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Mathematics: Reality And Infinity

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Mr. Speicher has temporarily overloaded my mental capacity in our conversation on physics, meaning: I've got some learning and thinking to do before I really get back into that conversation. But there are other issues that have been concerning me. So I'd like to turn to some of them now.

First off, the concept of infinity. I know that objectivism considers infinity to be a mathematical concept only (and not existing in physical reality), but what is the mathematical definition of infinity (or what are the definitions, if more than one)?

Second - and this question is directed primarily at you, Mr. Speicher, though others are welcome to chime in - what breaches exist between mathematics and physical reality? You 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?

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First off, the concept of infinity.  I know that objectivism considers infinity to be a mathematical concept only (and not existing in physical reality),

And, more specifically, infinity as a concept of method, a process representing a potential, not an actual.

(Note, however, that in recent years Harry Binswanger has made some rather persuasive arguments against infinity even as a concept of method. Taking the infinite as the simple process of always adding one to a number, the fact is that such a process is not, strictly speaking, even potentially always possible. One reason for this is that we have a psycho-epistemological limit on the degree and the amount of notation we can hold in our mind in order to represent such a process. There exists a point beyond which the very concept of 'number' becomes meaningless to a consciousness.

In addition, there is also a physically practical limit beyond which we cannot even represent numbers in _any_ notation. There will be some point which is reached where there are not even enough particles in the universe where the most compact notation could denote and delimit a number. [With apologies to HB for any misrepresentations of his position.] )

but what is the mathematical definition of infinity (or what are the definitions, if more than one)?

Hmm. I do not think you will like this one, since "infinity," just like some of the terms we discussed in physics, has several different meanings as used in different fields of mathematics. For instance, the "bread and butter" usage of "infinity," in modern real analysis, is, loosely speaking, an unbounded limit. But then in set theory, which you said you were starting to study, infinity wears several different hats. We can go as far back as Galileo, who rejected infinite sets, but who in the process also worked with the notion of one-to-one correspondence which is at the heart of modern definitions of infinite sets. But Galileo was in good company because some great mathematicians as the likes of Cauchy likewise rejected infinite sets. It was Bolzano, whose work was not published till a few years after he died in 1848, who advanced the notion of infinite sets and actually used a one-to-one correspondence to work with equivalent infinite sets.

But all this was before Cantor who was the true originator of a theory of infinite sets. Simply put, for Cantor a set was infinite if it could be put in a one-to-one correspondence with a part of itself. From the 1870s onward Cantor had set the stage, spread by other mathematicians such as the great Hilbert. But this was a different sense of infinity than that used in real analysis, and further notions of infinity were still to come.

I will (briefly) address the other question posed when I have some time tomorrow or later. I have been away for several days from here and will never catch up.

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Second - and this question is directed primarily at you, Mr. Speicher, though others are welcome to chime in - what breaches exist between mathematics and physical reality?  You 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?

I really do not have very much more to say about this, and I can offer no specific alternatives, but set theory is, at best, simply not fundamental enough to provide a complete foundation for mathematics, and, at worst, it is a rationalistic concoction.

Take Cantor's foundation for the equivalence of infinite sets, the one-to-one correspondence between elements of two sets. It is not at all clear what facts of reality are actually implied or connected with the one-to-one correspondence between, say, the set of all integers (positive and negative) and the set of positive integers alone. Or, for that matter, the one-to-one correspondence between the positive integers and the set of rational numbers. 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? The significance of a one-to-equivalence between finite sets is quite obvious and clear, but not so clear or obvious when it comes to infinite sets.

That sort of logic can be extended throughout the edifice built upon set theory, this despite that there are some valid and practical uses to which set theory can be put. Without offering an alternative I still place my bet on a more solid foundational aspect for mathematics. To get a glimpse of the seeds of such an alternative, not one that I necessarily agree with, you can look up some of the older issues that are probably still available from The Intellectual Activist. Ron Pisaturo, along with Glen Marcus for some of the work, wrote several articles in TIA around 1994 and again in 1998 or so, which dealt with these foundational issues. You might find them somewhat interesting.

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I know that objectivism considers infinity to be a mathematical concept only (and not existing in physical reality), but what is the mathematical definition of infinity (or what are the definitions, if more than one)?

You raise this question in the context of Objectivism, but it is not clear what you are trying to find out. To ask for a definition and explanation of the technical concept of infinity within mathematics is like asking about any other technical concept in a science. It can only be addressed directly knowing the context and purpose of the question and the extent of knowledge that can be relied on in a technical explanation.

Ayn Rand and Leonard Peikoff both made reference to mathematical concepts such as infinity for specific purposes in their writing, but Objectivism, as general philosophy, does not depend on technical concepts of advanced sciences such as mathematics or physics, and in particular does not depend on technical concepts of infinity in advanced mathematics. Nor does Objectivism have a specialized philosophy of mathematics or philosophy of any other science because Ayn Rand, not being a physical scientist or mathematician, did not have the specialized scientific knowledge required for that and properly did not try to rationalize a specialized philosophy of science without knowing the required facts.

Ayn Rand did, however, understand basic (but not technically advanced) mathematics and referred to it in discussing her epistemology. You should look up the few references to mathematics, and the concept of infinity in particular, in Introduction to Objectivist Epistemology (including the appendix on the epistemology seminars she held in the early 70's) and in Leonard Peikoff's OPAR.

