The Opposite of Nothing Is/Isn't Everything

[Dragonfly]

Tomorrow I'm going on vacation for 3 weeks, with no access to the Internet, but I trust that Ba'al, Merlin and Laure will hold the fort...

[Selene]

Have a great trip.

[Dragonfly]

Thanks!

[Michael Stuart Kelly]

Dragonfly,

Be well and have a wonderful trip.

I, for one, will miss you.

Michael

[merjet]

It seems to me the problem here is Roger Bissell and Thom making ontological interpretations based on counting and simple integer arithmetic which are easily demonstrated using pennies or pebbles or whatever, and then trying to apply such interpretations to mathematics as a whole. Of course, at least to Thom 0 pennies, 0 pebbles, 0 rows, 0 columns, etc. commit the fallacy of "reification of zero", which leads to bizarre ontological interpretations of 0. The result is a philosophy of mathematics limited to early grade school arithmetic.

This can also be seen for "multiplication by zero." Typically, we are taught that any number multiplied by 0 is 0. This is another misinterpretation of what is going on. In 5 x 3 = 15, you are multiplying 5 by 3, but in the expression "5 x 0," you are not ~multiplying~ 5 by ~zero~; you are ~not multiplying~ 5 by ~anything~. You are specifying that there ~zero~ multiples of 5. Considering that multiplication is just compressed addition, you can see this easily: 5 x 3 is 5 + 5 + 5, 3 multiples of 5. The number 5 must appear 3 times as the only addends, and the sum of those three multiples of 5 is 15. However, 5 x 0 is ~no~ multiplies of 5. The number 5 must appear 0 times, and there are no other addends, which means no addition (and hence no multiplication) is being performed. 0 is expressed as the product of 5 and 0, but this is not the expression of a multiplication operation, but what must be the situation when no such operation is performed!

I see as easily that 5*3 in terms of addition is 3+3+3+3+3. Then why isn't 5*0 in terms of addition 0+0+0+0+0? (0*5 can be interpreted the same way. Incidentally, see here.) Is it possible to physically show this? Of course, using jars and pennies, for example. Five jars, each empty, means there are 5*0 = 0 pennies in the jars. Then if Roger arrived and smashed all the jars, there would be (5-5)*0 = 0*0 = 0 pennies, not an "undefined" quantity.

Consider making 5*3 more concrete, using jars and pennies, including these in the calculations. Is it:

1. (5 jars)*(3 pennies/jar) = 5*3 pennies = 15 pennies, or

2. (3 pennies/jar)*(5 jars) = 3*5 pennies = 15 pennies?

Either is fine. Replace each 3 and 15 with 0 and both expressions are equally fine. Clearly the order makes no difference. But somehow Roger seems to believe it does.

Have a great trip, Dragonfly.

Edited June 1, 2009 by Merlin Jetton

[BaalChatzaf]

Tomorrow I'm going on vacation for 3 weeks, with no access to the Internet, but I trust that Ba'al, Merlin and Laure will hold the fort...

Bon Voyage! Three weeks an no internet. Wunderbar!

[tjohnson]

It seems to me the problem here is Roger Bissell and Thom making ontological interpretations based on counting and simple integer arithmetic which are easily demonstrated using pennies or pebbles or whatever, and then trying to apply such interpretations to mathematics as a whole. Of course, at least to Thom 0 pennies, 0 pebbles, 0 rows, 0 columns, etc. commit the fallacy of "reification of zero", which leads to bizarre ontological interpretations of 0. The result is a philosophy of mathematics limited to early grade school arithmetic.

Very similar to what I have been trying to say.

[merjet]

Shift gears a bit and you will see that quantity as a concept of method and quantity as an existent are slightly different. Same word. Different meanings.

Is zero or any number a concept of method? I don't think so.

