Jump to content
Objectivism Online Forum
Sign in to follow this  
IdentityCrisis

Mathematics: Reality And Infinity

Rate this topic

Recommended Posts

[...]You are using the concept of "duration" in a context where "duration" does not apply.[...]

Thanks a lot! Sorry, I didn't realize the issue has been discussed before and should have checked. I appreciate the article and your explanation, as they cleared up my misunderstandings.

Share this post


Link to post
Share on other sites
I appreciate the article and your explanation, as they cleared up my misunderstandings.

I'm glad for that. Alex is a young and very bright philosophy student and he has a marvelous way of formulating and communicating ideas. His essay also cleared up a few outstanding issues for me that I had on this fascinating subject.

Share this post


Link to post
Share on other sites
Alex is a young and very bright philosophy student and he has a marvelous way of formulating and communicating ideas.

I've noticed that as well--I'm in the same class as him at the OAC.

Share this post


Link to post
Share on other sites

I have issues with the essay Alex has written:

First there are the theretical disagreements about the nature of time and space in the universe, which naturally stem from controversial arguments such as his, and where truth just might actually lie on his side (I'll give it more thought, we'll see).

Second, from a practical point of view, I just don't see how his essay can be of any use to me in my life. I am able to derive no positive benefit from the complicated chains of reasoning, other than learning that my concepts are inadequate for describing the true nature of existence. (Sounds rather ominous when I put it that way, doesn't it?)

And third, is it just me or does he really say the following in his references:

this rebuts Dr. Binswanger's metaphysical argument against the mathematical concept of infinity. Whether his psycho-epistemological argument still holds or not is unknown to the author.

The way he writes, 'this rebuts Dr. Binswanger's arguments' is just a bit too casually callous for my liking. I don't know Alex at all, and I liked the organization of his paper, but I just found much in it that was disagreeable; some things were also disagreeable to me about his style of writing ("I am an expert in this field"), and casual comments about errors in arguments of an Objectivist who also happens to hold a doctorate in philosophy.

Share this post


Link to post
Share on other sites
I have issues with the essay Alex has written:

First there are the theretical disagreements about the nature of time and space in the universe, which naturally stem from controversial arguments such as his, and where truth just actually might lie on his side (I'll give it more thought, we'll see).

Okay, but since you have not presented any content for disagreement, I do not know why this is an "issue."

Second, from a practical point of view, I just don't see how his essay can be of any use to me in my life. I am able to derive no positive benefit from the convoluted chains of reasoning, other than learning that my concepts are inadequate for describing the true nature of existence. (Sounds rather ominous when I put it that way, doesn't it?)
Well, yes, it does sound rather ominous. But, more importantly, if you see that as the conclusion which he presents, then I would respectfully suggest re-reading the essay. Frankly, I had to read it several times myself before I was able to shake off some of my prior thinking on this subject, and see the issue more clearly from the perspective Alex presents.

And third, is it just me or does he really say the following in his references:

The way he writes, 'this rebuts Dr. Binswanger's arguments' is just a bit too casually callous for my liking. I don't know Alex at all, and I liked the organization of his paper, but I just found much in it that was disagreeable; some things were also disagreeable to me about his style of writing ("I am an expert in this field"), and casual comments about errors in arguments of an Objectivist who also happens to hold a doctorate in philosophy.

First, I find no justification for your judgment of "casually callous." Alex presented the facts, and his argument is, in fact, a rebuttal of the point that HB makes. You are free to argue against it, but it is a rebuttal nevertheless. And, in fact, Alex holds HB in the highest regard, and I think it is your interpretation that is askew, not Alex's comment.

Second, you indicate (yet again) that you disagree with Alex's arguments, but you have failed to offer any arguments or explanation as to why. What is the point of making such a declaration, if you do not support it with fact and argumentation?

Share this post


Link to post
Share on other sites
Okay, but since you have not presented any content for disagreement, I do not know why this is an "issue."
Probably like you at first, I've had much to disagree with in the paper's arguments, as I was reading it for the first time; it is all of these initial reactions of disagreement about the paper's content that I subsume under this category. It is not an issue of active disagreement yet, because I have a suspicion he may be right on many of claims he makes, despite my initial reactions; thus I leave them as an 'open question' for myself to think about later on. The reason it's included here at all is for the sake of completeness, so that it will not seem as if my sole reasons for not liking the paper are described in issues 2 and 3, and that I otherwise am in full agreement with what the author said.

Well, yes, it does sound rather ominous. But, more importantly, if you see that as the conclusion which he presents, then I would respectfully suggest re-reading the essay.
I will undoubtfully follow your advice and read the paper more than once more. But as a slight correction, it's not that the official conclusion of the paper is that our concepts are inadequate for understanding existence. It is more like: "concept A is inappropriate", "using concept B is fallacy of stolen concept", "concept C does not apply", so that at the end, what am I left with? When I try to add it all together and figure out what I can take away from all this, I don't see any new solutions or better explanations proposed, but all of my existing concepts (which I've used heretofore to understand this issue) are disqualified and dismissed.