Her purpose was primarily to illustrate certain epistemological points in Objectivist epistemology using particularly clear and precise concepts from school mathematics. She also thought that a mathematical approach would help to further advance an expanded science of epistemology (which she began working on near the end of her life only to the extent of embarking on learning more mathematics). And her use of simple mathematical concepts to help explain her epistemological concepts also provide important insights into how Objectivist epistemology should be applied in mathematics.

The mathematical concept of infinity referred to by Ayn Rand is the simple idea of the infinite integers ("an arithmetical sequence extend[ing] into infinity, without implying that infinity actually exists" [iOE]). The basic philosophical ideas she discussed in relation to this are 1) the idea of infinity as only a "potential" rather than an "actual" (as in Aristotle vs. Plato), i.e., as epistemological rather than metaphysical, and 2) the open-endedness of abstract concepts ("such extension means only that whatever number of units does exist, it is to be included in the same sequence" [iOE]).

The cognitive potential of counting beyond any explicitly or implicitly known "highest number" in some context (and across time into the future) does not imply an "actual" infinite (which would have no identity), but no specific limit on referents already known or referred to, either explicitly or implicitly, need be specified or implied in order to hold an open-ended concept. Finitude and specificity in all respects are necessary in any valid concept (and need not and should not be explicitly re-iterated as an essential or distinguishing characteristic for the concept), but that does not imply a universal bound within concepts of method.

(Note also that in referring to the mathematical concept of infinity as the "potential infinite", this is meant strictly in an epistemological sense, and does not mean -- as I once heard in a confused allegedly Objectivist argument against the mathematical infinite -- that something has the "potential" to become an "actual infinite", i.e., metaphysically.)

The same (correct) idea behind the concept of an infinity of the integers, although not described in Objectivist epistemology, also applies in mathematics to infinite subdivisions and limits (Ayn Rand understood this, for example as it applies to irrational numbers like the square root of two), basic geometry (such as infinitely long infinitely thin lines, infinitely small points, circles composed of infinitely thin perfectly smooth curves, etc.), the "point at infinity" in geometry and in the theory of conformal mapping (in which straight lines are regarded, for economy of thought, as a kind of "infinite circle" through the "point at infinity") in complex analysis, and much more.

But this should be regarded as a proper application of Objectivist epistemology (which is not unique in this regard), not an element of Objectivist philosophy or something on which Objectivism depends. As for all the uses of the "infinite" in mathematics, it pervades and is indispensable to most of mathematics.

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What breaches exist between mathematics and physical reality?
Properly conceived and applied, none. Mathematical concepts are abstract concepts of method used to measure reality. 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. What did you have in mind?

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You 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?
"Basis" for what purpose? Set theory is an indispensable methodological concept of modern mathematics that allows tremendous precision and economy of thought. But if you are looking for a philosophical foundation, it's most glaring deficiency is that it obviously isn't: it leaves out the entire cognitive nature and purpose of mathematics -- beginning by ignoring the distinction between a set and a concept. It's historical use as a "foundation" was a deliberate attempt to bypass epistemological issues for the sake of "precision" and therefore "certainty" through a purely mechanical means allegedly immune to human error by throwing out all references to human cognitive meaning. A certain amount can be done by focusing on "form" alone (analogous to the structural form of syllogisms in logic), but not everything. Russell's paradox was only intended to show that one particular approach (Frege's) was hopelessly self-contradictory as a consequence of its completely arbitrary notion of set creation (following Cantor literally); subsequent attempts tried to use more sophisticated rules for the manipulations while retaining the original goal.

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[

Take Cantor's foundation for the equivalence of infinite sets, the one-to-one correspondence between elements of two sets. It is not at all clear what facts of reality are actually implied or connected with the one-to-one correspondence between, say, the set of all integers (positive and negative) and the set of positive integers alone. Or, for that matter, the one-to-one correspondence between the positive integers and the set of rational numbers... The significance of a one-to-equivalence between finite sets is quite obvious and clear, but not so clear or obvious when it comes to infinite sets.
The abstract "correspondence" between equivalent sets represents a conceptual process, or method, closely related to the idea of counting through establishing a correspondence between a collection and the sequence of natural numbers, with the cognitive meaning of "quantity" inherent in actual counting with integers omitted, but with the abstract non-finite cardinals still measuring relative "sizes" ("cardinality of the set") in a specific way based on the correpsondence. For infinite sets the measurement is established inductively by finite means for identifying a specific extensible method of correspondence between the extensible sets.

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. (For the positive and negative integers, the procedure jumps back and forth between positive and negative for each number, and for the rationals it follows the "diagonal" pattern across the two dimensional array of fractions, skipping duplicate entries.)

That the reals cannot be so enumerated shows that the potential for their extension is "infinitely" greater, in a well-defined way, than those that can be enumerated with the positive integers. In that sense, the integers are "larger" than any finite set of specific elements, and the reals are "larger" than the integers. In the same way, a set (including any infinite set) is always "smaller" than its power set (the set of all its subsets) by a factor of an exponential (power of 2), and indicates an attribute of rates of convergence in combinatoric algorithms (which of course always are restricted to finite subsets in actual computing).

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, the conceptual utility of carrying on with this mentality in abstract mathematics and how far to go with it is a different question. The idea of a formal arithmetic and a whole algebra of infinite "cardinals" is already starting to stretch it. A "science of method" for what purpose? -- method for what, other than subjectivism pragmatically supported by government grants? For others, a normal response would be, "that's nice" -- followed by going off to do something else. 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.

