...So, here then is a modern problem: mathematics makes a claim that contradicts philosophy. More specifically, a branch of math, set theory--which is the basis of mathematical logic and many other offshoots dependent on it--this branch has an axiom stating that the complement of the empty set is the universal set. That is, the opposite of nothing is everything. Its corollary is that the complement of everything is nothing.

Now this mathematical axiom contradicts philosophy, namely, the Objectivist philosophy. In particular, it contradicts a basic philosophical axiom, the axiom of existence: that existence exists--and its corollary: that only existence exists. (For the full context, see 58-60.) It suffices to say informally, the opposite of existence is not nonexistence.

By the nature of the problem, both branches cannot stand apart in epistemological détente, if they purport to be knowledge. Being axioms, the repudiation of either one has fundamental ramifications for its respective branch, if not its destruction. One side must be true, but which one?

How is "nothing," or "nonexistence" distinguished from "does not exist?"

Ayn Rand in ITOE has a special place for mathematics, a science which she holds in high esteem. Mathematics in her view discovers and defines standards of measurement for all conceptual cognition. Since every formation of any concept depends on grasping similarities on the basis of measurements, since every piece of knowledge one has ever thought or communicated depends on such concepts, therefore by AAA-1, all of man's knowledge of reality depends on mathematics, the science of measurement. (ITOE 7)

1. Mathematics is not a science, since it has no empirical content. It is about objects that do not exist in the physical world (e.g. points, lines, numbers with infinite decimal expansions and such like).

2. Mathematics is the logos of abstract systems and structures, some of which can be used in measurement and some of which are not. I have been all through this with MSK

I agree with the importance of mathematics in human knowledge but I think it is better characterized as a language of relations, exact relations in particular. In mathematics relations are exact, when applied to a field they are only approximate because all factors are not included in science.

There's a conceptual symbol for "nothing" in most every culture and in every language. I can draw the character for "nothing" in Chinese. These symbols exist. A symbol that signifies the absence of everything. You need the language (and Rand needed the language) to denote "nothing" in order to discuss it in OE.

So, here then is a modern problem: mathematics makes a claim that contradicts philosophy. More specifically, a branch of math, set theory--which is the basis of mathematical logic and many other offshoots dependent on it--this branch has an axiom stating that the complement of the empty set is the universal set. That is, the opposite of nothing is everything. Its corollary is that the complement of everything is nothing.

Now this mathematical axiom contradicts philosophy, namely, the Objectivist philosophy. In particular, it contradicts a basic philosophical axiom, the axiom of existence: that existence exists--and its corollary: that only existence exists. (For the full context, see 58-60.) It suffices to say informally, the opposite of existence is not nonexistence.

By the nature of the problem, both branches cannot stand apart in epistemological détente, if they purport to be knowledge. Being axioms, the repudiation of either one has fundamental ramifications for its respective branch, if not its destruction. One side must be true, but which one?

Mathematics makes no claims about existence whatsoever. Please understand that.

More specifically, a branch of math, set theory--which is the basis of mathematical logic and many other offshoots dependent on it--this branch has an axiom stating that the complement of the empty set is the universal set. That is, the opposite of nothing is everything. Its corollary is that the complement of everything is nothing.

Now this mathematical axiom contradicts philosophy, namely, the Objectivist philosophy. In particular, it contradicts a basic philosophical axiom, the axiom of existence: that existence exists--and its corollary: that only existence exists. (For the full context, see 58-60.) It suffices to say informally, the opposite of existence is not nonexistence.

(snip)

Thom -

Complement and opposite are not the same thing. Complement is a well-defined notion, and clearly the complement of the universal set is the empty set. This in no way contradicts the fact that existence exists.

If you use words such as "complement" in such a casual way, you will end up in lots of trouble. But to treat complement as an exact synonym for opposite is a mistake.

Complement and opposite are not the same thing. [...]

Bill, thank you for highlighting my semantic bridge from mathematics to everyday philosophy. Yes, my equating of "complement" to "opposite" was deliberate. I wanted to emphasize the connection that mathematics has a role to play in philosophy, and vice versa, for the purpose of living in the world. While I do realize that the term "opposite" is equivocal in the realm of logic to mean either of two incompatible relationships, namely, contrariety and contradiction; I did not want to admix all these terms without the proper context. Rather, I would like for interested readers to understand the issue from the view of mathematics and then to be able to relate it to philosophy.

So, the issue here purports to be a contradiction between two branches of knowledge concerning the opposite of some notion to be some other notion. On the math side, "opposite" means "complement." On the philosophy side, I cited page references for readers to follow up to see which meaning of "opposite" Ayn Rand discusses in the present context.

How is "nothing," or "nonexistence," distinguished from "does not exist?"

Ian.

Ian,

"Nothing," and "nonexistence" are synonymous English words denoting a legitimate concept, which Christopher can also notate in Chinese. What that concept is, is central to the present issue, namely, the issue of the opposite of nothing. That is, while the concept does exist, the issue I am curious about is the nature of its opposite.

The predicate "does not exist" is straightforwardly the denial of existence. What is the opposite of "existence"? This is also central to the issue.

Complement and opposite are not the same thing. [...]

Bill, thank you for highlighting my semantic bridge from mathematics to everyday philosophy. Yes, my equating of "complement" to "opposite" was deliberate. I wanted to emphasize the connection that mathematics has a role to play in philosophy, and vice versa, for the purpose of living in the world. While I do realize that the term "opposite" is equivocal in the realm of logic to mean either of two incompatible relationships, namely, contrariety and contradiction; I did not want to admix all these terms without the proper context. Rather, I would like for interested readers to understand the issue from the view of mathematics and then to be able to relate it to philosophy.

