John Link
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Posts posted by John Link
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The rational concept is a total prohibition of the use of chemical weapons.
What's so rational about a ban on the use of chemical weapons? Does it really matter whether one is killed by a chemical weapon or a bullet? I find the entire idea of banning any sort of weapons to be completely absurd. "It's ok to wage ware with each other but only with certain sorts of weapons."
Here's a link to an article by Gavin de Becker: http://www.huffingtonpost.com/gavin-de-becker/syria-fooling-ourselves-into-war_b_3874266.html "The act of identifying one type of lethal weapon as being unacceptable carries with it the implicit endorsement of the other lethal weapons as acceptable."
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I would argue that we're better off toppling Assad first.
Toppling Assad is not a goal I heard expressed by anyone in front of the Senate Foreign Relations Committee. Instead the goal is to degrade (not eliminate) Assad's ability to deploy chemical weapons.
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Here's a link to a video of a great interview with Steve Jobs: http://movies.netflix.com/WiMovie/Steve_Jobs_The_Lost_Interview/70243590?trkid=13641907
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Last night on C-SPAN I watched the Senate foreign relations committee question John Kerry, Chuck Hegel, and Martin Dempsey about possible military action in Syria. Here's what I took away from the session:
Oops! That should be "Hagel", not "Hegel".
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Last night on C-SPAN I watched the Senate foreign relations committee question John Kerry, Chuck Hegel, and Martin Dempsey about possible military action in Syria. Here's what I took away from the session:
1) Obama has already decided that the U.S. will attack Syria whether or not Congress votes to support such action.
2) Obama and Kerry either believe, or want the citizens of the U.S. to believe, that the President has the Constitutional power to go to war.
3) Those who favor attacking Syria either believe, or want the citizens of the U.S. to believe, that attacking Syria is not going to war.
4) Those who favor attacking Syria either believe, or want the citizens of the U.S. to believe, that it is possible to put a limit on how long the military engagement with Syria will last.
Haven't we learned ANYTHING from our experience that includes both Viet Nam and Iraq? -
Two and one half minutes into the two-and-one-half-hour review it's already off topic.
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So the situation is only hypothetical?
My response to the original post is as follows: What would you do if you were the biological father instead of the step-father?People who get married can get fully divorced only if they never have children.
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Here's a link to a page where tickets are available:
http://www.AnthemthePlay.com/ -
aleph_1, I will respond to your most recent post later today but in the meantime would you please answer the question I've now asked several times? It was a compound question which I will now simplify by removing the second question.
Do you agree that exhibiting a construction of your function would not prove that Objectivism is consistent with the axiom of choice?A simple "yes" or "no" will suffice.
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John,
Explicit Construction of my function would be significant because it would show how to bypass AoC, I assert. I know three existence proofs of my function--my own and by two others. Each uses AoC. I don't believe it is possible to bypass AoC in the construction of a function so pathological. That pathology is, in my view, the nature of good applications of AoC.
I think it would be quite significant to show how to construct the function in question. It is certainly a pathological function!
My position on the Natural Numbers is, I believe, close to that of GrandMinnow. It is a simple matter to construct individual such numbers. However, the assumption of the existence of a "set" of all of them (minimal inductive set) is objectionable to me. Once you go down that path the splitting of infinities is inevitable. Perhaps, as alluded to by GrandMinnow, the problem is the application of the power set operation to infinite sets that is objectionable. That leads to sets that have actually distinct infinite cardinalities.
I understand all you say but do not share your objection to infinite sets or to the application of the power set operation to infinite sets. I consider them both obviously valid.
"Math induction" is a standard abbreviation in the profession for "mathematical induction". So is just "induction". You will see published articles that say, "by induction" meaning mathematical induction. I see no harm in it. By the way, it is such a standard method of proof that such proofs are seldom published. The mere assertion that the proof is by induction is sufficient. Any mathematician worth his salt should then be capable of constructing the proof on his/her own.
I haven't taken a mathematics course since 1981 when I finished my classwork for my Ph.D. in economics from Northwestern University. That might explain why I've never encountered the abbreviation. For whatever it's worth a google search for "math induction" yields about 17,000 results, while a google search for "mathematical induction" yields about 399,000 results.
