those ideas are of the primary qualities of things such as Rand’s leaf. Secondary qualities apparently of things, such qualities as colors, are ideas produced in us by the impingement of particles endowed with only primary qualities. In Locke’s day, there were no ways of measuring relations of sameness and difference in the degree of a secondary quality, though he ventures to suppose that greater intensities of primary qualities produce greater degrees of a secondary quality. There are presumably degrees of differences in whiteness that we cannot discern, degrees corresponding to fine degrees of differences in primary qualities that could produce them.

All the same, there are degrees of difference in secondary qualities we do perceive, and such intuitions suffice to found inferential knowledge beyond subjects such as geometry or mechanics. Where the difference in discerned difference in a secondary quality “is so great as to produce in the mind clearly distinct ideas, whose differences can be perfectly retained, there these ideas or colors, as we see in different kinds, as blue and red, are as capable of demonstration as ideas of number and extension” (EU 4.2.13). Where Locke has written “as capable of demonstration,” I think he means “as capable of use in demonstration” (though he could mean additionally what he argues elsewhere in the treatise: there is equally zero capability of any of these different kinds being discerned by demonstration).

Is a sheet of paper before Locke, a paper waiting for the first impression of his pen, all white? Even if it is everywhere white, is it anywhere also red (yet not pink) or black (yet not gray)? We have no direct verdict below our visual thresholds. How does Locke have us know for sure that if the sheet is all white, it is in no part also red or black?

Leaving rather faint the issue of how we know secondary qualities are always produced by primary qualities (cf. Ayers 2011, 146–51), Locke sinks intuitive knowledge concerning imperceptible secondary qualities into the bedrock of intuitive knowledge of primary qualities (EU 4.2.11–12; 4.3.11–13, 15; cf. Descartes 1632, 3–6; 1647–48, 255–56; on Locke’s distinction of primary-secondary, see EU 2.8.9–23; 2.30.2). Locke maintains that for any particular object whatsoever, at a particular time, its extension will be a particular extension, excluding all other particular extensions; its figure will be a particular figure, excluding all others; its motion will be a particular motion, excluding all others. He maintains furthermore that particles of light reflected from a definite part of an object to a particular place of a viewer cannot appear both yellow and azure. “For it is as impossible that the very same particle of any body should at the same time differently modify or reflect the rays of light, as that it should have two different figures and textures at the same time” (EU 4.3.15; see also 2.32.14; unique awareness from unique physical inputs had been embraced also by Descartes, supra; Des Chene 2001, 139–40). Locke generalizes these various sorts of particular exclusion: an object cannot have two exclusive degrees of a given quality simultaneously.

Locke realizes that the impossibility of a leaf being all red and all green is a case of the principle that it is impossible to be both A and non-A. After all, he takes the latter principle to be a generalization of such exclusions encountered in leaf color, fortified by underlying exclusivities in the characters of primary qualities (cf. EU 2.27.1, 4). To say the impossibility of a leaf being at once all red and all green is a case of the logical impossibility of being both A and non-A is not to say the impossibility of the former case derives from the impossibility of the latter general principle. That redness of entire leaf and greenness of entire same leaf are mutually exclusive is not shown by the general principle there are mutually exclusive attributes of entities in existence. Locke correctly recognizes that. “These particular instances, when well reflected on, are no less self-evident to the understanding than the general maxims [superfluously] brought to confirm them: and it was in those particular instances that the first discoverer found the truth, without the help of the general maxims: and so may any one else do, who with attention considers them” (EU 4.7.11[3]).

One weakness in Locke’s sensualist tendency to inductively warrant the necessity of noncontradiction by necessity of exclusions one encounters in sensory qualities is noted by Leonard Peikoff. Leaving the necessity of the principle of noncontradiction as only a necessity encountered in sensory qualities “seems to invite an immediate Humian type of refutation” (Peikoff 1985, 199). Locke’s theory of abstraction is inadequate to the task of delivering from perceptual bases the principle of noncontradiction, noncontradiction in particular and specific identity, noncontradiction in natured entities, their actions, and their attributes. In Locke’s view, we do not attain certain knowledge of the essential natures of physical entities, unlike the situations of our knowledge of simple sensory qualities (primary and secondary) and of mathematical entities. Staying within his view, there is no possibility of arriving at Rand’s absolute principles of identity and noncontradiction through the objects of the senses (EU 2.1.3–9, 22–26; 2.8.7–8; 2.10.6; 2.11.1–9; 2.18.6–7; 2.22.4–5; 2.25.9; 3.3.6–20, 28–38, 49; 4.4.1–6; 4.6.4–16; 4.7.4, 9–10, 16–19; 4.8; 4.11.13–14).

It is a defect of Locke’s view of color qualities that it is set on difference and sameness in sensed qualities as secondary and their tie to qualities as primary. According to Locke’s scheme, the former, such as the qualities red and green, are in us; they are modifications of our sense organs, thence appearing in our minds, wrought by impinging primary qualities. Rand is set, rather, on the leaf and its color nature. Yes, red and green are different things. Yes, part of the story of how we attain express understanding of the principle of noncontradiction is by prior learning of sameness, difference, and exclusivity encountered in experience. But the sense of noncontradiction Rand is deploying—the sense of identity and exclusion Rand is deploying—is of natured entities. One does not need to know anything about the travel of light to the eye, the physical nature of light, the physiology of the eye, or perceptual thresholds in order to know that it is of the nature, the identity, of leaves and our visual power that no leaf can be at once all red and all green (cf. EU 4.11.2).

