Between philosophy and pure mathematics there is a certain affinity, in the fact that both are general and a priori. Neither of them asserts propositions which, like those of history and geography, depend upon the actual concrete facts being just what they are. We may illustrate this characteristic by means of Leibniz's conception of many possible worlds, of which only one is actual. In all the many possible worlds, philosophy and mathematics will be the same; the differences will only be in respect of those particular facts which are chronicled by the descriptive sciences. Any quality, therefore, by which our actual world is distinguished from other abstractly possible worlds, must be ignored by mathematics and philosophy alike. Mathematics and philosophy differ, however, in in their manner of treating the general properties in which all possible worlds agree; for while mathematics starting from comparatively simple propositions, seeks to build up more and more complex results by deductive synthesis, philosophy, starting from data which are common to all knowledge, seeks to purify them into the the simplest statements of abstract form that can be obtained from them by logical analysis.
Bertrand Russell, Our Knowledge of the External World, Lecture 7
Retrieved from Project Gutenberg
I have areas of agreement and areas of disagreement with this post. I would put both mathematics and philosophy, as well as fields of study like constitutional law, largely in a class of knowledge referred to as formal knowledge. For me, this is knowledge derived from systems we set up, such as logic, uses propositions we assert to be true. In this sense, it is true mathematics, philosophy, or the law would be the same in any alternate world, as long as you hold the assumptions that they make to be unchangeable.
On the other hand, Russell wrote these lectures in 1914. long before the Kurt Gödel's incompleteness theorems and Paul Cohen had proved the independence of the Continuum Hypothesis in set theory. We can certainly talk about one possible world where the Continuum Hypothesis is true, and another where it is not true. In that case, we can't say mathematics will be identical in these two possible worlds. The would hold true for any branch of philosophy (or any other formal system). We will always come across unprovable statements, which may be true or false, and discuss possible worlds for each case. The assumptions of mathematics are not unchangeable, but instead, up to the decision of the mathematician.
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Showing posts with label Math. Show all posts
Showing posts with label Math. Show all posts
Wednesday, November 12, 2014
Thursday, July 28, 2011
Finding the delay of a clock without using the Lorentz transformations, part 2
In my previous post, I discussed how Jill would measure the time delay of a clock she was approaching at .6c without using the Lorentz transformations. I don't know if this has been done, but experiments like this could certainly serve as another validation of SR. However, in particular I'm trying to point out that Jill, regardless of whether she is moving, does not measure Jack's clock to be going faster.
I just edited the previous post to add some more information, including about how Jack can use the same reasoning to show the time delay in Jill clock is by a factor of .8, without using the Lorentz transform. This post will be about how Jill can make the same observation for clock1 (although the calculation is different), and an observer at clock1 would be able to make the reciprocal observation about Jill. Details are below the fold.
Since Jill and clock1 are moving away from each other, rather than toward each other, the diagram is different (and actually simpler). Jill still sees her clock move from 0 to 8 on her journey from clock1 to Jack, however, at the end of the trip cloc1 is read 4. That means Jill sees two of her own seconds to pass for every second that passes on clock1, or that the images of consecutive seconds on clock1 are two light-seconds apart for Jill.
So, if clock1 waits for t seconds between sending the image of 0 and sending the image of 1, the separation distance between the image of 0 and the image of 1 will be 1.6ct. Since 1.6ct = 2 ls, we get t=1.25 seconds. Thus, the fraction of seconds as measured by clock1 to seconds Jill measures for clock is 1/1.25, which is again .8. This means Jill measures clock1 to tick off 6.4 seconds on her trip between clock1 and Jack, the same as she measured for Jack.
In the reciprocal, when an observer (Jerry) at clock1 sees Jill pass Jack, Jerry has seen 8 seconds pass on Jill's clock, but 16 seconds pass on clock1 (the ten for the trip itself plus another 6 to see the image). So Jerry sees Jill's clock move at half the rate his clock does. Jerry can make the same calculations Jill does to measure Jill's clock ticking off .8 seconds for each of his.
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I just edited the previous post to add some more information, including about how Jack can use the same reasoning to show the time delay in Jill clock is by a factor of .8, without using the Lorentz transform. This post will be about how Jill can make the same observation for clock1 (although the calculation is different), and an observer at clock1 would be able to make the reciprocal observation about Jill. Details are below the fold.
Since Jill and clock1 are moving away from each other, rather than toward each other, the diagram is different (and actually simpler). Jill still sees her clock move from 0 to 8 on her journey from clock1 to Jack, however, at the end of the trip cloc1 is read 4. That means Jill sees two of her own seconds to pass for every second that passes on clock1, or that the images of consecutive seconds on clock1 are two light-seconds apart for Jill.
So, if clock1 waits for t seconds between sending the image of 0 and sending the image of 1, the separation distance between the image of 0 and the image of 1 will be 1.6ct. Since 1.6ct = 2 ls, we get t=1.25 seconds. Thus, the fraction of seconds as measured by clock1 to seconds Jill measures for clock is 1/1.25, which is again .8. This means Jill measures clock1 to tick off 6.4 seconds on her trip between clock1 and Jack, the same as she measured for Jack.
In the reciprocal, when an observer (Jerry) at clock1 sees Jill pass Jack, Jerry has seen 8 seconds pass on Jill's clock, but 16 seconds pass on clock1 (the ten for the trip itself plus another 6 to see the image). So Jerry sees Jill's clock move at half the rate his clock does. Jerry can make the same calculations Jill does to measure Jill's clock ticking off .8 seconds for each of his.
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Tuesday, July 26, 2011
Finding the delay of a clock without using the Lorentz transformations
For my readers who are not following the almost 2000-comment thread, aintnuthin and I are discussing Special Relativity. In particular, there is a question about whether a specific number is a prediction of what Jill measures, or a prediction of what Jill predicts Jack will measure.
The basic scenario: clock1 and Jack are at rest, sitting six light-seconds apart, and Jack has a clock (clock2) synchronized to clock1. Jill, holding a clock, passes by clock1 traveling inertially at .6c and synchronizes her clock to clock1 (so they now both read 0), and them passes by Jack. When Jill passes Jack, her clock reads 8 seconds and Jack's clock reads 10 seconds. If you use the Lorentz Transformations (LT) from the view that Jill's inertial state is the rest frame, Jill gets 6.4 seconds for clock2. The disagreement is over whether the 6.4 seconds is supposed to be what jack sees on his clock, as far as I can tell. My answer is below the fold.
My response is that the 6.4 seconds is the time Jill measures for clock2, not the time Jack measures for clock2. You can show it is the former with basic algebra. First, because of light-speed delay, Jill sees jack's clock to read -6 when Jill passes clock1. As Jill passes Jack, her clock has gained 8 seconds while Jack’s has gained 16 seconds. Jill can use that and her relative velocity of .6c to tell how much time passes on Jack's clock for her, without using the LT. I will load a diagram to help illustrate this.
This diagram is based on clock2 sending out an image reading -6 and then an image reading -4, and Jill receiving those images 1 second apart in time. Jill can measure how far apart the images were when they were sent, and therefore how much time passed in Jill's frame between when the first image was sent and when the second image was sent.