So, the issue here purports to be a contradiction between two branches of knowledge concerning the opposite of some notion to be some other notion. On the math side, "opposite" means "complement." On the philosophy side, I cited page references for readers to follow up to see which meaning of "opposite" Ayn Rand discusses in the present context.

Thom -

Boil down what you have said above, and it appears to be that you have chosen to equivocate and then pretend to see a contradiction - which "contradiction" is simply rooted in your equivocation, not in either mathematics or philosophy.

What is your intent in doing this? If I were to describe a contradiction between athletics and clothing because the word "run" is used in one way referring to an athletic activity, and another way to refer to a defect in hosiery, would that be a contradiction? Or just silliness on my part? I submit the latter - and it seems to be exactly the same thing you are doing.

Boil down what you have said above, and it appears to be that you have chosen to equivocate and then pretend to see a contradiction - which "contradiction" is simply rooted in your equivocation, not in either mathematics or philosophy.

What is your intent in doing this? If I were to describe a contradiction between athletics and clothing because the word "run" is used in one way referring to an athletic activity, and another way to refer to a defect in hosiery, would that be a contradiction? Or just silliness on my part? I submit the latter - and it seems to be exactly the same thing you are doing.

Bill P

Good rebuttal, Bill. Now consider another thing I have also said. If you accept the normative premise that your knowledge of reality must be an integrated whole, then my intent becomes apparent. I am not denying that ideas in respective systems of thought are already self-consistent (I am not asserting it either), but I deny that systems of thought can be separately free floating. Does the notion of "complement" in set theory have any use to the task of living on earth? What are its links to the rest of our knowledge? What are its links to reality? We can and should apply these same questions to our notions in philosophy. I invite you to proceed if you accept the normative premise. And if you then don't see a contradiction, I definitely want to know about it.

Boil down what you have said above, and it appears to be that you have chosen to equivocate and then pretend to see a contradiction - which "contradiction" is simply rooted in your equivocation, not in either mathematics or philosophy.

What is your intent in doing this? If I were to describe a contradiction between athletics and clothing because the word "run" is used in one way referring to an athletic activity, and another way to refer to a defect in hosiery, would that be a contradiction? Or just silliness on my part? I submit the latter - and it seems to be exactly the same thing you are doing.

Bill P

Good rebuttal, Bill. Now consider another thing I have also said. If you accept the normative premise that your knowledge of reality must be an integrated whole, then my intent becomes apparent. I am not denying that ideas in respective systems of thought are already self-consistent (I am not asserting it either), but I deny that systems of thought can be separately free floating. Does the notion of "complement" in set theory have any use to the task of living on earth? What are its links to the rest of our knowledge? What are its links to reality? We can and should apply these same questions to our notions in philosophy. I invite you to proceed if you accept the normative premise. And if you then don't see a contradiction, I definitely want to know about it.

Thom -

I think that if you want to get a serious and thoughtful response from people it would be best to not engage in equivocation to get their attention. Was that what you were attempting? I still don't understand what you were getting at with your original equivocation re "complement." Can you state specifically what you really meant?

I think that if you want to get a serious and thoughtful response from people it would be best to not engage in equivocation to get their attention. Was that what you were attempting? I still don't understand what you were getting at with your original equivocation re "complement." Can you state specifically what you really meant?

Bill P

Bill, here below are my integrations. The meaning of the latter will be obscured unless you go to the pages cited in ITOE.

From the branch of mathematics, the opposite of nothing is everything, and the opposite of everything is nothing.

From the branch of philosophy, neither can there be an opposite of nothing, nor can there be an opposite of everything.

...mathematics makes a claim that contradicts philosophy. More specifically, a branch of math, set theory--which is the basis of mathematical logic and many other offshoots dependent on it--this branch has an axiom stating that the complement of the empty set is the universal set. That is, the opposite of nothing is everything. Its corollary is that the complement of everything is nothing.

Now this mathematical axiom contradicts philosophy, namely, the Objectivist philosophy. In particular, it contradicts a basic philosophical axiom, the axiom of existence: that existence exists--and its corollary: that only existence exists. (For the full context, see 58-60.) It suffices to say informally, the opposite of existence is not nonexistence.

By the nature of the problem, both branches cannot stand apart in epistemological détente, if they purport to be knowledge. Being axioms, the repudiation of either one has fundamental ramifications for its respective branch, if not its destruction. One side must be true, but which one?

I'd say that philosophy is fundamentally correct (on the Aristotelian-Objectivist basis), while mathematics is "operationally" valid, but its principles (some of them) have been misinterpreted ontologically.

For instance, when you write out the equation 1 + 0 = 1, does that represent the ~addition~ of 0 to 1? That is the standard interpretation. But how can you add nothing to something? Actually, what you are doing is ~not~ adding ~anything~ to something. In other words, the notation ~really~ symbolizes that you ~are not~ adding anything to 1, not that you ~are~ adding 0 to it. The zero means the operation of adding IS NOT PERFORMED.

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!

A similar thing happens in regard to the "zero power," which is always 1 for any real number (except 0?). E.g., 5 to the zero power is 1, 100 to the zero power is 1, etc. Some people are mystified by this, wondering what it means ontologically. Well, its meaning is in the operation that ~is not~ being performed. (In that respect, a zero power is like a zero addend, as above.) See, the key to grasping what is going on with powers is to realize that the factor 1 is always the base to which the power multiplication is applied or not. E.g., 5 squared (i.e., to the second power) ~actually~ means the number one multiplied by the number 5 two times. 5 to the 3rd power means 1 multiplied by the number 5 3 times. 5 to the 0th power means 1 ~not~ multiplied by the number 5 ~any~ times. The zero means the operation of power multiplication on the factor 1 IS NOT PERFORMED. That is why any number to the zero power is always 1. Not because 5 is ~taken~ times ~itself~ zero times, but because 1 is ~not taken~ times 5 ~any~ times.