Alex presented the facts, and his argument is, in fact, a rebuttal of the point that HB makes.
If it was you who said it, then it'd be ok. Do you see the distinction I'm making here between someone like you saying you've got an argument that rebuts HB, and someone like Alex (or me, for that matter)? If all students at OAC were to take up the habit of regularly claiming to have successfully rebutted their professors in the latter's areas of expertise, it would end up being rather more like a circus than an august learning institution.

And if it truly happened that Alex has caught HB in a serious error somewhere, he should approach the subject as matter of great exception and rarity, almost with disbelief that such a thing occured; what he shouldn't do is approach it as a matter-of-fact thing ("Oh and by the way, I know I'm just your student, and you are a man twice my age and more than twice my knowledge and experience, but I just thought you should know that you're wrong here, here and here. And regarding your argument about psychological implications - I'll let it stand for now, until I turn my critical eye to that direction as well.") Maybe it's just me, but it appears to be a simple case of manners here, regardless of the validity of the particular argument HB was trying to make in that instance.

So, the way I understand the proper order of things, if Alex feels that he is entitled to make statements to the effect that he rebutted HB in some aspect of philosophy of science, and feels the right to state such discoveries in a matter-of-fact fashion, as an matter of course, and something that shouldn't really surprise anyone (which is how that comment comes off), then he should be a teacher at OAC by HB's side, not a student at OAC under HB's tutelage.

And I don't think that I'm blowing things out of proportion either, because the way the whole paper is worded suggests an author of great sagacity and knowledge of the subject matter, rather than a student of Objectivism just beginning to spread his wings and understanding of things.

Now as I said, I know nothing about Alex other than what I read in the paper. You obviously seem to have a lot more knowledge about him than can be derived only from the paper itself, so I will leave my initial impressions as a matter of record, put them aside for now, take Alex at your word, and give him the benefit of the doubt.

---

Anyway, I think we've gotten a bit astray from the original subject of the thread. My post was aimed to express not only a (tentative) disagreement with the actual arguments of the paper, but also with their manner of presentation. I'll read the paper a few more times and post again whether I finally see the light and understand what Alex proposes as the positive solutions for the problems he describes.

Share this post


Link to post
Share on other sites

I have a question regarding the essay by Alex

Quote from the essay

It recognizes that "Is the universe finite or infinite in time?" is a complex question; it assumes that the concept of time is applicable to existence.  But, it isn't applicable; so, neither is the question.  The universe isn't "in time," so it therefore isn't "finite in time."  ("Finite in time" and "in time," like "finite in size" and "in size," are equivalent statements.)
[Emphasis mine]

The essay says that the concept of time is not applicable to existence.

This means that time is not a property of existence, matter or universe.

Does this mean that time exists only in our perception? That time is a very useful and indispensable tool we use to reference to reality but time actually makes no difference to reality?

Share this post


Link to post
Share on other sites
I will undoubtfully follow your advice and read the paper more than once more. But as a slight correction, it's not that the official conclusion of the paper is that our concepts are inadequate for understanding existence. It is more like: "concept A is inappropriate", "using concept B is fallacy of stolen concept", "concept C does not apply", so that at the end, what am I left with? When I try to add it all together and figure out what I can take away from all this, I don't see any new solutions or better explanations proposed, but all of my existing concepts (which I've used heretofore to understand this issue) are disqualified and dismissed.

In 1962, when Miss Rand started The Objectivist Newsletter, one of the first essays dealt with the issue that since everything in the universe has a cause, what is the cause of the universe? The answer, in essence, was that the fallacy lies in the assumption that the universe as a whole requires a cause. The answer to this metaphysical issue, then, just like the epistemological status of your response to the essay by Alex, "disqualified and dismissed" the existing approach which stood in the way of actually grasping the issue.

What Alex has done in his essay is put some broader metaphysical issues into a proper perspective, such that it reveals the fallacies in our prior approaches which stood in the way of making sense of this fundamental issue. In other words, if the answer to the metaphysical problem lies primarily in using a false methodology towards finding a solution, then the clear identification of that fact is of enormous value.

As to this issue regarding HB and a rebuttal, I truly think you are blowing this way out of proportion. When Alex's essay was presented on HBL, HB found the essay of interest and discussed it in some detail. No offense was was taken, because no offense was meant. Even your characterization of the point is out of proportion; it is not the case that "Alex has caught HB in a serious error." HB's point was valid and consistent with the acceptance of a certain premise; given another premise, the one which Alex argues for in his essay, that then obviates one aspect of the point made by HB. It is not a "serious error," with all the drama associated with that terminology and as is with the rest of what you wrote that accompanies it. It was a rebuttal, plain and simple, and it stands or falls on the validity of Alex's premise. I really think you are creating a problem here which otherwise does not exist.