In between the times when clock2 reads -6 and clock2 reads -4, the image of clock2 reading -6 travels at c (as measured by Jill), while the clock2 itself travels at .6c. (Edit: adding sentnces) That means the rate of separation between the image of clock1 readin -6 and the actual clock 1 is c-.6c, or .4c. Thus, the ratio of the distance traveled by the image of clock1 reading -6 to the distance traveled by clock1 is c/.4c, or 1/.4 (End of edit). Since the image of clock2 reading -6 and the image of clock2 reading -4 are 1 light-second (ls) apart, the total distance from where clock2 generates the image of -6 and where that image is when clock2 generates the image of -4 is 1/.4 which is 2.5 ls. Since the image of clock2 travels at c, the image takes 2.5 seconds to travel 2.5 ls. So, Jill measures 2.5 seconds to pass while Jack’s clock moves from -6 to -4, or two seconds. 2/2.5 = .8, so for every second Jill observes on her clock, she observes .8 seconds to pass on Jack's clock, the amount predicted by the LT. Over the course of all 8 seconds Jill observes on her clock, this becomes 8 * .8 = 6.4 seconds. Again, this is what she observes to pass on the clocks, not a prediction of what Jack observes.
To forestall an objection, none of this is an explanation for the time delay. It is only a measure of the time delay that Jill can make without using the LT.
(Edit: adding sentnces) I also want to note that Jack can use the exact same process to measure the time delay for Jill, without using the Lorentz equations. Jack does not see the image of Jill passing clock1 until 6 seconds after it happens, when his clock read 6. So, Jack sees the entire trip in 4 seconds. In that time, Jill's clock moves from 0 to 8. Jack can use the same logic as above to measure Jill's clock to tick off .8 seconds for every seconds of his. (End of edit).
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The basic scenario: clock1 and Jack are at rest, sitting six light-seconds apart, and Jack has a clock (clock2) synchronized to clock1. Jill, holding a clock, passes by clock1 traveling inertially at .6c and synchronizes her clock to clock1 (so they now both read 0), and them passes by Jack. When Jill passes Jack, her clock reads 8 seconds and Jack's clock reads 10 seconds. If you use the Lorentz Transformations (LT) from the view that Jill's inertial state is the rest frame, Jill gets 6.4 seconds for clock2. The disagreement is over whether the 6.4 seconds is supposed to be what jack sees on his clock, as far as I can tell. My answer is below the fold.
My response is that the 6.4 seconds is the time Jill measures for clock2, not the time Jack measures for clock2. You can show it is the former with basic algebra. First, because of light-speed delay, Jill sees jack's clock to read -6 when Jill passes clock1. As Jill passes Jack, her clock has gained 8 seconds while Jack’s has gained 16 seconds. Jill can use that and her relative velocity of .6c to tell how much time passes on Jack's clock for her, without using the LT. I will load a diagram to help illustrate this.
This diagram is based on clock2 sending out an image reading -6 and then an image reading -4, and Jill receiving those images 1 second apart in time. Jill can measure how far apart the images were when they were sent, and therefore how much time passed in Jill's frame between when the first image was sent and when the second image was sent.
In between the times when clock2 reads -6 and clock2 reads -4, the image of clock2 reading -6 travels at c (as measured by Jill), while the clock2 itself travels at .6c. (Edit: adding sentnces) That means the rate of separation between the image of clock1 readin -6 and the actual clock 1 is c-.6c, or .4c. Thus, the ratio of the distance traveled by the image of clock1 reading -6 to the distance traveled by clock1 is c/.4c, or 1/.4 (End of edit). Since the image of clock2 reading -6 and the image of clock2 reading -4 are 1 light-second (ls) apart, the total distance from where clock2 generates the image of -6 and where that image is when clock2 generates the image of -4 is 1/.4 which is 2.5 ls. Since the image of clock2 travels at c, the image takes 2.5 seconds to travel 2.5 ls. So, Jill measures 2.5 seconds to pass while Jack’s clock moves from -6 to -4, or two seconds. 2/2.5 = .8, so for every second Jill observes on her clock, she observes .8 seconds to pass on Jack's clock, the amount predicted by the LT. Over the course of all 8 seconds Jill observes on her clock, this becomes 8 * .8 = 6.4 seconds. Again, this is what she observes to pass on the clocks, not a prediction of what Jack observes.
To forestall an objection, none of this is an explanation for the time delay. It is only a measure of the time delay that Jill can make without using the LT.
(Edit: adding sentnces) I also want to note that Jack can use the exact same process to measure the time delay for Jill, without using the Lorentz equations. Jack does not see the image of Jill passing clock1 until 6 seconds after it happens, when his clock read 6. So, Jack sees the entire trip in 4 seconds. In that time, Jill's clock moves from 0 to 8. Jack can use the same logic as above to measure Jill's clock to tick off .8 seconds for every seconds of his. (End of edit).
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Tuesday, July 27, 2010
Putting the Grim Philosopher argument to sleep
Now that we understand just what sort of thing a maximal collection of consistent truths must be, we can get around to discussing the attempt to form a contradiction based upon its possible existence, which argument The Maverick Philosopher (Dr. Vallicella) seems determined to support.
In fact, The Maverick Philosopher is so intent on preserving this argument that my comments pointing out a summarized version of my last post are no longer present. That is, I posted the first comment on 7/14, it was gone by 7/15, I posted another comment on 7/18 in the morning, I checked back on 7/18 in the afternoon to see it there, it was gone on 7/19, I sent an email on 7/19 asking why my comment was removed, and so far I have received no response to that email. I should not be surprised The Maverick Philosopher prefers to deal the disproof of his argument by covering up the evidence. After all, this is a man who, despite the presence of a dictionary definition that specifies "A strong fear, dislike, or aversion" as the definition of 'phobia', insists that "A phobia is an irrational fear. If you use the word in any other way you are misusing it. " Apparently avoiding the inconvenient is one of his preferred methods of dealing with reality. So, I'll have to produce the reality regarding his proof on my blog below the fold, where he can avoid it by not coming here at all. For someone who shakes in his boots at the idea of Muslims and thinks it rational, discretion is undoubtedly the better part of valor.
Addressing the 4 points in order:
Quite accurate, assuming the power set can be formed. That's part of the issue with this proof. When you get to a certain size, a collection goes from being a set to a proper class. Depending on the particular set theory, this means either that, for some proper class V, either P(V) cannot be constructed (NBG and ZFC) or it cannot be larger than V (MK). Either way, Cantor's Theorem does not apply.
As I demonstrated in the last post, the size of T is the size of the ordinal numbers, which is a proper class. So, there cannot be a greater cardinality in the class of all subclasses of T, even if such a subclass were to exist.
Since we have no disproof of the notion that there can be a class of all truths (such a class would exist under any typical rendition of set theory), it turns out that worlds can indeed be maximal collections of consistent truths.
This position requires at least some reason to differentiate between the category of consistent sets and the categroy of consistent conjunctive proposition, in that there are member of one which are not naturally a part of the other. However, there is an obvious and natural one-to-one relationship between the category of consistent sets and the category of consistent conjunctions. Namely, {pi}i ε I <---> &i ε I(pi). In fact, if you define a partial order on the category of consistent conjuction by sayin that &i ε I(pi) <= &j ε J(pj) whenever, for each i ε I, pi ε {pj}j ε J, this relationship even preserves the partial order of the set category. This means any maximal consisent conjunction would map to a maximal collection of consistent propositions. So, if you say the maximal collections of sets does not exist, perforce the maximal consistent conjuction does not exist, either.
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In fact, The Maverick Philosopher is so intent on preserving this argument that my comments pointing out a summarized version of my last post are no longer present. That is, I posted the first comment on 7/14, it was gone by 7/15, I posted another comment on 7/18 in the morning, I checked back on 7/18 in the afternoon to see it there, it was gone on 7/19, I sent an email on 7/19 asking why my comment was removed, and so far I have received no response to that email. I should not be surprised The Maverick Philosopher prefers to deal the disproof of his argument by covering up the evidence. After all, this is a man who, despite the presence of a dictionary definition that specifies "A strong fear, dislike, or aversion" as the definition of 'phobia', insists that "A phobia is an irrational fear. If you use the word in any other way you are misusing it. " Apparently avoiding the inconvenient is one of his preferred methods of dealing with reality. So, I'll have to produce the reality regarding his proof on my blog below the fold, where he can avoid it by not coming here at all. For someone who shakes in his boots at the idea of Muslims and thinks it rational, discretion is undoubtedly the better part of valor.