This reminds me of the old saw about evidence and justification: absence of evidence is NOT evidence of absence. Nothing is not something. In other words, I think Thom is onto something -- and it's not nothing!

The complement of a set is always understood in regard to some larger set, of which they both are subsets and together in relation to which they non-overlappingly comprise the total membership of the larger set. For instance, in regard to a set of six apples, the set comprised by two of those apples is the complement of the set comprised by the the other four of those apples. There is no problem understanding the meaning of "complement" here, nor of the union of a set and its complement in relation to a larger whole. But it is the fact a set and its complement are both subsets of a ~larger~ whole that rules out considering the "empty" set as the complement of the larger whole. To complement means to add to something in order to make a whole. But the six apples ~already~ are a whole six apples, and you cannot meaningfully add ~zero~ apples in order to make the six apples a whole, because they already ~are~ a whole. Zero apples is (are?) NO PART of six apples, and thus NO SUBSET of six apples.

You cannot speak of the ~union~ of something and nothing, so you cannot speak of the union of zero apples and six apples, any more than you can speak of adding 0 and 6. What you are doing is ~not~ adding ~anything~ to 6, because the 6 is already 6. You are ~not~ finding the union of ~anything~ with the set of six apples, because the set of six apples is already a set of six apples. The notation expressing a union of the null set with another set simply means that the operation of set union IS NOT PERFORMED.

That, IMHO, is the ontological meaning of operations conventionally taken to involve zero or null sets. The operations are actually being specified as not having been performed! In this way, a number of mathematical and logical expressions conventionally regarded as arbitrary premises in order to build a system of inference can instead be seen as specifying that zero and null sets are operation-blockers.

In the same way, the concept of "nothing" is also an operation-blocker. Nothing does not exist. You can't get inside it, outside of it, around it, underneath it, period. All that exists is Existence, and Existence is ~all~ that exists. It is a complete sum total. It cannot have a complement, because there isn't anything you can add to it. And you especially can't add Nothing to it, because Nothing isn't anything. So, Existence as the set or sum total of everything that exists cannot have a complement. Existence as a sum total ~must~ exist. It cannot go out of existence, so it has no "opposite" either--no whatever-it-is that there would be if Existence stopped existing (because it can't).

"Nothing" or "non-existence" only has meaning in relation to some specific thing that might or might not exist, but even then, it's an operation-blocker. If you look into a room that contains a table and chair, and someone asks you what you see, your perceptual mechanism finds the two objects to lock onto, and you report, "I see a table and chair." But if you look into an empty room, and someone asks you what you see, how do you reply? Do you say, "I see nothing there"? Perhaps, but what you are really saying is, "I ~don't see~ ~anything~ there." You are not ~seeing~ ~nothing~. You are ~not seeing~ ~anything~ (except a room). The absence of anything in the room is an operation-blocker. There isn't anything for your perceptual mechanism to lock onto (except for the room itself), so your entity-perceiving function is blocked.

So, Thom, I guess I'm on your side on this one. (I know I'm on ~my~ side, anyway. I hope this helps.

For instance, when you write out the equation 1 + 0 = 1, does that represent the ~addition~ of 0 to 1? That is the standard interpretation. But how can you add nothing to something? Actually, what you are doing is ~not~ adding ~anything~ to something. In other words, the notation ~really~ symbolizes that you ~are not~ adding anything to 1, not that you ~are~ adding 0 to it. The zero means the operation of adding IS NOT PERFORMED.

0 is the identity element of an additive group. If you like subtraction (the inverse operation to addition) you will love 0. 0 = a - a (by definition) a*0 = x*(a -a) = a*x - a*x (by distributive law, commutative law and associative law) = 0 (by definition of subtraction). In addition (pardon the pun!) a + 0 = a + (a - a) = 2a - a = (2 -1)*a = 1*a = a. So adding 0 to a is equivalent to adding a to a then subtracting a from the result which leaves you with what you started with. In short adding 0 to a is equivalent to doing something to a that does not change it.

The root of the identity element goes back to the notion of a group or a semi-group with identity (look up on wiki or in any standard textbook on abstract algebra). Ultimately every group is isomorphic to a subgroup of the permutation group on a (non-empty) set of elements (Cayle's Theorem). Now consider that permutation which maps every element onto itself. There, my friend, is your identity element. In the land of computer operations it is the famous "no op." which leaves the data storage of the computer unchanged. In the land of geometrical transformations it is the motion in the space that leaves every point fixed. And so on and so. In the land of arithmetic, if you think of addition as an operation that maps the set of integers into the integers, then adding 0 is the operation that leaves each integer fixed (a "no op.", as it were). The identity element of a group is NOT nothing. It is a very important something.

This perverse interpretation of 0 confuses the uneducated and blinds the eye of philosophers. Now, perhaps, you will know why Karl Friedrich Gauss thought little of the metaphysics practitioners of his time. It is a wonder that the mathematicians of the world are not all bald from tearing their hair out from the balderdash spouted by the philosophers.

This perverse interpretation of 0 confuses the uneducated and blinds the eye of philosophers. Now, perhaps, you will know why Karl Friedrich Gauss thought little of the metaphysics practitioners of his time. It is a wonder that the mathematicians of the world are not all bald from tearing their hair out from the balderdash spouted by the philosophers.

I submit you are unduly critical about the role of zero in math.