"Concept of method" is not a common term and Rand's use of it in ITOE2 raises questions. She writes, "Concepts of method are formed by retaining the distinguishing characteristics of the purposive course of action and of its goal, while omitting the particular measurements of both" (ITOE2, 36). In the next paragraph she says logic is the fundamental concept of method. In the Appendix of ITOE2 she calls infinity and the imaginary number i (square root of -1) concepts of method.

This suggests an inconsistency or more than one meaning. Logic is a method, but infinity and i are not methods. Nor are they "courses of action." One might say they arise using a method, but that implies a second meaning of "concept of method", one too broad to be useful IMHO.

[Michael Stuart Kelly]

Is zero or any number a concept of method? I don't think so.

Merlin,

That to me is like asking whether a letter of the alphabet is a concept of method.

Within use (the real concept of method), one number presumes that the other numbers exist and a relationship among them does also. A method emerges once that happens. Ditto for letters.

Try doing logic without the letters of the alphabet.

Michael

[thomtg]

[...]

"Concept of method" is not a common term and Rand's use of it in ITOE2 raises questions. She writes, "Concepts of method are formed by retaining the distinguishing characteristics of the purposive course of action and of its goal, while omitting the particular measurements of both" (ITOE2, 36). In the next paragraph she says logic is the fundamental concept of method. In the Appendix of ITOE2 she calls infinity and the imaginary number i (square root of -1) concepts of method.

This suggests an inconsistency or more than one meaning. Logic is a method, but infinity and i are not methods. Nor are they "courses of action." One might say they arise using a method, but that implies a second meaning of "concept of method", one too broad to be useful IMHO.

Merlin,

Do you recall a lecture some years ago by David Ross where he walked his listeners through the introspective process of arriving at the concept of "infinity"? The process of getting there required a purposive mental course of action. And so is i, which calls for a concise series of math operations toward the goal of a result. "Logic" is a concept of method because it calls for a certain mental course of action to arrive at noncontradictory identification of the facts of reality. "Recipe" is a concept of method because it calls for a certain existential course of action toward the goal of delicious foods. "Computer program" is another concept of method which encapsulates the frozen intelligence of men into the form of machine instructions, etc. And "etc." too is a concept of method because it calls for the reader to repeat some prior course of action toward an inductive conclusion. There is but one genus that covers all of these concepts: "method," or synonymously, "technique."

"Zero," I contend, is another such concept of method; its psycho-epistemological existence calls for a certain mathematical course of action, namely, to halt operating.

[tjohnson]

"Zero," I contend, is another such concept of method; its psycho-epistemological existence calls for a certain mathematical course of action, namely, to halt operating.

It could be. It could also be a label for an identity element under addition in a group.

[merjet]

Do you recall a lecture some years ago by David Ross where he walked his listeners through the introspective process of arriving at the concept of "infinity"? The process of getting there required a purposive mental course of action. And so is i, which calls for a concise series of math operations toward the goal of a result.

Maybe I heard Ross' lecture. Regardless, I don't remember it and it is not hard to imagine what he said. The fact that infinity and i require a course of action does not imply they are courses of action.

"Zero," I contend, is another such concept of method; its psycho-epistemological existence calls for a certain mathematical course of action, namely, to halt operating.

I disagree; it is a concept of quantity. I acknowledge that you and Bissell construe zero to mean "halt operating." I just think it is a bizarre notion that cripples commutativity in arithmetic and much else in mathematics, like I said here and here.

[Guyau]

David Ross on Imaginary Unit

Rand and Infinity

Rand on Concept of Nothing

[thomtg]

[...]

The fact that infinity and i require a course of action does not imply they are courses of action.

"Zero," I contend, is another such concept of method; its psycho-epistemological existence calls for a certain mathematical course of action, namely, to halt operating.

I disagree; it is a concept of quantity. I acknowledge that you and Bissell construe zero to mean "halt operating." I just think it is a bizarre notion that cripples commutativity in arithmetic and much else in mathematics, like I said here and here.

