The Opposite of Nothing Is/Isn't Everything

[Alfonso Jones]

At least Bob is not preaching the slaughter of everyone in Roger's neighborhood or city.

There has been some improvement over time...

Michael

Darn! You had to remind him!

Seriously, though: Bob has a serious point here, which should not be missed or ignored just because the wrapping may be a bit offputting.

Bill P (smiling)

[Roger Bissell]

At least Bob is not preaching the slaughter of everyone in Roger's neighborhood or city.

There has been some improvement over time...

Michael

If you'd broaden it to "Roger's state," I wouldn't mind, as long as I had 24 hours' notice!

reb

[Roger Bissell]

I don't care how serious Roger is. It is a case of the ignorant ignoring the mathematical facts. That is ignorance, not scholarship. He mocked and reviled rigor and clarity. And my response to this is a model of constraint. I am refraining from really venting on this.

Give Ba'al a gold star for his manly forbearance. But wait, why vent, when you can content yourself with misrepresenting and distorting? Feh.

Mocking and reviling rigor and clarity? Moi? It is I who pulled the covers off of Monsieur Goedel and his nonsensical slingshot argument by sidestepping all of his obfuscating symbolisms that threw up a big 20th century smokescreen around his sophistry.

No sir! I am not giving a free pass to nonsense and balderdash!

Ba'al Chatzaf

Would Ba'al's post be an example of the self-referential fallacy?

REB

[Roger Bissell]

As for the ontology of 0, I still maintain that there is no better way to consistently look at ~all~ powers, including 0, than to regard them as based on the unit 1 taken times a certain factor a certain number of times -- and that when the certain number of times is 0, you are basically left with the unit 1, to which you do not do anything! You all can stand on your heads (for all I care), manipulating the Associative and Commutative "laws" to show how the zeroth power is the nth power minus the nth power, but if you are not referring to a ~real~ mental operation that is not carried out ~any times~, you are NOT truly understanding the zeroth power.

REB

I love it! A philosopher telling the mathematicians they do not know what they are doing! And just how many solutions to problems have the philosophers come up with? Here is a natural application of the empty set.

If this weren't so ludicrous I would be annoyed. An ignoramus telling the learned that they know nothing. Jeeeeezus!

Ba'al Chatzaf

Bob -

I would suggest a little more courtesy as being appropriate.

In this instance, what would "a little more courtesy" consist of? A slightly less inflammatory bit of name-calling? Ba'al, after all, merely thinks he is speaking the truth, so he is justified in calling a spade a spade.

If the learned are immune from an amateur philosopher-mathematician telling them they are in error, then why is Ba'al so riled up? Is he afraid that anyone listening to me is going to be diverted from proper obeisance to the Gods of Modern Logic and Philosophy of Mathematics?

Modern civilization and technological progress has many, far greater enemies than yours truly! I'm just trying to get people to think straight and realistically about logic and mathematics, instead of swallowing all the arbitrary goop that has been strewn about for the past century and a half.

Excuse me if I don't shrivel in abject shame and intimidation over being told I don't know what I'm talking about.

REB

[thomtg]

Some people would prefer to set up an arbitrarily defined set of rules ("axioms") for manipulating symbols and then play with them, occasionally exclaiming in great surprise when their arbitrarily based manipulations produce a pattern that applies to the real world -- than to acknowledge that mathematics is an ~abstraction from~ the real world, and that, to be valid, every rule and procedure must be based on or ultimately derivable from a concrete mental operation directed toward real objects and their attributes, actions, and relations.

One of the most formidable manipulators of symbols was Goedel, and I took pains to lay bare the equivocation in his slingshot argument that all facts are a single fact. Goedel's reliance on Russell's tottery notion of "definite descriptions" is about as reassuring as Libertarians and Objectivists relying on Greenspan's tottery understanding of the free market, the money system, and capitalism! Spare me!

As for the ontology of 0, I still maintain that there is no better way to consistently look at ~all~ powers, including 0, than to regard them as based on the unit 1 taken times a certain factor a certain number of times -- and that when the certain number of times is 0, you are basically left with the unit 1, to which you do not do anything! You all can stand on your heads (for all I care), manipulating the Associative and Commutative "laws" to show how the zeroth power is the nth power minus the nth power, but if you are not referring to a ~real~ mental operation that is not carried out ~any times~, you are NOT truly understanding the zeroth power.

REB

I agree that axioms need to be grounded. True sentences (axioms, theorems, etc.) are true by virtue of some standard of truth. Objectivists take a version of the correspondence theory of truth to be that standard. And by this standard, axioms in science (including math, philosophy, chemistry, etc.) cannot be arbitrarily defined. They must have a basis in facts of reality. Mathematics in particular is a science of measurement. On this conception, the philosophy of mathematics needs to take cognizant both of man the measurer and of that which can be measured, in establishing criteria for determining truths in mathematics. Hence, I agree that "mathematics is an ~abstraction from~ the real world, and that, to be valid, every rule and procedure must be based on or ultimately derivable from a concrete mental operation directed toward real objects and their attributes, actions, and relations."