Suppose you are given a series of numbers to sum such as 17 - 3 - 8 + 4 - 9 - 1 + 5. Per your interpretation you would be unable to complete the task in the given order because the first 6 addends sum to 0.

That is why any number to the zero power is always 1. Not because 5 is ~taken~ times ~itself~ zero times, but because 1 is ~not taken~ times 5 ~any~ times.

No, the reason x^0=1 is that (x^n)/(x^n) = x^0 obviously equals 1.

In short the role of 0 is to make the math complete, not to invite bizarre metaphysical interpretations.

It's similar for the null set.

That, IMHO, is the ontological meaning of operations conventionally taken to involve zero or null sets. The operations are actually being specified as not having been performed!

If you were asked to find the intersection of {1, 3, 5} and {2, 4, 6}, would you:

(1) perform and answer nothing, i.e. the null set, and then insist you didn't do anything, or

(2) say you can't answer because answering would imply some bizarre metaphysical interpretations?

In short the role of 0 is to make the math complete, not to invite bizarre metaphysical interpretations.

Bizzare metaphysical interpretations is what the metaphysicists produce. It is well that mathematics has long been snatched away from the clutches of philosophy. That is one of the better legacies of the 18th and 19th century mathematicians.

Bizzare metaphysical interpretations is what the metaphysicists produce. It is well that mathematics has long been snatched away from the clutches of philosophy. That is one of the better legacies of the 18th and 19th century mathematicians.

Whatever nothing isn't. One thing for sure. The empty set is something. Among other things it is the intersection of the set of odd integers with the set of even integers. It is also the intersection of the set of positive perfect square integers with the set of positive perfect cube integers and an instance of many other things. The notion that the empty set is nothing is a notion only a philosopher could come up with.

"While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians.

"The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something. This can be a stumbling block. If so, the following homely figure of speech may be helpful. Think of a set as a bag, and its members as being the contents of the bag. An empty bag undoubtedly still exists."

Do you mean the contrary or do you mean the negation. The negation of Nothing is Something. Any particular in the world is something. In set theory we talk about sets and their complements. Let A be a set. Then A-comp is the set of elements not in A. If A is empty then any element is a member of A-comp. The general practice when using sets is to identify some non-empty set as the universal set. Let is call it U. Then the complements are taken relative to U. Let A be a subset of U. The A-comp relative to U is usually written U - A which is the set of elements in U not in A. The (relative) complement of 0 is U. The relative complement of U ( U - U) is 0. So when we are dealing sets of real numbers, the universal set is taken to be the set of all real numbers. That pins the meanings of the sets down better.

...mathematics makes a claim that contradicts philosophy. More specifically, a branch of math, set theory--which is the basis of mathematical logic and many other offshoots dependent on it--this branch has an axiom stating that the complement of the empty set is the universal set. That is, the opposite of nothing is everything. Its corollary is that the complement of everything is nothing.

Now this mathematical axiom contradicts philosophy, namely, the Objectivist philosophy. In particular, it contradicts a basic philosophical axiom, the axiom of existence: that existence exists--and its corollary: that only existence exists. (For the full context, see 58-60.) It suffices to say informally, the opposite of existence is not nonexistence.

By the nature of the problem, both branches cannot stand apart in epistemological détente, if they purport to be knowledge. Being axioms, the repudiation of either one has fundamental ramifications for its respective branch, if not its destruction. One side must be true, but which one?

I'd say that philosophy is fundamentally correct (on the Aristotelian-Objectivist basis), while mathematics is "operationally" valid, but its principles (some of them) have been misinterpreted ontologically.

For instance, [...]

[...]

This reminds me of the old saw about evidence and justification: absence of evidence is NOT evidence of absence. Nothing is not something. In other words, I think Thom is onto something -- and it's not nothing!

The complement of a set is always understood in regard to some larger set, of which they both are subsets and together in relation to which they non-overlappingly comprise the total membership of the larger set. For instance, [...]

[...]

That, IMHO, is the ontological meaning of operations conventionally taken to involve zero or null sets. The operations are actually being specified as not having been performed! In this way, a number of mathematical and logical expressions conventionally regarded as arbitrary premises in order to build [aka to make math complete (per Merlin's)] a system of inference can instead be seen as specifying that zero and null sets are operation-blockers.

In the same way, the concept of "nothing" is also an operation-blocker. Nothing does not exist. You can't get inside it, outside of it, around it, underneath it, period. All that exists is Existence, and Existence is ~all~ that exists. It is a complete sum total. It cannot have a complement, because there isn't anything you can add to it. And you especially can't add Nothing to it, because Nothing isn't anything. So, Existence as the set or sum total of everything that exists cannot have a complement. Existence as a sum total ~must~ exist. It cannot go out of existence, so it has no "opposite" either--no whatever-it-is that there would be if Existence stopped existing (because it can't).

"Nothing" or "non-existence" only has meaning in relation to some specific thing that might or might not exist, but even then, it's an operation-blocker. If you look into a room that contains a table and chair, and someone asks you what you see, your perceptual mechanism finds the two objects to lock onto, and you report, "I see a table and chair." But if you look into an empty room, and someone asks you what you see, how do you reply? Do you say, "I see nothing there"? Perhaps, but what you are really saying is, "I ~don't see~ ~anything~ there." You are not ~seeing~ ~nothing~. You are ~not seeing~ ~anything~ (except a room). The absence of anything in the room is an operation-blocker. There isn't anything for your perceptual mechanism to lock onto (except for the room itself), so your entity-perceiving function is blocked.

So, Thom, I guess I'm on your side on this one. (I know I'm on ~my~ side, anyway. I hope this helps.

REB

Roger, this is a major philosophical insight about arithmetic and set theory! There is no humility here, no "IMHO" about it, kudos for having the courage to express it!

