But what does .9999999999… MEAN?

[Grames]

See Grames , I don't consider this a cosmological (special science)distinction. It's an epistemic one based on the axioms. To be clear I was not referring to fundamental constituent of matter. My response is directed at the discussion that began with Alepho on the bounded universe. What I consider fundamental is boundedness. We don't run a test to see what we mean by "entity" or universe.

Using an epistemic principle, even an axiomatic one, to lay down the law on what is possible in metaphysics gets the hierarchy backwards.

Consider also that "the universe" is not an existent but an abstraction, an epistemological construct defined as the sum of all that exists. The universe is only an entity in the extended sense of a collective noun. 'Entity' is an axiomatic concept, 'universe' is not. The universe is different from other entities.

The universe is eternal in the sense of timeless, that it does not exist within time rather time exists within the universe. It is also true that the universe does not exist within space, space is a property of the universe. What is the word that is the spatial counterpart to 'eternal', connoting 'spaceless'? That is the concept that we need.

[Plasmatic]

Using an epistemic principle, even an axiomatic one, to lay down the law on what is possible in metaphysics gets the hierarchy backwards.

Consider also that "the universe" is not an existent but an abstraction, an epistemological construct defined as the sum of all that exists. The universe is only an entity in the extended sense of a collective noun. 'Entity' is an axiomatic concept, 'universe' is not. The universe is different from other entities.

The universe is eternal in the sense of timeless, that it does not exist within time rather time exists within the universe. It is also true that the universe does not exist within space, space is a property of the universe. What is the word that is the spatial counterpart to 'eternal', connoting 'spaceless'? That is the concept that we need.

I'm only laying down the "law" of Identity. I concede though that it would have been better to just say it is a philosophical point instead of simply "epistemic". Regardless the axioms, and I contend, my point , can be validated by simply opening ones eyes. A multiplicity of bounded particulars as far as the eye can see. I see my point on universe as the same as rejecting the possibility of discovering an existent without identity.

I use the term universe as Rand does in ITOE, a synonym of existence.

Concepts are mental existents, and their referents certainly exist as well.

Aren't eternal and space incummensurate characteristics?

Can an axiomatic concept have a genus?

Edited August 1, 2010 by Plasmatic

[Marc K.]

No, you're evading! And if you say you're not, then you're evading that you're an evader!

"The lady doth protest too much, methinks."

Sounds like I hit a nerve and of course what is the tactic? Stomping of the feet, misdirection, avoid answering the question, anything except address the argument before you, ... more evasion. You are practiced in these techniques, you've used them before, but they can't change the fact that you haven't addressed the issue.

If you don't accept the law of identity or its axiomatic status, then have the integrity to say it straight out instead of wasting all of our time. It wouldn't surprise me if you reject the law of identity since you have a penchant for denying the undeniable -- you don't even accept that we possess volition.

If you do accept the law of identity then please explain your understanding of it, why it is true, and if it is an axiom.

[AlexL]

Following Schmarksvillian's private objection to some of the statements contained in this post of mine, I think some clarifications are needed. They will also answer aleph_0's objections.

I wrote:

"Most of mathematics doesn't really need the concept infinity."

"This concept ("concept") is incompatible with arithmetic: operations with infinity are inconsistent and mostly undefined." Etc.

It is true that mathematics extensively uses conceptual constructs with an unlimited number of elements.

However, in my statements above I was speaking about something else, namely about the illegitimacy of operating with the number of such elements as if it were similar to a normal number.

It was Cauchy (and also Weierstrass) who, about 200 years ago, put the analysis on a more rigorous basis by precisely defining limit, convergence, continuity, derivative and other concepts, thus eliminating the so called "infinitesimals", and also infinities in the above sense.

A quantity without limit is still a specific quantity ...

It is rather a "specific quantity" without a specific quantity !

Now on a more serious note:

In order for your concept of infinity (=quantity w/o limit) to have at least a mathematical (if not real) existence, it is necessary for it to be

defined in or by a consistent theory dealing with exact concepts. Constructs typical of fuzzy or inconsistent theories are devoid of mathematical existence: they do not differ essentially from mythological characters (M. Bunge).

Therefore, what is the exact definition of "infinity" as a quantity, what properties does it have and how does it fit in other mathematical theories, for example how does it behave with respect to arithmetic operation - being a quantity, arithmetic is surely relevant?

Alex

[aleph_0]

This is no good. The counting numbers are all epistemological abstractions, they are not metaphysical. They are justified and objective, but calling them real or describing them as entities goes too far.