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?
The idea of a set being in "correspondence" with an infinite subset of itself as a characterization of infinite size captures the fact that this is not possible for a finite set of specific elements. You can "count" or "enumerate" the rationals or the integers (positive and negative together) with the positive integers (which are skipping at a slower rate through themselves as a subset of the set being enumerated) only because both sets have the potential to be extended as you keep going. If you try that with a finite set you run off the end for one of them at a different point than where you are in the other (within the subset).

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Without offering an alternative [to set theory]I still place my bet on a more solid foundational aspect for mathematics. To get a glimpse of the seeds of such an alternative, not one that I necessarily agree with, you can look up some of the older issues that are probably still available from The Intellectual Activist. Ron Pisaturo, along with Glen Marcus for some of the work, wrote several articles in TIA around 1994 and again in 1998 or so, which dealt with these foundational issues. You might find them somewhat interesting.
In "The Foundations of Mathematics" (TIA July & Sept. 1994) and "Mathematics in One Lesson" (TIA Sept. & Oct. 1998) Ron Pisaturo and Glenn Marcus explicitly applied their understanding of Objectivist epistemology to elementary mathematical concepts and principles. They 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.

That series of articles, however, was by design restricted to very elementary concepts of school mathematics, in particular arithmetic concepts. It is not an explanation of the epistemology of higher mathematics and its vast abstractions, which cannot be understood either philosophically or mathematically at the cognitive level of counting marbles. Nor does it attempt to deal with the "mainstream" philosophical approaches common over about the last century and still dominant today. Also, even at this elementary level the authors erroneously rejected geometry as a branch of mathematics because it did not fit certain key ideas they used to explain arithmetic, and the explanation of the concept of an irrational number was strained and incomplete, containing both mathematical and epistemological elements of truth, but fishing for an epistemological explanation for the concept which was never found.

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[The abstract "correspondence" between equivalent sets represents a conceptual process, or method, closely related to the idea of counting through establishing a correspondence between a collection and the sequence of natural numbers, with the cognitive meaning of "quantity" inherent in actual counting with integers omitted, but with the abstract non-finite cardinals still measuring relative "sizes" ("cardinality of the set") in a specific way based on the correpsondence. For infinite sets the measurement is established inductively by finite means for identifying a specific extensible method of correspondence between the extensible sets.

Sorry, but this is mostly gobbledygook. Bijective functions are more sensible than that. I will not pursue this further.

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You raise this question in the context of Objectivism, but it is not clear what you are trying to find out.  To ask for a definition and explanation of the technical concept of infinity within mathematics is like asking about any other technical concept in a science.  It can only be addressed directly knowing the context and purpose of the question and the extent of knowledge that can be relied on in a technical explanation.

Perhaps my opening comment was misleading. I should clarify.

I am not using objectivism as a context. But I do realize that my audience consists largely of objectivists and I anticipated that the first response of many of them would probably be to mention that infinity is valid as an epistemological concept only. I simply wanted to forstall that and avoid wasting everybody's time (including my own).

As for my purpose in asking... That's a little complicated. My long-term goal is the creation of true artificial intelligence; computer programs that can take sense data and organize and theorize about it in the same way that a human can.

The premise I am working on is that humans get away with not consciously knowing the precise meaning of every word they use or the exact steps involved in coming to a conclusion because we have a piece of "black box" technology that does that for us: the subconscious. The subconscious handles the details and then presents my conscious mind with the results, but not necessarily with the steps leading to those results.

Unfortunately, if I'm going to program a computer to think it's not enough that I have such a "black box" myself. I need to know exactly how it works so that I can design a software equivalent. Thus, I need to know exactly what the irreducible concepts are that my mind uses, exactly what each one means and exactly what rules govern their combination into more complex concepts. Kind of like Aristotle's categories, but taken to the next (and final) level.

To this end, I'm having to go even deeper than Rand did in identifying basic concepts. To give you an idea of the sort of thing I'm doing, I've been going through lists of words we use all the time but never really think about the meaning of - "I," "the," "and," "one," etc. - and trying to explicitly identify what I mean by each one and if I can completely define any of them in terms of others on the list (establishing that they're reducible concepts after all).

Right now I'm working on basic logic, and in addition to a couple of other rather interesting discoveries I finally managed to bridge the gap between logic and basic mathematics just recently. I've derived the concepts "one," "plus," "equality,"

"inequality," "greater than" and "lesser than" from basic logical operators. I'm still a long way from differential equations - I haven't even defined "zero" or "minus" yet - but I've definitely got a solid foundation laid down.

However, I'm going to need to know how to define "infinity" for various areas of my work. Without that, I'll be stuck before long. That's why I'm asking about it; I don't expect that any one definition out there will be what I want, but I'm hoping that by considering various definitions it might help me to abstract out the element that I'm looking for.

Is it clearer now?

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Sorry, but this is mostly gobbledygook. Bijective functions are more sensible than that. I will not pursue this further.

In pursuit of brevity, my explanation was evidently too terse, but you don't have to discuss it further.

For others who may be interested let me add in the way of a summary of basic principles that there is no such thing as a completed infinity in mathematics or anywhere else. The mathematical infinite only has meaning in terms of the "open-ended", with a specific means required to identify new elements. There is no such thing as a 'size' of an infinite set (qua abstraction) because it is open-ended. When infinite sets are compared, it must be done with respect to their specific means of extension so that at any point one is always dealing (inductively) with the finite. Further, mathematical abstractions do not refer directly to 'things' in reality; the facts that give rise to a new mathematical abstraction are a combination of known mathematical facts on which it depends and its purpose as a concept of method. In these terms, one can make sense of such notions as 'correspondence' between infinite sets and the cardinality (not literally a quantitative 'size') of a set for specific purposes of comparison.

