Others in this discussion treat pure math and applied math as if the issue were content, not intent, especially when they claim there is no correspondence between pure math and reality.

That is precisely what I question. If there were no correspondence, pure math only applies to reality in some cases by accident, but not by correspondence.

I believe this kind of reasoning is flawed.

I am surprised you have not noticed this.

Michael

Well, there will always be the reality that it is a person doing the mathematics! But as I said, we study relations in nature and we study relations in mathematics but in mathematics they are exact relations. We even have ways of writing numbers that never end, like pi.

Pi=4 - 4/3 + 4/5 - 4/7 + ....

Even though this can be computed indefinitely it may still be considered an exact representation since we can compute to as many decimal places as we wish. In applying the value of pi we always round it off to an inexact value which serves our purpose because in reality we don't need exactness - only close enough for government work.

5. 0^n/0^n = 1 multiplied by n groups of things taken 0 at a time, and divided by 1 multiplied by n groups of things taken 0 at a time (there are no such things, so 1 is not multiplied by anything)--i.e., 0^n/0^n = 1/1 = 1

now, suppose n = 0; then:

6. 0^0/0^0 = 1 NOT multiplied by 0 groups of things taken 0 at a time, and divided by 1 NOT multiplied by 0 groups of things taken 0 at a time--i.e., 0^0/0^0 = 1/1 = 1.

This could only be so if 0^0 is NOT undefined. Indeed, I have defined/interpreted it above as: 1 NOT multiplied by 0 groups of things taken 0 at a time--that is to say, 0^0 = 1.

This is how our mental processes ~really~ work. The standard, traditional conception leads to paradox and obfuscates rather than clarifying how our minds do arithmetic (including exponentials).

REB

This really does make a lot of sense from a process-model perspective. Drawing from Chapter 8 of ITOE "Consciousness and Identity," I would add that in identifying facts of reality, including the ontology of numbers, we must take cognizant of not only the existents but also the nature of how the mind works. Your explanation respects both aspects.

Another example is the notion of something being "undefined" in mathematics. Does it mean, as Laure suggests in Post #110, that anything that is computed variously and produced incompatible results should be considered undefined? No, that simply means that something is being defined incompatibly. And in a contradictorily incompatible relationship, both relata cannot be true; at least one is false. It leaves the possibility that one or the other may be true. So, I would say that "undefined" is not as defined, and that it should be interpreted as "not operable or calculable by any process."

In the case of 0^0/0^0, it surely is calculable to 1.

I agree with all of this, Thom.

[...]

Now, Roger, I have a question for you. On this proposal, 1*0 is different from 0*1. The former is blocked; the latter is zero. Is this correct?

That is correct, yes, but misleading--TO CLARIFY (5/5/13): both are blocked and both are nonetheless zero. Neither multiplication takes place, so 0 is not the PRODUCT of an operation. But 0 ~is~ the COUNT of what you have, with no operation being possible. So, 0 is the RESULT OF ATTEMPTED IMPOSSIBLE MULTIPLICATION. So, we read the equation 1 * 0 = 0 as: "Since there is no such thing as a group containing no items, the attempt to count the number of items in such a group gives the result of 0 items" -- and the equation 0 * 1 = 0 as: "Since there is no group containing 1 item, the attempt to count the number of items has no group to count them in, which gives the result of 0 items."

The same reasoning, by the way, applies to 1 + 0 = 1 and 0 + 1 = 1. The additive identity of 0 is preserved not because nothing (0) ~is~ added to the other number, or the other number is being added to nothing (0), but because there ~isn't anything~ that is being added to the other number, and the other number ~isn't being~ added to anything. The identity of the other number remains unchanged, because there isn't anything being done to it, and it isn't doing anything to something else.

In addition, you can add one element to the empty set, but you cannot add zero to a set containing one element. In the former case, you are definitely doing something real. In the second case, you aren't doing anything.

I disagree with and have deleted the rest of my comments here about having a group not containing anything vs. not having zero groups containing one thing (There aren't zero-membered groups, any more than there are zero groups with members.) and about division being a complete inverse of multiplication (There is no division by ~or~ of 0, any more than there is multiplication by ~or~ of 0, but nonetheless there is a ~result~ of 0 for both multiplications and for division of 0 by non-zero numbers, while there is ~no~ result for attempted division ~by~ 0. All of this is as per standard arithmetic, but reinterpreted ontologically by my zero-blocker perspective.)

Other than what we have covered so far, I think we can content ourselves that all financial spreadsheets continue to be perfectly sound. We just have to remember what the numbers stand for ontologically. There is no escape from metaphysics.

Unfortunately, people manage to do it all the time and get by just fine, more or less. As Corvini notes, each advance in mathematics was done with a lot of protest, but ultimately adopted not because it was properly understood conceptually, but because it ~worked~. We have a lot of wonderful things in our lives because mathematicians, scientists, and technicians found things that worked, even while not properly grasping the foundation in reality for the discoveries they made and applied. As long as they don't blow us up, that's OK--but in principle, I don't like the brandishing of the Holy Cross of conventional wisdom when people like us try to reduce abstractions to their base in the real world. http://www.objectivistliving.com/forums/public/style_emoticons/#EMO_DIR#/no.gif

Anyway, thanks for your good questions and suggestions and comments, Thom. It's nice to read comments from a clear thinker such as yourself, who is willing to think outside the box of received dogma.