Thanks to a presentation of Paul Churchland’s, I have experienced afterimages that seem to be entirely of two distinct colors at once. That experience has now become available to me (more tenuously) in Churchland’s chapter on chimerical colors in Neurophilosophy at Work (2007). I follow the viewing procedure, and in figure 9.11, I get to see fading afterimage discs that are at once mauve and black or at once blue and black and so forth. The point of constructing this figure is for the experience of what Churchland calls “impossibly dark” afterimage colors (the scare quotes are his), not to show patches of afterimages that are two colors at once. I leave it for the reader to dig into this important book’s purposes and their fulfillment.

Afterimages are always fading. What I am experiencing as mauve and black “at once” occurs while mauve is turning to black. Whether this experience is rightly a case of seeing a portion of surface, screen or page, as two colors could be reasonably disputed. Suppose it qualifies as such a case. It can be taken under wing in Rand’s picture by saying that leaves have their color nature, afterimages have theirs. The same sort of assimilation could be taken were it the case in the future we learn (i) that canonical reflectance profiles of a surface (Churchland 2007, Ch. 10) can be made two canonicals at once by some treatment of the surface and (ii) our visual system can be made to discern them distinctly when artificially and appropriately altered by, say, an electronic apparatus. Were leaves susceptible to said treatment giving them two canonical reflectance profiles, then in Rand’s conception of noncontradiction in attributes, we should say untreated leaves have their color nature, treated leaves have theirs.

In the experience of Rand or Locke, it was not only the surfaces of leaves that were not anywhere at once both red and green. The same was found of any physical surface whatever. It was found likewise for volumetric color. Locke, and Rand too, might recall a certain liquid which when poured into another produces two colors; but the orange and azure produced are regional in the volume, not both simultaneously throughout (EU 2.11.3).*

Locke, like all of us, would want to look into the particulars of the physiology that make afterimage “impossibly dark” colors possible. Understanding of this physiology indeed enabled prediction of this previously unknown effect (Churchland 2007, Ch. 9). Locke could be pleased to see that afterimage colors have their bases in some primary qualities. These colors, like all experienced colors so far as we know, are patterns of nervous activities transforming retinal patterns of activity. Afterimage effects are artifacts of a visual system adaptive in its evolution to color perception of the world. The exclusion character of different canonical surface-reflectance profiles is present in the exclusions of the pattern of nervous activities that are color. But the particulars of this transport of exclusivity shows that Locke’s reasoning from exclusions among primary qualities to exclusions among secondary qualities was spurious.

Returning to Locke’s sensualist epistemology more generally, afterimage experience of two colors everywhere in a region and my futuristic technological scenario quake Locke’s claim that we know the essence of colors just by knowing red is not green and not blue and so forth. Our knowing distinct simple ideas of qualities, secondary or primary, and our putting distinct names on these distinct ideas is insufficient to capture their exclusionary character, their essential natures as attributes of entities. Let the quake shake off that pretension of closure. It remains for Locke and everyone that “red is green” is a contradiction of experience or a contradiction of correct labels. Experience supporting “red is not green” can still support a principle of noncontradiction that recognizes there are distinct items in the world and distinct names to keep them straight. These remains are, however, not distinctive of attributes. They fall short of recognizing the radical general dependence of attributes on their entities and recognizing the full structure of noncontradiction as it applies to attributes.

There is more trouble for Locke. We have seen he held we grasp the exclusivity of red and green by self-evident perception, which he called intuition. Though we become ready to grasp the general exclusivity of A and non-A by such previous sensory experiences of exclusivity, the former is self-evident and self-evident in the same way as the latter (EU 4.7.9–10). If the intuition that nothing can be at once all red and all green were self-evident in the same way as red not being green is self-evident, then fallibility of self-evidence in the proposition that nothing is at once red and green all over shows fallibility of self-evidence for the principle that nothing is at once A and non-A. My afterimage experience and my futuristic, scientifically informed scenario indicate fallibility in the “self-evidence” of the proposition that nothing is at once red and green all over. Time to check premises, for the result that contradictions are not perfectly self-evidently false is patently false.

We may have also some trouble for Rand, but as suggested already, it can be skirted without significant alteration of her metaphysics. In saying a leaf cannot be at once all red and all green, she may have been relying on an Aristotelian sort of dynamical contrariety among colors themselves. Then regardless of what surface or medium may sport colors, the colors dynamically exclude each other at a given place. Her language is pretty strongly against this interpretation, for she speaks of leaf color, though that could by slim chance indicate merely that the dynamical contrariety of colors holds for all surfaces, including leaf surface. Any such line of thought about attributes can be omitted from Rand’s metaphysics. There remain more than enough riches of identity to get beyond Locke.

(To be continued.)


References

Ayers, M. 2011. Primary and Secondary Qualities in Locke’s Essay. In Primary and Secondary Qualities. L. Nolan, editor. Oxford.

Churchland, P. 2007. Neurophilosophy at Work. Cambridge.

Descartes, R. 1632. Treatise on Light. In Gaukroger 1998.
——. 1647–48. Description of the Human Body. In Gaukroger 1998.

De Chene, D. 2001. Spirits & Clocks – Machine & Organism in Descartes. Cornell.

Gaukroger, S., editor, 1998. Descartes – The World and Other Writings. Cambridge.