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As for my purpose in asking... That's a little complicated. My long-term goal is the creation of true artificial intelligence; computer programs that can take sense data and organize and theorize about it in the same way that a human can.

I don't know what else you have in mind for how to make your project work, but based on this opening description, it sounds dubious. Human thought and consciousness cannot be reduced to an algorithm and reproduced in software. That principle isn't new, but a recent good article on it from an Objectivist perspective is in the April 2004 issue of Rob Tracinski's monthly journal, "The Intellectual Activist": "Mindless Intelligence: Machine Thinking and Contemporary Philosophers' Rejection of the Mind". Have you seen this?

The premise I am working on is that humans get away with not consciously knowing the precise meaning of every word they use or the exact steps involved in coming to a conclusion because we have a piece of "black box" technology that does that for us: the subconscious. The subconscious handles the details and then presents my conscious mind with the results, but not necessarily with the steps leading to those results.
People may "get away" with not having clear definitions of their concepts, but only at their peril. To think properly (and therefore also to understand proper thinking) requires requires that concepts be defined explicitly by the subject. You are trying to take care of this in your "black box", but I think you misunderstand the role of the subconscious. A person's subconscious consists of what he puts into it, in effect by "programming" it with what he understands, or once understood, for better or worse at a conscious level. The subconscious does not take half-baked ideas and turn them into properly defined concepts to be used to provide material to a half-bumbling conscious mind thereafter! It is true that much of the details of what we learned are forgotten at the conscious level and that much of the mental processing that goes into writing and speaking is at the subconscious level, but even at its best, the subconscious is not infallible; the results it gives you have to be validated consciously. Furthermore, the process used by the subconscious is not the same as the conscious process of validation. A software implementation of the subconscious, even if it could be done, would not account for human intelligence and thinking.

Ayn Rand describes the role and proper use of the subconscious in her lectures on writing (published a few years ago as two books on writing fiction and non-fiction), and one of the Objectivist monthlies in the 60's reviewed and recommended Arthur Koestler's book, The Act of Creation, which also emphasizes the role of the subconscious. A more specialized work of particular interest here that describes the stages of creation and problem solving in mathematics, which in some ways serves as an example of the highest form of problem solving in abstract thought, is Jacques Hadamard's small book, The Psychology of Invention in the Mathematical Field. You may have already read and considered these, but if not you should read them.

Unfortunately, if I'm going to program a computer to think it's not enough that I have such a "black box" myself. I need to know exactly how it works so that I can design a software equivalent. Thus, I need to know exactly what the irreducible concepts are that my mind uses, exactly what each one means and exactly what rules govern their combination into more complex concepts. Kind of like Aristotle's categories, but taken to the next (and final) level.

To this end, I'm having to go even deeper than Rand did in identifying basic concepts. To give you an idea of the sort of thing I'm doing, I've been going through lists of words we use all the time but never really think about the meaning of - "I," "the," "and," "one," etc. - and trying to explicitly identify what I mean by each one and if I can completely define any of them in terms of others on the list (establishing that they're reducible concepts after all).

What do you mean by "irreducible"? First level concepts based on perception? All the ones you listed are higher level abstractions. Do you expect a program to compute all higher level abstractions and structure them hierarchically in a (mathematical) lattice? If so, under what "motivation" for the formation of specific concepts? To know how the subconscious works, which no one now does, will take a lot more than defining concepts "programmed" into it.

Right now I'm working on basic logic, and in addition to a couple of other rather interesting discoveries I finally managed to bridge the gap between logic and basic mathematics just recently. I've derived the concepts "one," "plus," "equality," "inequality," "greater than" and "lesser than" from basic logical operators. I'm still a long way from differential equations - I haven't even defined "zero" or "minus" yet - but I've definitely got a solid foundation laid down.
How is this different than Logicism (one of the traditional and not very successful modern attempts at the foundations of mathematics)? You haven't described how you "derive" mathematical concepts from logic, but based on the traditional history alone it sounds dubious.

For the conceptual foundations of the most basic mathematics, this is one area in which you should definitely read the articles in the The Intellectual Activist by Ron Pisaturo and Glenn Marcus mentioned by Stephen and I above: "The Foundations of Mathematics" (July & Sept. 1994) and "Mathematics in One Lesson" (Sept. & Oct. 1998).

However, I'm going to need to know how to define "infinity" for various areas of my work. Without that, I'll be stuck before long. That's why I'm asking about it; I don't expect that any one definition out there will be what I want, but I'm hoping that by considering various definitions it might help me to abstract out the element that I'm looking for.

Is it clearer now?

Partially. Presuming that your project is feasible at all, why do you need to incorporate definitions of advanced mathematical technical concepts that the vast majority of people do not know and get along without very well in their thinking? -- arguably better than some mathematicians who do know them :-) More relevant to understanding the nature of concepts is the whole topic of "open-endedness" already mentioned in the post above and emphasized in Ayn Rand's IOE. What else do you need the "mathematically infinite" for?

But to the extent you need to better understand the valid meaning of the infinite as exemplified in basic mathematics, first review Aristotle and the brief discussions by Ayn Rand in IOE. Then turn to the actual history of the concept's proper development in mathematics where it is shown how to deal with it exclusively in terms of precise, finite processes and meanings, particularly in the 19th century work of Weierstrass, which is now used to define mathematical concepts such as limits, continuity, irrational numbers, maxima and minima of curves, etc. in any good calculus book. There is an obvious relation between the "epsilon-delta" approach to infinite limits and the Objectivist ideas of measurement and measurement omission in concept formation.