On the ontology of the number zero. I would like for us to remember the basic philosophical question: what in reality is the nature of the number zero?

Now, in agreement with the topic of this thread, Roger is proposing the number zero to be a blocker of mathematical operations. (See Post #14.) As I see it, there are three legitimate ways to falsify this proposal: Find a counterexample, reject an implicit premise, or propose an alternate more informative proposal. The zeroth power is one example being discussed. Let it be the test case for the proposal.

Since the proposal is an attempt to answer a philosophical question, any rebuttal, in my view, is necessarily philosophical. Anyone who counterproposes does so therefore in his capacity as a philosopher, and not as anything else, be he mundanely a mathematician, politician, lawyer, botanist, businessman, chemist, teacher, housewife, or trombonist.

[Roger Bissell]

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

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

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

Point 1. 0 is NOT nothing. It is the identity element of an additive commutative group.

Point 2. a * 0 = 0 is a consequence of the distributive law and the definition of subtraction.

a * 0 = a * ( b - b ) = ab - ab = 0.

Point 3. a^0 (a to the zeroth power) is a^(n + -n) = a^n * a^( -n ) = a^n / a^(-n) = 1 provided a is not 0. Otherwise we would get 0/0 which is undefined. And that is why 0^0 is undefined whereas 0^n, n not 0 is 0. At no point does the notion of "nothing" enter into the calculation.

In the wonderful world of computers the "no-op" which leaves the data store unchanged is not "nothing" The program or location counter is advanced by the length of the "no-op" machine code and it does have the aforementioned side effect. So even the "no-op" is something.

Only a philosopher could confuse 0 and "nothing", but not a mathematician or a computer programmer.

I think you are in need of a refresher course in genuine mathematics.

Ba'al Chatzaf

Point 1 response: in an additive communitive group, 0 is the "identity element," because when 0 is "added to" some other number, the other number is left unchanged. "Adding 0" is ~not~ doing something to the other number. It is NOT DOING ANYTHING to that number except AFFIRMING that you are not doing anything to it. Of course affirming that you are not doing anything to a number is ~something~, but it is NOT ADDING. (This pertains to the "no-op" paragraph remarks, too.) 0 is the "additive identity" because what you start with is identical to what you end up with. You have not done anything to it quantitatively; all that you have done to it is to MENTALLY AFFIRM that you have not done anything to it quantitatively. That is the real meaning of x + 0 = x.

Point 2 response: 4 * 2 means 2 groups containing 4 units each, so a * 0 means an absence of groups containing a units each. Since there aren't any groups, there are not any units to count either. Zero, zip, nada. Unlike multiplication by a non-zero number, you are not counting groups with units when you "multiply by" zero. There are ~not any~ multiples of a number when it is "multiplied" by zero. You have NOT DONE ANYTHING quantitative to the number; all that you have done to it is to MENTALLY AFFIRM that you have not counted any groups containing that number of units. That is the real meaning of x * 0 = 0.

Point 3 response: I disagree with Ba'al's claim that 0^0 is undefined. There is a long history of controversy over this issue (See Wikipedia: http://en.wikipedia.org/wiki/Exponentiation and http://en.wikipedia.org/wiki/Empty_product), and I take the other side of it, namely, that 0^0 is 1. Follow me on this, from my starting premise and approach to dealing with exponents. If (as I propose) a^n = 1*(...n factors of a...), then a^0 = 1*(...0 factors of a). You don't multiply 1 by 0 groups of a things, but instead you ~don't~ multiply 1 by ~any~ groups of a things. That is the real meaning of a^0 = 1. You have NOT DONE ANYTHING quantitative to the 1; all that you have done to it is to MENTALLY AFFIRM that you have not multiplied it by any groups of a things.

Specifically in regard to 0^0, as said above, I hold that it is not undefined, but = 1. 0^0 is 1 ~not~ multiplied by any groups of 0 things. There are no such things as ~groups~ of 0 things, any more than there are 0 ~things~, so again you have NOT DONE ANYTHING quantitative to the 1 that is the implicit factor of all exponented numbers; all that you have done to it is to MENTALLY AFFIRM that you have not multiplied it by any groups of 0 things. If this seems at all paradoxical (recall our earlier discussion about propositions about things that don't exist), think of it this way: you can just as easily not be surrounded by a square circle as by a circle. (It's being ~surrounded~ by a square circle that is...somewhat...problemmatic!

In fairness, I will address (i.e., deconstruct) Ba'al's purported proof that 0^0 is undefined. He writes (as above):

a^0 (a to the zeroth power) is a^(n + -n) = a^n * a^( -n ) = a^n / a^(-n) = 1 provided a is not 0. Otherwise we would get 0/0 which is undefined. And that is why 0^0 is undefined whereas 0^n, n not 0 is 0.

