Jump to content
Objectivism Online Forum

A brain teaser on the notion of "absolute".

Rate this topic


Recommended Posts

Hang on Nelly, what makes you think that the arbitrary is the non-essential?

Nothing I posted implies that I hold that Objectivism claims that what is arbitary is identical with what is non-essential, and nothing I posted implies that I claim that what is arbitrary is identical with what is non-essential.

Can you connect the dots for me: I have no idea how you got this connection between the arbitrary and an essential characteristic.

From your post:

We're speaking of Objectivist epistemology, and as I said, the closest analog is the concept. Concepts are defined in terms of essential similarities (and omitting nonessential differences), so the members of the "set" would have some defining characteristic (such as color, size, use, whatever...). No concept identifies both actual dogs and the concept "dog" (they have no similarities). It is irrelevant that set theory allows arbitrary collections to be called "a set" -- as I pointed out, the concept "set" refers to reality only in a limited set of cases, and it excludes the type of set that's required for RP to be a "paradox".

I take it that you are arguing that the formation of sets in set theory is arbitrary while the Objectivist method of defining concepts is not, since the Objectivist method depends on using essential similiarities. If that is not your argument, and you do not intend to assert a connection between essence and arbitrariness, then I welcome your explaining what you had in mind in the above quote.

An essential characteristic is one which is not omissable in identifying the extension of a concept. Hmm... I'm being redundant.

Yes, what is omissible and non-omissible? Perhaps the extension of a concept can be identified in two different ways, so that the characteristics chosen are non-omissible for each identificaion onto itself. By what method do you determine that one identification is correct and uses "the" or even "an" essential characteristic but the other identification does not? (Rand offers only scant help, and not as to identification of extensions but as to defintition of concpets, as I recall, in ITOE.) Also, I would not presume to claim that Objectivistism holds that the extension of a concept is all there is to a concept. But you've shifted from defintion of concepts to identification of extensions. I should think there is much more at stake here than identification of extensions rather than definitions of concepts (though they are related), since mere extensionality, such as that of set theory, does quite well without a criteria of essentialness.

Sure, I'll reciprocate when you do.

I should reciprocate by not suggesting that your arguments are "Platonist" and such? I had already accomplished that.

I've not only given you an adequate definition for our purposes

You've given no defintion of 'essential' that can be used for mathematics in the context of difficulties such as Russell's paradox.

What I'd like is your proof that the Objectivist concept "arbitrary" is incomprehensible without a definition of "essential", and that the concept "essential" is incomprehensible without a definition of "arbitrary".

First, nothing I posted implies a claim that 'essential' can only be defined in terms of 'arbitrary'. Second, if you can define 'arbitrary' without eventually going up the chain of definitions to 'essential' (or synonymous), then I'll reconsider whether the notion of 'arbitrary' depends on that of 'essential'. I do recognize another, but related, sense of 'arbitrary' used by Objectivism, but that does not detract from the kinds of connections between these that you suggested in your own post. If you do insist that 'arbitrary, as you used the term, does not depend on 'essential', then, as I said, I welcome hearing how your argument in that quote is really supposed to work. In any case, Objectivism does depend on the notion of 'essential'. A lucid and cogent explication of the Objectivist sense of 'essential' (which, please now, is not just the everyday sense, but is philosophical) is needed.

If it helps you to understand Objectivist epistemology, "concept" is closer to being axiomatic than either "essential" or "arbitrary".

I know that. So what? I didn't claim that a definition of 'essential' should not use 'concept'. Edited by LauricAcid
Link to comment
Share on other sites

Does OE accept proofs of the negative type?

For instance:

Zero is defined as the number (marked "0") which satisfies: (Ax: (0*x = 0)) AND (Ax: (0+x = x)).

Claim: In any given set with well-defined and useful (+,*) operators, If zero exists - it`s unique.

("useful" means commutative and associative)

Proof: Assume that two different numbers "0" != "O" qualify as zeroes.

We know that 0*O=0, since O is a zero,

Also we know that 0*O=0 since 0 is a zero and * is commutative.

Hence, 0*O=0=O -> 0=O -> *BOOM*

How about this one:

Claim: There`s an infinite number of integers.

Proof: Assume that the claim is false, that there`s only a finite number of integers.