Locke, J. 1690. An Essay concerning Human Understanding. A. C. Fraser, editor. 1894. Dover.

Peikoff, L. 1985. Aristotle’s “Intuitive Induction.” The New Scholasticism 59(2):185–99.

Rand, A. 1957. Atlas Shrugged. Random House.

~~~~~~~~~~~~~~~~


References

Ayers, M. 2011. Primary and Secondary Qualities in Locke’s Essay. In Primary and Secondary Qualities. L. Nolan, editor. Oxford.

Thanks, looks very interesting. Amazon link.
On that topic: my article.



#7 Stephen Boydstun

Stephen Boydstun

    $$$$$$

  • Members
  • 1,767 posts
  • Gender:Male
  • Location:Virginia
  • Interests:Metaphysics; Theory of Concepts and Predication; Philosophy of Science and Mathematics; Philosophy of Mind; Foundations of Ethics; Physics; Mathematics; Biology; Cognitive Science

Posted 29 April 2012 - 07:13 AM

—Exclusions of Non-Contradiction: Attributes

Part 2 – Leibniz I

Leibniz responded to Locke’s Essay concerning Human Understanding (EU – 1690)* with New Essays on Human Understanding (NEU – 1704).* Leibniz maintained we have some true foundational ideas that could not have been received from the senses. Locke was wrong about that. He was wrong, too, in imagining young children or anyone can think without using, and implicitly knowing, the maxims of identity and noncontradiction (NEU 76). The necessary truths of reason and pure ideas in general stand in contrast with images of sense and truths of fact. Arithmetic and geometry are not learned truly from experience or teachers, but by “attending carefully and methodically to what is already in our minds” (77). To be roused from their dark depths in mind, geometric ideas such as surface and figure require sight or touch of such things as leaves. The necessary truths of logic, arithmetic, and geometry, however, “are proved by what lies within, and cannot be established by experience as truths of fact are” (79).

It cannot be denied that the senses are inadequate to show their necessity, and that therefore the mind has a disposition (as much active as passive) to draw them from its own depths; though the senses are necessary to give the mind the opportunity and the attention for this, and to direct it towards certain necessary truths rather than others. . . . The fundamental proof of necessary truths comes from the understanding alone, and other truths come from experience or from observations of the senses. Our mind is capable of knowing truths of both sorts, but it is the source of the former; and however often one experienced instances of a universal truth, one could never know inductively that it would always hold unless one knew through reason that it was necessary. (NEU 80; also 1714, §5)

The human mind has a special affinity to the truths of reason. It is preformed such that the truths of reason are derivable with necessity from itself (NEU 80). Leibniz allows “the senses can hint at, justify and confirm these truths, but can never demonstrate their infallibly and perpetual certainty” (80; also 1702, 188). The senses are, to be sure, cohorts of all our thinking. Thought and sensation are together in us and always in us, but in us also “we always have all our pure or distinct ideas independently of the senses” (NEU 119). Leibniz is a post-Scholastic inclining towards Plato (e.g. Leibniz 1702, 189; Mercer 2001, Ch. 5).

As a young man, in the 1660’s, Leibniz allegiance had been won by Aristotle and by the modern mechanical philosophy and physics. He envisioned reducing Aristotelian concepts such as matter and form to modern primitives such as extension, figure, and impenetrability. He tried to reconcile the mechanical philosophy with theism. Not all properties of a body in all their particulars can be inferred from its fundamental properties. Their particulars come about through particular interactions with other bodies and all these together from previous ones for which eventually there must be an incorporeal creating and ruling mind, which is God (Garber 2009, 5–13; cf. later Leibniz 1695a, 126; 1710, 127; 1714, §8). To refuge the further mystical doctrines of transubstantiation and resurrection of the body, young Leibniz supplemented modern concepts of body with Aristotelian concepts of substance and substantial form (Garber 2009, 40–43, 47–52). Starting in the late 1670’s, with crystallization by the middle ’80’s, Leibniz puts substantial form as (unintelligent) mentality to work for bringing some basic animate structure into even inanimate bodies so as to endow them with enough identity to exist and to be unified individuals (ibid., 62–81; Gaukroger 2010, 121–25; Leibniz 1686a, 78–80; 1687, 82, 85–89; 1695a, 118–20; 1695b, 139–40; 1699, 172; 1703, 175, 177; 1710, 360–61, 365; 1712, 262–64; 1714, §§1–4; contrast with Descartes Meditations [M], Sixth Replies, 298–300; Garber 2001, 266–73). All material bodies are necessarily infinitely divisible, and there are infinitely many substantial forms giving unity and individual identity to every bit.

Locke had observed that there are not only logical and mathematical propositions that will be immediately affirmed upon being understood. “That ‘two bodies cannot be in the same place’ is a truth that nobody any more sticks at than at these maxims, that ‘it is impossible for the same thing to be and not to be’, that ‘white is not black’, that ‘a square is not a circle’, that ‘bitterness is not sweetness’” (EU 1.1.18). Leibniz distinguishes types within these propositions to which rational beings are allegedly compelled to assent.