There are two arguments for the finitude of the universe. Deriving finitude from the Law of Identity applied to the universe as a whole is problematic, first because as the universe cannot have a boundary and second because of the difficulty in pinning down what exactly would be contradicted in the case of a universe of infinite extent. The better argument for regarding the universe as finite is simply that it would be to entertain the arbitrary. Unlike the case of crows where we know feathers occur in different colors on other birds so a white crow is plausible, there is no known instance of an actual infinity.

This later argument is perhaps the most persuasive, though I am not certain we will never find evidence that the universe is infinite. For instance, we have evidence to suggest that the cosmic background radiation has a kind of reverberation due to the Big Bang, and the direction of the ripples indicates the place where the bang occurred. If we were to find that the reverberation had the same frequency at every point in the universe, this could plausibly suggest that the "beginning" of the universe is infinitely far away from every point, and from this we could possibly argue--if we satisfied certain other logical conditions for the argument--that there are at least aleph_0 electrons in the universe. But until any such evidence is found, I am merely stating that we cannot rule out the possibility, which is what has been at issue all along.

It is defined. Limits are no big deal. But, can you deny that the purpose of declaring, outside a specifically mathematical context, that .999... = 1 is intended to confuse and provoke, based on what you call "suggestive notation?" You're amusing yourself with a trick question, by choosing a context in which its "suggestive notation" will be misunderstood.

-- Mindy

I can deny it--it's intended, here, to understand why they should be identified with each other in mathematics. I chose the phrase "suggestive notation" because we use the phrase all the time in mathematics, such as when we discuss some particular field like integers modulo 5, and denote 0 as the additive identity and 1 as the multiplicative identity. Even though we are not dealing with the usual 0 and 1, we suggest that they share a kinship because when you "add" the "0-element" of the system to any other element, your operation just returns the other element.

Here the notation suggests that .999... (where this is understood as the limit of a sequence) can be approximated by arbitrarily many decimals, and that the limit of a particular sequence is 1. There is nothing sneaky about it.

"Universe" is the sum of all existents. If even a single existent was unbounded, there could be no other existents whatever.

Multiplicity is fundamental, ontologically, as far as I'm concerned. I see no way around this. To eschew a finite universe, is to contradict identity. To reject the idea of a jar "with no specific amount of marbles in it", is not a rejection simply of the arbitrary, but of invalid concepts. I submit the willingness of some to do otherwise, is to attempt to preserve other premises of the special sciences held to be true.

Why would this be true? Why couldn't there be an infinity of many things? Why can I not simultaneously hold the assertion "A equals A", which is the statement of the law of identity, and the assertion "There is an infinity of electrons," which is a very different statement with seemingly no logical connection between them? The connection between the two is distant at best, so to treat it as if the latter is somehow obviously false because of the former is plainly ridiculous. Yes, a thing's identity has to be defined, but nobody has given any reason to suspect that infinity is not defined any less than "finite" is defined.

Note that we are not talking about a jar when we talk about the universe because the meaning of "a jar" precludes its being infinite and so having finitely many things with some finite volume (I assume electrons have some non-zero volume, but if not we could easily amend the example to talk about marbles). Part of the concept of a jar is that it encompasses a finite space. The universe is just the collection of all things that exist, and there is nothing inherent in that notion which precludes an infinite quantity of electrons contained therein.

Also note that finite does not mean "no specific amount". It means "without limit to its amount", and so implies that there is an injection from the set of natural numbers to any infinite set. No notion here is ill-defined, every term is specific.

Okay, here's a question. Do you mean 'objects' when you say that or all existents, ALL of them? You do realize that between any two objects there are numerous relationships, and between those objects and their relationships there are further more relationships, and that the process of relating relations and the objects or other relations that gave rise to them is non-terminating, don't you? Sure some relationships may be arbitrarily constructed but they nonetheless exist.

Or, if I may use numbers here, if the number 1 exists, then the number 2 exists, and so on and so forth. There are an infinite number of finite numbers. Unless you want to claim that numbers don't exist, you have to accept that the universe contains an infinite number of elements. In fact, set-theoretically, you can't even construct a universal set from the axioms, there are paradoxes which make it contradictory to the axioms. So you can't talk about a universe as being the sum of all that exists, unless you're just talking about objects.

I was hoping to avoid this discussion by talking about the number of some particular kind of object like electrons or marbles.

[aleph_0]

"The lady doth protest too much, methinks."