I'll read the paper a few more times and post again whether I finally see the light and understand what Alex proposes as the positive solutions for the problems he describes.

Fair enough.

Share this post


Link to post
Share on other sites
Does this mean that time exists only in our perception? That time is a very useful and indispensable tool we use to reference to reality but time actually makes no difference to reality?

Time is a relationship, and relationships have an objective existence just as do physical entities. Relationships are different from entities in that they do not have an independent existence -- a relationship depends upon the entities it relates -- but nevertheless they are parts of reality.

Share this post


Link to post
Share on other sites

Regarding Alex's essay, his whole thesis hinges on the premise that the universe as a whole is Euclidean. I would have liked to see more in his essay as to why he accepts this as a premise.

I think that such a premise is a mistake. Euclidean geometry, in my opinion, only applies to parts of the universe, not the universe as a whole. Every frame of reference that we choose to describe with Euclidean geometry can only be described by Euclidean geometry within a limited range. I do not believe it makes sense to think of lines, for example, that extend outwards in a never-ending fashion.

It does not make sense, by my way of thinking, to think of the universe extending outwards in all directions in a never-ending fashion, Alex's intriguing essay notwithstanding.

Share this post


Link to post
Share on other sites
Regarding Alex's essay, his whole thesis hinges on the premise ...

thinkonaut is not qualified to properly identify the premises upon which Alex's essay rests. In the past thinkonaut has consistently misunderstood, misrepresented, and generally distorted the facts regarding Alex's essay in particular, and metaphysics and epistemology in general. See the entirety of this thread for the evidence and judge for yourself.

Share this post


Link to post
Share on other sites
...Maybe it's just me, but it appears to be a simple case of manners here, regardless of the validity of the particular argument HB was trying to make in that instance...

Besides what Stephen Speicher has said in response to this alleged rudeness of mine, I wanted to expose the many unwarranted assumptions that Free Capitalist has made about the context in which I created my essay.

In HB's 1987 lecture "Selected Topics in the Philosophy of Science," he expressed uncertainty about how exactly the universe could be both finite and unbounded; and after trying to explain it by likening the universe to a sphere, he said: "So, I leave it to you to try and do better than that." His saying this was my impetus for starting my thinking on this issue.

Then, when I came up with the fundamental thesis for my essay, I was taken aback at how simple and true it seemed, and how I had never heard anyone give my particular account before. So, I took the burden of proof, and for the next 10 months, I thought of every objection to my theory that I could come up with, and read and listened to everything in the Objectivist corpus I could find to try to find out how my ideas could be defeated. I came up empty, and I only became more confident in my thesis. On one level, it felt odd that I would come up with a solution that HB hadn't thought of; but the idea that I should squash this confidence simply because I don't have my Ph.D., and in defiance of all the evidence and knowledge that I had spent months accumulating, is nothing short of bizarre (and quite self-effacing, might I add).

What Free Capitalist refers to as a lack of "manners" in my essay is nothing other than a manifestation of my earned confidence that my ideas were and are correct.

--Alex

Share this post


Link to post
Share on other sites
thinkonaut is not qualified to properly identify the premises upon which Alex's essay rests. In the past thinkonaut has consistently misunderstood, misrepresented, and generally distorted the facts regarding Alex's essay in particular, and metaphysics and epistemology in general. See the entirety of this thread for the evidence and judge for yourself.

thinkonaut is not qualified to properly identify the premises upon which Alex's essay rests. In the past thinkonaut has consistently misunderstood, misrepresented, and generally distorted the facts regarding Alex's essay in particular, and metaphysics and epistemology in general. See the entirety of this thread for the evidence and judge for yourself.

Thanks for your participation, Stephen, I only wish you (or Alex) would address the objection that I raise. Why should we assume the universe to be Euclidean? I don't think it makes sense to do so.

One of the cardinal features of Objectivism is the recognition that existence is primary to consciousness. Regardless of whether or not someone *perceives* that I am qualified to discuss something, the essential question is: What is the claim? Is it logical? What is the evidence? What counterevidence might exist?

I would rather not spend my time on this forum addressing non-essential questions. Whether anyone judges me to be qualified to do anything should be of secondary interest to members of this forum and should not rise to the level of being important enough to comment on in posts.

Coupling such assertions that I am not qualified with a link to a thread in order to "look for yourself" does not help. You should reference some specific claim that I have made, and then provide the link.

The content of that of which I speak speaks for itself. And that's one of the many pleasures of being an Objectivist.

Since I don't want to address nonessential questions on this forum, I won't try to defend my qualifications on this forum. Anyone interested in facts which show that I am qualified is invited to contact me by private e-mail at thinkonaut------at-----yahoo-----dot-----com.