Merlin,

I illustrated for you a facsimile of "infinity." I gave you "etc." Whenever you say "et cetera" or "and so forth," you are bidding a consciousness to repeat some course of action to arrive at your point. You are designating by the word or phrase a mental existent. "Infinity" refers not to metaphysical existents but to mental courses of action. Ask a child to add 1 to the biggest number he can think of. He will get a bigger number. Ask him again to add 1 to that number to get another, bigger number. Ask him to repeat the operation again and again. Soon he will grasp the concept that the process of getting the biggest number never stops. This process of repeating operations is an instance of the concept of "infinity."

I do not see how you can argue that an absence of a quantity is another quantity. Explain that and we'll see if we can explain your derivative objections.

[Michael Stuart Kelly]

Stephen,

I don't have time to digest Ross's comments right now (too many references to things and works I am not too familiar with), but I do have a comment on your Rand posts.

As with you, I fully agree that nothing is a relative concept. It is not a primary one.

On Rand's metaphysical delimitation of infinity to not really being infinite, I don't see how this can be claimed as fact without observation. It is only a supposition deduced from an axiomatic concept (identity).

And even so, I don't see how "something not limited by anything" (Rand's words) are "not definable" (Rand's words again). Once you use the word "anything," metaphysical infinity becomes a relative concept. That's not only definable, it has a quite distinct identity.

Michael

[merjet]

I do not see how you can argue that an absence of a quantity is another quantity. Explain that and we'll see if we can explain your derivative objections.

Explain what you mean by "an absence of a quantity" and maybe I will. Examples might help.

Meanwhile:

1. 5 +

2. 5 + 0

The first example has an "absence of a quantity." The addition can't be done, and this "absence of a quantity" is NOT another quantity.

In the second example the 0 denotes an absence of another number, and the addition can be done. This makes it a quantity as good as any other number.

Meanwhile, here is a task for you. Multiply 103*700 using long multiplication. If you stick to your premise that 0 is an operation blocker, then you cannot get an answer. On the other hand, if you do get the correct answer, you must reject your notion that 0 is an operation blocker (except when a divisor).

[thomtg]

I do not see how you can argue that an absence of a quantity is another quantity. Explain that and we'll see if we can explain your derivative objections.

Explain what you mean by "an absence of a quantity" and maybe I will. [...]

Merlin,

I think we are repeating ourselves. I refer you to my Post #240 where I referred you to your dictionary definition for "zero" and then to my interpretation of this relational concept "absence of" in the math context.

So accept it: an absence of a quantity is not another quantity.

[thomtg]

David Ross on Imaginary Unit

Rand and Infinity

Rand on Concept of Nothing

Stephen,

David Ross's explanation of the transformations of complex numbers reminds me of Robert Heinlein's technological weapon of projecting whole worlds 90 degrees into some phantom zone. (See his Have Spacesuit Will Travel.) The explanation is really good and thoroughly understandable. It affirms the view that i indeed designates a technique.

Some choice excerpts:

...

If complex numbers can be thought of as transformations in a plane, then of course, in a general way, complex numbers do represent definite quantities. However, a complex number represents

two

quantities, the coordinates of the corresponding point in the 2-dimensional plane. Calling a complex number a number is somewhat like calling a vector a number, and I am against this. [104]

...

The basic point is, when the layperson is told that sqrt(-1) is a number, and he balks, he is right. Of the entities he integrated under the concept he calls

number

, none has the property of becoming negative upon being squared. He has identified an objectively valid cognitive category. It seems to me that given this, to insist, over the protests of zillions of laypeople, that sqrt(-1) is a number is either to hold an intrinsicist view of concepts or to refuse to communicate out of stubbornness. [105]

...

There are a few other items in the Ross remark about wave, electromagnetic wave, and probability wave in relation to complex numbers. Thanks for the interesting read, Stephen.

[Christopher]

Rereading Robert's online article of Piaget http://hubcap.clemson.edu/~campber/piaget.html sheds more light on the mental processes going on in mathematics:

[Reflecting abstraction] is responsible for the bigger leaps that take place during development.