Sounds like I hit a nerve and of course what is the tactic? Stomping of the feet, misdirection, avoid answering the question, anything except address the argument before you, ... more evasion. You are practiced in these techniques, you've used them before, but they can't change the fact that you haven't addressed the issue.

If you don't accept the law of identity or its axiomatic status, then have the integrity to say it straight out instead of wasting all of our time. It wouldn't surprise me if you reject the law of identity since you have a penchant for denying the undeniable -- you don't even accept that we possess volition.

If you do accept the law of identity then please explain your understanding of it, why it is true, and if it is an axiom.

Oh noes, it's Evader Hollyfiled! No, I'm not protesting, I'm just mocking you to show the silliness of your tactic.

The tactic is as I stated: You cannot support your claims, and so you call someone an "evader". You might as well just say, "Nuh-uh! Noo! NNNOOOOO!"

If you cannot provided an argument--and I'm certain you can't--then don't ruin the forum by resorting to name-calling and ad hominem accusations. Now stop evading your evasion and recognize that you're an evader! Darth Evader.

However, in my statements above I was speaking about something else, namely about the illegitimacy of operating with the number of such elements as if it were similar to a normal number.

That wasn't even the point at issue, and at best is done in the surreal numbers, though even then it's not clear that's what is going on. The point at issue was why .999... = 1. We've now side-tracked into whether there can be infinitely many things, but I have explained what it means to say that a thing has an infinite quantity--namely, there being a one-to-one correspondence with the natural numbers--so the only question is whether this is possible, and I've seen no statement that indicates a negative answer.

It was Cauchy (and also Weierstrass) who, about 200 years ago, put the analysis on a more rigorous basis by precisely defining limit, convergence, continuity, derivative and other concepts, thus eliminating the so called "infinitesimals", and also infinities in the above sense.

Calculus has been alternately constructed using infinities and infinitesimals to produce all of the exact same results, i.e. there is an isomorphism between the mathematical structures, so the difference between the two is name only. Moreover, other branches of mathematics have employed the use of infinities and infinitesimals.

It is rather a "specific quantity" without a specific quantity !

Whoa, going off the deep end a bit? Are you ready to deny the law of excluded middle in order to hand on to the idea that the universe is finite?

In order for your concept of infinity (=quantity w/o limit) to have at least a mathematical (if not real) existence, it is necessary for it to be

Here is the exact definition: Standing in a injective relationship with the set of the closure of 0 with the successor function, i.e. that there is an injection from the natural numbers to the other set (the set that we are calling infinite).

This has the property that, for any natural number n, there is no injection from the set {1, ..., n} to the infinite set, nor is there a surjection from the infinite set to the set {1, ..., n}. It has the property that, given a homomorphic map, injectivity does not imply surjectivity or vice versa (this is not the case with finite sets of equal size). It has many others. I'm not sure if there's one in particular that you're looking for.

for example how does it behave with respect to arithmetic operation - being a quantity, arithmetic is surely relevant?

Arithmetic is defined only on the natural numbers, and aleph_0 is not a natural number, so you would have to extend the system of arithmetic in order to account for such a number. So arithmetic is used only for operations on finite quantities. This in no way implies that infinite quantities do not exist.

However, here are some properties that may answer your question: For any set S with cardinality aleph_c which is greater than or equal to aleph_0, and any element r which is not in S, {r} union S has cardinality aleph_c. (This is analogous to addition since the new set has "one extra".) A similar statement is made for any element s in S and the set S \ {s}, which is analogous to subtraction. However, the power set of any set is strictly larger, thus the cardinality of P(S), which is analogous to raising 2 to the power of the cardinality, is aleph_c+1.

Edited August 1, 2010 by aleph_0

[Mindy]

"Universe" is the sum of all existents. If even a single existent was unbounded, there could be no other existents whatever.

"Unbounded" doesn't mean extending without limit in all directions and dimensions. A single direction will do. Think of a line made up of two rays with shared end-points. The one, infinite, ray doesn't prevent there being another.

-- Mindy

[Grames]

I'm only laying down the "law" of Identity. I concede though that it would have been better to just say it is a philosophical point instead of simply "epistemic". Regardless the axioms, and I contend, my point , can be validated by simply opening ones eyes. A multiplicity of bounded particulars as far as the eye can see. I see my point on universe as the same as rejecting the possibility of discovering an existent without identity.

I use the term universe as Rand does in ITOE, a synonym of existence.

Concepts are mental existents, and their referents certainly exist as well.

Aren't eternal and space incummensurate characteristics?

