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are they infinite numbers ?

eldorado
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Posted 07/22/09 - 05:12 AM: Subject: are they infinite numbers ?	#1
Hi, this is my first post. from this topic : http://www.math.utoronto.ca/mathnet/answers/existence.html I understand that numbers are abstract concepts ( exist in our brains only ). So , are they infinite numbers ? I think the answer should be "no" , because I have finite set of numbers in my brain at any time. Is that right ? Thanks. Edited by Caldwell on 07/28/09 - 12:49 AM. Reason: caps

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1 of 1 people found this post helpful Posted 07/22/09 - 07:41 AM:	#2
Don't think of numbers and individual abstract symbols, (like a vocabulary), but a system of describing quantities, (like a grammar). The grammar of producing numerical values, (like natural numbers), is potentially infinite, (although there are some problems with that on the very fringe). But remember, "infinite" really means "no limit", it isn't a specific value. Your brain can contain an infinite set represented by a grammar to produce that set.
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Posted 07/22/09 - 07:44 AM:	#3
Pi is probably the most famous infinite number.
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1 of 1 people found this post helpful Posted 07/22/09 - 09:50 AM:	#4
Pi has an infinite decimal expansion which may be generated by a rule. "Pi", the word, is the symbolic representation of both its rule and approximations to the potentially infinite decimal expansion. That is, the word "Pi", in ordinary mathematical parlance, is used to represent two fundamentally kind of things. Moreover, in a strict logical sense, as it happens, it is a synecdochic term because the "logical entity" which is the concept of Pi (the whole) itself is referred to by the same linguistic item as that which refers to the parts of the concept. Namely, we refer to "3.14" by the term "Pi" and we refer to "3.14159265 and so on" by the term "Pi." The mathematical description, which is the latter use, should never be mistaken for the former. Moreover, one must bear in mind that it makes no sense to describe that which is in principle imperceptible. Thus, we may as well give up the idea that we are "describing a mathematical entity." Pi is both a substantive: the former use (when it is a finite decimal expansion) and a rule: the latter use (one cannot give the full necessary and sufficient conditions for a rule, as it holds for every possible case; therefore, one cannot give a complete description of it, or rather, one cannot one infer from the substantive case that one as sufficiently described the rule). So, your question boils down to: Are there actual infinite sets (taken through the route of number theory -- actual infinites in the ontology of Numbers)? Given our understanding of the infinite, are there conditions under which one could be said to have arrived at an actual infinite in the way, say, that we arrive at a mountain or a chair?
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Posted 07/22/09 - 10:17 AM:	#5
despinozist wrote: Pi has an infinite decimal expansion which may be generated by a rule. "Pi", the word, is the symbolic representation of both its rule and approximations to the potentially infinite decimal expansion. But we don't have to limit ourselves to such constructivistic views. That is, 'pi' is also a name of a specific real number that has, as do all real numbers, a representation as a denumerable sequence of rational numbers. The range of that sequence is an infinite set. Moreover, the ordinary construction of real numbers is either by Dedekind cuts or by equivalence classes of Cauchy sequences. In either case then, each real number is itself an infinite set, though, admittedly, that is an "artifact" of the construction. As to the general question whether there exist infinite numbers, in such foundational theories as Z set theory, there exist infinite sets, and (aside from the situation I mentioned about real numbers, which applies in a similar way for certain other kinds of numbers) there are infinite ordinal numbers and infinite cardinal numbers. despinozist wrote: Namely, we refer to "3.14" by the term "Pi" and we refer to "3.14159265 and so on" by the term "Pi." Maybe such a 'we' includes you, it sure does not include me. I definitely do not refer to 3.14 as 'pi' despinozist wrote: one must bear in mind that it makes no sense to describe that which is in principle imperceptible. Perhaps it makes no sense to you. It makes fine sense to me. Mathematical objects are abstractions. despinozist wrote: Thus, we may as well give up the idea that we are "describing a mathematical entity." I describe mathematical entities often. I don't have to give up doing that. despinozist wrote: Given our understanding of the infinite, are there conditions under which one could be said to have arrived at an actual infinite in the way, say, that we arrive at a mountain or a chair? I don't know what you mean by "arrive". Meanwhile, there's no mathematical question that I happen to have these days that depends on whatever notion you have of "arriving". Edited by MoeBlee on 07/22/09 - 11:59 AM

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Posted 07/22/09 - 10:31 AM:	#6
despinozist wrote: Namely, we refer to "3.14" by the term "Pi" and we refer to "3.14159265 and so on" by the term "Pi." I don't, and I doubt any mathematicians do. EDIT: Oh Moblee, you're just too fast...