I've asked a question that you haven't answered (or I missed the answer) so I will ask it again: Do you agree that exhibiting a construction of your function would not prove that Objectivism is consistent with the axiom of choice, and that such a construction would only show that the axiom of choice is not needed to prove the existence of your function?
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I still think that math induction requires the Infinity Axiom. This axiom asserts the existence of a collection of objects that is not finite. It is an actual set that is actually infinite. This set, the Natural Numbers, is required to have math induction.
Concerning my function, you are right. However, you will never be able to demonstrate construction of such a function.
I don't understand what you are saying I'm right about. Do you mean to say that you agree with my statement that exhibiting a construction of your function would not prove that Objectivism is consistent with the axiom of choice, and that such a construction would only show that the axiom of choice is not needed to prove the existence of your function?
I don't expect to be able to construct your function.
Do you reject the existence of the natural numbers (i.e., the validity of the concept of the natural numbers)?
Why do you use the phrase "math induction"? I've never heard anyone say that, only "mathematical induction".
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You mean, you actually believe that the code sequence "C-H-A-I-R" is a pattern that resembles, mimetically represents, or looks like the actual physical object that it means, in the same sense that your mold formed a pattern that resembled and mimetically represented the Virgin Mary?
Pitiful.
It's no wonder you don't grasp modern biochemistry. You can't even grasp the idea of the word "CHAIR."
Shall we take this as an example of an effective way to carry out intellectual discourse?
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Math induction is a consequence of the Well Ordering Principle (WOP). WOP is equivalent to the Axiom of Choice. AoC is not really consistent with Objectivist thought, in my opinion, since it pertains to infinite collections of sets, and specifies the existence of a choice function without specifying how to find or construct it. The choice function exists because we want it to, not because reality has constrained us to assert its existence. it exists even if we do not know what it is and it may even be impossible to find it.
I used the AoC to prove the existence of a function that maps every nontrivial subinterval of [0,1] onto all of [0,1], I do not know how to construct this function, only that it exists. If anyone here thinks Oism is consistent with AoC, you can prove this by exhibiting a construction of my function.
Do I understand correctly that you think Objectivism would not accept the existence of a function (or any other mathematical entity) unless one could construct it? If so, why do you think that?
I do not see how exhibiting a construction of your function would prove that Objectivism is consistent with the axiom of choice. Such a construction would only show that the axiom of choice is not needed to prove the existence of your function.
I understand from your bio on this website that you have a Ph.D. in mathematics and that you are a professor of mathematics. Is that correct?
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It's been a few years since I've had to explain a mathematical proof. It's been longer since I've done a mathematical proof.
I've regained an interest in mathematics after some recent discussions with some friends of mine, and it occurred to me that many abstractions are introduced to students without proper foundations in previous knowledge.
It's fine and well to study mathematical induction after studying discrete math, but I had to have a sense of mathematical induction when I was learning series and then again when I was dealing with proofs of some methods in linear algebra. Had people talked about discrete math first, it would have been too much of a learning curve.
I take no issue with *understanding* something less abstract by means of something more abstract. But I want to clarify how what is more abstract depends on previous knowledge of what is less abstract.
You seem to suggest that there is nothing to do besides the step by step learning between what I already know and discrete math.
I don't think I suggested that, and I'm not even sure of what it means. I suggested that you now study one or more proofs that employ that method, but you still might question the validity of the proof. So let's get right to the heart of the matter:
Imagine that there is a ladder with an infinite number of rungs. In order to show that you can step on every rung of the ladder it is sufficient to prove the following:
1) That you can step on the first rung of the ladder
2) That if you are standing on a rung of the ladder then you can step up to the next rung of the ladder
That's all there is to mathematical induction!
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The wikipedia article on mathematical induction includes a proof of mathematical induction:
http://en.wikipedia.org/wiki/Mathematical_induction#Proof_of_mathematical_induction
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I am looking for how to to validate abstract concepts of method.
I am particularly interested in how to validate the concepts of method by which we validate other concepts.
Consider: it would be improper to validate deductive proof by means of a deductive proof. You must resort to other kinds of validation.
My questions were aimed at integrating an idea with its referents by attending the levels of abstraction in the reverse order of what was needed to reach the knowledge. My purpose was to clarify the abstract idea beyond what a mere encyclopedia entry would describe. I'm happy to mine encyclopedias to obtain a definition, but I don't normally do mathematical proofs as part of my day. Without that context, I would end up sketching loose associations among ideas only to find out that's I'm missing something very important for knowing that mathematical induction "works" and must work.
