The Opposite of Nothing Is/Isn't Everything

[Brant Gaede]

Very good perspective Chris. I have some experience teaching mathematics and getting people to use the abstract part of the brain is very difficult at first. This is why there is such widespread dislike for mathematics I suspect. Most people when presented with very abstract formulations will respond by saying "what use will this ever be?" or something and, in most cases, probably it will be of no use in practice. But there is another reason to learn mathematics besides applying it to engineering and science and that is because it represent the highest level of human thinking ever achieved and can provide a basis for critical thinking in general.

I could agree with "the highest level of human thinking," but not the highest level of mental functioning.

--Brant

[tjohnson]

I could agree with "the highest level of human thinking," but not the highest level of mental functioning.

--Brant

Ah...are you going to keep me in suspense or tell me what the highest level of mental functioning is??

[Brant Gaede]

I could agree with "the highest level of human thinking," but not the highest level of mental functioning.

--Brant

Ah...are you going to keep me in suspense or tell me what the highest level of mental functioning is??

Is has to do with sex and the transformation of mere ejaculation into orgasm.

Just joking, unless you do it in an airplane.

It's creativity. That may include creativity in math, but not merely using math. That is, Newton invents calculus, not Newton uses calculus.

--Brant

Edited May 14, 2009 by Brant Gaede

[kiaer.ts]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

REB

I am sorry I missed this the first time around, very interesting.

[BaalChatzaf]

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

[Laure]

Amen, Ba'al!

(but, shouldn't a^n / a^(-n) be a^n / a^n ?)

[kiaer.ts]

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

Oh, please.

Are you denying, Bob, that "add nothing" is the equivalent of "do not add anything"?

You say "Only a philosopher could confuse 0 and "nothing", but not a mathematician or a computer programmer." That's cute. You might as well say that a type setter would not confuse the symbol "0" with the seven-letter word n-o-t-h-i-n-g.

Yes, the computer instruction to add zero to an amount is an instruction, and does take up memory, or "advance the tape" and hence is not "nothing" in that sense.

But this is mere sophistry, a refusal to understand the underlying meaning, and to confuse the means of saying something for the idea communicated by the words said. Of course, you stand in good stead, with the likes of goedel and others who obfuscate and equivocate.

There are plenty of people here worth being rude to, Roger is not one of them.

[BaalChatzaf]

But this is mere sophistry, a refusal to understand the underlying meaning, and to confuse the means of saying something for the idea communicated by the words said. Of course, you stand in good stead, with the likes of goedel and others who obfuscate and equivocate.

No sophistry. 0 is the identity element of an additive semi-group. That is something, not nothing. And I was not being rude. I was suggesting that Roger learn some mathematics rather than make foolish statements about something he does not know sufficiently well. If you think that arithmetic 0 has anything to do with nothing, it is you who are confused. I pay attention to the postulates of the system and the rules of inference. What do you pay attention to?

And Goedel did not obfuscate anything. He discovered a genuine limitation of first order theories that are capable of expressing the axioms of arithmetic. The limitation is that provability cannot catch up with truth with respect to a model. Goedel blew up Hilbert's attempt to reduce mathematics to an algorithmic formalism.

My math is sound. Is yours?

Ba'al Chatzaf

[Alfonso Jones]

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

Oh, please.

Are you denying, Bob, that "add nothing" is the equivalent of "do not add anything"?

You say "Only a philosopher could confuse 0 and "nothing", but not a mathematician or a computer programmer." That's cute. You might as well say that a type setter would not confuse the symbol "0" with the seven-letter word n-o-t-h-i-n-g.

Yes, the computer instruction to add zero to an amount is an instruction, and does take up memory, or "advance the tape" and hence is not "nothing" in that sense.

But this is mere sophistry, a refusal to understand the underlying meaning, and to confuse the means of saying something for the idea communicated by the words said. Of course, you stand in good stead, with the likes of goedel and others who obfuscate and equivocate.

There are plenty of people here worth being rude to, Roger is not one of them.

Ted - - -

I'll take up just one item from your post, first:

Can you indicate precisely where and in what way Goedel obfuscated, and where and in what way he equivocated? Please provide specific technical terms (and of course the paper/book references to them in Goedel's writing) Goedel used in an equivocal way, if that is your meaning by "equivocate." So that we don't pass like ships in the night, please advise re your mathematical background (so I don't use terms or allude to issues and concepts in the foundations of mathematics which would be intelligible if you also have a PhD in mathematical sciences and have studied foundations, but would require more detailed laying out otherwise). Likewise, where did Goedel obfuscated - specifically?

