The Opposite of Nothing Is/Isn't Everything

[Brant Gaede]

The opposite of nothing is something.

--Brant

[Roger Bissell]

Merlin, a partial reply. (Busy household, dangling projects...)

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.

Merlin, I am not trying to dispense with zero in calculations entirely. I am just saying that it cannot be an ~operator~. It can certainly be ~operated on~.

In the example you gave, after five additions, you arrive at 0 + 5. You can add 5 to 0 perfectly well. You can add five chairs to an empty room. But you cannot add zero chairs to a room with five chairs in it. You cannot add 0 to 5.

0 ~as an operator~ is an operation-blocker.

Suppose you had given me this series to sum: 17 - 3 - 8 + 4 - 9 - 1 + 5 + 0. Per my interpretation, my calculations would end after the first six additions (the last one being where I added 5 to 0 to get 5). What more is there to do, after I have reached 5 after six additions? You cannot add 0 to 5, so any further operation is blocked, and the result you get with six additions is your final result.

Suppose you had given me: 17 - 3 - 8 + 4 - 9 - 1 + 5 + 0 + 6. I can't use the 0 as an operator on the preceding sum, but I ~can~ use the 6, and my final result is 11.

Honestly, if you saw a zero in a string of calculations like these, wouldn't you just ~ignore~ it and deal with what's left? Don't you, in fact, in your actual mental operation, simply pass by the 0 and ~not do anything~ with it? Don't you, in fact, automatically think: don't perform any operations with this? (In subtraction, multiplication, or powers, what you are not doing something to with the zero differs in specifics, but the general principle is the same. You just have to get clear on what it is that you are doing nothing to, especially in powers. You don't understand, yet, the breakthrough nature of what I said about 1 being the base of all power calculations. I'll try again later.)

As an operator in addition, 0 says "don't add anything to the preceding." In the series of numbers you gave me, once you reach zero as a preliminary result, it is ~not~ an operator. It is that which is operated on by the 5, like the empty room to which you add the 5 chairs.

When I have time, I'll comment on the rest of your remarks. But I want you to see that I am ~not~ making "bizarre metaphysical interpretations." I am making an ~observation~ of what we ~in fact~ do mentally, operationally, when we use zero as an operator, whether as an added, subtrahend, multiplier, or power. I use the examples of the empty room and the room with 5 chairs in order to concretize what I am saying about the addition operations that can or cannot be done involving zero. To put it another way, you can do something ~to~ zero, but you can't do anything ~to something else~ with zero.

And Ba'al's remark about doing something to something that leaves it unchanged is about as bizarre a statement as I have ~ever~ heard, metaphysical or otherwise. If you have not changed something, you have not done anything to it! (He wants to say you have done ~nothing~ to it, as though nothing were some special kind of something that has no effect in addition, rather than nothing, period.)

REB

[Roger Bissell]

...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.

Not bad for an amateur, huh. I'm glad you see the merit in my perspective, Thom, and I appreciate your additional comments, linking mathematics, logic, and epistemology (viz., concept-formation).

I first came up with this angle a little over 12 years ago in David Kelley's cyberseminar on propositions. They were wondering what was the ontological interpretation of x to the zero power. I told them it had to do with an operation that was not performed on the unit 1. (If Ba'al agreed with this, he'd still probably want to say: "an operation that performed no times on the unit 1." I have a couple of essays on my web site about zero powers and such, but I won't bother with links at this time. Those interested can prowl around on www.rogerbissell.com and find it easily enough.

Anyway, I'm grateful for this particular discussion, because it has crystallized my thinking more generally about zero as not an operator, but a blocker of operations.

I agree with you that in order to properly orient mathematics (back) to the real world, a lot of re-interpretation is necessary. Some of that re-interpretation has to do with getting clear about exactly what operations we are or are not performing. So, it's not all metaphysical/ontological, but the fact that the operations of mathematics can be performed on things in the real world means that such homely little examples I used such as the empty room vs. the room with 5 chairs can actually shed light on what operations can and cannot be performed mentally, mathematically or logically or otherwise.

REB

[Roger Bissell]

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?

Merlin, I have clarified a couple of posts ago that zero can be a result, and it can be what you start from, but it cannot be an ~operator~. So, ~arriving~ at the null set as the result of intersecting two incompatible sets is not a problem. Also, dumping a non-empty set into the null set is not a problem either, because you are "adding" something to nothing, like adding 5 chairs to an empty room. But it makes no sense to dump the null set into a non-empty set, because you cannot "add" nothing to something, any more than you can add 0 chairs to a room with 5 chairs already in it. (Clearly from this, I reject the notion of commutativity with respect to 0 and the null set, because 0 and the null set cannot be operators; they can only be that which is operated on, or that which results from an operation.)

So, I pick (3): perform and answer "the null set," and acknowledge that ~did~ do something: I attempted to intersect two sets with no members in common, and found that it could not be done, that the result was a set with no members.

REB

[Roger Bissell]

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.

Merlin, I'm sure you are aware that your reasoning breaks down when x = 0. By convention, 0^0 is ~not~ "obviously" equal to 1, but instead is undefined. You must have forgotten to mention it.

You are arguing that: x^0 = x^(1 - 1) = (x^1)/(x^1) = 1. But clearly this cannot be the case when x = 0, because (0^1)/(0^1) = 0/0, which is also undefined.