One of the reasons I quit the Circular Time thread was because I don't appreciate being shadowed by critic(s) who don't address the essential issues. The fact that such extraneous and counterproductive talk is allowed onto this forum (or to remain here) shows a severe flaw in the editorship herein and I just won't be able to participate with such continued shadowing of that nature. I must insist that people address the issues, otherwise I will withdraw my participation. Such shadowing is a form of argument by intimidation, pure and simple (or "intellectual bullying" to use plain talk) and it won't wash in the eyes of the savvy participants of this group.

All that being said, Stephen, I find much of what you write to be very informative and interesting, and you have, in prior posts, made comments which have helped me to correct or augment my views, and I appreciate it. You're a bright guy and much more knowledgeable than me on many topics. I admire you for it.

Merry Christmas.

Share this post


Link to post
Share on other sites
The content of that of which I speak speaks for itself.

Exactly, which is why at least a couple of us gave up trying to communicate with you on that thread several months ago. You make naive use of technical words you do not understand, like your bizarre "hyperspheroid" universe, and you misrepresent and distort the ideas of others and then demand that they answer to your misrepresentations and distortions.

Since I don't want to address nonessential questions on this forum ...
That is exactly how some of us feel, which is why I wish you would refrain from your gratutious remarks. You have been absent since that last thread some four or five months ago, and now you reappear to take a gratuitous swipe at the mention of Alex's essay, demanding a response to a portion of the same nonsense that was addressed before we gave up trying to communicate with you. You simply do not know when to leave well-enough alone. You did the same thing with me privately. I told you that private email from you was unwelcome, that I wanted absolutely nothing to do with you at all, but you persisted and invaded my privacy against my explicit wishes. As if this were not enough, when I continued to ignore you, you then sought out my wife in email as a conduit to me. Your actions are despicable, only matched by the degree of your rationalism and distortions of fact.

I must insist that people address the issues, otherwise I will withdraw my participation.

I for one welcome that day.

Share this post


Link to post
Share on other sites

IdentityCrisis asked "... what is the mathematical definition of infinity (or what are the definitions, if more than one)?".

A set is infinite if it is not finite.

There are several definitions of finite. These are equivalent if the 'Axiom of Choice' is true.

1. A set is finite if it can be placed into a one-to-one correspondence with the set of natural numbers (non-negative integers) less than some specific natural number (the cardinality of the set).

2. A set if finite if it has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time.

3. A set is finite if it can be given a total ordering which is both well ordered forwards and backwards. That is, a set is finite if every non-empty subset has both a least and a greatest element in the subset (these are the same if the subset is a singleton).

4. A set is finite if every function from the set one-to-one into itself is onto.

5. A set is finite if every function from the set onto itself is one-to-one.

Share this post


Link to post
Share on other sites

"You [stephen_speicher] said that you don't consider set theory to be the correct basis for mathematics... I'd like some more detail. Is it just because Russell's Paradox and other contradictions can be derived from it (not that that isn't enough), or is there something more?" [identityCrisis]

Zermelo set theory (or any equivalent axiomatization or any extension proven to be relatively consistent with it or similar theories such as Bernays class theory or even Quine's NF) has not been shown to lead to Russell's paradox.

"[...] set theory is, at best, simply not fundamental enough to provide a complete foundation for mathematics, and, at worst, it is a rationalistic concoction." [stephen_speicher]

If we demand that whether a sequence is a proof be effectively decidable ('effective' per Church's thesis) then, as I surmise you know, Godel proved that there is no conistent and complete axiomatization of arithmetic. So of course set theory is not complete. So what theorem of mathematics is not proven by set theory that is proven by some system that you prefer?

As to fundamental, ZF set theory rests upon four fundamentals: first order predicate logic; identity theory; aside from equality, one primitive 2-place predicate; and six straightforward axioms (even if you view one or more of them as opposing your ontology, they're still straigtforward, and ontologically offensive only if you insist that they have any ontological import at all). That's pretty fundamental, I would say. (Additional axioms such as the axiom of choice and the axiom of regularity are needed for certain other results, especially certain results about infinite sets or to prove certain theorems in other mathematical studies. But the axiom of choice, which is the most controversial axiom, is not needed for set theory to develop arithmetic and seems to be unavoidable for certain mathematics, no matter what the foundation.)

That said, what system do you propose in place of set theory? That is, do you adopt some other logic than first or higher order predicate logic and what are your non-logical axioms?

And set theory is an axiomatization in a formal language, so its use does not carry with it a presumption of philosophical rationalism or any other ism. As to set theory being a concoction, yes, any particular presentation of it is something devised by man.

Edited by LauricAcid

Share this post


Link to post
Share on other sites

stephen_speicher

Rereading your post 4 I see that you say that you don't offer an alternative to set theory, so my question about that was unnecessary. However, you do mention some articles by other writers, so I am curious as to what kind of system might be countenanced in the hints given by those writers.