(As Laure pointed out, the next to last term in his serial equation should have read a^n/a^n.)

Now, using my approach to interpreting exponential numbers and tracing/interpreting the steps of Ba'al's argument:

1. a^0 = 1 not multiplied by any groups of things taken a at a time

2. a^(n + -n) = 1 multiplied by (n + - n) groups of things taken a at a time

3. a^n*a^(-n) = 1 multiplied by n groups of things taken a at a time multiplied by (-n) groups of things taken a at a time

4. a^n/a^n = 1 multiplied by n groups of things taken a at a time, and divided by 1 multiplied by n groups of things taken a at a time

now, suppose a = 0; then:

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

[Roger Bissell]

Some people would prefer to set up an arbitrarily defined set of rules ("axioms") for manipulating symbols and then play with them, occasionally exclaiming in great surprise when their arbitrarily based manipulations produce a pattern that applies to the real world -- than to acknowledge that mathematics is an ~abstraction from~ the real world, and that, to be valid, every rule and procedure must be based on or ultimately derivable from a concrete mental operation directed toward real objects and their attributes, actions, and relations.

One of the most formidable manipulators of symbols was Goedel, and I took pains to lay bare the equivocation in his slingshot argument that all facts are a single fact. Goedel's reliance on Russell's tottery notion of "definite descriptions" is about as reassuring as Libertarians and Objectivists relying on Greenspan's tottery understanding of the free market, the money system, and capitalism! Spare me!

As for the ontology of 0, I still maintain that there is no better way to consistently look at ~all~ powers, including 0, than to regard them as based on the unit 1 taken times a certain factor a certain number of times -- and that when the certain number of times is 0, you are basically left with the unit 1, to which you do not do anything! You all can stand on your heads (for all I care), manipulating the Associative and Commutative "laws" to show how the zeroth power is the nth power minus the nth power, but if you are not referring to a ~real~ mental operation that is not carried out ~any times~, you are NOT truly understanding the zeroth power.

REB

I agree that axioms need to be grounded. True sentences (axioms, theorems, etc.) are true by virtue of some standard of truth. Objectivists take a version of the correspondence theory of truth to be that standard. And by this standard, axioms in science (including math, philosophy, chemistry, etc.) cannot be arbitrarily defined. They must have a basis in facts of reality. Mathematics in particular is a science of measurement. On this conception, the philosophy of mathematics needs to take cognizant both of man the measurer and of that which can be measured, in establishing criteria for determining truths in mathematics. Hence, I agree that "mathematics is an ~abstraction from~ the real world, and that, to be valid, every rule and procedure must be based on or ultimately derivable from a concrete mental operation directed toward real objects and their attributes, actions, and relations."

On the ontology of the number zero. I would like for us to remember the basic philosophical question: what in reality is the nature of the number zero?

Now, in agreement with the topic of this thread, Roger is proposing the number zero to be a blocker of mathematical operations. (See Post #14.) As I see it, there are three legitimate ways to falsify this proposal: Find a counterexample, reject an implicit premise, or propose an alternate more informative proposal. The zeroth power is one example being discussed. Let it be the test case for the proposal.

Since the proposal is an attempt to answer a philosophical question, any rebuttal, in my view, is necessarily philosophical. Anyone who counterproposes does so therefore in his capacity as a philosopher, and not as anything else, be he mundanely a mathematician, politician, lawyer, botanist, businessman, chemist, teacher, housewife, or trombonist.

Just to clarify: there are also ~female~ mathematicians, politicians, lawyers, botanists, businessmen(?), chemists, teachers, and trombonists. And male housewives(?).

Whatever Ba'al is in his mundane identity, I think it's amusing that he will have to function philosophically (logically, rationally) in addressing this question in order to deal with it validly and credibly. In other words, to stop all the name-dropping and name-calling. Quite a challenge, Ba'al. Are you up to it?

REB

[tjohnson]

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).

Speak for yourself, this is how your mental processes work. Have fun with that!

[BaalChatzaf]

On the ontology of the number zero. I would like for us to remember the basic philosophical question: what in reality is the nature of the number zero?

In reality, which means in fact, 0 is the additive identity of the semi group of integers under addition. That is what 0 is. Now what are the integers? This was rigorously answered by Peano.

The integers are a set of elements S and a function s from S onto S with a distinguished element 0.

1. s maps S into S, 1 to 1. That is if x != y then s(x) != s(y)

2. For each element x in S, s(x) != x

3.1 For each element in x which is not 0 there exists x* in S such that s(x*) = x

3.2 s(s(x*)) = s(x)

4. For any non empty subset P contained in S if

4.1 0 in P

4.2 x in P implies s(x) in P

then

4.3 P = S

(Postulate 4 is the postulate of Mathematical Induction, not to be confused with empirical induction)

These are the well-known Peano Postulates for integers (whole numbers).