You should also look at the principle of mathematical induction and how it was used by Peano to formulate an axiomatic theory of the infinite integers using recursion and strictly finite means, which is presented in modern books on abstract algebra. All of these mathematical concepts show how to properly conceptualize infinite processes in strictly finite terms and without invoking any notion of a "completed infinity" for either the processes or the collection of numbers to which they apply (although it is often not presented that way when dealing with infinite sets).

I would not bother at all with Cantor's theories of infinity through his transfinite numbers and infinite sets. You have to already understand the mathematically infinite before you can make any sense of Cantor at all, which requires knowing when to re-interpret what he did and when to reject it. He adds nothing to a basic conceptual understanding.

Keep in mind that while these mathematical concepts and theories explain basic concepts like number and continuity, they are not the way people begin to think about and use these and other basic mathematical concepts, which are required before anyone can move on to the more advanced technical concepts of mathematical science (which most people do not). You will find, for example, that even Peano, in his theoretical, formal treatment of the axioms of arithmetic, said explicitly that his subject matter was the ordinary numbers that people already understand conceptually. On the other hand, Ayn Rand believed that a more specifically mathematical approach to epistemology could be very helpful for understanding and advancing the science of epistemology. To the extent that you can accomplish anything in artificial intelligence using better epistemological principles, that will be the case for you, too. I don't believe you will succeed with a software implementation of human intelligence in the way you appear to be trying, but you may be able to do something new of worth once you understand the limitations. of software AI.

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I don't know what else you have in mind for how to make your project work, but based on this opening description, it sounds dubious. Human thought and consciousness cannot be reduced to an algorithm and reproduced in software.
Not all aspects of it, no. I realize that I can't recreate volition that way. But then again, I'm not interested in things like volition.

That principle isn't new, but a recent good article on it from an Objectivist perspective is in the April 2004 issue of Rob Tracinski's monthly journal, "The Intellectual Activist": "Mindless Intelligence: Machine Thinking and Contemporary Philosophers' Rejection of the Mind". Have you seen this?

I'm not subscribed. I know that they have a free trial, but since it only lasts 30 days I want to wait until the semester is over and use the trial period over the break.

People may "get away" with not having clear definitions of their concepts, but only at their peril. To think properly (and therefore also to understand proper thinking) requires requires that concepts be defined explicitly by the subject...

...It is true that much of the details of what we learned are forgotten at the conscious level and that much of the mental processing that goes into writing and speaking is at the subconscious level, but even at its best, the subconscious is not infallible; the results it gives you have to be validated consciously.

We don't consciously validate - or even define - everything our subconscious gives us to infinite precision. For example, you walk into a room and see a patch of color that you identify as a man. Do you stop and consciously start taking measurements and comparing them to your concept of "man?" Could you even give a mathematical description of what a man looks like? Of course not. And even if you could, stopping to consciously confirm that every bit of sensory data that comes your way is exactly what you think it is would take so much time that you would starve to death if you tried to function that way.

Past a certain point, we don't stop to analyze in that kind of detail unless we have reason to.

Furthermore, the process used by the subconscious is not the same as the conscious process of validation. A software implementation of the subconscious, even if it could be done, would not account for human intelligence and thinking.

Why would it have to? I don't need my calculator to account for human mathematical thinking, I just need it to come up with useful answers faster than I can myself.

What do you mean by "irreducible"?
There are a finite number of concepts in my mind at any one time. Some are primary, or "atomic." They can't be broken down further. Concepts of sensation (like "blue") fall into this category, for example. Other concepts result from combining these atomic concepts according to certain rules.

I'm trying to identify which concepts are atomic and the rules for their combination, and apply that. Think of it as "linguistic housecleaning," if you will.

First level concepts based on perception? All the ones you listed are higher level abstractions.

We'll see, eventually. I can only tackle one level at a time.

Do you expect a program to compute all higher level abstractions and structure them hierarchically in a (mathematical) lattice? If so, under what "motivation" for the formation of specific concepts?
Under whatever motivation I decide to program into it. You don't expect a calculator to be self-motivated, you expect it to do what you tell it to do. Unit-reduction, as described by Rand in ITOE, would be a good use.

To know how the subconscious works, which no one now does, will take a lot more than defining concepts "programmed" into it.

Doubtless. But while that may not be the only step, it is a step. One thing at a time.

How is this different than Logicism (one of the traditional and not very successful modern attempts at the foundations of mathematics)? You haven't described how you "derive" mathematical concepts from logic, but based on the traditional history alone it sounds dubious.
Of course not. I would have to write a paper, not a short post. And I only accomplished this in the last week. I'm still proving properties. I'm keeping notes in a text file (call it the bastard love child of an outline and a rough draft), but it's going to be a while before I can turn that into a full essay.

And every new discovery or invention defies "traditional history." So that doesn't worry me.

Partially. Presuming that your project is feasible at all, why do you need to incorporate definitions of advanced mathematical technical concepts that the vast majority of people do not know and get along without very well in their thinking?

...More relevant to understanding the nature of concepts is the whole topic of "open-endedness" already mentioned in the post above and emphasized in Ayn Rand's IOE. What else do you need the "mathematically infinite" for?

First off, I'm not so sure that it's going to prove to be all that "advanced." I realize that this claim won't mean anything to you since I haven't posted/published anything yet, but the definitions of the basic mathematical concepts I cited (one, plus, etc.) were far simpler than I expected them to be and I have reason to suspect that "infinity" will follow the same pattern. Simply put, the complexity lies in defining the unit, not the quantity.