Also, you mention the definition of a set being infinite if it can be put in a one-to-one correspondence with a part of itself. As you know, that is more precisely put: A set is infinite iff there is a 1-1 function from the set onto a proper subset of itself. But as far as I know, the proper subset definition is shown equivalent to the 'not finite' definition only by assuming the axiom of choice. For this reason, I prefer the 'not finite' definition.

"What facts of reality are contained in the premise that an infinite set can be put into a one-to-one correspondence with another set which contains the original set?"

In a strict sense, 'infinite' in set theory is a 1-place predicate symbol that is introduced with a definition. In this sense, set theory does need to account for any facts of reality just to assign a definition to a symbol. If the ontological connotations of 'infinite' are not acceptable to you, then, personally, I'd say: Okay, forget 'infinite'. Instead, we're taking a symbol, say, 'H', as defined by 'Hx iff x not finite'. Then we'll watch proofs of theorems from the axioms that are made shorter to read by using 'H' but never depend on it since anything we say using 'H' can always be said with just the 2-place predicate symbol 'element_of'. Further, with this theory we can formalize virtually all of mathematics and meta-mathematics without known inconsistency. A pretty good deal, I'd say.

/

ewv

You wrote:

"If mathematics is thought of in terms of Platonism or in terms of subjectivist symbol manipulation, it is an invalid concept and is entirely divorced from reality by its nature."

Who advocates that mathematics is subjectivist symbol manipulation?

"[...] one particular approach (Frege's) was hopelessly self-contradictory as a consequence of its completely arbitrary notion of set creation (following Cantor literally) [...]"

Since it is not clear to me what you mean by 'completely arbitrary' in this context, I'm not necessarily taking exception with you here, but it should be kept in mind that in one sense the unrestricted approach of Cantor and Frege was the opposite of arbitrary since by stipulating no existence restrictions, there was no arbitrariness.

"For infinite sets the measurement is established inductively by finite means for identifying a specific extensible method of correspondence between the extensible sets."'

I don't know why you use the word 'inductively' there. The definition of 'equinumerous' is: x is equinumerous to y iff there is a 1-1 function from x onto y. To prove equinumerosity is to prove that there is such a 1-1 function from x onto y. All proofs are within a deductive system. For that matter, mathematical induction (which is still deduction) is not necessarily employed in any such proof.

"The method of establishing the correspondence between the rationals and the positive integers, or all the integers and the positive integers, shows how these sets can be enumerated in a process that (contrary to Cantor) does not entail a completed, or actual, infinity, yet serves the purpose of establishing an ordering between relative "size" of the open-ended sets: In both examples, one relies on a specific procedure for exactly how the mathematically infinite sets have the potential to be extended from any finite stage."

Perhaps that's the case in some system you haven't specified, but not in set theory. First, set theory says nothing about 'complete' or 'actual' anything. There are no predicates, primitive or defined, that correspond to 'complete' or 'actual' except the meta-theoretic term 'completeness', which is unrelated to your remarks. Second, set theory says nothing about such a procedure as you describe it. Instead, as I mentioned, one proves equinumerosity by proving the existence of a 1-1 onto function between the sets.

"Beyond the basic identification of key ideas such as cardinality, enumeration, and the Cantor diagonalization method all relying on the idea of rates of growth of extension [...]"

Cardinality, enumeration, diagonalization. None of these depend on ideas of rates of growth of extension or on ideas of rates of growth or on any ideas of rates of any kind.

"The idea of a formal arithmetic and a whole algebra of infinite "cardinals" is already starting to stretch it."

There's no stretching past what is provable from the axioms. The arithmetic of cardinals is logically implied by the axioms. (The axiom of choice is needed too.)

"A "science of method" for what purpose? -- method for what, other than subjectivism pragmatically supported by government grants?"

What subjectivism? What result in set theory do believe is not a logical consequence of its axioms? It is an objective fact that theorems of set theory are indeed theorems from the axioms. If you don't like the axioms, then set theorists have no objection to anyone offering a different set of axioms, as long as they'll get the job done of axiomatizing mathematics. In fact, set theorists do prove equivalences and differences among different axiomatizations. A quite objective program, wouldn't you say?

As to purpose, mathematics has always had both practical and non-practical motivation. One could scold not just set theory but many other mathematical endeavors for not having sufficiently immediate practical application. However, set theory and foundational mathematics have had a profound influence on the field of computing, and I barely doubt that this influence has been crucial in quite practical ways. And set theory and mathematical logic (as they are inextricable from each other) provide a venue for human reason at a stupendously high level of abstraction and rigor. Do you think that the conceptions of a pioneer of computing like Von Neuman did not benefit from his meditations in set theory?