Now, I disagree with the conventional view. By ~my~ perspective on zero as a non-operator, 0^0 in fact ~does~ = 1.

1. x^n is the unit 1 multiplied n times by the factor x.

2. where n = 0, x^0 is the unit 1 ~not multiplied any times~ by the factor x.

3. where n = 0 and x = 0, 0^0 is the unit 1 ~not multiplied any times~ by the factor 0.

4, i.e., 0^0 = 1.

By my argument x^0 = 1 for any number x ~including~ zero.

By your argument x^0 = 1 for any number x ~except~ zero, where it is undefined.

By yet another argument x^0 = 1 for any number x ~except~ zero, where it is = 0 (because 0 to any other power than 0 is 0, so 0 to the 0 power should be zero, too).

Which is right? I would suggest that ~mine~ is correct, because it gives a completely consistent result for all values of x, and it does not rely on ~undefined results of operations~, merely on an operation ~not being performed~ on the unit 1.

A very interesting web site gives several other arguments why 0^0 should be = 1 (rather than 0 or undefined). Check out Zero to the Zero Power -- Mudd Math Fun Facts You will note that while the author agrees with you, Merlin, that 0^0 is undefined, if it "should" be defined, there are a number of reasons why it "should" be = 1. In particular, take a look at the fourth of the five reasons. It vaguely resembles my approach, though it is very sketchy.

What is important to realize is that ~all~ power operations should be defined in terms of the unit 1 being multiplied (or not!) by some factor x a total of n times. This allows us to see that any number to the zero power is 1, without having to invoke undefined results of operations to explain away the exception (0^0).

REB

[Dragonfly]

Merlin, I have clarified a couple of posts ago that zero can be a result, and it can be what you start from, but it cannot be an ~operator~.

No one claims that zero or the null set (or any other number or set) are operators; operators are things like addition, multiplication, exponentiation etc. Now the additive group of real numbers is isomorphic to the multiplicative group of positive real numbers, where the 0 of the first group is equivalent to the 1 of the second group, being the unit element of the group. Without unit element the whole edifice of group theory (and with it a large part of mathematics) would be destroyed. Well, if that is what philosophy has to contribute to mathematics... thanks, but no thanks!

[tjohnson]

(zero to the zeroth power) itself is undefined. The lack of a well-defined meaning for this quantity follows from the mutually contradictory facts that is always 1, so should equal 1, but is always 0 (for ), so should equal 0. The choice of definition for is usually defined to be indeterminate, although defining allows some formulas to be expressed simply (Knuth 1992; Knuth 1997, p. 57).

http://mathworld.wolfram.com/Power.html

[merjet]

Merlin, I am not trying to dispense with zero in calculations entirely. I am just saying that it cannot be an ~operator~. It can certainly be ~operated on~.

In the example you gave, after five additions, you arrive at 0 + 5. You can add 5 to 0 perfectly well. You can add five chairs to an empty room. But you cannot add zero chairs to a room with five chairs in it. You cannot add 0 to 5.

0 ~as an operator~ is an operation-blocker.

Like Dragonfly I wasn't aware that anyone in this thread has characterized 0 as an operator. I certainly have not and do not. Elsewhere I have seen 0 described as the "identity operator in addition" and 1 as the "identity operator in multiplication", which I take to be "shortened speech." The first is short for something like "when adding 0, the before and after states are identical." The second is short for something like "when multiplying by 1, the before and after states are identical."

I think "adding 0 chairs" is loose or odd speech, equivalent to "adding no chairs", but I am not alarmed by it. Can you "add -2 chairs" to a room? That sounds pretty odd, but you can "remove 2 chairs" from a room having 2 or more.

Honestly, if you saw a zero in a string of calculations like these, wouldn't you just ~ignore~ it and deal with what's left? Don't you, in fact, in your actual mental operation, simply pass by the 0 and ~not do anything~ with it? Don't you, in fact, automatically think: don't perform any operations with this?

I do not ignore zeros, but recognize that adding zero doesn't affect my result. If you were designing the operation of additon for a computer, would you design it to "not operate" when it encounters a zero or handle it in the same manner as any other number?

And Ba'al's remark about doing something to something that leaves it unchanged is about as bizarre a statement as I have ~ever~ heard, metaphysical or otherwise.

Really? Put on a rubber glove and use your finger to barely touch a brick wall. I added the bit about the glove because using a bare finger you would probably say you left some oil.

Merlin, I'm sure you are aware that your reasoning breaks down when x = 0. By convention, 0^0 is ~not~ "obviously" equal to 1, but instead is undefined. You must have forgotten to mention it.

You are arguing that: x^0 = x^(1 - 1) = (x^1)/(x^1) = 1. But clearly this cannot be the case when x = 0, because (0^1)/(0^1) = 0/0, which is also undefined.

Yes, I am well aware of the exception and neglected to mention it.

But I want you to see that I am ~not~ making "bizarre metaphysical interpretations."

Assuming others on this thread treat 0 or the null set as operators strikes me as bizarre.

[Roger Bissell]

Merlin, I have clarified a couple of posts ago that zero can be a result, and it can be what you start from, but it cannot be an ~operator~.

No one claims that zero or the null set (or any other number or set) are operators; operators are things like addition, multiplication, exponentiation etc.