Bill P

[tjohnson]

Again it's the confusion of applying math and pure math. The symbol zero as used in a+(-a)=0, for example, is needed to define the additive inverse of an element. This is purely abstract and has nothing to do with the idea of having "zero objects", which is not mathematics but instead counting things, a type of applied math..

[tjohnson]

Additive and multiplicative identity - There exists an element of F, called the additive identity element and denoted by 0, such that for all a in F, a + 0 = a. Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all a in F, a · 1 = a. For technical reasons, the additive identity and the multiplicative identity are required to be distinct.

Additive and multiplicative inverses - For every a in F, there exists an element −a in F, such that a + (−a) = 0. Similarly, for any a in F other than 0, there exists an element a⁻¹ in F, such that a · a⁻¹ = 1. (The elements a + (−b) and a · b⁻¹ are also denoted a − b and a/b, respectively.) In other words, subtraction and division operations exist.

[thomtg]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

REB

I am sorry I missed this the first time around, very interesting.

Thank you, Ted, for seeing the import of this perspective.

[Roger Bissell]

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

Oh, please.

Are you denying, Bob, that "add nothing" is the equivalent of "do not add anything"?

You say "Only a philosopher could confuse 0 and "nothing", but not a mathematician or a computer programmer." That's cute. You might as well say that a type setter would not confuse the symbol "0" with the seven-letter word n-o-t-h-i-n-g.

Yes, the computer instruction to add zero to an amount is an instruction, and does take up memory, or "advance the tape" and hence is not "nothing" in that sense.

But this is mere sophistry, a refusal to understand the underlying meaning, and to confuse the means of saying something for the idea communicated by the words said. Of course, you stand in good stead, with the likes of goedel and others who obfuscate and equivocate.

There are plenty of people here worth being rude to, Roger is not one of them.

Ted - - -

I'll take up just one item from your post, first:

Can you indicate precisely where and in what way Goedel obfuscated, and where and in what way he equivocated? Please provide specific technical terms (and of course the paper/book references to them in Goedel's writing) Goedel used in an equivocal way, if that is your meaning by "equivocate." So that we don't pass like ships in the night, please advise re your mathematical background (so I don't use terms or allude to issues and concepts in the foundations of mathematics which would be intelligible if you also have a PhD in mathematical sciences and have studied foundations, but would require more detailed laying out otherwise). Likewise, where did Goedel obfuscated - specifically?

Bill P

I think that a very grotesque and (potentially) destructive obfuscation and equivocation by Goedel is his "slingshot" argument, which purports to prove that all true sentences stand for the same thing. Check out the wikipedia entry on "slingshot argument" for background on this kind of argument. I took part in a discussion on the Analytic list on the Enlightenment website back in September of 2000, and I deconstructed the Modern Logic approach to dealing with the slingshot by using plain language instead. It seemed clear to me where the equivocation and obfuscation was, and that you didn't need special symbolic machinery to uncover it, just a sharp eye and a refusal to be taken in by thimbleriggery. Perhaps some of the readers on this list can see it, too, as I lay it out. I'm sorry that the link to the Analytic list no longer works, or I could point you toward the initial heavily-symbolized essay by Bryan Register and the reply by Thomas Radcliffe, both of which motivated the following essay and followup comments.

REB

Goedel’s “Slingshot” Error:

Begging the Question—or Equivocation?

by Roger E. Bissell

The following material was originally written for and posted on the Analytic list

on 9/22/00 and 9/25/00. Comments are welcome.

===================================================

Dear Analytic members:

I have some comments on the Sep. 20 exchange between Thomas Radcliffe and Bryan Register.

I agree with Thomas that "the slingshot fails at a very early stage," and I agree that the crucial element in its failure is its claim that "the same fact" makes true all identity statements with co-referring terms. And I reject Bryan's and his logician friend's suggestion that there is anything that need concern us about 3, 6, and 7 being made true by the same fact (for they are). The damage is already done long before that point. However, I disagree with Thomas as to the nature of the error that invalidates the slingshot. As I will try to show, the slingshot suffers not from begging the question ("the slingshot has its conclusion very nearly built into its premise"), but instead from one of the oldest fallacies in the book (obscured by the notational blizzard): reasoning with an invalid middle term by means of equivocation on the term "the same fact."

Now, at the risk of aggravating or distressing those who paid good money (and lots of it!) to learn how to make things more difficult than they need to be, I'm going to eschew the use of what to me is needlessly confusing symbolic notation. Instead, I'm going to try to clarify matters by stripping away some of the alphabet soup and examine the slingshot in terms of ordinary language and a specific example.

I'm going to set a as apple, b as banana, F as red thing, and G as yellow thing. Then the first three true slingshot sentences become:

1) An apple is a red thing.

2) A banana is a yellow thing.