The proposition that two bodies cannot be in the same place (at the same time) is not a self-evident identity, in the view of Leibniz. This proposition is, I notice, part of what is infirm in Locke’s attempt to conclude exclusivity of colors from exclusivity in primary qualities proposed as their objective source. Leibniz finds falsehood and confusion in Locke’s view that the hardness of bodies is a simple idea of sense. What we have in sense, according to Leibniz, is degree of firmness in contrast to degree of softness or fluidity.[1] Hardness is a notion related to firmness, and we can say in an approximate way, hardness occurs in nature. But we have a notion of absolute hardness, such as in the atoms of Epicurus, and this is an intellectual notion, one that Leibniz thinks impossible to be physically real (NEU 125; 1686a, 81; 1687, 89; 1695b, 164–65; 1712, 264; also, Leibniz three decades before NEU, in Garber 2009, 19, 47, 64–65; further, Garber 1995, 321–25; 2009, 81–90, 288–89).

Locke was correct in thinking that hardness, as extreme firmness, is given in sense experience. However, supposing hardness to be inherent in conjectured finest particles composing macroscopic hard matter, rather than an emergent property of such matter, turned out to be a mistake. We explain hardness not by atomic hardness, but by electromagnetic forces, fermi quantum exclusions, chemical bonds, and thermodynamic conditions.

The remainder of the propositions in Locke’s list are identical propositions or nearly so, according to Leibniz. Those are the propositions “White is not black” and “A square is not a circle” and “Bitterness is not sweetness.” They do not admit of proof. Among these, the ones pertaining to what the senses provide “merely apply the general maxim of identity to particular cases” (NEU 82). Negations of an idea by an opposed idea are applications of the principle “It is impossible for the same to be and not be.” In Leibniz’ analysis, Locke errs in thinking that the principle of contradiction has its origin in prior grasp that the same is not different, or A is not B. Rather, “It is because B contains non-A that A is prevented from being B” (82).

Leibniz and Locke are each partly right and partly wrong. Contrary to Locke, if one has grasped the proposition that A is not B, then one has also grasped at least implicitly that A is A and cannot be anything not A. Locke is right and Leibniz is wrong concerning prelinguistic, nonpropositional thought. One can grasp the particular facts that instance Round is not flat (Ball is not floor) or that instance Light is not dark without implicitly grasping those or any other generals as generals. It is not the principle of identity and lack of contradiction that makes white not black, and prelinguistic grasp of the latter does not require grasp of the former. It remains that the proposition stating the fact that white is not black is an application of and entails the proposition A is not non-A (as well as the proposition Existence is identity *). Moreover, one able to grasp the proposition White is not black will normally have the potential to grasp explicitly the axiom of identity (NEU 412).

Leibniz concedes that we grasp particular truths before we grasp general ones. That is the order of learning. He maintains that in the order of nature simpler ideas, such as identity, come first, and that “the reasons for particular truths rest wholly on the more general ones of which they are mere instances” (NEU 83; see also 127, on priority of abstract space to concrete space). That is incorrect by Rand and me and by Locke. Leibniz says further that the simple axiom of identity “is in us implicitly, before all awareness” (83). All consciousness is identification, as Rand said, but that does not show the general principle of identity is implicitly in our nonlinguistic awareness any more than principles of grammar are implicitly in such awareness, which is to say not at all for either.

Leibniz’ picture has it that as the prelinguistic infant is grasping a particular present difference, the particular difference is being made possible by the general difference and by the principle of identity, and in the grasping of the particular present difference, the general difference and the principle of identity are implicitly in that grasp. All of that is incorrect.

“The ideas of being, possible, and same . . . enter into all our thoughts and reasoning, and I regard them as essential to our minds” (NEU 102). For fully symbolic minds, those with language, that is true enough, and it is in the thinking of such minds about the world that we must consider the logical status of “A leaf cannot be all red and all green at the same time.”

Leibniz says that propositions such as The bitter is not sweet or Red is not green apply the axiom of identity to sensible truths. “But as for the proposition The square is not a circle, . . . in thinking it one applies the principle of contradiction to materials which the understanding itself provides” (NEU 83). Yes and no. That building blocks stack and balls roll is learned by the infant prior to language. Incompatible shapes are available to see and handle. Leibniz will allow that. He allows also that a child having language can know what are a square and its diagonal without yet knowing a square’s diagonal is incommensurable with its side (102; cf. Descartes’ M, Reply to Objection I, 229, and IV, 273–75; Arnauld’s Objection IV to M, 268; TF 167). To grasp the perfectly exact figures and relations that enter geometry—such as the incommensurability of a square’s diagonal with its side or the equality of the sum of angles in a triangle to two right angles—requires a high level of conceptual understanding. Leibniz errs, however, in thinking perfectly exact figures and relations are only from abstract thought. They are partly taken from the world, they may obtain perfectly in physical space, and without mind.

Recall that Locke held we have some simple sensory apprehensions such as of whiteness, warmth, softness, or sweetness. These are among what Locke called simple ideas. Leibniz remarks:

It can be maintained, I believe, that these sensible ideas appear simple because they are confused and thus do not provide the mind with any way of making discriminations within what they contain; just like distant things which appear rounded because one cannot discern their angles, even though one is receiving some confused impression from them. It is obvious that green, for instance, comes from a mixture of blue and yellow; which makes it credible that the idea of green is composed of the ideas of those two colours, although the idea of green appears to us as simple as that of blue, or as that of warmth. So these ideas of blue and of warmth should also be regarded as simple only in appearance. I freely admit that we treat them as simple ideas, because we are at any rate not aware of any divisions within them; but we should undertake the analysis of them by means of further experiments, and by means of reason in so far as they can be made more capable of being treated by the intellect. (NEU 120; also 1682–84, 285–87; 1684, 24, 27; 1702, 187–88; 1710, 109–10, 158, 303, 339; 1714, §§13, 17)


Leibniz is correct to dispute the idea that simplicity in appearance guarantees irreducibility to other simplicities in appearance. All perceived simplicities should be held as of appearances whose simplicities are prima facie only. Malebranche had remarked, further, in Elucidation Sixteen of The Search after Truth (ST –1678), with high praise for Newton’s optical experiments, that green can be shown to be sometimes simple, sometimes not, and that for the latter case it is not surprising that the brain receiving at once two frequencies of nervous vibrations should combine them into an intermediate frequency. Malebranche had remarked also that though white light appears simple, it is never so (ST E16.19; also Leibniz 1682–84, 286).

Notice that Leibniz trusts also the resolution of sense by empirical investigation for the attribute shape. Reducing the distance between object and eye will reveal distinct angles in the object that were obscure in the distant view. In the present study, we should put Rand’s application of the law of identity to attributes under the intellectual microscope not only by her case of leaf color, but by the case of leaf shape. A leaf cannot be three-lobed and five-lobed at the same time.

We shall need to assess the logical status of such truths from Malebranche: “A piece of wax is incapable of simultaneously receiving several perfect and distinct figures at the same time” (ST 3[1].3[1]). “A piece of wax cannot be both square and round at the same time, but only half round and half square, and as the more different figures it has, the less perfect and distinct it will be” (3[1].2[1]; also 6[1].5).

It is Leibniz’ view that colors, Locke’s simple “ideas” of color, consist in a je ne sais quoi. Such is not the case for the simple quality extension (NEU 127). Extension and figure are among the simple ideas that Locke thinks are gotten from more than one sense. Leibniz thinks they “come rather from the common sense, that is, from the mind itself; for they are ideas of the pure understanding (though ones which relate to the external world and which the senses make us perceive), and so they admit of definitions and of demonstrations” (128; also 1682–84, 286).[2]

Locke had thought of all simple ideas as being adequate ideas.[3] They are truths we grasp in their entirety. Simple ideas of secondary qualities such as color and odor of a mint leaf are nothing but the effects in us of definite, if often unknown, external causes. Primary qualities such as “solidity and extension, and the termination of it, figure, with motion and rest, whereof we have [simple] ideas would be really in the world as they are, whether there were any sensible being to perceive them or no” (EU 2.31.2). A mint leaf itself has the shape we see it has. Minute particles of various sorts, endowed with primary qualities, travel from the leaf to our receptors, causing mint color and mint odor to come into our awareness.

Since Leibniz regards color as consisting in a je ne sais quoi, we should hardly expect him to accede to Locke’s position that simple ideas of secondary qualities are adequate ideas (NEU 267). Malebranche had it that God causes us to have the sensation of greenness when perceiving a mint leaf, but that in perceiving the leaf our minds are in the Architect’s mind understanding univocally the idea leaf there, though only to a partial extent of the divine understanding. I should point out that for Malebranche, God’s understanding is really distinct and autonomous vis-à-vis God’s will. The nature of the leaf is an objective matter in God’s mind and in ours because of his.

Like the rationalists who preceded him, Locke appeals to the goodness of God to assure that same and different secondary qualities in our sensations reliably shadow sameness and difference in their physical causes, which are primary qualities. But Locke is letting go of God in comparison to Malebranche, for he has us getting to perception of the leaf only through our experience of secondary and primary qualities, not by participation in the mind of God (further, Yolton 1984, 90–98). Plato would recognize Malebranche as belonging to the school that insists “true being is certain nonbodily forms that can be thought about,” whereas Locke (like Arnauld) belongs to the school that “drags everything down to earth from the heavenly region of the invisible” (Sophist 246a–b).

Locke would have it that a mint leaf’s oblong shape with fine serration on its perimeter is primary physical quality perfectly, completely known. That the leaf is not lobed, but oblong (somewhat elongated with approximately parallel sides), is likewise perfectly known. Leaf shape is a primary quality, and among our simple, adequate ideas. Locke thought of the exact figures of geometry, though they exist in nature, as not simple primary qualities. They are our voluntary assemblies, “without reference to any real archetypes, or standing patterns existing anywhere” (EU 2.31.3; cf. Descartes M V; Carriero 2009, 299–300). Such assemblies Locke calls ideas of modes.