I think you are right. Your presentation of the misinterpretations of the philosophy behind mathematics in this context deserves serious study. If we take the Aristotelian-Objectivist approach as the correct line of sight, then Objectivists have much work ahead of us to turn the ship of Math toward the right direction.

Let me distill three of your points that are directly pertinent to the topic. From the excerpt, I think you are saying :

That (1) the conventional interpretation of the set operator "complement" is mistaken in applying this operation to a content that is either the "empty" set or the set of everything. Other set operators--"union," "intersection," "product," "difference," etc.--may be similarly misinterpreted with respect to {} and U, but this is irrelevant in the current scope.

That (2) the proper context for performing the "complement" operation is on a set within U (absolute complement) or within a "larger" set (relative complement), and the result of which must be within the larger set, complementing that which is being complemented.

That (3) the processor must take account of both the action and the content for the result to be valid.

To all of these, I agree. Why?

Your first and second points translate "complement" from the cognitive process of isolation and classification. I can isolate something in a field of perception and make that the focus, and I can leave the rest, its complement, in the background. At the conceptual level, if I classify drinking cups into cup-sizes small, medium, and large; and if I prefer the mediums, then I will have opposed them against the sizes small and large as one aggregate subclass.

Now, with cups, as an entire class, I can always oppose it within a larger classification of objects, such as drinking vessels, from which an opposite aggregate subclass of noncups would include glasses, cans, and bottles. I can do the same cognitive operation with drinking vessels at the next level of classification, and so on, until I come to that axiomatic class of everything. But here, the context of a "larger" class for my cognitive operation has reached its maximum. Therefore, I'm done. I have no context to try to isolate, to subclass, i.e., to oppose, the class of everything. At this level, there can be no valid result from the operation. The context "blocks" this action. It is not that the opposite of everything is nothing, but that there can be no opposite of everything.

At the fundamental level, the nature of consciousness is such that every conscious state entails some content independent of it. Without a content, there is no conscious awareness. To be conscious is to be conscious of something. A conscious awareness of nothing is a contradiction in terms. (And a self-conscious awareness of nothing but itself is a contradiction in terms.) That said, a perception without that which is perceived is a contradiction; an isolation or classification without that which is be isolated or classified is inconceivable. Thus, your third point applies here. A "complement" of nothing cannot get off the ground. It is not that the opposite of nothing is anything or even everything, but that there can be no opposite of nothing.

---

[...] it is the fact a set and its complement are both subsets of a ~larger~ whole that rules out considering the "empty" set as the complement of the larger whole. [...]

The same fact implies two further points. (1) Three distinct sets, all nonempty, are involved in the complement operation: the set to be operated on, the context-set for the operation, and the result-set from the operation. And (2) the to-be-operated-on set is always smaller--not smaller or equal to, but smaller--than the context-set. Adding your above, (3) the to-be-operated-on set cannot be the "empty" set.

If mathematics and epistemology ever need to concur, a point of concurrence must be here. Since epistemology sets criteria of investigation and criteria of determining truth for all branches of knowledge, with math being one such branch, it is epistemology that arbitrates among branches of knowledge whenever conflicts arise. Thus, just as you don't form a more abstract concept (the larger-scale integration) from a less abstract one plus an alleged other that is vacuous in meaning or referent, so you don't form a new set by taking an old set and a "set" that is empty. A concept is always an integration of at least two or more units which are isolated from other units commensurably measured. This applies to concepts per se. So should it apply to sets per se.

---

Roger,

[...]

That, IMHO, is the ontological meaning of operations conventionally taken to involve zero or null sets. The operations are actually being specified as not having been performed!

If you were asked to find the intersection of {1, 3, 5} and {2, 4, 6}, would you:

(1) perform and answer nothing, i.e. the null set, and then insist you didn't do anything, or

(2) say you can't answer because answering would imply some bizarre metaphysical interpretations?

I would answer Merlin's particular objection about the handling of a set intersection, when based on the presented interpretation, as follows:

There can be no such thing as an "empty" set. A "set" with no element is not a set. Stipulating by definition or by axiom that it be so does not make it so. In what sense then does the notation "{}" or the phrase "the empty set" denote? It denotes nothing more than the absence of everything in whatever context-set one operates on. All sets denote their elements, i.e., their units. When a "set" fails in denotation, it fails to be a set. So, "{}" is not a set. It denotes not a presence of elements but an absence of elements. (See ITOE for the full context.)

Now, to the intersection question specifically, the answer is exactly to be expected: {}. But the interpretation is that "{}" is not some set to be plugged into another operation. In effect, "{}" denotes the absence of any set. It denotes that the intersection operation produces an undefined result, requiring the cessation of any subsequent operation.

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22 posts

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## Selene

"...that existence exists--and its corollary: that only existence exists."

or that non-existence does not exist.

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## Ian

How is "nothing," or "nonexistence" distinguished from "does not exist?"

Ian.

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## BaalChatzaf

1. Mathematics is not a science, since it has no empirical content. It is about objects that do not exist in the physical world (e.g. points, lines, numbers with infinite decimal expansions and such like).

2. Mathematics is the logos of abstract systems and structures, some of which can be used in measurement and some of which are not. I have been all through this with MSK

Ba'al Chatzaf

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## tjohnson

I agree with the importance of mathematics in human knowledge but I think it is better characterized as a language of relations,

relations in particular. In mathematics relations are exact, when applied to a field they are only approximate because all factors are not included in science.exact## Link to comment

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## Christopher

There's a conceptual symbol for "nothing" in most every culture and in every language. I can draw the character for "nothing" in Chinese. These symbols exist. A symbol that signifies the absence of everything. You need the language (and Rand needed the language) to denote "nothing" in order to discuss it in OE.