Yes. 0 is an operation stopper. You cannot "do" zero to anything. Starting with 0, you can't add 0 to it, multiply it by 0, divide it by zero, or subtract 0 from it. 0 + 0 IF we start with zero, then (as they say in "Jolly Holiday") THERE WE STOP! Viz., you cannot subtract 0*n from 0*n, so your attempted application of the distributive law breaks down in the left-hand side of the attempted chain of equations.

What if, instead of speaking about operations, we spoke about associations? So when we have the pair of numbers (1,2) we associate this pair with the number 3. Likewise;

(2,3)=>5

(3,-2)=>1

(6,-8)=>-2

(2,0)=>2

In this scenario we are not performing an operation on a quantity we are merely associating 2 numbers with a third number. In this example, when the symbol '0' is one of the pair the result is always the other number and so it is called an identity element.

Others in this discussion treat pure math and applied math as if the issue were content, not intent, especially when they claim there is no correspondence between pure math and reality.

That is precisely what I question. If there were no correspondence, pure math only applies to reality in some cases by accident, but not by correspondence.

I believe this kind of reasoning is flawed.

I am surprised you have not noticed this.

Michael

I have noticed it and registered my disagreement here ( near the end) and other threads as well. It just hasn't been my focus.

Distributive law, huh? I think you're being a bit careless with it. The way I learned it: (x + y) * (z + w) = (xz + xw) + (yz + yw), and you then perform the operations ~inside~ the parenthesis, only then combining the two results inside parenthesis. :yes:

Applying this to the present case: (1 - 1) * (1 - 1) = (1 - 1) + (- 1 + 1) = 0 + 0. Oops, oh my, here we are with another undefined (in RogerLand) expression. Can't add 0 to 0 any more than you can multiply 0 by 0. :no:

Sorry, Laure. The contradiction is not in my system, but in your misapplication of the Distributive Law.

Laure applied the law fine. The purpose of parentheses is to clarify the order of operations. Operations inside parentheses are performed before operations outside the parentheses (most embedded ones first when one set is found inside another). The usual priority order is exponents, multiply/divide, add/subtract. (The programming language APL is an exception). But if you want to nitpick, it is you who misapplied the distributive law.

Newton's law is for the kinetic energy of a ~moving~ body. If a body isn't moving, it DOESN'T HAVE ANY kinetic energy. Kinetic energy is a concept that only applies to (is only defined for) ~moving~ entities.

Not true. Kinetic energy is relative to a frame of reference. If you are riding in a car, you have positive kinetic energy relative to the ground and zero kinetic energy relative to the car.

Here are 2 sets (groups) of numbers: {1, 3, 5} {2, 4, 6, 8}

How many (a number) groups contain the number 7?

Merlin,

The question asked can be unpacked procedurally as,

1. Is 7 an element of the first set {1, 3, 5}? False

2. Is 7 an element of the second set {2, 4, 6, 8}? False

3. How many times was the operation "an element of" performed? 2

4. How many times did it fail to yield the result True? 2

5. How many times did it succeed? 2 - 2 = 0

Would you not agree that it is epistemology that establishes the criteria for postulating anything in any science, including mathematics? If at one time it was thought that concepts could in reality be formed without referentce or linkage to reality, and then at a later time it was rethought otherwise; wouldn't you say that that epistemological change in the standard of the unit would cause a tectonic change in every science that depended on it?

I have a question and I know this is not traditional math jargon, but Objectivist-style jargon instead.

Isn't the reality referent for zero a placeholder that can be filled at any time with an identity-bearing something, the criteria for which are defined by the context where zero is used?

Roger, "for all x, y, z, (x+y)+z = (x+z)+y" is a postulate in my math. "For all x, y, x+y = y+x" is another postulate. So, (1-1) + (1-1) = (1+(1-1)) -1 = ((1-1)+1)-1 = 0+1-1 = 1-1 = 0.

Let's keep it real simple. Roger says x+0 is incalculable. Roger, would you agree that if you see "x+0" in an equation, that you can simply substitute "x", since the "+0" does nothing? Well, when we say that "x+0=x", we are saying precisely that "we can substitute 'x' for 'x+0'". That is what it MEANS.

"x+0" is synonymous with "x".

Another example, this idea that 0*1 is 0 but 1*0 is undefined. Let's go back to Montessori Preschool for a moment. If we want to show what 2*3 is, we can lay out some pennies in 2 rows and 3 columns. If we want to show what 3*2 is, we can lay out the pennies in 3 rows and 2 columns. Now let's try 0*1. Roger says that's OK. 0 rows and 1 column. 0 pennies. How about 1*0? Oh, we could have 0 rows of pennies, but 0 columns is verboten! What if we go round to the side? So Roger, if 1 row of 0 columns is verboten, so is 0 rows of 1 column. So you can't have it both ways. If you object to 1*0, you can't "do" 0*1 either.

The empty set means a set with no elements. Saying that there is no such thing as the empty set is as silly as objecting to the use of the word "nothing".