Define N as the largest integer.

Since the method of generating integers is unbounded, there exists an integer larger than N -> *BOOM*

The problem here is: In what sense do these integers "exist"?

Surely these don`t "exist" in the same way as apples or socialists...

Do thoughts exist?

Link to comment
Share on other sites

I missed this earlier. May as well point to a few of the more significant errors and confusions...

I take it that you are arguing that the formation of sets in set theory is arbitrary while the Objectivist method of defining concepts is not, since the Objectivist method depends on using essential similiarities.
I see where you're going. I thought you were speaking of the epistemological concept "arbitrary". My unfortunate use of "arbitrary" in reference to sets was intended to point to random or (intentionally) undefined sets, i.e. "just any old garbage".
Yes, what is omissible and non-omissible?
For which concept? The point which you stalwartly refuse to get is that concepts are not Platonic abstractions: they are fundamentally cognitive. They are a relation between a particular mental existent and the external world. This is an empirical question of experimental psychology.
By what method do you determine that one identification is correct and uses "the" or even "an" essential characteristic but the other identification does not?
If you want a very broad account, we're looking at a discrimination experiment. You identify test properties and vary them systematically, to see how minds make identifications. Beyond that, we're really in the domain of how to perform a valid experiment in psychology, and I generally oppose shattershot testing. If you want to know whether percentage of hair the body being covered by hair is an essential characteristic that distinguishes man from chimp, we can do the test. You don't get scientific results a priori or by cooking up one grand experiment -- you get results piece by piece.

So for example, due to a tragic accident at a nuclear power plant, massive radiation is unleashed on the neighbors, and they suffer genetic mutations that create an extensible air sack, allowing their descendants to croak. Are such beings then identified as "men"? I would guess they would be, that our concept of "man" has evolved past crude morphology -- we no longer think that a man who has lost an eye or an arm has ceased being a man. If these croaking beings are actually identified as "frogs" and not "men", then you'd know that "not croaking" was an essential characteristic of the concept "man". There's only one way to find out (well, maybe there is no way to find out, at least at present).

But you've shifted from defintion of concepts to identification of extensions. I should think there is much more at stake here than identification of extensions rather than definitions of concepts (though they are related), since mere extensionality, such as that of set theory, does quite well without a criteria of essentialness.
Part of the reason for the shift, which is just a shift in emphasis, is that we don't have any efficient means of directly representing the correct definition of certain concepts, especially ones that are ostensively defined. Definitions are mental entities, and are harder to display, unlike gross anatomical entities like brains and spleens. The other reason for paying closer attention to the extension is that it is the basis for the concept -- things exist, and we then identify them. Defining a concept without there being any referents is an invalid misapplication of concept formation.
Link to comment
Share on other sites

May as well point to a few of the more significant errors and confusions...

Yes, you are confused:

My unfortunate use of "arbitrary" in reference to sets was intended to point to random or (intentionally) undefined sets, i.e. "just any old garbage".

What you call an "unfortunate" use by yourself is a one that is right in line with Objectivism: That neither concepts nor definitions are to be formed arbitrarily. So, still, the question is what distinguishes non-arbitrary, correct concept formation and definition from arbitrary, incorrect concept formation and definition? Aside from having reference, Rand declares that correct concepts and definitions are had through identification of essential properties, while essential properties, per a particular concept, are declared to be those properties upon which the greater number of other properties depend (or a word similar to 'depend', as I don't recall the exact word she uses). Thus, the question cannot be deferred: What does it mean for properties to depend on one another and how is it determined that some properties are more depended upon than other properties?

The point which you stalwartly refuse to get is that concepts are not Platonic abstractions

This is the second time you've pulled this trick. And this is the second time I've asked you not to do this. I haven't posted anything that even suggests Platonism, so you're setting up a strawman with this Platonism business. Second request: Please don't.

[concepts] are fundamentally cognitive. They are a relation between a particular mental existent and the external world. This is an empirical question of experimental psychology. If you want a very broad account [of determing essential properties], we're looking at a discrimination experiment. You identify test properties and vary them systematically, to see how minds make identifications. Beyond that, we're really in the domain of how to perform a valid experiment in psychology, and I generally oppose shattershot testing. If you want to know whether percentage of hair the body being covered by hair is an essential characteristic that distinguishes man from chimp, we can do the test. You don't get scientific results a priori or by cooking up one grand experiment -- you get results piece by piece.

