Jump to content
Objectivism Online Forum

Reduction of Mathematical Induction

Rate this topic


Vik

Recommended Posts

1. To what does the concept of mathematical induction refer to in reality?

2. What facts give rise to the need for such a method?

3. What did we need to discover before we could form the concept?

4. What systematic course of action is designated by the concept of mathematical induction?

5. What goals does this method help achieve?

6. What is the role of the subconscious?  Are there any psychological actions necessary for successfully doing a mathematical induction?

7. What particular measurements are omitted?  They are measurements of the purposive course of action and its goal.  But what are they?

 

Link to comment
Share on other sites

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.

Edited by John Link
Link to comment
Share on other sites

Vic,

 

I would say that concepts logically presupposed by measurement methods are not analyzable in terms of the omission of measurements. How far concepts of measurement, including geometry, require a form of mathematical induction would be good to look into. I know that in my practical measurements, I’m not presuming the principle of mathematical induction. The point of contact with mathematical induction would be in measurement theory, with conceptual analysis of measurement, such as in Foundations of Measurement.*

 

To pull away the logical and mathematical presuppositions of measurement theory leaves a great deal of substance to Rand’s proposal. That all other concepts are in principle analyzable in terms of measurement-omission, even if only ordinal measurement (or ordered geometry) in some cases, is a very substantial claim about the nature of those remaining universal concepts.

 

John,

 

Thank you for sharing your achievement. 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.

 

–Stephen

Link to comment
Share on other sites

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.

Link to comment
Share on other sites

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.

Edited by John Link
Link to comment
Share on other sites

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.

Link to comment
Share on other sites

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.

Link to comment
Share on other sites

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!

Edited by John Link
Link to comment
Share on other sites

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.

 

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.

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.

Link to comment
Share on other sites

As an addendum to my previous post, the axiom of choice is rendered moot if one other axiom is rejected. You need never consider AoC if you reject the Infinity Axiom. Only AoC is necessary to have math induction. One would have to take a non-standard approach to constructing the Natural numbers without the Infinity Axiom. Without the traditional Natural numbers, there is no need for math induction.

My point stands: If you want to hold on to the notion of number as constructed by nineteenth century mathematicians and at the same time claim to be an Oist, you must demonstrate a construction of my connectivity preserving nowhere continuous function.

Link to comment
Share on other sites

"Math induction is a consequence of the Well Ordering Principle (WOP). WOP is equivalent to the axiom of choice." - aleph_1

 

It is correct that in, for example, Z set theory (perforce, for example, ZF set theory), the well ordering principle is equivalent to the axiom of choice.

 

And usually mathematical induction pertains to sets well ordered by a successor relation. But the well ordering principle is that EVERY set has a well ordering; yet there are many sets that are well ordered by a successor relation without our having to adopt the well ordering principle itself. In particular, the form of mathematical induction most discussed in this thread (weak induction on the natural numbers) does not require adoption of the well ordering principle.

 

And it is correct that all three - weak induction, strong induction, and well ordering - are equivalent: If one of those holds for a set and relation then the other two hold also. But, again, for many sets, their well ordering is provable without having to adopt the well ordering principle (which, again, is that EVERY set has a well ordering).

 

"You need never consider AoC if you reject the Infinity Axiom." - aleph_1

 

That depends on what you mean by 'reject'. If by 'reject' you mean merely to drop the axiom, then, no, the axiom of choice still has clout, since in this situation it is undetermined whether or not there exist infinite sets. But if by 'reject' you mean dropping the axiom AND adopting its negation, then, yes, the axiom of choice then is superfluous, since in this situation every set is finite and we know that every finite set has a choice function (which, by the way, we prove by mathematical induction on finite sets). Note, though, that this pertains to ZF (where the axiom of infinity is equivalent to the statement that there exists an infinite set (when I say 'infinite' I mean "Tarski infinite" (i.e. not finite) as opposed to "Dedekind infinite" (i.e. 1-1 with a proper subset)) but not Z (where the axiom of infinity implies there exists an infinite set, but the statement that there exists an infinite set does not imply the axiom of infinity); so to make the axiom choice superfluous to Z, we would have to drop the axiom of infinity and adopt an axiom that states that there exists an infinite set.

 

"One would have to take a non-standard approach to constructing the Natural numbers without the Infinity Axiom." - alpeh_1

 

We don't need the axiom of infinity to construct each natural number. The axiom of infinity though is needed to have a set that has all the natural numbers as members. And the axiom of infinity is not needed to prove the principle of mathematical induction for natural numbers as long as the principle is stated schematically for the property of being a natural number.