3) An apple is not a banana (and a banana is not an apple).

The next slingshot sentence, which supposedly is made true by the same fact that makes 1) true, is:

4) An apple is a thing that is an apple and a red thing, i.e., An apple is an apple and a red thing.

To verify that I have not lost anything important in my reduction to ordinary language, compare 1) and 4). To paraphrase Bryan, surely an apple's being a red thing is the same state of affairs as a thing's being an apple and a red thing. Fine, so far. But note well: there are ~two~ facts that make 4) true: an apple's being an apple and an apple's being a red thing. The former fact is merely the self-identity of the apple, and it makes only 4) true; it is the latter fact, the apple's being a red thing, that makes ~both~ 1) and 4) true. To put it another way: 1) is true simply by virtue of the fact of an apple's being a red thing, while 4) is true by virtue of ~two~ facts: an apple's being a red thing and an apple's being an apple. For clarity, let's label them as:

Fact A1: an apple's being a red thing

Fact A2: an apple's being an apple

Likewise, the next slingshot sentence, supposedly made true by the same fact that makes 2) true, is:

5) A banana is a thing that is a banana and a yellow thing, i.e.,

A banana is a banana and a yellow thing.

Again, surely a banana's being a yellow thing is the same state of affairs as a thing's being a banana and a yellow thing. And again, there are ~two~ facts that make 5) true: a banana's being a banana and a banana's being a yellow thing. The former fact is merely the self-identity of the banana, and it makes only 5) true; it is the latter fact, the banana's being a yellow thing, that makes ~both~ 2) and 5) true. That is, 2) is true simply by virtue of the fact of a banana's being a yellow thing, while 5) is true by virtue of ~two~ facts: a banana's being a yellow thing and a banana's being a banana. For clarity, let's label them as:

Fact B1: a banana's being a yellow thing

Fact B2: a banana's being a banana

Next comes the crucial step, in which the groundwork is laid for the equivocation that is used to supposedly prove that there are no facts that make propositions true.

From 3), we can derive:

6) An apple is a thing that is an apple and that is not a banana, i.e.,

An apple is an apple and not a banana.

7) A banana is a thing that is a banana and that is not an apple, i.e.,

A banana is a banana and not an apple.

Again, to paraphrase Bryan: surely an apple's not being a banana is the same state of affairs as an apple's being an apple and not being a banana -- and a banana's not being an apple is the same state of affairs as a banana's being a banana and not being an apple.

There are two facts that make 6) true: an apple's being an apple and an apple's not being a banana -- and there are two facts that make 7) true: a banana's being a banana and a banana's not being an apple. In this case it is the self-identity of the apple that makes ~both~ 4) and 6) true -- and the self-identity of the banana that makes ~both~ 5) and 7) true -- while it is the fact that an apple is not a banana that makes ~only~ 6) and 7) true (and not 4) and 5), also). Let's label this fact, too:

Fact AB: an apple's not being a banana, and a banana's not being an apple.

Now, we can see how the whole house of cards collapses:

The argument goes:

P1: The fact that makes 1) true is the same fact that makes 4) true.

P2: The fact that makes 4) true is the same fact that makes 6) true.

C1: Therefore, the fact that makes 1) true is the same fact that makes 6) true.

P3: The fact that makes 6) true is the same fact that makes 7) true.

C2: Therefore, the fact that makes 1) true is the same fact that makes 7) true.

P4: The fact that makes 7) true is the same fact that makes 5) true.

P5: The fact that makes 5) true is the same fact that makes 2) true.

C3: Therefore, the fact that makes 7) true is the same fact that makes 2) true.

C4: Therefore, the fact that makes 1) true is the same fact that makes 2) true.

Now, let's analyze the components of the argument, step by step:

P1: Fact A1 is the same as Fact A1 or Fact A2. True.

P2: Fact A1 or Fact A2 is the same as Fact A2 or Fact AB. True.

C1: Therefore, Fact A1 is the same as Fact A2 or Fact AB. FALSE!!!

P3: Fact A2 or Fact AB is the same as Fact B2 or Fact AB. True

C2: Fact A1 is the same as Fact B2 or Fact AB. FALSE!!!

P4: Fact B2 or Fact AB is the same as Fact B2 or Fact B1. True

P5: Fact B2 or Fact B1 is the same as Fact B1. True

C3: Fact B2 or Fact AB is the same as Fact B1. FALSE!!!

C4: Fact A1 is the same as Fact B1. FALSE!!!!!!!!

As my friends in Tennessee would say: that slingshot don't hunt! :-) Rather than being "a disaster for the correspondence theory of truth," the slingshot is a disaster for those who try to avoid/evade Aristotelian logic by the terminally obfuscatory tactics of modern symbolic logic.