Dragonfly and Merlin, addition, multiplication, etc. are not operatORS, they are operatIONS. An operatION is an ACT. An operatOR is what you perform the operation WITH UPON something else. If I perform the operatION 2 + 3, 3 is what I perform the operation WITH UPON 2.

Now, you may be thinking of the ~symbols~ for those operations, e.g., +, /, -, etc. Those symbols are technically referred to in mathematics and logic as "operators" to indicate the operation to be performed. I think this is confusing, because the symbol doesn't do any operating, just ~indicating~ of an operation. On the other hand, e.g., in 3 + 2, the 2 additively operates on the 3 (courtesy of my adding mental operation). That is why I think it is clearer to refer to the second number in the expression as the "operator."

You may think this is bizarre, non-standard usage, but consider that in multiplication, e.g., 2 x 3, 2 is referred to as the "multiplicand" and 3 is referred to as the "multiplier," THAT WITH WHICH you multiply 2. Similarly, in division, e.g., 8/2, 8 is the "dividend," and 2 is the "divisor." This terminology makes it clear what I am referring to when I use the term "operator" to refer to the second number in an operation.

Now the additive group of real numbers is isomorphic to the multiplicative group of positive real numbers, where the 0 of the first group is equivalent to the 1 of the second group, being the unit element of the group. Without unit element the whole edifice of group theory (and with it a large part of mathematics) would be destroyed. Well, if that is what philosophy has to contribute to mathematics... thanks, but no thanks!

Dragonfly, don't you mean that 1 is the ~identity~ element of the multiplicative group of positive real numbers? 1 is the unit element of ~both~ the additive and multiplicative groups, but you are wanting to focus, are you not, on the element in each group that, when combined with another element in that group, leaves that element unchanged? That would be the ~identity~ element.

Now, in regard to 0 as the identity element of the additive group. I am NOT advocating that we dispense with it, any more than I am advocating that we dispense with 1 as the identity element of the multiplicative group. I am ONLY advocating that, while we can legitimately add (and speak of adding) any real number ~to 0~, we cannot properly add ~0 to~ any real number.

In other words, I recognize 0 + x = x as a valid statement of the additive identity property of 0 -- but NOT x + 0 = 0. [EDIT: As Dragonfly surmises in a subsequent post, I meant to say: x + 0 = x. Sorry!] In other words, I am denying additive commutativity to zero. You can introduce elements to a situation where you have nothing, but you can't introduce nothing to a situation where you have elements. You are not ~introducing (adding) nothing~. You ~aren't introducing~ (adding) ANYTHING!

These seem like simple, sensible modifications of standard practice in order to keep mathematics and ontology aligned with one another, and mathematics and our actual mental operations aligned with one another, rather than treating mathematics like an enormous spider-web of intellectual relations bearing no necessary connection to the real world. ~That~ is what philosophy "contributes to mathematics," not (as you fear) a bizarre, invalid jettisoning of the additive identity element.

REB

[Brant Gaede]

It is precisely because "0" is not an operator that its invention--an act of pure abstraction with no concrete referent--was one of genius ranking right up there with the invention of the wheel. As these simplicities have been discovered and used genius has had to come up with things more complex to match up, like Calculus and human rights or whatever it was that Einstein did. I'd throw in Maxwell too. As I so poorly understand it, Einstein's mind and Maxwell's sort of operated the same creatively.

--Brant

[Roger Bissell]

Merlin, I am not trying to dispense with zero in calculations entirely. I am just saying that it cannot be an ~operator~. It can certainly be ~operated on~.

In the example you gave, after five additions, you arrive at 0 + 5. You can add 5 to 0 perfectly well. You can add five chairs to an empty room. But you cannot add zero chairs to a room with five chairs in it. You cannot add 0 to 5.

0 ~as an operator~ is an operation-blocker.

Like Dragonfly I wasn't aware that anyone in this thread has characterized 0 as an operator. I certainly have not and do not. Elsewhere I have seen 0 described as the "identity operator in addition" and 1 as the "identity operator in multiplication", which I take to be "shortened speech." The first is short for something like "when adding 0, the before and after states are identical." The second is short for something like "when multiplying by 1, the before and after states are identical."

Merlin, wouldn't you then say: "when adding anything OTHER THAN 0, the before and after states ARE NOT identical? And if 0 is considered the "identity operator in addition," why wouldn't ALL OTHER real numbers be considered NON-IDENTITY operators? In other words, why wouldn't ANY number when added (or attempted to be added!) to another number be considered an "operator"?

However, I would only refer to the ~second~ number in each case as the "operator." That means that in 1 + 0, 0 is the identity operator (that blocks the operation of addition, thus leaving the 1 unchanged), while in 0 + 1, not 0 but ~1~ is the operator (viz., a non-identity operator, since it does NOT leave the 0 unchanged).

I think "adding 0 chairs" is loose or odd speech, equivalent to "adding no chairs", but I am not alarmed by it. Can you "add -2 chairs" to a room? That sounds pretty odd, but you can "remove 2 chairs" from a room having 2 or more.

Merelin, we haven't talked much about the ontology of subtraction, or the ontology of adding negative numbers, but Pat Corvini does an excellent job of explaining it, and the mental operations and epistemological relations that go along with it, in one of her ARI lecture courses. I think it's "Two, Three, Four, and All That." (Considering the signal lack of success I've had in digesting previous ARI lecture courses in discussions here, I will forego that and simply give a high recommendation of these lectures to anyone interested.)