Addition (+) is defined by recursion (actually primitive recursion).

x + 0 = x (definition)

x + sy = s(x + y) (definition)

(NB: if you are not heavy in math you will have to struggle with recursion for a little bit before you get it).

The handy number 1 is defined to be s(0). It is very handy since it turns out to be (this can be proved) the identity of the semi-group of integers under multiplication. If X is a non empty set and when ever x, y in X x = y. That is to say there do not exist x, x' in X such that x != x'. The handy number 1 can be identified with the cardinality of X. (This is hard to prove so it won't be done here)

For background see http://en.wikipedia.org/wiki/Peano_postulates

and http://en.wikipedia.org/wiki/Primitive_recursion

Be sure to follow the pointers in the above articles a little ways in.

These postulates with the aid of primitive recursion can be used to define addition (+) (see above) multiplication * and a linear ordering < under which S is well ordered. 0 is the least element with respect to < which is to say for all x in S which are not 0, 0 < x. Also x < s(x) (part of the primitive recursive definition of < )

Among the things we can prove is that (S, +, 0) is a semi-group with identity 0. We can algebraically extend S to a larger set S* by constructing negative numbers (so-called). In this set S* the equation a + x = b where a, b in S* has a unique solution x which we can write

x = b - a. Under this extension of the operation +, (S*, +, 0) is a group with identity element 0. It is a special kind of group called a commutative group which mean x + y = y + x for all x, y in S*. Both S and S* have interesting properties. S has a smallest element (under <, namely 0 ) but no largest element S. The counting numbers ( including 0 ) satisfy the Peano Postulates. Just an interesting side note here. The Peano Postulates are satisfied by the integers, but there exist an infinite class of other sets which also satisfy them. This was proved by Thoralf Skolem back in the early 1930's. Hoodathunkit? So the Peano Postulates apply to the integers (in the intuitive sense) but do not uniquely and categorically define them.

From this humble beginning all of the arithmetic of integers can be derived including elementary number theory ( integers number theory which does not rely on the theory of real numbers ).

And now a general remark. Search the physical world from one end of the cosmos to the other and you will not find an integer as a physical or material entity. Integers exist only in our heads or in the thinking organs of sufficiently intelligent sentient beings if there are such other than humans. In a sense, all numbers are Imaginary Numbers. If every sentient being in the cosmos ceased to exist, there would be no numbers.

Ba'al Chatzaf

[Laure]

Roger, the simplest way to explain why 0^0 should be thought of as undefined is to note that for nonzero n, 0^n = 0, and n^0 = 1. You'll accept that, right? So, what happens at zero? Is 0^0 = 0 or is 0^0 = 1? It depends which formula you use. Since it can't be both zero and one, it must be undefined.

Also, you've got a contradiction in your post 106. First you say,

Point 2 response: 4 * 2 means 2 groups containing 4 units each, so a * 0 means an absence of groups containing a units each. Since there aren't any groups, there are not any units to count either. Zero, zip, nada. Unlike multiplication by a non-zero number, you are not counting groups with units when you "multiply by" zero. There are ~not any~ multiples of a number when it is "multiplied" by zero. You have NOT DONE ANYTHING quantitative to the number; all that you have done to it is to MENTALLY AFFIRM that you have not counted any groups containing that number of units. That is the real meaning of x * 0 = 0.

Then you say,

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

Here, you are taking (1 * (0^n)) / (1 * (0^n)) and you say 0^n is "nothing", i.e. 0. So, you are here assuming that 1 * 0 = 1, when above, you said that x * 0 = 0.

*edit* Also, defining 0/0 as 1, as you want to do, can allow you to "prove" contradictions, and that's not a good thing! Here's a link to a good example of this: *Algebra Quandry*.

I think you're trying to do math with the right side of your brain.

Question for you: explain to me what a number like, say, 8 to the 0.4325 power corresponds to in reality?

Edited May 24, 2009 by Laure

[Roger Bissell]

On the ontology of the number zero. I would like for us to remember the basic philosophical question: what in reality is the nature of the number zero?

In reality, which means in fact, 0 is the additive identity of the semi group of integers under addition. That is what 0 is. Now what are the integers? This was rigorously answered by Peano.

The integers are a set of elements S and a function s from S onto S with a distinguished element 0.

1. s maps S into S, 1 to 1. That is if x != y then s(x) != s(y)

2. For each element x in S, s(x) != x

3.1 For each element in x which is not 0 there exists x* in S such that s(x*) = x

3.2 s(s(x*)) = s(x)

4. For any non empty subset P contained in S if

4.1 0 in P

4.2 x in P implies s(x) in P

then

4.3 P = S

(Postulate 4 is the postulate of Mathematical Induction, not to be confused with empirical induction)

These are the well-known Peano Postulates for integers (whole numbers).

Addition (+) is defined by recursion (actually primitive recursion).

x + 0 = x (definition)

x + sy = s(x + y) (definition)

(NB: if you are not heavy in math you will have to struggle with recursion for a little bit before you get it).