So, if you are cataloging approaches to refuting the slingshot, you can add to Neale's Russellian strategy and Thomas' early Fregean strategy this, Roger's moldy-oldie Aristotelian strategy.

==================================================

Dear Analytic List Members:

Bryan Register found enough other aspects of my post that were unacceptable to him, that he asked me to do a major revision of my original post...[...]

But rather than re-writing my original post, I am instead going to work with Thomas' example. As far as I can tell, it satisfies all of Bryan's strictures, and is nice and concrete so that I can comfortably work with it, and in good conscience impose on you readers my admittedly unprofessional analysis of it. :-)

Thomas Radcliffe wrote:

> In response to Bryan's critique of Roger's non-symbolic

> form of the slingshot, I've repeated my example here,

> which addresses all of the comments Bryan made. Is

> there a problem with this example that prevents us from

> working with it as a concrete, English instantiation of

> the slingshot for people who don't like symbolic logic?

I see no such problem, Thomas, so here goes!

> Taking:

>

> a = Nut the cat

> b = Bolt the cat

> F = has two eyes

> G = has one eye

>

> 1) F(a)

>

> Nut the cat has two eyes

>

> 2) G(B)

>

> Bolt the cat has one eye

>

> 3) ~(a = B)

>

> Nut the cat is not identical to Bolt the cat.

>

> 4) a = (x such that)([x = a] & F(x))

>

> Nut the cat is identical to any cat at all such that

> that cat is identical to Nut the cat and that cat has two eyes.

>

> 5) b = (y such that)([y = b] & G(y))

>

> Bolt the cat is identical to any cat at all such that that

> cat is identical to Bolt the cat and that cat has one eye.

>

> 6) a = (x such that)([x = a] & ~(x = B))

>

> Nut the cat is identical to any cat at all such that that

> cat is identical to Nut the cat and that cat is not

> identical to Bolt the cat.

>

> 7) b = (y such that)([y = b] & ~(y = a))

>

> Bolt the cat is identical to any cat at all such that

> that cat is identical to Bolt the cat and that cat is

> not identical to Nut the cat.

I am going to run the very same analysis that I ran last time, and I hope it will be not only clear to the reader (as I think it was last time) but also admissable by the strict logician that I have detected the fallacy of equivocation in the slingshot argument. Specifically, the equivocation turns on the ambiguity in the phrase "the same fact."

Remember, the point of the slingshot is to prove that there are no facts that propositions refer to and which thus make them true if they do it correctly, and that the slingshot proposes to do this by showing that every true proposition is made true by the same fact, which would mean that they all refer to the same fact, which is impossible. This reductio argument falls flat on its face, because it can't even keep straight what it means by "the same fact." I maintain that the reason people are so mystified by the slingshot is that it is couched in all the standard symbolic trappings of modern logic, which is why I am so eager to re-cast the slingshot in simple English. I'm glad that Thomas agrees, and I gladly avail myself of his example to try again to state my case...

On Thomas' example, the first three true slingshot sentences become:

1) Nut the cat has two eyes.

2) Bolt the cat has one eye.

3) Nut the cat is not identical to Bolt the cat.

The next slingshot sentence, which supposedly is made true by the same fact that makes 1) true, is:

4) Nut the cat is identical to any cat at all such that that cat is identical to Nut the cat and that cat has two eyes. (I take this to be equivalent to: Nut the cat is Nut the cat, and Nut the cat has two eyes. It is a compound sentence and thus clearly refers to two facts that are its truth-conditions; it has two "truth-makers," so to speak. But the standard form serves to ~obscure~ this fact, so bear this point in mind as we proceed.)

Compare 1) and 4). To paraphrase Bryan, surely Nut the cat's having two eyes is the same state of affairs as Nut the cat's being Nut the cat and Nut the cat's having two eyes. Fine, so far. But note well: there are ~two~ facts that make 4) true: Nut the cat's being Nut the cat, and Nut the cat's having two eyes. The former fact is merely the self-identity of Nut the cat, and it makes ~only~ 4) true; it is the latter fact, Nut the cat's having two eyes, that makes ~both~ 1) and 4) true. To put it another way: 1) is true simply by virtue of the fact of Nut the cat's having two eyes, while 4) is true by virtue of ~two~ facts: Nut the cat's having two eyes, and Nut the cat's being Nut the cat. For clarity, let's label them as:

Fact A1: Nut the cat's having two eyes

Fact A2: Nut the cat's being Nut the cat

Likewise, the next slingshot sentence, supposedly made true by the same fact that makes 2) true, is:

5) Bolt the cat is identical to any cat at all such that that cat is identical to Bolt the cat and that cat has one eye.