But it seems clear to me that "loose or odd speech" is ~exactly~ what "adding 0 chairs" or "adding no chairs" is. You mean precisely to say "not adding ANY chairs." It refers NOT to ~carrying out a process of addition~ but to NOT carrying out such a process.

Honestly, if you saw a zero in a string of calculations like these, wouldn't you just ~ignore~ it and deal with what's left? Don't you, in fact, in your actual mental operation, simply pass by the 0 and ~not do anything~ with it? Don't you, in fact, automatically think: don't perform any operations with this?

I do not ignore zeros, but recognize that adding zero doesn't affect my result. If you were designing the operation of additon for a computer, would you design it to "not operate" when it encounters a zero or handle it in the same manner as any other number?

Depends on where the zero is, Merlin. If it were in the first position -- 0 + x -- I'd design the computer NOT to stop when it reached the zero, because then x would not be introduced. But if it were in the second position -- x + 0 -- I'd design the computer to STOP when it reached the zero and not perform any operation on x.

And Ba'al's remark about doing something to something that leaves it unchanged is about as bizarre a statement as I have ~ever~ heard, metaphysical or otherwise.

Really? Put on a rubber glove and use your finger to barely touch a brick wall. I added the bit about the glove because using a bare finger you would probably say you left some oil.

Tut-tut,Merlin. Absence of (perceivable) evidence is not evidence of absence! The demons of micro-determinism will surely seek their revenge on you if you persist in saying that a interaction of "barely" touching a brick wall with a rubber glove leaves ~either~ the rubber glove or the brick wall unchanged. Arthur Koestler taught us long ago in "The Act of Creation" that--as modern physics has demonstrated/proved--entities are not the static, unchanging things of traditional, quasi-Platonic physics, but instead are seething cauldrons of imperceptible activity, with atoms frequently being exchanged with other entities and the surrounding environment. (And you thought that when you nudged that other car in the parking lot, you didn't "steal" some of his car's paint!

There is NO interaction without physical consequence, however small.

Where there is action, there is change. Where there is interaction, there is SOME change in each of the interacting entities. Where there is no change whatsoever, there has been no action. And when there is no change in interacting entities, there has been no interaction. This is simple metaphysical fact, and the alternative to this (viz., Ba'al's remark) is truly bizarre. An action with no consequence is impossible, and so is an operation with no effect.

REB

[Dragonfly]

Dragonfly and Merlin, addition, multiplication, etc. are not operatORS, they are operatIONS. An operatION is an ACT. An operatOR is what you perform the operation WITH UPON something else. If I perform the operatION 2 + 3, 3 is what I perform the operation WITH UPON 2.

Now, you may be thinking of the ~symbols~ for those operations, e.g., +, /, -, etc. Those symbols are technically referred to in mathematics and logic as "operators" to indicate the operation to be performed. I think this is confusing, because the symbol doesn't do any operating, just ~indicating~ of an operation. On the other hand, e.g., in 3 + 2, the 2 additively operates on the 3 (courtesy of my adding mental operation). That is why I think it is clearer to refer to the second number in the expression as the "operator."

No, a number is NOT an operator, what you call an operator is a number in combination with an operation, the operator is "add 2" and the operand is 3. Or in other words, it is a function f(x) = x + 2.

Dragonfly, don't you mean that 1 is the ~identity~ element of the multiplicative group of positive real numbers? 1 is the unit element of ~both~ the additive and multiplicative groups, but you are wanting to focus, are you not, on the element in each group that, when combined with another element in that group, leaves that element unchanged? That would be the ~identity~ element.

See here : [identity] Also called identity element, unit element, unity. an element in a set such that the element operating on any other element of the set leaves the second element unchanged.

Now, in regard to 0 as the identity element of the additive group. I am NOT advocating that we dispense with it, any more than I am advocating that we dispense with 1 as the identity element of the multiplicative group. I am ONLY advocating that, while we can legitimately add (and speak of adding) any real number ~to 0~, we cannot properly add ~0 to~ any real number.

In other words, I recognize 0 + x = x as a valid statement of the additive identity property of 0 -- but NOT x + 0 = 0.

I suppose you mean "x + 0 = x". You cannot deny the commutative property of 0 in an abelian group as the whole building of mathematics would be destroyed, and that only for some nebulous metaphysics! Moreover, due to the isomorphism with the multiplicative group for positive numbers, this would entail that x * 1 = x would not be a valid statement either.

These seem like simple, sensible modifications of standard practice in order to keep mathematics and ontology aligned with one another, and mathematics and our actual mental operations aligned with one another, rather than treating mathematics like an enormous spider-web of intellectual relations bearing no necessary connection to the real world.

And that is exactly what mathematics is. Ontology has nothing to do with mathematics.

~That~ is what philosophy "contributes to mathematics," not (as you fear) a bizarre, invalid jettisoning of the additive identity element.

But you are jettisoning commutativity of the identity element, which is fatal and makes nonsense of mathematics!

[Laure]

(I have got to jump in here, even though Merlin and Dragonfly are handling things ably themselves, and even though I vowed I wouldn't post here anymore because I really don't like being accused of "not thinking with my own mind.")

Roger, you're very mixed up about mathematics, and I urge you to take a break from posting about it and go back and study it some more.