Addressing the 4 points in order:
1. Cantor's Theorem states that for any set S, the cardinality of the power set P(S) of S > the cardinality of S. ... But the proof needn't concern us. It is available in any standard book on set theory.
Quite accurate, assuming the power set can be formed. That's part of the issue with this proof. When you get to a certain size, a collection goes from being a set to a proper class. Depending on the particular set theory, this means either that, for some proper class V, either P(V) cannot be constructed (NBG and ZFC) or it cannot be larger than V (MK). Either way, Cantor's Theorem does not apply.
2. Suppose there is a set T of all truths, ... But according to Cantor's Theorem, the power set of T is strictly larger than T. So there will be more of those truths than there are truths in T. It follows that T cannot be the set of all truths.
As I demonstrated in the last post, the size of T is the size of the ordinal numbers, which is a proper class. So, there cannot be a greater cardinality in the class of all subclasses of T, even if such a subclass were to exist.
3. Given that there cannot be a set of all truths, the actual world cannot be the set of all truths. This implies that possible worlds cannot be maximally consistent sets of propositions. I learned the Cantorian argument that there is no set of all truths from Patrick Grim. I don't know whether he applies it to the question whether worlds are sets.
Since we have no disproof of the notion that there can be a class of all truths (such a class would exist under any typical rendition of set theory), it turns out that worlds can indeed be maximal collections of consistent truths.
4. As far as I can see, the fact that possible worlds cannot be maximally consistent sets does not prevent them from being maximally consistent conjunctive propositions.
This position requires at least some reason to differentiate between the category of consistent sets and the categroy of consistent conjunctive proposition, in that there are member of one which are not naturally a part of the other. However, there is an obvious and natural one-to-one relationship between the category of consistent sets and the category of consistent conjunctions. Namely, {pi}i ε I <---> &i ε I(pi). In fact, if you define a partial order on the category of consistent conjuction by sayin that &i ε I(pi) <= &j ε J(pj) whenever, for each i ε I, pi ε {pj}j ε J, this relationship even preserves the partial order of the set category. This means any maximal consisent conjunction would map to a maximal collection of consistent propositions. So, if you say the maximal collections of sets does not exist, perforce the maximal consistent conjuction does not exist, either.
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Thursday, July 22, 2010
A primer on transfinitie recursion
I am going to discuss a transfinite recursion in this post, as a prelude to filling in the argument from last July pointing out that the attempted argument by Dr. Vallicella against the possible existence of a maximal collection of truths, to which I wrote a response last July. Today, that response seems inadequate, so I intend to do the proof more comprehensively. That way, in the future I can just link to the new post whenever Dr. Vallicella decides to raise this argument again, as he did last week.
A good background in ordinal numbers generally will be very helpful for understanding this post. I will not attempt to improve on the three-part series penned by Mark Chu-Carroll over at Good Math, Bad Math. For better or worse, I'll assume the reader understands that material and go from there below the fold.
Transfinite recursion is the process of defining a function over all of the ordinals. Recursive definitions are fairly common at the level of the finite. For example, the factorial function (identified by a bang, for example, 3!) comes up in pretty much any college level math class that mentions probability. Factorials can be defined recursively. You start off saying 0! = 1. Then, for any integer n > 0, n! = n * (n-1)!. Thus,
With transfinite recursion, the same basic idea applies: for any particular ordinal ω, you use the definitions of some or all the ordinals β where β < ω to define the function on ω. If you can do that as a single definition, so much the better, but often these definitions consist of three steps: define the function for the first ordinal (0), define the function for each ordinal that has a direct predecessor (the are called successor ordinals), and define the function for ordinals that have predecessors, but no direct predecessor (these are called limit ordinals). As you might have noticed, the first two steps are the same as we saw for the definition of factorial. I will be using transfinite recursion to start with a single truth t, and generate from that a maximal collection of consistent truths T that derive from t. "Maximal" means: given any collection Tx of truths derived from t, Tx is a subset of T.
The first step of the transfinite recursion is simple: T0 = {t}. T0 is very obviously derived from t.
Edit: In my discussion in the next paragraph, I glected to account for the fact that Tβ+1 will contain every truth in Tβ even without being unioned. This is corrected.
The next step will be to define Tβ+1 given some Tβ derived from t. Of course, I will use the definition offered by Dr. Vallicella in his proof, otherwise there would be no point to this. Tβ will consist of truths, {tβ,1, . . . , tβ,i, tβ,i+1, . . ., tβ,ω} that derive from t. Consider the power set P(Tβ) of Tβ. The truth tβ,1 in Tβ will be a member of some of Tβ's subsets but not of others. Thus, tβ,1 ε {tβ,1, tβ,2}, and tβ,1 ~ε { } are both truths. In general, for each subset s in the power set P(Tβ) there will be a truth of the form tβ,1 ε s or tβ,1 ~ε s. You can denote whichever of these statements is true by q(tβ,1,s). We then define Tβ+1 = {q(tβ,i,s) | tβ,i ε Tβ and s ε P(Tβ)} union {t}, and say that Tβ+1 is also derived from t. This is the process Dr. Vallicella uses to generate a larger set. Note that since every statement of Tβ is repeated in Tβ+1, Tβ is a subset of Tβ+1.
Before we move on, lets look at T1 and T2. T0 has two subsets, {} and {t}, and one element, {t}. t ε {t} (we can call this truth t1,1) and t ~ε {} (we can call this truth t1,2). Then, T1 = {t, t1,1, t1,2}. T1 has eight subsets: {}, {t}, {t1,1}, {t1,2}, {t, t1,1}, {t, t1,2}, {t1,1, t1,2}, and {t, t1,1, t1,2}. With three elements of T1 and eight subsets of T1, we can generate 24 true statements of the form t2,j, 1<= j <= 24, two of which were in T1 (left as an exercise for the interested reader). T2 will consist of these 24 statements, and t, for twenty-five truths. In case you are curious, there are 25*2^25+25 (i.e. 838,860,825) statements in T3.
The third step is simple. For any limit ordinal ω, Tω is the union of all the Tβ where β < ω. Since all the statements of each Tβ are derived from t by the process in the previous paragraph, every element of Tω is derived from t.
The (proper) class of all of all ordinals is commonly called Ω. So, for each ω ε Ω, we have defined Tω. T is the union of all the Tω over all the ω ε Ω. T is now the maximal collection of truths derived from t. Since for any ordinal ω, #(Tω) >= #(ω) (that is, Tω at least as many elements as ω), T has more elements than any ordinal. Therefore, T must be a proper class. I'll discuss what this means in my next post.
Read more!
A good background in ordinal numbers generally will be very helpful for understanding this post. I will not attempt to improve on the three-part series penned by Mark Chu-Carroll over at Good Math, Bad Math. For better or worse, I'll assume the reader understands that material and go from there below the fold.
Transfinite recursion is the process of defining a function over all of the ordinals. Recursive definitions are fairly common at the level of the finite. For example, the factorial function (identified by a bang, for example, 3!) comes up in pretty much any college level math class that mentions probability. Factorials can be defined recursively. You start off saying 0! = 1. Then, for any integer n > 0, n! = n * (n-1)!. Thus,
1! = 1 * 0! = 1 * 1 = 1.This is not the only way to define n!, but it is a useful way. This definition consisted of two steps: the definition for 0, and the definition for any number given the value for the immediately previous number.