As to government grants, unless set theory benefits from these more than any other academic field (I'm pretty sure it doesn't), then I don't see the significance of targeting set theory for special reproof.

"Cantor, who was a mystic, of course thought that he had demonstrated a whole world of actual infinities in a regular "order" extending all the way to the infinity of all infinities at the end, which was supposed to be God."

And nothing in set theory depends on Cantor's mysticism. The correct mathematics that he performed is subsumed by revised systems; the inconsistent assumption he made leading to paradox is not adopted in the revised systems; and none of this has to do with mysticism.

"[Pisaturo and Marcus] emphasize the concept of interchangeable "unit" to patiently develop the basic concepts of counting, arithmetic and elementary algebra starting with the role of physical reality and the cognitive, measurement-based purposes of mathematics -- not "arbitrary" definitions and postulates. The exposition corresponds to the way many of us understood it as it was taught in school courses on arithmetic and elementary algebra (or thought it was the way it was intended to be understood) but with an explicitly philosophical emphasis on the conceptual relationship of mathematics to reality and how elementary mathematics is developed through increasingly complex concepts."

Mathematics itself is not the study of human cognition. So of course set theory doesn't take a tour through that subject. Yet concerns in the philosophy of mathematics are important, but if you are to provide an alternative axiomatization of mathematics, then how would your alternative avoid arbitrary definitions and postulates? Complaining about arbitrariness in axiomatization doesn't ring much of a bell unless you can show a non-arbitrary axiomatization or at least show that one is possible. And if your axiomatization is non-arbitrary, then how is it different from logicism? Perhaps your axiomatization derives from certain propositions you consider to be epistemological truths. That's fine, as long as the math is good. But where is it?

Also you mention Peano arithmetic as not invoking infinite sets. If all set theory was meant to support was Peano arithmetic, then the axioms would be just those of Peano arithmetic. But Peano arithmetic can't express the mathematics of the reals, so, yes, Peano arithmetic does not require infinite sets, but Peano arithmetic is not called to do the job that set theory is.

"You have to already understand the mathematically infinite before you can make any sense of Cantor at all [...]"

Actually, set theory (such as Z set theory) presents a series of definitions that are quite lucid and require no presumptions about infinity. The axiom of infinity may be presented just after having defined the successor operation and the closure of a set under the successor operation. It's quite straightforward.

"[...] mathematics is not reduceable to logic alone except as a rationalistic exercise. Logical formalism is not how we learn or understand even basic mathematical concepts, starting with numbers, counting and basic arithmetic.

If mathematics were reducible to logic alone, then it wouldn't be an exercise, it would be an intellectual breakthrough of tremendous magnitude. Aside from that, are you speaking of formalism as a philosophy of mathematics or of mathematical formalization itself? That formalization does not emulate the cognitive processes of learning and understanding is no mark against formalization. This is not the purpose of formalization and its advantages (its virtual necessity) are not reckoned in terms of explicating cognition.

"Ignore notions of the "completed infinite", which is an invalid concept, and concentrate on the idea of open-endedness and the finitary methods used to specify it with precision in mathematics."

'finitary method' is a term that may not have a precise mathematical definition, but I wonder where you've seen it used that supports the way you're using it here.

As to open-endedness and precision, please cite the mathematical work that defines 'infinity' as open-ended in a mathematically precise way. That is, please cite your mathematically precise definitions of the predicates 'infinite' and 'open-ended'. Meanwhile, please cite any imprecision whatsoever in formalized set theory.

/

IdentityCrisis:

You wrote that your program is:

"1) Fully and exactly identify all of the concepts and rules at the base of deductive logic, since I've found that Aristotelean deductive logic is not complete.

2) Use them - and ONLY them - to reconstruct the language and tools of logic (such as "if...then", "all...are", etc.).

3) Construct the concepts at the base of mathematics ("one", "plus", the process of counting, etc.) from the above."

Step 3, without non-logical axioms, is the logicist program. As far as I know, it failed about a hundred years ago and has not had much luck since. Some approaches have come close, but still, no cigar. Perhaps you'd benefit by researching the various approaches, beginning with Frege, then Russell, up to and including Quine's NF.

If you believe you have achieved step 3 without non-logical axioms, then you should consult a mathematician expert in foundations, either to confirm your claim or to have explained to you how you've erred.

Edited by LauricAcid

Share this post


Link to post
Share on other sites

LauricAcid (Post #43):

For your information, Stephen Speicher has left this forum and now has his own forum.

Russell's paradox and the many similar paradoxes of self-reference are the result of two contradictory assumptions:

1. one is free to define new mathematical objects at will; and

2. there is a completed totality of such objects which can be referred to in the definition.

When Stephen said "complete" (referring to set theory), he may not have meant complete in the sense of being able to prove or disprove each sentence in the language.