Can an axiomatic concept have a genus?

All concepts exist in the form of mental existents, but only for valid concepts are we certain the referents exist. An invalid concept may be an improper integration of incommensurate units or it may refer to things that do not exist.

Yes, eternal (time) and space are incommensurate characteristics. Looking for a term analogous to eternal but for space does not assert, imply, or entail that space and time are commensurable. If they were commensurable the same word could be re-used. I answer my own question by stating that the universe is boundless, meaning literally having no boundary or border. How the universe can be both boundless and finite is a quandry within the realm of metaphysics, especially if there is an insistence on not resorting to tricks from the special sciences such as curvature of space or some version of locality.

The axiomatic concept 'entity' has the genus 'existent'.

[Mindy]

Aleph_0, in post 105:

"I can deny it--it's intended, here, to understand why they should be identified with each other in mathematics. I chose the phrase "suggestive notation" because we use the phrase all the time in mathematics, such as when we discuss some particular field like integers modulo 5, and denote 0 as the additive identity and 1 as the multiplicative identity. Even though we are not dealing with the usual 0 and 1, we suggest that they share a kinship because when you "add" the "0-element" of the system to any other element, your operation just returns the other element.

Here the notation suggests that .999... (where this is understood as the limit of a sequence) can be approximated by arbitrarily many decimals, and that the limit of a particular sequence is 1. There is nothing sneaky about it."

1)"Sneaky" : Who needed enlightenment as to the math of limits of sequences? And was this example the best way to begin the discussion?

2) You are admitting that "suggestive notation" is required because there is a difference, "not dealing with the usual 0 and 1," they are not the same.

3) If the limit of the sequence .9, .99, .999, ... is 1, it doesn't need approximation. You allow yourself to say that .999... is the limit of this sequence, and that 1 is its limit. If .999... is a name for the limit of the sequence .9, .99, .999, ..., why the approximation, we don't need to approximate names, signs, and symbols, only quantities.

I understand that the notational conventions are to set them equal. Here, precision isn't necessary in math (despite its mis-placed vanity,) but when you set out to discuss things, you do have to be precise enough to get your meaning across. Unless, of course, your purpose isn't to get your meaning across.

Still looks to me like a trick, putting it up here.

-- Mindy

[aleph_0]

1)"Sneaky" : Who needed enlightenment as to the math of limits of sequences? And was this example the best way to begin the discussion?

The original poster seemed confused about the matter, thinking it of philosophical significance. And indeed, while the most immediate answer to the question is purely mathematical and not philosophical, the answer gives rise to a natural philosophical question.

2) You are admitting that "suggestive notation" is required because there is a difference, "not dealing with the usual 0 and 1," they are not the same.

Well, that depends on what you think the usual 0 and 1 are. If you mean the integers, then in a sense I would agree with you. As I've said repeatedly, the natural numbers are for counting (finite quantities). However, I think most mathematicians think of 0 and 1 as complex numbers. So it kind of depends.

3) If the limit of the sequence .9, .99, .999, ... is 1, it doesn't need approximation.

What do you mean, "need"? The point is that you can approximate it, and that this relatively trivial example extends to less trivial cases like .333..., .314..., and others.

You allow yourself to say that .999... is the limit of this sequence, and that 1 is its limit. If .999... is a name for the limit of the sequence .9, .99, .999, ..., why the approximation, we don't need to approximate names, signs, and symbols, only quantities.

If you don't have any need for approximating it, then don't. A great thing about math is, you not do it if you don't want to. Don't care what 0 + 0 is? Fine. Don't add them.

However, since the real numbers are often used to measure, it can be quite valuable to know and easy to calculate the limit of a sequence. For instance, you may know that the function f(x) describes the acceleration of an object very well, and you would like to know [f(x) - f(x_0)]/[x - x_0] at some point x_0, for values of x arbitrarily close to x_0, i.e. the derivative of the function at the point, which describes the instantaneous velocity, you will want to take the limit as x goes to x_0. This can be understood as finding the value of the fraction at points x_1, x_2, x_3, ... where this sequence approaches x_0. Perhaps x_0 is 1 in this scenario, and good choices for x_1, x_2, x_3, ... happen to be .9, .99, .999, ... Well that's no problem, because luckily we now know that .999... converges to 1. Whew! Good for us.

I understand that the notational conventions are to set them equal. Here, precision isn't necessary in math (despite its mis-placed vanity,) but when you set out to discuss things, you do have to be precise enough to get your meaning across. Unless, of course, your purpose isn't to get your meaning across.