Again, surely Bolt the cat's having one eye is the same state of affairs as Bolt the cat's being Bolt the cat and Bolt the cat's having one eye. And again, there are ~two~ facts that make 5) true: Bolt the cat's being Bolt the cat and Bolt the cat's having one eye. The former fact is merely the self-identity of Bolt the cat, and it makes ~only~ 5) true; it is the latter fact, Bolt the cat's having one eye, that makes ~both~ 2) and 5) true. That is, 2) is true simply by virtue of the fact of Bolt the cat's having one eye, while 5) is true by virtue of ~two~ facts: Bolt the cat's having one eye, and Bolt the cat's being Bolt the cat. For clarity, let's label them as:

Fact B1: Bolt the cat's having one eye

Fact B2: Bolt the cat's being Bolt the cat

Next comes the crucial step, in which the groundwork is laid for the equivocation that is used to supposedly prove that there are no facts that make propositions true.

From 3), we can derive:

6) Nut the cat is identical to any cat at all such that that cat is identical to Nut the cat and that cat is not identical to Bolt the cat. (I would simply say: Nut the cat is Nut the cat and not Bolt the cat. What is gained by making it so much more complicated other than the opportunity for such fallacious arguments as the slingshot?)

7) Bolt the cat is identical to any cat at all such that that cat is identical to Bolt the cat and that cat is not identical to Nut the cat. (I.e., Bolt the cat is Bolt the cat and not Nut the cat.)

Again, to paraphrase Bryan: surely Nut the cat's not being Bolt the cat is the same state of affairs as Nut the cat's being Nut the cat and Nut the cat's not being Bolt the cat -- and surely Bolt the cat's not being Nut the cat is the same state of affairs as Bolt the cat's being Bolt the cat and Bolt the cat's not being Nut the cat.

There are two facts that make 6) true: Nut the cat's being Nut the cat and Nut the cat's not being Bolt the cat -- and there are two facts that make 7) true: Bolt the cat's being Bolt the cat and Bolt the cat's not being Nut the cat.

In this case it is fact of the self-identity of the Nut the cat that makes ~both~ 4) and 6) true -- and the fact of the self-identity of Bolt the cat that makes ~both~ 5) and 7) true -- while it is the fact that Nut the cat is not Bolt the cat that makes ~only~ 6) and 7) true (and not 4) and 5), also). Let's label this fact, too:

Fact AB: Nut the cat's not being Bolt the cat (and Bolt the cat's not being Nut the cat)

Now, we can see how the whole house of cards collapses:

The argument goes:

P1: The fact that makes 1) true is the same fact as the fact that makes 4) true.

P2: The fact that makes 4) true is the same fact as the fact that makes 6) true.

C1: Therefore, the fact that makes 1) true is the same fact as the fact that makes 6) true.

P3: The fact that makes 6) true is the same fact as the fact that makes 7) true.

C2: Therefore, the fact that makes 1) true is the same fact as the fact that makes 7) true.

P4: The fact that makes 7) true is the same fact as the fact that makes 5) true.

P5: The fact that makes 5) true is the same fact as the fact that makes 2) true.

C3: Therefore, the fact that makes 7) true is the same fact as the fact that makes 2) true.

C4: Therefore, the fact that makes 1) true is the same fact as the fact that makes 2) true.

Now, let's analyze the components of the argument, step by step:

P1: Fact A1 is the same fact as Fact A1 or Fact A2. (Nut the cat's having two eyes is the same fact as Nut the cat's having two eyes or Nut the cat's being Nut the cat.) True...

P2: Fact A1 or Fact A2 is the same fact as Fact A2 or Fact AB. (Nut the cat's having two eyes or Nut the cat's being Nut the cat is the same fact as Nut the cat's being Nut the cat or Nut Nut the cat's not being Bolt the cat.) True...

C1: Therefore, Fact A1 is the same fact as Fact A2 or Fact AB. (Nut the cat's having two eyes is the same fact as Nut the cat's being Nut the cat or Nut the cat's not being Bolt the cat.) FALSE!!! (This is an equivocation. We could stop here...)

P3: Fact A2 or Fact AB is the same fact as B2 or Fact AB. (Nut the cat's being Nut the cat or Nut the cat's not being Bolt the cat is the same fact as Bolt the cat's being Bolt the cat or Bolt the cat's not being Nut the cat.) True...

C2: Therefore, Fact A1 is the same fact as Fact B2 or Fact AB. (Nut the cat's having two eyes is the same fact as Bolt the cat's being Bolt the cat or Bolt the cat's not being Nut the cat.) FALSE!!!