The handy number 1 is defined to be s(0). It is very handy since it turns out to be (this can be proved) the identity of the semi-group of integers under multiplication. If X is a non empty set and when ever x, y in X x = y. That is to say there do not exist x, x' in X such that x != x'. The handy number 1 can be identified with the cardinality of X. (This is hard to prove so it won't be done here)

For background see http://en.wikipedia.org/wiki/Peano_postulates

and http://en.wikipedia.org/wiki/Primitive_recursion

Be sure to follow the pointers in the above articles a little ways in.

These postulates with the aid of primitive recursion can be used to define addition (+) (see above) multiplication * and a linear ordering < under which S is well ordered. 0 is the least element with respect to < which is to say for all x in S which are not 0, 0 < x. Also x < s(x) (part of the primitive recursive definition of < )

Among the things we can prove is that (S, +, 0) is a semi-group with identity 0. We can algebraically extend S to a larger set S* by constructing negative numbers (so-called). In this set S* the equation a + x = b where a, b in S* has a unique solution x which we can write

x = b - a. Under this extension of the operation +, (S*, +, 0) is a group with identity element 0. It is a special kind of group called a commutative group which mean x + y = y + x for all x, y in S*. Both S and S* have interesting properties. S has a smallest element (under <, namely 0 ) but no largest element S. The counting numbers ( including 0 ) satisfy the Peano Postulates. Just an interesting side note here. The Peano Postulates are satisfied by the integers, but there exist an infinite class of other sets which also satisfy them. This was proved by Thoralf Skolem back in the early 1930's. Hoodathunkit? So the Peano Postulates apply to the integers (in the intuitive sense) but do not uniquely and categorically define them.

From this humble beginning all of the arithmetic of integers can be derived including elementary number theory ( integers number theory which does not rely on the theory of real numbers ).

And now a general remark. Search the physical world from one end of the cosmos to the other and you will not find an integer as a physical or material entity.

The three people currently residing in my apartment find this rather amusing. Someone asked to find an example of three in my apartment would immediately point to me, my wife, and my daughter. Or our three computers. Or any three objects. An integer is not ~an~ entity, but ~some~ (a group of) entities, however many that integer is. There are no integers (or groups) apart from entities--but there are no entities that are not an integer in number, whether as individuals (one) or as part of a number of individuals. I am a proud individual (one) AND a member of an indefinitely large number of groups of different numbers of things. That is WHAT we "construct" the numbers we use mentally FROM.

Any other basis for number systems is PARASITIC UPON this natural basis of numbers. The Peano Concerto you just performed is a prime example.

Integers exist only in our heads or in the thinking organs of sufficiently intelligent sentient beings if there are such other than humans. In a sense, all numbers are Imaginary Numbers. If every sentient being in the cosmos ceased to exist, there would be no numbers.

Ba'al Chatzaf

Correction: there would be no ~numbered~ (counted) things. But there are three human beings in my apartment now, whether or not any one numbers (counts) them. That ~is~ how many human beings there ~are~ in my apartment, and that remains true (an "objective" fact), whether or not there is anyone to perceive and count them. Even if every sentient being in the cosmos suddenly died, there would still be three dead humans in my apartment now. You are confusing number (a quantity) with count (a measurement of quantity).

Number/quantity is intrinsic to reality. There are no things that do not have ~some~ number/quantity, apart from human or other awareness of it. Count/measurement is "objective"--in the Randian sense of: the product of a consciousness being aware of number/quantity.

In confusing number/quantity with count/measurement, you are conflating the intrinsic with the objective, just as surely as if you were ignoring attributes, which exist independent of consciousness, and focusing only on qualities, which are our awareness of independently existing attributes. Certainly qualities and count/measurement do not exist apart from sentient beings. But attributes and number/quantity do.

If a tree fell in the forest and crushed three deaf humans, but there were no one there to hear it and to count the three dead humans, would it still have made a sound and would there still be three dead deaf humans? Yes. There would have been no experienced sensory quality of sound and no counting of the three dead deaf humans, that's all.

REB

[BaalChatzaf]

Correction: there would be no ~numbered~ (counted) things. But there are three human beings in my apartment now, whether or not any one numbers (counts) them. That ~is~ how many human beings there ~are~ in my apartment, and that remains true (an "objective" fact), whether or not there is anyone to perceive and count them. Even if every sentient being in the cosmos suddenly died, there would still be three dead humans in my apartment now. You are confusing number (a quantity) with count (a measurement of quantity).

There are three people in your apartment but not the number three, which is my point.

In a one sense all numbers are nothing (i.e. nothing material or physical) and in another sense they are all something (something thought of). And that includes zero.

Ba'al Chatzaf

[Roger Bissell]

Roger, the simplest way to explain why 0^0 should be thought of as undefined is to note that for nonzero n, 0^n = 0, and n^0 = 1. You'll accept that, right? So, what happens at zero? Is 0^0 = 0 or is 0^0 = 1? It depends which formula you use. Since it can't be both zero and one, it must be undefined.