I suspect you don't understand that the equality is not a convention, it's a genuine result. And I'm not sure what "precision" you're referring to. If you mean the approximation of 1 by a sequence, even the approximation is precise in the sense that it converges to the number 1, and even does so monotonically and we have very precise bounds for the difference between the nth term of the sequence and the point to which it converges. But you seem to be talking about precision in meaning, and on that account everything is perfectly precise. So I'm not sure what you want. Every term is defined, every operation explicit.

[Edit: added my penultimate paragraph.]

Edited August 2, 2010 by aleph_0

[Plasmatic]

The axiomatic concept 'entity' has the genus 'existent'.

I asked because I know you agree with me on the definition of entity. The problem is if it has a genus it has a definition that is not simply ostensive and not irreducible as such. Been on this topic myself for a bit now....

Since axiomatic concepts are identifications of irreducible primaries, the only way to define one is by means of an ostensive definition—e.g., to define “existence,” one would have to sweep one’s arm around and say: “I mean this.”

Edited August 2, 2010 by Plasmatic

[Grames]

I asked because I know you agree with me on the definition of entity. The problem is if it has a genus it has a definition that is not simply ostensive and not irreducible as such. Been on this topic myself for a bit now....

But damn near everything has the genus of 'existent'. Would that imply there are no axiomatic concepts besides 'existent'? If so, that can't be right.

What does Rand mean by 'reducible' and 'irreducible'? The clue comes in the phrase perceptual self-evidencies. What is irreducible is a first level concept, what is reducible is a higher level concept. Chairs, tables, men are all used as examples of first level concepts and they all have a genus, differentia and definition. Having a genus is no bar to being irreducible or being ostensive.

[Plasmatic]

But damn near everything has the genus of 'existent'. Would that imply there are no axiomatic concepts besides 'existent'? If so, that can't be right.

What does Rand mean by 'reducible' and 'irreducible'? The clue comes in the phrase perceptual self-evidencies. What is irreducible is a first level concept, what is reducible is a higher level concept. Chairs, tables, men are all used as examples of first level concepts and they all have a genus, differentia and definition. Having a genus is no bar to being irreducible or being ostensive.

[The] underscoring of primary facts is one of the crucial epistemological functions of axiomatic concepts. It is also the reason why they can be translated into a statement only in the form of a repetition (as a base and a reminder): Existence exists—Consciousness is conscious—A is A. (This converts axiomatic concepts into formal axioms.)

Existence exist

Consciousness is conscious

A is A

Entities are existents with a physical boundary

Big difference.

Damn near everything is not axiomatic.

If something has a genus it is reducible and cannot be axiomatic . I'm stunned by yor last, after the quote I posted. Anyway we are probably distracting too much here.

Edited August 2, 2010 by Plasmatic

[Grames]

Existence exist

Consciousness is conscious

A is A

These are axioms, not axiomatic concepts.

If something has a genus it is reducible and cannot be axiomatic.

Trivial counterexample: Consciousness is an axiomatic concept which is also an existent.

You are mistaken to look to the definitions to find what is reducible and what is not. The meaning of a concept is its referent, not its definition. Look to the referent for reducibility. If the referent is given directly by perception it is not reducible.

Reducible is used as an epistemological term referring to conceptual hierarchy. Metaphysical reducibility would mean composed of parts, which is not the context of her statement. There is no other sense of reducibility remotely applicable here.

[Plasmatic]

These are axioms, not axiomatic concepts.

Trivial counterexample: Consciousness is an axiomatic concept which is also an existent.

You are mistaken to look to the definitions to find what is reducible and what is not. The meaning of a concept is its referent, not its definition. Look to the referent for reducibility. If the referent is given directly by perception it is not reducible.

Reducible is used as an epistemological term referring to conceptual hierarchy. Metaphysical reducibility would mean composed of parts, which is not the context of her statement. There is no other sense of reducibility remotely applicable here.

These are axioms, not axiomatic concepts.

Trivial counterexample: Consciousness is an axiomatic concept which is also an existent.

You are mistaken to look to the definitions to find what is reducible and what is not. The meaning of a concept is its referent, not its definition. Look to the referent for reducibility. If the referent is given directly by perception it is not reducible.

Reducible is used as an epistemological term referring to conceptual hierarchy. Metaphysical reducibility would mean composed of parts, which is not the context of her statement. There is no other sense of reducibility remotely applicable here.

I thought of the same counter on consciousness after I posted my last. It only adds to the problem I think.

