The Opposite of Nothing Is/Isn't Everything

[BaalChatzaf]

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.

Question: what is the square of 0? It is 0*0. And what is 0*0 ? It is 0. In fact 0^n, for n > 0 is 0.

Proof: 0^1 = 0 so the thing is true for n = 1.

Suppose 0^n = 0. Then 0^(n+1) = (0^n)*(0^1) = 0*0 = 0

The induction completes the proof. For all n >= 1 0^n = 0

Ba'al Chatzaf

Edited May 24, 2009 by BaalChatzaf

[Roger Bissell]

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?

No, I am NOT saying 0 = 1. And I am not saying that 0 * 0 is 1. 0 * 0 is undefined. [Clarification on 5/5/13: by "undefined," I mean to say that there is no PRODUCT for 0 * 0, that 0 blocks the operation of multiplication from taking place. There is still a ~count~ of units that constitutes the RESULT of the ATTEMPTED OPERATION OF MULTIPLICATION, just as there is for 0 * n and n * 0, and that result is 0 units. Viz. here, when you try to count no groups containing no items, there aren't any groups of items and there aren't any items, so there is no count, which is represented as 0. The result is the same as in conventional arithmetic, but the result of the non-operation, rather than a product of an operation, is based on my perspective of 0 as an operation blocker.]

If 0 * 0 = 0, as you doubtless believe, then the inverse operation 0/0 would have to = 0, too. But it is undefined, as we are all taught in grade school. I am asserting that not only is 0/0 undefined, but also 0 * 0 is undefined, as is ANY number times zero. [Clarification 5/5/13: Although 0/0 and 0 * 0 are "undefined," i.e., do not have a QUOTIENT or a PRODUCT, they both have RESULTS, viz., 0. Any other number n/0 does not have a determinate result, but 0/0 ~must~ have the result of 0, if 0 * 0 has the result of 0. I provide the argument for this in my forthcoming e-book, "How the Martians Discovered Algebra."]

[Clarification 5/5/13: I have deleted the rest of this comment, because I no longer hold that 0 to any and every power is 1. 0^0 is 1, because it is the unit 1 NOT multiplied by ANY factors of 0, which is simply 1. But 0 to any other power is the unit 1 MULTIPLIED by n (non-zero) factors of 0. And though 0 being a factor is an operation blocker, multiplication is being attempted (unlike for 0^0 where the 0 power specifies NOT multiplying), and attempted multiplication of 1 by 0 yields no PRODUCT, but a COUNT of 0, because it is attempting to count the units in 1 group that does not contain any items, which does not exist, and so the RESULT or count is 0.]

And no, it's not Objectivist mathematics, but I believe it's consistent with the Objectivist metaphysics, and with Aristotle's metaphysics. It's my own view. Any Objectivist who agrees with it does so at his/her own risk! http://www.objectivistliving.com/forums/public/style_emoticons/#EMO_DIR#/poke.gif

REB

[Roger Bissell]

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!

Bingo, my pasty white ass.

Read my reply to Dragonfly just above.

Yours truly, Flagondry

[Roger Bissell]

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.

...for an amateur philosopher with rusty mathematics skills? I appreciate the compliment. And I'm glad that I've been able to help some bewildered students get a better handle on exponents.

And thanks for the suggested readings. I do have the Bell book, but it is deep in a pile of boxes of books in the garage. I have some other more recent books on history of mathematics, so I will check them out in relation to your comments.

Also, I would recommend Pat Corvini's ARI-sponsored lectures on Eudoxus and Archimedes etc.

All 4 now,

REB

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

[Roger Bissell]

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.

Question: what is the square of 0? It is 0*0. And what is 0*0 ? It is 0. In fact 0^n, for n > 0 is 0.

Proof: 0^1 = 0 so the thing is true for n = 1.

Suppose 0^n = 0. Then 0^(n+1) = (0^n)*(0^1) = 0*0 = 0

The induction completes the proof. For all n >= 1 0^n = 0

Ba'al Chatzaf

0 does not have a square. 0*0 is undefined. 0^n is NOT 0*0 n times. It is 1 * n factors of zero. There is no number that corresponds to n factors of zero. 0*0 n times is just as undefined as 0*0 is. So 0^n = 1.