If we have 5 chairs and add none, we still have 5 chairs. Why is it "forbidden" to say that we have added none? Just because Rand made a comment disparaging the "reification of the zero"? She was just complaining about philosophers who glommed onto the concept "zero" and tried to give it some mystical interpretation. You say "there isn't any kinetic energy", and we say "the kinetic energy is zero". It's the same thing! There is nothing to be gained by avoiding the word "zero" like a taboo!

Here are the equality principles and postulates of my math:

Show me yours. I think what you'll find (best-case scenario) is that if you create a new symbol for your special "I can't say 'zero' so let's say 'undefined'", (0*0=#) you will find that that symbol "#" is redundant, and is equivalent to "0". Worst-case, you end up with a system that contains a contradiction.

My unpacking would be simpler. I count the number of sets containing 7, and the answer is 0. There is no need to count how many sets there are nor to subtract.

My answer to your first question is I agree, tentatively. I add 'tentative' because I don't know what you are trying to achieve by asking the question. What you are trying to achieve with your second question is even less clear. What do "change in the standard of the unit" and "tectonic change" refer to?

Isn't the reality referent for zero a placeholder that can be filled at any time with an identity-bearing something, the criteria for which are defined by the context where zero is used?

Isn't the reality referent for zero a placeholder that can be filled at any time with an identity-bearing something, the criteria for which are defined by the context where zero is used?

zero -- the symbol or numeral 0, representing the complete absence of any quantity or magnitude

Example: Ayn Rand had 0 children.

If I were going to express this in standard propositional form -- for doing deductive logic -- I would replace "had" with a form of "to be," and I would express the predicate as being in the same category (entity or person) as the subject. Viz., Ayn Rand was not a person having children, or Ayn Rand was not a parent.

But leaving it in the verbal form you use, I'd still rewrite it as Ayn Rand did not have any children.

You can't ~have~ ~zero~ children. You can't ~have~ a "complete absence of any quantity or magnitude" of ~something~. You simply ~don't have~ "any quantity or magnitude" of something.

This reificiation of zero infects even propositional speech and logic--if you let it.

Isn't the reality referent for zero a placeholder that can be filled at any time with an identity-bearing something, the criteria for which are defined by the context where zero is used?

zero -- the symbol or numeral 0, representing the complete absence of any quantity or magnitude

Example: Ayn Rand had 0 children.

0 is the least unsigned integer in the linear ordering of the integers. Also it is the identity element of the additive semi-group of unsigned integers. Also it is the cardinal number of the empty set. The empty set is something not nothing. For instance it is the intersection of the set of even integers and the set of odd integers. Zero is also the midpoint of the numerical line separating the negative numbers from the positive numbers. It is also the projection of a vector v upon another vector w where v is orthogonal to w. It is also the determinant of a matrix with two identical rows or columns. It is also the Borel measure of the set of rational numbers in the unit interval of real numbers. It is also the integral of the cosine function between 0 and pi radians. That is something and not nothing. It is also a place holder in the positional representation of integers with respect to a base. It zero (the digit) separates powers of b, where b is the base. It is how we distinguish 1 from 10 from 100 for example. It is also the limit of the sequence {1/n} as n increase indefinitely from 1, where n is an integer.

That an awful lot of something for a nothing, don't you think?

So a "true" mathematical zero would be expressed without a context ... as in zero is non-a thru z?

This is why these discussions on "0" has always made my hair hurt.

If some of you do provide a response, please use small words for us "equationally challenged". :sweat: :frantics:

Adam

Well,... yeah! The whole idea of mathematics is to abstract away the units, or the referents in reality, so that we can just manipulate the symbols. That's why we can say 2+2=4 and it doesn't matter 2 of what. We don't have to stop and think, "OK, if I take 2 apples and add 2 apples, I have 4 apples. But, what if I had 2 oranges and add 2 oranges?? Gee, what could the answer be?"

Distributive law, huh? I think you're being a bit careless with it. The way I learned it: (x + y) * (z + w) = (xz + xw) + (yz + yw), and you then perform the operations ~inside~ the parenthesis, only then combining the two results inside parenthesis. :yes:

Applying this to the present case: (1 - 1) * (1 - 1) = (1 - 1) + (- 1 + 1) = 0 + 0. Oops, oh my, here we are with another undefined (in RogerLand) expression. Can't add 0 to 0 any more than you can multiply 0 by 0. :no:

Sorry, Laure. The contradiction is not in my system, but in your misapplication of the Distributive Law.

Laure applied the law fine. The purpose of parentheses is to clarify the order of operations. Operations inside parentheses are performed before operations outside the parentheses (most embedded ones first when one set is found inside another). The usual priority order is exponents, multiply/divide, add/subtract. (The programming language APL is an exception). But if you want to nitpick, it is you who misapplied the distributive law.

(x + y) * (z + w) = x*(z + w) + y*(z + w)

Let's go back to the womb, shall we? No, wait, let's go back to pre-conception, when "we" were non-existents--wait, I mean before "we" were existents...oh, hell, let's just get on with it! (I was briefly caught up in Laure's helpful, respectful efforts to try to drag me back first to high school, then to Montessori pre-school. Please forgive me.)

Well, two comments about your comment:

1. Using ~your~ form of the distributive law, Laure's example (trying to show the contradiction of my claim that 0 * 0 is undefined) would parse out like this:

(1 - 1) * (1 - 1) = 1*(1 - 1) + (-1)*(1 - 1) = (1 * 0) - (1 * 0). Subtracting one undefined expression from another. Good one. And the way this avoids my conclusion that 0 * 0 is undefined is what again?