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Posted 07/22/09 - 12:14 PM:	#7
Good! This should be fine. I should hope to learn something from this. For a while, I was a very strict finitist in the vein of Brouwer. Still, even to this day "all" statements give me shudders. MoeBlee wrote: But we don't have to limit ourselves to such constructivistic views. I should comment on this, but I'm not exactly sure of the tone in which you make this statement. What do you mean by "limiting ourselves"? It wouldn't be unreasonable for me to assume a "pessimistic" quality to your making this statement. Or, more directly, that I am somehow proposing a philosophy which suffocates mathematical practice or thinking. Whether I rightly show that talk of infinite sets to be absurd is one question, whereas whether we should engage in, generally speaking, talk of or talk with nonsense is another. I'm sure you agree that nonsense is dangerous to discourse of any kind. Moreover, "limiting ourselves" from nonsense is what my position seeks to achieve. Now I don't see a point to this initial remark, other than that it may only be a red herring. Unless, of course, what exactly we are limited from doing just is your argument. But that remains to be seen. That is, 'pi' is also a name of a specific real number that has, as do all real numbers, a representation as a denumerable sequence of rational numbers. This seems circular. "is specific" is just the mathematical predicate which is in question. When one says "Look, it's 3.14, which is a specific representation of Pi" is this the same as saying "Look, she's generating out Pi; boy that will go on for ages..." What are the conditions for saying that one is specifically generating Pi? My issue is that: We say of concrete entities: This is specifically φ. But you've just called Pi a "specific real number." What does this mean? In what sense do you mean "specific"? Do you mean "a specific rule" or "a specific algorithm"? The range of that sequence is an infinite set. Moreover, the ordinary construction of real numbers is either by Dedekind cuts or by equivalence classes of Cauchy sequences. In either case then, each real number is itself an infinite set, though, admittedly, that is an "artifact" of the construction. Right. We're concerned with the ontological status of ranges, whether they exist like tables and chairs prior to the construction of them through mathematical proof. Maybe such a 'we' includes you, it sure does not include me. I definitely do not refer to 3.14 as 'pi' I'm not arguing for a personal view. I'm concerned with what we could possibly mean; "we" including all of us. Naturally my arguments come off as presumptive. But what about in everyday language? Nonmathematicians? Nonphilosophers? Do we constantly add in the caveat "bear in mind, this is an approximation to Pi." Moreover, this question has a lot to do with the pragmatics of reference ( a topic in philosophy of language ). I'm not at all saying that "3.14" successfully refers to Pi. Obviously it is an approximation. You do not at all need to intimate that I might not know this. I am fully aware of it. What I am concerned with is whether or successfully refers to Pi by way of the term "3.14", say, in non-academic settings. And how do these questions of the pragmatics of mathematical language influence questions of the pure mathematical questions with which we are concerned? Perhaps it makes no sense to you. It makes fine sense to me. Mathematial objects are abstractions. Again: The playground we're in is sense and senselessness, or at least I'm playing there. And, if I am playing alone, and you should wish not to join with me in discussing the possibility of sense, or sense (all that one could mean), then there's no reason for us to continue. What I am saying is that talk of infinite sets is nonsense. It is because to confuse intensional terms with extensional terms yields you nonsense. My essential claim is that Pi is a rule, or a kind of rule, for generating Pi-like decimal expansions. Naturally, you may agree depending on your own understanding of what it means to be an "abstraction." But where we indeed must part ways is if you hold that "abstractions" have reality in the way that tables and chairs do (actual objects). Pi is an actual rule like the Bill of Rights is an actual (set of) rule(s). I describe mathemtical entities often. I don't have to give up doing that. Depends on what you mean by entity. That's largely what this entire issue is about: mathematical ontology. We have entities as laws, entities as chairs and tables, etc. The former are not "actual substantives," so we don't describe them in the same way, in principle, because: What would it mean to say that I've approximately described the chair to you? We would have to presume that chairs have essences of some kind that we implicitly fail to apprehend in our description of that chair. So when we describe Pi, how do we go about it? I'd say a "description" of Pi is nothing more than the computation of the decimal expansion it yields (of course, as I suppose this just is constructivism). I don't know what you mean by "arrive". Meanwhile, there's no mathematical question that I happen to have these days that depends on whatever notion you have of "arriving". I said what I meant: Like arriving at tables and chairs, tripping over them, walking into them. This sense of "actual" is one we cannot permit to numbers and rules. That said, the answer to the question: Are there infinite numbers? is an easy one: Yes, but not like actual tables and chairs, but like actual rules and algorithms.