No induction necessary here.

REB

[BaalChatzaf]

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

Everything that exists is utterly dependent on reality. According to Wittgenstein the World is the Totality of Facts (Tractatus Prop 1.1)

The relationship of abstract mathematics to the real world has been a puzzling question to many. Why should an abstract theory like the theory of 4 x 4 tensors and non-euclidean manifolds (Riemannian Geometry) lead to a model of gravitation? Why should unit vectors in a Hilbert Space model quantum states? Why should the theory of knots be useful in physics? (See http://www.nytimes.com/1989/02/21/science/...to-physics.html )

See

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

which is Wigner's Essay: The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Wigner does not answer the question, by the way. Nobody knows so far.

Ba'al Chatzaf

[Dragonfly]

0 does not have a square. 0*0 is undefined. 0^n is NOT 0*0 n times. It is 1 * n factors of zero. There is no number that corresponds to n factors of zero. 0*0 n times is just as undefined as 0*0 is. So 0^n = 1.

Let's see how this works in practice. In Newtonian mechanics the kinetic energy of a mass m with velocity v equals 1/2 mv^2. So for very small velocities the kinetic energy is almost zero (sounds logical, doesn't it?). But if the velocity becomes zero, in your theory the kinetic energy suddenly jumps to 1/2 m! In other words, an object that doesn't move at all has more kinetic energy than the same object moving slowly. Hmmm, I still think that the old-fashioned mathematics, which is not consistent with Objectivist metaphysics, is more practical...

[tjohnson]

Wigner does not answer the question, by the way. Nobody knows so far.

Korzybski has a theory. Briefly, mathematics is similar in structure to the world and to the human nervous system.

[BaalChatzaf]

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.

Question: what is the square of 0? It is 0*0. And what is 0*0 ? It is 0. In fact 0^n, for n > 0 is 0.

Proof: 0^1 = 0 so the thing is true for n = 1.

Suppose 0^n = 0. Then 0^(n+1) = (0^n)*(0^1) = 0*0 = 0

The induction completes the proof. For all n >= 1 0^n = 0

Ba'al Chatzaf

0 does not have a square. 0*0 is undefined. 0^n is NOT 0*0 n times. It is 1 * n factors of zero. There is no number that corresponds to n factors of zero. 0*0 n times is just as undefined as 0*0 is. So 0^n = 1.

No induction necessary here.

REB

Your math is incorrect.

0*n = 0 when n is not 0. O.K. 0*n - 0*n = 0 since 0 - 0 = 0. But by the distributive law

0*n - 0*n = 0*(n - n) = 0 * 0 . So 0*0 = 0. You do like the distributive law, don't you? In a ring (a ring is an algebraic structure with + and * that is a group with respect to + and * distributes across +) 0*anything = 0. Are you saying that 0 - 0 is not defined?

Ba'al Chatzaf

[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, just as any number to the zero power is 1.

Question: what is the square of 0? It is 0*0. And what is 0*0 ? It is 0. In fact 0^n, for n > 0 is 0.

Proof: 0^1 = 0 so the thing is true for n = 1.

Suppose 0^n = 0. Then 0^(n+1) = (0^n)*(0^1) = 0*0 = 0

The induction completes the proof. For all n >= 1 0^n = 0

Ba'al Chatzaf

0 does not have a square. 0*0 is undefined. 0^n is NOT 0*0 n times. It is 1 * n factors of zero. There is no number that corresponds to n factors of zero. 0*0 n times is just as undefined as 0*0 is. So 0^n = 1.

No induction necessary here.

REB

*sigh* Roger, now you're saying that 0*0 is undefined, but 0^n = 1. But 0^2 is just another way of writing 0*0. So, if 0*0 is undefined, then 0^2 must be undefined; it's the SAME THING. Are you saying that 0^n is undefined for n=2, but 1 for n not equal to 2??

Here's another grade-school example of why that just doesn't work.

What's (1-1) * (1-1) ? If we simplify what's inside the parentheses first, we get 0 * 0, which in RogerLand is Undefined. But, if we use the good old "FOIL" method to distribute it out, we get 1 - 1 - 1 + 1. Compute it left to right, we get 0 - 1 + 1, or -1 + 1, or 0. So (1 - 1) * (1 - 1) both equals zero and is undefined. In your math, can something equal zero AND be undefined? Or don't you believe that (x + y)*(z + w) = x*z + x*w + y*z + y*w?