2. But suppose we ~fully~ apply the distributive law and not stop where you did:

(x + y) * (z + w) = x*(z + w) + y*(z + w) = (xz + xw) + (yz + yw). That is the way I used it, and I ended up with 0 + 0, another undefined expression.

Once you accept, even provisionally, the premise that any number times 0 is undefined, you ~cannot~ through sleight of hand with the distributive (or any other) law, prove otherwise! In other words, your reductio attempts are not working!

I showed you my equality principles and postulates. Show me yours, Roger. Let's nail down the specifics of your system, and then we will see whether it is logically inconsistent or just consistent and useless!

This reificiation of zero infects even propositional speech and logic--if you let it.

REB

It has even infected my CASIO fx-115MS calculator. I punched in 0x0= and

I got 0.00000000 . Should I take the calculator back to the store and get a refund? Is my calculator the victim of some Kantian plot? I also got 0 + 0 = 0 and 0 - 0 = 0. Oh my!

reify - to treat (an abstraction) as substantially existing, or as a concrete material object

If we are guilty of reifying zero, then you are equally guilty of reifying whenever you use "no", "not", "never", "don't have", "childless", "colorless", "odorless", etc. These are all various ways of expressing an absence of something.

I have a question and I know this is not traditional math jargon, but Objectivist-style jargon instead.

Isn't the reality referent for zero a placeholder that can be filled at any time with an identity-bearing something, the criteria for which are defined by the context where zero is used?

Michael

Michael,

Welcome to the ontology of the zero. You pose a legitimate question concerning the nature of 0. To what facts of reality does 0 refer? This is a philosophical question. And the answer must be philosophical, not mathematical. Any mathematician who tries to answer it must answer it in his capacity not as a professional mathematician but as an amateur philosopher.

You are the first person to counterpropose philosophically another view of 0 to Roger's. (See Post #105.)

If I understand you, you are saying that 0 identifies a relation to any identity-bearing something, in relation to that something's absence. In which case, I do not see any incompatibility whatsoever with what you express and what Roger has proposed. Where Roger's proposal identifies the role of 0 in math operations, yours identifies its role in the math quantities being operated. Since a quantity as a relational existent presupposes other existents (entities, attributes, etc.), it follows that some math operations require some presumed quantity on the understanding and operational context that a relation to some existents must exist but that it may exist (thereafter) in any quantity. So 0 in this strict sense is not a quantity per se but denotes a placeholder (and/or a result) in some math operations for the absence of a presumed, existing quantity. Thus, like "i", which was discovered much later in history, the concept "0" is a methodological concept. (For the full context of my analysis, see ITOE Ch. 4 on introspection and the Appendix on numbers.)

Ontologically speaking, as nothing is not another something, so zero is not another quantity. To think so is to commit the fallacy of Reification of the Zero. (ITOE 60)

Welcome to this "side" of the proposal, Michael. :flowers:

Roger, "for all x, y, z, (x+y)+z = (x+z)+y" is a postulate in my math. "For all x, y, x+y = y+x" is another postulate. So, (1-1) + (1-1) = (1+(1-1)) -1 = ((1-1)+1)-1 = 0+1-1 = 1-1 = 0.

Let's keep it real simple. Roger says x+0 is incalculable. Roger, would you agree that if you see "x+0" in an equation, that you can simply substitute "x", since the "+0" does nothing? Well, when we say that "x+0=x", we are saying precisely that "we can substitute 'x' for 'x+0'". That is what it MEANS.

"x+0" is synonymous with "x".

I never said otherwise! In fact, I said just that (or its equivalent) a few (or a few dozen) posts ago. (If not, then the following is the real deal...)

I do NOT deny that x + 0 = x. I just deny that you are ADDING. THERE IS NO OPERATION BEING PERFORMED. Just a recognition that when you do not do anything (adding zero is not doing anything) to a number, the number remains unchanged. That is why 0 is the "additive identity." It does not change the numerical identity of the number to which you ~try~ to add it, because you ~cannot~ add it.

As a purported addition, the equation x + 0 = 0 ~means~ that THE SUM OF X and 0 IS 0. But the equation is NOT an addition. There is NO SUM of x and 0, because you are NOT ADDING, so the SUM of x and 0 is undefined. However, the equation x + 0 DOES = x. Not because x PLUS 0 is x, but because x is x. When you include + 0 in a statement, it is as if you had not said anything after the number preceding it.

The same reasoning, of course, applied to x * 1 = x. You are NOT multiplying. You are just stating that x is x. 1 is the multiplicative identity. There is no operation being performed, just a recognition that when you do not do anything to a number (multiplying by one is not doing anything), the number remains unchanged.

Another example, this idea that 0*1 is 0 but 1*0 is undefined. Let's go back to Montessori Preschool for a moment. If we want to show what 2*3 is, we can lay out some pennies in 2 rows and 3 columns. If we want to show what 3*2 is, we can lay out the pennies in 3 rows and 2 columns. Now let's try 0*1. Roger says that's OK. 0 rows and 1 column. 0 pennies. How about 1*0? Oh, we could have 0 rows of pennies, but 0 columns is verboten! What if we go round to the side? So Roger, if 1 row of 0 columns is verboten, so is 0 rows of 1 column. So you can't have it both ways. If you object to 1*0, you can't "do" 0*1 either.