P4: Fact B2 or Fact AB is the same fact as Fact B2 or Fact B1. (Bolt the cat's being Bolt the cat or Bolt the cat's not being Nut the cat is the same fact as Bolt the cat's being Bolt the cat or Bolt the cat's having one eye.) True...

P5: Fact B2 or Fact B1 is the same fact as Fact B1. (Bolt the cat's being Bolt the cat or Bolt the cat's having one eye is the same fact as Bolt the cat's having one eye.) True...

C3: Therefore, Fact B2 or Fact AB is the same as Fact B1. (Bolt the cat's being Bolt the cat or Bolt the cat's not being Nut the cat is the same fact as Bolt the cat's having one eye.) FALSE!!!

C4: Therefore, Fact A1 is the same fact as Fact B1. (Nut the cat's having two eyes is the same fact as Bolt the cat's having one eye.) FALSE!!!!!!!!

Now, C4 is precisely the kind of conclusion -- wiping out the distinction between different facts -- that the slingshot wants us to accept, and thereby to jettison the fundamental idea that propositions are true if and when they correctly refer to facts. Having shown that the slingshot arrives at a palpable contradiction, and having exposed the equivocations that cause it, I rest my case -- and renew my insistence that we keep our logical analyses anchored to the real world (assuming that Nut and Bolt are real cats :-), especially when analyzing sophistical arguments such as Goedel's slingshot!

[BaalChatzaf]

especially when analyzing sophistical arguments such as Goedel's slingshot!

Clear statement of Goedel's first incompleteness theorem (excerpted from Wiki)

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory.

What slingshot? Goedel uses ordinary mathematical reasoning to produce the theorem.

Ba'al Chatzagf

[thomtg]

[...]

Can you indicate precisely where and in what way Goedel obfuscated, and where and in what way he equivocated? Please provide specific technical terms (and of course the paper/book references to them in Goedel's writing) Goedel used in an equivocal way, if that is your meaning by "equivocate." So that we don't pass like ships in the night, please advise re your mathematical background (so I don't use terms or allude to issues and concepts in the foundations of mathematics which would be intelligible if you also have a PhD in mathematical sciences and have studied foundations, but would require more detailed laying out otherwise). Likewise, where did Goedel obfuscated - specifically?

Bill P

I think that a very grotesque and (potentially) destructive obfuscation and equivocation by Goedel is his "slingshot" argument, which purports to prove that all true sentences stand for the same thing. Check out the wikipedia entry on "slingshot argument" for background on this kind of argument. I took part in a discussion on the Analytic list on the Enlightenment website back in September of 2000, and I deconstructed the Modern Logic approach to dealing with the slingshot by using plain language instead. It seemed clear to me where the equivocation and obfuscation was, and that you didn't need special symbolic machinery to uncover it, just a sharp eye and a refusal to be taken in by thimbleriggery. Perhaps some of the readers on this list can see it, too, as I lay it out. I'm sorry that the link to the Analytic list no longer works, or I could point you toward the initial heavily-symbolized essay by Bryan Register and the reply by Thomas Radcliffe, both of which motivated the following essay and followup comments.

REB

Goedel’s “Slingshot” Error:

Begging the Question—or Equivocation?

by Roger E. Bissell

The following material was originally written for and posted on the Analytic list

on 9/22/00 and 9/25/00. Comments are welcome.

[...]

C4: Therefore, Fact A1 is the same fact as Fact B1. (Nut the cat's having two eyes is the same fact as Bolt the cat's having one eye.) FALSE!!!!!!!!

Now, C4 is precisely the kind of conclusion -- wiping out the distinction between different facts -- that the slingshot wants us to accept, and thereby to jettison the fundamental idea that propositions are true if and when they correctly refer to facts. Having shown that the slingshot arrives at a palpable contradiction, and having exposed the equivocations that cause it, I rest my case -- and renew my insistence that we keep our logical analyses anchored to the real world (assuming that Nut and Bolt are real cats :-), especially when analyzing sophistical arguments such as Goedel's slingshot!

Roger, I just read both versions of your refutation of the slingshot argument, which was first suggested by Kurt Gödel (1944). If both versions are correct, and I think they are, then a whole field of mathematical logic needs to be mowed down!

All of our knowledge--all of our true sentences together--is alleged to be about one single "Fact"? Preposterous! Yet this conclusion has been taken seriously since 1944. Even now, from Google search, ivory-tower intellectuals continue to take it seriously. Here is a fascinating book review in Notre Dame Philosophical Review on the slingshot argument (Neale, Stephen, Facing Facts, Oxford University Press, 2001). More current is this 2007 sidebar article in Stanford Encyclopedia of Philosophy "The Slingshot Argument," claiming, "There is a famous argument of Davidson's to the effect that if true statements correspond to facts, then they all correspond to the same Great Fact."