[52] In his 1970 essay, titled simply "Piaget's theory," Piaget says that reflecting abstraction "is the general constructive process of mathematics: it has served, for example, to construct algebra out of arithmetic, as a set of operations on operations" [note 13]. It abstracts from, and generalizes over, your prior ways of coordinating your actions. It is distinct from, and opposed to empirical abstraction, which ranges over the properties of objects out in the environment.

[53] Here's what Piaget considered a rather simple example of reflecting abstraction. Why is multiplication harder to understand than addition? They're both operations on numbers, after all. Piaget's analysis was that to understand multiplication it is not enough to center your thinking "on the objects that are being put together with other objects, and thus on the result of this union. Multiplication also involves isolating the number of times that the objects are being brought together; it means enumerating operations as such, not just the results of those operations (i.e., the number of objects transferred each time)" [note 14]."

I left more at the end for those who want to read a specific example. Reflecting abstractions is a step out of reality and into thoughts. "Zero" at the level of empirical abstraction (if I understand the terms correctly) refers to the absence of a quantity; whereas, zero at the level of multiplication is a reflecting abstraction of no operations being done. I think this is especially relevant in addressing Merlin's post, but also Merlin - 700 is a quantity, with zeroes as symbols, whereas 0 alone is not necessarily the same cognitive symbol as 700.. although I guess it could be.

Anyway, when we do math using the term "0," we physically denote an absence in symbolic representation of whatever unit is being added. But unless those units have some identity (# of apples, # of oranges, etc.), then numbers themselves are abstractions of quantity, of measurement... and necessarily measurement has a zero-point. And when we do multiplication for example, zero denotes not a quantity of units but rather no operations being done.

I also wanted to add that conceptualizing infinity is a reflecting abstraction (according to Piaget), which is not in the same realm of empiricism... it is a concept that arises from operations and not entities.

Chris

[54] Piaget and his students asked children to make two rows (sometimes two towers) of poker chips, with one restriction--they had to add chips to row A two at a time, and to row B three at a time. So, did they realize that if they did this, they could get equal rows? Did they know how they could get equal rows?

[55] Here is how Pat responds (Piaget used three-letter codes to indicate subjects in his studies). Pat is 5 1/2 years old and functions at Level IA.

Pat... adds 2 As and 3 Bs until she discovers to her astonishment that she ends up with 6 chips of each color: "Both have the same amount!" "How did that happen?" "I don't know." "Could you do it again?" "No, I don't think so." "Let's try" (same procedure). "Again, they're both the same amount!" "How did you do that?" "I counted 6 there and 6 there" (pure imagination!). "What did you do to make your pile?" "I took 2 (at a time)." "And to make my pile?" "I took 3." "How many times did you take 3?" "I don't remember any more." "And (how many times did you take) 2?" "I don't remember either." [note 15]

[56] Pat obviously thought that adding 2 at a time to one row and adding 3 at a time to other were going to produce unequal outcomes. And she had no idea how many times she carried out the additions. At what Piaget calls Level IB (average age around 6), children eventually notice how many times they added sets of 2 or 3 chips when the rows came out equal. Piaget concludes that they have become conscious of the number of times they added, so reflecting abstraction is starting to take place. But the children can't predict in advance that adding another 3 sets of 2 and 2 sets of 3 will make the rows equal again; they just have to try and see what happens.

[57] At Level IIA (around 7 or 8) children predict that they can make the rows equal, but without being able to figure out in advance how to do it. Whereas at Level IIB they do figure it out in advance:

VAS (9;11) on the chip problem immediately sets out 3 pairs of As, then 2 triplets of Bs. "I made each one have 6." "Could you do it with smaller piles?" "No, because you have to take 2 As at a time, or 3 Bs. If you make a pile of 3 chips, that'll work for the Bs but not for the As. And if you take 4, that'll work for the As and not for the Bs." "And bigger piles?" "Yes, 12 for example."