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## tjohnson

Mathematics makes

. Please understand that.no claims about existence whatsoever## Link to comment

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## Alfonso Jones

Thom -

Complement and opposite are not the same thing. Complement is a well-defined notion, and clearly the complement of the universal set is the empty set. This in no way contradicts the fact that existence exists.

If you use words such as "complement" in such a casual way, you will end up in lots of trouble. But to treat complement as an exact synonym for opposite is a mistake.

Bill P

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## thomtg

AuthorBill, thank you for highlighting my semantic bridge from mathematics to everyday philosophy. Yes, my equating of "complement" to "opposite" was deliberate. I wanted to emphasize the connection that mathematics has a role to play in philosophy, and vice versa, for the purpose of living in the world. While I do realize that the term "opposite" is equivocal in the realm of logic to mean either of two incompatible relationships, namely, contrariety and contradiction; I did not want to admix all these terms without the proper context. Rather, I would like for interested readers to understand the issue from the view of mathematics and then to be able to relate it to philosophy.

So, the issue here purports to be a contradiction between two branches of knowledge concerning the opposite of some notion to be some other notion. On the math side, "opposite" means "complement." On the philosophy side, I cited page references for readers to follow up to see which meaning of "opposite" Ayn Rand discusses in the present context.

Ian,

"Nothing," and "nonexistence" are synonymous English words denoting a legitimate concept, which Christopher can also notate in Chinese. What that concept is, is central to the present issue, namely, the issue of the opposite of nothing. That is, while the concept does exist, the issue I am curious about is the nature of its opposite.

The predicate "does not exist" is straightforwardly the denial of existence. What is the opposite of "existence"? This is also central to the issue.

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## Alfonso Jones

Thom -

Boil down what you have said above, and it appears to be that you have chosen to equivocate and then pretend to see a contradiction - which "contradiction" is simply rooted in your equivocation, not in either mathematics or philosophy.

What is your intent in doing this? If I were to describe a contradiction between athletics and clothing because the word "run" is used in one way referring to an athletic activity, and another way to refer to a defect in hosiery, would that be a contradiction? Or just silliness on my part? I submit the latter - and it seems to be exactly the same thing you are doing.

Bill P

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## thomtg

AuthorGood rebuttal, Bill. Now consider another thing I have also said. If you accept the normative premise that your knowledge of reality must be an integrated whole, then my intent becomes apparent. I am not denying that ideas in respective systems of thought are already self-consistent (I am not asserting it either), but I deny that systems of thought can be separately free floating. Does the notion of "complement" in set theory have any use to the task of living on earth? What are its links to the rest of our knowledge? What are its links to reality? We can and should apply these same questions to our notions in philosophy. I invite you to proceed if you accept the normative premise. And if you then don't see a contradiction, I definitely want to know about it.

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## Alfonso Jones

Thom -

I think that if you want to get a serious and thoughtful response from people it would be best to not engage in equivocation to get their attention. Was that what you were attempting? I still don't understand what you were getting at with your original equivocation re "complement." Can you state specifically what you really meant?

Bill P

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## thomtg

AuthorBill, here below are my integrations. The meaning of the latter will be obscured unless you go to the pages cited in ITOE.

From the branch of mathematics, the opposite of nothing is everything, and the opposite of everything is nothing.

From the branch of philosophy, neither can there be an opposite of nothing, nor can there be an opposite of everything.

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## Roger Bissell

I'd say that philosophy is fundamentally correct (on the Aristotelian-Objectivist basis), while mathematics is "operationally" valid, but its principles (some of them) have been misinterpreted ontologically.

For instance, when you write out the equation 1 + 0 = 1, does that represent the ~addition~ of 0 to 1? That is the standard interpretation. But how can you add nothing to something? Actually, what you are doing is ~not~ adding ~anything~ to something. In other words, the notation ~really~ symbolizes that you ~are not~ adding anything to 1, not that you ~are~ adding 0 to it. The zero means the operation of adding IS NOT PERFORMED.

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!

A similar thing happens in regard to the "zero power," which is always 1 for any real number (except 0?). E.g., 5 to the zero power is 1, 100 to the zero power is 1, etc. Some people are mystified by this, wondering what it means ontologically. Well, its meaning is in the operation that ~is not~ being performed. (In that respect, a zero power is like a zero addend, as above.) See, the key to grasping what is going on with powers is to realize that the factor 1 is always the base to which the power multiplication is applied or not. E.g., 5 squared (i.e., to the second power) ~actually~ means the number one multiplied by the number 5 two times. 5 to the 3rd power means 1 multiplied by the number 5 3 times. 5 to the 0th power means 1 ~not~ multiplied by the number 5 ~any~ times. The zero means the operation of power multiplication on the factor 1 IS NOT PERFORMED. That is why any number to the zero power is always 1. Not because 5 is ~taken~ times ~itself~ zero times, but because 1 is ~not taken~ times 5 ~any~ times.

This reminds me of the old saw about evidence and justification: absence of evidence is NOT evidence of absence. Nothing is not something. In other words, I think Thom is onto something -- and it's not nothing!

The complement of a set is always understood in regard to some larger set, of which they both are subsets and together in relation to which they non-overlappingly comprise the total membership of the larger set. For instance, in regard to a set of six apples, the set comprised by two of those apples is the complement of the set comprised by the the other four of those apples. There is no problem understanding the meaning of "complement" here, nor of the union of a set and its complement in relation to a larger whole. But it is the fact a set and its complement are both subsets of a ~larger~ whole that rules out considering the "empty" set as the complement of the larger whole. To complement means to add to something in order to make a whole. But the six apples ~already~ are a whole six apples, and you cannot meaningfully add ~zero~ apples in order to make the six apples a whole, because they already ~are~ a whole. Zero apples is (are?) NO PART of six apples, and thus NO SUBSET of six apples.