Second, the plan is to use the concepts and processes of my own mind as the model for AI. To do that, I first have to understand the model. Hence the linguistic housecleaning. If I can't define the concept precisely enough in my own mind, I can't replicate it in software.

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I still don't know any more about what you expect to accomplish or what you need a concept of infinity for to do it. Your project sounds dubious with respect to your goals, based on what you have said, because your description, such as it is, has relied on solely traditional terms and without indicating any understanding of Objectivist epistemology or new insights, or how it would be used in a new and unique way.

There is much more to both the conscious and subconscious processes than you seem to be aware. If you have not read the books by Koestler, Hadamard and Rand cited earlier on the creative process and the role of the subconscious in particular, you should look them up. The reference to "Mindless Intelligence: Machine Thinking and Contemporary Philosophers' Rejection of the Mind" is in the monthly print version of TIA; that and the Pisaturo/Marcu articles are ordered as back issues, not part of a new trial (but you might try asking for the most recent of them that way).

Knowledge is a grasp of reality in the form of concepts formed with a cognitive purpose. This is a creative process using both the conscious and the subconscious working together striving for understanding in the face of new facts. Concepts are not formed by telling your subconscious to perform a "unit reduction" in a vacuum, combining existing concepts in accordance with hidden rules. Formation and expansion of knowledge is not a matter of "linguistic housekeeping".

There is a lot more to the human process of concept formation and application than an act of "volition" -- which is only the basic choice to focus (without specifying on what, with what emphasis on what aspects, etc.) -- followed by a result expressed in some structural representation formed with some degree of speed regarded as in competition with software running on a computer. Some mental activities can be reduced to "calculation", but based on abstract concepts and principles established in advance. That this can be done does not mean that everything beyond "volition" can be implemented in software like a calculation.

You do have to consciously define your concepts or live with the resulting chaos. The subconscious is not a substitute for that. In any creative process you must validate conclusions suggested by the subconsciousness.

We don't do anything to "infinite precision" (which is an example of the invalid concept of the actual infinite), but we also do not form basic concepts like "man" out of extensive numerical measurements and mathematical descriptions. That is not what is meant by "measurement" in the Objectivist theory of concepts. Once we have a concept, it can be extended and refined through more precise definitions to account the growth in our scope and depth of knowledge, but we don't start with a "mathematical description of what a man looks like" and neither does our subconscious provide one behind the scenes. Nor do we (or could we) consciously reprocess sensory data as such; that is not what it means to validate conceptual knowledge.

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I still don't know any more about what you expect to accomplish or what you need a concept of infinity for to do it.

Alright, let me see if I can spell my intentions out in a little more detail:

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.

4) Once I have the root of mathematics (counting natural numbers), I will then construct the foundations of various branches of mathematics (arithmatic, calculus, etc.) from the above. I don't intend to exhaustively recreate all of mathematics single-handedly, but I do intend to demonstrate how it can be done.

The essentials of steps 1-3 are finished, though I still have a lot of little details to attend to. Step 4, however, will involve my having to define even more concepts, the most troublesome of which is that of infinity. Since calculus (for example) deals with infinity, I must have a definition for infinity to be able to deal with calculus.

Does that clarify things?

Your project sounds dubious with respect to your goals, based on what you have said, because your description, such as it is, has relied on solely traditional terms and without indicating any understanding of Objectivist epistemology or new insights, or how it would be used in a new and unique way.
I told you, to go into detail about my "new insights" I would have to write a paper, not a post.

First I intend to finish the above, then I will program the basic components of my system (those identified in step 1) into a computer and use them - and ONLY them - to reconstruct the rest. That will be my "acid test;" my proof that my system really is functional and complete, and that I'm not just unwittingly glossing over any details.

Then I will write a paper on all of this, once I've proven to myself the validity of my work.

Oh, and as far as my "indicating any understanding of Objectivist epistemology" goes, I have read ITOE and OPAR (though it's been a while). But as I said, I'm not working in an objectivist context. Rand and Aristotle were my starting points, but I do not take every word out of their mouths as gospel. I question them too.

There is much more to both the conscious and subconscious processes than you seem to be aware.

...And that is precisely what I am investigating.

Formation and expansion of knowledge is not a matter of "linguistic housekeeping".
No, but explicitly identifying the roots of those concepts you've already formed is.

That this can be done does not mean that everything beyond "volition" can be implemented in software like a calculation.

Please quote the part where I claim that everything beyond volition can be, because I don't remember saying that.

but we also do not form basic concepts like "man" out of extensive numerical measurements and mathematical descriptions.
That's certainly not all that there is to the concept of "man," no. But it is a component. If you see a patch of color and - based on that alone - classify the object you're looking at as a "man," then how did you know that it was a man and not, say, a chair?

If you're comparing, say, the shapes of a man and a chair to the object you're looking at, then those shapes can be descibed using geometry. And if you're not comparing shapes, colors, etc. and still somehow ariving at the conclusion that you're looking at a man, how are you doing it without divorcing your conclusions from sense data?

but we don't start with a "mathematical description of what a man looks like" and neither does our subconscious provide one behind the scenes.

When it comes to the task of identifying what is and is not a man by shape and color, I suspect that we do.

Earlier you said, "To know how the subconscious works, which no one now does, will take a lot more than defining concepts "programmed" into it." (emphasis mine). Now you seem to be claiming precisely that sort of knowledge. Would you care to elaborate, or is this a contradiction?