I just want to make a brief point about your idea that we don't add zero to something, we just "don't do the addition at all." Someone made the point that you wouldn't design a computer that way, and he's absolutely right. If you look in a computer hardware textbook about how an adder is designed, it doesn't have logic on the front-end to say that if one of the inputs is zero, you bypass the adder. You don't check, you just run your two inputs through the adder out pops a result. Nothing wrong with putting the zero through the adder. Or to bring it to a level that the layman would be more familiar with (or not!), if you're writing a computer program to calculate the total amount due on an order, it'd probably be something like this:

total = Price(item) + Tax(item, state_of_residence)

NOT,

tax = Tax(item, state_of_residence)

if (tax > 0)

total = Price(item) + tax

else

total = Price(item)

[tjohnson]

I think Roger's difficulty comes from restricting numbers to something invented to count things, which is true. The problem is that when you investigate deeper you see that what numbers really represent is relations. If you substitute values of x in the equation y=x+1 and make a value table like this;

x | y

1 | 2

2 | 3

3 | 4

etc.

The numbers in the left hand column are related to the ones in the right hand column and the equation y=x+1 expresses the relation. The case of y=x+0 simply results in two identical columns so it expresses an identity relation. One could also write simply y=x instead.

[BaalChatzaf]

And Ba'al's remark about doing something to something that leaves it unchanged is about as bizarre a statement as I have ~ever~ heard, metaphysical or otherwise. If you have not changed something, you have not done anything to it! (He wants to say you have done ~nothing~ to it, as though nothing were some special kind of something that has no effect in addition, rather than nothing, period.)

REB

A mapping abstractly is not a physical action (else, Newton's third law might apply). It is an association . The identity mapping i: A->A is an association that associates each x in A with x. The result of applying this association to an element produces the element itself.

Looking at geometric transformations one might think of motion or moving. What about moving or motion that is less than any measurable. What is its result. It leaves whatever was so moved in the same place or the same state. What is the result of turning or rotating a circle about its center by angle zero. The result is that each point on the circle (meaning the circumference) is left right where it is. The identity transformation is that which does not change what it acts on. In terms of association (which is what a function really is) it associates each point of the figure with itself.

Do you have a problem with that? If so, what is the problem?

In the world of computers the famous "no-op" (null operation or no operation) only advances the program counter to the location of the next instruction but leaves the data store of the computer unchanged. Is the "no-op" an instruction? It surely is, since the machine executes it. Its only side effect is to advance the program counter.

Do you have a problem with that? If so, what is the problem?

[BaalChatzaf]

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.

If the Objectivists get a hold of mathematics (Reality forbid!) they will turn it right into the rocks. Some O'ists got a hold of physics and produced the Theory(???) of Elementary Waves; pure crackpot and crank nonsense. I shudder to think what they will do to math.

Progress in mathematics went at warp speed once it was liberated from metaphysics.

Ba'al Chatzaf

[thomtg]

[...]

These seem like simple, sensible modifications of standard practice in order to keep mathematics and ontology aligned with one another, and mathematics and our actual mental operations aligned with one another, rather than treating mathematics like an enormous spider-web of intellectual relations bearing no necessary connection to the real world.

And that is exactly what mathematics is. Ontology has nothing to do with mathematics.

~That~ is what philosophy "contributes to mathematics," not (as you fear) a bizarre, invalid jettisoning of the additive identity element.

But you are jettisoning commutativity of the identity element, which is fatal and makes nonsense of mathematics!

Dragonfly,

If I understand Roger correctly, it is the fact that "zero" is the absence of any quantity, not that it is the identity element for the addition operation, that blocks the commutativity of the addition operation.

So, I do not think Roger is arguing for jettisoning commutativity of the identity element, say, in multiplication.

On the other hand, given Roger's interpretation, I think he would conclude "0 x X = 0" as a valid statement--but not "X x 0 = 0".

Edited May 12, 2009 by Thom T G

[merjet]

There can be no such thing as an "empty" set. A "set" with no element is not a set.

I'm a little late replying to this because I didn't see it earlier, but I disagree. Firstly, see post #23. Secondly, defining a set requires some criteria for membership, e.g. people in a room less than 6 feet tall. But suppose the criteria were people in a room greater than 10 feet tall. It is a set in the sense of having a criteria for membership. It's simply that none of the people qualify as a member. It's an empty set.

[BaalChatzaf]

[...]

These seem like simple, sensible modifications of standard practice in order to keep mathematics and ontology aligned with one another, and mathematics and our actual mental operations aligned with one another, rather than treating mathematics like an enormous spider-web of intellectual relations bearing no necessary connection to the real world.

And that is exactly what mathematics is. Ontology has nothing to do with mathematics.

~That~ is what philosophy "contributes to mathematics," not (as you fear) a bizarre, invalid jettisoning of the additive identity element.

But you are jettisoning commutativity of the identity element, which is fatal and makes nonsense of mathematics!

Dragonfly,

If I understand Roger correctly, it is the fact that "zero" is the absence of any quantity, not that it is the identity element for the addition operation, that blocks the commutativity of the addition operation.

So, I do not think Roger is arguing for jettisoning commutativity of the identity element, say, in multiplication.

In the set of integers, rationals, reals and complex numbers both addition and multiplication are commutative.

Ba'al Chatzaf

[Roger Bissell]

Dragonfly and Merlin, addition, multiplication, etc. are not operatORS, they are operatIONS. An operatION is an ACT. An operatOR is what you perform the operation WITH UPON something else. If I perform the operatION 2 + 3, 3 is what I perform the operation WITH UPON 2.