You cannot speak of the ~union~ of something and nothing, so you cannot speak of the union of zero apples and six apples, any more than you can speak of adding 0 and 6. What you are doing is ~not~ adding ~anything~ to 6, because the 6 is already 6. You are ~not~ finding the union of ~anything~ with the set of six apples, because the set of six apples is already a set of six apples. The notation expressing a union of the null set with another set simply means that the operation of set union IS NOT PERFORMED.

That, IMHO, is the ontological meaning of operations conventionally taken to involve zero or null sets. The operations are actually being specified as not having been performed! In this way, a number of mathematical and logical expressions conventionally regarded as arbitrary premises in order to build a system of inference can instead be seen as specifying that zero and null sets are operation-blockers.

In the same way, the concept of "nothing" is also an operation-blocker. Nothing does not exist. You can't get inside it, outside of it, around it, underneath it, period. All that exists is Existence, and Existence is ~all~ that exists. It is a complete sum total. It cannot have a complement, because there isn't anything you can add to it. And you especially can't add Nothing to it, because Nothing isn't anything. So, Existence as the set or sum total of everything that exists cannot have a complement. Existence as a sum total ~must~ exist. It cannot go out of existence, so it has no "opposite" either--no whatever-it-is that there would be if Existence stopped existing (because it can't).

"Nothing" or "non-existence" only has meaning in relation to some specific thing that might or might not exist, but even then, it's an operation-blocker. If you look into a room that contains a table and chair, and someone asks you what you see, your perceptual mechanism finds the two objects to lock onto, and you report, "I see a table and chair." But if you look into an empty room, and someone asks you what you see, how do you reply? Do you say, "I see nothing there"? Perhaps, but what you are really saying is, "I ~don't see~ ~anything~ there." You are not ~seeing~ ~nothing~. You are ~not seeing~ ~anything~ (except a room). The absence of anything in the room is an operation-blocker. There isn't anything for your perceptual mechanism to lock onto (except for the room itself), so your entity-perceiving function is blocked.

So, Thom, I guess I'm on your side on this one. (I know I'm on ~my~ side, anyway. I hope this helps.

REB

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## BaalChatzaf

0 is the identity element of an additive group. If you like subtraction (the inverse operation to addition) you will

love0. 0 = a - a (by definition) a*0 = x*(a -a) = a*x - a*x (by distributive law, commutative law and associative law) = 0 (by definition of subtraction). In addition (pardon the pun!) a + 0 = a + (a - a) = 2a - a = (2 -1)*a = 1*a = a. So adding 0 to a is equivalent to adding a to a then subtracting a from the result which leaves you with what you started with. In short adding 0 to a is equivalent to doingsomethingto a that does not change it.The root of the identity element goes back to the notion of a group or a semi-group with identity (look up on wiki or in any standard textbook on abstract algebra). Ultimately every group is isomorphic to a subgroup of the permutation group on a (non-empty) set of elements (Cayle's Theorem). Now consider that permutation which maps every element onto itself. There, my friend, is your identity element. In the land of computer operations it is the famous "no op." which leaves the data storage of the computer unchanged. In the land of geometrical transformations it is the motion in the space that leaves every point fixed. And so on and so. In the land of arithmetic, if you think of addition as an operation that maps the set of integers into the integers, then adding 0 is the operation that leaves each integer fixed (a "no op.", as it were). The identity element of a group is NOT nothing. It is a very important

something.This perverse interpretation of 0 confuses the uneducated and blinds the eye of philosophers. Now, perhaps, you will know why Karl Friedrich Gauss thought little of the metaphysics practitioners of his time. It is a wonder that the mathematicians of the world are not all bald from tearing their hair out from the balderdash spouted by the philosophers.

Ba'al Chatzaf

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## tjohnson

Hear! Hear!

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## merjet

Roger,

I submit you are unduly critical about the role of zero in math.

Suppose you are given a series of numbers to sum such as 17 - 3 - 8 + 4 - 9 - 1 + 5. Per your interpretation you would be unable to complete the task in the given order because the first 6 addends sum to 0.

No, the reason x^0=1 is that (x^n)/(x^n) = x^0 obviously equals 1.

In short the role of 0 is to make the math

complete, not to invite bizarre metaphysical interpretations.It's similar for the null set.

If you were asked to find the intersection of {1, 3, 5} and {2, 4, 6}, would you:

(1) perform and answer nothing, i.e. the null set, and then insist you didn't

doanything, or(2) say you can't answer because answering would imply some bizarre metaphysical interpretations?

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## BaalChatzaf

Bizzare metaphysical interpretations is what the metaphysicists produce. It is well that mathematics has long been snatched away from the clutches of philosophy. That is one of the better legacies of the 18th and 19th century mathematicians.

Ba'al Chatzaf

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## Dragonfly

Hear! Hear!

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## Selene

"Hear! Hear!"

Ver? Ver?

quoting Kenneth Mars in Springtime For Hitler!

lol

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## bradbradallen

Simply put, the opposite of nothing is anything.

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## BaalChatzaf

Whatever nothing isn't. One thing for sure. The empty set is something. Among other things it is the intersection of the set of odd integers with the set of even integers. It is also the intersection of the set of positive perfect square integers with the set of positive perfect cube integers and an instance of many other things. The notion that the empty set is nothing is a notion only a philosopher could come up with.