In Descartes modes were traits inhering in substance, where variation in the traits leave the substance constant in kind (Principles [P] I 56; M Reply to Objection I [231]; see also Normore 2010). Size, shape, position, divisibility of its parts, and motion are distinct modes of corporeal substance (P 48, 61). These modes may be thought of as residing simply in extension, which is the principal attribute of corporeal substance, the attribute constituting its essence (53, 65; see further, Garber 1992, 63–70). Square, triangle, and sphere are modes of extension. Locke is following somewhat the rationalist lead of Descartes, Spinoza (E 1D5, 2P13, 17; Bennett 1984, 92–94), Malebranche (ST 1.1[1], 1.16[4], 3[2].1[1], 4.2[4], E10.3), and Arnauld (TF 53, 56, 66, 71–73, 99–100, 117–18, 124–26, 132–33, 134, 172) in characterizing the status of geometric objects. Though he takes these objects to be modes of extension, he refuses the rationalist perspective in which the specific natures of geometric objects are simply objective givens to the mind, natures accessible through demonstration or requiring none (M V). For Locke, those natures are constructs of the mind, notwithstanding their fixity.

Leibniz is partly in step with those earlier rationalists in taking at least some attributes of objects to be modes of extension. “Extended is what has size and situation [situs]. Size is the mode by which all the parts of a thing, or all the entities by means of which the thing can be understood, are determined. Situation is the mode of determining with which qualities a thing can be perceived. / . . . / The situation of parts among themselves is called figure” (Leibniz 1682–84, 277–78).

Unlike Locke and like the other rationalists, Leibniz takes Euclid’s figures and their specific natures to be objective givens in the mind (Leibniz 1675, 1–2). We have seen that in the view of Leibniz extension and its geometric modes, though they are related to physical objects such as leaves, come forth in the mind and are therefore definite objects and relations suited for entering demonstrations. No material objects have shapes so precise and determinate as geometric objects (Leibniz 1687, 86–87; 1689 [?], 34; 1704 or 1705, 183). The boundaries of figures we draw on paper in a geometric proof are not exactly the boundaries and figure in mind for the proof (further, NEU 360; TF 99–100; Norman 2006, Ch. 6; Azzouni 2004*), and while the former, as with all matter, are compositions, the latter are not (Leibniz 1695c, 146–47).

“Extension is nothing but simultaneous continuity” (Leibniz 1699, 171). For extension of a leaf, “something must always be assumed which is either continued or diffused” (ibid.), such as its color and the resistance of its matter (1695a, 118, 130; 1704 or 1705, 183; 1712, 261–62). “The notion of extension is not a primitive one, but is resolvable. For an extended being implies the idea of a continuous whole in which there is a plurality of things existing simultaneously” (1692, 390).

Unlike the other rationalists, and unlike Locke, Leibniz thinks the extension of a leaf is not only that of which shape is a mode. The extension of a leaf is itself a mode of something deeper, specifically a dull passive, resistive force in leaf matter, as in all matter (Leibniz 1683–86, 365; 1686b, §12; 1695a, 118, 130; further, Garber 2009, 155–64, 296–99; also Adams 2010, 55–69, and Garber 2010, 74–78; for comparison with Newton, see Garber 2009, 172–79, and Gaukroger 2010, 115–20).

That a leaf cannot be three-lobed and five-lobed at the same time is a truth about all shapes in Euclidean space. Leibniz would say such shapes are put upon the leaf by our imaginations (our minds) as ideal limits to which maple leaves, for example, tend in an infinite sequence of them. That a bounded surface in Euclidean space cannot as a whole figure be three-lobed and five-lobed is a necessary geometric truth. Because the limit is true of some classes of real, physical leaves, the necessity that a leaf cannot be both three-lobed and five-lobed at the same time is geometrical necessity.

When it comes to the color of a leaf, it too is something brought to the leaf from our response to the leaf. However, by Leibniz, color is a perceptual response lacking definite intelligible character lain upon it, unlike the case of shape; although as Leibniz could see, we shall be making color more intelligible in our scientific investigations of it.

It seems to me there are implications for the character of necessity in “A leaf cannot be all red and all green at the same time” within Leibniz’ conception of matter and mind. Matter is infinitely divisible in his view. In that it is like geometric extension. Every figure in geometry, no matter how small, can be divided into yet smaller surfaces. Color in leaf surface will always be an aggregate effect of sub-portions for every portion of leaf surface. In respect of a given quality, aggregate resultants are uniquely determined (Leibniz 1682–84, 287, 289). Leibniz can say such phenomena not manifestly intelligible are “confused,” but at a single time, they are single aggregate extensions all the same, just as extension and its modes in leaf, approximating geometrical modes, are single aggregate expressions of underlying passive force. No matter how finely one divides the surface of a leaf, it would seem there is a metaphysical necessity, in Leibniz’ philosophy, that if color still obtains for certain fineness of portion the color of that portion is one color and not another. A surface of white paper too, where the resultant white arises from the full spectrum of colors, is a single resultant, even if not irreducibly simple (cf. Leibniz 1712, 263). In sum the exclusion of one color by another in leaf is metaphysical, and the exclusion of one shape by another for leaf is both metaphysical and geometric.