Now, you may be thinking of the ~symbols~ for those operations, e.g., +, /, -, etc. Those symbols are technically referred to in mathematics and logic as "operators" to indicate the operation to be performed. I think this is confusing, because the symbol doesn't do any operating, just ~indicating~ of an operation. On the other hand, e.g., in 3 + 2, the 2 additively operates on the 3 (courtesy of my adding mental operation). That is why I think it is clearer to refer to the second number in the expression as the "operator."

No, a number is NOT an operator, what you call an operator is a number in combination with an operation, the operator is "add 2" and the operand is 3. Or in other words, it is a function f(x) = x + 2.

So, Dragonfly, in 2 x (times) 3, the operand is 2, and the operator is "multiply by 3"?? How do you square this with the terminology we all learned in grade school, that the multiplicand is 2, and the multiplier is 3? Or are you saying that all of our textbooks and teachers were wrong, and that the multiplier was actually "multiply by 3"? You're not making a lot of sense here -- or perhaps it is higher math that is not consistent with arithmetic terminology.....

OK, I did a Google search, and the second hit was the wikipedia entry on "operator," and it defines it as "a function which operates on or modies another function." I am using it in an analogous sense, as "a number which operates on or modifies another number."

The article goes on, citing Oliver Heaviside, to say: "Often, an "operator" is a function which acts on functions to produce other functions." Again, analogously, I am using it to mean: "a number which acts on numbers to produce other numbers." It is by this definition that I hold that 0 is not an additive operator, whereas all other real numbers are.

Dragonfly, don't you mean that 1 is the ~identity~ element of the multiplicative group of positive real numbers? 1 is the unit element of ~both~ the additive and multiplicative groups, but you are wanting to focus, are you not, on the element in each group that, when combined with another element in that group, leaves that element unchanged? That would be the ~identity~ element.

See here : [identity] Also called identity element, unit element, unity. an element in a set such that the element operating on any other element of the set leaves the second element unchanged.

~Of course~ the additive identity element, zero, "leaves the second element unchanged," Dragonfly! It ~must~, because it is not capable of doing anything to the second element! I.e., we are not capable of doing anything with zero to the second element. (And by "second element," we simply mean it here in the same sense as "any other element." We are actually talking about the ~first~ number in the expression x + 0.)

Now, in regard to 0 as the identity element of the additive group. I am NOT advocating that we dispense with it, any more than I am advocating that we dispense with 1 as the identity element of the multiplicative group. I am ONLY advocating that, while we can legitimately add (and speak of adding) any real number ~to 0~, we cannot properly add ~0 to~ any real number.

In other words, I recognize 0 + x = x as a valid statement of the additive identity property of 0 -- but NOT x + 0 = 0.

I suppose you mean "x + 0 = x". You cannot deny the commutative property of 0 in an abelian group as the whole building of mathematics would be destroyed, and that only for some nebulous metaphysics! Moreover, due to the isomorphism with the multiplicative group for positive numbers, this would entail that x * 1 = x would not be a valid statement either....you are jettisoning commutativity of the identity element, which is fatal and makes nonsense of mathematics!

1. Thom is correct. I do not propose to deny the commutative property of the ~multiplicative~ identity, 1.

2. I'm sorry you regard metaphysics -- or at least the metaphysics of number -- as nebulous. I don't see it that way at all, any more than I regard the metaphysics of perception as "nebulous." When I perceive that the number of chairs in a room is 5, and then, continuing to steadily perceive the room, I later note that that number has not changed, I do not infer...somehow!...that "zero chairs have been added." Instead, I simply conclude that no chairs have been added to the room. This does not seem to present a difficulty, either ontological or mathematical, does it?

3. Your draconian assessment sounds disturbingly like Global Warming Syndrome to me, Dragonfly. What catastrophic result, what mathematical meltdown, would occur if the "commutative property of 0 in an abelian group" is denied? Perhaps a concrete example of the dire consequences of my position would help.

REB

[Laure]

I don't know what silly things some Objectivists may have been expounding in mathematics, but I didn't read anything in ITOE that contradicted what I learned in math. I don't know where Roger gets this stuff about numbers being operators! "Plus" is an operator. A number is an operand.

Example in pseudocode:

Plus(a,b )

{

return a+b

}

Plus3(a)

{

return a+3

}

In the second one, 3 is a constant, or you can think of the function as being y = x + 3. The operator is "plus" and it is acting on the operands x and 3.

Roger, you seem to be saying that 0 + x does not equal x + 0. If it doesn't, then what IS x + 0? If you're chugging along trying to solve an equation, and you get it down to:

x = 12 + 14 - 14 ...

x = 12 + 0 ...

Is your next step "Ah Ah Ah, x is undefined because you cannot add zero to anything"?

Or do you say x = 12?

[thomtg]

There can be no such thing as an "empty" set. A "set" with no element is not a set.

I'm a little late replying to this because I didn't see it earlier, but I disagree. Firstly, see post #23. Secondly, defining a set requires some criteria for membership, e.g. people in a room less than 6 feet tall. But suppose the criteria were people in a room greater than 10 feet tall. It is a set in the sense of having a criteria for membership. It's simply that none of the people qualify as a member. It's an empty set.