No, I will NOT accept that. 0^n = 1, just as n^0 = 1. See below.

Also, you've got a contradiction in your post 106. First you say,

Point 2 response: 4 * 2 means 2 groups containing 4 units each, so a * 0 means an absence of groups containing a units each. Since there aren't any groups, there are not any units to count either. Zero, zip, nada. Unlike multiplication by a non-zero number, you are not counting groups with units when you "multiply by" zero. There are ~not any~ multiples of a number when it is "multiplied" by zero. You have NOT DONE ANYTHING quantitative to the number; all that you have done to it is to MENTALLY AFFIRM that you have not counted any groups containing that number of units. That is the real meaning of x * 0 = 0.

Then you say,

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

Here, you are taking (1 * (0^n)) / (1 * (0^n)) and you say 0^n is "nothing", i.e. 0. So, you are here assuming that 1 * 0 = 1, when above, you said that x * 0 = 0.

*edit* Also, defining 0/0 as 1, as you want to do, can allow you to "prove" contradictions, and that's not a good thing! Here's a link to a good example of this: *Algebra Quandry*.

Correction: I do NOT define 0/0 as 1, because I do not define 0^n as 0! 0^n does not equal 0. 0^1 does not equal 0. 0 to any power is 1, just as any number to the zero power is 1.

I think you're trying to do math with the right side of your brain.

I would think that would be better than trying to do it with the ~wrong~ side of my brain! :poke:

Question for you: explain to me what a number like, say, 8 to the 0.4325 power corresponds to in reality?

What does 4325 correspond to in reality? If you know that, then I don't think it takes much longer to figure out the answer to your own question, ~if~ you take my approach.

Yes, it looks rather formidable--tedious, at the very least. But remember, if any mathematical expression is valid, it must be derived from and ultimately reducible to something in reality or some mental procedure in regard to reality. Also remember: in general (I claim) powers mean: multiply (or don't multiply) the unit 1 by some specified number of factors of a certain number. Just as we learn that 4325 means 4 groups of 1000 things + 3 groups of 100 things + 2 groups of 10 things + 5 things, I think that the power 0.4325 can be (ultimately) unpacked in the same general way. Please don't make me go through the steps on this. My head hurts already. :no:

*edit. Oh, all right...Here is the general principle: For any real number, r, a positive rational exponent, m/n, indicates that the unit 1 is to be considered as having been multiplied by one of r's n equal factors a total of m times. In your example, the positive rational exponent is 4325/10,000. That means that 8^.4325 is 1 muiltiplied by one of 8's 10,000 equal factors a total of 4325 times. If this sounds bizarre, it is in principle no more bizarre than saying (as we learn in high school) that 8^2/3 is the square of the cube root of 8.

Oh, you wanted me to ~compute~ the answer? Sorry, the "right side" of my brain doesn't provide details, just method! For further details, see this: essay on fractional exponents

REB

[BaalChatzaf]

Any other basis for number systems is PARASITIC UPON this natural basis of numbers. The Peano Concerto you just performed is a prime example.

By the way it is not a Concerto. That is a musical form, not a mathematical form.

An interesting way of putting the origin of mathematical systems. Of course all mathematical systems originate from experiences. Every thought we have is PARASITIC upon our experience. The Peano Arithmetic is a rigorous and logical grounding for arithmetic (one of several). It is part of a progression from the heuristic/empirical to the logical/theoretical. What started as a system for counting head of cattle or pebbles or rocks is now the basis for electronic computers.

Non-euclidean geometry is PARASITIC on experience in limited nearly flat areas and the Euclidean Geometry which originated in Greece. What this PARASITE has given us is, among other things, the GPS which is derived from Einstein's General Theory of Relativity which is a non-Euclidean (semi-Reimannian to be exact) system of Geometry for organizing physical events in a four dimensional space-time manifold. In Einstein's hands, physics is geometry. All advances are PARASITIC upon their predecessors. All beginnings are hard and spare and are necessary to achieve greater and better things.

Rome was not built in a day and arithmetic was not perfected in two thousand years.

Ba'al Chatzaf

[BaalChatzaf]

*edit. Oh, all right...Here is the general principle: For any real number, r, a positive rational exponent, m/n, indicates that the unit 1 is to be considered as having been multiplied by one of r's n equal factors a total of m times. In your example, the positive rational exponent is 4325/10,000. That means that 8^.4325 is 1 muiltiplied by one of 8's 10,000 equal factors a total of 4325 times. If this sounds bizarre, it is in principle no more bizarre than saying (as we learn in high school) that 8^2/3 is the square of the cube root of 8.

Oh, you wanted me to ~compute~ the answer? Sorry, the "right side" of my brain doesn't provide details, just method! For further details, see this: essay on fractional exponents

REB

The Greeks could do square roots (ruler and compass) and could do some cube roots (but not all). Proving that 8 had a 10,000 th root could not be done until the 19th century. One thing the philosophers never gave us was logarithms.