Edited by GrandMinnow
Link to comment
Share on other sites

"The choice function exists because we want it to, not because reality has constrained us to assert its existence." - aleph_1

 

In what sense has reality constrained us to assert the existence of the image under a set of a function class (replacement), a set closed under successor (infinity), pairs, unions, or power sets?

Edited by GrandMinnow
Link to comment
Share on other sites

"The choice function exists because we want it to, not because reality has constrained us to assert its existence." - aleph_1

In what sense has reality constrained us to assert the existence of the image under a set of a function class (replacement), a set closed under successor (infinity), pairs, unions, or power sets?

Ah, GrandMinnow! It is good to see that I have stirred you from the briny realms mon ami.

Let us make a little distinction that is not always made. Call the Well Ordering Theorem the axiom that Every set has a well ordering and call the Well Ordering Principle the idea that every subset of the natural numbers has a least element. WOT is equivalent to the AoC while WOP is what you have claimed may be proved for the Natural Numbers.

Concerning the Infinity Axiom, my position is that it is not possible to apply the principle of reduction to such an idea and hence it cannot have the same meaning as, say, the notion of two-ness. Instead, it has the standing of the notion "dragon". What is more, this position makes the application of AoC to only reducible concepts trivial and hence it loses some standing.

Now, you might have noticed that my moniker, Aleph_1, is a stick in your eye, but putting that aside, it is not my intention to justify nineteenth century constructions but rather to explore the philosophical validity of these ideas. It seems to me that AoC and the Infinity Axiom are the source of a great deal of confusion and meaninglessness. An alternative construction of numbers would be helpful. Fortunately, there are some such that are out there but not popular. We shall ever be struggling to overcome the burden of Cantor, Dedekind, ZF, etc. So be it but let us understand our own ideas!

Reduction has consequences for our understanding of number. The construction of number in the nineteenth century was based on nineteenth century philosophies. Let us recognize that and develop our own understanding of number.

I have a question for you. Is it consistent with your philosophy that math, as currently founded, can demonstrate the existence of that for which it cannot even provide a model for? I can understand the Intermediate Value Theorem since models of curves taking on intermediate values may be constructed, so that is not what I'm talking about. Do you find it consistent with your philosophy that there exists f:[0,1]->[0,1] such that for each subinterval (a,b ) of the domain, f((a,b ))=[0,1]? AoC implies it's existence but I have no idea how to exhibit such a function. I cannot even make a model for it. Nor can I make a model for how you could go about constructing it. I can prove that it exists and appreciate that it is a beautiful counterexample to the converse of the theorem that continuous function are connectivity preserving. Being nowhere continuous it shows that connectivity preserving functions need not be continuous at any point. But it is also absurd. We are playing with dragons.

Respect for authority of nineteenth century mathematicians must be overcome. Reduction has consequences. I think it is constructive to pursue a math that is consistent with reduction.

You should be aware that some concepts are conventional, meaning they involve arbitrary choices. That does not mean that those choices are not necessary. The color green is conventional. That is a consequence of assigning a discrete number of color names to a continuum of colors. Vagueness and arbitrariness are necessary. We have choices in how we construct the notion of number, but just as there are objectively different colors, so there are objectively different numbers. However, there are theories about number that are not objective.

Edited by aleph_1
Link to comment
Share on other sites

To answer your first question, for first order theories, the model existence theorem (aka the completeness theorem) ensures that any consistent theory has a model. So, in that context, my answer to your question is 'no'. For other kinds of theories, I could not answer without specific context.

 

As to the intermediate value theorem, while some proofs may use choice, if I recall correctly, the intermediate value theorem is provable without choice (check me on that; my memory is not perfect and I don't have my notes with me now). 

 

As to the greater issue, of course I do recognize that choice is not constructive and leads even to such conundrums as that there is a well ordering of the reals but even in principle no specific well ordering of the reals can be defined.

 

As to axioms, I don't take them as statements that govern ALL mathematical situations, but rather I see a set of axioms as only governing those models in which the axioms are true. For example, the axioms for first order group theory are true for all groups but are false for many other mathematical situations. So too the axioms of ZFC are true for all models of ZFC but are false for many other mathematical situations. In other words, for me, an axiomatization is a "description" of a certain kind of mathematical context; so the axioms do govern in that context but don't govern in other contexts. 

 

In that regard, it doesn't make sense to ask me whether mathematical theorems are consistent with my philosophy. Mathematical theorems are not things I would even compare for consistency with my philosophical inclinations. For me, mathematics (putting aside for the moment consideration of applied mathematics, etc.) regards (1) the formal deductions themselves and (2) purely abstract structures. Neither of those are mediated by my philosophical inclinations.