Merlin, though I saw your Post #23, and had assumed your endorsement of the quoted passages, I did not include it in the discussion on Post #25 which you excerpted above. But since you now bring it up explicitly, I have to ask you whether you are endorsing the dropping of context in the definition of "empty set" in that cited article. My context for claiming the absence of an "empty" set is grounded in the relational aspect of a concept and its referents from the perspective of the human consciousness that isolates the latter to form the former.

Tell me if there can be a concept without the units from which their integration results into a concept with an affixed word. If you appeal to the fact that there are plenty of people who use legitimate, everyday words without knowing their meanings, or referents, I will grant you that. But then those words as they use them are not concepts for them. They are uttering sounds and writing scribbles, not words, and not concepts.

The relation between a concept and its referents cannot be severed. The identities of the relata depend on the existence of the relation. We are here discussing the ontology of the primacy of entities with respect to relations and their attributes. Don't you tell me that it doesn't apply to math, if it is the math needed for living on earth. One can no more say that one is a parent without having had at least one child, and one can no more claim to be a straight husband without the correlative existence of a wife, than one can somehow claim the existence of a set without the correlative existence of its elements.

A correction to mathematics from the Aristotelian-Objectivist perspective is exactly what is desperately needed today to clear up the widespread errors in all the sciences that depend on standards of measurement. Invalidating the "empty" set is only the beginning.

Now, to address your second point,

[Whoops, gotta go. I'll finish this later.]

[ --- --- resumption -- I'll have you know that I'm missing American Idol (live) for this. (But this is worth it.) --- --- ]

Now, to address your second point, you are wondering whether it is possible to form a set with criteria so strict that no element qualifies for membership.

Let me translate this problem into epistemological terms. what is the cognitive problem to be solved here for a volitional consciousness? It is the problem of identifying a qualified instance of a concept. "Qualified instance" of a concept, as defined by Ayn Rand (ITOE 23) is a category of existents identified by means of integrating known concepts with more intensive knowledge. It is a subdivision of a concept on the basis of a fixed range of variations within the measurements specified that distinguishes the units to be integrated from other existents.

If it is the volitional goal of a person to isolate and integrate many concretes (mental or existential) into one unit for further integrations and identifications, i.e., the goal of unit economy, then an ad hoc qualified instance is the perfect unit to identify some aspects of reality, and a descriptive phrase from known words, specifying the criteria, is the perfect linguistic vehicle to convey this ad hoc unit to consciousness for thought and speech.

But notice that, with any unit of integration, whether it be an ad hoc qualified instance or a permanent concept, the epistemological requirement for its identification is still axiomatically operant; namely, for every action of consciousness there must be a content.

For a qualified instance to be valid, therefore, the units must exist for abstraction. Said another way, the qualified instance must be reducible down to perceptual reality, if it is to be valid and not a floating abstraction.

Of course, a volitional consciousness doesn't have to choose to do this. And even if it does so choose, unlike automatic perception, the process of forming a qualified instance from intensive knowledge can still be error prone. These are the reasons why a rational being needs to have objectivity in the cognitive process. And this is summarized earlier in Post #25 in a one-line principle:

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

So, yes, a set does require criteria for membership, but the process of forming it requires actual contents and more intensive knowledge. And crucially, the product of cognition, the qualified instance, i.e., the set, cannot be dissociated from the objectivity of the cognitive action. If what you want is a valid unit for further integrations and identifications, you can't be subjective; you can't make it up by whim.

Now, let's take a look at your counterexample:

Suppose you are standing at the door looking into a room full of people. You want to make several integrations and identifications, and you want to do them objectively. These are your cognitive goals. Let's now activate your consciousness and put it in focus:

QI1. "People in a room less than 6 feet tall."

QI2. "People in a room greater than 10 feet tall."

PI1. "QI1 are stylish dressers."

PI2. "QI2 are drinkers of frobscottle."

I would say that QI1 qualifies as a valid unit. QI2 is not a unit. It is not that QI2 is empty, but that there can be no QI2 qua unit/set/QI. Linguistically, we can still say it, but saying so doesn't make it so, a floating abstraction. And analogous to the invention of the zero, we can likewise invent a symbol, say, "{}", for the sake of further processing, i.e., of further integrations and identifications, to denote that QI2 is not at all a unit.

Thus, PI1 qualifies as a propositional identification, subject to further verification. But PI2 is not a unit of identification, not because it isn't grammatical or meaningful in some context (see The BFG, by Roald Dahl), but because there is now the fact that QI2 is {} and is not a unit and cannot qualify as the subject in the propositional structure. This judgment, i.e., this identification, i.e., a conscious action, requires a subject, i.e., a content. Therefore, upon encountering QI2 during your cognitive processing of PI2, I would hope that you, the cognitive processor, will adhere to objectivity and halt its processing. And if you can't process it, you can't verify subsequently whether it is true or false.

Of course, on encountering any absence of a unit, you the volitional consciousness, can choose to halt processing--as dictated by the principle of action+content--or: you can choose to proceed with the misintegration and misidentification--as dictated by whim.

Edited May 13, 2009 by Thom T G

[Roger Bissell]

(I have got to jump in here, even though Merlin and Dragonfly are handling things ably themselves, and even though I vowed I wouldn't post here anymore because I really don't like being accused of "not thinking with my own mind.")

Roger, you're very mixed up about mathematics, and I urge you to take a break from posting about it and go back and study it some more.

Laure, thank you for sharing your opinion about my understanding of mathematics. You're in good company...