Although I personally have no objection to set theory. I think that the Objectivists are concerned that it is a floating abstraction. This vagueness is apparent in the fact that there maybe multiple non-isomorphic models of set theory, even if one requires that they be truly well-founded.

Share this post


Link to post
Share on other sites

Thanks for the info on the poster.

/

"Although I personally have no objection to set theory. I think that the Objectivists are concerned that it is a floating abstraction. This vagueness is apparent in the fact that there maybe multiple non-isomorphic models of set theory, even if one requires that they be truly well-founded."

Wait! Hold it right there!

There are people who object to set theory on the basis that it's not categorical? Please tell me where I can read these objections.

If set theory is consistent, then it's not categorical, just as number theory is not categorical if it's consistent. That is, there is NO consistent, categorical theory for even plain old arithmetic. Objecting to set theory not being categorical is unreasonable since NO consistent theory for arithmetic is categorical.

(There may be non-classical logics that hold a consistent and categorical account of arithmetic, but I'm not familiar with them (so I am open to learning more about them if they exist).)

Also, what do you mean by 'truly well-founded'? Well foundedness is a property of sets. Unless there is some other special sense I'm not aware of (that very well might be), a well founded model would be one in which the universe is well founded. In that sense it is true that the axiom of regularity does not ensure the categoricalness of set theory. But I don't know if that's what you mean, nor do I know what it means for a set to be truly well founded as distinct from well founded.

/

'define new mathematical objects at will' needs to be couched more precisely to evaluate its import for Russell's paradox.

A formal set theory with an unrestricted axiom of comprehension allows the existence of any set that can be defined by a predicate expressible in the formal language. So formal expressibility is a restriction from the start. Further, set theory does not violate the principles of definition: eliminability and non-creativity. So that's a fundamental restriction also. So 'define at will' may be too broad a way of describing the fault of an unrestricted axiom of comprehension, unless the qualifications I mentioned are well understood.

As to 'there is a completed totality of such objects which can be referred to in the definition', set theory needs no predicates 'completed', 'totality', or even 'objects', so Russell's paradox does not hinge of any of them.

Also, don't forget that how one takes 'existence' is not specified by a formal language except by the fairly ontologically neutral specification of a structure. The word 'exists' in this context is an English locution used to make it easy to convey the symbols of a formula that is a syntactical object. One may argue that the formal semantics of the language demand certain ontological, metaphysical, or philosophical tenets, but one can also argue that the particular and supposedly irrational tenets that set theory is often saddled with by certain detractors are not entailed by even the formal semantics, so that certain of these philosophical charges against set theory are strawman arguments. However, it is true that some set theorists do hold certain ontological or philosophical beliefs that are part of how these set theorists understand the meaning of the formulas. So the beliefs of those mathematicians are fair game for critique. But such critique should not be conflated with a critique of set theory itself. Also, even to the extent that one does intend for the existential quantifier to really convey the notion of existence, logic still does not mandate that this existence be physical, non-physical, concrete, abstract, ideal, platonic, phenomenological, empirically known, concept, mental construct, or of any other form. Some logicians may argue for such mandates, but there is no codification in classical logic for them.

Anyway, Russell's paradox is more fundamental than many people realize. We can see the mechanics of the paradox in first order logic alone, since, for any binary predicate symbol S, the following is a theorem:

~Ex(Sxx <-> ~Sxx)

Anecdotally speaking: There is no x such that x shaves itself if and only if x does not shave itself.

One can view Russells' paradox in set theory as arising from overlooking that the above is a theorem of logic and that, therefore, existence axioms cannot be allowed to contradict it.

/

CORRECTION: In my previous post I said that ZF has only six initial axioms. But the axiom of replacement is an axiom schema, so it is an infinite set of axioms. Had I been more careful I would have said (correctly) that ZF starts with five axioms and one axiom schema.

Edited by LauricAcid

Share this post


Link to post
Share on other sites

LauricAcid:

You said "There are people who object to set theory on the basis that it's not categorical?".

The people that I am talking about would not even know what "categorical" (in the sense of model theory) means.

They are concerned that ZF is not firmly tied down to physical reality. I was trying to convey to you (who are clearly very mathematical in orientation) the features of ZF which tend to create that feeling on their part.

You said "If set theory is consistent, then it's not categorical, just as number theory is not categorical if it's consistent.".

Yes, they are not categorical. (Aside to others: categorical means having only one model modulo isomorphisms.)

But there is only one STANDARD model of number theory modulo isomorphisms. That is, there is only one for which "0" means 0, "successor" means successor, and there are no "natural numbers" which are not really natural numbers.

The same cannot be said of ZF set theory. Standard models of set theory (assuming at least one exists) vary, including variation among models of the same rank (= class of ordinals).

You asked "Also, what do you mean by 'truly well-founded'?".

A model of set theory is well-founded iff every nonempty collection of elements of the model contains at least one element which does not "contain" any other element in the collection. This is a second-order property which cannot be expressed in the first-order language of ZF (the axiom of regularity is the closest approximation).