2! = 2 * 1! = 2 * 1 = 2.
3! = 3 * 2! = 3 * 2 = 6.
4! = 4 * 3! = 4 * 6 = 24.
5! = 5 * 4! = 5 * 24 = 120.
With transfinite recursion, the same basic idea applies: for any particular ordinal ω, you use the definitions of some or all the ordinals β where β < ω to define the function on ω. If you can do that as a single definition, so much the better, but often these definitions consist of three steps: define the function for the first ordinal (0), define the function for each ordinal that has a direct predecessor (the are called successor ordinals), and define the function for ordinals that have predecessors, but no direct predecessor (these are called limit ordinals). As you might have noticed, the first two steps are the same as we saw for the definition of factorial. I will be using transfinite recursion to start with a single truth t, and generate from that a maximal collection of consistent truths T that derive from t. "Maximal" means: given any collection Tx of truths derived from t, Tx is a subset of T.
The first step of the transfinite recursion is simple: T0 = {t}. T0 is very obviously derived from t.
Edit: In my discussion in the next paragraph, I glected to account for the fact that Tβ+1 will contain every truth in Tβ even without being unioned. This is corrected.
The next step will be to define Tβ+1 given some Tβ derived from t. Of course, I will use the definition offered by Dr. Vallicella in his proof, otherwise there would be no point to this. Tβ will consist of truths, {tβ,1, . . . , tβ,i, tβ,i+1, . . ., tβ,ω} that derive from t. Consider the power set P(Tβ) of Tβ. The truth tβ,1 in Tβ will be a member of some of Tβ's subsets but not of others. Thus, tβ,1 ε {tβ,1, tβ,2}, and tβ,1 ~ε { } are both truths. In general, for each subset s in the power set P(Tβ) there will be a truth of the form tβ,1 ε s or tβ,1 ~ε s. You can denote whichever of these statements is true by q(tβ,1,s). We then define Tβ+1 = {q(tβ,i,s) | tβ,i ε Tβ and s ε P(Tβ)} union {t}, and say that Tβ+1 is also derived from t. This is the process Dr. Vallicella uses to generate a larger set. Note that since every statement of Tβ is repeated in Tβ+1, Tβ is a subset of Tβ+1.
Before we move on, lets look at T1 and T2. T0 has two subsets, {} and {t}, and one element, {t}. t ε {t} (we can call this truth t1,1) and t ~ε {} (we can call this truth t1,2). Then, T1 = {t, t1,1, t1,2}. T1 has eight subsets: {}, {t}, {t1,1}, {t1,2}, {t, t1,1}, {t, t1,2}, {t1,1, t1,2}, and {t, t1,1, t1,2}. With three elements of T1 and eight subsets of T1, we can generate 24 true statements of the form t2,j, 1<= j <= 24, two of which were in T1 (left as an exercise for the interested reader). T2 will consist of these 24 statements, and t, for twenty-five truths. In case you are curious, there are 25*2^25+25 (i.e. 838,860,825) statements in T3.
The third step is simple. For any limit ordinal ω, Tω is the union of all the Tβ where β < ω. Since all the statements of each Tβ are derived from t by the process in the previous paragraph, every element of Tω is derived from t.
The (proper) class of all of all ordinals is commonly called Ω. So, for each ω ε Ω, we have defined Tω. T is the union of all the Tω over all the ω ε Ω. T is now the maximal collection of truths derived from t. Since for any ordinal ω, #(Tω) >= #(ω) (that is, Tω at least as many elements as ω), T has more elements than any ordinal. Therefore, T must be a proper class. I'll discuss what this means in my next post.
Read more!
Wednesday, August 26, 2009
The foundations of non-skeptical thinking
I was looking over some of the web sites attached to the members of Intelligent Reasoning when I came across the blog More Than Words by team member David Anderson, and in particular noticed a label he had for Mathematics. It turned into an opportunity to blog on skepticism, mathematics, and philosophy in general, an irresistible combination to me. Continued below the fold.
The particular topic is the general notion of whether mathematics, and similar sorts of activities, are describing something that is real. I have touched upon this topic in previous posts. In some ways, this argument is at the very heart of the worldviews that assert there can be some absolute truth, moral principles, authority, etc. You can't hang an absolute morality or absolute knowledge on a logical foundation that is arbitrary, at least not without some cognitive discomfort.
So, n Anderson's post on mathematics, we find Anderson arguing for how mathematicians feel mathematics is a feature of reality. All quotes following are from his post, unless otherwise noted.
Yes, the good old 2+2 = 5 argument (it never occurs to such people that it is perfectly reasonable in some circumstances to say 2 + 2 = 1). Now, I am sure there are mathematicians who think there is some big design they are uncovering, and that belief does not prevent them from being fine mathematicians. It's probably even a positive when you are working closely with computer scientists, engineers, or physicist in pushing the boundaries of mathematics for direct applications. However, having enjoyed some 66 semester hours of undergraduate mathematics and another 36 at the graduate level, I can say that I ran into more than a few instructors and fellow students who treated mathematics exactly like an arbitrary toy, something you could play with, take apart, and build to order. Most of them, to my knowledge, would not have cared if there were a philosophical position that reflected this attitude, but I find fictionalism to be a decent approximation. We were playing in a universe we created for our benefit, and occasionally something useful would pop out.
Actually, mathematics is highly receptive to humans treating it like an arbitrary invention. It's why we have Euclidean, Lobachevkian, and Riemannian geometries to describe different sorts of space. It's we we have intuitionist, constructivist, and para-consistent logics and their mathematical descendants. We add or remove axioms at our pleasure, look at the results, and call it fun (and occasionally useful).
Now, this is a great example of fuzzy-headed thinking, an absolute truth that is right except where it is wrong. If I smash two apples against two apples, I will quite possibly have over 100 bits of apple. If I pour two liters of water into two liters of alcohol, I will have under 3.9 liters of fluid. Except in very limited circumstances, numbers are not conserved!
Here's an interesting question: how the fundamental nature of an ideal reality depend upon a human mind? Personally, I don't think is can or does. Perhaps Anderson mistyped, and meant that the reality is revealed to the mind, or perhaps it was a Freudian slip. I do think that the reality of any particular five-dimensional construction we create depends on our mind. That makes it not a fundamental property of nature.
The notion of twoness, and similar notions, is the pattern that we humans impose upon our world to make it simpler. Since we are identifying a pattern, it is unsurprising that no one instance of the pattern will form a complete rendition.
I can create a two-dimensional shape that is approximately (but never exactly - because we live in a world of discrete atoms and molecules) equal to the one in the equation of the graph I was using. This is all simple enough.
Anderson references quantum mechanics later on. Perhaps it's my ignorance, but wouldn't it be much more correct to say we live in a worlds of fuzzy atoms and molecules behaving in probabilistic ways? Even here, the need to have absolutes alters the mindset.
I'm not sure why this is particularly breathtaking. While we play with mathematics as a toy, we also do like to make it useful. So, a mathematician, playing around with equations that are useful descriptions of quantum mechanics, comes across a new feature which also has utility, based upon deductively playing with something already utilized. I don't see why that would be surprising. I don't think it would be particularly shocking to Anderson if an engineer predicted that a steel beam would collapse under a certain weight load, and was correct.
Anderson very obligingly illustrated my point for me. This notion of real versus useful fictions is not just semantics and playing with words. It's a part of how and why believers believe, one of the pillars they use to prop up their world view.
I'll offer one final link to a rant by a mathematician, offered because he views mathematics as an art. I don't endorse everything he says about education, but his view of mathematics fits much more closely with the mathematicians I learned from and with.
Read more!