In the respectful spirit in which your and others' comments were intended, I hereby return the favor. I urge you and they to avail yourselves of some excellent resources about the ~epistemological~ grounding of mathematics. First and foremost, I heartily recommend the excellent lectures (several series) by Pat Corvini, which are available through the Ayn Rand Bookstore, a subsidiary of the dreaded Ayn Rand Institute. :poke: Secondly, I recommend with some reservation the fine book by George Lakoff and Rafael E. Nunez: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being (Basic Books, 2000).

I just want to make a brief point about your idea that we don't add zero to something, we just "don't do the addition at all." Someone made the point that you wouldn't design a computer that way, and he's absolutely right. If you look in a computer hardware textbook about how an adder is designed, it doesn't have logic on the front-end to say that if one of the inputs is zero, you bypass the adder. You don't check, you just run your two inputs through the adder out pops a result. Nothing wrong with putting the zero through the adder. Or to bring it to a level that the layman would be more familiar with (or not!), if you're writing a computer program to calculate the total amount due on an order, it'd probably be something like this:

total = Price(item) + Tax(item, state_of_residence)

NOT,

tax = Tax(item, state_of_residence)

if (tax > 0)

total = Price(item) + tax

else

total = Price(item)

That's ~precisely~ how I'd set it up. That is how we set up ~division~ (i.e., by bifurcating the cases), so why not make it consistent across the four arithmetical operations, to reflect the fact that 0 is not an operator for any of the operations?

quotient = dividend/divisor (if divisor not = 0)

else, quotient is undefined

(There ~is no~ quotient, because there is no operation. You can't divide SOMETHING by NOTHING.)

product = multiplicand*multiplier (if multiplier not = 0)

else, product is undefined

(There ~is no~ product, because there is no operation. You can't multiply SOMETHING by NOTHING. The expression x*0 means "no multiples of x," and you ~aren't finding~ any multiples of the number x, so you have nothing, 0, as a result. But that is not a product; there is no operation.)

difference = minuend - subtrahend (if subtrahend not = 0)

else, difference is undefined

(There ~is no~ difference, because there is no operation. You can't subtract NOTHING from SOMETHING.)(See below.)

sum = augend + addend (if addend not = 0)

else, sum is undefined

(There ~is no~ sum, because there is no operation. You can't add NOTHING to SOMETHING. The expression x + 0 means "x plus nothing," and you ~aren't adding~ anything to the number x, so you have the number, x, as a result. But that is not a sum; there is no operation.)

I'm still waiting for someone to show me the flaw in this. Most of you are so convinced I'm an ignoramus. Please show me something I don't know that refutes the above, or that renders it fatal to the existence of mathematics.

REB

[Roger Bissell]

There can be no such thing as an "empty" set. A "set" with no element is not a set.

I'm a little late replying to this because I didn't see it earlier, but I disagree. Firstly, see post #23. Secondly, defining a set requires some criteria for membership, e.g. people in a room less than 6 feet tall. But suppose the criteria were people in a room greater than 10 feet tall. It is a set in the sense of having a criteria for membership. It's simply that none of the people qualify as a member. It's an empty set.

Merlin, though I saw your Post #23, and had assumed your endorsement of the quoted passages, I did not include it in the discussion on Post #25 which you excerpted above. But since you now bring it up explicitly, I have to ask you whether you are endorsing the dropping of context in the definition of "empty set" in that cited article. My context for claiming the absence of an "empty" set is grounded in the relational aspect of a concept and its referents from the perspective of the human consciousness that isolates the latter to form the former.

Tell me if there can be a concept without the units from which their integration results into a concept with an affixed word. If you appeal to the fact that there are plenty of people who use legitimate, everyday words without knowing their meanings, or referents, I will grant you that. But then those words as they use them are not concepts for them. They are uttering sounds and writing scribbles, not words, and not concepts.

The relation between a concept and its referents cannot be severed. The identities of the relata depend on the existence of the relation. We are here discussing the ontology of the primacy of entities with respect to relations and their attributes. Don't you tell me that it doesn't apply to math, if it is the math needed for living on earth. One can no more say that one is a parent without having had at least one child, and one can no more claim to be a straight husband without the correlative existence of a wife, than one can somehow claim the existence of a set without the correlative existence of its elements.

That's correct, Thom. You can't be the parent of zero children, and you can't be a husband, straight or otherwise, with zero wives! Nor can you ~add~ zero wives to your life, nor add zero children to your household. If you don't get married or have children, then you ~are not adding~ wife or children to your life or household.

This seems so patently obvious, we ~must~ be missing something that these other, superior beings are privy to. (Not!)

REB

[Roger Bissell]

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.

If the Objectivists get a hold of mathematics (Reality forbid!) they will turn it right into the rocks. Some O'ists got a hold of physics and produced the Theory(???) of Elementary Waves; pure crackpot and crank nonsense. I shudder to think what they will do to math.

Progress in mathematics went at warp speed once it was liberated from metaphysics.

Ba'al Chatzaf

Ba'al, I'm still waiting to hear how denying the commutativity of zero in abelian groups will cause the catastrophic meltdown of mathematics.

In the meantime, I would like two or three concrete historical examples of rapid advances in mathematics that were made possible by the jettisoning of metaphysics. Please pick at least two that you think are decisive illustrations of your point.

Thanks in advance.

REB