Ba'al Chatzaf

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## merjet

"While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians.

"The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something. This can be a stumbling block. If so, the following homely figure of speech may be helpful. Think of a set as a bag, and its members as being the contents of the bag. An empty bag undoubtedly still exists."

Source: http://en.wikipedia.org/wiki/Empty_set

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## BaalChatzaf

Do you mean the contrary or do you mean the negation. The negation of Nothing is Something. Any particular in the world is something. In set theory we talk about sets and their complements. Let A be a set. Then A-comp is the set of elements not in A. If A is empty then any element is a member of A-comp. The general practice when using sets is to identify some non-empty set as the universal set. Let is call it U. Then the complements are taken relative to U. Let A be a subset of U. The A-comp relative to U is usually written U - A which is the set of elements in U not in A. The (relative) complement of 0 is U. The relative complement of U ( U - U) is 0. So when we are dealing sets of real numbers, the universal set is taken to be the set of all real numbers. That pins the meanings of the sets down better.

Have a look here for some background:

http://www.britannica.com/EBchecked/topic/...9/universal-set

Not all set theories have universal sets. For example Zormelo-Frankel set theory does not, but set theory based on Quine's New Foundations does.

You also might want to look at this:

http://en.wikibooks.org/wiki/Discrete_mathematics/Set_theory

and

http://en.wikipedia.org/wiki/Set_theory

Ba'al Chatzaf

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## thomtg

AuthorRoger, this is a major philosophical insight about arithmetic and set theory! There is no humility here, no "IMHO" about it, kudos for having the courage to express it!

I think you are right. Your presentation of the misinterpretations of the philosophy behind mathematics in this context deserves serious study. If we take the Aristotelian-Objectivist approach as the correct line of sight, then Objectivists have much work ahead of us to turn the ship of Math toward the right direction.

Let me distill three of your points that are directly pertinent to the topic. From the excerpt, I think you are saying :

That (1) the conventional interpretation of the set operator "complement" is mistaken in applying this operation to a

contentthat is either the "empty" set or the set of everything. Other set operators--"union," "intersection," "product," "difference," etc.--may be similarly misinterpreted with respect to {} andU, but this is irrelevant in the current scope.That (2) the proper context for performing the "complement" operation is on a set within

U(absolute complement) or within a "larger" set (relative complement), and the result of which must be within the larger set, complementing that which is being complemented.That (3) the processor must take account of both the action and the content for the result to be valid.

To all of these, I agree. Why?

Your first and second points translate "complement" from the cognitive process of isolation and classification. I can isolate something in a field of perception and make that the focus, and I can leave the rest, its complement, in the background. At the conceptual level, if I classify drinking cups into cup-sizes small, medium, and large; and if I prefer the mediums, then I will have opposed them against the sizes small and large as one aggregate subclass.

Now, with cups, as an entire class, I can always oppose it within a larger classification of objects, such as drinking vessels, from which an opposite aggregate subclass of noncups would include glasses, cans, and bottles. I can do the same cognitive operation with drinking vessels at the next level of classification, and so on, until I come to that axiomatic class of everything. But here, the context of a "larger" class for my cognitive operation has reached its maximum. Therefore, I'm done. I have no context to try to isolate, to subclass, i.e., to oppose, the class of everything. At this level, there can be no valid result from the operation. The context "blocks" this action.

It is not that the opposite of everything is nothing, but that there can be no opposite of everything.At the fundamental level, the nature of consciousness is such that every conscious state entails some content independent of it. Without a content, there is no conscious awareness. To be conscious is to be conscious of something. A conscious awareness of nothing is a contradiction in terms. (And a self-conscious awareness of nothing but itself is a contradiction in terms.) That said, a perception without that which is perceived is a contradiction; an isolation or classification without that which is be isolated or classified is inconceivable. Thus, your third point applies here. A "complement" of nothing cannot get off the ground.

It is not that the opposite of nothing is anything or even everything, but that there can be no opposite of nothing.---

The same fact implies two further points. (1) Three distinct sets, all nonempty, are involved in the complement operation: the set to be operated on, the context-set for the operation, and the result-set from the operation. And (2) the to-be-operated-on set is always smaller--not smaller or equal to, but smaller--than the context-set. Adding your above, (3) the to-be-operated-on set cannot be the "empty" set.

If mathematics and epistemology ever need to concur, a point of concurrence must be here. Since epistemology sets criteria of investigation and criteria of determining truth for all branches of knowledge, with math being one such branch, it is epistemology that arbitrates among branches of knowledge whenever conflicts arise. Thus, just as you don't form a more abstract concept (the larger-scale integration) from a less abstract one plus an alleged other that is vacuous in meaning or referent, so you don't form a new set by taking an old set and a "set" that is empty. A concept is always an integration of at least two or more units which are isolated from other units commensurably measured. This applies to concepts per se. So should it apply to sets per se.

---

I would answer Merlin's particular objection about the handling of a set intersection, when based on the presented interpretation, as follows:

There can be no such thing as an "empty" set. A "set" with no element is not a set. Stipulating by definition or by axiom that it be so does not make it so. In what sense then does the notation "{}" or the phrase "the empty set" denote? It denotes nothing more than the absence of everything in whatever context-set one operates on. All sets denote their elements, i.e., their units. When a "set" fails in denotation, it fails to be a set. So, "{}" is not a set. It denotes not a presence of elements but an absence of elements. (See ITOE for the full context.)

Now, to the intersection question specifically, the answer is exactly to be expected: {}. But the interpretation is that "{}" is not some set to be plugged into another operation. In effect, "{}" denotes the absence of any set. It denotes that the intersection operation produces an undefined result, requiring the cessation of any subsequent operation.

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