Mortimer Adler's little book Ten Philosophical Mistakes (1985) comes to mind. What began as a kernel of error grew and seeded a whole field. Your presentation, Roger, really highlights the value of learning how to read and write plain prose instead of relying on the abstractness of symbols. There is a bias toward symbolic representation of truths--supposedly to remove misinterpretation--that I think is completely unwarranted. This slingshot error could have and should have been snipped long ago.

We get back again to the source of the error: Misunderstanding through misreading. The slingshot argument misreads what facts are, in relation to truths. Here on this forum thread, history is repeating itself: a misreading "that 'add nothing' is the equivalent of 'do not add anything' " (Post #82). There seems to be a vested interest in maintaining misunderstanding. Why? I don't know. But I for one wish to see a proposed new idea being treated and debated more seriously among self-professed rational individuals.

[Dragonfly]

From Wikipedia: "As Gödel (1944) observed, the slingshot argument does not go through if Bertrand Russell's famous account of definite descriptions is assumed."

Gödel was no dummy, in fact he was very probably smarter than anyone on this forum. So don't presume to criticize him if you don't know what he really wrote.

[thomtg]

From Wikipedia: "As Gödel (1944) observed, the slingshot argument does not go through if Bertrand Russell's famous account of definite descriptions is assumed."

Gödel was no dummy, in fact he was very probably smarter than anyone on this forum. So don't presume to criticize him if you don't know what he really wrote.

Ah, yes, but I have mentioned elsewhere that Russell's theory of definite descriptions is faulty (here and here). And if it is faulty, Gödel's slingshot argument comes to the fore, which since the 1970s, the latter has indeed come to take up the field in mathematical logic.

I don't think anyone (else) here presumes he is smarter than anyone else, dead or alive. The issue is not smartness or intelligence. By the way, I don't think I am that smart. I know there are others whose native intelligence is greater than mine, as I know there are people who are physically stronger, faster, etc. But I can spot an error, even if it comes from one smarter than me.

Have you read Roger's discussion on the ontology of the zero (Post #14)? Have you read his refutation of the slingshot argument (Post #88)? Can you spot an error therein?

[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

[BaalChatzaf]

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

[Alfonso Jones]

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. I'm working on my own response to Roger (and I think that Roger's position is wrong) but I don't see any need to express positions in this discourteous fashion. Roger is a serious person who contributes substantially in these discussions. He is not a "drive-by troll" who is unwilling to learn or seriously interact. He has demonstrated substantial knowledge in many of the subjects discussed on OL. So - I urge more restraint here - let's avoid the personalities, and stick to the issues.

Save the insults for the trolls, if you must insult somebody.

I note that you have in fact insulted all philosophers. For what purpose?

Bill P

[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. I'm working on my own response to Roger (and I think that Roger's position is wrong) but I don't see any need to express positions in this discourteous fashion. Roger is a serious person who contributes substantially in these discussions. He is not a "drive-by troll" who is unwilling to learn or seriously interact. He has demonstrated substantial knowledge in many of the subjects discussed on OL. So - I urge more restraint here - let's avoid the personalities, and stick to the issues.

Save the insults for the trolls, if you must insult somebody.

I note that you have in fact insulted all philosophers. For what purpose?

Bill P

Perhaps he's hanging out his shingle, maybe because he heard that Peikoff is retiring soon and figured that Objectivism needed a new hit-man to bad-mouth academic philosophers. (Have you read the ghastly introduction to Peikoff's OPAR?)

But thanks, Bill, for your support in re courtesy -- even as you prepare to mete me my fate (so you think) in re my position.

And before you go too far down that road, consider this:

From the standard perspective, in dealing with explaining the meaning of exponents, you have to say that any number x to some power n is that number x multiplied by itself n times, unless the power is 0, in which case x^n is 1.

From my perspective, all I have to say is: any number x to some power n is the unit 1 taken times x n times, unless n is 0, in which case the unit 1 is not taken time x ~any~ times. For my money, this is much more elegant and integrated--single-premised, and it appears less arbitrary (where did the answer 1 come from on the standard perspective? you have to do a lot of manipulation to show it). Also, think of the analogy between my unit-1 foundation for exponents and scientific notation (e.g., 100 = 1.0 x 10^2, 3952 = 3.952 x 10^3). We already have the precedent of analyzing multiple-digit numbers in terms of a single-digit (with decimals) number multiplied by factors of 10. Well, why not analyze power numbers in terms of a single-digit number (1) multiplied (or not!) by factors of the number that is being powered?

If I were King of All Mathematics, I would institute this way of conceptualizing powers immediately! It might forestall a lot of perplexity in young math students. (I have gotten quite a number of grateful emails from high school and college students who understood zero powers, negative powers, and fractional powers for the first time from my online essays.)