My point is NOT look at the definition. My point is a thing gets defined a certain way because of what IT is. It, being the referent once the concept is formed and the definition made.

By the way the quote I posted was from the "axiomatic concepts" section of the Lexicon. The "they" in the second sentence refers to the "axiomatic concepts" in the first sentence. Your distinction about a is a not being an axiomatic concept is obviously wrong. However I do agree that perceptual self evidency is where one would avoid a contradiction, along with the "have to use it to refute it" principle. We are still left with an inconsistency as far as I can tell.

Your comment about metaphysical hierarchy relates to nothing I've said that I know of.

[brian0918]

Your distinction about a is a not being an axiomatic concept is obviously wrong.

"A is A" is certainly not an "axiomatic concept" - it is a statement, consisting of concepts. All the axioms you quoted are statements consisting of concepts. That is not the same as an "axiomatic concept".

Axioms - which are statements - relate axiomatic concepts to one another. Axioms are not axiomatic concepts themselves - that doesn't even make any sense.

The first and primary axiomatic concepts are “existence,” “identity” (which is a corollary of “existence”) and “consciousness.”

Edited August 2, 2010 by brian0918

[Plasmatic]

"A is A" is certainly not an "axiomatic concept" - it is a statement, consisting of concepts. All the axioms you quoted are statements consisting of concepts. That is not the same as an "axiomatic concept".

Axioms - which are statements - relate axiomatic concepts to one another. Axioms are not axiomatic concepts themselves - that doesn't even make any sense.

Yes "statements consisting of". Was at a red light when posting. My point still stands. Review the context to see what it was. (regarding axiomatic concepts and "repetition")

Edited August 2, 2010 by Plasmatic

[Grames]

I thought of the same counter on consciousness after I posted my last. It only adds to the problem I think.

My point is NOT look at the definition. My point is a thing gets defined a certain way because of what IT is. It, being the referent once the concept is formed and the definition made.

Whatever you may think your point is, if you rely on the fact that the mere presence of a definition establishes reducibility that is proof that you do look at the definition.

Your comment about metaphysical hierarchy relates to nothing I've said that I know of.

The relation is that I was establishing my identification of reducible as referring to a concept's conceptual hierarchy by considering and eliminating the only other plausibly relevant possibility. My identification of what reducibility refers to is correct, yours is wrong. This is the reason you have been stymied in understanding it. Ostensive does not mean indefinable, those words are not contraries. Every first level concept is ostensive and nearly all of them can also be placed in relation to something else as a genus (example: every species of living organism).

Another point of confusion may be conflating reducible with deducible. It is true that every deduced conclusion can be reduced to its premises, but a genus is not a premise of an argument. One cannot start with 'animal' and deduce the existence of ants, cats, men and unicorns. A genus is itself a product of a perceived similarity, presupposing the existence of the things that are similar. A genus is an integration, a concept unto itself. A genus is necessarily a wider integration but it is not necessarily either a chronologically earlier or a logically hierarchically prior integration. Knowing cats, dogs and birds goes before knowing 'animal' both logically and chronologically. First level concepts originate with ostensive definitions because a genus is simply unavailable until another concept is learned that has the necessary degree and type of similarity. Definitions are contextual and change over time according to the other knowledge gained, so a concept at first ostensively defined can later gain a conceptual definition. Ostensive and conceptual definitions are not contradictory or mutually exclusive.

[Mindy]

[Edit: added my penultimate paragraph.]

Yes, I can't make you understand my points.

[dream_weaver]

Zeno was excited. “Notational system for numbers! This makes things so much easier for adding and subtracting than using the Greek alphabet.”

I had to agree with Zeno. My experience with Roman numerals was enough to discourage me from developing the ability to manipulate numbers outside the familiar Arabic system.

Archimedes quickly agreed. “Yes. It had never occurred to me to use number to count numbers before. With each notational column being 10 of the units in the column to its right, condenses numbers in a way I had never imagined before. Why it is even easier to convey the concept of infinity using them. If we run out of numbers, we can generate more by simply adding another column to the notation. What a method! What a concept!”

Zeno pondered for a moment. “And the handling of ratio numbers is much easier to grasp with the discovery of decimal fractions. Why even my paradox has a much more elegant way of being expressed.

It was time to cut the grass. A tedious task made enjoyable at times as an opportunity to reflect on different things. For instance, cutting the grass was considered a leisure time activity. Only the people of means could afford the time to participate in this activity. It was considered a status symbol, brought about the time freed up via the industrial revolution. Even the Luddites could be found with immaculately kept lawns.