The empty set means a set with no elements. Saying that there is no such thing as the empty set is as silly as objecting to the use of the word "nothing".

I'm not sure that the rows and columns approach is any more helpful than length and width or any other two-dimensional approach. I prefer to take it as groups of things taken so many at a time. This is more "primitive" and more simple, and it is how we learn, especially if our parents were not affluent enough to afford Montessori Preschool!

2*3 is three groups of things taken two at a time, and 3*2 is two groups of things taken three at a time.

0*1 is one group of things taken zero at a time, and1*0 is zero groups of things taken one at a time.

Let's use pennies. 0 pennies * 1 = 1 group of pennies taken zero at a time. This is a group without any pennies. The null or empty set. 0. I don't object to this, and if I said I did earlier, I take it back! All I am saying is that you cannot perform operations WITH zero. You can certainly perform them ON zero.

1 penny * 0 = 0 groups of pennies taken 1 at a time. This is NO GROUP. Not the null group or 0. But NO GROUP. YOU HAVE NOT MENTALLY PERFORMED A GROUPING OPERATION, SO THERE IS NO GROUP, NOT EVEN THE NULL GROUP, or 0. (And when I say "there is no group," I ~mean~ "there is not any group." )

If we have 5 chairs and add none, we still have 5 chairs. Why is it "forbidden" to say that we have added none? Just because Rand made a comment disparaging the "reification of the zero"? She was just complaining about philosophers who glommed onto the concept "zero" and tried to give it some mystical interpretation. You say "there isn't any kinetic energy", and we say "the kinetic energy is zero". It's the same thing! There is nothing to be gained by avoiding the word "zero" like a taboo!

Laure, I don't have a problem with the word or concept "zero." I just don't think you can use it to ~do~ things. It is simply a place-holder for what is not there, or for what is not done to something else. :yes:

When you state that "kinetic energy is zero" or "the number of chairs in the room is zero," you cannot be stating a measurement or count. You cannot count or measure what is not there. :no: At most, what you mean--as you acknowledge--is that there ~isn't~ any kinetic energy, or there ~aren't~ any chairs in the room, or you ~haven't added~ any chairs to the room.

I can assure you that my dear wife would much prefer that I assure her that I ~haven't~ slept with ~any~ other women, than to tell her that I ~have~ slept with ~zero~ women. Not only would she find the latter a bit odd, but also probably disquieting. For one thing, if you want to reassure someone about what you ~haven't~ done, you probably shouldn't start out by saying that you ~have~ done something and then quantify it as "zero." ;)

Here are the equality principles and postulates of my math:

Show me yours. I think what you'll find (best-case scenario) is that if you create a new symbol for your special "I can't say 'zero' so let's say 'undefined'", (0*0=#) you will find that that symbol "#" is redundant, and is equivalent to "0". Worst-case, you end up with a system that contains a contradiction.

I haven't seen the contradiction yet, and a lot of you seem to be trying very hard to display one, with little to show for it yet! :poke:

Your post #194 is fatally flawed. You tried to mix your premises, ones I disagree with, in with mine. 1*0 = 0. It being undefined is your premise, not mine.

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## tjohnson

Well, there will always be the reality that it is a person doing the mathematics! But as I said, we study relations in nature and we study relations in mathematics but in mathematics they are

relations. We even have ways of writing numbers that never end, like pi.exactPi=4 - 4/3 + 4/5 - 4/7 + ....

Even though this can be computed indefinitely it may still be considered an exact representation since we can compute to as many decimal places as we wish. In applying the value of pi we always round it off to an

value which serves our purpose because in reality we don't need exactness - only close enough for government work.inexact## Link to comment

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## Roger Bissell

[...]

That is correct, yes, but misleading--TO CLARIFY (5/5/13): both are blocked and both are nonetheless zero. Neither multiplication takes place, so 0 is not the PRODUCT of an operation. But 0 ~is~ the COUNT of what you have, with no operation being possible. So, 0 is the RESULT OF ATTEMPTED IMPOSSIBLE MULTIPLICATION. So, we read the equation 1 * 0 = 0 as: "Since there is no such thing as a group containing no items, the attempt to count the number of items in such a group gives the result of 0 items" -- and the equation 0 * 1 = 0 as: "Since there is no group containing 1 item, the attempt to count the number of items has no group to count them in, which gives the result of 0 items."

The same reasoning, by the way, applies to 1 + 0 = 1 and 0 + 1 = 1. The additive identity of 0 is preserved not because nothing (0) ~is~ added to the other number, or the other number is being added to nothing (0), but because there ~isn't anything~ that is being added to the other number, and the other number ~isn't being~ added to anything. The identity of the other number remains unchanged, because there isn't anything being done to it, and it isn't doing anything to something else.

In addition, you can add one element to the empty set, but you cannot add zero to a set containing one element. In the former case, you are definitely doing something real. In the second case, you aren't doing anything.