I used the word "truly" to emphasis that I was talking about a statement ABOUT the model, not merely a statement in the language of the model.

My explanation of the paradoxes was necessarily vague because I was trying to address them all at once and they are expressed in different languages. Also I was not speaking in the language of set theory, but rather about set theory (and other theories).

I was trying to give a heuristic which could be used to help predict whether a proposed theory is likely to suffer from such a paradox.

Let me put it another way. To avoid paradoxes, it is helpful to have something like TIME in the theory. When the axioms require the existence of an object, the time when it is introduced should be later than the times of all the things upon which it depends. In ZF, this time is the rank of the set in question (or an upper-bound on the rank). In logic, it is called the type (or order).

You said "One can view Russell's paradox in set theory as arising from overlooking that the above is a theorem of logic and that, therefore, existence axioms cannot be allowed to contradict it.".

This is too narrow. Suppose g[f[x]]=x for all x in the model, then:

R={x|~(x e g[x])} implies

(f[R] e R) <=> ~(f[R] e R) a contradiction.

Share this post


Link to post
Share on other sites

Okay, standard models. That is a much needed qualification. Now I see your point about why some people are averse to set theory. Yes, there is something so basic in our thinking that makes the standard model of arithmetic ineluctable in a way that probably no model of set theory could be.

Why do you say well foundedness is second order and only approximated by the axiom of regularity? As you just defined 'well founded', the axiom of regularity says that every set is well founded - in exactly the same way you defined well foundedness. Or is your point that since the element symbol does not necessarily have to be assigned to the element relation in a given structure, it's possible for the domain to be not well founded and the structure still model ZFR?

About Russell's paradox, quite literally, if we ensure that the axioms of a theory don't imply the formula I mentioned, then the theory is consistent. On the other hand, if at first glance we don't see how the axioms do entail the formula, then that is not enough to assure us that they don't. But it is at least a good start to recognize that the formula can pop up as a theorem quite easily and that therefore we should at least start by avoiding axioms that obviously imply the formula. I did not suggest that this triage is sufficient for consistency. And my main point was that we should not overlook that Russell's paradox is lurking to strike not just set theory. This is to give some perspective for those averse to set theory due to a perception that there's something basically fishy about set theory because of Russell's paradox.

In your example, I didn't see how you derived the biconditional, and what's the moral of the story?

Edited by LauricAcid

Share this post


Link to post
Share on other sites

LauricAcid:

You asked "Why do you say well foundedness is second order and only approximated by the axiom of regularity?".

Well-foundedness talks about collections of elements of the model. The axiom of regularity talks about "sets" which are themselves elements of the model. Not every subcollection of the model is represented by a "set" in the model. So it is possible to have models which satisfy the axiom of regularity and are not well-founded. Such non-standard models must use an "element" relation which is not the real element relation.

You said "In your example, I didn't see how you derived the biconditional, and what's the moral of the story?".

Suppose g[f[x]]=x for all x in the model.

Notice that the union of the singleton of x is x. So for example, f[x] could be the singleton of x, and g[y] could be the union of y.

Further suppose that R={x|~(x e g[x])} exists.

This means: x e R <=> ~(x e g[x]) for all x in the model.

Take x to be f[R].

Thus f[R] e R <=> ~(f[R] e g[f[R]]).

But g[f[R]]=R by replacing R for x in g[f[x]]=x.

So f[R] e R <=> ~(f[R] e R) which is a contradiction.

Moral: There are non-obvious variations on Russell paradox which one must also avoid.

Share this post


Link to post
Share on other sites

The problem with set theory is that it's utterly dependent on the early 20th century idea that mathematics somehow needs a 'foundation'. It doesnt. There is simply no advantage that I can see coming from the axiomization of a particular branch of mathematics, and your claim that formalization allows proofs to be easily 'checked by computer' is straight out of fantasy land. If it were even remotely possible to reduce a fairly complex proof to a direct deduction from the ZFC axioms in this manner, then it wouldnt have taken significant numbers of highly talented mathematics several years to check the proofs of (for instance) Fermat's Last Theorem or the Poincare conjecture. In any case, any 'uncertainty' about the validity of a proof will simply be replaced by uncertainty regarding how well it has been formalised within a system - there is always going to be a possibility that an accepted proof can turn out to be wrong, and set theory does nothing to change this.

Mathematics isnt really a deductive discipline in practice, and I'm unclear what advantages there are to any axiomatic/set-theoretical approaches. Mathematics managed to function quite well before ZFC despite the objections of philosophers, and I'm sure it would continue to function in its absence.

Edited by Hal

Share this post


Link to post
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...
Sign in to follow this  

  • Recently Browsing   0 members

    No registered users viewing this page.

×
×
  • Create New...