> question 7
I did not literally mean measurements. Higher levels of abstraction omit details as if they were measurements. Concepts of method pertain to the products of consciousness and are formed by retaining the distinguishing characteristics of the purposive course of action and of its goal, while omitting the particular measurements of both.
If you don't do mathematical proofs as part of your day, then why are you exploring the questions you posed? Have you ever created a proof with mathematical induction? Have you ever fully understood of proof by mathematical induction well enough to explain the proof to someone else? If the answer to either of those questions is "no" (and I now suspect it may be), then I suggest you do whatever you need to do to change the answer to "yes". Otherwise you are wasting your time, "sketching loose associations among ideas only to find out that's [you're] missing something very important".
For anyone not familiar with mathematical induction, an excellent place to start would be with the following theorem:
For any integer n greater than or equal to 0:
0 + 1 + 2 + 3 + . . . + (n-1) + n = (n*(n+1))/2
I.e., the sum of the integers from 0 to n is equal to (n*(n+1))/2.
You can find a proof here, but it would be interesting to see whether you can create a proof yourself.
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John,
Thank you for sharing your achievement.
That's quite an overstatement. I'd say that I simply referred to my achievement. Sharing it would require posting the proof of the theorem, which to understand would require reading and understanding most (maybe all!) of my dissertation up to the point at which the theorem is proved.Are some results provable only by mathematical induction? Is there some sort of proof or disproof of that conjecture. Even if there is no such proof, I’d be interested to know simply if there are results that have only been proven using mathematical induction so far as you know.
I'm not aware of any proof that some results can be proved only by mathematical induction. A disproof of the conjecture that there are some results able to be proved only by mathematical induction would need to show that every provable theorem can be proved without mathematical induction. I do not know if there are any theorems that have been proved only with mathematical induction, but I would guess that there are.
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Mathematical induction is an effective proof strategy in mathematics, one that I used to prove one of the theorems in my Ph.D. dissertation, "The Existence of the Maximum-Likelihood Estimate in Log-Linear Probability Models".
I believe that questions 1 through 5 above are all answered by this wikipedia article: http://en.wikipedia.org/wiki/Mathematical_induction
To question 6 I would answer "none" and "no". However, it was a dream that led me to realize that mathematical induction would let me prove a theorem I was attempting to prove, so the subconscious can be involved in mathematical proofs (of any sort, not just those using mathematical induction) even if, strictly speaking, it is not logically necessary for it to be involved.
I have no idea why question 7 is included in the list above. Neither do I know to what it refers. Therefore I have no answer to offer.
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I make my living as a teacher of the Feldenkrais Method and consider its inclusion in the table totally inappropriate. I also note that it is not included in the latest version of the table: http://crispian-jago.blogspot.com/2013/03/the-periodic-table-of-irrational.html
Also, the description of Rolfing seems more appropriate to the Alexander Technique, given that the former was developed by Ida Rolf (a woman, not a chap) and the latter by F.M. Alexander (an Australian man, and therefore a chap). Neither Rolfing nor the Alexander Technique ought to be included in the table.
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It seems it can be moral to use if it's for a productive purpose, and so forth, it's contextual. Even smoking I think Peikoff said, if it's something that can get you through. I'm not sure what podcast I heard that one in.
Given what's known about the dangers of smoking any defense of the morality of smoking would necessarily include a huge heap of dishonesty. I certainly hope that nobody would take such a defense seriously just because Leonard Peikoff made it.
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The assertion that "it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrats, or generally any power except a square into two powers with the same exponent." would be a theorem once it has been proven.
Oops! The statement quoted above has been proven and therefor is a theorem.
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As previously mentioned, I've never dealt with math proofs with the exception of coming up with formulas that do what they are intended to do. Fermat's margin stated
"On the other hand, it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrats, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvelous proof of this, which, however, the margin is not large enough to contain."
Is this considered a theorem?
The assertion that "it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrats, or generally any power except a square into two powers with the same exponent." would be a theorem once it has been proven.
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I'm still waiting for a statement of a theorem, including all the necessary definitions and examples, and then a proof.
Syria Intervention
in Current Events
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If the ban on chemical weapons is rational, then let's have a ban on all weapons, since such a ban would put an end to war.