Your system has a contradiction. Face up to it.

Edited May 24, 2009 by Laure

[BaalChatzaf]

In your math, can something equal zero AND be undefined? Or don't you believe that (x + y)*(z + w) = x*z + x*w + y*z + y*w

The distributive law wrapped up in a pretty package.

Ba'al Chatzaf

Edited May 24, 2009 by BaalChatzaf

[BaalChatzaf]

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

"Derivative" would have been a better word. It has less of a negative or hostile connotation.

Ba'al Chatzaf

[thomtg]

[...]

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

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.

[...]

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.

[...]

0 * 0 is undefined.

If 0 * 0 = 0, as you doubtless believe, then the inverse operation 0/0 would have to = 0, too. But it is undefined, as we are all taught in grade school. I am asserting that not only is 0/0 undefined, but also 0 * 0 is undefined, as is ANY number times zero.

What I am saying that 0 TO ANY POWER is 1. Not 0 * 0 * 0 * 0 (n times) = 1, because that is undefined -- but 1 * (0 factors of zero), which means that 1 is NOT MULTIPLIED BY ANYTHING.

And no, it's not Objectivist mathematics, but I believe it's consistent with the Objectivist metaphysics, and with Aristotle's metaphysics. It's my own view. Any Objectivist who agrees with it does so at his/her own risk! :poke:

:thumbsup:

[...]

*sigh* Roger, now you're saying that 0*0 is undefined, but 0^n = 1. But 0^2 is just another way of writing 0*0. So, if 0*0 is undefined, then 0^2 must be undefined; it's the SAME THING. Are you saying that 0^n is undefined for n=2, but 1 for n not equal to 2??

Here's another grade-school example of why that just doesn't work.

What's (1-1) * (1-1) ? If we simplify what's inside the parentheses first, we get 0 * 0, which in RogerLand is Undefined. But, if we use the good old "FOIL" method to distribute it out, we get 1 - 1 - 1 + 1. Compute it left to right, we get 0 - 1 + 1, or -1 + 1, or 0. So (1 - 1) * (1 - 1) both equals zero and is undefined. In your math, can something equal zero AND be undefined? Or don't you believe that (x + y)*(z + w) = x*z + x*w + y*z + y*w?

Your system has a contradiction. Face up to it.

Laure,

Let me try to channel Roger's spirit and answer your questions.

[Channeling Roger:]

If we accept the proposal and interpret the number zero ontologically to stand for a mental blocker of mathematical operations (from

Post #105

), then indeed:

1. Any number multiplied by 0 must mean that that number is

not multiplied

[period]. The operation is blocked. The operation is not processed. It is in this sense that the product is "undefined." (See above.) So, 0*0 is undefined, as 44*0 is undefined.

2. Now, for the power operation x^y, the explicit form of the operation is 1 * x^y, meaning 1 times y groups of things taken x at a time. Thus, 0^n becomes 1 * n groups of things taken 0 at a time. But because the latter is an operation blocker, 1 is

not multiplied

[period]. So, the result remains 1. Thus 0^n = 1.

3. The transformation of 0*0 == 0^2 is invalid. The former had no base number to begin; it's stopped before it's started. The latter has the identity element for multiplication 1 at the start; and it ends there.

4. For the grade-school example (1-1) * (1-1), each parenthetical group (1-1) does proceed operationally, and the result for each is nothing, as represented by 0. When looked at by a mental processor, 0 * 0 is a non-operation, for there is nothing to process. Mentally, nothing is multiplied [period].

[Channelling done]

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?

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.

[BaalChatzaf]

[...]

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

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.

[...]

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.

[...]

0 * 0 is undefined.

If 0 * 0 = 0, as you doubtless believe, then the inverse operation 0/0 would have to = 0, too. But it is undefined, as we are all taught in grade school. I am asserting that not only is 0/0 undefined, but also 0 * 0 is undefined, as is ANY number times zero.

What I am saying that 0 TO ANY POWER is 1. Not 0 * 0 * 0 * 0 (n times) = 1, because that is undefined -- but 1 * (0 factors of zero), which means that 1 is NOT MULTIPLIED BY ANYTHING.