I disagree with and have deleted the rest of my comments here about having a group not containing anything vs. not having zero groups containing one thing (There aren't zero-membered groups, any more than there are zero groups with members.) and about division being a complete inverse of multiplication (There is no division by ~or~ of 0, any more than there is multiplication by ~or~ of 0, but nonetheless there is a ~result~ of 0 for both multiplications and for division of 0 by non-zero numbers, while there is ~no~ result for attempted division ~by~ 0. All of this is as per standard arithmetic, but reinterpreted ontologically by my zero-blocker perspective.)

Unfortunately, people manage to do it all the time and get by just fine, more or less. As Corvini notes, each advance in mathematics was done with a lot of protest, but ultimately adopted not because it was properly understood conceptually, but because it ~worked~. We have a lot of wonderful things in our lives because mathematicians, scientists, and technicians found things that worked, even while not properly grasping the foundation in reality for the discoveries they made and applied. As long as they don't blow us up, that's OK--but in principle, I don't like the brandishing of the Holy Cross of conventional wisdom when people like us try to reduce abstractions to their base in the real world. http://www.objectivistliving.com/forums/public/style_emoticons/#EMO_DIR#/no.gifAnyway, thanks for your good questions and suggestions and comments, Thom. It's nice to read comments from a clear thinker such as yourself, who is willing to think outside the box of received dogma.

REB

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## tjohnson

What if, instead of speaking about

, we spoke aboutoperations? So when we have the pair of numbers (1,2) we associate this pair with the number 3. Likewise;associations(2,3)=>5

(3,-2)=>1

(6,-8)=>-2

(2,0)=>2

In this scenario we are not performing an operation on a quantity we are merely associating 2 numbers with a third number. In this example, when the symbol '0' is one of the pair the result is always the other number and so it is called an

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## merjet

I have noticed it and registered my disagreement here ( near the end) and other threads as well. It just hasn't been my focus.

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## merjet

Laure applied the law fine. The purpose of parentheses is to clarify the order of operations. Operations inside parentheses are performed before operations outside the parentheses (most embedded ones first when one set is found inside another). The usual priority order is exponents, multiply/divide, add/subtract. (The programming language APL is an exception). But if you want to nitpick, it is you who misapplied the distributive law.

(x + y) * (z + w) = x*(z + w) + y*(z + w)

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## merjet

Not true. Kinetic energy is relative to a frame of reference. If you are riding in a car, you have positive kinetic energy relative to the ground and zero kinetic energy relative to the car.

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## merjet

Here are 2 sets (groups) of numbers: {1, 3, 5} {2, 4, 6, 8}

How many (a number) groups contain the number 7?

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## thomtg

AuthorMerlin,

The question asked can be unpacked procedurally as,

1. Is 7 an element of the first set {1, 3, 5}?

False2. Is 7 an element of the second set {2, 4, 6, 8}?

False3. How many times was the operation "an element of" performed?

24. How many times did it fail to yield the result True?

25. How many times did it succeed?

2 - 2 = 0Would you not agree that it is epistemology that establishes the criteria for postulating anything in any science, including mathematics? If at one time it was thought that concepts could in reality be formed without referentce or linkage to reality, and then at a later time it was rethought otherwise; wouldn't you say that that epistemological change in the standard of the unit would cause a tectonic change in every science that depended on it?

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## Michael Stuart Kelly

I have a question and I know this is not traditional math jargon, but Objectivist-style jargon instead.

Isn't the reality referent for zero a placeholder that can be filled at any time with an identity-bearing something, the criteria for which are defined by the context where zero is used?

Michael

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## Laure

Roger, "for all x, y, z, (x+y)+z = (x+z)+y" is a postulate in my math. "For all x, y, x+y = y+x" is another postulate. So, (1-1) + (1-1) = (1+(1-1)) -1 = ((1-1)+1)-1 = 0+1-1 = 1-1 = 0.

Let's keep it real simple. Roger says x+0 is incalculable. Roger, would you agree that if you see "x+0" in an equation, that you can simply substitute "x", since the "+0" does nothing? Well, when we say that "x+0=x", we are saying precisely that "we can substitute 'x' for 'x+0'". That is what it MEANS.

"x+0" is synonymous with "x".

Another example, this idea that 0*1 is 0 but 1*0 is undefined. Let's go back to Montessori Preschool for a moment. If we want to show what 2*3 is, we can lay out some pennies in 2 rows and 3 columns. If we want to show what 3*2 is, we can lay out the pennies in 3 rows and 2 columns. Now let's try 0*1. Roger says that's OK. 0 rows and 1 column. 0 pennies. How about 1*0? Oh, we could have 0 rows of pennies, but 0 columns is verboten! What if we go round to the side? So Roger, if 1 row of 0 columns is verboten, so is 0 rows of 1 column. So you can't have it both ways. If you object to 1*0, you can't "do" 0*1 either.

The empty set means a set with no elements. Saying that there is no such thing as the empty set is as silly as objecting to the use of the word "nothing".

If we have 5 chairs and add none, we still have 5 chairs. Why is it "forbidden" to say that we have added none? Just because Rand made a comment disparaging the "reification of the zero"? She was just complaining about philosophers who glommed onto the concept "zero" and tried to give it some mystical interpretation. You say "there isn't any kinetic energy", and we say "the kinetic energy is zero". It's the same thing! There is nothing to be gained by avoiding the word "zero" like a taboo!