[58] Piaget says that "at Level IIB the multiplicative operation 'n times x' is finally understood as constituting the product of reflecting abstraction from additions of additions. Now n is no longer the number of 'packets' that had to be assembled to reach the goal, but rather the number of operations that constituted these classes" [note 17]. Piaget adds that what has happened here is a step beyond plain vanilla reflecting abstraction--it's reflecting abstraction to the 2nd power, or reflected abstraction.

Edited June 2, 2009 by Christopher

[merjet]

I think we are repeating ourselves. I refer you to my Post #240 where I referred you to your dictionary definition for "zero" and then to my interpretation of this relational concept "absence of" in the math context.

I refer you to post #198. There you said, in some sense, that 0 is "the absence of a presumed, existing quantity." So please explain to us why it is fine for you to say 0 is the absence of a quantity, but there is something wrong if I say it.

Have you tried the long multiplication problem I gave you? Which horn of the dilemma did you choose?

Edited June 2, 2009 by Merlin Jetton

[thomtg]

I think we are repeating ourselves. I refer you to my Post #240 where I referred you to your dictionary definition for "zero" and then to my interpretation of this relational concept "absence of" in the math context.

I refer you to post #198. There you said -- in some sense, presumably abstract -- that 0 is "the absence of a presumed, existing quantity." So please explain to us why it is fine for you to say 0 is the absence of a quantity, but there is something wrong if I say it.

Have you tried the long multiplication problem I gave you? Which horn of the dilemma did you choose?

1. "Zero" is a quantity............................ [ Qo, given o="Zero" ]

2. "Zero" is an absence of a quantity..... [ (z)(Qz > Aoz) ]

They are not the same, Merlin. I reject 1 and endorse 2.

There is no problem at all with 103x700: drop down 0, drop down 0, 1 carry 2, (7x0 halt; 0 add 2) 2, 7. Writing in reverse order, the product is 72100.

Edited June 3, 2009 by Thom T G

[BaalChatzaf]

1. "Zero" is a quantity. .......................... [ Qo, given o="Zero" ]

2. "Zero" is an absence of a quantity..... [ (z)(Qz > Aoz) ]

They are not the same, Merlin. I reject 1 and endorse 2.

You are in error. Doing arithmetic operations on quantities yield quantities. 1 is a quantity. What is 1 - 1? It is 0 a quantity. Also be careful of 0 in positional notation. Positional notation is a way of naming numbers. Zero within the positional names of numbers is a place holder. For example in the number 100 the two zeros indicate that 1 is a multple of one hundred (the number). Positional notation is nice because certain symbolic operations on the names of numbers yield the correct number named. 20 plus 40 yield 60 by applying the rules you learned in second grade. It so happens that 60 is the right name for the number which is the sum of twenty and forty (the numbers).

Ba'al Chatzaf

[Rich Engle]

I always thought that the opposite of "nothing" is "something," not "everything."

But then you still have a world to live in where you have something, but nothing becomes everything else that isn't something.

Then you might start wondering about the part where the nothing part becomes everything else.

Then you have a thing of somethings and everything else, but you know that is all of it.

This is why people can teach Phil 101 courses at community colleges: easy to confuse the locals.

Good discussion, been following.

[Selene]

Rich:

Seems like you are correct -

no thing

Main Entry: nothing

Part of Speech: noun

Definition: emptiness

Synonyms: annihilation, aught, bagatelle, blank, cipher, crumb, diddly, duck egg, extinction, fly speck, goose egg, insignificancy, naught, nihility, nix, no thing, nobody, nonbeing, nonentity, nonexistence, not anything, nothingness, nought, nullity, obliteration, oblivion, scratch, shutout, trifle, void, wind, zero, zilch, zip, zippo, zot

Antonyms: something

Definitions of terms - so critical to intelligent discourse.

[Rich Engle]

I can do this stuff, it just gives me a headache. I like to watch the real guys discourse.

It's been a great discussion.