And no, it's not Objectivist mathematics, but I believe it's consistent with the Objectivist metaphysics, and with Aristotle's metaphysics. It's my own view. Any Objectivist who agrees with it does so at his/her own risk! :poke:

:thumbsup:

[...]

*sigh* Roger, now you're saying that 0*0 is undefined, but 0^n = 1. But 0^2 is just another way of writing 0*0. So, if 0*0 is undefined, then 0^2 must be undefined; it's the SAME THING. Are you saying that 0^n is undefined for n=2, but 1 for n not equal to 2??

Here's another grade-school example of why that just doesn't work.

What's (1-1) * (1-1) ? If we simplify what's inside the parentheses first, we get 0 * 0, which in RogerLand is Undefined. But, if we use the good old "FOIL" method to distribute it out, we get 1 - 1 - 1 + 1. Compute it left to right, we get 0 - 1 + 1, or -1 + 1, or 0. So (1 - 1) * (1 - 1) both equals zero and is undefined. In your math, can something equal zero AND be undefined? Or don't you believe that (x + y)*(z + w) = x*z + x*w + y*z + y*w?

Your system has a contradiction. Face up to it.

Laure,

Let me try to channel Roger's spirit and answer your questions.

[Channeling Roger:]

If we accept the proposal and interpret the number zero ontologically to stand for a mental blocker of mathematical operations (from

Post #105

), then indeed:

1. Any number multiplied by 0 must mean that that number is

not multiplied

[period]. The operation is blocked. The operation is not processed. It is in this sense that the product is "undefined." (See above.) So, 0*0 is undefined, as 44*0 is undefined.

2. Now, for the power operation x^y, the explicit form of the operation is 1 * x^y, meaning 1 times y groups of things taken x at a time. Thus, 0^n becomes 1 * n groups of things taken 0 at a time. But because the latter is an operation blocker, 1 is

not multiplied

[period]. So, the result remains 1. Thus 0^n = 1.

3. The transformation of 0*0 == 0^2 is invalid. The former had no base number to begin; it's stopped before it's started. The latter has the identity element for multiplication 1 at the start; and it ends there.

4. For the grade-school example (1-1) * (1-1), each parenthetical group (1-1) does proceed operationally, and the result for each is nothing, as represented by 0. When looked at by a mental processor, 0 * 0 is a non-operation, for there is nothing to process. Mentally, nothing is multiplied [period].

[Channelling done]

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?

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.

My dear friends here at OL. This is what you get when Objectivists try to do mathematics from their philosophical basis. You get utter nonsense.

For example what is 0*0 (* means multiplication)? Let x = 1/n where n is an intenger != 0

what is limit (n->oo) of 1/n*1/n ? It is lim 1/n^2 as n->oo which is 0[*]. It is also

lim (n->oo)1/n * lim (n->0) 1/n = 0*0 since lim (n->oo) 1/n[**] = 0.

So 0*0 = 0

* 1/n^2 becomes arbitrarily small as n becomes arbitrarily great. More rigorously given any small number epsilon > 0 there exists an integer N(epsilon) such that n > N(epsilon) implies 1/n^2 < epsilon.

** 1/n becomes arbitrarily small as n becomes arbitrarily great. look at 1/2, 1/3/, 1/4, .... what do you notice? They get smaller and smaller and approach 0 as a limit.

Here is the bottom line. Objectivists could not have (consistent with their philosophical ground) have invented calculus. Hell, they could not have even invented algebra.

I have to say it is damned weird recommending that when one has an algebra problem that he consult ITOE. Very weird. That is like Mary Baker Eddy telling a person running a 105 degree fever to read her book or to go to a Christian Science Reading Room instead of the emergency department of a hospital.

If you have any serious calculation to do, do not hire Roger or Thom as consultants.

Most likely this is the last I will post to this thread. I would sooner teach my pet goldfish differential equations than try to teach an Objectivist simple algebra.