The particular topic is the general notion of whether mathematics, and similar sorts of activities, are describing something that is real. I have touched upon this topic in previous posts. In some ways, this argument is at the very heart of the worldviews that assert there can be some absolute truth, moral principles, authority, etc. You can't hang an absolute morality or absolute knowledge on a logical foundation that is arbitrary, at least not without some cognitive discomfort.
So, n Anderson's post on mathematics, we find Anderson arguing for how mathematicians feel mathematics is a feature of reality. All quotes following are from his post, unless otherwise noted.
Mathematics is also a very interesting field if you have an interest in philosophy and questions about design in nature. Almost all mathematicians are in practice realists - they believe that as they make progress in their field they are involved in discovering and not in inventing. (See here for more on this distinction). That is, they act and research as if there is already a transcendent, pre-existing mathematical universe "out there" that is waiting for us to find and explore it. The opposite of that is behaving as if mathematics is our arbitrary toy, to be played with, deconstructed and rebuilt as we please. Shall we adopt the convention that 2+2 = 5 from now on and see where that takes us?
Yes, the good old 2+2 = 5 argument (it never occurs to such people that it is perfectly reasonable in some circumstances to say 2 + 2 = 1). Now, I am sure there are mathematicians who think there is some big design they are uncovering, and that belief does not prevent them from being fine mathematicians. It's probably even a positive when you are working closely with computer scientists, engineers, or physicist in pushing the boundaries of mathematics for direct applications. However, having enjoyed some 66 semester hours of undergraduate mathematics and another 36 at the graduate level, I can say that I ran into more than a few instructors and fellow students who treated mathematics exactly like an arbitrary toy, something you could play with, take apart, and build to order. Most of them, to my knowledge, would not have cared if there were a philosophical position that reflected this attitude, but I find fictionalism to be a decent approximation. We were playing in a universe we created for our benefit, and occasionally something useful would pop out.
In my view, the atheist materialists who have tried to explain their view of reality are in an exceptionally weak position when they seek to explain mathematics in non-transcendent terms. Mathematics resists, at multiple levels, any attempt to treat it as an arbitrary invention of the human mind. Almost at every turn it cries out "I was here before you, and I am bigger than you!". Maths is a very theistic subject!
Actually, mathematics is highly receptive to humans treating it like an arbitrary invention. It's why we have Euclidean, Lobachevkian, and Riemannian geometries to describe different sorts of space. It's we we have intuitionist, constructivist, and para-consistent logics and their mathematical descendants. We add or remove axioms at our pleasure, look at the results, and call it fun (and occasionally useful).
I can of course always add two apples to two apples and will always get four apples (an inconvenient truth for the atheists who want to argue that mathematical truths are not transcendent!) - but as I do so I'm conscious that there is a notion of "two-ness" or "four-ness" that goes far beyond the tasty bits of fruit and is independent of them.
Now, this is a great example of fuzzy-headed thinking, an absolute truth that is right except where it is wrong. If I smash two apples against two apples, I will quite possibly have over 100 bits of apple. If I pour two liters of water into two liters of alcohol, I will have under 3.9 liters of fluid. Except in very limited circumstances, numbers are not conserved!
The more complicated the mathematics gets, the more obvious this becomes. I can move from the simple adding of objects to a dimension up and do calculus to work out the area under a graph. I can then accelerate to five or six dimensional spaces and work out their corresponding concept of volume. I can work out the properties of completely theoretical objects. you get the idea. Mathematics speaks to us of an ideal reality which depends on the mind.
Here's an interesting question: how the fundamental nature of an ideal reality depend upon a human mind? Personally, I don't think is can or does. Perhaps Anderson mistyped, and meant that the reality is revealed to the mind, or perhaps it was a Freudian slip. I do think that the reality of any particular five-dimensional construction we create depends on our mind. That makes it not a fundamental property of nature.
Whilst it depends on the mind, mathematics also seems to have an unbreakable link to the physical world. In the most simple example, there's something about those two oranges that has the notion of two-ness. The notion of two-ness is contained, but not exhausted, by them.
The notion of twoness, and similar notions, is the pattern that we humans impose upon our world to make it simpler. Since we are identifying a pattern, it is unsurprising that no one instance of the pattern will form a complete rendition.
I can create a two-dimensional shape that is approximately (but never exactly - because we live in a world of discrete atoms and molecules) equal to the one in the equation of the graph I was using. This is all simple enough.
Anderson references quantum mechanics later on. Perhaps it's my ignorance, but wouldn't it be much more correct to say we live in a worlds of fuzzy atoms and molecules behaving in probabilistic ways? Even here, the need to have absolutes alters the mindset.
What is more breath-taking, though, is to understand that correspondences between abstract mathematics and the physical world have also been discovered in far more complicated cases. In some areas, mathematicians discovered new theorems in highly abstract areas that nobody thought would ever turn out to have a practical application - but in fact they actually perfectly described physical phenomena observed decades later. Do you get that? Away in his dusty study somewhere, the mathematician was working on a problem that was thought to be far too abstract to have any real application. Some time later, a physicist realised that this bit of mathematics was the key to something that he was observing. Quantum physics provides a number of illustrations of this.
I'm not sure why this is particularly breathtaking. While we play with mathematics as a toy, we also do like to make it useful. So, a mathematician, playing around with equations that are useful descriptions of quantum mechanics, comes across a new feature which also has utility, based upon deductively playing with something already utilized. I don't see why that would be surprising. I don't think it would be particularly shocking to Anderson if an engineer predicted that a steel beam would collapse under a certain weight load, and was correct.
Observation one: Mathematics has its seat in minds. Observation two: We also now know that mathematics is also embedded at a fundamental and essential in physical reality. Inescapable conclusion: Physical reality is the product of a mind.
Anderson very obligingly illustrated my point for me. This notion of real versus useful fictions is not just semantics and playing with words. It's a part of how and why believers believe, one of the pillars they use to prop up their world view.
I'll offer one final link to a rant by a mathematician, offered because he views mathematics as an art. I don't endorse everything he says about education, but his view of mathematics fits much more closely with the mathematicians I learned from and with.
Read more!
Sunday, July 12, 2009
My oversight, apologies, and thoughts
After last week's post concerning the Cantorian argument against possible worlds being maximally consistent sets of propositions posted by Dr. Vallicella, the Maverick Philosopher, I sent an email to Dr. Vallicella for his thought, and his initial response was that I missed the point of the argument entirely. He was kind and patient through out the exchange, and entirely correct instating that I had missed the point. Using "e" to represent the element relation, I had entirely missed the point that the truths being constructed were of the form t1 e {t1, t2} and ~(t1 e {t2, t3}. This means that Dr. Grim/Dr. Vallicella have created a set of size T X P(T) of truths from set T. Why I missed this, and what it means for my analysis of the proof, is below the fold.
As for why I missed it, it just seems to be a blind spot with me. Ii will probably not make this error again in the next month or two, but it will happen eventually, because I see a fundamental difference in the type of between saying "the earth orbits the sun" and "t1 e {t1, t2}", and I don't really see the latter belonging in T. The latter sort of statement I think of as being a formal truth, one that it true based on the initial definitions, definition that we select arbitrarily because of their usefulness. My understand of Dr. Vallicella's worldview (and if I mis-characterise him here, I apologize) is that there is no relevant difference between the two statements. As long as I carry the first view in my brain, I no doubt will fall prey to the same blindness about arguments founded in the second view in the future. I beg the patience of any readers in this regard.
Well, after being so kind and professorial in our exchange, Dr. Vallicella certainly deserves to have Grim's/his analysis validated, and it would have been my pleasure to do so. Unfortunately, I can't do this with honesty, because it turns out that the proof will still fail. One of the unfortunate side effects of mixing formal truths and non-formal truths is the formal truths have a tendency to grow past any reasonable size. When you include all the formal truths of the type "t1 e {t1, t2}", you wind up with T being a proper class, and that means |T| does not exist, and to the degree P(T) can be defined, |P(T)| does not exist either, so obviously you can't have |P(T)| > |T|.