 

As to construction of the natural numbers, again, neither the axiom of infinity nor choice need be involved. In set theory we may define the predicate 'natural number' and construct any particular natural number we wish without ever invoking infinity or choice. Indeed, using the von Neumann method, and even in greater generality, we can do the job with extensionality and three existence principles: that there exists an object, that for any object there is the set whose only member is that object, and that for any two sets there is the union of them. Finitistic and constructive.

Link to comment
Share on other sites

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?

Edited by John Link
Link to comment
Share on other sites

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?

John,

I believe that reduction is important to establishing the validity of concepts. Without it you end up able to define concepts that correspond to nothing in existence and hence, by Rand's Razor, should be rejected as irrelevant. Dragon" is such a concept. A dragon is a flying reptile that breathes fire. It is definable but refers to nothing in existence. God is even worse since that concept cannot even be defined. "Social justice" is also a concept that cannot be defined. These concepts are not worthy of a rational being, just as special pants that help you lose weight or video games that make you smarter are contemptible.

I believe that the notion of number arises from observable properties of physical objects. However, there are aspects of mathematical theory that were introduced through convenience in constructing a suitable theory of limits. These axioms do not correspond to anything observable and hence have the same standing as that of "dragon". That's better than "god" but worse than "two". Reduction is an essential element of Objectivism and should not be disregarded when constructing a valid system of concepts.

The success of mathematical theory at modeling everything we observe may be taken as a justification of it's founding principles. Who cares whether or not they are "true" as long as they are useful? That position is, in my view, intellectual laziness.

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.

My Ph.D. Is in math, awarded in 1995 from UCDavis. Please do not blame them for my shortcomings mathematically. My errors are my own. I am a math professor in Southeast Texas and am fortunate to have such a position. My area is not set theory, but that is where the issues lie in this thread. Perhaps geometric constructions of numbers would have immediate reducibility and hence spare us these discussions concerning the number of angels that can dance on the head of a pin.

Link to comment
Share on other sites

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".

Link to comment
Share on other sites

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".

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.

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.

"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.

Link to comment
Share on other sites

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?

Edited by John Link
Link to comment
Share on other sites

John,

My specific assertion is that there are mathematical axioms, like AoC and the Infinity Axiom, that allow the existence of objects that are not reducible to perception and hence these axioms are not consistent with Oist philosophy. I believe that my function is not reducible to perception or even amenable to construction. Demonstration of a construction of my function would demonstrate that I am wrong about the non-reducibility associated with it, or/and it might show that this function does not depend on AoC as you suppose. It is my firm belief that this function depends in an essential way on AoC and I would be interested in a demonstration otherwise.

Concerning your assertion to the "obvious" validity of infinite sets, my reply is that you have been well indoctrinated into modern mythologies. There is no reducibility to these notions and so ockham's razor implies they and what they are founded upon should be expunged from an efficient theory. I thought this was an essential element of Oism: Reduction to perception for purposes of validation. Concepts for which this is not possible are treated contemptibly. We should not contaminate our system of concepts with notions that are not reducible.

Link to comment
Share on other sites

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.

Link to comment
Share on other sites

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.

John,

Demonstration of a construction of my function would show that AoC is not inconsistent with Oism. It would not show that Oism is consistent with AoC. AoC is a short-cut around reduction. It allows for assertions of existence with no consideration for reduction. In this sense it is antithetical to Oism. A construction of my function would show that we may retain hope that the consequences of AoC may still be amenable to later construction. My purpose in presenting the challenge of an explicit construction of my function was to vividly illustrate via an interesting example the existence without reduction property of AoC. That AoC is inconsistent with Oism is apparent, in my view.

I hope that you were not offended by my comment that you are well indoctrinated into modern mythologies. No offense was intended. I was just attempting to alert you to the idea that much of what we deem obviously valid may depend on questionable premises. As an economist, I am sure you are sensitive to this.

By the way, an old graduate school friend of mine studied ag econ at UCD. I pictured cows and walnuts but he would jerk my chain by asking about rank, genus and order of complex analytic functions--notions I did not learn until graduate complex analysis. I am sure of your competence at math. However, there are important questions that underlie it's development.

My short answer is, No. My long answer is that AoC is an end-run around reduction and is hence antithetical to Oism whether you construct my function or not. Construction of my function would convince me that AoC is not inconsistent with Oism in the sense that there would be end-runs around AoC.

Edited by aleph_1
Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Reply to this topic...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
  • Recently Browsing   0 members

    • No registered users viewing this page.
×
×
  • Create New...