Ba'al Chatzaf

Edited May 26, 2009 by BaalChatzaf

[tjohnson]

Roger and Thom are not able to make the conceptual leap from applied to pure mathematics, no amount of arguing will convince them. It reminds me of the situation of Einstein and contemporary physicists because Einstein was deducing experimental results from general theories derived mathematically! This was unheard of, you are supposed to observe first and then figure out the theory, right? Well, not always.

[Roger Bissell]

Roger and Thom are not able to make the conceptual leap from applied to pure mathematics, no amount of arguing will convince them. It reminds me of the situation of Einstein and contemporary physicists because Einstein was deducing experimental results from general theories derived mathematically! This was unheard of, you are supposed to observe first and then figure out the theory, right? Well, not always.

For a hitherto unrevealed insight into Einstein's deductive E = mc^2 discovery, see my essay (OK, short story): How the Martians Discovered Algebra

REB

[kiaer.ts]

There is no such thing as "pure mathematics" as in mathematics not instantiated by a human mind. It is the fact that humans can attempt to realize contradictions that leads to the so called problems of pure mathematics.

The flaw in goedel's "proof" is that no matter how one formulates the statement "this statement is false" it only has meaning when some mind tries to apply the ambiguous term "this" to some statement. The formulation doesn't do that on its own without some mind attempting to realize it. Ultimately all such attempts will be both arbitrary and empty.

Mathematics isn't pure. It doesn't do itself. In reality, "this statement is false" is not a contradiction. It is squiggles on a page.

Of course, some confused fools think their inabilty to grasp this fact is proof of their superior wisdom.

[Michael Stuart Kelly]

I can't resist the quip:

Nobody has any problem with unapplied pure mathematics, that is until you have to apply it to something.

Michael

EDIT: Then the pure math becomes quantum—it changes depending on who's looking...

[Brant Gaede]

[...]

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

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.

[...]

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.

[...]

0 * 0 is undefined.

If 0 * 0 = 0, as you doubtless believe, then the inverse operation 0/0 would have to = 0, too. But it is undefined, as we are all taught in grade school. I am asserting that not only is 0/0 undefined, but also 0 * 0 is undefined, as is ANY number times zero.

What I am saying that 0 TO ANY POWER is 1. Not 0 * 0 * 0 * 0 (n times) = 1, because that is undefined -- but 1 * (0 factors of zero), which means that 1 is NOT MULTIPLIED BY ANYTHING.

And no, it's not Objectivist mathematics, but I believe it's consistent with the Objectivist metaphysics, and with Aristotle's metaphysics. It's my own view. Any Objectivist who agrees with it does so at his/her own risk! :poke:

:thumbsup:

[...]

*sigh* Roger, now you're saying that 0*0 is undefined, but 0^n = 1. But 0^2 is just another way of writing 0*0. So, if 0*0 is undefined, then 0^2 must be undefined; it's the SAME THING. Are you saying that 0^n is undefined for n=2, but 1 for n not equal to 2??

Here's another grade-school example of why that just doesn't work.

What's (1-1) * (1-1) ? If we simplify what's inside the parentheses first, we get 0 * 0, which in RogerLand is Undefined. But, if we use the good old "FOIL" method to distribute it out, we get 1 - 1 - 1 + 1. Compute it left to right, we get 0 - 1 + 1, or -1 + 1, or 0. So (1 - 1) * (1 - 1) both equals zero and is undefined. In your math, can something equal zero AND be undefined? Or don't you believe that (x + y)*(z + w) = x*z + x*w + y*z + y*w?

Your system has a contradiction. Face up to it.

Laure,

Let me try to channel Roger's spirit and answer your questions.

[Channeling Roger:]

If we accept the proposal and interpret the number zero ontologically to stand for a mental blocker of mathematical operations (from

Post #105

), then indeed:

1. Any number multiplied by 0 must mean that that number is

not multiplied

[period]. The operation is blocked. The operation is not processed. It is in this sense that the product is "undefined." (See above.) So, 0*0 is undefined, as 44*0 is undefined.

2. Now, for the power operation x^y, the explicit form of the operation is 1 * x^y, meaning 1 times y groups of things taken x at a time. Thus, 0^n becomes 1 * n groups of things taken 0 at a time. But because the latter is an operation blocker, 1 is

not multiplied

[period]. So, the result remains 1. Thus 0^n = 1.