(To be continued.)

Notes

1. Leibniz would not deny that the technical niches we have since awarded hardness as scratch hardness and as dent hardness are evident in sense. His comments are on the general common concept hardness in the late seventeenth to early eighteenth century.

2. We should notice that for Rand and most moderns the elementary form of the idea existence, one of Locke’s simple ideas (EU 2.7.1, 7), is gotten from perception of the world (including one’s body). Leibniz protests the view that the senses could “convince us of the existence of sensible things without help from reason” (NEU 129). I counter that our road to existence is by existent, whose primary and salient form is entity, more particularly object. Schematic and iconic representation of objects, whose instances have been encountered in sensory experience, are the forerunners of indexical representations and fully symbolic, linguistic representations of objects (kind or particular) and object in general (cf. Carey 2009, Ch. 3). The human animal reaches symbolic cognition of existents in perception knowing full well their existence. No convincing is needed that one’s waking world exists stock full of existing things, perceptually encountered as existing, until the mind has become susceptible to degradation by mystical and skeptical conceits. The preventative and corrective will be reason, in Rand’s sense of it.

3. The notion of adequate signification is found in fourteenth century realists about universals. There, adequacy amounted to complete identity. For Locke and Leibniz, adequate idea was a conception received as it had been rendered by Descartes (Discourse §II [119–20]; M Replies to Objections I [231–32], II [237], IV [270–73]), Arnauld (Objection IV to M [266–69]; TF 167), and Spinoza. According to Spinoza, writing in Ethics (E – 1677), the human mind is first constituted by having the idea of a singular thing that actually exists. Such knowledge is partial, which is called inadequate, for its occasion of finite subject and object is only a part of God’s infinite knowledge (E 2P11c, P24–31). “So long as the human Mind perceives things from the common order of nature, it does not have an adequate, but only a confused and mutilated knowledge of itself, of its own Body, and of external bodies” (2P29c). Commonalities among diversities that are equally in the part and the whole of their diverse members are conceived adequately (2P38). That is, A in an external body affecting A in our body, where A is equally in the part and whole of both bodies will also be adequate in the mind (2P39). Man is such an A, considered apart from particular colors and sizes (etc.) of men (2P40s1).

Adequate knowledge is clear, distinct, whole, and perfectly true. An adequate idea is one that in itself, considered apart from its object, has the “denominations of a true idea” (E 2D4). An adequate idea in one’s mind is an adequate, perfect idea in God, therefore a true idea perfectly fitting its object (2P34). Adequate and true ideas are their own standard; by having them, we know they are adequate and true (2P43).

References

Adams, R. M. 2010. Continuity and Development of Leibniz’ Metaphysics of Body. The Leibniz Review 20:51–71.

Ariew, R., and D. Garber, editors and translators, 1989. G. W. Leibniz – Philosophical Essays. Hackett.

Arnauld, A. 1641. Objection IV to Descartes’ Meditations. In Wilson 1976.
——. 1683. On True and False Ideas. S. Gaukroger, translator. 1990. Manchester.

Bennett, J. F. 1984. A Study of Spinoza’s Ethics. Hackett.

Carey, S. 2009. The Origin of Concepts. Oxford.

Carriero, J. 2009. Between Two Worlds – A Reading of Descartes’s Meditations. Princeton.

Azzouni, J. 2004. Proof and Ontology in Euclidean Mathematics. In New Trends in the History and Philosophy of Mathematics. T. H. Kjeldsen, S. A. Pedersen, and L. M. Sonna-Hansen. Southern Denmark.

Descartes, R. 1637. Discourse on the Method. In Wilson 1976 (W).
——. 1641. Meditations on First Philosophy. (W)
——. 1646. Principles of Philosophy. (W)

Garber, D. 1992. Descartes’ Metaphysical Physics. Chicago.
——. 1995. Leibniz: Physics and Philosophy. In The Cambridge Companion to Leibniz. N. Jolley, editor. Cambridge.
——. 2001. Descartes Embodied – Reading Cartesian Philosophy through Cartesian Science. Cambridge.
——. 2009. Leibniz: Body, Substance, Monad. Oxford.
——. 2010. Reply to Robert Sleigh and Robert Adams. The Leibniz Review 20:73–79.

Gaukroger, S. 2010. The Collapse of Mechanism and the Rise of Sensibility. Oxford.

Leibniz, G. W. 1775. Letter to Foucher. In Ariew and Garber 1989 (AG).
——. 1682–84. On the Elements of Natural Science. In Loemker 1969 (L).
——. 1683–86. On the Method of Distinguishing Real from Imaginary Phenomena. (L)
——. 1684. Meditations on Knowledge, Truth, and Ideas. (AG)
——. 1686a. Letter to Arnauld, 28 November/8 December. (AG)
——. 1686b. Discourse on Metaphysics. (AG)
——. 1687. Letter to Arnauld, 30 April. (AG)
——. 1689(?). Primary Truths. (AG)
——. 1692. Critical Thoughts on the General Part of the Principles of Descartes. (L)
——. 1695a. A Specimen of Dynamics. (AG)
——. 1695b. A New System of Nature. (AG)
——. 1695c. Note on Foucher’s Objection. (AG)
——. 1699. Letter to de Volder, 24 March/3 April. (AG)
——. 1702. Letter to Queen Sophie Charlotte of Prussia. (AG)
——. 1703 Letter to de Volder, 20 June. (AG)
——. 1704. New Essays on Human Understanding. P. Remnant and J. Bennett, translators. Cambridge.
——. 1704 or 1705. Letter to de Volder. (AG)
——. 1710. Theodicy. E. M. Huggard, translator. 1985. Open Court.
——. 1712. Conversation of Philarète and Ariste. (AG)
——. 1714. Principles of Nature and Grace, Based on Reason. (AG)

Locke, J. 1690. An Essay concerning Human Understanding. A. C. Fraser, editor. 1894. Dover.