As I walked through my kitchen to grab the keys to the shed, I glanced at the clock on the stove. It read 12:00. I turned my attention to the wall by the door, reached up and grabbed the small ring of keys hanging on the nail to the right of the door. Unlocking the door, I stepped out into the porch and made way for the door leading out onto the driveway. Crossing the apron, I selected the key to the shed, inserted it into the keyhole and proceeded to unlock the door. I opened it as far as it would travel, and placed the small stone against the direction of travel to hold it in place.

I thought about the conversation I had listened to earlier, while I returned the keys to the nail. Their mathematical discoveries were legendary.

I retrieved my 19” electric lawn mower from the shed. My mind began to focus on the job ahead of me. As I unwound my 100ft extension cord, I reflected on the fact that it had been done so many times before it was simply a matter of route. Plugging the terminal of the cord into the wall outlet, I turned to walk back to where the lawn mower sat awaiting to cut the 1/3rd of an acre that surround my home.

I flipped the toggle switch that I had installed to replace the factory-installed lever that had ceased to work several years prior. The lawn mower handle shifted slightly to the left as the motor applied its torque to the blade attached under the molded plastic deck. Grabbing the handle with both hands, I pushed the lawn mower into motion, off of the apron, onto the lawn.

Over half a billion blades of grass are contained within 43560 square feet of lawn. Curiosity had driven me earlier to do a search on the web for such a tidbit of trivia. I can only imagine that the data had been extrapolated from a smaller sampling.

I glanced over at the three blue spruce pine trees belonging to my neighbor. I had been able to determine there were three of them simply by looking when I first moved in. Three lampposts in the front yard had stood out in the same way. The two mailboxes sitting on the post by the road had puzzled me. Yes I could see there were two, without even counting, but why were there two of them installed? The four wheels on the lawn mower had been adjusted to the highest height, cutting the grass to the length of about 3 inches.

I thought about Ayn Rand’s illustration of Crow Epistemology, from Introduction to Objectivist Epistemology. Two mailboxes. Three trees. Three lampposts. Four wheels. At a glance, I could see and grasp and differentiate these quantities. I recalled Pat Corvini’s lecture on number being the relationship between a group and one of its members taken as a unit.

There are over a half-billion blades of grass contained on an acre of property. How was that number determined? Did someone count them all? It did not seem likely. Was there an idiot savant who looked at a field and identified the quantity as such? Was it verified?

Perhaps someone took a measuring tape out into their yard, and marked of a square inch, and counted the blades of grass therein. A pair of scissors, a ruler, snip the grass and count blades. If they grouped the grass clippings into groups of ten, they could count the groups and any that might be left over. Eight groups of ten blades of grass with one group of five blades of grass left over. Eighty-five blades of grass with over six million square inches of area in an acre, is much easier to calculate than to count.

It seems so elegant. A group of ten things treated as a unit. Count the groups. It is a means of using number to count numbers. This should be fine as long as you do not forget what you are counting. In the process of using number to count numbers, we can come up with new condensations. We can use the condensations we come up with to establish more condensations. We could exhaust the supply of paper available to us for noting what the condensations stood for. We could run out of materials to build super computers to manage these condensations and still not run out of numbers to count. Even if the universe, by its nature, were a computer dedicated to enumerating numbers, being eternal as it is, would still be unable to establish the quantity of this marvelous abstraction the human mind has been able to identify and define: infinity. Multiplicity: the relationship of a group to one of its members, taken as a unit. 1, 2, 3, 4, etc, ad hoc, ad infinitum. We should be fine, as long as we don’t forget what we are counting,

It was a sunny afternoon. The thermometer indicated the temperature is 85° Fahrenheit. I flipped the toggle switch to the off position and started to walk toward the house as I listened to the sound of the blade and motor wind down from the friction and removal of electric current to a stop. The yard nearly half cut. I was thirsty.

I opened the storm door to the house and proceeded to enter. The contrast of the air-conditioned air felt refreshingly cool while the warmth behind me quickly dissipated as the two closers functioned as designed. Making my way to the kitchen, I grabbed the 8 fluid ounce bottle of water off of the bamboo butcher block, twisted the lid off of the container and raised the sought after prize to my lips. The water was as satisfying as the coolness of the house, in a similar, differing way. I crushed the plastic container in one hand and threaded the cap back on, compacting it so to increase the efficiency of the space available in the trashcan.