This is easy to see when you consider what must be included in T. For simplicity sake, let's start with a universe with one non-formal truth, t0, and call the universe itself T0. We can use the very process describe to create R0 containing two truths: t1 saying t0 e {t0} and t2 saying ~(t0 e {}). Defining T1 as the union of R0 and T0, T1 = {t0, t1, t2}. Applying the same process to T1, you get 24 elements in T2, 402,653,184 elements in T3, etc. Then, you need to combine the contents of all the Tn into Taleph-0. From Taleph-0 you can build (assuming without loss of generality the generalized continuum hypothesis)Taleph-1, Taleph-2, etc., so that there will be a version of Tx greater than any specific cardinal number. This means that, when you finally union all these Tx to create T, T has the size of a proper class.
I want to thank Dr. Vallicella for his cooperation and gentlemanly behavior.
Read more!
As for why I missed it, it just seems to be a blind spot with me. Ii will probably not make this error again in the next month or two, but it will happen eventually, because I see a fundamental difference in the type of between saying "the earth orbits the sun" and "t1 e {t1, t2}", and I don't really see the latter belonging in T. The latter sort of statement I think of as being a formal truth, one that it true based on the initial definitions, definition that we select arbitrarily because of their usefulness. My understand of Dr. Vallicella's worldview (and if I mis-characterise him here, I apologize) is that there is no relevant difference between the two statements. As long as I carry the first view in my brain, I no doubt will fall prey to the same blindness about arguments founded in the second view in the future. I beg the patience of any readers in this regard.
Well, after being so kind and professorial in our exchange, Dr. Vallicella certainly deserves to have Grim's/his analysis validated, and it would have been my pleasure to do so. Unfortunately, I can't do this with honesty, because it turns out that the proof will still fail. One of the unfortunate side effects of mixing formal truths and non-formal truths is the formal truths have a tendency to grow past any reasonable size. When you include all the formal truths of the type "t1 e {t1, t2}", you wind up with T being a proper class, and that means |T| does not exist, and to the degree P(T) can be defined, |P(T)| does not exist either, so obviously you can't have |P(T)| > |T|.
This is easy to see when you consider what must be included in T. For simplicity sake, let's start with a universe with one non-formal truth, t0, and call the universe itself T0. We can use the very process describe to create R0 containing two truths: t1 saying t0 e {t0} and t2 saying ~(t0 e {}). Defining T1 as the union of R0 and T0, T1 = {t0, t1, t2}. Applying the same process to T1, you get 24 elements in T2, 402,653,184 elements in T3, etc. Then, you need to combine the contents of all the Tn into Taleph-0. From Taleph-0 you can build (assuming without loss of generality the generalized continuum hypothesis)Taleph-1, Taleph-2, etc., so that there will be a version of Tx greater than any specific cardinal number. This means that, when you finally union all these Tx to create T, T has the size of a proper class.
I want to thank Dr. Vallicella for his cooperation and gentlemanly behavior.
Read more!
Sunday, July 5, 2009
Cantor offers nothing useful on maximally consistent worlds
I fully acknowledge that I am a philosophical amateur, and no doubt from time my thought and questions reflect this. However, I know enough to know that people who have studied philosophy professionally, but not mathematics, make ludicrous mathematical arguments. Such is the case with an attempt by Dr. Vallicella to export a standard Cantorian argument from mathematics to philosophical constructs. While you could say logic is the grammar of mathematics, the difference in vocabulary makes any sort of transition of proofs from one venue to the other a difficult procedure, and not one to be done casually.
So, lets say we have this maximal set T of true statements (truths, for short). {t1, . . . , ti, ti + 1, . . .}. In particular, let's consider {t1 ,t2}. The first question we need to answer: Is this subset itself a truth? If this subset is not a truth, then the entire argument from the creation of the power set is meaningless, because the power set does not consist of truths, but of collections of truths, and there is no reason to presume the cardinality of all collections of truths would be the same as the cardinality of all truths. In fact, since the proof relies on looking at elements of P(T) as if there were in T, for Dr. Vallicella's argument to be cogent, we need to apply the standard that {t1, t2} is itself a truth. We are not given a definition for this truth, unfortunately, and how it relates to t1 or t2, possibly via some sort of truth table. At least, since we are using sets, we know that {t1} = {t1, t1} = {t1, t1, t1}.
However, that leads us to another area of fuzzy definition. Is {t1} the same truth as t1? Is {t1} the same truth as {{t1}}? Will {t1 ,t2} be the same truth as {t1 ,t2,{t1 ,t2}}? Basically, can we remove all internal braces (except for the empty set)?
If we we allow the removal of all internal braces, then the proof falls apart, because all of the elements of P(T) will already be elements of T, after removing the internal braces and reducing the duplications. For example, let's look at a world of one atomic truth (that is, truths that not sets of other truths). T = {{}, t1}. Then P(T) = {{},{{}},{ t1},{{},t1}} = {{},{}, t1,{},t1} = {{}, t1} = T.
So let us consider the construction where we can not remove internal braces. Now, since we have a valid method of creating a new truth from previously existing truths, by inclusion in sets, that means for any set Q of atomic truths, we find the power set of that set of truths, and the power set of that first power set of truths, and the power set of that second power set of truths, etc., already in T. How far can we go? Do we allow for there to be (loosely speaking) an infinite number of brace levels? If we do not allow that, then we know either |T| (the cardinality of T) = |{1, 2, 3, 4, ...}|, that is, T is countable, whenever Q is finite or countable, otherwise |T| = |Q|. However, this has the defect of removing much of P(T) from being eligible to be in T, because P(T) will include elements with an infinite number of braces.
In fact, if we place any limit M at all on the number of braces, we find that |Q| being less than or equal to M means |T| = M, otherwise |T| = |Q|, and either way P(T) will have elements that are not capable of being in T. So, the only way around this is place no restriction on the number of levels of inclusion. This has the side effect of making T a proper class even when |Q| = 1, so P(T) does not even exist.
So, it would seem regardless of set-up we are left with a choice of P(T) = T, P(T) having elements that do not qualify to be in T, or P(T) not existing. Regardless, the attempted proof fails.
Read more!
So, lets say we have this maximal set T of true statements (truths, for short). {t1, . . . , ti, ti + 1, . . .}. In particular, let's consider {t1 ,t2}. The first question we need to answer: Is this subset itself a truth? If this subset is not a truth, then the entire argument from the creation of the power set is meaningless, because the power set does not consist of truths, but of collections of truths, and there is no reason to presume the cardinality of all collections of truths would be the same as the cardinality of all truths. In fact, since the proof relies on looking at elements of P(T) as if there were in T, for Dr. Vallicella's argument to be cogent, we need to apply the standard that {t1, t2} is itself a truth. We are not given a definition for this truth, unfortunately, and how it relates to t1 or t2, possibly via some sort of truth table. At least, since we are using sets, we know that {t1} = {t1, t1} = {t1, t1, t1}.
However, that leads us to another area of fuzzy definition. Is {t1} the same truth as t1? Is {t1} the same truth as {{t1}}? Will {t1 ,t2} be the same truth as {t1 ,t2,{t1 ,t2}}? Basically, can we remove all internal braces (except for the empty set)?
If we we allow the removal of all internal braces, then the proof falls apart, because all of the elements of P(T) will already be elements of T, after removing the internal braces and reducing the duplications. For example, let's look at a world of one atomic truth (that is, truths that not sets of other truths). T = {{}, t1}. Then P(T) = {{},{{}},{ t1},{{},t1}} = {{},{}, t1,{},t1} = {{}, t1} = T.