Loemker, L. E., editor and translator, 1969 [1954]. Gottfried Wilhelm Leibniz – Philosophical Papers and Letters. 2nd ed. Kluwer.

Malebranche, N. 1678. The Search after Truth. T. M. Lennon and P. J. Olscamp, translators. Cambridge.

Mercer, C. 2001. Leibniz’s Metaphysics – Its Origins and Development. Cambridge.

Norman, J. 2006. After Euclid: Visual Reasoning and the Epistemology of Diagrams. CSLI.

Normore, C. G. 2010. Accidents and Modes. In The Cambridge History of Medieval Philosophy. Volume II. R. Pasnau, editor. Cambridge.

Plato c. 428–348 B.C. Sophist. In Plato – Complete Works. J. M. Cooper, editor. 1997. Hackett.

Spinoza, B. 1677. Ethics. In The Collected Works of Spinoza. E. Curley, translator. 1985. Princeton.

Wilson, M. D., editor, 1976. The Essential Descartes. Meridian.

Yolton, J. W. 1984. Perceptual Acquaintance from Descartes to Reid. Minnesota.

#8 Stephen Boydstun

Stephen Boydstun

    $$$$$$

  • Members
  • 1,767 posts
  • Gender:Male
  • Location:Virginia
  • Interests:Metaphysics; Theory of Concepts and Predication; Philosophy of Science and Mathematics; Philosophy of Mind; Foundations of Ethics; Physics; Mathematics; Biology; Cognitive Science

Posted 29 April 2013 - 12:46 PM

Randian Axioms and Postulates in Metaphysics

 

Basics / Identity with Measurement

 

I. Exclusions of Non-Contradiction: Entities
Part 1 – Identity as to Kind
Part 2 – Identity as to Particularity
 

II. Exclusions of Non-Contradiction: Actions

 

III. Exclusions of Non-Contradiction: Attributes
Part 1 – Locke
Part 2 – Leibniz I
Part 3 –
Part 4 –



#9 Stephen Boydstun

Stephen Boydstun

    $$$$$$

  • Members
  • 1,767 posts
  • Gender:Male
  • Location:Virginia
  • Interests:Metaphysics; Theory of Concepts and Predication; Philosophy of Science and Mathematics; Philosophy of Mind; Foundations of Ethics; Physics; Mathematics; Biology; Cognitive Science

Posted 25 January 2014 - 12:13 PM

My own philosophy has been so rapidly coming together lately that I’m writing my first book on it. I’m not yet sharing the book’s title nor the name of the new philosophy. Hopefully it can be completed within two years.

 

As it has worked out, for my own philosophy, there are axioms—axiomatic propositions and concepts—additional to Rand’s set dissected in this thread. Each axiom, old or new, requires argument as to its axiomatic standing, as ever. That done, it is possible that propositions not deducible from the smaller set of axioms are deducible from the larger set, hence are to be seen as not only universal postulates, but as with the necessity had by axioms.

 

Writing of the book is taking ever more of my writing time. The issues of this thread, as well as those in the thread on objective analyticity and those in the thread on truth in geometry, will be completed within my book exclusively.

 

Those three threads I’ll stop where they are now. The axioms of the new philosophy reverberate through theory of value, most importantly through ethical theory. Metaphysics, epistemology, and theory of ethical value are the scope of the book. Esthetics will not get substantial attention therein, and I’ll try to continue making installments to the thread “Beauty, Goodness, Life” in the coming months, though probably with a reduced frequency.



#10 BaalChatzaf

BaalChatzaf

    $$$$$$

  • Members
  • 11,360 posts
  • Gender:Male
  • Location:Currently residing in New Jersey, the Bad-a-Bing State.
  • Interests:mathematics, physics, alternative energy sources.

    I am also involved in preparing recorded books for blind and dyslexic folks.

Posted 25 January 2014 - 02:34 PM

My own philosophy has been so rapidly coming together lately that I’m writing my first book on it. I’m not yet sharing the book’s title nor the name of the new philosophy. Hopefully it can be completed within two years.

 

As it has worked out, for my own philosophy, there are axioms—axiomatic propositions and concepts—additional to Rand’s set dissected in this thread. Each axiom, old or new, requires argument as to its axiomatic standing, as ever. That done, it is possible that propositions not deducible from the smaller set of axioms are deducible from the larger set, hence are to be seen as not only universal postulates, but as axioms.

 

Writing of the book is taking ever more of my writing time. The issues of this thread, as well as those in the thread on objective analyticity and those in the thread on truth in geometry, will be completed within my book exclusively.

 

Those three threads I’ll stop where they are now. The axioms of the new philosophy reverberate through theory of value, most importantly through ethical theory. Metaphysics, epistemology, and theory of ethical value are the scope of the book. Esthetics will not get substantial attention therein, and I’ll try to continue making installments to the thread “Beauty, Goodness, Life” in the coming months, though probably with a reduced frequency.

Good health to you!!! I will be one of your first customers.   


אויב מיין באָבע האט בייצים זי וואָלט זיין מיין זיידע

#11 Michael Stuart Kelly

Michael Stuart Kelly

    $$$$$$

  • Root Admin
  • 20,197 posts
  • Gender:Male

Posted 25 January 2014 - 05:58 PM

Go for it. Stephen.

 

And may you contribute long and hard to mankind's wisdom.

 

I'm one of your cheerleaders.

 

Michael


Know thyself...


#12 Roger Bissell

Roger Bissell

    $$$$$$

  • VIP
  • 2,158 posts
  • Gender:Male
  • Location:Antioch, Tennessee
  • Interests:philosophy, psychology, genealogy, fiction

Posted 25 January 2014 - 06:05 PM

Congratulations on your exciting project, Stephen! I wish for you to have smooth sailing throughout the process and a few unexpected but delightful discoveries along the way. :-)


Objectivism, properly used, is a tool for living, not a weapon with which to bash those one disagrees with.




0 user(s) are reading this topic

0 members, 0 guests, 0 anonymous users