How about 8 to the pi power. Or e to the pi power where e is Euler's Number (approx 2.7182818... )

Read up on how Eudoxus broke numbers out of the ratio box and made any ratio sensible. If the Greeks only had algebra (1600 years in Eudoxus' future) they would have found the (so-called) real numbers.

Ba'al Chatzaf

[tjohnson]

If every sentient being in the cosmos ceased to exist, there would be no numbers.

There would still be the relations that numbers represent. Numbers are symbols we invented to express relations, like the relation of the circumference to the diameter of a circle.

[Dragonfly]

Correction: I do NOT define 0/0 as 1, because I do not define 0^n as 0! 0^n does not equal 0. 0^1 does not equal 0. 0 to any power is 1,

Oh, that's interesting... so 0^1 = 0^2 = 1 or in other words, 0 = 1 and 0 * 0 = 1 ? That is Objectivist mathematics?

[Laure]

Correction: I do NOT define 0/0 as 1, because I do not define 0^n as 0! 0^n does not equal 0. 0^1 does not equal 0. 0 to any power is 1,

Oh, that's interesting... so 0^1 = 0^2 = 1 or in other words, 0 = 1 and 0 * 0 = 1 ? That is Objectivist mathematics?

Bingo! Roger, you have just stated that 0 = 1. You can certainly come up with your own mathematical system where 0 = 1, but I thought you liked math to be useful in practical applications!

[BaalChatzaf]

Oh, you wanted me to ~compute~ the answer? Sorry, the "right side" of my brain doesn't provide details, just method! For further details, see this: essay on fractional exponents

REB

I read your essay. Not bad. The approach given therein is symptomatic of the the limitations of Greek mathematics. The Greeks had no way to generalize exponentiation to irrational numbers. So finding the square root of 2th root of 2 was impossible for them. Eudoxus inadvertently prepared a path out of that box, but it required algebra to exploit the Eudoxian insight to ratios. Eudoxus was one of the mathematical Greats of all time. If he could have been brought to his Future in a time machine and brought up to speed on modern algebraic and analytic techniques he would have become a Champion in the field of mathematics. Ditto for Archimedes. Eudoxus almost invented the analysis of real numbers, and Archimedes almost invented infinitesimal calculus. All they lacked was algebra which was invented by Arab Mathematicians around 1100 or 1200 c.e. and a decent positional systems for number arithmetic.

The bottom line is that the philosophical approach to mathematics leads to some real road blocks. One needs an analytical method (method of limits) to pole vault over these barriers. The Greeks simply did not have the tools. A book you should read on how mathematics flourished through the centuries is: The Development of Mathematics by E.T.Bell. It was first published in 1940 and Dover Books has a nice edition at a reasonable price.

It is interesting to note that using the Eudoxian treatment of ratios of irrational quantities, the real numbers can be derived without first producing the algebra of rational numbers and using either Dedekind Cuts or Cauchy Sequences on them.

See The Eudoxus Real Numbers by R.D. Arthan

arXivmath/0405454v1 [math.HO] 24 May 2004

A real nifty paper.

Ba'al Chatzaf

[Michael Stuart Kelly]

That is Objectivist mathematics?

Dragonfly,

Objectivism was not being discussed and I seriously doubt Roger is engaged in trying to set up a tribal mathematics system and calling it Objectimath or Objectifractions or something.

(How about Objectizero for the present discussion? )

I think Roger is just discussing math and using his own honest independent thinking.

Roger.

Not Objectivism.

Whether you agree with him or not, or whether you think he is in error or not, it is a huge mistake to conclude that he has only been studying math within the Objectivist orbit.

Your bias is showing...

I thought objectivity meant zero bias. Apparently here is one place where zero does not equal zero.

Michael

[BaalChatzaf]

That is Objectivist mathematics?

Dragonfly,

Objectivism was not being discussed and I seriously doubt Roger is engaged in trying to set up a tribal mathematics system and calling it Objectimath or Objectifractions or something.

(How about Objectizero for the present discussion? )

I think Roger is just discussing math and using his own honest independent thinking.

Roger.

Not Objectivism.

Whether you agree with him or not, or whether you think he is in error or not, it is a huge mistake to conclude that he has only been studying math within the Objectivist orbit.

Your bias is showing...

I thought objectivity meant zero bias. Apparently here is one place where zero does not equal zero.

Michael

It is fortunate for both mathematics and Objectivism that there is no such thing as Objectivist Mathematics.

Ba'al Chatzaf

[Alfonso Jones]

(snip)

No, I will NOT accept that. 0^n = 1, just as n^0 = 1. See below.

(snip)

REB

Roger -

Please clarify if the material I quote from you above is a typo, or is what you actually mean.

Bill P

[tjohnson]

The Development of Mathematics by E.T.Bell.

FYI, E.T. Bell proofread Korzybski's sections on mathematics.