So let us consider the construction where we can not remove internal braces. Now, since we have a valid method of creating a new truth from previously existing truths, by inclusion in sets, that means for any set Q of atomic truths, we find the power set of that set of truths, and the power set of that first power set of truths, and the power set of that second power set of truths, etc., already in T. How far can we go? Do we allow for there to be (loosely speaking) an infinite number of brace levels? If we do not allow that, then we know either |T| (the cardinality of T) = |{1, 2, 3, 4, ...}|, that is, T is countable, whenever Q is finite or countable, otherwise |T| = |Q|. However, this has the defect of removing much of P(T) from being eligible to be in T, because P(T) will include elements with an infinite number of braces.
In fact, if we place any limit M at all on the number of braces, we find that |Q| being less than or equal to M means |T| = M, otherwise |T| = |Q|, and either way P(T) will have elements that are not capable of being in T. So, the only way around this is place no restriction on the number of levels of inclusion. This has the side effect of making T a proper class even when |Q| = 1, so P(T) does not even exist.
So, it would seem regardless of set-up we are left with a choice of P(T) = T, P(T) having elements that do not qualify to be in T, or P(T) not existing. Regardless, the attempted proof fails.
Read more!
Sunday, January 4, 2009
The reality of sets
I checked the blog of the Maverick Philosopher, aka Dr. William Vallicella, over the course of the last week, and once again find he has provided material to which I would respond. In fact, in writing two posts about the existence of sets, he has touched on a subject near to my heart. In fact, I discuss this very topic, in a much less formal manner, in every single Geometry class. There is a very real distinction between a formal system, a construct of the mind that we use to generalize reality and draw deductive conclusions, and the underlying reality itself. Mathematics is first and foremost a formal system. I think Vallicella provides an excellent example of how the eagerness that believers have to create these non-materials things of real existence spills over into assigning reality to all sorts of fanciful concoctions, much as we see supporters of one type of woo are often more likely to pickup or support other sorts of woo. Vallicella's comments are italicized.
Set theory can be done either naively or axiomatically. In the standard axiomatic approach to the subject, that of Zermelo-Fraenkel, the existence of the null set is posited in a special axiom.
Two errors in two sentences. One of the primary reasons that set theory was put into an axiomatic format is that, when doing set theory naively, you come across both logical and semantic contradictions that are avoided by the axiomatic process. Also, while Zermelo's original approach had a special axiom for the empty, the standard approach to the Zermelo-Fraenkel axiom schema contains no such axiom. Rather, the empty set is the derived result of the axiom schema of replacement, which in the short form says that functions defined on classes will map sets into sets.
Intuitively, the existence of a set depends on the existence of its members. The set consisting of me and my cat cannot exist unless both man and cat exist: if either of us should cease to exist, the set would cease to exist. It exists because we exist, not vice versa. (This is of course not a causal use of 'because.') In the case of the null set, however, there is nothing on which the null set can depend for its existence.
The truth is that sets don't exist, at all. The set of Vallicella and his cat is a mental shortcut that we use, but it has no reality unto itself. When I talk about the red books on my bookshelf, they are a set of books for only as long as any person considers them to be a set of books. There is no ontological status the property of their being a collection. After all, how is the nature of the books changed if I say that I have carefully designed them to be a set, or if I say they are just a bunch of book I have in that place? I can tell Person A the one and Person B the other, they can't compare notes, examine the books, and thereby discern which account is more truthful. So, it seems the nature of being a set has no reality behind it. In particular, why should the empty set have any more reality?
No working mathematician is likely to lose any sleep over this, however. He will tell us that the null set is convenient, computationally useful and ought to be judged by its practical fruits.
Working mathematicians generally seem aware that almost everything they do is based upon assumption that are help arbitrarily, because we have found them useful. The very grammar of mathematics, predicate logic, is based upon the highly arbitrary notion notion that every statement is true or false. The arbitrariness of this decision is shown by the success of the intuitionists and the various alternate logical evaluation schemata in creating perfectly functional mathematical systems. They know that sets exists only to the degree that we think of the collection of various objects as a set, and so one more useful, non-real entity is not going to cause consternation.
From the first post:
So on the one hand, the null set is useful and well-motivated from within the circle of set-theoretical ideas, but on the other hand, it appears philosophically to be a creature of darkness. Is there a way to get rid of the darkness?
By my lights, the philosopher aims at a degree and a type of clarity that the mathematician qua mathematician -- not to mention other nonphilosophers -- does not care about. Of course, I am not saying that he should care about it. He is within his rights in simply dismissing concerns like the one raised in this post as irrelevant to his concerns or unimportant given his goals and priorities. What is intolerable, though, is the mathematician who gives a lousy philosophical answer to a philosophical question, especially if he is only half-aware that it is a philosophical question.
Of course, philosophy itself is also a formal system. When the mathematician chooses to ignore the needs and desires of one formal system in favor of another, they do so 2with the understanding that all formal systems are arbitrarily chosen based upon their perceived usefulness.
He is then like the neuroscientist who, refusing to stick to his subject-matter, says silly things about mind and consciousness, all the while oblivious to the philosophical problems to which he gives silly answers.
Naturally, we see one sort of woo dissolving into another.
From the second post:
A set in the mathematical (as opposed to commonsense) sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-over-many: it is one single item despite its having six distinct members each of which is distinct from the set, and all of which, taken collectively, are distinct from the set.
Or, since the very notion of set is in and of itself a mental construct, and not a reality, it is certainly different from the shoes themselves, which presumably are real enough that Vallicella puts them on his feet once in a while.
Vallicella goes on for quite a while being very clear that sets are created as a product of thinking. Yet, somehow he does not seem to understand this means they are just as unreal as Romeo or a modern-day Casanova. Just as we can use the conventions and shortcuts of our understanding of Shakespeare's play or a historical personage to create mental constructs of what it means to be Romeo of a moder-day Casanova, we use conventions and shortcuts to understand what a set of size six means, and assign contents to sets at need.
Your objection does not show that sets cannot be mental constructions; it shows that sets cannot be mental constructions of a finite mind. If there were an infinite, necessarily existent mind, then sets could be the constructs of such a mind. If you maintain that there is no such mind, then you should also maintain that there are no sets. If, however, you hold that there are mathematical sets, and that nothing contradictory can exist, then you should hold that they are the mental constructs of an infinite mind. If you deny that there is an infinite mind, but hold that there are sets, then you owe us an alternative explanation of how a set can be both one and many.
Left unaddressed is the reason we should think sets exist at all.
Read more!
Set theory can be done either naively or axiomatically. In the standard axiomatic approach to the subject, that of Zermelo-Fraenkel, the existence of the null set is posited in a special axiom.
Two errors in two sentences. One of the primary reasons that set theory was put into an axiomatic format is that, when doing set theory naively, you come across both logical and semantic contradictions that are avoided by the axiomatic process. Also, while Zermelo's original approach had a special axiom for the empty, the standard approach to the Zermelo-Fraenkel axiom schema contains no such axiom. Rather, the empty set is the derived result of the axiom schema of replacement, which in the short form says that functions defined on classes will map sets into sets.
Intuitively, the existence of a set depends on the existence of its members. The set consisting of me and my cat cannot exist unless both man and cat exist: if either of us should cease to exist, the set would cease to exist. It exists because we exist, not vice versa. (This is of course not a causal use of 'because.') In the case of the null set, however, there is nothing on which the null set can depend for its existence.