REB

[BaalChatzaf]

Bob -

I would suggest a little more courtesy as being appropriate. I'm working on my own response to Roger (and I think that Roger's position is wrong) but I don't see any need to express positions in this discourteous fashion.

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.

Over 150 years has passed since Boole and Frege liberated mathematics and logic from thrall of metaphysics. The result has been a hyper-exponential increase in scope and capability. The philosophers want to go back and dwell in caves. I prefer sky-scrapers. The philosophers want to go about in ox-carts. I prefer jet planes. The philosophers, with regard to physics and mathematics, have been reactionary and retrograde with some notable exceptions. If they produced some useful results they might claim a right to their backward opinions, but what have they produced? What have they given us? Bupkis, Kaduchis, Nada, Zip, Zero. Which you may very well interpret as Nothing. More and better mathematics has been produced in the past 150 years than in the previous 10,000 years. The same could be said for the natural science of motion and matter, i.e. physics.

Just a little hint here. Doing the identity mapping on an infinite set (I: N -> N into and onto) is an infinite number of mental operations, not Nothing. Consider the identity on the integers:

0 mapped to 0. 1 mapped to 1, 2 mapped to 2 etc. That is a lot of work, not Nothing. Adding 0 is the identity operation which is not Nothing. It is Something. The mapping exists, it has properties and can be combined with other operations. That is a lot more than Nothing.

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

Ba'al Chatzaf

[Alfonso Jones]

Bob -

I would suggest a little more courtesy as being appropriate. I'm working on my own response to Roger (and I think that Roger's position is wrong) but I don't see any need to express positions in this discourteous fashion.

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.

Over 150 years has passed since Boole and Frege liberated mathematics and logic from thrall of metaphysics. The result has been a hyper-exponential increase in scope and capability. The philosophers want to go back and dwell in caves. I prefer sky-scrapers. The philosophers want to go about in ox-carts. I prefer jet planes. The philosophers, with regard to physics and mathematics, have been reactionary and retrograde with some notable exceptions. If they produced some useful results they might claim a right to their backward opinions, but what have they produced? What have they given us? Bupkis, Kaduchis, Nada, Zip, Zero. Which you may very well interpret as Nothing. More and better mathematics has been produced in the past 150 years than in the previous 10,000 years. The same could be said for the natural science of motion and matter, i.e. physics.

Just a little hint here. Doing the identity mapping on an infinite set (I: N -> N into and onto) is an infinite number of mental operations, not Nothing. Consider the identity on the integers:

0 mapped to 0. 1 mapped to 1, 2 mapped to 2 etc. That is a lot of work, not Nothing. Adding 0 is the identity operation which is not Nothing. It is Something. The mapping exists, it has properties and can be combined with other operations. That is a lot more than Nothing.

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

Ba'al Chatzaf

Bob -

Showing common courtesy is not equivalent to giving a free pass to error.

In the field of statistics, the Royal Statistical Society has a long tradition of presentation of papers with discussion. In this, the discussion is often quite vigorous and spirited. It is also typically done without abusive personality-type discussions.

I do remember an occasion when one distinguished mathematical statistician presented his paper.

Sir Ronald Alymer Fisher (who had laid the foundations which the paper being presented dealt with) was a discussant. As I recall his comments, they began as .. (I am paraphrasing - don't have my library at hand)...

"It is a great pleasure to hear such a distinguished mathematician as Professor X speak. It is a pity that he could not have selected a topic on which he was more knowledgeable." He then went on to describe in broad brush and then in detail what he saw as the major errors of the presenter.

Now, to my mind, that is a lot more polite and appropriate.

Bill P

[Alfonso Jones]

(snip)

Perhaps he's hanging out his shingle, maybe because he heard that Peikoff is retiring soon and figured that Objectivism needed a new hit-man to bad-mouth academic philosophers. (Have you read the ghastly introduction to Peikoff's OPAR?)

But thanks, Bill, for your support in re courtesy -- even as you prepare to mete me my fate (so you think) in re my position.

(snip)

I am familiar with Peikoff's Preface in OPAR. (I assume you are referring to "Like any proper work of general philosophy, this book is written not for academics, but for human beings (including any academics who qualify)." I find that to be an extremely rude comment by Peikoff.)

I think Peikoff is still, decades later, smarting about failure to get tenure.

Regarding that comment, I think Peikoff could have better said:

"Like most proper works of general works of general philosophy, this book is aimed at a general audience, not solely at academics."

That way, the comment would have been true, and the rude comment avoided.

Bill P

[Michael Stuart Kelly]

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

There has been some improvement over time...

Michael