Nor do we (or could we) consciously reprocess sensory data as such; that is not what it means to validate conceptual knowledge.

Again, would you care to elaborate? My school library has copies of OPAR and ITOE, so if you want to at least give page references...

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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.

4) Once I have the root of mathematics (counting natural numbers), I will then construct the foundations of various branches of mathematics (arithmatic, calculus, etc.) from the above. I don't intend to exhaustively recreate all of mathematics single-handedly, but I do intend to demonstrate how it can be done.

The essentials of steps 1-3 are finished, though I still have a lot of little details to attend to. Step 4, however, will involve my having to define even more concepts, the most troublesome of which is that of infinity. Since calculus (for example) deals with infinity, I must have a definition for infinity to be able to deal with calculus.

Does that clarify things?

Relational forms of deduction in addition to Aristotle's syllogisms were identified long ago by British logicians such as Boole and DeMorgan and are now commonly used, especially in mathematics. You should look into that history, too, if the topic interests you. But concept formation is primarily an inductive process.

You still haven't said how what you are doing is different than logicism, which is essentially what you are describing, as your approach to the foundations of mathematics. Are you familiar with the history of logicism and formalism in the technical foundations of mathematics over the last century or so? How do you see what you are doing as related to that?

But 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.

You had said earlier that you are concentrating on the most basic "atomistic" concepts at the root of knowledge (although you haven't answered what you mean by that in relation to first level concepts based on perception) and said that you might later get to abstractions after you get the basics down. Yet all the examples of concepts and kinds of knowledge you mention of current interest for your project are high level abstractions, especially the mathematics. That distinction is part of why it so hard to follow what you are talking about. Calculus, in particular, is very high level in the conceptual hierarchy, depending on prior abstract concepts of functions, algebra, real and rational numbers, integers and counting. It also depends on sophisticated scientific knowledge of facts that motivate the need for the abstract mathematical concepts. Even basic counting is relatively abstract in comparison with the most elementary knowledge.

But to the extent you have an interest in understanding concepts of infinity in calculus and other mathematical subjects, you already have the answer for what to look at. 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. Look at how the infinite is actually used in theoretical formulations of mathematics with respect to 1) the size of a set such as the integers (using the inductive definitions as formulated by Peano) and 2) infinite limits (including both unboundedness and subdivisibility) in analysis. You should not attempt to epistemologically explain abstract mathematics without a thorough understanding of how the number systems are constructed both in their elementary conceptual formulation and in advanced mathematics, and how limits are defined and worked with in calculus and higher analysis. Have you not encountered these subjects? Even when they are introduced in elementary courses it is easy to overlook their significance because of a common lack of motivation.

... as far as my "indicating any understanding of Objectivist epistemology" goes, I have read ITOE and OPAR (though it's been a while). But as I said, I'm not working in an objectivist context. Rand and Aristotle were my starting points, but I do not take every word out of their mouths as gospel. I question them too.
You ought to go back and reread IOE, including the appendix on the epistemolog seminars. You will get a lot more out of it on a second reading, especially since you say it has been a while since you read it. You appear to be missing a lot. No one has suggested that you take anything as gospel or not question anything. That has no place here; this is a science discussion. Remember that using a body of knowledge as a true "starting point" for further work requires fully understanding it. That's not the same thing as a rough understanding based on a casual or mostly forgotten first reading.

That's certainly not all that there is to the concept of "man," no. But it is a component. If you see a patch of color and - based on that alone - classify the object you're looking at as a "man," then how did you know that it was a man and not, say, a chair? If you're comparing, say, the shapes of a man and a chair to the object you're looking at, then those shapes can be descibed using geometry. And if you're not comparing shapes, colors, etc. and still somehow ariving at the conclusion that you're looking at a man, how are you doing it without divorcing your conclusions from sense data?

When it comes to the task of identifying what is and is not a man by shape and color, I suspect that we do.
At an advanced level you could do that explicitly, but most people don't have the means to do that at all. We begin concept formation on a much more elementary level using key perceptual characteristics that include perceptions of shape -- which you later can identify as a concept of geometry -- but which are not explicitly and consciously mathematical in an advanced technical sense. Our earlier, crude concepts are often later refined and made more precise, but most people do not know much mathematics and do not think explicitly in technical mathematical terms such as those of "geometry". The formation of most concepts is not done by rigorous "mathematical descriptions"; it is "mathematical" only in the crudest sense of recognizing familiar shapes and limits on size grasped perceptually. Remember that mathematics is itself highly abstract. People can certainly form concepts of and distinguish between "man" and "chair" without using geometry and "mathematical descriptions".

But that is not to say that you can't formulate a more mathematical theory of epistemology itself, and use that in explaining more about concepts without claiming that people think explicitly in mathematical terms or mathematical descriptions.

Please quote the part where I claim that everything beyond volition can be, because I don't remember saying that.
You said that you seek to implement human intelligence and knowledge, including the creation of knowledge, which is both an unconscious and conscious inductive process, in software ("the creation of true artificial intelligence; computer programs that can take sense data and organize and theorize about it in the same way that a human can"). That is a very sweeping goal. The only aspect you mentioned that you are not trying to incorporate and are not interested in is "volition".

Earlier you said, "To know how the subconscious works, which no one now does, will take a lot more than defining concepts "programmed" into it." (emphasis mine). Now you seem to be claiming precisely that sort of knowledge. Would you care to elaborate, or is this a contradiction?
A good deal is known about the role and function of the subconscious and the kinds of things it covers in creative, inductive processes, as described in the references cited earlier. But no one knows how the subconscious actually works. That is the distinction, which is not a contradiction. It is clear that there is a lot more involved in its processes than the meaning, let alone only definitions, of concepts programmed into it.