The truth is that sets don't exist, at all. The set of Vallicella and his cat is a mental shortcut that we use, but it has no reality unto itself. When I talk about the red books on my bookshelf, they are a set of books for only as long as any person considers them to be a set of books. There is no ontological status the property of their being a collection. After all, how is the nature of the books changed if I say that I have carefully designed them to be a set, or if I say they are just a bunch of book I have in that place? I can tell Person A the one and Person B the other, they can't compare notes, examine the books, and thereby discern which account is more truthful. So, it seems the nature of being a set has no reality behind it. In particular, why should the empty set have any more reality?
No working mathematician is likely to lose any sleep over this, however. He will tell us that the null set is convenient, computationally useful and ought to be judged by its practical fruits.
Working mathematicians generally seem aware that almost everything they do is based upon assumption that are help arbitrarily, because we have found them useful. The very grammar of mathematics, predicate logic, is based upon the highly arbitrary notion notion that every statement is true or false. The arbitrariness of this decision is shown by the success of the intuitionists and the various alternate logical evaluation schemata in creating perfectly functional mathematical systems. They know that sets exists only to the degree that we think of the collection of various objects as a set, and so one more useful, non-real entity is not going to cause consternation.
From the first post:
So on the one hand, the null set is useful and well-motivated from within the circle of set-theoretical ideas, but on the other hand, it appears philosophically to be a creature of darkness. Is there a way to get rid of the darkness?
By my lights, the philosopher aims at a degree and a type of clarity that the mathematician qua mathematician -- not to mention other nonphilosophers -- does not care about. Of course, I am not saying that he should care about it. He is within his rights in simply dismissing concerns like the one raised in this post as irrelevant to his concerns or unimportant given his goals and priorities. What is intolerable, though, is the mathematician who gives a lousy philosophical answer to a philosophical question, especially if he is only half-aware that it is a philosophical question.
Of course, philosophy itself is also a formal system. When the mathematician chooses to ignore the needs and desires of one formal system in favor of another, they do so 2with the understanding that all formal systems are arbitrarily chosen based upon their perceived usefulness.
He is then like the neuroscientist who, refusing to stick to his subject-matter, says silly things about mind and consciousness, all the while oblivious to the philosophical problems to which he gives silly answers.
Naturally, we see one sort of woo dissolving into another.
From the second post:
A set in the mathematical (as opposed to commonsense) sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-over-many: it is one single item despite its having six distinct members each of which is distinct from the set, and all of which, taken collectively, are distinct from the set.
Or, since the very notion of set is in and of itself a mental construct, and not a reality, it is certainly different from the shoes themselves, which presumably are real enough that Vallicella puts them on his feet once in a while.
Vallicella goes on for quite a while being very clear that sets are created as a product of thinking. Yet, somehow he does not seem to understand this means they are just as unreal as Romeo or a modern-day Casanova. Just as we can use the conventions and shortcuts of our understanding of Shakespeare's play or a historical personage to create mental constructs of what it means to be Romeo of a moder-day Casanova, we use conventions and shortcuts to understand what a set of size six means, and assign contents to sets at need.
Your objection does not show that sets cannot be mental constructions; it shows that sets cannot be mental constructions of a finite mind. If there were an infinite, necessarily existent mind, then sets could be the constructs of such a mind. If you maintain that there is no such mind, then you should also maintain that there are no sets. If, however, you hold that there are mathematical sets, and that nothing contradictory can exist, then you should hold that they are the mental constructs of an infinite mind. If you deny that there is an infinite mind, but hold that there are sets, then you owe us an alternative explanation of how a set can be both one and many.
Left unaddressed is the reason we should think sets exist at all.
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Tuesday, January 29, 2008
Probability, and a stupid error by me
In case anyone is reading this who doesn't know the background, I was engaged in a lengthy discussion concerning statistics and the types of inferences that were or were not valid from them. During this discussion, another person proposed a simple probability problem, and I made an error in evaluating the answer. I think we disputed on this in well over a hundred posts before I realized the mistake I made. It wasn’t the first time, and I have been on the correct side of such exchanges more often the wrong side, historically. That’s about all I expect.
Recently when referring back to this discussion, the other poster said I had made a specific kind of error. Since I had made a completely different error, I thought I should be correct about the type of oversight I had made. Rather than possibly do another hundred posts on probability in a message board actually devoted to the Utah Jazz (JazzFanz), I decided to bring that discussion over here.
The problem: your partner flips two coins in secret, and you choose one at random and it is heads. What is the probability the other one is heads?
My incorrect reasoning: there were four possible outcomes from the flip: TT, HT, TH, and HH. All four outcomes were equally likely. However, you know that TT did not occur. Therefore, among the three remaining possible flips, two of them would have tails, and one would have heads, so it is twice as likely the other coins is tails as heads. The probability is 1/3. Yes, it was stupid of me.
His (correct) reasoning: we have information only about one coin, and no information about the other. The probability is 1/2.
However, he seems to believe the problem was with my methodology, rather than the implementation of that methodology, so I want to be clear that the correct methodology is this: you have to adjust by the probability that heads would be shown for each type of flip. I did that for TT, by saying it was not possible, but I did not do so for the HT or TH groups. In each case, the odds of revealing heads at random (H) is 1/2, so you have to adjust for that in the calculation of the probabilities. Basically, you are twice as likely to get a head from a HH combination as from HT or TH. In terms of odds, it's [1/2 * P(HT) + 1/2 * P(TH)] : 1 * P(HH), or 1:1, making the probability 1/2.
If his method is easier, why did I use mine? Habit, I suppose. Looking at all the possibilities lets you answer a deeper range of questions than a mere comparison of knowledge method. For example, you can use it to evaluate the following question.
You get an offer to play a game. The rules are you flip two coins. You must reveal one coin, and it must be head if you have one. If the other player guesses the value of the second coin, he gets $1. For the game to be fair, how much should you get when he is wrong?
Edited to remove vertical bars and for clarification.
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Recently when referring back to this discussion, the other poster said I had made a specific kind of error. Since I had made a completely different error, I thought I should be correct about the type of oversight I had made. Rather than possibly do another hundred posts on probability in a message board actually devoted to the Utah Jazz (JazzFanz), I decided to bring that discussion over here.
The problem: your partner flips two coins in secret, and you choose one at random and it is heads. What is the probability the other one is heads?
My incorrect reasoning: there were four possible outcomes from the flip: TT, HT, TH, and HH. All four outcomes were equally likely. However, you know that TT did not occur. Therefore, among the three remaining possible flips, two of them would have tails, and one would have heads, so it is twice as likely the other coins is tails as heads. The probability is 1/3. Yes, it was stupid of me.
His (correct) reasoning: we have information only about one coin, and no information about the other. The probability is 1/2.
However, he seems to believe the problem was with my methodology, rather than the implementation of that methodology, so I want to be clear that the correct methodology is this: you have to adjust by the probability that heads would be shown for each type of flip. I did that for TT, by saying it was not possible, but I did not do so for the HT or TH groups. In each case, the odds of revealing heads at random (H) is 1/2, so you have to adjust for that in the calculation of the probabilities. Basically, you are twice as likely to get a head from a HH combination as from HT or TH. In terms of odds, it's [1/2 * P(HT) + 1/2 * P(TH)] : 1 * P(HH), or 1:1, making the probability 1/2.
If his method is easier, why did I use mine? Habit, I suppose. Looking at all the possibilities lets you answer a deeper range of questions than a mere comparison of knowledge method. For example, you can use it to evaluate the following question.
You get an offer to play a game. The rules are you flip two coins. You must reveal one coin, and it must be head if you have one. If the other player guesses the value of the second coin, he gets $1. For the game to be fair, how much should you get when he is wrong?
Edited to remove vertical bars and for clarification.
Read more!
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