Then that's your view of essentiality. But I'd be surprised to read any defense of it as consistent with Objectivistism. And, with regard to mathematical concepts, particularly with regard to problems such as Russell's paradox, which is the original context of my remarks, the view you espouse, that essential properties are discovered by experimental psychology is pretty much irrelevant.

we don't have any efficient means of directly representing the correct definition of certain concepts, especially ones that are ostensively defined.

Mathematical systems and theories, such as those related to Russell's paradox, don't use ostensive definitions. It seems to me that eventually, at some meta-level, mathematical logic, does resort to ostensive definition or explanation, such as when all formalization has been performed yet we are still left to ask about things like how we regard the most fundamental judgments as to the identity of symbols that are thought to be only mentioned by, not identical with, the actual marks formed with ink on paper. However, this burr does not seem to bear directly upon the kinds of things we've been talking about regarding concepts, sets, and Russell's paradox.

More importantly, it's reasonable for a philosophy to rely upon a certain number of ostensive definitions in the earlier stages of the development, but it would not be reasonable for a philosophy, especially one that declares itself to offer a precise and hierarchical system of principles, to resort to ostensiveness as just an escape hatch when up against crucial questions. And I don't take Objectivism to be resorting to ostensiveness in this way. I expect that if Objectivism asserts that there are essential properties and non-essential properties based on some system of comparison of properties, then Objectivism can give a precise, or even just lucid, formulation of this principle, without relying on arm waving justification by ostensiveness or mere example.

Anyway, it's fine to talk about extensions, especially in context of set theory, which, of course, rests upon the axiom of extensionality. But your remarks about extensionality do not answer the question about the Objectivist notion of essentiality and non-arbitrariness. If one keeps insisting that correct concepts cannot be arbitrary, then a precise criterion of non-arbitrariness needs to be provided. Especially, if a philosophy, such as Objectivism steps into mathematics to declare certain mathematical formulations as incorrect on account of being arbitrary, then said philosophy needs to give precise criteria of non-arbitrariness that can be formulated so that they are substantive for mathematical theories and use.

Link to comment
Share on other sites

YGoldenberg, aside from Objectivism, in your example about zero, if you declare that 0 is the y, such that for all x, y*x = y, then there is no need of a uniqueness proof, since your declaration, by using 'the', is an axiom that declares uniqueness. On the other hand, if you simply declare that 0 is such that, for all x, 0*x = 0, then indeed a uniqueness proof is required to show that 0 is the only y such that, for all x, y*x = y.

This is to say that you need to be very clear about what your axioms are and what your definitions are. Generally, the introduction of any constant symbol, such as '0', carries with it a commitment to that constant naming a unique object in whatever interpretation of the theory you give. Whether that unique object also is unique in being, say, an identity element for an operation, such as addition, then requires proof such as you gave, unless the uniqueness is given by an axiom, in which case the proof is simply to restate the axiom.

As to integers and infinity, I cannot meet your question properly, since I don't know what theory you have in mind for terms such as 'generate', 'unbounded', and 'infinite'. However, while the question of mathematical existence is not an easy one to answer, at least mathematics does provide a reduction, which if not a satisfactory philosophical explanation, does suggest a lucid starting point for one. Most basically, "existence" is asserted by a sentence that begins with an existential quantifier whose variable is that of a free variable in the formula following the quantifier. Granted, that is merely syntatical. But the semantical upshot is given under meta-theoretic evaluation by a structure ("interpretation") for the language. Very broadly put, perhaps whether one is at heart a formalist or a Platonist (not to exclude other possible views too) might be characterized by whether one is inclined to call off the chicken vs. egg escalation of meta at a syntactical stage or a semantical stage, respectively.

Edited by LauricAcid
Link to comment
Share on other sites

  • 1 month later...
The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself. However, suppose that we can form a class of all classes (or sets) that, like the null class, are not included in themselves. The paradox arises from asking the question of whether this class is in itself. It is if and only if it is not.

Have fun.

Ah, that one. That's a fun thing.

Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Reply to this topic...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
  • Recently Browsing   0 members

    • No registered users viewing this page.
×
×
  • Create New...