Here are the equality principles and postulates of my math:

Show me yours. I think what you'll find (best-case scenario) is that if you create a new symbol for your special "I can't say 'zero' so let's say 'undefined'", (0*0=#) you will find that that symbol "#" is redundant, and is equivalent to "0". Worst-case, you end up with a system that contains a contradiction.

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## merjet

Thom,

My unpacking would be simpler. I count the number of sets containing 7, and the answer is 0. There is no need to count how many sets there are nor to subtract.

My answer to your first question is I agree, tentatively. I add 'tentative' because I don't know what you are trying to achieve by asking the question. What you are trying to achieve with your second question is even less clear. What do "change in the standard of the unit" and "tectonic change" refer to?

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## merjet

From http://www.yourdictionary.com/zero:

zero -- the symbol or numeral 0, representing the complete absence of any quantity or magnitude

Example: Ayn Rand had 0 children.

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## Michael Stuart Kelly

Merlin,

But in this example, doesn't zero refer to

children?Or can the statement, "Ayn Rand had 0 children" also mean "Ayn Rand had 0 giraffes"? I don't see that. I see it as meaning "0 children."

If it means children, you have a correspondence to reality right there that is contextual, and it is good for any placeholder worthy of the name.

The referent of zero comes from the context in which it is used. But the referent is something that exists.

Michael

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## Roger Bissell

If I were going to express this in standard propositional form -- for doing deductive logic -- I would replace "had" with a form of "to be," and I would express the predicate as being in the same category (entity or person) as the subject. Viz., Ayn Rand was not a person having children, or Ayn Rand was not a parent.

But leaving it in the verbal form you use, I'd still rewrite it as Ayn Rand did not have any children.

You can't ~have~ ~zero~ children. You can't ~have~ a "complete absence of any quantity or magnitude" of ~something~. You simply ~don't have~ "any quantity or magnitude" of something.

This reificiation of zero infects even propositional speech and logic--if you let it.

REB

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## BaalChatzaf

0 is the least unsigned integer in the linear ordering of the integers. Also it is the identity element of the additive semi-group of unsigned integers. Also it is the cardinal number of the empty set. The empty set is something not nothing. For instance it is the intersection of the set of even integers and the set of odd integers. Zero is also the midpoint of the numerical line separating the negative numbers from the positive numbers. It is also the projection of a vector v upon another vector w where v is orthogonal to w. It is also the determinant of a matrix with two identical rows or columns. It is also the Borel measure of the set of rational numbers in the unit interval of real numbers. It is also the integral of the cosine function between 0 and pi radians. That is something and not nothing. It is also a place holder in the positional representation of integers with respect to a base. It zero (the digit) separates powers of b, where b is the base. It is how we distinguish 1 from 10 from 100 for example. It is also the limit of the sequence {1/n} as n increase indefinitely from 1, where n is an integer.

That an awful lot of something for a nothing, don't you think?

Ba'al Chatzaf

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## Selene

Folks:

So a "true" mathematical zero would be expressed without a context ... as in zero is non-a thru z?

This is why these discussions on "0" has always made my hair hurt.

If some of you do provide a response, please use small words for us "equationally challenged". :sweat: :frantics:

Adam

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## Laure

Well,... yeah! The whole idea of mathematics is to abstract away the units, or the referents in reality, so that we can just manipulate the symbols. That's why we can say 2+2=4 and it doesn't matter 2 of what. We don't have to stop and think, "OK, if I take 2 apples and add 2 apples, I have 4 apples. But, what if I had 2 oranges and add 2 oranges?? Gee, what could the answer be?"

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## Roger Bissell

Let's go back to the womb, shall we? No, wait, let's go back to pre-conception, when "we" were non-existents--wait, I mean before "we" were existents...oh, hell, let's just get on with it! (I was briefly caught up in Laure's helpful, respectful efforts to try to drag me back first to high school, then to Montessori pre-school. Please forgive me.)

Well, two comments about your comment:

1. Using ~your~ form of the distributive law, Laure's example (trying to show the contradiction of my claim that 0 * 0 is undefined) would parse out like this:

(1 - 1) * (1 - 1) = 1*(1 - 1) + (-1)*(1 - 1) = (1 * 0) - (1 * 0). Subtracting one undefined expression from another. Good one. And the way this avoids my conclusion that 0 * 0 is undefined is what again?

2. But suppose we ~fully~ apply the distributive law and not stop where you did:

(x + y) * (z + w) = x*(z + w) + y*(z + w) = (xz + xw) + (yz + yw). That is the way I used it, and I ended up with 0 + 0, another undefined expression.

Once you accept, even provisionally, the premise that any number times 0 is undefined, you ~cannot~ through sleight of hand with the distributive (or any other) law, prove otherwise! In other words, your reductio attempts are not working!

REB

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## Laure

I showed you my equality principles and postulates. Show me yours, Roger. Let's nail down the specifics of your system, and then we will see whether it is logically inconsistent or just consistent and useless!

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## BaalChatzaf

It has even infected my CASIO fx-115MS calculator. I punched in 0x0= and

I got 0.00000000 . Should I take the calculator back to the store and get a refund? Is my calculator the victim of some Kantian plot? I also got 0 + 0 = 0 and 0 - 0 = 0. Oh my!

Ba'al Chatzaf

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## merjet

Nobody here has reified zero, just as nobody here except you has incorrectly called a number an operator.