Again, would you care to elaborate? My school library has copies of OPAR and ITOE, so if you want to at least give page references...

We do not consciously process or validate sensory information, which is automatic. Our concepts start with the perceptual level of awareness, which already integrates sensations. You really should reread IOE in its entirety. That will answer a lot of your questions and provide a lot of new insights. OPAR is very good, too, but purposely did not include as much detail on concept formation in particular because that is one area of Ayn Rand's philosophy that she had already addressed comprehensively and in detail in her monograph.

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Here some mathematical definitions of infinities (actually the term is "transfinite numbers"), and yes there are more than one infinities:

aleph-0 : the set of all natural numbers {0,1,2,3,...}

c: the power set of aleph-0 (that is the set of all subsets of aleph-0) [this is the same as the set of all real numbers]

It can be shown that c does not equal aleph-0, that is to say there is no bilinear mapping between aleph-0 and c (colloquially the reals have a higher cardinality than the natural numbers). In general the power set B of a set A has a higher cardinality than does the set A itself. So we can take the power set of c and produce another higher "infinity" and so on.

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Here some mathematical definitions of infinities (actually the term is "transfinite numbers"), and yes there are more than one infinities:

aleph-0 :  the set of all natural numbers {0,1,2,3,...} ...

These are not "mathematical definitions of infinities," but rather characterizations of the cardinality of infinite sets.

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These are not "mathematical definitions of infinities," but rather characterizations of the cardinality of infinite sets.

"Cardinality" is the property that represents the set theorists view on what "magnitude" or "quantity" is in a number. "Cardinality" represents "how many elements are in the set".

Two quick definitions for onlookers:

Ordinality -> the property of numbers which specifies their position in the series of numbers:

first, second, third, fourth,...

so if a number x occurs later in the sequence of numbers than a number y then x has a higher ordinality than y

Cardinality -> the property of numbers which specifies their relative size to each other:

one, two, three, four,...

so if a number x is larger than a number y then x has a higher cardinality than y

For two finite numbers x,y the following holds:

x has a higher ordinality than y if and only if x has a higher cardinality than y

For this reason it is natural for people to use the notions of "ordinality" and "cardinality" interchangably. This property fails for transfinite numbers. However cardinality embodies the property of "moreness", so there are in a well-definied sense "more" items in a set of higher cardinality than in the set of lower cardinality in the same sense that there are more items in a set of three things than in a set of two things.

It is perfectly fine to define a sequence of sets with increasing ordinality (for infinite set W):

W, W+1, W+2, W+3,....

Even though the sets have the same cardinality.

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punk, I think Stephen meant that aleph-0 etc are not definitions of infinite, but rather specific examples of transfinite cardinal numbers (in the same sense that 2 and 3 are examples of integers, rather than being a definition of an integer). A mathematical definition of infinity would be something along the lines of "a set S is infinite if the elements of a proper subset of S can be put into one-to-one correspondence with the elements of S" (from mathworld) or "greater than any specific integer".

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My bad.

I guess I've been spending too much time pondering the transfinite number which is the power set of the continuum and functional analysis. I got all excited that something on my mind actually came up.

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I realize this is off the exact subject of set theory you guys are talking about, but I think it fits the subject of infinity still.

Since the universe has always existed, hasn't an infinite amount of time already passed? Meaning, doesn't time go back forever, to infinity? True--we would never be able to represent an infinite length of time in any compact notation created by counting objects or their arrangements (since only so many objects and arrangements exist), but doesn't infinity still exist in how much time has gone by in the past?

I don't think an infinity with respect to time would cause the problems implicit in an infinity applied to physical reality. From ITOE:

"Infinity" in the metaphysical sense, as something existing in reality, is another invalid concept. The concept "infinity," in that sense, means something without identity, something not limited by anything, not definable.  Therefore, the measurements omitted here are all measurements and all reality.

Time, though, would still be definable, since it is a measurement between two points. Saying an infinity exists doesn't mean that there aren't any two points--but that a "first point" doesn't exist (no creation). I don't think this would be omitting all measurements or all reality, since the claim being made isn't that one can go back infinitely in time, or that one can go forward infinitely in time. It just means that, given that time passes at a constant rate and nothing comes into or goes out of existence, everything must always have existed (infinite time). To say this somehow undermines measurements of time or reality would be a stolen concept.

To apply this back to the ITOE quote above, it is not being said to "exist in reality" since an interval of time doesn't "exist" in the same sense that physical objects do.

Is this consistent with Objectivism? Is it correct to say that an infinity is applicable to reality in the sense that there is not a finite limit to the amount of time that has elapsed before now? If not, could anyone please help me understand the error I'm making in my thought process?

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... Since the universe has always existed, hasn't an infinite amount of time already passed?  ... Is this consistent with Objectivism?  Is it correct to say that an infinity is applicable to reality in the sense that there is not a finite limit to the amount of time that has elapsed before now?  If not, could anyone please help me understand the error I'm making in my thought process?

This has been discussed to death in several prior threads. Briefly, your error, in essence, lies in attempting to apply time to the universe, but the universe does not exist within time. You are using the concept of "duration" in a context where "duration" does not apply. Time exists within the universe; the universe does not exist within time.

Alex wrote this excellent essay that I have referenced before. This should help you get a grasp on the finite/infinite issue and understand why we cannot apply much at all to the universe as a whole.

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