No satisfactory justification has ever been given for connecting in any way the

consequences of mathematical reasoning with the physical world. (22) E. T. BELL

That is precisely what common sense is for, to be jarred into uncommon sense. One of the chief

services which mathematics has rendered the human race in the past century is to put 'common

sense' where it belongs, on the topmost shelf next to the dusty canister labeled 'discarded nonsense.'

(23) E. T. BELL

Traditionally, mathematics was divided into two branches: one was called

arithmetic, dealing with numbers; the other was called geometry, and dealt with

such entities as 'line', 'surface', 'volume', . Once Descartes, lying in bed ill,

watched the branches of a tree swaying under the influence of a breeze. It occurred

to him that the varying distances of the branches from the horizontal and vertical

window frames could be expressed by numbers representing measurements of the

distances. An epoch-making step was taken: geometrical relations were expressed

by numerical relations; it meant the beginning of analytical geometry and the

unification and arithmetization of mathematics.

Further investigation by the pioneers Frege, Peano, Whitehead, Russell, Keyser,

and others has revealed that 'number' can be expressed in 'logical terms'—a quite

important discovery, provided we have a valid 'logic' and structurally correct non-el

terms.

Traditionally, too, since Aristotle, and, in the opinion of the majority, even

today, mathematics is considered as uniquely connected with quantity and

measurement. Such a view is only partial, because there are many most important

and fundamental branches of mathematics which have nothing to do with quantity

or measurement—as, for instance, the theory of groups, analysis situs, projective

geometry, the theory of numbers, the algebra of 'logic', .

Emphasis mine.

[Roger Bissell]

(snip)

No, I will NOT accept that. 0^n = 1, just as n^0 = 1. See below.

(snip)

REB

Roger -

Please clarify if the material I quote from you above is a typo, or is what you actually mean.

Bill P

Bill, the material you quoted from me above is ~not~ a typo. It is precisely what I mean. [Clarification on 5/5/13: However, the first part of it is incorrect. See below...]

Here are the proofs, based on my interpretation of exponented numbers.

1. 0^n = 1 multiplied by n factors of 0, which means 1 multiplied by 0 n times. You cannot mutiply 1 by 0 any times, so 1 is NOT multiplied, so 0^n = 1. [Correction 5/5/13: this is a sloppy misinterpretation of what is going on with 0^n, and is partially incorrect. It has to be split into two cases: n = 0 and n not= 0. When n not= 0, 0^n means 1 multiplied by n factors of 0, which means 1 multiplied by 0 n times. You cannot multiply 1 by 0 any times, so 1 is NOT multiplied, but you still have a COUNT of items in 1 group not containing any items (1 * 0) to contend with. This RESULT is 0, which is then multiplied by any additional factors of 0 specified in the exponent. 0 * 0 again has no PRODUCT, but there is a COUNT of the number of items where you don't have any groups not containing any items, which is 0. When n = 0, however, 0^n = 0^0, which means 1 NOT multiplied by ANY factors of 0, which is simply 1, by my perspective of 0 as an operation blocker and of exponented numbers actually being fully expressed as the unit 1 multiplied by some number some number of times.]

2. n^0 = 1 multiplied by 0 factors of n, which means 1 is NOT multiplied, so n^0 = 1.

Is this starting to sink in yet, Ba'al? It's not rocket science, and I don't think it will get in the way of rocket science either! http://www.objectivistliving.com/forums/public/style_emoticons/#EMO_DIR#/laugh.gif

REB

[Roger Bissell]

Any other basis for number systems is PARASITIC UPON this natural basis of numbers. The Peano Concerto you just performed is a prime example.

By the way it is not a Concerto. That is a musical form, not a mathematical form.

That was a JOKE, a PUN. <arrrgh>

An interesting way of putting the origin of mathematical systems. Of course all mathematical systems originate from experiences. Every thought we have is PARASITIC upon our experience. The Peano Arithmetic is a rigorous and logical grounding for arithmetic (one of several). It is part of a progression from the heuristic/empirical to the logical/theoretical. What started as a system for counting head of cattle or pebbles or rocks is now the basis for electronic computers.

Non-euclidean geometry is PARASITIC on experience in limited nearly flat areas and the Euclidean Geometry which originated in Greece. What this PARASITE has given us is, among other things, the GPS which is derived from Einstein's General Theory of Relativity which is a non-Euclidean (semi-Reimannian to be exact) system of Geometry for organizing physical events in a four dimensional space-time manifold. In Einstein's hands, physics is geometry. All advances are PARASITIC upon their predecessors. All beginnings are hard and spare and are necessary to achieve greater and better things.

Rome was not built in a day and arithmetic was not perfected in two thousand years.

Ba'al Chatzaf

The point of my using the term "parasitic," was to underscore abstract mathematics' utter dependence on reality, and to stress that its validity is tied to its traceable derivation from reality. If that relationship is not traceable, then any application it has to the real world is accidental.

REB