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Posted 07/22/09 - 02:29 PM:	#8
MoeBlee wrote: But we don't have to limit ourselves to such constructivistic views. despinozist wrote: I should comment on this, but I'm not exactly sure of the tone in which you make this statement. What do you mean by "limiting ourselves"? It wouldn't be unreasonable for me to assume a "pessimistic" quality to your making this statement. Or, more directly, that I am somehow proposing a philosophy which suffocates mathematical practice or thinking. Whether I rightly show that talk of infinite sets to be absurd is one question, whereas whether we should engage in, generally speaking, talk of or talk with nonsense is another. I'm sure you agree that nonsense is dangerous to discourse of any kind. Moreover, "limiting ourselves" from nonsense is what my position seeks to achieve. Now I don't see a point to this initial remark, other than that it may only be a red herring. Unless, of course, what exactly we are limited from doing just is your argument. But that remains to be seen. You're making too much of my remark. I simply meant that aside from your viewpoint, there are other viewpoints. MoeBlee wrote: 'pi' is also a name of a specific real number that has, as do all real numbers, a representation as a denumerable sequence of rational numbers. despinozist wrote: This seems circular. "is specific" is just the mathematical predicate which is in question. When one says "Look, it's 3.14, which is a specific representation of Pi" is this the same as saying "Look, she's generating out Pi; boy that will go on for ages..." What are the conditions for saying that one is specifically generating Pi? Specific can be made perfectly formal by saying that there is a definining existence and uniqueuness theorem. There is no circularity in that. And I've not said anything about, nor do I depend on a notion of, 'generating'. despinozist wrote: My issue is that: We say of concrete entities: This is specifically φ. But you've just called Pi a "specific real number." What does this mean? In what sense do you mean "specific"? Do you mean "a specific rule" or "a specific algorithm"? I mean there is an existence and uniqueness theorem that permits definition of the symbol 'pi'. And, with any of the ordinary definitions of 'pi', it follows that pi is a real number. MoeBlee wrote: The range of that sequence is an infinite set. Moreover, the ordinary construction of real numbers is either by Dedekind cuts or by equivalence classes of Cauchy sequences. In either case then, each real number is itself an infinite set, though, admittedly, that is an "artifact" of the construction. despinozist wrote: Right. We're concerned with the ontological status of ranges, whether they exist like tables and chairs prior to the construction of them through mathematical proof. Maybe that is your concern. It is not mine. MoeBlee wrote: Maybe such a 'we' includes you, it sure does not include me. I definitely do not refer to 3.14 as 'pi' despinozist wrote: I'm not arguing for a personal view. I'm concerned with what we could possibly mean; "we" including all of us. Naturally my arguments come off as presumptive. But what about in everyday language? Nonmathematicians? Nonphilosophers? Do we constantly add in the caveat "bear in mind, this is an approximation to Pi." I only remarked that whatever your 'we' means, it doesn't include me. despinozist wrote: Moreover, this question has a lot to do with the pragmatics of reference ( a topic in philosophy of language ). I'm not at all saying that "3.14" successfully refers to Pi. Obviously it is an approximation. You do not at all need to intimate that I might not know this. I am fully aware of it. What I am concerned with is whether or successfully refers to Pi by way of the term "3.14", say, in non-academic settings. And how do these questions of the pragmatics of mathematical language influence questions of the pure mathematical questions with which we are concerned? Fine that those are among your concerns. MoeBlee wrote: Perhaps it makes no sense to you. It makes fine sense to me. Mathematial objects are abstractions. despinozist wrote: Again: The playground we're in is sense and senselessness, or at least I'm playing there. And, if I am playing alone, and you should wish not to join with me in discussing the possibility of sense, or sense (all that one could mean), then there's no reason for us to continue. You said someting doesn't make sense (or that it doesn't make sense to you). I only said that that something you mentioned does make sense to me. I've not thereby obligated myself to join in discussing such questions as to "possibility of sense". And if you don't wish to comment ("continue") then that is your choice. despinozist wrote: What I am saying is that talk of infinite sets is nonsense. Yes, I gathered that. despinozist wrote: It is because to confuse intensional terms with extensional terms yields you nonsense. I don't confuse extensionality with intensionality. despinozist wrote: My essential claim is that Pi is a rule, or a kind of rule, for generating Pi-like decimal expansions. Naturally, you may agree depending on your own understanding of what it means to be an "abstraction." Yes, I got that that is your view; I happen not to share that view. despinozist wrote: But where we indeed must part ways is if you hold that "abstractions" have reality in the way that tables and chairs do (actual objects). Obviously, abstract objects are not like concrete objects, otherwise I wouldn't say they are abstract and instead I'd say they are concrete. despinozist wrote: Pi is an actual rule like the Bill of Rights is an actual (set of) rule(s). I understand such views. I just happen not to share your view. MoeBlee wrote: I describe mathemtical entities often. I don't have to give up doing that. despinozist wrote: Depends on what you mean by entity. I mean it in its most primitive sense. I probably wouldn't even try to define it. Mathematically, though, there's the old Quine saying "To be is to be the value of a bound variable". despinozist wrote: That's largely what this entire issue is about: mathematical ontology. Maybe for you it is. Not so much for me. If I am willing to endure the awkwardness, I can even speak without using such terms as 'object'. despinozist wrote: We have entities as laws, entities as chairs and tables, etc. The former are not "actual substantives," so we don't describe them in the same way, in principle, because: What would it mean to say that I've approximately described the chair to you? We would have to presume that chairs have essences of some kind that we implicitly fail to apprehend in our description of that chair. So when we describe Pi, how do we go about it? I'd say a "description" of Pi is nothing more than the computation of the decimal expansion it yields (of course, as I suppose this just is constructivism). But there are other descriptions (indeed, definitions even) of pi that don't resort to whatever its decimal expansion may be. These are found in many textbook on real analysis (by the way, I'm quite rusty on certain of those particulars, so please don't ask me to produce here one of these definitions; I'm confident you can find them on the Internet if you're not already familiar with them). MoeBlee wrote: I don't know what you mean by "arrive". Meanwhile, there's no mathematical question that I happen to have these days that depends on whatever notion you have of "arriving". despinozist wrote: I said what I meant: Like arriving at tables and chairs, tripping over them, walking into them. Then, of course, I don't view that there is an arrival upon mathematical objects. despinozist wrote: This sense of "actual" is one we cannot permit to numbers and rules. That said, the answer to the question: Are there infinite numbers? is an easy one: Yes, but not like actual tables and chairs, but like actual rules and algorithms. And again, I'm pointing out that there is another view that differs from yours: a view that there are infinite sets and infinite cardinal and ordinal numbers that are not defined as themselves rules or algorithms. Edited by MoeBlee on 07/22/09 - 02:35 PM

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Posted 07/22/09 - 10:18 PM:	#9
despinozist wrote: My issue is that: We say of concrete entities: This is specifically φ. But you've just called Pi a "specific real number." What does this mean? In what sense do you mean "specific"? Do you mean "a specific rule" or "a specific algorithm"? If you believe there should be some (real) number associated with the area of any collection of points in 2 space then it seems to me, it's pretty hard to deny the existence of pi. Putting aside questions of convention, in your personal view, what exactly should the area of the unit disk be?
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Posted 07/23/09 - 12:04 AM:	#10
It's been on the order of years since I last posted here, it's good to see some old names - MoeBlee et al. Anyhow, I've been studying set theory non-stop for the last month, so when I see someone saying that they don't think infinite sets exist, it gives me pause for thought. An observation occurred to me and I thought I could interject it without drawing myself too far into the debate: despinozist, you say you don't believe in infinite sets; is it for one of the following reasons: a) you reject all of modern mathematics altogether. Nearly every branch of mathematics invokes the idea of infinite sets, and moreover infinite sets of various sizes (some are bigger than others). Modern mathematics has an explicit expression in a first-order symbolic language which defines what it means to be an infinite set, and this is (more or less) the same language which all of mathematics is done in. Every branch of modern mathematics uses infinite sets, and regards them as such, not as "processes" or anything like that. If you look at specific questions mathematicians try to answer and honestly try to rephrase them in terms of "processes," it would be impossible. Does everything need to be overhauled, does mathematics need to go back to its synthetic roots where we only talk about things that can be immediately represented by physical objects, and where the language of mathematics is merely natural language? b) You accept modern mathematics the way it is, but you're an anti-platonist, and thus regard it as a big game of consistent symbolic manipulation. In this case infinite sets don't exist, but neither do finite sets. Indeed in this case no mathematical objects exist, they're all just figments of our imaginations used to try to give some semantic/intuitive content to the purely syntactic manipulations that mathematics really is. c) You accept modern mathematics and are a platonist, but somehow reject the axiom of infinity.
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