3. The transformation of 0*0 == 0^2 is invalid. The former had no base number to begin; it's stopped before it's started. The latter has the identity element for multiplication 1 at the start; and it ends there.

4. For the grade-school example (1-1) * (1-1), each parenthetical group (1-1) does proceed operationally, and the result for each is nothing, as represented by 0. When looked at by a mental processor, 0 * 0 is a non-operation, for there is nothing to process. Mentally, nothing is multiplied [period].

[Channelling done]

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?

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.

My dear friends here at OL. This is what you get when Objectivists try to do mathematics from their philosophical basis. You get utter nonsense.

For example what is 0*0 (* means multiplication)? Let x = 1/n where n is an intenger != 0

what is limit (n->oo) of 1/n*1/n ? It is lim 1/n^2 as n->oo which is 0[*]. It is also

lim (n->oo)1/n * lim (n->0) 1/n = 0*0 since lim (n->oo) 1/n[**] = 0.

So 0*0 = 0

* 1/n^2 becomes arbitrarily small as n becomes arbitrarily great. More rigorously given any small number epsilon > 0 there exists an integer N(epsilon) such that n > N(epsilon) implies 1/n^2 < epsilon.

** 1/n becomes arbitrarily small as n becomes arbitrarily great. look at 1/2, 1/3/, 1/4, .... what do you notice? They get smaller and smaller and approach 0 as a limit.

Here is the bottom line. Objectivists could not have (consistent with their philosophical ground) have invented calculus. Hell, they could not have even invented algebra.

I have to say it is damned weird recommending that when one has an algebra problem that he consult ITOE. Very weird. That is like Mary Baker Eddy telling a person running a 105 degree fever to read her book or to go to a Christian Science Reading Room instead of the emergency department of a hospital.

If you have any serious calculation to do, do not hire Roger or Thom as consultants.

Most likely this is the last I will post to this thread. I would sooner teach my pet goldfish differential equations than try to teach an Objectivist simple algebra.

Ba'al Chatzaf

Since you are so contemptuous of Objectivists are you here only for the reason you aren't welcome anywhere else? Do you have any admiration for Objectivists? Did they get anything right? Or do you acquire stature by standing on the shoulders of Objectivists while shitting on them?

--Brant

[tjohnson]

There is no such thing as "pure mathematics" as in mathematics not instantiated by a human mind. It is the fact that humans can attempt to realize contradictions that leads to the so called problems of pure mathematics.

The flaw in goedel's "proof" is that no matter how one formulates the statement "this statement is false" it only has meaning when some mind tries to apply the ambiguous term "this" to some statement. The formulation doesn't do that on its own without some mind attempting to realize it. Ultimately all such attempts will be both arbitrary and empty.

Mathematics isn't pure. It doesn't do itself. In reality, "this statement is false" is not a contradiction. It is squiggles on a page.

Of course, some confused fools think their inabilty to grasp this fact is proof of their superior wisdom.

The adjective 'pure' in pure mathematics does not mean independent of humans, as you imply here. It means capable of representing exact relations, the kind that can only be found in the human mind. In nature we only find approximate relations but in mathematics we have a model of exact relations that we may apply to natural relations whenever possible. Numbers (including 0) are used to represent exact relations in pure mathematics but when applied to actual counting and measuring they are subject to physical restraints.

[merjet]

Oh, my. I'm away for almost two weeks and the bizarre interpretations of 0 continue? Now even 0^2 = 0*0 is undefined per Roger.

Well, to add a little more fuel to the fire, 0! (zero factorial) = 1. Why? Because it makes sense in combinatorics (link) and elsewhere. If k = n or 0 in the binomial coefficient C(n,k), then 0! appears in the denominator and evaluating it as 1 gives the correct result. There is one way to pick 0 or n elements from n elements.

This and Laure's excellent post suggest an important principle when considering math. The handling of 0, and other mathematical expressions containing 0, is context dependent. It should be handled so that the relevant math is as complete and consistent as possible. You cannot remove 5 chairs from a room containing only 3 chairs. However, that does not make 3 - 5 = -2 illegitimate. Indeed, the equation makes perfect sense with temperature (Celsius or Fahrenheit) or net worth.

Leave the bizarre (and incoherent) interpretations to the likes of George Berkeley (who saw "ghosts of departed quantities" in Newton's calculus) and Roger Bissell!