From the same dictionary:

reify - to treat (an abstraction) as substantially existing, or as a concrete material object

If we are guilty of reifying zero, then you are equally guilty of reifying whenever you use "no", "not", "never", "don't have", "childless", "colorless", "odorless", etc. These are all various ways of expressing an absence of something.

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## thomtg

AuthorMichael,

Welcome to the ontology of the zero. You pose a legitimate question concerning the nature of 0. To what facts of reality does 0 refer? This is a philosophical question. And the answer must be philosophical, not mathematical. Any mathematician who tries to answer it must answer it in his capacity not as a professional mathematician but as an amateur philosopher.

You are the first person to counterpropose philosophically another view of 0 to Roger's. (See Post #105.)

If I understand you, you are saying that 0 identifies a relation to any identity-bearing something,

in relation tothat something's absence. In which case, I do not see any incompatibility whatsoever with what you express and what Roger has proposed. Where Roger's proposal identifies the role of 0 in math operations, yours identifies its role in the math quantities being operated. Since a quantity as a relational existent presupposes other existents (entities, attributes, etc.), it follows that some math operations require some presumed quantity on the understanding and operational context that a relation to some existents must exist but that it may exist (thereafter) in any quantity. So 0 in this strict sense is not a quantity per se but denotes a placeholder (and/or a result) in some math operations forthe absence of a presumed, existing quantity. Thus, like "i", which wasdiscoveredmuch later in history, the concept "0" is a methodological concept. (For the full context of my analysis, see ITOE Ch. 4 on introspection and the Appendix on numbers.)Ontologically speaking, as nothing is not another something, so zero is not another quantity. To think so is to commit the fallacy of Reification of the Zero. (ITOE 60)

Welcome to this "side" of the proposal, Michael. :flowers:

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## Roger Bissell

I never said otherwise! In fact, I said just that (or its equivalent) a few (or a few dozen) posts ago. (If not, then the following is the real deal...)

I do NOT deny that x + 0 = x. I just deny that you are ADDING. THERE IS NO OPERATION BEING PERFORMED. Just a recognition that when you do not do anything (adding zero is not doing anything) to a number, the number remains unchanged. That is why 0 is the "additive identity." It does not change the numerical identity of the number to which you ~try~ to add it, because you ~cannot~ add it.

As a purported addition, the equation x + 0 = 0 ~means~ that THE SUM OF X and 0 IS 0. But the equation is NOT an addition. There is NO SUM of x and 0, because you are NOT ADDING, so the SUM of x and 0 is undefined. However, the equation x + 0 DOES = x. Not because x PLUS 0 is x, but because x is x. When you include + 0 in a statement, it is as if you had not said anything after the number preceding it.

The same reasoning, of course, applied to x * 1 = x. You are NOT multiplying. You are just stating that x is x. 1 is the multiplicative identity. There is no operation being performed, just a recognition that when you do not do anything to a number (multiplying by one is not doing anything), the number remains unchanged.

I'm not sure that the rows and columns approach is any more helpful than length and width or any other two-dimensional approach. I prefer to take it as groups of things taken so many at a time. This is more "primitive" and more simple, and it is how we learn, especially if our parents were not affluent enough to afford Montessori Preschool!

2*3 is three groups of things taken two at a time, and 3*2 is two groups of things taken three at a time.

0*1 is one group of things taken zero at a time, and1*0 is zero groups of things taken one at a time.

Let's use pennies. 0 pennies * 1 = 1 group of pennies taken zero at a time. This is a group without any pennies. The null or empty set. 0. I don't object to this, and if I said I did earlier, I take it back! All I am saying is that you cannot perform operations WITH zero. You can certainly perform them ON zero.

1 penny * 0 = 0 groups of pennies taken 1 at a time. This is NO GROUP. Not the null group or 0. But NO GROUP. YOU HAVE NOT MENTALLY PERFORMED A GROUPING OPERATION, SO THERE IS NO GROUP, NOT EVEN THE NULL GROUP, or 0. (And when I say "there is no group," I ~mean~ "there is not any group." )

Laure, I don't have a problem with the word or concept "zero." I just don't think you can use it to ~do~ things. It is simply a place-holder for what is not there, or for what is not done to something else. :yes:

When you state that "kinetic energy is zero" or "the number of chairs in the room is zero," you cannot be stating a measurement or count. You cannot count or measure what is not there. :no: At most, what you mean--as you acknowledge--is that there ~isn't~ any kinetic energy, or there ~aren't~ any chairs in the room, or you ~haven't added~ any chairs to the room.

I can assure you that my dear wife would much prefer that I assure her that I ~haven't~ slept with ~any~ other women, than to tell her that I ~have~ slept with ~zero~ women. Not only would she find the latter a bit odd, but also probably disquieting. For one thing, if you want to reassure someone about what you ~haven't~ done, you probably shouldn't start out by saying that you ~have~ done something and then quantify it as "zero." ;)

I haven't seen the contradiction yet, and a lot of you seem to be trying very hard to display one, with little to show for it yet! :poke:

REB

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## merjet

Roger,

Your post #194 is fatally flawed. You tried to mix your premises, ones I disagree with, in with mine. 1*0 = 0. It being undefined is your premise, not mine.

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