I left the house through the porch egress to unplug the extension cord from the outlet. The 100 foot of 12-gage extension cord had served me well. It was long enough to reach the perimeter of the yard from three outlets. 100 feet. A foot. At one time, it was the king’s foot that was used as the standard unit of length. Measurement. Mathematics is the science of measurement. Measurement is the identification of a relationship—a quantitative relationship established by means of a standard that serves as a unit.

I plugged the cord into the outlet and walked back to the lawn mower. Flipping the toggle switch back to the on position, I paused to allow the blade to come up to speed. Gripping the handle with both hands I resumed where I had left off.

I own 1/3rd of an acre. If one hundred one million, six hundred forty thousand blades of grass have been cut, that would leave the same. Dividing the next portion in half would leave fifty million, eight hundred twenty thousand blades of grass to cut. When Zeno’s paradox gets to 1/268435456 there should only be one blade of grass left to cut. If I wanted to apply Zeno’s method here, would I cut half way thru then?

With a 19” lawn mower, it will have been pushed 9170 feet to cover the lawn. At 1/131072 there will be one foot left to cut. If I wanted to use Zeno’s method, would I have to convert to inches at this point?

I discovered that I push the lawn mower at 1 foot per second. At 1/16384 using Zeno’s method, what should I do with the remaining second?

Within the realm of mathematics, the method to develop Zeno’s paradox is contained. As with other concepts of method, we can continue to extend it in with regard to continuous quantity as needed. It is still important to remember what is referring to.

After putting the equipment away, I headed back into the house. I grabbed another bottle of water and began to drink. My thoughts began to wander. One half of eight ounces is four ounces. What happens when I get down to the last drop? What about the last molecule? If I put a molecule of water on my tongue, would I be able to taste it?

The end.

Please note, the mathematics in this story were truncated at the decimal point via an excel sheet.

[Plasmatic]

Grames since nothing in your last post addresses my points and mostly proposes positions I do not and have not maintained, we can just table this.

[aleph_0]

Yes, I can't make you understand my points.

Well if you want to get snippy about it, to be honest, your argument is to blame. Write me when you have something intelligible, and lacking so many errors.

Zeno was excited. “Notational system for numbers! This makes things so much easier for adding and subtracting than using the Greek alphabet.”

If this is intended as a response to me, I'm sorry, but I'm not going to read something that makes its point in 10+ paragraphs of children's historical fiction, when it could be made in a paragraph of argument. If this isn't directed at me, then nevermind this post.

[Mindy]

Well if you want to get snippy about it, to be honest, your argument is to blame. Write me when you have something intelligible, and lacking so many errors.

I'm snippy? You said you were closing out the discussion...

To be honest...sigh, how good it feels to write that, to be honest, you don't seem to have any idea of what my argument has been. (Yes I do! no you don't! yes I do!...)

Mindy

[TuringAI]

Okay, let's skip all the drama.

The decimal 0.(9) is referring to a 'completed' infinity. The most important question to be asked is this: Since you can't measure something with infinite accuracy, does infinity itself have a valid meaning in a purely mathematical concept?

This touches the idea that there can exist things in mathematics that do not DIRECTLY apply to anything in reality. I don't accept Ayn Rand basing ALL of mathematics on measurement. Rather I think that the mere notion of an entity as having an independent existence implies the notion of wholeness, or of something corresponding to an integer. Furthermore, math is purely deductive. It tells you, IF you take certain things as premises, where those premises lead.

So really what we should be asking is this: Is math any more fundamental than Ayn Rand's definition implies? If you have dealt with any advanced mathematics you have no choice but to consider infinity as something with some semblance of identity, and that contradict Ayn Rand's theory.

If someone closes this thread at least split this off. Everybody here is attacking method but nobody's actually going to convince anyone of anything by doing so.

Please consider my offer to start a new debate about where mathematics ACTUALLY fits into the world versus where Ayn Rand put it in her philosophy.

[Schmarksvillian]

I don't accept Ayn Rand basing ALL of mathematics on measurement.

She allows allows concepts of method, and she allows mathematics such as complex numbers if they are useful.

If you have dealt with any advanced mathematics you have no choice but to consider infinity as something with some semblance of identity

I don't know exactly what you mean by "infinity as some semblance with identity", but in advanced mathematics there are approaches that do reject that there exist infinite sets. There are various ultra-finitistic approaches in advanced mathematics. It's true that most mathematicians accept the existence of infinite sets, but there are some, even if only relatively a few, who work in advanced mathematics without accepting that notion, or outright rejecting it, or even claiming that it is meaningless.