Ba'al's judgment about Objectivists is overstated. Many do not agree with Roger, me included.

[BaalChatzaf]

Leave the bizarre (and incoherent) interpretations to the likes of George Berkeley (who saw "ghosts of departed quantities" in Newton's calculus) and Roger Bissell!

Berkeley's critique of infinitesimals as used by Newton and Leibniz was dead on accurate. In his essay "The Analyst" Berkeley show that infinitesimals lacked a logical basis and their use could lead to contradictions.. His criticism was so on point it shamed the mathematicians into coming up with a proper rigorous theory of limits and convergence which August Cauchy did in 1821 and others followed. This put the analysis of real variables on a proper basis. Cauchy and Dedikind provided a proper foundation for the theory of (so-called) real numbers. Infinitesimals made a comeback in the the late 1950's with the creation of non-standard analysys and hyper-reals based on (of all things) logic and the compactness property. Infinitesimals, after 250 years were rendered kosher and logically fit.

See http://en.wikipedia.org/wiki/Non-standard_analysis

Ba'al Chatzaf

[merjet]

My reference to George Berkeley wasn't that his criticism of infinitesimals was entirely off-base, but his added metaphysical interpretations. For example:

"Berkeley regarded his criticism of calculus as part of his broader campaign against the religious implications of Newtonian mechanics – as a defence of traditional Christianity against deism, which tends to distance God from His worshippers." (source)

Also, Berkeley did not offer "solutions" that were worse than the problems.

Edited May 26, 2009 by Merlin Jetton

[Dragonfly]

There is no such thing as "pure mathematics" as in mathematics not instantiated by a human mind. It is the fact that humans can attempt to realize contradictions that leads to the so called problems of pure mathematics.

The flaw in goedel's "proof" is that no matter how one formulates the statement "this statement is false" it only has meaning when some mind tries to apply the ambiguous term "this" to some statement. The formulation doesn't do that on its own without some mind attempting to realize it. Ultimately all such attempts will be both arbitrary and empty.

Mathematics isn't pure. It doesn't do itself. In reality, "this statement is false" is not a contradiction. It is squiggles on a page.

Of course, some confused fools think their inabilty to grasp this fact is proof of their superior wisdom.

Wrong. The fact that we (or any other intelligent beings) need some symbols and physical tools to express mathematical truths or contradictions doesn't mean that those truths/contradictions are dependent on the existence of those symbols and tools and the conventions how to use them. For example the number π and the number e have respective values that are independent of any mind, even if a mind is needed to express their definitions and those values with physical means. Of course, some confused Objectivist fools think their inability to graps this is proof of their superior wisdom.

[kiaer.ts]

There is no such thing as "pure mathematics" as in mathematics not instantiated by a human mind. It is the fact that humans can attempt to realize contradictions that leads to the so called problems of pure mathematics.

The flaw in goedel's "proof" is that no matter how one formulates the statement "this statement is false" it only has meaning when some mind tries to apply the ambiguous term "this" to some statement. The formulation doesn't do that on its own without some mind attempting to realize it. Ultimately all such attempts will be both arbitrary and empty.

Mathematics isn't pure. It doesn't do itself. In reality, "this statement is false" is not a contradiction. It is squiggles on a page.

Of course, some confused fools think their inabilty to grasp this fact is proof of their superior wisdom.

Wrong. The fact that we (or any other intelligent beings) need some symbols and physical tools to express mathematical truths or contradictions doesn't mean that those truths/contradictions are dependent on the existence of those symbols and tools and the conventions how to use them. For example the number π and the number e have respective values that are independent of any mind, even if a mind is needed to express their definitions and those values with physical means. Of course, some confused Objectivist fools think their inability to graps this is proof of their superior wisdom.

Fine. In the statement "this statement is false" what objective reality independent of anyone's attribution does the word "this" refer to? Sorry, deictics are always context dependent. The ratio of a circle's circumference to its diameter is universal. "This" is radically relative to an asserting mind.

Compare:

(1) The value of pi is greater than e.

(2) This is greater than that.

Sorry, "this" always means that to which someone is referring. (Statements are also always statements made by or assented to by someone, but that's just overkill.)